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Low-field permanent magnets for industrial process and quality control
Accepted Manuscript Low-Field Permanent Magnets for Industrial Process and Quality Control J. Mitchell, L.F. Gladden, T.C. Chandrasekera, E.J. Fordham PII: DOI: Reference:
S0079-6565(13)00088-5 http://dx.doi.org/10.1016/j.pnmrs.2013.09.001 JPNMRS 1376
To appear in:
Progress in Nuclear Magnetic Resonance Spectroscopy
Received Date: Accepted Date:
17 July 2013 19 September 2013
Please cite this article as: J. Mitchell, L.F. Gladden, T.C. Chandrasekera, E.J. Fordham, Low-Field Permanent Magnets for Industrial Process and Quality Control, Progress in Nuclear Magnetic Resonance Spectroscopy (2013), doi: http://dx.doi.org/10.1016/j.pnmrs.2013.09.001
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Low-Field Permanent Magnets for Industrial Process and Quality Control J. Mitchella,b , L.F. Gladdena,∗, T.C. Chandrasekeraa , E.J. Fordhamb a Department
of Chemical Engineering and Biotechnology, University of Cambridge, Pembroke Street, Cambridge CB2 3RA, United Kingdom. b Schlumberger Gould Research, High Cross, Madingley Road, Cambridge CB3 0EL, United Kingdom.
Keywords: Low-field, Permanent magnets, Processing, Quality control, Industry PACS: 07.57.Pt, 76.60.Pc, 76.60.-k, 82.56.-b Contents 1
Introduction
2
Which Field Strength?
11
2.1
NMR magnets and industrial environments . . . . . . . . . . . .
12
2.2
A rough guide to selecting a magnet . . . . . . . . . . . . . . . .
14
3
5
Low-Field Spectrometers
20
3.1
Permanent magnet configurations . . . . . . . . . . . . . . . . . .
20
3.1.1
Bench-top magnets . . . . . . . . . . . . . . . . . . . . .
20
3.1.2
Ex situ NMR . . . . . . . . . . . . . . . . . . . . . . . .
22
3.1.3
Well-logging tools . . . . . . . . . . . . . . . . . . . . .
25
Radio frequency pulses . . . . . . . . . . . . . . . . . . . . . . .
27
3.2
∗ Corresponding
author. Tel: +44 (0)1223 334762 Email address: [email protected] (L.F. Gladden)
Preprint submitted to Progress in Nuclear Magnetic Resonance Spectroscopy September 26, 2013
4
3.2.1
Elementary circuit theory . . . . . . . . . . . . . . . . . .
29
3.2.2
Spin dynamics during pulse transients . . . . . . . . . . .
32
3.3
Digital filtering . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
3.4
Maximizing signal-to-noise ratio . . . . . . . . . . . . . . . . . .
46
Pulse Sequences and Experiments
50
4.1
Relaxation time measurements . . . . . . . . . . . . . . . . . . .
50
4.1.1
Longitudinal relaxation . . . . . . . . . . . . . . . . . . .
50
4.1.2
Transverse relaxation . . . . . . . . . . . . . . . . . . . .
51
4.2
Diffusion and flow . . . . . . . . . . . . . . . . . . . . . . . . .
54
4.3
Multi-dimensional correlations . . . . . . . . . . . . . . . . . . .
58
4.3.1
Relaxation time correlations . . . . . . . . . . . . . . . .
59
4.3.2
Diffusion-relaxation correlations . . . . . . . . . . . . . .
60
4.3.3
Exchange rate measurements . . . . . . . . . . . . . . . .
62
Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
62
4.4.1
Frequency encoding . . . . . . . . . . . . . . . . . . . .
63
4.4.2
Phase encoding . . . . . . . . . . . . . . . . . . . . . . .
65
4.4.3
Slice selection . . . . . . . . . . . . . . . . . . . . . . .
65
4.4.4
Multi-dimensional imaging . . . . . . . . . . . . . . . . .
67
4.4.5
Pure phase-encoded imaging . . . . . . . . . . . . . . . .
69
Stray field experiments . . . . . . . . . . . . . . . . . . . . . . .
74
4.5.1
Relaxation time measurements . . . . . . . . . . . . . . .
74
4.5.2
Diffusion measurements . . . . . . . . . . . . . . . . . .
78
4.5.3
Stray field imaging . . . . . . . . . . . . . . . . . . . . .
79
4.6
Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . .
80
4.7
Fast field cycling . . . . . . . . . . . . . . . . . . . . . . . . . .
81
4.4
4.5
2
4.8 5
7
83
Data Analysis
84
5.1
Pre-processing for optimum results . . . . . . . . . . . . . . . . .
85
5.1.1
Zeroth-order phase correction . . . . . . . . . . . . . . .
85
5.1.2
Data compression . . . . . . . . . . . . . . . . . . . . . .
87
5.1.3
First-order phase correction . . . . . . . . . . . . . . . .
88
5.2
Time-domain fitting for spectroscopy . . . . . . . . . . . . . . . .
89
5.3
Time-domain fitting for relaxation and diffusion . . . . . . . . . .
94
5.4
Distributions of relaxation and diffusion (1D) . . . . . . . . . . .
95
5.4.1
Data fitting . . . . . . . . . . . . . . . . . . . . . . . . .
95
5.4.2
Kernel functions . . . . . . . . . . . . . . . . . . . . . .
98
5.4.3
Limitations and considerations . . . . . . . . . . . . . . .
99
5.5
6
Ultra-low field and Earth’s field . . . . . . . . . . . . . . . . . . .
Distributions of relaxation and diffusion (2D) . . . . . . . . . . . 101 5.5.1
Data fitting . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.5.2
Kernel functions . . . . . . . . . . . . . . . . . . . . . . 103
Interpretation
105
6.1
Surface-to-volume ratio . . . . . . . . . . . . . . . . . . . . . . . 106
6.2
Surface interaction strength . . . . . . . . . . . . . . . . . . . . . 112
6.3
Diffusive or chemical exchange . . . . . . . . . . . . . . . . . . . 114
6.4
Droplet sizing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
6.5
Internal gradients . . . . . . . . . . . . . . . . . . . . . . . . . . 123
Industrial Applications 7.1
129
Food . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 7.1.1
Meat . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 3
7.2
7.3
8
9
7.1.2
Fish . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
7.1.3
Dairy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
The built environment . . . . . . . . . . . . . . . . . . . . . . . . 140 7.2.1
Cement . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
7.2.2
Wood . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
7.2.3
Conservation . . . . . . . . . . . . . . . . . . . . . . . . 147
Petrophysics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 7.3.1
Well-logging . . . . . . . . . . . . . . . . . . . . . . . . 149
7.3.2
Laboratory core analysis . . . . . . . . . . . . . . . . . . 161
7.3.3
Hydrocarbon characterization . . . . . . . . . . . . . . . 165
Future Directions
168
8.1
Hardware . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
8.2
Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
8.3
Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 8.3.1
Bayesian inference . . . . . . . . . . . . . . . . . . . . . 171
8.3.2
Compressed sensing . . . . . . . . . . . . . . . . . . . . 173
Summary
175
10 Acknowledgments
177
Appendix A
Nomenclature
256
Appendix B
Abbreviations
264
4
1. Introduction Nuclear magnetic resonance (NMR) is now widely used as a process and quality control tool in a numerous industries including food [1], pharmaceuticals [2, 3], and chemicals manufacturing [4]. Increasingly, a diversity of NMR measurements is exploited. For example, whilst magnetic resonance imaging (MRI) has traditionally been associated with clinical medicine, presenting a non-destructive diagnosis tool and a powerful technique for in situ biomedical investigations [5–8], it is now well established in applications to materials science and is gaining popularity for probing dynamics in optically opaque media [9–14]. High-resolution NMR spectroscopy provides a method of structural determination for complicated molecules such as proteins [15, 16], and enables studies of molecular interactions including enzyme activity [17]. Solid state spectroscopy is useful for materials characterization, such as determining the structures of catalyst metals and supports [18–21]. Finally, NMR measurements of molecular diffusion or advection are possible with pulsed field gradient (PFG) techniques, enabling the determination of droplet size distributions in emulsions [22, 23], dissolution of pharmaceutical tablets [24], degree of polymerization [25–27], or structure-transport relations in porous media [14, 28–30]. In this review we focus on the technology associated with low-field NMR. We present the current state-of-the-art in low-field NMR hardware and experiments, considering general magnet designs, radio frequency (rf) performance, data processing and interpretation. We provide guidance on obtaining the optimum results from these instruments, along with an introduction for those new to low-field NMR. The applications of low-field NMR are now many and diverse. Furthermore, niche applications have spawned unique magnet designs to accom5
modate the extremes of operating environment or sample geometry. Trying to capture all the applications, methods, and hardware encompassed by low-field NMR would be a daunting task and likely of little interest to researchers or industrialists working in specific subject areas. Instead we discuss only a few applications, to highlight uses of the hardware and experiments in an industrial environment. For details on more particular methods and applications, we provide citations to specialized review articles. Let us start by defining “low field”. The terms low field, intermediate field, and high field do not carry any strict meaning. We choose to define low field as the range of magnetic field strengths corresponding to B0 = 10 mT to 1 T; for 1 H,
this range is equivalent to Larmor frequencies of ν0 = 425 kHz to 42.5 MHz
and corresponds to typical bench-top and portable NMR devices based on permanent magnet technology. A summary of the most common experiments and applications within the remit of low-field NMR is given in Fig. 1. Below our low field range exists “ultra-low field” NMR, corresponding to B0 1 mT. The ultralow field [31] applications often involve superconducting quantum interference devices (SQUIDs) [32], required to detect the very weak NMR signals encountered at such small polarizing magnetic fields. Earth’s field NMR [33] is a special case of ultra-low field technology wherein the NMR signal is detected using a conventional rf tuned circuit but polarized to provide sufficient magnetization for detection. The Earth’s magnetic field varies depending on location; it is typically in the range B0 ≈ 30 − 40 µT, corresponding to a 1 H frequency of ν0 ≈ 2 kHz. Above the low field range we define intermediate field (B0 = 1 − 3 T) associated primarily with clinical medicine and full-body MRI scanners. Further still we define high field (B0 > 3 T) as the domain of high-resolution spectroscopy.
6
Earth’s field
ultra low field
intermediate field
low field
40 µT
50 mT
0.2 T
0.5 T
2 kHz
2 MHz
7.5 MHz
20 MHz
well−logging
quality control
1T
1.5 T
45 MHz
60 MHz
B0 ν0
!1 " H
process monitoring
SQUIDs petrophysics
relaxation
clinical diagnosis
imaging
relaxation
spectroscopy
Figure 1: Summary of magnetic field strengths – primary applications and measurements using permanent magnets. The range of B0 encompassed by the term “low field” is considered to extend from B0 = 10 mT (ν0 = 425 kHz) to B0 = 1 T (ν0 = 42.5 MHz). Permanent magnets can provide stronger field strengths up to B0 = 2.1 T (ν0 = 90 MHz), considered to fall in the “intermediate range”. Ultra-low field strengths encompass the range B0 < 1 mT; NMR measurements at “ultralow field” are typically conducted using SQUIDs, although Earth’s field NMR (a special case of low-field NMR where B0 ≈ 30 − 40 µT, ν0 ≈ 2 kHz) is achieved using a polarizing coil to generate a detectable signal. A wide range of applications utilize regular permanent magnet technology, from oil reservoir well-logging to process (chemical reaction) monitoring. Relaxation, diffusion, and imaging are possible across the entire range of low-field NMR. Petrophysical and quality control applications rely mainly on relaxation time analysis. Spectroscopy is possible only at intermediate field over small sample volumes where a homogeneous magnetic field is achievable with permanent magnets. Images of a selection of low-field magnets designed for different purposes are shown at the top (left to right): a Magritek Earth’s field NMR system (B0 ∼ 40 µT), an Oxford Instruments bench-top rock core analyzer (B0 = 50 mT), a Siemens C-shaped whole body MRI system (B0 = 0.35 T), a Bruker bench-top analyzer (B0 = 0.5 T), and a Magritek bench-top spectroscopy magnet (B0 = 1 T). Permanent magnet designs are often tailored to specific applications and can be open-access or single-sided.
7
The maximum field strength that can be currently achieved with a bench-top permanent magnet is around B0 ∼ 1.5 T, corresponding to ν0 ∼ 60 MHz. This field strength is sufficient for basic spectroscopy, and we are seeing a resurgence in spectroscopic measurements at such modest field strengths for in-line process monitoring. Spectroscopic resolution is achievable with intermediate-field permanent magnets on small samples (5 mm diameter tubes), though most low-field instrumentation provides a magnetic field that is insufficiently homogeneous for 1H
spectroscopy. Within NMR research and application in general, there remains
a drive for stronger magnets to enable improved signal-to-noise ratio (SNR) and spectroscopic resolution. Commercial spectroscopy systems are available up to about B0 ≈ 23.5 T (ν0 = 1 GHz for 1 H) and full-body MRI systems up to B0 = 7 T
(ν0 = 300 MHz for 1 H) and even higher for clinical research (combined MRI and
spectroscopy) [8, 34]. However, with the rising cost of liquid helium required for superconducting magnets, plus the safety and economic factors of installing such systems in industrial environments, there is a persistent interest in low-field technology. Low-field bench-top magnets (typically B0 ≈ 0.5 T, ν0 ≈ 20 MHz for 1 H) are popular in the food and agriculture industry for product screening and quality control [35]. Bench-top magnets operating at lower frequencies (ν0 = 2 MHz for 1 H)
are preferred for studies of petrophysical rock and environmental soil sam-
ples [14]. Bench-top MRI systems are available for biomedical studies of rodents [36]. Full-body MRI scanners based on permanent magnet technology are also available [37]; these C-shaped magnets offer the advantages of (1) open access to the subject, (2) improved patient experience (being less claustrophobic than an enclosed superconducting magnet), (3) treatment during scans, and (4) reduced safety hazards. Smaller C-shaped magnets are used to observe tissue damage in
8
body extremities such as hands and feet [38–40]. A similar magnet design was implemented recently for examining in situ water transport in trees [41]. Portable low-field NMR systems are available for specialized applications, such as monitoring moisture transport in skin [42], analysis of car tyres [43], measurements of cement or soils [44], and oil well-logging [45, 46]; the last example is perhaps the most significant commercial application of low-field NMR. Portable magnets are often single-sided (unilateral) devices that project an inhomogeneous B0 field, allowing the user to “see” a short distance into a sample. A popular commercial implementation is the hand-held Mobile Universal Surface Explorer (MOUSE) [47]. The so-called stray field of unilateral devices can also be used to acquire one-dimensional (1D) profiles – a technique known as stray field imaging (STRAFI) [48]. A further advantage is that the very large magnetic field gradient associated with the B0 field of unilateral devices can provide a pixel resolution over small samples that is difficult to achieve with conventional MRI methods. A specialized magnet design with a Gradient at Right Angles to the Field (GARField) [49] optimizes the STRAFI technique with shaped magnet pole pieces; this system is ideal for studying drying and curing of surface coatings. Profiling with a NMR-MOUSE is achieved by moving the magnet, and hence the resonant volume, through the sample. Well-logging tools operate on a similar principle, acquiring signal from a defined volume within the rock formation but near the well bore [45]. NMR instruments are complicated systems often capable of performing many different measurements, such as relaxation time analysis, diffusion, imaging, and spectroscopy. Such diverse instrumentation requires a skilled operator and this necessity has, to some extent, limited the commercial application of NMR in in-
9
dustrial environments. The use of low-field NMR for process monitoring has led to the development of automated spectrometers, capable of self-calibration and designed to perform a single type of experiment. These simplified systems are adequate for basic quality control, but will not necessarily provide optimum performance. For the skilled experimentalist, there are many factors to consider when installing and operating a low-field NMR system. First and foremost is the magnet: in Section 2 we consider the advantages and disadvantages of low-field NMR hardware compared to superconducting magnets. We also offer generic guidelines for selecting an appropriate magnetic field for a particular application in Section 2.2. Common hardware (magnet and rf probe) designs are considered in Section 3, including mobile NMR devices. Detailed aspects of the rf pulse performance and signal reception are also discussed. In Section 4 we summarize the experiments that are most commonly conducted at low field. Longitudinal T1 and transverse T2 relaxation times, or selfdiffusion coefficient D, are used as proxies for chemical sensitivity when spectroscopic resolution is unavailable. Multi-dimensional relaxation and diffusion experiments are popular because they provide more information than their onedimensional counterparts, notably detection of diffusive or chemical exchange. Careful data analysis is required at low field, especially when the SNR is poor; data processing techniques are considered in Section 5 where we focus on time domain analysis. We note that even in spectroscopic measurements, time-domain fitting can provide more robust identification of chemical species than traditional Fourier transform spectroscopy at low field. Many of the industrial samples investigated using low-field NMR fall within the remit of “porous media”. Heterogeneous, fluid-saturated materials are studied
10
using low-field NMR techniques as the magnetic susceptibility contrast between the solid and fluid in such systems results in pore-scale magnetic field gradients (so-called “internal gradients”) which distort the NMR signal. These internal gradients are known to scale with magnetic field strength, and so the use of a low-field spectrometer is one method for reducing sample-dependent magnetic field inhomogeneities. Porous materials that have been studied include cellular structures in biomedicine and food production, cement-based construction materials, reservoir rocks in petrophysics, and catalysts in chemical engineering. Interpretation of the data obtained from such systems can be complicated and we offer advice for converting NMR parameters into sample properties in Section 6. We consider three industrial applications of low-field NMR in Section 7: food, construction, and petroleum. Finally, in Section 8, we consider the changing world of low-field NMR and look to possible future developments. The majority of low-field experiments interrogate 1 H (proton) spins; the high natural abundance and large gyromagnetic ratio of the hydrogen nucleus provides a detectable magnetization even with low-field instrumentation. However, heteronuclear NMR is possible at low field; other detectable nuclei include 13 C,
23 Na,
and/or 31 P, although their lower natural abundance and gyromagnetic ratios
make detection difficult except at the upper end of the low field range. Unless otherwise stated, all techniques described herein pertain to 1 H detection. 2. Which Field Strength? There has been a continual drive in NMR and MRI for the manufacture of stronger magnets. A higher magnetic field strength provides better SNR, enabling enhanced spectroscopic resolution and improved image resolution, or shorter ac11
quisition times. In MRI, the construction of high field imaging magnets enables in situ spectroscopy, providing localized studies of tissue chemistry. However, very strong magnetic fields are not always desirable in an industrial context. In this Section we consider the advantages and disadvantages of high and low-field magnet technology, and offer suggestions for selecting an appropriate low-field magnet for industrial applications. 2.1. NMR magnets and industrial environments There are several disadvantages to the installation of a superconducting magnet in an industrial environment: • The stray magnetic field presents a serious safety hazard. • High economic cost (an average system is around $500, 000). • Supply and containment of cryogens (liquid helium and nitrogen). Cryogenic cooling presents an additional operating cost and hazard. The worldwide supply of liquid helium is limited and non-renewable although modern magnet designs require only small volumes of cryogenic liquids, have low boil-off rates, and include mechanical refrigeration to reduce or even eliminate helium losses. Beyond the practical limitations of installing a high-field magnet in an industrial environment, strong magnetic fields are inappropriate for the study of fluids in many heterogeneous (porous) materials such as oil-field reservoir rocks, cementbased building materials, and some biological systems, e.g., lungs. The magnetic susceptibility contrast between the solid and fluid phases results in large, porescale magnetic field gradients (so-called “internal gradients”) that distort measure12
ments of relaxation time, diffusion, and chemical spectra, or can prevent detection of a NMR signal entirely due to rapid dephasing of the spin ensemble [50]. Low-field permanent magnets can provide a suitable compromise between magnetic field strength and experimental versatility for installation in industrial environments and the investigation of heterogeneous materials. Several advantages of low-field instruments over their high-field superconducting counterparts are: • No cryogenic cooling. • Lower economic cost (a typical bench-top system is around $100, 000). • Reduced safety considerations (negligible stray field). • Versatility of design including the possibility of open-access and singlesided magnet arrangements. A particular disadvantage of low-field permanent magnets is the field stability. Thermal fluctuations in the magnet poles will cause the B0 field to vary and hence cause a shift in the Larmor frequency. To improve temperature stability, permanent magnets are typically thermostatted to super-ambient temperatures such as 35 − 40◦ C, depending on the local climate. The presence of a large thermal mass, such as an iron yoke, improves the temperature regulation. Magnetic materials with opposite temperature coefficients may be combined to reduce field drift when the temperature fluctuates. Small temperature drifts are unavoidable and methods exist to compensate for such inevitable fluctuations [51]. For example, the Larmor frequency may be measured and corrected between scans. Alternatively, a frequency lock can be installed including a B0 shim coil, to automatically 13
and continuously maintain the correct Larmor frequency. A low cost solution to improving temperature performance is to surround the magnet with insulation, such as a layer of expanded polystyrene or similar material; this simple approach is particularly effective where air-conditioning units produce circulating currents of cold air. Small shifts in frequency can be accommodated in relaxation or diffusion experiments when the signal is acquired in the time domain and summed over all frequencies. However, frequency shifts can be much more problematic during imaging or spectroscopy measurements. Excellent thermal regulation is essential on permanent magnet systems designed for such experiments. 2.2. A rough guide to selecting a magnet Selecting an appropriate low-field magnet for a given application will be a compromise between numerous factors: • Sample size. • Magnetic field strength. • Magnetic field homogeneity. • Experimental capability. • Logistical limitations (size, weight, portability). • Cost. Many process monitoring and quality control tasks rely on basic relaxation time analysis of bulk liquid or soft-solid samples. For such industrial applications, a standard bench-top magnet is appropriate. These systems have a typical field strength of B0 = 0.5 T (ν0 ≈ 20 MHz) and rf probe bore diameters in the range 14
10 − 20 mm. Stronger permanent magnets can be manufactured by the addition of more magnetic material, although this will increase the cost, size, and weight of the instrument. For a given magnet size there is an inverse relationship between pole gap (which determines the rf probe size and hence sample volume) and magnetic field strength. In addition to the magnet, a basic relaxometry system requires only a spectrometer (including acquisition computer) and rf amplifier. These days, fully digital spectrometers are available, capable of direct synthesis and acquisition of rf signals up to about ν0 = 40 MHz, thereby reducing the complexity of the hardware. Such digital spectrometers were originally based on the technology (chip sets) manufactured by the mobile telecommunications industry. For heterogeneous materials or larger sample volumes it is necessary to adopt a lower field strength such as B0 ∼ 0.24 T (ν0 ≈ 10 MHz). The pole gap can be increased to accommodate rf probe diameters of 50 mm within the confines of a bench-top system. Studies of extremely heterogenenous samples will require field strengths of B0 < 100 mT (ν0 < 4 MHz) in order to extract quantitative signal intensities. Low-field NMR systems are appropriate for studies of large sample volumes. At low frequencies, solenoid rf probes with small bore diameters can be difficult to tune due to their low inductance. A larger bore typically leads to an increased inductance which is easier to tune. For more complicated NMR experiments such as diffusion, flow, and droplet sizing, a 1D magnetic field gradient coil is required. Most commercial benchtop magnet designs can accommodate a single-axis gradient without modification, although the maximum bore diameter, and hence rf probe bore diameter, will be reduced. The addition of even a 1D gradient set increases the complexity of the extraneous hardware, requiring a high-powered gradient (audio) amplifier and
15
sometimes a gradient plate cooling system (typically a circulating water chiller). The strength of the gradient produced will depend on the design of the gradient coils; the orientation (gradient axis) may also influence the gradient strength. Again, depending on design, it may be possible to obtain a stronger gradient by changing the orientation. This approach is useful for bulk diffusion studies. A 1D gradient is most usefully aligned parallel to the long axis of the rf probe bore, allowing flow measurements and 1D profiling to be implemented, though this orientation may not provide the maximum achievable gradient strength. Full three-dimensional (3D) imaging obviously requires a 3D gradient set plus three audio amplifiers. Considering each audio amplifier will have a typical cost of $50, 000, the change from a basic relaxometry spectrometer (no gradients) to a 3D imaging magnet effectively doubles the cost of the entire system. Most bench-top magnet designs can be upgraded to accommodate a 3D gradient set; the potential value of 3D gradients should be considered carefully. Obviously, applications in MRI for clinical diagnosis, biomedicine, and pharmaceuticals (e.g., imaging of drug tablet dissolution [24]) demand this capability. Very low field applications, such as petrophysics, may not benefit: the SNR per voxel will be inherently poor, necessitating long acquisition times (many repeat scans) and low image resolution [14]. For such applications a 1D gradient coil for diffusion and profiling is sufficient. Spectroscopy systems have the very different requirement of excellent magnetic field homogeneity. Useful 1 H spectra can be recorded at field strengths of B0 = 1 − 1.5 T. The limitation in permanent magnet technology is the volume over which the magnetic field can be made sufficiently homogeneous for subparts per million (ppm) resolution. In current bench-top designs the sample vol-
16
ume is restricted to a 5 mm diameter tube. A particularly important consideration when purchasing a bench-top spectroscopy system is thermal stability; the results are susceptible to slight drifts in magnetic field (or shim) caused by temperature fluctuations. As with other bench-top systems, a 1D gradient set can be advantageous for chemical identification through diffusion ordered spectroscopy (DOSY) measurements, as well as enabling the usual diffusion-based applications such as droplet sizing. Spectroscopic resolution may also be obtained on more conventional magnet designs operating at lower magnetic field strengths by using rf micro-coils to examine very small sample volumes [52], or even on unilateral devices with careful design of the magnetic field profile [53–55]. Another factor to consider is portability. Numerous designs for portable NMR devices can be found in the literature although only a few, such as the NMRMOUSE [27, 47, 56], are commercially available. Of course the concept of portability is open to interpretation: early “portable” NMR systems were large and heavy, having to be transported on a vehicle. Most bench-top NMR systems can be made portable under the same criteria, although locating or transporting a suitable power supply, notably for gradient amplifiers, can be a limitation. Here we refer to portable systems as magnets conceptualized specifically to be taken to the sample: the NMR-MOUSE is effectively a hand-held device that is used outside the laboratory for studies of art and cultural heritage [57, 58], the “Surface GARField” magnetic is used to study concrete in the built environment [44, 59], and the “tree-hugger” magnet is designed to examine water transport in trees [41]. Oil well-logging tools are another example of specialized mobile NMR instrumentation [45, 46]. These unilateral portable systems acquire signal from an external (stray) magnetic field. Such stray fields are known to be characterized by large
17
and spatially variant magnetic field gradients. The signal is therefore detected from a defined sensitive volume: a plane or volume over which the magnetic field is constant. The shape and position of the sensitive volume will be determined by the design of the magnet and also the interaction of the B0 and B1 fields; further discussion is given in Section 3. Therefore, the sensitive volume can be adjusted by altering the rf excitation and detection frequency; this approach is used in some well-logging tool designs [60], see Section 3.1.3. Unilateral NMR devices offer the possibility of high-resolution 1D profiling by moving the sensitive volume through the sample. In additional, both the Surface GARField and Profile NMRMOUSE [61] provide spatial (depth) resolution in surface studies [62]. However, unilateral devices typically provide poor SNR compared to conventional benchtop systems due to the small resonant volume within a sample and hence the poor rf coil filling factor. For laboratory applications, the presence of a large magnetic field gradient can cause other complications: if a sample is enclosed by fluorinated plastic or composite materials, commonly used due to their low 1 H content, the resonance condition for 19 F will be met somewhere in the magnetic field. Therefore, signal will be obtained simultaneously from the required 1 H sensitive volume and also a different volume sensitive to 19 F. Unilateral NMR devices are best used in remote locations, away from sources of rf interference (noise). To summarize, there are three simple questions that should be answered before purchasing a low-field permanent magnet: • Where will the NMR device be located? For most laboratory or industrial process control applications, a bench-top instrument will be appropriate. The lack of a stray magnetic field on these instruments makes them suitable for location in industrial environments without the safety concerns associ18
ated with superconducting magnets. If the system is to be used outside a laboratory or factory, or if the sample is an unusual size or shape, then a unilateral (mobile) NMR device should be considered. • What magnetic field strength is required? The magnetic field strength will be determined largely by the sample. Heterogeneous (porous) materials containing large quantities of paramagnetic or ferromagnetic impurities (e.g., rocks, catalysts) will benefit from very low field strengths (B0 < 100 mT) to minimize internal gradients and allow quantitative signal intensities to be obtained. Otherwise, higher field strengths are preferable (improved SNR) although a compromise between sample volume and B0 will exist. Field strengths of B0 > 1.5 T can be achieved with permanent magnets but only by significantly increasing the volume of magnetic material. • Which experiments are to be conducted? Most NMR experiments (relaxation, diffusion, imaging) can be conducted at any field strength; the exception is spectroscopy which requires an intermediate-strength homogeneous magnetic field (B0 ∼ 1 T). Basic relaxation time (T1 and T2 ) distributions can be acquired on any bench-top system. Diffusion-based experiments and profiling requires a 1D gradient. Imaging will require 2D or 3D gradients. If a system is to be used only for a single, defined application – as is typical in an industrial environment – the minimum required specification should be chosen to simplify operation. For research applications it is desirable to over-specify the instrument capabilities to provide versatility for future projects.
19
3. Low-Field Spectrometers In this Section we consider the hardware behind low-field NMR experiments with permanent magnets. First, we examine the most common arrangements of magnets and rf probes currently used in commercial NMR spectrometers. Then we consider the optimization of the tuned rf circuit and the recent phenomenon of digital filtering in low-field spectrometers. Finally we discuss noise sources and hardware modifications to maximize the SNR achievable in low-field experiments. 3.1. Permanent magnet configurations 3.1.1. Bench-top magnets There are two designs that currently dominate commercial bench-top magnets for use in laboratory environments. A simple and robust design consists of two magnetic pole pieces (parallel plates) mounted in an iron yoke, see Fig. 2(a). The iron yoke minimizes the stray field. The amount and type of magnetic material used in the pole pieces, plus the pole gap, determines the field strength B0 between the poles. Typically the pole pieces are formed from neodymium iron boron (NdFeB). An alternate permanent magnetic material used in some designs is samarian-cobalt (SmCo) which has an inverse temperature coefficient compared to NdFeB. Therefore, magnet designs incorporating both SmCo and NdFeB magnetic material can provide a magnetic field that is insensitive to temperature fluctuations; a similar effect is achieved with an alloy of SmCo and GdCo. Temperature regulation in the parallel plate magnet is achieved by heating the iron yoke to superambient temperatures, usually 35 − 40◦ C. The B0 homogeneity may be improved by the inclusion of shim coils; gradient coils can also be added to enable diffusion and imaging experiments. The direction of the B0 field across 20
the pole gap allows the rf antenna to be constructed as an inductive solenoid, unlike in superconducting magnets where the rf antenna is a birdcage or saddle coil arrangement. Solenoid rf coils are simpler to design and tune, and provide improved B1 homogeneity compared to the alternatives. The B1 homogeneity can be further improved by adjusting the loop spacing at either end of the solenoid to compensate for the finite length of the inductor. The iron yoke and rf probe housing act as effective screens to external noise sources. The parallel plate magnet design shown in Fig. 2(a) can be converted to an open access (so-called C-shaped magnet) by the removal of one end face of the iron yoke. The parallel plate design is therefore straightforward to construct and versatile in implementation. An alternative magnet design that has become popular for bench-top NMR is the cylindrical Halbach permanent magnet [63], see Fig. 2(b). A series of polarized magnetic blocks are placed in a circular pattern, where the direction of polarization of each block is rotated slightly relative to its immediate neighbors. In the limit that an infinite number of blocks are used, a uniform B0 field is generated across the ring of magnets with zero stray field. Identical rings are stacked to form a cylindrical magnet. Obviously the practical constraints of building these Halbach magnets ensure that the B0 field is not perfectly homogeneous and there will be some small stray field. Notwithstanding, careful construction can provide a very good approximation to the ideal design and the Halbach magnet can provide improved B0 homogeneity compared to the parallel plate magnet design in Fig. 2(a). Constructing a good Halbach magnet does rely on the selection of nearidentical magnetic blocks [63]; heterogeneities in the B0 field of Halbach magnets are considered harder to correct by shimming than in their parallel plate counterparts, and temperature regulation is more difficult due to the lack of an iron
21
yoke. Gradient coils and the rf coil are located inside the Halbach magnet ring. Again, the alignment of the B0 field across the long axis of the magnet, as shown in Fig. 2(b), enables the use of a solenoid rf coil. Halbach magnets can be designed for open access by placing a second Halbach cylinder of magnets around the first: by rotating the outer cylinder, the magnetic field of the inner cylinder can be negated, allowing the magnets to be separated with minimum resistance. This open-access Halbach concept is embodied in the NMR-CUFF [64], used to monitor water transport in plant stems. Both the parallel plate magnet and the Halbach magnet can be constructed with a wide range of magnetic field strengths for most laboratory applications. Parallel plate and Halbach magnets are readily available at field strengths ranging from B0 = 50 mT (ν0 ≈ 2 MHz) for rock core analysis [14] to B0 = 1 T (ν0 ≈ 42.5 MHz) for spectroscopy [65, 66]. 3.1.2. Ex situ NMR Portable magnet designs are required for in situ measurements of materials in industrial environments. The NMR-MOUSE is one of the simplest and most robust NMR devices available [67]. It was designed to allow NMR surface measurements to be conducted on arbitrarily large samples [56, 61, 68]. The original NMR-MOUSE was designed around a bar magnet and surface rf antenna. More complicated versions of the MOUSE have been constructed over the years with increasing numbers of permanent magnetic components: the two-pole MOUSE (also called the U-shaped MOUSE) is illustrated in Fig. 3 where the B0 field is generated by two anti-parallel polarized permanent magnets mounted on an iron yoke to increase the magnetic flux on the upper surface [43]. Improved B0 homogeneity can be achieved using multiple magnets (four or more) [69], or by affixing 22
a B1
N
S y z
B0 b
B0
x z
Figure 2: Schematics of standard bench-top (laboratory) permanent magnet configurations: (a) parallel plate magnet and (b) cylindrical Halbach magnet. In (a) the B0 magnetic field is generated on the z-axis between the pole pieces (light gray). Shim and gradient plates (medium / dark gray) can be included. The sample is placed in a solenoid rf coil (blue) which generates a B1 field in the y-axis. In (b) the magnetic pole pieces are arranged in a rotating pattern (indicated by arrows) to form a cylindrical Halbach magnet with a B0 field in the z-axis. Shim and gradient coils (dark gray) can be included. The sample is placed in a solenoid rf coil (blue) which generates a B1 field in the y-axis. Note the reference axes are different in (a) and (b).
23
shaped pole pieces molded in soft iron to the pole faces [70, 71]. The rf antenna, usually of a 2D surface design (e.g., printed circuit board) is located between the permanent magnet pole pieces. Signal is detected from a sensitive volume in the stray field of the magnet where the inhomogeneous B0 and B1 fields are perpendicular [62]. This sensitive volume is located a short distance (∼ 10 mm) above the face of the magnet. Hence the MOUSE can be used to “see inside” a planar object. As the strength of the B0 field decays away from the surface of the magnets, the exact height of the sensitive volume on the y-axis is determined by the chosen Larmor frequency. Each position in the B0 field will also have an associated magnetic field gradient that can be used for diffusion measurements. The NMR-MOUSE concept is particularly useful for studying molecular reorientations in soft solids such as polymeric rubber [72]. sensitive volume
B0 B1 N
rf antenna S
y z iron yoke
Figure 3: Schematic of the two-pole NMR Mobile Universal Surface Explorer (MOUSE). In this basic configuration, two permanent magnet pole pieces are mounted on an iron yoke. The B0 field is approximately parallel to the z-axis above the center of the magnet. A surface rf antenna generates a B1 field approximately parallel to the y-axis above the center of the magnet. Signal is obtained from a sensitive volume defined by the exact shapes of the B0 and B1 fields. More complicated MOUSE designs have been presented with larger numbers of pole pieces to provide better definition of the sensitive volume.
24
3.1.3. Well-logging tools The history of NMR well logging dates back almost to the inception of the NMR experiment: a patent was filed in 1952 by Russell Varian on the use of the Earth’s field for NMR logging. The Nuclear Magnetic Logging (NML) tool was developed by Chevron and Schlumberger, although it did not achieve great commercial success. However, continued investment in that technology led to the development of successful logging tools by Schlumberger and NUMAR (now Halliburton). Schlumberger launched its series of Nuclear Magnetism Tools (NMT) in 1978; these were superseded in the 1990’s by the Combinable Magnetic Resonance (CMR)1 tool. This design had the important capability of being combined with a multitude of other logging tools on a single tool string (as indicated by its name). The CMR tool contains permanent magnets arranged in such a way as to generate a defined “sweet spot” of known volume from which the NMR signal is detected, as illustrated in Fig. 4(a). The resonant frequency of the sweet spot is nominally ν0 = 2 MHz. An alternative design, adopted by NUMAR, is a gradienttool implementation where the magnetic field decays continuously away from the pole face of the magnets; this design is akin to the NMR-MOUSE concept. Different resonant “shells” are excited by varying the Larmor frequency, each with a unique position and associated magnetic field gradient strength. Halliburton now operate the Magnetic Resonance Imaging Logging (MRIL)2 tool [73] – here, “imaging” refers to the (vertical) spatial resolution achieved as the tool moves up the well-bore measuring continuously as well as to the ability to acquire signal at different depths into the formation. The MRIL tool operates at a nominal reso1 Mark
of Schlumberger. is a registered trade mark of Halliburton.
2 MRIL
25
nance frequency of ν0 = 1 MHz. Schlumberger offer a similar tool design called the Magnetic Resonance Scanner (MRScanner)1 , illustrated in Fig. 4(b), which operates at ν0 = 1.4 MHz or less, depending on the resonant shell selected [46]. Logging tool designs are reviewed in [74]. a
well bore bowspring
S
N
S
N
sensitive volume rf antenna
b
N
S
well bore bowspring
multiple resonant shells rf antenna
Figure 4: Cross-section schematics of Schlumberger’s magnetic resonance logging tool sensor: (a) the Combinable Magnetic Resonance (CMR) tool [45] and (b) the Magnetic Resonance Scanner (MRScanner) [46]. The CMR tool (a) has magnet pole pieces designed to provide a defined “sweet spot” of known volume. The MRScanner (b) has a decaying magnetic field gradient away from the tool, so varying the resonance frequency of the rf field enables detection of signal from different resonant “shells”. Each resonant shell has an associated magnetic field gradient that is used for diffusion measurements. In both tools the bowspring keeps the face of the tool in contact with the well bore.
The industry standard NMR logging measurement is a T2 relaxation time distribution and all logging tools offer this capability. The MRIL and MRScanner 26
tools offer the ability to measure diffusion coefficients utilizing the magnetic field gradient that is approximately constant across the resonant volume [46, 75]. CMR, on the other hand, is not appropriate for diffusion measurements because the field gradient varies across the resonant volume; however, because the sweet spot volume is defined precisely, the CMR provides calibrated measures of liquid volume and hence porosity in the vicinity of the well bore [45]. MRScanner is the tool of choice in monitoring operations with cased wells to determine long-term changes in a reservoir, whereas CMR is the tool of choice in piloting and exploration wells as it can be run simultaneously with other logging tools. Logging tool designs are modified to enhance the sensitivity and reliably of acquired data, and hence improve the interpretation. One area of ongoing development is the implementation of logging-while-drilling (LWD), in which an NMR logging tool is mounted on the drill string to provide immediate feedback on formation and enable informed decisions for directional drilling. 3.2. Radio frequency pulses Theoretical descriptions of spin physics are often based on ideal rf behaviour, where the spin ensemble is assumed to experience an applied oscillating field of the amplitude, duration, and frequency specified. However, this situation is never encountered in reality and consideration must be given to non-ideal rf transmission. This is especially important at low field where a rf pulse may contain only a few cycles. The Larmor (precession) frequency of the spins is defined by the magnet according to ωL = −γ B0 . The rf probe is tuned so its impedance is entirely resistive at the angular frequency ω0 ; tuning such that ωL = ω0 is usual but not necessarily accurately achieved. Furthermore, the probe (as for any linear resonant system) 27
will have a free-ringing frequency ωr that differs slightly from ω0 . This is of practical consequence where rf pulses have a significant period of free ringing, after the rf drive from the transmitter amplifier is removed. Ideally, the condition
ωr = ωL would be satisfied; however the tuning is unlikely to be either accurate or stable enough for this to be achieved in practice. The experimentalist defines a forcing (driving) frequency ωd to be produced by the spectrometer and amplified prior to transmission to the probe. Depending on the fidelity of the rf amplifier stage, the excitation pulse may or may not be a good representation of an ideal sine wave constrained within the envelope of the rf pulse. The forcing frequency is typically set to ωd = ωL , although in broadline samples spins will precess at a range of frequencies and excitation will never be uniform across the spin ensemble; the CPMG experiment is particularly sensitive to off-resonance spin dynamics [76, 77]. Furthermore, in imaging and spectroscopy, slice- or chemically-selective soft rf pulses are applied off-resonance, with the forced frequency chosen specifically so that ωd 6= ωL . In these cases, one assumes that the quality factor Q (a measure of the width of the resonance curve) of the probe is sufficiently low that the excitation pulse is transmitted without significant distortion. Theoretically, then, the condition ωr ' ωd ' ωL would always be met. Clearly this is not practical, or necessarily desirable. It is important to realize that a mismatch between the forcing frequency and the free-ringing frequency of the probe may lead to unexpected transient responses, which in turn leads to complicated spin dynamics. The duration of the transients is approximately 2Q/ω0 ; hence, at high field (typically Q ∼ 100 but also very large ω0 ) the transient may be shortlived compared to the duration of the rf pulse and if so can be ignored. However,
28
at very low fields (a smaller Q ∼ 10 but also much smaller ω0 ) the transient can be a significant fraction of the rf pulse. It is therefore important to understand the origin and influence of pulse transients in low-field experiments. We now present the theoretical expressions governing pulse transients in order to highlight the interplay between the forcing, free-ringing, and resonance frequencies. In Section 3.2.2 we provide a visual demonstration of theoretical pulse transients in low-field experiments. 3.2.1. Elementary circuit theory Transient effects in NMR excitation pulses can result in phase errors in the spin dynamics [78, 79]. These transients occur if: 1. The forcing frequency is very different from the free ringing frequency of the probe (ωd 6= ωr ). 2. Pulse switching (on/off) does not occur at zero-crossings of the carrier wave. 3. Pulse switching is not coherent with Larmor periods. Let us consider a simple rf probe tuned so that the impedance is real at the resonance frequency, i.e., ω0 = ωL . Conventional rf probes are constructed around the tuned series-parallel circuit illustrated in Fig. 5(a). Discussion of rf resonance circuits (tank circuits) appropriate for low-field probes is given in [80]. More complicated rf circuit designs are considered in [81]. Analysis of the pulse transients is conducted by representing our rf probe as an equivalent series LCR (inductor-capacitor-resistor) circuit, see Fig. 5(b). This series circuit representation is possible for any linear system, and has sufficient generality in its impedance to cover any actual single-resonance probe constructed from linear passive circuit elements. 29
LP
a CS
CP RP
b LS
CS
RS
Figure 5: Basic schematics for (a) series-parallel LCR circuit on which the designs for conventional low-field solenoid rf probes are based and (b) the equivalent series LCR circuit. The subscripts S and P indicate series or parallel components, respectively.
The time-varying current in the rf probe I(t) is proportional to the magnetic field B1 generated within the coil, and will be represented as the imaginary part of a complex current I(t) = ℑ{i(t)}, following the arbitrary convention in [78]. Given a complex driving voltage v(t) (such that the physical voltage V (t) = ℑ{v(t)}), the current obeys the differential equation dv di d2 i LC 2 + RC + i(t) = C , dt dt dt
(1)
and the driving voltage is harmonic with a driving (angular) frequency ωd , an abrupt start at t = 0, and initial phase angle θ : V (t) = 0
for t < 0
(2)
= V0 exp { j (ωdt + θ )} for t ≥ 0. The series circuit differential equation is conveniently represented as d2 i dv di + ∆ω + ω02 i(t) = ω02C , 2 dt dt dt
(3)
where ω02 = 1/LC and ω0 /Q = R/L. At the angular frequency ω0 , the circuit impedance is purely real (resistive), and ω0 /Q is a measure of the half-width of the circuit resonance written in terms of the circuit quality factor. It is shown in 30
elementary texts that the free-ringing frequency ωr , arising from solutions of the homogeneous equation d2 i ω0 di + + ω02 i(t) = 0, dt 2 Q dt
(4)
differs from the real-impedance frequency ω0 according to
ω02 − ωr2 =
ω02 . 4Q2
(5)
The steady-state solution to the forced problem, eq. (1) given eq. (2), is found by standard complex impedance methods, and given by is (t) = I0 exp { j (ωdt + θ − ϕ )} ,
(6)
with the current amplitude I0 =
V0 cos ϕ , R
(7)
and the phase lag ϕ from the forcing voltage given by ! Q ωd2 − ω02 . tan ϕ = ωd ω0
(8)
It can be shown [78] that the transient response that occurs when the driving current is switched on is ih (t) = I0 e−ω0t/Q [(A − 1) exp {− j(ωrt + −θ + ϕ )} −A exp {+ j(ωrt + θ − ϕ )}] ,
(9)
where 1 A= 2
ωd + ωr ω0 ω0 exp {+ jϕ } −j . +j ωr 2ωr Q 2ωr Q cos ϕ
(10)
The last term in A (neglected in [78]) describes the impulsive part of the forcing function. When the rf pulse is applied, if other than at a zero-crossing, there will 31
be an impulse response corresponding to the step change in driving voltage. The current in the rf coil will be i(t) = is (t) + ih (t) over the time interval 0 ≤ t ≤ tω , where tω is the duration of the rf pulse: the time for which the forcing frequency is applied. There will be a similar transient response when the driving current is turned off. If the rf pulse has a duration of tω then the transient response after t = tω is obtained from eq. (10) by the simple substitutions t → t 0 +tω and θ → θ 0 − ωdtω . The current in the coil after the forcing frequency has been removed will be −ih (t) for t > tω . Barbara et al. [79] extended the theory to include the response of the tuning and matching capacitors in the rf probe. They also demonstrated that such theoretical descriptions of phase transients do not precisely match experimental observations; they attributed the differences to treating the rf amplifier as an ideal non-reactive voltage source in their calculations. As we will discuss in the next section, the finite response time of the rf amplifier can reduce both the severity of phase transients and the sensitivity to voltage phase when the pulse is turned on or off. 3.2.2. Spin dynamics during pulse transients A steady-state current of amplitude I0 applied to the rf probe results in a steady-state magnetic field of magnitude B1 . If we define B1 as being on the y-axis (rotating frame) then the spins experience a rotation about the y-axis of angle γ B1tω , where the spin precession frequency around the applied magnetic field is ω1 = −γ B1 . However, the transient response of the rf pulse will generate a time-varying magnetic field in the xy-plane, and hence the spins will experience a time-varying axis of precession. The full effect of a non-ideal rf pulse on the spin 32
dynamics is therefore complicated, and most readily determined by simulation [79]. Here, we present three examples of rf pulses under different conditions. In all cases the driving frequency ωd (supplied by the spectrometer) is 4π × 103 rad s−1
higher than ωL and the tuned frequency of the probe ω0 is 4π × 103 rad s−1 lower than ωL . Practically, it is easy to set both the probe tuning and driving frequencies closer to the Larmor frequency than ±4π × 103 rad s−1 , but these values were chosen to provide a clear visual demonstration of the effects of such experimental imperfections on the spin dynamics. The first two examples are relevant for very low-field experiments with ν0 = 2 MHz. In Fig. 6, the performance of a moderate Q = 15 rf probe is simulated. The current in the probe, Fig. 6(a), is dominated by the slow rise and fall times (> 10 µs) of the transient responses when the rf pulse starts (t = 0 µs) and stops (t = 14.62 µs). The current is set to zero whenever the corresponding driving voltage amplitude V0 < 0.2 V to simulate crossed diodes to ground in the LCR circuit. The B1 field generated by the rf pulse is shown in Fig. 6(b); again, the slow transient response of the probe is evident in the magnitude of the magnetic field. The angle of phase rotation of the B1 vector relative to the y-axis is also shown in Fig. 6(b). There is a significant deviation in φrot during the transient intervals; these notable phase rotations have little impact on the spin dynamics, shown in Fig. 6(c), due to the corresponding low magnitude of the B1 field. The transient behavior is sensitive to errors in the probe tuning frequency ωp of ∼ 10 kHz relative to the Larmor frequency. A small yet continual drift in the B1 phase angle is observed throughout the rf pulse, caused by the offset between the driving and the Larmor frequencies. This change in phase angle shifts the
33
magnetization vector away from the x-axis. Although small, such phase errors will accumulate in multi-pulse sequences, notably CPMG. The steady-state behavior is sensitive to errors in the forcing frequency ωd of ∼ 1 kHz relative to the Larmor frequency. It is therefore extremely important, especially in CPMG-like experiments, to set the driving frequency correctly. An example of a CPMG echo train using an imperfect pulse as shown in Fig. 6, is shown in Fig. 7, where the effects of the imperfect rf pulses are clearly visible. A “zig-zag” phenomenon is present as a beating oscillatory pattern in the components of the magnetization vector. This simplistic simulation neglects two relevant physical properties of the experiment: (1) spin relaxation that leads to complications in CPMG experiments due to off-resonance spin dynamics, and (2) the transient response of the rf amplifier. Depending on the response time of the rf amplifier, the transmitted pulse may be subject to significant pulse transients; logically, trying to generate a large, instantaneous voltage (e.g., θ = π rad, equivalent to a step from zero to maximum voltage) will provide a transient response that will not be present if the rf pulse starts at a zero-crossing in the waveform. To some extent, the damping effects of finite amplifier response time appear to minimize the influence of start-up condition on the rf pulse. Paradoxically, impulse response effects may be more pronounced with high-specification amplifiers than with slow-response devices. It is found empirically that improved CPMG echo train performance is obtained with either coherent pulsing (fixed θ ) or the use of shaped (e.g., trapezoidal) rf pulses to control the rise and fall times. The performance of very low-field rf probes (ν0 < 5 MHz) can be improved using more sophisticated rf circuit designs. For example, Oxford Instruments’ GeoSpec instruments, shown in Fig. 1, include “Q-Sense technology” based on
34
a
4
={i(t)} / A
2 0 −2 −4 0
5
10
15
20
25
30
time / µs b 0.5 B1 / mT
0.5
0.3 0 0.2 −0.5
0.1 0 0
5
10
15
20
φrot / rad
1
0.4
−1 30
25
time / µs
M(t)/M(0)
c
1 0.8
Mx My Mz
0.6 0.4 0.2 0 0
5
10
15
20
25
30
time / µs
Figure 6: Simulated transient response of a very low-field rf probe with modest Q = 15. In (a) the probe current (solid blue line) is calculated with a driving frequency offset of +2 kHz and a tuning frequency offset of −2 kHz relative to the Larmor frequency of 2 MHz (red dotted line), and assuming a 500 W rf amplifier driving into a matched 50 Ω load. The corresponding magnetic field (b) is obtained for an ideal solenoid of length L = 10 cm and N = 10 turns. The spin dynamics (c) indicate that the chosen pulse length of tω = 14.62 µs provides a 90◦ rotation (Mz → Mx ). The solid black line in each plot indicates the end of the driving current at time tω = 14.62 µs.
the works of Hoult [82]. A combination of passive circuit components provides a low Q ' 3 probe with the SNR of a higher Q circuit. In the general LCR series circuit, Q is reduced by increasing R since Q = ω0 L/R. A low Q probe has a much faster transient response, as demonstrated by the simulation in Fig. 8. The transient response now lasts only 2 µs, hence providing a significant reduction in the physical probe dead time after the end of the rf pulse. The overall spin dynamics in Fig. 8(c) are unchanged, with the obvious exception that the rotation of the magnetization vector is almost complete when the forcing current stops. As a rule-of-thumb, the transient response lasts approximately Q cycles of
35
1
M(t)/M(0)
0.8 0.6
Mx My Mz
0.4 0.2 0 −0.2 −0.4 0
50
100
150
200
250
300
350
400
450
500
number of echoes
Figure 7: Simulation of magnetization at spin echo maxima in a standard CPMG pulse sequence with rf pulses of the quality shown in Fig. 6. Imperfect refocusing results in a regular oscillation in the components of the magnetization vector. This simulation ignores spin relaxation, other offresonance effects, and phase cycling schemes that are usually employed to minimize such artifacts.
the rf frequency. Hence the influence of the rf probe quality factor decreases as the Larmor frequency increases. The third rf pulse simulation, Fig. 9, shows the response of a rf pulse on a ν0 = 20 MHz system with a modest Q = 30 probe. Despite the higher Q compared to the previous examples, the transient response only accounts for a small proportion of the total rf pulse, with rise and fall times of < 2 µs. A higher Larmor frequency also reduces the sensitivity to offsets in the tuning and driving frequencies, as reflected in the final state of the magnetization vector in Fig. 9(c); the phase shift in the xy-plane is notably smaller than in the earlier examples (Fig. 6 and 8). Incidentally, precise setting of the tuning frequency becomes more important as the Q is increased as an rf pulse applied away from ω0 will suffer significant electrical loses, thereby requiring longer rf pulses to achieve the desired tip angle. 3.3. Digital filtering Modern low-field spectrometers now have the same digital filtering capabilities as their high-field counterparts. Digital filtering is important for low-field experiments as careful design and implementation of filters can provide a significant reduction in amount of noise observed in a time-domain signal. In spectroscopy 36
a
4
={i(t)} / A
2 0 −2 −4 0
5
10
15
20
25
30
time / µs b 0.5 B1 / mT
0.5
0.3 0 0.2 −0.5
0.1 0 0
5
10
15
20
φrot / rad
1
0.4
−1 30
25
time / µs
M(t)/M(0)
c
1 0.8
Mx My Mz
0.6 0.4 0.2 0 0
5
10
15
20
25
30
time / µs
Figure 8: Simulated transient response of a very low-field rf probe with low Q = 3, consistent with Oxford Instruments’ Q-Sense technology. In (a) the probe current (solid blue line) is determined with a driving frequency offset of +2 kHz and a tuning frequency offset of −2 kHz relative to the Larmor frequency of 2 MHz (red dotted line), and assuming a 500 W rf amplifier driving into a matched 50 Ω load. The corresponding magnetic field (b) is obtained for an ideal solenoid of length L = 10 cm and N = 10 turns. The spin dynamics (c) indicate that the chosen pulse length of tω = 14.62 µs provides a 90◦ rotation (Mz → Mx ). The solid black line in each plot indicates the end of the driving current at time tω = 14.62 µs.
and imaging, filters are useful for removing baseline artifacts, plus aliased signals and noise. In the case of low-field experiments, 1 H observation benefits from 19 F signal suppression achieved with careful filtering. A detailed description of digital filter design and operation has been presented by Moskau [83]. Here, we focus on the significance of the digital filter stage on data analysis. Our aim is to apply a filter that discriminates efficiently between the signal and the noise whilst leaving the signal unchanged. In general, this means any filter can only be designed to suppress noise outside the frequency range associated with the signal. Separa-
37
a
4
={i(t)} / A
2 0 −2 −4 0
2
4
6
8
10
12
time / µs b
1 0.5 0.4
0
0.2 0 0
−0.5 2
4
6
8
φrot / rad
B1 / mT
0.6
−1 12
10
time / µs
M(t)/M(0)
c
1 0.8
Mx My Mz
0.6 0.4 0.2 0 0
2
4
6
8
10
12
time / µs
Figure 9: Simulated transient response of a low-field rf probe with Q = 30. The probe current (a) is determined with a driving frequency offset of +2 kHz and a tuning frequency offset of −2 kHz relative to the Larmor frequency of 20 MHz, and assuming a 300 W rf amplifier driving into a matched 50 Ω load. The corresponding magnetic field (b) is obtained for an ideal solenoid of length L = 10 cm and N = 20 turns. The spin dynamics (c) indicate that the chosen pulse length of tω = 8.99 µs provides a 90◦ rotation (Mz → Mx ). The solid black line in each plot indicates the end of the driving current at time tω = 8.992 µs.
tion of signal and noise acquired at the same frequency is achieved through data processing, which will be discussed in Section 5. The process by which an analogue signal detected in the rf probe is converted into a digital signal is represented in Fig. 10. The analogue signal is initially amplified and filtered using a fast electronic (analogue) low-pass filter to remove frequency components and harmonics above the Nyquist limit (2ω0 ) of the resonance frequency. This filtering stage is necessary to prevent aliasing in the analogue to digital converter (ADC). The digital signal is then mixed with sine and cosine waveforms to provide a complex signal. Digital multipliers do not intro-
38
input
analogue filter digital digital multipliers filters
ADC preamplifier
complex signal
Figure 10: Schematic of NMR signal detection. The analogue signal passes through a preamplifier and an analogue low-pass filter before being converted to a digital signal in the ADC. Digital multipliers are then used to obtain a quadrature signal which is filtered digitally. The output is a complex digital signal.
duce gain or phase errors into the signal, unlike analogue multipliers [84]. The digital filtering is then applied before the final (complex) NMR signal is stored. 40
passband
gain / dB
0 −40 −80
stopband attenuation
transition band
−120
stopband −160 0
100
200
300
400
500
frequency / kHz
Figure 11: Schematic of a digital filter function, indicating the parameters used to define the filter performance. Frequency and gain scales are shown as an example of typical ranges. The filter shown has a total bandwidth of fsw = 1 MHz (hence a dwell time tdw = 1 µs) but the passband has a width of fpb = 400 kHz, so any frequency components outside the range ν0 ± fpb /2 will be suppressed. The passband and stopband ripple has been exaggerated for clarity.
Digital NMR filters are usually formed as a chain of digital filters that start with very high sampling rates and apply successive levels of data decimation over several stages. They are usually of the finite impulse response (FIR) class (also known as causal or nonrecursive filters) [83]. The exact form of the digital filter will depend on the receiver hardware design. However, the entire digital filter 39
chain can be described by one overall set of characteristics, mostly determined by the final filter stage, which in some instruments is user-programmable. Most digital filters will be designed to operate as low-pass filters operating on the detected, or heterodyned, signal centered on the Larmor frequency. An example of an oversampling filter is given in Fig. 11, showing the performance of the filter at frequencies > ν0 ; an antisymmetric filter function exists for frequencies < ν0 . Only signal components at frequencies within the passband, i.e., ν0 ± fpb /2, will be retained by the filter. Any signal or noise in the stopband will be suppressed; an input attenuated by 120 dB (10−6 in power) is considered to have zero amplitude. The transition band describes the manner in which the filter changes from “pass” to “stop” and has important consequences for the NMR experimentalist. On either side of the transition band, there will be some ripple in the gain; passband ripples are significant as these will distort the NMR signal. a 0
−120
gain / dB
b 0
−120
c 0
−120
frequency
Figure 12: Examples of filter designs appropriate for (a) noise suppression at very low field, (b) imaging, and (c) spectroscopy. In each case, the dotted vertical lines indicate the transition band. Exact frequency ranges will depend on sample line width and experimental parameters.
An ideal filter function will have minimal passband ripple, be close to 0 dB 40
gain over the entire passband, and have a narrow transition band. However, any filter design will always be a compromise between quality (shape) in the frequency domain and response time in the time domain. Filter selection should be determined primarily by the application. The impulse response of the filter is characterized by the group delay tGD , the time required for a signal to pass through the filter. Any signal recorded after filtering will be time-shifted by tGD ; we consider the implications of this time shift on the NMR experiment in Figs. 13 and 14. There is also a corresponding phase shift, but this is easily removed at the processing stage. The group delay is important as it defines the rise time of the filter. Data passing through the filter should not be stored until after the rise time (or dead time) of the filter tDE . As a rule-of-thumb, tDE ≈ 2tGD . The group delay is determined from the Fourier reciprocal of the filter function, and hence a high quality filter (narrow transition band) will have a long group delay and vice versa. The rf resonator itself has the properties of a linear filter, measured conventionally by the quality factor Q, introduced in Section 3.2, and so by extension we note that the rf probe has a physical group delay tGD = 2Q/ω0 that should be included in the experimental timing. This becomes more significant at low Larmor frequencies. Examples of digital filter functions for different applications are given in Fig. 12. A noise suppression filter, Fig. 12(a), suitable for very low-field applications, will have a narrow passband (determined by the spectral line width of the sample or magnet heterogeneity) and a “lazy” transition band, leading to a short impulse response time. A short response time and group delay is important when studying broadline samples or acquiring CPMG echo trains with short echo spacings. An imaging filter, in contrast, needs a flat passband to prevent image distortion, see Fig. 12(b). The transition band should be narrow (preventing aliasing of the
41
image) which will lead to a long response time. Most imaging sequences, particularly in medical applications, do not require very short acquisition times and so a long filter dead time can be accommodated. Filters for spectroscopic applications tend to be of intermediate quality, see Fig. 12(c). Again, the passband must be flat over the frequency range of interest, and the transition band should be as short as possible within the limits of the dead time imposed by the FID acquisition. transmitter output, V0 / kV
a probe group delay (transmit), 2Q/ω0
probe current, I0 / A or rf field, B1 / T
b probe dead time
magnetization, Mxy / A m−1
c probe group delay (receive), 2Q/ω0
d
e
probe dead time
induced voltage, VM / V
filter group delay digitized signal, S / arb. units
filter dead time
time / µs
total group delay
Figure 13: Timing diagram for a FID experiment. The spectrometer provides the rf transmission pulse in (a) and the experimentalist expects to acquire the signal shown by the dotted line. However, the actual recorded signal (e) is delayed. The timebase of the observed FID must be shifted by a time equal to the total group delay of the system to match the timebase of the transmitted rf pulse in (a). The intermediate stages (b – d) are discussed in the text.
Correct assimilation of the digital filter into the data acquisition process is extremely important. When implementing or designing a pulse sequence it is necessary to be aware of the time delays that are incurred by the filtering stages and also the rf probe. A schematic showing the various delays in a FID experiment 42
transmitter output, V0 / kV
te
a τ
τ
probe dead time
b
probe current, I0 / A or rf field, B1 / T
τ
probe group delay (transmit), 2Q/ω0
magnetization, Mxy / A m−1
c probe group delay (receive), 2Q/ω0
d probe group delay (total), 4Q/ω0
e
filter group delay
filter dead time
induced voltage, VM / V
digitized signal, S / arb. units
total group delay
time / µs
Figure 14: Timing diagram for a spin echo experiment. The spectrometer provides the rf transmission pulses (a) and the experimentalist expects to acquire the echo shown by the dotted line. However, the actual recorded echo (e) is delayed. The timebase of the observed echo must be shifted by a time equal to the total group delay of the system to match the timebase of the transmitted rf pulses in (a). The intermediate stages (b – d) are discussed in the text.
is given in Fig. 13. The spectrometer issues a 90◦ rf pulse at time t = 0, Fig. 13(a); the pulse has a duration of t90 . The resonant circuit of the rf probe does not respond immediately. The B1 field experienced by the spins is time shifted by the physical group delay of the probe, equal to 2Q/ω0 , Fig. 13(b). It follows that the probe cannot be used to detect the evolution of the magnetization Mxy until the rf pulse has decayed, as described in Section 3.2.2, leading to a dead time associated with the probe, Fig. 13(c). Once the probe is able to detect the magnetization, there is a further group delay of 2Q/ω0 on detection, Fig. 13(d). The observed signal will then be time shifted by the group delay of the digital filter tGD . Furthermore, the dead time of the filter prevents the initial response of the signal from being 43
detected, and so data are collected only once the filter has settled, Fig. 13(e). On the spectrometer time-base (relative to the issued rf pulse), the zero time origin of the data is actually t = t90 + 4Q/ω0 + tGD . Therefore, if the data are to be fitted to determine the total signal amplitude S0 , the time base of the acquired data must be adjusted accordingly. Failure to do so will result in a significant over-estimation of the total signal. The dead time of the probe and filter result in phase errors in a direct FT of the signal; methods for extracting an undistorted spectrum from time-shifted data will be discussed in Section 5.2. A similar situation arises in the case of spin echoes. A schematic showing the various delays in a spin echo experiment is given in Fig. 14. The spin echo forms between a pair of 180◦ rf pulses, each of duration t180 , Fig. 14(a). The detected signal is delayed relative to the issued rf pulses by 4Q/ω0 which is the total group delay of the probe on transmit and receive, Fig. 14(b,c,d). If the half-echo time
τ is sufficiently long, the dead time of the probe can be neglected. However, this time may become significant if the rf pulse separation is small and the ringdown duration of the rf pulses is long. Similarly, the dead time of the filter can be neglected for fast filters and sufficiently long echo times. The actual echo center will be displaced on the spectrometer time base by the total group delay of the system tGD + 4Q/ω0 , Fig. 14(e). It is important to know the exact temporal placement of the echo in order to acquire the data correctly. If the acquisition command is not delayed relative to the rf transmission time base, the echo will be off-center in the acquisition window. Such an error in timing is more critical when only the echo amplitude is recorded (single point acquisition): the echo maximum will not be observed, thereby leading to a significant error in the measurement. Similarly, problems can arise with very short echo times where the acquisition
44
window overlaps with the next rf pulse. In such circumstances it is necessary to design fast digital filters to suit the experimental parameters. On the spectrometer time base, the echo center occurs at a time t180 /2 + τ2 + tGD + 4Q/ω0 after the start of the 180◦ refocusing pulse. amplitude / arb. units
3 2 1 0 0
5 signal / arb. units
a
4
0.2 0.4 0.6 0.8 time / ms
1
c
4 3 2 1 0 0
0.2 0.4 0.6 0.8 time / ms
1
b
0.8 0.6 0.4 0.2 0 −500 −250 0 250 frequency / kHz
amplitude / arb. units
signal / arb. units
5
1
1
500
d
0.8 0.6 0.4 0.2 0 −500 −250 0 250 frequency / kHz
500
Figure 15: Simulation of digital filtering. The FID signal from a liquid sample (a), acquired with a dwell time tdw = 1 µs, is inhomogeneously broadened by heterogeneities in the magnetic field; SNR ≈ 50. The FT of the signal (b) clearly shows that the passband of the filter covers the full ±500 kHz range. If a carefully chosen digital filter is used for the data acquisition instead (c), then the noise observed in the time domain is reduced, whilst the SNR, defined in the frequency domain (d), remains constant. This filter has a passband approximately corresponding to ±20 kHz, and so the noise is suppressed outside this frequency band.
The difference between timing in the FID, Fig. 13, and spin echo experiments, Fig. 14, is subtle and easy to overlook. In the FID acquisition, there is no need to account for the group delay in the pulse sequence. The data are acquired immediately following the probe and filter dead times. However, the time base of the data must be adjusted at the processing stage to account for the group delays 45
in the system. In the spin echo experiment, the acquisition must be shifted by the total group delay of the system relative to the expected position of the echo center. The observed echo can then be considered to have reached a maximum midway between the two 180◦ rf pulses as issued on the spectrometer time base and no further adjustment of the time base is necessary during processing. An example highlighting the significance of careful digital filter design in lowfield experiments is presented in Fig. 15. The FT of an FID provides a spectrum in which noise has been acquired across the full frequency range, Fig. 15(a,b). The frequency range is determined by the dwell time tdw used in the acquisition. However, if a digital filter is applied, it is possible to suppress much of the noise in the spectral window without adversely affecting the shape of on-resonance components, Fig. 15(d). This technique is especially useful at low field where the entire 1 H proton signal is expected to be on, or close to, the Larmor frequency. Furthermore, the digital filter is useful in suppressing signal contamination from 19 F,
often present in the rf probe body or sample container. Although the SNR
of the data acquired at the Larmor frequency is unchanged, the digital filter has suppressed noise that would otherwise be integrated into the time-domain signal, Fig. 15(c). The noise has therefore been reduced in the time domain, allowing a more robust interpretation of the time domain data. 3.4. Maximizing signal-to-noise ratio Low-field NMR is considered to have an inherently poor SNR compared to experiments conducted at intermediate and high field. The rule-of-thumb that SNR ∝ ω0 demonstrates why low-field NMR is associated with poor SNR, although Hoult has argued that the exact theoretical SNR owes as much to the performance of the rf probe (both in transmission and reception due to the principle 46
of reciprocity) as to the static field strength [85]. It is important to realize that SNR is a property of the measurement and not of the instrument. There are a few straightforward steps that can be taken when designing a lowfield experiment to maximize the available SNR. Wherever possible, the strongest static magnetic field should be used to maximize the inherent SNR of the experiment. There is one exception to this rule: heterogeneous materials (such as porous media, dense composite materials, or any systems with gas/liquid/solid interfaces) will be prone to the formation of internal gradients due to the magnetic susceptibility contrast in the sample [50]. These magnetic field gradients scale with B0 , and so a higher field strength will result in a larger distortion in the static magnetic field. Strong internal gradients can lead to very rapid signal decay within the dead-time of the rf probe, preventing the detection of any signal. Therefore, depending on the properties of the sample, improved SNR may be achieved at lower field strengths. Capacitive coupling between the probe and the sample can lead to significant changes in the probe tuning, particularly if the sample is electrically conductive, such as a high salinity brine. Low Q circuits will be robust to small changes in sample composition (conductivity) during the course of an experiment. Maximizing the coil filling factor, i.e., the fraction of the sensitive volume of the rf probe occupied by the sample, is also important. For bench-top solenoid coils, the filling factor is easily calculated and can be close to unity (volume of sample is equal to volume of probe). However, due to the cylindrical geometry of solenoid rf coils, small reductions in sample radius result in a significant reduction in filling factor. We note that it is advisable to leave some space between rf coil and sample as local B1 “hot spots” will be present in close proximity to the wire. Filling factor
47
is more difficult to estimate for surface coils used on single-sided NMR devices; the filling factor will always be less than
1 2
because the sample exists only on one
side of the rf probe. Elimination of external rf sources is significant for optimizing the SNR. In bench-top magnet systems, as illustrated in Fig. 2, the outer frame of the magnet (combined with the rf probe housing) acts to screen the antenna from external sources. However, this screening can do little to reduce noise sources conducted directly into the system along cables. Unfortunately, gradient power cables and temperature sensors (thermocouples or PRT’s) are obvious transmission routes both for rf noise and coherent interference. White, Gaussian noise is not a particular problem for the experimentalist: the SNR can be improved by increasing the √ number of repeat scans Ns (where SNR ∝ Ns ) and signal may be extracted or fitted in the presence of white noise by careful data analysis as discussed in Section 5. Interference that is coherent with the receiver will sum at the same rate as the signal and is not reduced by signal averaging. Furthermore, robust separation of signal and interference can be difficult to achieve at the analysis stage. It is important, therefore, to remove sources of interference whenever possible. In the laboratory environment these sources can be identified using a network analyzer tuned to the Larmor frequency of the magnet and a simple pick-up coil (a loop of copper wire is sufficient). Switch-mode power supplies, now common in electrical devices, are particularly egregious sources of rf interference. It is usual to ground the NMR equipment using the laboratory Earth line, although in some cases this may be a source of interference. If the building ground is particularly noisy, the NMR equipment may be powered through an isolation transformer and grounded independently using a local Earth. Similarly, laboratory Ethernet connections can
48
be interference sources and computers should be isolated with wireless or optical Ethernet connections. Furthermore, all coaxial cables used on the NMR system should be shielded, with the shield grounded only at one end (preferably as close to the magnet housing as possible); this mode of grounding prevents the formation of earth loops. A main route of interference into the NMR system is via the gradient connections; at modest frequencies (e.g., B0 > 20 MHz) it is possible to filter the gradient lines without adversely affecting gradient amplifier performance, although at lower frequencies this is not a practical solution. The other common routes of noise injection are temperature sensors either on the sample or in the gradient unit. For superambient measurements, electronic sensors could be replaced with temperature-sensitive fiber-optic Bragg gratings that are interrogated remotely and are pure electrical insulators. With more traditional devices it is important to keep the temperature sensor away from the rf coil (preferably outside the probe housing) to limit rf interference unless considerable efforts are made to effectively ground the sensor. Grounding of fluid flow lines during NMR measurements is also important as discussed in [86]. The issue of noise and interference sources is quite different for single-sided NMR instruments where the rf probe is unshielded. Devices such as the NMRMOUSE suffer from poor SNR in “noisy” laboratory environments and it may be necessary to construct a Faraday cage around the instrument and sample. However, when using single-sided devices in remote locations (construction sites or oil wells) the sample itself will act as a rf shield and SNR performance can be significantly better than expected based on laboratory tests. Again, the ideas of minimizing or eliminating local sources of rf interference apply. Overall, it is possible to optimize low-field hardware to provide a usable SNR
49
within a practical acquisition time. Careful data processing can allow information to be extracted even when the SNR is poor and so for many industrial process or quality control applications, the perceived SNR performance should not be considered a reason to avoid low-field instrumentation. 4. Pulse Sequences and Experiments 4.1. Relaxation time measurements Relaxation time analysis is the most common technique used in industrial applications of low-field NMR. Distributions of relaxation time are used to characterize fluid content, fluid type, and pore size. Here, we consider the basic pulse sequences for determining longitudinal T1 and transverse T2 relaxation times. These relaxation time measurements can be combined, or performed in conjunction with diffusion measurements, to provide multi-dimensional relaxation time and diffusion-relaxation correlations. Additionally, relaxation time measurements are incorporated within MRI protocols to provide relaxation weighted images. 4.1.1. Longitudinal relaxation Recovery of the spin ensemble to equilibrium is governed by the exponential time constant T1 . Accordingly, the observed signal (magnetization) b at time τ1 is described by b (τ1 ) τ1 = 1 − c exp − . beq (∞) T1
(11)
The change in magnetization process is typically monitored using either the inversion recovery or saturation recovery method [87]; eq. (11) applies to both methods. Pulse sequence schematics for both inversion and saturation methods are shown in Fig. 16. Inversion recovery (c = 2) begins with the spin ensemble 50
aligned along −z, whereas saturation recovery (c = 1) begins with the spin ensemble fully saturated, i.e., zero net magnetization, on the x-y plane. Inversion recovery provides twice the dynamic range in the data compared to saturation recovery but relies on the application of a good 180◦ inversion rf pulse. Furthermore, the spin ensemble must be at equilibrium on +z prior to the inversion pulse. To guarantee this condition, the recycle delay tRD between scans must be long and satisfy tRD ≡ 5 × T1 . Saturation of the spin ensemble is easier to achieve when B0 and B1 are non-uniform, or in broadline samples. The saturation recovery experiment does not require a long tRD as effective saturation is achieved regardless of the initial state of the spin ensemble. The saturation recovery experiment is therefore much faster than the inversion recovery experiment when long T1 times are observed (i.e., when T1 > 1 s). A rapid double-shot T1 measurement is described in [88]; at present this pulse sequence has not been implemented at low field due to the reliance on small tip angle rf pulses and hence reduced SNR. 4.1.2. Transverse relaxation The observed NMR signal amplitude following a single excitation pulse (nominally 90◦ ) decays over time as the spin ensemble loses phase coherence in the x-y plane due to local magnetic field fluctuations. These fluctuations arise from dipolar interactions, heterogeneities in the background magnetic field, and other terms in the nuclear spin Hamiltonian. The observed free induction decay (FID) [89] is governed by the exponential time constant T2∗ , as [90] 1 1 ≈ + γ ∆B0 , ∗ T2 T2
(12)
51
τ1
a
τ1
b ns time
Figure 16: Pulse sequence schematic for T1 relaxation time measurements via (a) inversion recovery and (b) saturation recovery [87]. In all cases, the thin and thick vertical bars represent 90◦ and 180◦ rf pulses, respectively. The saturation comb of 90◦ pulses is repeated nc times. Homospoil gradients (gray trapezoids) may be included between the rf pulses to eliminate unwanted coherence pathways. The initial magnetization b at time τ1 = 0 is set to (a) b(0) = −beq or (b) b(0) = 0. The spin ensemble is allowed to recover on the z-axis for a time τ1 before being interrogated by a 90◦ excitation pulse. The signal amplitude immediately after the excitation pulse is determined by fitting the FID. At intermediate field, the FID can be Fourier transformed to obtain a chemical spectrum. The experiment is repeated for a range of τ1 times to determine the T1 recovery time.
where ∆B0 is a characteristic measure of magnetic field variation. T2∗ is inversely related to the full-width half-maximum (FWHM) of the spectral line, such that 1 + γ ∆B0 ≈ πFWHM. T2
(13)
When using intermediate-field spectroscopy magnets, ∆B0 ≈ 0 in bulk liquids and
so T2∗ → T2 , where T2 is the transverse relaxation time determined by spin inter-
actions. However, when using low-field permanent magnets with poor B0 homogeneity, ∆B0 is large and the FID will be characterized by T2∗ T2 . A similar situation arises when measuring heterogeneous samples (such as liquids confined in porous media) because the magnetic susceptibility contrast between the solid and liquid distorts the static magnetic field. To measure T2 relaxation in the presence of magnetic field inhomogeneities, the spin ensemble is refocused in the x-y plane to form a spin echo [91] by a 180◦ rf 52
pulse applied a time τ2 after the initial excitation pulse. The spin echo, comprising two FIDs back-to-back, has a maximum at time te = 2τ2 after the excitation pulse. By repeatedly applying 180◦ rf pulses, separated in time by te , a train of n echoes is formed [92]. The method of forming this echo train has been refined into the standard Carr-Purcell Meiboom-Gill (CPMG) experiment [93]. The envelope of the echo maxima decays exponentially with T2 as nte b (t) . = exp − b (0) T2
(14)
τ2
n
time
Figure 17: Schematic of the standard CPMG pulse sequence [92, 93] for T2 relaxation time measurements. In all cases, the thin and thick vertical bars represent 90◦ and 180◦ rf pulses, respectively. The echo time te = 2τ2 ; the echo maximum forms at a time τ2 after the 180◦ pulse. The repeated application of 180◦ refocusing rf pulses generates a train of n echoes. The maximum signal intensity of each echo is acquired at time nte to provide an entire echo train in a single scan. At intermediate field, the FID (latter half of final echo) is acquired after n 180◦ rf pulses and Fourier transformed to provide a chemical spectrum. The experiment is repeated to obtain a spectrum for a range of n echoes.
It is important to note that the CPMG sequence does not refocus J-coupling interactions, and so at intermediate field the signal decay will be modified when studying chemical species with multiple functionality. The J-coupling interaction still exists at low field, but in general the B0 and B1 fields are sufficiently inhomogeneous as to mask scalar coupling processes. At ultra-low fields, J-coupling is used for chemical identification in lieu of conventional spectroscopy [94].
53
4.2. Diffusion and flow Pulsed field gradient (PFG) diffusion measurements are used to determine coefficients of self-diffusion, restricted diffusion, and advection (flow) [95–98]. A magnetic field gradient g is applied for a time tδ to provide the spins with a phase shift proportional to their position r(0). After an observation time t∆ , a second gradient pulse of equal area is applied to remove the phase shift. If the spins have not moved, their net phase shift is zero. However, if the spins have moved to a new position r(t∆ ), then the spins have a net phase shift equal to
ϕ = γ g · [r(t∆ ) − r(0)] .
(15)
This phase shift can be simplified to ϕ = 2πq · R where q = γ tδ g/2π is the magnetization wave vector and R = r(t∆ ) − r(0) is the displacement vector. The signal decay due to diffusion is then b(g) = exp −4π2 q2 Dt∆ , b(0)
(16)
where q is the magnitude of q and D is the diffusion coefficient. It is usual to apply the gradient along a single axis so we need only consider the gradient magnitude g. The diffusion coefficient is readily determined by repeating the measurement for a range of g. If all the timings remain constant, the only contribution to the normalized signal decay is diffusion. There are exceptions to this condition, notably when g is fixed by the hardware as discussed in Section 4.5.2. The basic principles of PFG diffusion measurements are the same regardless of the acquisition method. However, the details of q in eq. (16) will depend on the exact form of the experiment. For example, the pulsed gradient spin echo (PGSE) experiment [95] with arbitrary gradient pulse shape gives n h io tδ b(g) 2 2 2 + A (ε ,tδ ) , = exp −γ g D tδ t∆ − b(0) 3 54
(17)
τse g
a tδ
t∆
τse g
b tδ
t∆ time
τse tδ1
c tδ /2
tδ2
g
t∆
Figure 18: Basic diffusion pulse sequence schematics for (a) PGSE, (b) PGSTE, and (c) APGSTE. Thin and thick vertical lines indicate 90◦ and 180◦ rf pulses, respectively. Timings are specified following the notation of Tanner [97]: in each case, the spin echo time is τse and the observation time (defined as the time between leading gradient edges) is t∆ . Trapezoidal gradient pulses are shown with effective duration tδ and amplitude g. The gradient amplitude is incremented in successive experiments. Homospoil gradients are indicated by gray trapezoids. The delay between the rf pulse and leading gradient edge is tδ1 , and the delay between the trailing gradient edge and the rf pulse is tδ2 . At intermediate field the measured FID may be Fourier transformed to provide a spectrum, as in the DOSY experiment.
55
where A (ε ,tδ ) is an approximate correction term in the Stejskal-Tanner equation that depends only on the shape of the gradient pulse, following the notation of Price and Kuchel [99]. For example, a trapezoidal pulse requires
ε 3 tδ ε 2 A (ε ,tδ ) = − , 30 6
(18)
where ε is the duration of the gradient ramp (assuming the rise and fall rates are equal). Other correction terms for a variety of pulse shapes are given by Price [100, 101], and exact exponents in the Stejskal-Tanner equation for a selection of pulse shapes are given by Sinnaeve [102]. It can be helpful to use shaped gradient pulses (e.g., trapezoids or half-sine pulses) to provide robust and reproducible gradients. If the ramp time is short compared to the pulse duration (i.e., approximate rectangular gradient pulses), such that ε 2 tδ , we may assume A (ε ,tδ ) → 0. The basic PGSE pulse sequence is shown in Fig. 18(a). The maximum amount of signal obtained in the PGSE sequence will depend on the ratio of T2 to t∆ . In heterogeneous materials, T1 > T2 (even at low field [103]) and so it is desirable to use a pulsed gradient stimulated echo (PGSTE) experiment instead [97], see Fig. 18(b); eqs. (17) and (18) still apply to this pulse sequence. A stimulated echo [91] stores the spin ensemble on the z-axis during the observation time, so the signal decays only with T1 , enabling longer observation times to be employed. It is worth noting that we have neglected the relaxation terms in eq. (17) on the assumption that all pulse timings are constant and so these fixed pre-factors are encompassed in the signal normalization b(g)/b(0), and this assumption is sufficient when determining diffusion coefficients. However, if quantitative signal intensities are required, terms describing T2 relaxation (during spin echoes) and T1 relaxation (during stimulated echoes) need consideration; the exact form of these terms will depend on the pulse sequence. It is also necessary to note that a 56
stimulated echo returns only half the initial signal amplitude, whereas a spin echo returns the entire signal amplitude. As well as the applied gradient pulses, distortions in the B0 field will contribute to diffusive attenuation. In the presence of a constant background gradient g0 , the signal attenuation observed in a PGSTE experiment (assuming rectangular gradient pulses) will be [104] n b(g) tδ 2 g = exp −γ 2 D tδ2 t∆ − b(0) 3 !2 2tδ2 2 + tδ 2τset∆ − − tδ (tδ1 + tδ2 ) − tδ1 + tδ2 gg0 3 τse 2 o 2 g0 . + τse t∆ − 3
(19)
Cotts et al. [104] devised several alternating pulsed gradient stimulated echo (APGSTE) sequences to reduce the influence of background gradients in diffusion experiments. The popular version is the Condition I “13-interval” APGSTE pulse sequence illustrated in Fig. 18(c). Here, the gradient pulses are split into positive and negative amplitude lobes around a 180◦ refocusing pulse. Each lobe has an area gtδ /2, so that the spins experience a total applied gradient of gtδ (as in the simpler PGSE and PGSTE sequences) due to the refocusing pulse. Assuming rectangular pulses, this 13-interval APGSTE experiment provides a signal that varies as 2 t tδ 2 b(g) 2 = exp −γ D δ 4t∆ − 2τse − g b(0) 4 3 4 3 2 , + τsetδ (tδ1 − tδ2 ) gg0 + τse g0 3
(20)
such that the cross term in gg0 vanishes when tδ1 = tδ2 . This condition on the pulse sequence timing is the ideal mode of operation and should only be waived 57
when very short τse times are required. Assuming tδ g τse g0 , the overall signal attenuation is dominated by the applied gradient pulses and not the internal gradients. Improved compensation for background gradients can be achieved using the “17-interval” APGSTE pulse sequence [104] or higher order versions [105]. These pulse sequences are designed to remove the cross term in gg0 regardless of the ratio of tδ1 to tδ2 , and provide additional reduction of the term in g20 relative to that in g2 . However, the increased complexity of implementation and greater relaxation attenuation associated with these long pulse sequences leads to them not being widely used. The diffusion pulse sequences in Fig. 18 are also used to acquire a probability distribution of advective displacement (the so-called “flow propagator”) [98], averaged over all possible starting positions of the spins in the volume of interest. The only modification required to the aforementioned pulse sequences is that the gradient amplitude is ramped between ±g(max) in successive scans. Flow propagators have been used to assess pre-asymptotic Stokes’ flow in bead packs [106–117], catalysts and chemical reactors [98, 118, 119], biofilms [120–122], foams [123, 124], and rocks [86, 115–117, 125–136]. However, to date the implementation of the propagator experiment on low-field hardware has been limited [86]. 4.3. Multi-dimensional correlations Two-dimensional (2D) correlations are becoming increasingly popular at low field. The 2D correlations of relaxation time or diffusion coefficient can provide robust chemical identification when spectroscopic resolution is not available. A notable example is the separation of oil and water signals in well-logging using D58
T2 correlations as will be discussed in Section 7.3.1. Other applications, including monitoring of hydration in cement-based materials (Section 7.2.1), are to be found in the literature. Higher order correlations, such as three-dimensional T1 -D-T2 correlations, are possible, although the experiment duration and complexity of data processing have thus far limited the use of these experiments. Diffusive or chemical exchange processes can be monitored by including a defined “storage time” between similar encoding intervals, such as the D-D or T2 -T2 experiments. We note that at intermediate field, all these experiments can be acquired with chemical resolution. However, doing so greatly increases the acquisition time. Examples of chemically resolved correlations can be found in the literature (e.g., [137]) although they are rare. Here, we consider only the standard low-field implementations without chemical resolution. 4.3.1. Relaxation time correlations T1 -T2 correlations have been applied to rocks [138–140] and to dairy products [141] where the additional information in the 2D distribution simplifies discrimination of the oil and aqueous signals [142]. T1 -T2 correlations also provide the relaxation time ratio T1 /T2 which is related to the strength of surface interaction between the imbibed liquid and a solid pore matrix [143–145]. The ratio T1 /T2 is therefore useful as an indicator of wettability [144]. The relaxation behavior observed in the 2D data array is exponential in a linear combination of terms in T1 and T2 . Therefore, the NMR data have the form nte b (nte , τ1 ) τ1 = exp − 1 − c exp − , b(0, 0) T2 T1
(21)
where the constant c again describes the scaling of the initial magnetization perturbation as in eq. (11) with c = 1 for saturation recovery and c = 2 for inversion 59
recovery. τ1
τ2
a n τ1
τ2
b ns n
time
Figure 19: Pulse sequence schematic for T1 -T2 relaxation time measurements via (a) inversion recovery [141] and (b) saturation recovery [146, 147]. In all cases, the thin and thick vertical bars represent 90◦ and 180◦ rf pulses, respectively. The saturation comb of 90◦ pulses is repeated nc times. Homospoil gradients may be included between the pulses in the T1 encoding portion to eliminate unwanted coherence pathways. The initial magnetization M at time t = 0 is set to (a) b(0) = −b(max) or (b) b(0) = 0. The spin ensemble is allowed to recover on the z-axis for a time
τ1 before being interrogated by a CPMG echo train. The experiment is repeated for a range of τ1 times to encode T1 , followed in each case by n echoes separated in time by te = 2τ2 to encode T2 .
4.3.2. Diffusion-relaxation correlations A combined PGSE and CPMG experiment was introduced for diffusion imaging [148]. At low field the PGSTE sequence is preferable for bulk liquid samples, with the APGSTE sequence being used predominantly in studies of porous media. In such applications to porous media, the 13-interval Cotts sequence is combined with the standard CPMG experiment [107, 134] as shown in Fig. 20. An additional z-storage delay of duration tstore may be incorporated between the diffusion encoding and acquisition sections of the experiment to provide eddy current stabilization prior to the CPMG sequence [149] or ensure the echo time in the CPMG sequence is constant for all echoes [134]. The quality of the diffusion data is improved by including dummy gradient pulses for eddy current stabilization at the start of the sequence [150]. 60
τse
tδ /2
te
g
t∆
ts
n time
Figure 20: A schematic of the APGSTE-CPMG pulse sequence [107, 134], based on the “13interval” sequence of Cotts et al. combined with the standard CPMG echo train. Thin and thick vertical lines indicate 90◦ and 180◦ rf pulses, respectively. Timings on the diffusion encoding portion are specified following the notation of Tanner [97]: the echo time is tse = 2τse and the observation time (defined as the time between leading gradient edges) is t∆ . Trapezoidal gradient pulses are shown with duration tδ and amplitude g; the gradient amplitude is incremented in successive experiments. Homospoil gradients are indicated by gray trapezoids. Data are acquired only in the CPMG each train comprising n echoes separated in time by te . An extra z-storage (stimulated echo) delay of tstore is included prior to the CPMG sequence.
Of course a CPMG echo train may be appended to any of the diffusion sequences illustrated in Fig. 18. A summary of the pulse sequence combinations appropriate for D-T2 experiments has been presented in [151]. The 2D data obtained from the APGSTE-CPMG experiment shown in Fig. 20 will be of the form nte + 4τse b (nte , g) = exp − b(0, 0) T2 n h io tδ 2 2 × exp −g D tδ t∆ − (22) + A (ε ,tδ ) , 3 with the approximate correction term for arbitrary gradient ramps given by A (ε ,tδ ), see Section 4.2. An additional (fixed) time is present in the T2 exponent to account for transverse relaxation during the spin echoes in the diffusion encoding portion of the sequence. An additional T1 relaxation term t∆ − 2τse + tstore exp − , T1 must also be included if the signal intensity is to be quantitative. 61
(23)
4.3.3. Exchange rate measurements Diffusive or chemical exchange processes can be monitored using T2 -T2 experiments [152–154], wherein a pair of CPMG echo trains are separated by a z-storage interval of duration tstore . A similar exchange measurement can be obtained with the D-D experiment [155, 156], although to the best of our knowledge this pulse sequence has not been implemented at low field. The T2 -T2 pulse sequence, as applied at low field to cements [143, 152] and rocks [50, 157], is illustrated in Fig. 21. Data are acquired only in the second CPMG train. We use (1)
the convention that the first (direct) dimension is T2
and the second (indirect)
(2)
dimension is T2 . The experimental data have the form ( ) ( ) b (nte , mte ) nte mte = exp − (2) exp − (1) . b(0, 0) T T 2
(24)
2
The total signal amplitude is reduced due to T1 relaxation during the storage interval, so a factor exp {−tstore /T1 } needs to be included when comparing T2 -T2 data acquired with different storage intervals. Methods for determining exchange rates from these data are discussed in Section 6.3. 4.4. Imaging Spatial position is encoded in the reciprocal k-space signal b(k) and the image ˆ amplitude b(r) is obtained from the FT of b(k); k-space is therefore the Fourier inverse of real space. The Larmor frequency ω (r) of the spins at position r is related to the applied magnetic field gradient vector g (in the laboratory reference frame) by
ω (r) = − (γ B0 + γ g · r) .
(25)
62
te
tstore
m
n
time
Figure 21: A schematic of the T2 -T2 exchange experiment pulse sequence. This sequence consists of two CPMG echo trains separated by a z-storage delay (stimulated echo) of duration tstore . The first CPMG echo train has a total duration mte ; the number of echoes m is incremented over sequential experiments to construct a 2D data array. The second CPMG train has a total duration of nte . The usual implementation consists of a single-scan acquisition of the echo amplitudes in the second CPMG train, although chemical resolution can be retained by incrementing n over separate experiments and recording the final half-echo. To determine an exchange rate, the storage time tstore is incremented in separate experiments.
When k-space data are acquired on resonance in the rotating frame, the voxel intensity is obtained by the 3D FT of b(k) such that ˆ b(r) =
ZZZ
b(k) exp [+i2πk · r] dk.
(26)
Complete introductions to k-space and imaging are to be found in [8, 90, 158]. 4.4.1. Frequency encoding In medical MRI it is typical to acquire a single dimension in multi-dimensional imaging using a frequency-encoded gradient echo [8]. As the amplitude of a gradient echo depends on T2∗ , it is desirable when imaging materials to include a coincident frequency-encoded spin echo (amplitude depends on T2 ); this modification requires only the addition of a refocusing rf pulse [90]. A frequency-encoded spin echo provides a single line of k-space points. Therefore, frequency encoding has the advantage of providing a profile in a single scan. It is important to realise that the shape of a frequency-encoded profile is convolved with the spectral lineshape. Therefore the profile resolution is determined by T2∗ . 63
A frequency-encoded spin-echo profile is obtained by applying a magnetic field “read” gradient during data acquisition such that excited nuclear spins precess with a frequency determined by their position. If the magnetic field gradient is applied for a time tr , then the range of wavenumbers spanned is given by ±k(max) = ±
γ gtr , 4π
(27)
where kmax is the maximum deviation from k(0), the central k-space position corresponding to ω0 . Each point in k-space will have a width defined by ∆k =
γ gtdw . 2π
(28)
The field of view (FOV) of the profile is then determined by the reciprocal relation FOV = 1/∆k such that the spatial resolution is ∆z = FOV/m where m is the number of complex data points acquired. When using digital filters, the practical FOV will be defined by the filter passband fpb rather than the reciprocal dwell time. Frequency-encoded profiles must be acquired with sufficient bandwidth to exceed the frequency range ∆ν0 =
γ gL . 2π
(29)
A frequency-encoded profile can be measured only in a single direction in any one experiment or acquisition interval. Images can be generated by acquiring read profiles in multiple directions and then reconstructing with a back-projection algorithm [158]. In modern MRI, it is far more common to use FT imaging reconstruction by combining frequency, phase, and slice techniques [8, 90]. Complicated k-space trajectories may be defined using time-varying read gradients, although such acquisition techniques have not yet been applied at low field. Rather, a single frequency-encoded echo is acquired in the direction requiring the highest 64
spatial resolution; frequency encoding is used since it provides the most efficient sampling of a large number of k-space points (pixels). 4.4.2. Phase encoding To select a k-space position using phase encoding, a “phase” gradient of variable amplitude g is applied for a time tp . T2∗ blurring is negated when using phase encoding because the acquisition time (i.e., time after initial excitation) of each datum is fixed. The k-space point sampled at a particular gradient strength is determined by k=
γ gtp . 2π
(30)
By adjusting the gradient amplitude between ±g(max) over m subsequent scans, a line of k-space is traversed. The corresponding FOV is FOV =
mπ
γ g(max)tp
,
(31)
and the pixel size is ∆z =
π
γ g(max)tp
.
(32)
The phase-encoded gradient can be ramped in multiple directions (x, y, or z) in subsequent experiments, so that k-space may be traversed in one, two, or three dimensions. When all the image dimensions are acquired in this manner, the technique is referred to as a “pure phase-encode method”. Details of pure phaseencoding experiments will be given in Section 4.4.5. 4.4.3. Slice selection A spatial slice is selected by applying a shaped rf pulse in the presence of a gradient [159]. Slice selection differs from frequency and phase encoding in that 65
the obtained signal intensity is directly proportional to the spin density in the slice. A profile may therefore be obtained by moving the location y0 of the resonant slice sequentially through the sample; no further data processing is required. The width of the slice ∆ys is determined by the frequency bandwidth ∆ν of the rf pulse and the gradient amplitude g as ∆ys =
2π∆ν , γg
(33)
where the rf pulse bandwidth is a property of the pulse shape. The position of the slice is determined by the frequency offset, γ gy0 /2π. The inverse relationship between frequency and time means a narrow slice width is achieved only with a long rf pulse duration, often 100 µs when using a modest gradient amplitude. For low flip angles, the slice shape is approximately the Fourier inverse of the rf pulse shape, so an ideal boxcar slice is obtained from a sinc pulse with an infinite number of oscillations. The practical quality of the slice will be influenced by the truncation of the rf pulse width to a realistic duration, coupled with the ability of the spectrometer and rf amplifier to reproduce the required shape. It is often easier to define a Gaussian rf pulse and accept that the slice shape will also be a Gaussian; the simplified waveform provides better reproducibility and control for the experimentalist. The slice-selection gradient of amplitude g is applied for a time ts , ideally equal to the duration of the rf pulse. During this time, the excited portion of the spin ensemble will acquire a phase shift. It is therefore necessary to apply a rephasing gradient lobe of approximate area gts /2 after the slice-selection gradient to remove the phase shift. The exact area of the rephase gradient lobe relative to the slice-selection gradient is dependent on the details of the rf pulse shape [8].
66
Slice selection can be inappropriate when imaging short T2∗ materials due to the long rf pulse durations. 4.4.4. Multi-dimensional imaging The primary application of MRI has been in medicine, where anatomical images of humans and rodents are used for clinical diagnosis and biomedical studies, respectively. The concept of applying MRI to medical diagnosis was suggested by Damadian [160]. The practical implementation of MRI was demonstrated by Mansfield and Grannell [161], and Lauterbur [162]. The first full-body MRI scanners were constructed nearly a decade later [163]. There has since been a drive in magnet technology to attain higher field strengths for improved SNR (leading to improved image resolution) and spectroscopic resolution for studies of tissue chemistry. Imaging of materials (soft solids, porous media, solids) presents different challenges due to the short relaxation times associated with these samples. The earliest example of MRI of materials was presented by Gummerson et al. [164] who profiled 1D spontaneous ingress of water into stones and cement based materials using a low-field magnet. High resolution 3D imaging is possible with the spin-warp pulse sequence [165, 166]. The standard implementation utilizes a frequency-encoded echo in the read dimension, combined with two phase-encoded gradients to achieve full 3D k-space sampling. There is inherent relaxation contrast in the images generated by the spin warp method, although the timing of each k-space acquisition is constant, so the final image has a uniform weighting within the limitations of the frequencyencoded spin echo acquisition (i.e., T2∗ ). The spin warp sequence does not provide the temporal resolution required in medical imaging, although it is suitable for materials which do not change with time. The basic spin warp pulse sequence is 67
rf
te read(z)
phase(y)
phase(x)
time
Figure 22: 3D spin warp imaging sequence utilizing one read gradient and two phase-encode dimensions. The thin and thick vertical lines represent 90◦ and 180◦ rf pulses, respectively. The spin echo is centered at time te after the initial rf excitation pulse. The phase-encoded gradients are incremented individually in successive experiments.
shown in Fig. 22. The acquisition of a full isotropic image can take several hours using the spin warp sequence, and therefore rapid imaging techniques are often chosen for studies of materials. Quantitative MRI is achievable if the relaxation weighting associated with each voxel is determined. Majors et al. demonstrated a 1D pulsed gradient version of the multi-echo imaging sequence [167], illustrated in Fig. 23. A series of spin echoes, each acquired in the presence of a read gradient, provides a CPMG decay for each pixel in the profile. These decay curves are fitted to obtain b(0), the projected signal at zero time, and a T2 distribution. The multi-slice multi-echo (MSME) sequence detailed in [8] provides a method for obtaining relaxation time maps in 2 or 3D. The phase-encoded gradient is ramped over subsequent scans to generate an echo train that decays with a mixture of T1 and T2 for each voxel. Relaxation time maps are again obtained by fitting the decay curves. A pure T2 relaxation time measurement is achieved by adding spoiler gradients to shift the stimulated echo contribution out of the acquisition window; note that this modifi68
cation reduces the acquired signal intensity. MSME offers quantitative 3D imaging using conventional frequency, phase, and slice encoding techniques. τ2 rf
n read(z)
g
tr
time
Figure 23: Pulse sequence schematic for multi-echo, spatially resolved T2 mapping [167]. The thin and thick vertical lines represent 90◦ and 180◦ rf pulses, respectively. A series of n CPMG-like echoes are acquired in the presence of a read gradient of amplitude g and duration tr . A T2 decay is associated with each k-space point (and hence pixel after Fourier transformation). The read gradient can be applied on any one axis; multiple directions must be scanned in separate experiments. Acquiring nD T2 maps requires the addition of phase-encoded gradients. The addition of bipolar phase-encoded gradients extends the multi-echo sequence to 2D. A third dimension is added by making the rf pulses slice-selective in the MSME sequence [8].
An alternative method of acquiring quantitative images is pre-conditioning of the spin ensemble [168, 169]. Rf pulses applied prior to the main imaging sequence are used to encode relaxation time or diffusion. The image acquisition is repeated for different pre-conditioning states, so that the voxel intensity changes as a function of T1 , T2 or D over a series of images. If the relaxation rate is slow relative to the image acquisition time, this approach may be used to generate quantitative images from fast imaging protocols such as the rapid acquisition with relaxation enhancement (RARE) sequence [170, 171]. 4.4.5. Pure phase-encoded imaging Another method of obtaining quantitative images is single point imaging (SPI) [172]. In SPI, each k-space point is acquired at a fixed time tp after a rf excitation pulse in the presence of a phase-encoded gradient; see Fig. 24(a). Accordingly, 69
the T2∗ relaxation weighting is constant across k-space and therefore does not distort the profile. If the acquisition time tp of each k-space point is short compared to T2∗ , then the resultant profile is considered quantitative. As such, SPI is well suited to the measurement of broadline samples. SPI is time consuming because each kspace point is acquired in a separate scan. Accordingly, SPI has been improved in the form of single point ramped imaging with T1 enhancement (SPRITE) [173] which uses repeated applications of small tip angle rf pulses to acquire a complete k-space raster in a single scan; see Fig. 24(b). This pulse sequence relies on short T1 relaxation times to continually restore a fraction of the spin ensemble to the z-axis, thus providing signal over a long acquisition window. SPRITE offers a significant reduction in experiment duration compared to SPI, although the required high gradient duty cycle does mean some commercial hardware struggles with the necessary gradient waveforms, which can last for several seconds. Similarly, commercial hardware may fail to deliver suitably short rf probe response times (probe dead time); ideally, the time between rf excitation and detection should be tp ∼ 10 µs. As SPI and SPRITE data are acquired in the presence of a phase-encoded gradient, the frequency bandwidth limit given in eq. (29) applies. SPRITE is a robust and reliable method for imaging a wide range of materials [13, 173] although application to very low magnetic field strengths (B0 < 0.1 T) is restricted by the inherent poor SNR arising from the the small tip angle rf pulses and wide frequency bandwidth. SPRITE is inherently insensitive to sample chemistry due to the lack of relaxation time weighting in the images. Methods for introducing chemical sensitivity into SPRITE through pre-conditioning have been explored for separation of oil and water signals [174], although these techniques are not straightforward to implement.
70
rf
a
tp phase(x, y, z)
time tp rf
n
b phase(x, y, z)
Figure 24: Pulse sequence schematics for (a) SPI and (b) SPRITE. The rf excitation in (a) is via a 90◦ pulse, and in (b), via a small tip angle pulse. The usual implementation involves the acquisition of a single point at time tp after the rf pulse. Multiple points may be acquired and adjusted to a constant FOV using a z-chirp transform. The phase-encoded gradients are ramped independently between ±g(max) in either 1, 2, or 3D. For (b) SPRITE, the gradients are incremented on each loop to sample n k-space positions in a single scan.
The pure phase-encoded nature of SPRITE lends itself readily to k-space sampling schemes other than rectilinear grids. For example, SPRITE data may therefore be acquired with spiral (2D) or conical (3D) sampling schemes [175]. The majority of the intensity in an image is acquired at low k positions. Therefore, under-sampling of high k positions reduces both the image fidelity and acquisition time but does not reduce the quantitative nature of the image. In materials imaging there is a bias towards achieving quantification, as opposed to high spatial resolution, which favors this type of undersampling. The SNR of SPRITE is improved by sampling the origin of k-space first, as encompassed in the 1D centric (double half-k) SPRITE acquisition, where the phase-encoded gradients are ramped from g = 0 to +g(max) in one scan, and g = 0 to −g(max) in the subsequent scan. When using complicated sampling schemes (such as 3D cones) the origin of k-space may be sampled multiple times, provid-
71
ing a more accurate total image intensity. Furthermore, multiple points may be sampled after each rf excitation in both SPI and SPRITE, as indicated in Fig. 24. However, as each point has a different associated tp , a series of images – each with a different field of view determined by tp according to eq. (31) – is obtained. The fields of view are standardized using the z-chirp transform [176–178]. Noise observed in the time domain signal can also be reduced by customizing the digital filter bandwidth used to sample each k-space location [179], although this method can be difficult to implement depending on the available hardware. a
τ2
rf
n
phase(x, y, z)
tp
b
τ2
τ20
tp
n−1
rf
g phase(x, y, z)
time
Figure 25: Pulse sequence schematics for (a) the original SESPI sequence and (b) the modified SESPI sequence. The thin and thick vertical bars represent 90o and 180o rf pulses, respectively. The phase-encoded gradients are ramped independently in either 1, 2, or 3 dimensions. In (a) the bipolar phase-encoded gradients may be incremented on each echo to sample a line of k-space in a single scan (turbo SESPI), or incremented on sequential scans to provide a T2 map. The modified SESPI sequence (b) only provides a T2 map.
An alternative to SPI is the spin echo single point imaging (SESPI) sequence 72
[180, 181], shown in Fig. 25(a). Bipolar phase-encoded gradients of amplitude ranging over ±g(max) are applied for a time tp around each echo. Only the echo amplitude is acquired in the SESPI experiment to provide a single k-space point. As the signal is not acquired in the presence of a gradient, there is no FOV limitation on the minimum frequency bandwidth that can be used. A narrow-band digital filter provides an additional noise reduction over SPI. The advantage of this SESPI sequence is the simplicity of the spin dynamics: the spin ensemble is coherent during the rf transmit and receive intervals. However, imaging artifacts will occur if the positive and negative gradient lobes are not matched precisely. Like SPI, the phase-encoded gradients may be applied on any axis to generate a 2 or 3D image. If the phase-encoded gradients are incremented on every echo, a line of k-space is traversed in a single scan; this acquisition mode is referred to as turbo-SPI [180, 182]. Alternatively, the phase-encoded gradients may be incremented on each scan to provide a CPMG-like decay, and hence T2 distribution, for each voxel [181]; 2D T2 mapping by this method is discussed in [183]. A modification to the SESPI sequence for short T2 samples is described in [184]. The modified SESPI sequence, Fig. 25(b), has only a single phase-encoded gradient of variable amplitude ranging over ±g(max) and duration tp between the leading excitation and refocusing rf pulses. The position of the gradient pulse allows the subsequent echo amplitudes in the CPMG-like echo train to be acquired with a reduced echo time te0 = 2τ20 . However, the spin dynamics are more complicated than in the standard SESPI sequence. To overcome the phase shifts that accumulate by repeated reversal of the incoherent spin ensemble, the xy-16 phase cycle [185] is applied to the 180o rf pulses along the echo train. Consequently, only the decay described by every 16th echo is a good approximation of the T2
73
relaxation behavior. The effective echo time for the determination of accurate T2 relaxation rates therefore becomes te0 = 32τ20 if accurate T2 relaxation rates are to be determined. Additionally, composite rf pulses can be used to improve the quality of the echo shape [184, 186]. These long rf pulses further limit the minimum echo time achievable. Notwithstanding, the direct implementation, as shown in Fig. 25(b), provides a reasonable T2 decay when fitting all the echo amplitudes. The greater SNR achieved with SESPI sequences (compared to SPI and SPRITE) makes them ideal for implementation at low field. 4.5. Stray field experiments Stray field measurements of relaxation time, diffusion, and image profiles are similar to those conducted in homogeneous field, bench-top instruments. However, there are a few important differences to the pulse sequences and analysis that need to be considered when using stray field devices. Of general significance we note that 1. All rf pulses are inherently slice-selective in the presence of the field gradient, resulting in a SNR reduction due to the poor rf probe filling factor. 2.
19 F
signal contamination will occur in 1 H experiments if fluorinated com-
ponents are used for sample containment or as part of the magnet assembly. Here we describe pulse sequences appropriate for use in stray fields; data analysis will be discussed in Section 5. 4.5.1. Relaxation time measurements When the CPMG pulse sequence, as illustrated in Fig. 17, is applied in the presence of an inhomogeneous magnetic field the accuracy of the T2 measurement is influenced by off-resonance spin dynamics [76, 77, 187–189]. This is 74
because the application of the rf pulses required to generate a CPMG echo in a grossly inhomogeneous magnetic field results in multiple coherence pathways that contribute to the observed signal. For off-resonance spins, the magnetization following the excitation 90◦ pulse will contain contributions from spins with a phase shift that is approximately proportional to their offset frequency ∆ω0 . The phase shift is removed by allowing the spins to precess for a time equal to [190] cos (ω1t90 ) − 1 2t90 →− , ω1 sin (ω1t90 ) π
(34)
assuming all spins experience the nominal rf field such that ω1t90 = π/2. Consequently, the time between the leading 90◦ and 180◦ pulses in a CPMG sequence applied in a grossly inhomogeneous field should be τ2 − 2t90 /π; the timing be-
tween subsequent 180◦ pulses remains at 2τ2 . Depending on the quality of the B1
field, the exact time shift can be slightly different from the theoretical prediction and should be calibrated empirically. H¨urlimann demonstrated that the modified timing results in a 1.2 dB (26 %) improvement in SNR. The performance of the CPMG sequence in the stray field is also improved with the use of composite pulses [186, 191]. Another important consideration when using CPMG experiments on a stray field system is the influence of diffusion. For static (or very slowly diffusing) solids, the CPMG sequence provides an accurate T2 measurement [192]. However, most samples measured in stray field experiments are liquid or liquid-like, so an effective transverse relaxation time T2,eff is observed. Diffusion in a weak (background) gradient of amplitude g0 during a CPMG experiment results in a signal nte 1 2 2 b (nte ) 3 = exp − exp − γ g0 Dnte . b(0) T2 12 75
(35)
However, in the presence of a strong gradient off-resonance effects ensure that n coherence pathways contribute to the nth echo amplitude; Goelman and Prammer showed that in this case the contribution from diffusion in the nth echo will be [192] n
1 n 2 2 3 ∑ exp − 2 3 + n − 1 γ g0Dte . 1
(36)
It is clear, therefore, that determining the expected signal decay over an entire CPMG echo train is extremely complicated. H¨urlimann and Griffin provided a generalized form of eq. (36), viz [77]: Z∞ nte b (nte ) = exp − dg0 f (g0 ) b(0) T2 −∞ 1 2 2 3 × exp − E γ g0 Dnte , 12
(37)
where f (g0 ) is the distribution of background gradients and E is an “efficiency factor” that depends on the off-resonance effects observed empirically as a reduction in the signal intensity. On stray field instruments the influence of diffusion on the observed magnetization decay is significant, especially for bulk samples of rapidly diffusing species such as water. Under these circumstances, accurate T2 relaxation time measurements are difficult, if not impossible. There are alternatives to the CPMG sequence for stray field measurements. A similar multi-echo measurement of relaxation time is achieved with a quadrature echo train using the Ostroff-Waugh pulse sequence [193]. The quadrature echo sequence, illustrated in Fig. 26, is based on the “solid echo” 90◦x −90◦y −echo pulse sequence that forms an echo by refocusing dipolar interactions [194]. In a stray field experiment, any two rf pulses will generate an echo due to the distribution of B0 and B1 . The quadrature echo can provide improved signal compared to CPMG 76
τ2
n
time
Figure 26: Schematic of the quadrature echo (or solid echo) pulse sequence for T2 relaxation time measurements in short T2∗ systems. The pulse sequence contains only 90◦ rf pulses, denoted by the thin vertical bars. The echo time te = 2τ2 ; the echo maximum forms a time τ2 after the 90◦ pulse. The repeated application of 90◦ refocusing rf pulses generates a train of n echoes. When studying solids (or other short T2∗ materials), a FID (latter half of the nth echo) is acquired and Fourier transformed to provide a chemical spectrum; the process is repeated for a range of n. When using the quadrature echo pulse sequence to overcome B0 inhomogeneities (e.g., stray-field measurements), where chemical resolution is not available, a few data are acquired at the top of each echo (or encompassing each echo, depending on line width) to determine the maximum signal intensity. Following this protocol, an entire echo train is interrogated in a single scan.
in very strong gradients as long as τ2 ≤ T2,eff . Like the CPMG sequence, a shorter
τ2 provides a slower effective relaxation decay; however, the signal intensity in the quadrature echo train is determined by a mixture of T2 and T1ρ , where T1ρ is the longitudinal relaxation time in the rotating frame (i.e., relaxation in the B1 field) [195]. Therefore, quantification of signal amplitudes obtained from quadrature echo sequences can be unreliable. In general the T2∗ associated with the measurement of materials using unilateral NMR devices will be very short regardless of sample. As an echo comprises two FID decays, back-to-back, narrow echoes are observed. It is therefore important to calculate the rf timing accurately in these experiments. Failure to account for, say, filter group delays (see Section 3.3), can result in the echo being located outside of the acquisition window.
77
4.5.2. Diffusion measurements The diffusion pulse sequences described in Section 4.2 utilize fixed experimental timings and variable gradient amplitude. In stray field instruments, the inherent magnetic field gradient ∇B0 is used to measure diffusion. We refer to these pulse sequences as fixed field gradient (FFG) experiments. To provide signal attenuation, the spin echo time τse is incremented, thereby increasing both the observation time t∆ and the gradient duration tδ . However, the relaxation weighting also increases and this needs to be considered in the data analysis. Carr and Purcell noted that spin echoes acquired in the presence of a weak gradient can be used to determine a diffusion coefficient [92] according to eq. (35). However, as indicated by eq. (37), the contribution from strong, spatially variant gradients (as found in many STRAFI devices) complicates the interpretation of such diffusion data. Rather, it is useful to consider stray field diffusion experiments in terms of the D-T2 correlation described in Section 4.3.2 in order to separate the contributions from diffusion and relaxation [139, 196, 197]. By acquiring a CPMG echo train (fixed echo time) after the diffusion encoding (variable echo time), the effects of relaxation and diffusion are decoupled. In practice, a double spin echo is acquired in the so-called “FFG-2SE” experiment to provide motion compensation; the pulse sequence is illustrated in Fig. 27. A comparable PFG-2SE diffusion experiment, appropriate for use on bench-top instruments, is available [198]. The only significant difference is found in the CPMG acquisition: T2,eff is measured in the presence of the FFG, whilst T2 is observed in the PFG case. Data acquired using the FFG-2SE experiment will have the form b (tse , nte ) = b(0, 0)
Z∞
−∞
1 3 dg0 f (g0 ) exp − γ 2 g20 Dtse 6
78
τse
te g0
tδ
t∆
tδ + te /2
n
time
Figure 27: Pulse sequence schematics for the stray field diffusion-relaxation correlation FFG2SE experiment. A background (fixed) gradient of amplitude g0 is present at all times; the initial echo interval tse ≡ t∆ (tse /2 ≡ tδ ) is incremented in successive scans to encode diffusion. The spin echoes are narrowed by the presence of the gradient and the CPMG echo train measurements T2,eff .
2tse + nte , × exp − T2,eff
(38)
assuming a background gradient distribution, where it is clear that both the relaxation and diffusion contributions depend on the initial echo time separation, tse . Therefore, as we shall see in Section 5, the analysis of FFG D-T2,eff correlation experiments differs from that used for PFG D-T2 correlations. 4.5.3. Stray field imaging Stray field imaging (STRAFI) [199] is achieved in one of two ways. If the field gradient is uniform across the sensitive volume, acquired echoes are considered to be frequency-encoded, see Section 4.4.1, and a profile is obtained by Fourier transform. This is the method of choice for studying planar samples in the GARField magnet [49]. Alternatively, if the sensitive volume is well defined, a profile may be obtained by moving either the sample or magnet in order to step the sensitive volume through the region of interest. A single data point (or projected total signal from a CPMG-like decay) is obtained at each spatial location and so a profile is constructed directly from the data; no FT processing stage is required. This SPI-like acquisition is the method used with the Profile MOUSE [61] and Surface GARField [44]. In both cases it is usual to acquire an echo train, thereby providing the T2,eff relaxation time as a function of spatial location. STRAFI is 79
ideally suited to 1D profiles, and can achieve a higher spatial resolution across a small sample than is readily achievable with standard low-field pulsed gradient imaging technology. To perform 2D or 3D imaging with STRAFI requires the sample to be rotated relative to the direction of the magnetic field gradient [48]. However, as the primary use for STRAFI in industrial process design has been the investigation of thin films and coatings, a 1D profile is usually sufficient [62]. 4.6. Spectroscopy In homogeneous magnetic fields, the Fourier transform (FT) of the FID provides a chemical spectrum. Individual nuclei within a molecule experience distortions in the background magnetic field created by their local electronic environment. This chemical shielding results in resonant spins at different positions within a molecule precessing at subtly different frequencies. The spectral line associated with each unique local environment is therefore observed to be shifted relative to the spectrometer reference frequency. For example, the 1 H spectrum of ethanol will contain lines associated with the CH2 , CH3 , and OH functional groups. The size of the chemical shift (in Hz) is determined by the magnet field strength. Furthermore, each of the spectral lines will be split into multiplets due to the J-coupling (indirect dipolar coupling, or scalar coupling) interaction between NMR active nuclei. For example, the CH3 line of ethanol is split into a triplet due to the presence of the neighbouring hydrogen nuclei. Unlike chemical shifts, the J-coupling interaction is independent of the background magnetic field strength. Many excellent texts are available on the subject of chemical spectroscopy; see for example [4, 200, 201]. In non-uniform magnetic fields, 1 H spectra are typically dominated by the line broadening caused by the ∆B0 term in eq. (12) and so spectroscopy of large samples is not practical with current permanent mag80
net technology. Permanent intermediate-field magnets capable of spectral resolution over small liquid samples (diameter ∼ 5 mm) are available [4], based on the standard designs for bench-top instruments shown in Fig. 2. Spectroscopic resolution has also been achieved with specialized single-sided magnets based on the MOUSE concept [53–55]. The quality of chemical spectra acquired with permanent magnet technology will suffer from poor SNR compared to similar experiments conducted at high field. It is recommended that time-domain fitting techniques (Section 5.2), rather than a conventional FT analysis, be applied to the data. 4.7. Fast field cycling Relaxation times are dependent on magnetic field strength and so information on processes that influence spin relaxation, notably molecular dynamics, can be accessed by variable field measurements. Fast field cycling (FFC) technology provides a method of determining the frequency dependence of the longitudinal relaxation time, T1 (ω0 ); a plot of T1 versus Larmor frequency is known as a T1 dispersion measurement. Commercial FFC hardware is based on electromagnet technology where a B0 coil is used to vary the fixed field between, typically, B0 ∼ 10 µT and 1 T. The minimum field will depend on the local environment and can be improved (reduced) with magnetic shielding. Acceptable SNR is achieved by polarization and detection of the spin ensemble near the upper limit of the magnetic field strength, corresponding to 1 H frequencies of ν0 ∼ 10 MHz. A typical FFC T1 relaxation time pulse sequence is illustrated in Fig. 28. The observed relaxation decay is described by τ1 b (τ1 ) = beq (∞) + b (0) − beq (∞) exp − , T1 81
(39)
where beq is the equilibrium magnetization in the Brel field as τ1 → ∞, b(0) is the magnetization at the start of the relaxation time (end of the polarization time), and
τ1 is the variable relaxation time in the Brel field. In practice, eq. (39) should be modified slightly to account for the finite switching times of the field as discussed in [202]. rf time τ1 Bpol B0
Bacq Brel
Figure 28: Schematic of a standard fast field cycling pulse sequence. The B0 field strength is varied during the course of the experiment: initially, the spin ensemble is polarized at Bpol . Then the field is shifted and the spins are allowed to relax towards equilibrium in Brel . After time τ1 , the field is shifted again to Bacq , at which point an rf excitation pulse is applied and a FID recorded. The experiment is repeated for different values of τ1 to obtain the T1 relaxation rate in the Brel field. The polarization Bpol and acquisition Brel are the same in each experiment. Brel is varied over different experiments to obtain the T1 (ω0 ) dispersion curve.
Field cycling is used to study the relaxation effects of paramagnetic species and molecular dynamics. As the T1 relaxation time is sensitive to molecular motions at the Larmor frequency, a hindrance of motion through adsorption results in a change in the relaxation rate; therefore, field cycling is of particular use for studying liquids in porous materials [203]. We have chosen to mention FFC here as a special case of low-field relaxation experiment, although, being a specialized technique requiring unique hardware, it is not widely used in industrial process control. Comprehensive reviews on FFC and its applications are available elsewhere [202–205]. 82
4.8. Ultra-low field and Earth’s field Another specialized area is ultra low-field NMR using superconducting quantum interference devices (SQUIDs) to detect signals in microtesla fields. The extreme sensitivity of SQUIDs enables the detection of NMR signals in such weak magnetic fields. In much of the literature, SQUIDs are referred to simply as “lowfield” instruments. The potential of SQUIDs as pre-clinical instruments has been explored and a review of the applications to medical MRI is presented in [32]. The properties of SQUIDs suggest they would be appropriate for use in industrial environments, but the requirement for cryogenic cooling counters one of the advantages of NMR with permanent magnets. To date, the use of SQUIDs for process control has been limited. Using the definition of ultra-low field in Section 1, this range also includes Earth’s field NMR, where signal is detected directly in the local magnetic environment [33]. The magnitude of the Earth’s magnetic field varies depending on latitude, being stronger at the poles than at the equator, and will have a value on the order of a few tens of microtesla. An extreme yet common use of Earth’s field NMR is the detection of groundwater aquifers [206]. A more practical approach using conventional rf probe technology requires that the spin ensemble first be polarized in a stronger field (typically a few millitesla) generated using a polarizing coil [207]. Earth’s field NMR has the advantage that the local field is (ideally) very homogeneous, although in practice it is difficult to experience the Earth’s field in a laboratory environment, especially when other NMR magnets are nearby [207]. Earth’s field NMR has achieved success in two distinct arenas: (1) applications in remote locations [208–210] and (2) J-coupling spectroscopy [94, 211–213]. The very weak, yet uniform, magnetic field yields narrow spectral lines but chemical
83
shifts (being field dependent) are negligible. However, as J-coupling interactions are independent of field strength, line splitting is still observed, allowing structural information to be obtained [33, 94]. J-coupling spectroscopy is of course well known at zero-field [214–216], and at near-zero-field whereby the application of small magnetic fields restores the loss of nuclear species information encountered at zero-field [217]. Low-resolution imaging is also a possibility with Earth’s field NMR [218]. Despite the obvious advantages of NMR without a permanent magnet, Earth’s field NMR has not achieved great success in industrial applications; for example, attempts at well-logging in the Earth’s field did not develop into a commercial service. The poor SNR achievable, even with pre-polarization, has thus far limited the use of Earth’s field NMR in industrial process control, and it is largely viewed as a novelty instead of a serious replacement for high-field magnets: Magritek sell a commercial Earth’s field NMR system primarily as an educational demonstration of NMR [207]. Notwithstanding, recent advances in data processing (such as Bayesian analysis, see Section 8.3.1) are enabling useful experiments on Earth’s field instruments [219] and may provide a route to industrial applications. 5. Data Analysis Early low-field spectrometers were crude analogue systems. The data quality did not warrant intensive processing and so many experimentalists often resorted to studying only the magnitude of the NMR signal. Nowadays, low-field spectrometers are complicated digital systems, fully capable of reproducing both the experiments and data quality of their high-field counterparts. Careful pulse sequence timing and appropriate use of digital filtering, as discussed in Section 3, 84
will yield time domain data that deserves equal care in processing to obtain the most from the experiments. Here we describe various stages of data processing and fitting necessary for reliable interpretation of low-field results. 5.1. Pre-processing for optimum results There are several stages of initial processing that can be applied to low-field NMR data prior to fitting. The exact choice of pre-processing method will depend on the data acquired. We list possible pre-processing stages and note where they become important. 5.1.1. Zeroth-order phase correction All NMR data contain a zeroth -order phase shift. Ideally the phase of the receiver would match that of on-resonance spins, so the NMR data are acquired entirely in the real channel. However, it is common for the spectrometer acquisition chain to introduce small (constant) phase shifts. These phase shifts can be attributed, for example, to filters [83] and lengths of cable, and manifest as a portion of the signal rotated into the quadrature channel. The total phase shift (angle)
θ (in radians) is simply the angle of the data in the complex plane. The phasecorrected signal is therefore obtained by multiplying the acquired signal with the phase rotation exp {− jθ }; this correction can be applied to any data in the time domain. The phase correction has the effect of rotating the signal into the real channel. In the case of FIDs or CPMG decays, the phase angle is obtained from an average of the first few data points. An example of a zeroth -order phase rotation applied to CPMG data is demonstrated in Fig. 29(a) and (c). In imaging, the phase angle must be obtained at k = 0 m−1 to ensure the correct zeroth -order phase shift is applied. There are other examples where the data are phase-sensitive, such as 85
inversion recovery data where the zeroth order phase shift must be obtained from the data acquired with the longest recovery time and applied consistently to all the other data; otherwise the correct inversion will not be observed. signal / arb. units
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Figure 29: Simulation of zeroth -order phase correction applied to CPMG data. The recorded echo amplitudes (a) include a zeroth -order phase shift, so some of the signal is observed on the quadrature channel. If the magnitudes of the data are used (b), the noise adds to form a nonzero baseline (dashed horizontal line). However, if the data are phase rotated (c), the real channel contains all the signal and has a baseline with zero mean (dashed horizontal line). The (real) CPMG data may be compressed using window sums (d) to reduce the computational requirements of curve fitting.
Historically, it has been convenient on low-field instruments to store the magnitude of the complex data. When qualitative images are sufficient, magnitude data may be used. For quantification it is necessary to apply a zeroth -order phase correction. The importance of phase correcting relaxation data is highlighted by the simulation in Fig. 29: the magnitude of a CPMG decay (b) contains a non-zero baseline generated by summing over the noise. Fitting the magnitude data will result in an apparent long T2 component due to the baseline offset. By applying a 86
phase correction and fitting the real channel in Fig. 29(c) the mean of the baseline is now zero (assuming Gaussian white noise). A fit to the phase-corrected data will provide an accurate T2 relaxation time. 5.1.2. Data compression We have taken the opportunity to include an example of window averaging (data compression) on the simulated CPMG data in fig. 29. Window averaging is a useful technique when data are noisy and reduces the computation required for fitting large quantities of data. The phase-corrected real channel of the data, Fig. 29(c), is selected and divided into bins whose sizes increase logarithmically [198]. The data points in each bin are averaged to provide the compressed data shown in Fig. 29(d). In this example, an input of n = 1024 echoes is compressed to n˜ = 16 elements (where ˜ represents a compressed value). It is often necessary to adjust the width of the initial bins to accommodate an integer number of data. The SNR of the compressed data varies from point to point due to the increasing widths of the bins. The compressed data are accompanied by a weighting vector w that describes the variation in SNR across the data; the weighting vector is required to fit the data. We note that if the echoes are narrow (such that the entire envelope of each echo is observed), as in STRAFI experiments, then the intensity of each echo needs to be determined as part of the pre-processing stage. Depending on the SNR, simply determining the value of the highest signal intensity datum within an echo may not be sufficient; similarly, if the echo is narrow and the dwell time comparatively long, there may not be a single datum corresponding to the peak intensity of the echo. In such cases, it is better to fit each echo to determine the intensity. If the echoes happen to be described well by a simple function, e.g., a 87
Gaussian, then a direct fit to each echo is practical. Alternatively, the exact functional (or analytic) form of the echo could be derived to provide an appropriate fitting function [77, 220]. When the echo shape is complicated, or when SNR is low, direct fitting of each echo becomes impractical. Under these circumstances a generic echo shape is defined by summing over the first few echoes in an echo train. The relative intensity of each echo in the train is then determined by vector division of the echo with the generic echo shape. This straight-forward method will fail if the position of each echo within the acquisition window changes along the echo train; such timing errors can be eliminated through careful pulse sequence design. 5.1.3. First-order phase correction A first-order phase shift observed in the frequency domain corresponds to a time shift in the time domain data. First-order phase shifts manifest when data are Fourier transformed. The most common source of a first-order phase shift in NMR data occurs when the initial data point is not acquired at time t = 0 s. An extreme example is presented in Fig. 30: (a) a frequency-encoded spin echo has been acquired in the presence of a read gradient and stored with the timebase running sequentially from t = 0 to 1 ms. The FT of these data is illustrated in Fig. 30(b) where a first-order phase shift (oscillation) is apparent throughout the top-hat profile. If the data are time shifted, Fig. 30(c), so that t = 0 ms corresponds to the center of k-space, the FT in Fig. 30(d) is a top-hat entirely in the real channel (no first-order phase shift). Less extreme first-order phase shifts are encountered commonly in frequency-encoded spin echoes as it is difficult to obtain a datum corresponding to k = 0 m−1 . Such small phase shifts can be mediated by interpolating the data to provide an approximation for the k = 0 m−1 datum. A 88
similar problem arises in spectroscopy when a FID does not start at t = 0 s. Simply assuming that the initial data point corresponds to zero time will introduce a first-order phase shift in the resulting spectrum and prevent quantitative determination of peak areas. It is preferable to project the data back to zero time using concepts of linear prediction, which we will discuss in Section 5.2. In the case of relaxation time measurements, first-order phase shifts can be neglected as the data are fitted directly on the correct time base. 0.4
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Figure 30: Simulation of the Fourier transform of a frequency-encoded spin echo for a top-hat profile. The echo (a) is stored with the initial data point at t = 0. When the FT is applied (b), a phase shift is observed in the profile. However, if the echo is time shifted (c) so that the echo maximum corresponds to t = 0, the FT (d) provides a profile with the correct phase. The echo is simulated with SNR = 100; in each case the blue line indicates the real channel and the red line indicates the quadrature channel.
5.2. Time-domain fitting for spectroscopy Liquid phase spectroscopy relies on the Fourier transform of FIDs into chemical spectra. It is typical in high-field experiments for the SNR to be good and 89
probe plus filter dead times to be short. Small zeroth - and first-order phase corrections, plus baseline corrections, are applied to the spectrum after the FT stage. However, intermediate-field spectroscopy systems have relatively poor SNR and probe dead times may be 100 µs or more, resulting in significant spectral phase distortions that are difficult to correct. Examples of these situations are presented in Fig. 31(a – f). An intermediate-field measurement (with “poor” SNR) is equivalent to Fig. 31(a,b). If a FT is applied to recorded data including a significant dead time in the FID, Fig. 31(c), then the spectrum is distorted Fig. 31(d); this distortion manifests as oscillations in the baseline. However, if the initial data are simply discarded, Fig. 31(e), the spectrum in Fig. 31(f) contains a significant firstorder phase shift that is difficult to correct at a post-processing stage. The areas under the peaks are no longer quantitative. An alternative to the FT is a direct fit to the data in the time domain [221]. An example FID and spectrum reconstructed from the fit parameters are shown in Fig. 31(g,h). We proceed to describe one method of time domain fitting for spectroscopy data. Time-domain fitting is based on the concepts of linear prediction (LP), used to project FID decays forward and backward to overcome baseline and phase artifacts in the corresponding spectrum. Numerous LP methods are available in the literature; a review covering several LP methods is given in [222]. Although several possible methods exist, Hankel total least squares (HTLS) [223] provides a robust, computationally efficient solution that is appropriate for intermediatefield data, which we will summarize here. This non-iterative method is based on the Hankel singular value decomposition (HSVD) [224] method but provides enhanced performance through an additional filtering stage. Ideally, only the signal contained within the data is fitted and not the noise. The FID is modeled as a sum
90
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Figure 31: Demonstration of time domain fitting of simulated intermediate-field spectroscopy data, adapted from [222]. The entire FID (a) would ideally be acquired (SNR ≈ 100) and Fourier transformed to provide a real spectrum (b). In practice there is a dead time on the FID acquisition (c); here tDE = 100 µs. This dead time results in baseline distortions in the spectrum (d). The FID can be time shifted (e) so the initial zero amplitude points are ignored, but this introduces a firstorder phase shift in the spectrum (f) and results in incorrect peak intensities. A better approach is to fit the data in (c) in the time domain using the concepts of linear prediction to determine the frequency, T2∗ , and intensity of each component, from which a FID (g) and spectrum (h) may be reconstructed. In each plot the blue line indicates the real channel and the red line indicates the quadrature channel.
91
of sinusoids, damped exponentially, according to D
bn =
∑ {ad exp [ j (ϕ0 + ϕd )]} exp [(−rd + j2π fd )tn] ,
(40)
d=1
for the number of data n = 0 . . . N − 1. D is the total number of sinusoidal components (model order), fd are the component frequencies, rd are the component decay rates (effectively 1/T2∗ ), ad are the component amplitudes, and the component phases are described by (ϕ0 + ϕd ), for d = 1 . . . D. Typically, ϕd is zero and the phases of all components can be described simply by the zeroth order phase shift ϕ0 . The data are uniformly sampled at times t = (n + n0 ) ∆t, n = 0 . . . N − 1,
(41)
where ∆t is the constant sampling interval (equivalent to dwell time) and n0 ∆t describes the time-shift of the first datum relative to the origin (the dead time). This model enforces the assumption that all the spectral lines are Lorentzian. This assumption is generally valid for liquid and liquid-like samples generally measured in spectroscopy; if broadline components (solids or macromolecular species) are known to be present, multiple Lorentizan peaks will be fitted; these can be summed by the experimentalist. If the spectral lines are known to be Gaussian or Voigt (sum of Lorentizan and Gaussian) peaks, an alternative fitting approach should be adopted where a different model may be assumed. A review of time-domain fitting methods has been presented in [225]. The choice of model order D is a crucial step in time domain fitting. The model order is selected using a model information criterion, such as the Akaike information criterion (AIC) [226] which is appropriate for FIDs acquired with a large number of data points, although alternatives can be found in the literature [227]. In SVD-based analyses, the model order corresponds to the number of 92
significant singular values (those singular values corresponding to signal rather than noise) in the SVD of the Hankel data matrix b0 b1 b2 · · · bO−1 b1 b2 · HM×O = b2 , · .. .. .. . . . bM−1 · · · bN−1
(42)
consisting of the measured data bn , n = 0 . . . N −1, with dimensions M, O > D and
satisfying M + O = N + 1. Note that H contains complex data. The SVD yields H = USV† ,
(43)
where † denotes a conjugate transpose and S is a diagonal matrix containing the singular values in descending order. Regularization is achieved by truncating the SVD matrices to D significant values, as determined by the chosen information criterion. If N is large, the SVD stage may be achieved efficiently with a truncated eigenvalue decomposition of H† H [223, 228]. ˆ is obtained (where ˆ denotes an estimated or fitted paraNext, the solution O meter) that satisfies + U− K O ≈ UK ,
(44)
− where U+ K and UK are obtained from UK by omitting either the first or last row of
ˆ is obtained in a total least squares sense, then the SVD matrix, respectively. If O ˆ = −V12 V−1 where the matrices V12 and V−1 are determined from the SVD of O 22 22 − + the augmented data matrix UK , UK as described in [223]. The eigenvalues lˆd of ˆ are O
! lˆd = exp −ˆrd + j2π fˆd ∆t , d = 1 . . . D, 93
(45)
from which the estimates of frequency fˆd and decay rates rˆd are obtained for the D components. Finally, the corresponding amplitude terms aˆd exp [ jϕ0 ] are obtained from a least squares fit of the damped sinusoids to the data. The estimated fre∗ , and amplitudes aˆ can be used to reconstruct quencies fˆd , decay rates rˆd = 1/T2,d d
a spectrum if required, noting that aˆd are the equal to the areas under the spectral peaks, not the heights. A simulated result is given in Fig. 31. 5.3. Time-domain fitting for relaxation and diffusion When relaxation time or diffusion data are expected to contain only a few components (between 1 and 3) then the most reliable way to the fit these data is with a straightforward least-squares fit to the expected exponential function. Appropriate functions are given throughout Sections 4 and 5 with a summary given in Table 1 (located in Section 5.4.2). An alternative approach that has been used for relaxation time analysis is an exponential “stretch” fit [229]. For example, the fitting function for CPMG data would be modified from eq. (14) to ( ) b(t) nte ζ = exp − , b(0) T2
(46)
where ζ defines the shape of the relaxation curve relative to a perfect exponential (ζ = 1). This fit is useful for monitoring changes in a sample property, particularly when the data are non-exponential, such as in the presence of a stray field gradient [230]. If the relaxation time or diffusion data contain numerous components, it is better to fit a distribution of component amplitudes. This situation is likely to occur in studies of porous materials or emulsion droplets. We discuss distribution fitting methods next.
94
5.4. Distributions of relaxation and diffusion (1D) 5.4.1. Data fitting To generate a distribution of relaxation time or diffusion coefficients it is necessary to solve the Fredholm integral equation that models the expected relaxation behavior. In general, the NMR signal is described by the 1st kind Fredholm integral equation h(t) = h(0)
Z∞
K0 (u,t) p (u) du,
(47)
0
where h(t) is the noiseless signal acquired at an experimental time t, p is the (probability) density distribution governed by the time constants in u, and K0 is the model function (kernel) describing the signal. However, it is typical to solve for u on the log10 scale in the presence of noise e(t) so the acquired data are modeled by b(t) = b(0)
Z∞
K0 (u,t) f (u) d (log10 u) + e (t) .
(48)
−∞
The numerical inversion provides the distribution f (u) rather than the actual density distribution p(u). It is typical in NMR acquisitions for the data to contain insufficient information to determine a unique solution to the distribution in the presence of noise. Consequently, the numerical inversion is an ill-posed problem and so the determination of f (u) is difficult. A more stable result can be obtained by fitting the normalized signal b(t)/b(0), rather than scaling the expected form to match the data. To provide quantitative results, the integral area under the fitted distribution fˆ(u) is rescaled to equal b(0) at the end of the fitting process. Practically, b(0) is rarely measured, and so the datum with maximum signal cor-
95
responding to b(t → 0), is used for scaling instead. The rescaled distribution area
is then b(t → 0) fˆ(u) ≈ b(0), equal to the signal amplitude at the time origin. R
In 1D, the statistical approach of Wilson [231] for solving this first-kind Fredholm integral equation is adopted. In reality, our data consist of N discrete points. We achieve a stable inversion in the presence of noise using regularization, whereby we assume the distribution f (u) is always smooth and positive, and the integral over log10 u is constrained. Accordingly, we wish to determine a functional form of f (u) that minimizes the residuals 2 Z N R = ∑ bn − K0 (u,t) f (u) d (log10 u) n=1
+ α
Z
(H f ) (u) d (log10 u) ,
(49)
where H is the smoothing function and α is a constant (smoothing parameter) that defines the trade-off between smoothness and consistency with the data. Equations (48) and (49) are restated in vector-matrix notation as b = K0 f + e,
(50)
where lower case letters represent vectors, and upper case letters represent matrices. We solve for f using regularization by choosing a solution ˆf ≥ 0 that minimizes the residuals matrix (E + α L) where L is the smoothing operator and the weighted sum of squared errors is E = (g − Kf)† W (g − Kf) ,
(51)
where W is a diagonal precision matrix, required if the data have different weights (e.g., when using window sums of echoes as described in Section 5.1.2). Otherwise, W becomes the identity matrix I and can therefore be ignored in subsequent 96
calculations. An appropriate smoothing function for 1D regularization is the integral over the square of the second derivative of f (u). This choice of smoothing term forces the end-points of the distribution to be zero. As demonstrated by Wilson [231], the corresponding smoothing operator L ≡ f† Qf where Q is symmetric, with dimensions equal to the length of f, and having the form 6 −4 1 0 ··· −4 6 −4 1 · · · 1 Q= 1 −4 6 −4 · · · , 3 [δ (log10 u)] 0 1 −4 6 · · · .. .. .. .. . . . . . . .
(52)
residuals matrix then becomes [231] E + α L = −2b† WKf + f† K† WK + α Q f.
(53)
where δ (log10 u) is the constant width of each interval in the log10 u scale. The
Therefore, a solution ˆf is found that satisfies the quadratic programming problem (QPP)
ˆfα ≡ arg min 1 f† K† WK + α Q f − b† WKf, (54) f≥0 2 for a given value of α ; ˆfα can be determined using an appropriate medium-scale QPP solver. It is also necessary to optimize the value of α to provide a suitable balance between smoothness and accuracy in the solution. Generalized cross validation (GCV) [232] is considered an appropriate method, wherein a GCV score ΦGCV (α ) is calculated such that the minimum score corresponds to the optimum smoothing parameter αopt . An initial guess for α is obtained from Tr K† WK , α1 = Tr [Q] 97
(55)
where Tr[· · ·] denotes the trace of a matrix. An exact GCV score for the constrained minimization of eq. (54) is given in [233]; however, this calculation is computationally inefficient and an approximate GCV score may be obtained instead, based on the unconstrained minimization of eq. (54), as ! ! 1 ˆfα † W b − Kˆfα b − K n ΦGCV (α ) = h i2 , −1 1 1 − n Tr K (K† WK + α Q) K† W
(56)
where n is the number of data. An appropriate update formula for the estimate of
α is α←
h ! −2 † i † K W −Tr K K WK + α Q f† K−1 f
ΦGCV (α ) .
(57)
5.4.2. Kernel functions The kernel functions describe the expected form of the data and are therefore experiment-specific. For NMR experiments in general, the kernel function will be an exponential decay of the form K0 (u,t) = exp {−ut} ,
(58)
where u might be a relaxation rate or diffusion coefficient. A list of kernel functions for the 1D experiments described in Section 4 is given in Table 1. The kernel matrix K0 is generated by inputing all combinations of u and t into the appropriate function. In this way, the kernel matrix provides the expected (normalized) signal amplitude for any set of measurement times and fit parameters. The fitting process outlined in Section 5.4.1 effectively compares these expected values to the actual data in order to determine the most likely set of (smooth) component amplitudes that recreates the measured signal.
98
Experiment
Kernel function
Fitted
Section
K0
parameter
number
T1
4.1.1
T1
4.1.1
T2
4.1.2
D
4.2
n o t Inversion recovery 1 − 2 exp − T1 o n t Saturation recovery 1 − exp − T1 n o CPMG exp − Tt2 n o ! PFG diffusion exp − (γ gtδ )2 D t∆ − t3δ
Table 1: Example kernel functions for 1D relaxation and diffusion experiments. The Section in which experimental details are given is noted in each case.
5.4.3. Limitations and considerations It is important to note that when obtaining a distribution of relaxation time or diffusion coefficients following the method presented here (or related inversion techniques), the result is sensitive to the mathematics of the method. For example, the appropriate degree of smoothing is determined primarily by the SNR of the data and noisy data are smoothed more than “clean” data. However, a very high SNR > 1000 can result in under-smoothing and instabilities in the solution; that is, the fit becomes sensitive to non-idealities in the spin dynamics (imperfect rf pulses, motion/diffusion artifacts) and unphysical peaks will be introduced into the distribution. An ideal SNR is ∼ 100 [143] and white noise may be added to the raw data to provide a smooth but stable solution. To compare the shapes of distributions, it is necessary to ensure that the raw data have been acquired with a consistent SNR. It is better practice to compare peak positions and areas than shape. At low SNR, the noise may provide discrete peaks in a distribution. It is therefore sensible to ignore low intensity peaks (e.g., < 10 % of the maximum intensity) when interpreting relaxation time or diffusion coefficient distributions 99
unless there is a physical reason to expect such a small component to be present. Another factor influencing the resulting distribution is the choice of fitting parameters, specifically the range and number of relaxation time or diffusion coefficients being fitted. Common artifacts are peaks of significant area appearing at the extremities of the fitting range. These peaks arise due to non-exponential behavior in the first or last points of the raw data (such as a baseline offset). To determine whether these components are artifacts, the fit must be repeated with a wider fitting range; artifact peaks will move and remain at the end of the range; these peaks should be ignored in the interpretation. A stable and reliable inversion will be obtained with reasonable SNR and a sufficient number of data points. For the types of dataset encountered in low-field NMR, a useful rule of thumb is to acquire 32 points, appropriately spaced on the time axis [234]. For example, if fitting a sum of exponentials, the temporal separation of the data points should increase logarithmically. In general, fewer data points will give a poor fit, whilst additional points will not improve the quality of the distribution but will slow the computation. For experiments where every point is acquired in a separate scan, such as conventional T1 or D measurement, 32 experiments must be performed. For single-shot CPMG where 1000’s of echoes may be accrued rapidly, it is advisable to window sum the echo train (see Section 5.1.2) in order to reduce the number of data to 32; this pre-processing stage reduces the computational requirements of the QPP without influencing the result. Care should be taken not to over-interpret the data obtained from these fitting procedures (such as assigning significance to every low intensity component [235]) and it is important to understand the limitations of the numerical inversion algorithms. The same artifacts and limitations will apply to the mathematics of
100
2D inversions that we discuss next. 5.5. Distributions of relaxation and diffusion (2D) 5.5.1. Data fitting The process of fitting 2D distributions of relaxation time or diffusion coefficient is conceptually identical to that employed in Section 5.4.1 for 1D data, although a slightly different mathematical approach is required to handle the larger matrices. Full details of current 2D fitting methods are given in a recent review [236] and a discussion of errors in 2D inversions is given in [237]. The 2D data are described by b (t1 ,t2 ) = b(0, 0)
Z∞ Z∞
d (log10 u1 ) d (log10 u2 )
0 0
× K0 (u1 ,t1 , u2 ,t2 ) f (u1 , u2 ) + e (t1 ,t2 ) ,
(59)
where the subscripts 1, 2 denote the first and second dimensions of the experiment, respectively. The 2D data are stacked to form a vector (conversely the fitted distribution is a vector rearranged to form a 2D matrix) so the 2D Fredholm integral equation (59) is expressible in vector-matrix notation according to eq. (50). However, the corresponding K0 matrix will be very large unless the data are compressed (e.g., using window sums) and the solution size limited. The complete K0 matrix can be generated on a 32-bit desktop PC if both the data matrix and solution matrix are reduced to 32 × 32 elements. A precision matrix will be required to account for the data compression and so the optimization problem becomes
2
1/2 ˆf = arg min (60)
W0 (g − K0 f) , f≥0
where k· · ·k indicates the `2 (Euclidean) norm of the vector. 101
A computationally efficient method of solving the 2D inverse problem utilizes the Kronecker product structure of the kernel matrix. If the two dimensions of the experiment are separable (i.e., do not share a common time base) then we may write K0 = K2 ⊗ K1 . The much smaller K1,2 matrices can be computed and handled separately, so there is no need to generate the full K0 matrix. Now the 2D Fredholm integral equation is expressed in matrix notation as B = K1 FKT2 + E,
(61)
where T denotes a matrix transpose. It is usual to compress only the first dimension of the experiment (CPMG echo train) and so we only need include a W1 precision matrix associated with K1 . The optimization problem is
2
1/2 ! Fˆ = arg min W1 G − K1 FKT2 , F≥0
(62)
where k· · ·k now indicates the Frobenius norm of the matrix.
A common solution for reducing the size of the minimization problem in 1/2
eq. (60) is to project the data onto a truncated singular value basis of the W0 K0 matrix. This reduction is achieved by determining the significant singular values 1/2
of the kernel matrix. A singular value decomposition of W0 K0 is performed, equivalent to eq. (43) and detailed in [236]. The number of significant singular values is selected based on the SNR of the data. If the kernel matrix exhibits Kronecker product structure, the SVD of the smaller K1,2 matrices is performed instead, as described in [238]. Following truncation of the SVD, the reduced data ˜ 0. (stacked as a vector) are b˜ and the reduced kernel matrix is K We solve the reduced optimization problem using Tikhonov regularization by introducing a smoothing parameter α , forming the QPP ˆfα ≡ arg min 1 f† K ˜ †K ˜ 0 + α I 0 f − b† K ˜ 0 f. 0 f≥0 2 102
(63)
The reader will notice the similarity between this QPP and eq. (54), except here our smoothing matrix is just the identity matrix I0 . This large-scale QPP is solved using the method of Butler, Reeds, and Dawson [239]. For a given α , we make the substitution ˜ † c, f=K 0
(64)
where c is a vector of fitting parameters such that c=
˜ 0f b˜ − K . α
(65)
The fitting vector is determined by an unconstrained minimization [238, 239] from which the stacked vector ˆf is interpolated. Again, non-negativity is enforced in ˆ As with the the solution. The 2D distribution is obtained by reshaping ˆf → F. 1D inversion, numerous methods are available for determining the optimum value of α . The GCV method has been found to provide a robust, reliable, and fast estimate of αopt ; formulas for the initial estimate of α , GCV score, and α update appropriate for 2D optimization are given in [236]. 5.5.2. Kernel functions The kernel functions that describe 2D data can be grouped into two categories: separable and non-separable. For convenience, we first consider separable kernels. Most 2D relaxation and diffusion measurements fall into this category. The complete kernel functions are obtained by multiplying the 1D functions that describe the expected form of the data in each dimension of the experiment. A list of separable 2D relaxation kernel functions is given in Table 2. The complete kernel matrix K0 provides the expected (normalized) signal amplitude for every combination of the measurement time t1 ,t2 and fitting parameter u1 , u2 . 103
Experiment
Direct kernel Indirect kernel
function K2 n o t T1 -T2 (inversion) exp − T2 n o t T1 -T2 (saturation) exp − T2 n o T2 -T2 exp − Tt2
function K1 n o t 1 − 2 exp − T1 n o t 1 − exp − T1 n o exp − Tt2
Section number 4.3.1 4.3.1 4.3.3
Table 2: Example kernel functions for separable 2D relaxation correlation and exchange experiments. The Section in which experimental details are given is noted in each case.
The separable kernel functions for diffusion-relaxation experiments are a little more complicated to define due to the inherent relaxation that occurs during the diffusion encoding portion of the pulse sequence. If the diffusion timings remain constant, the kernel function is still separable. For example, the APGSTE-CPMG experiment, as shown in Fig. 20, has separable kernel functions (assuming rectangular gradient pulses) of 1 t∆ + tstore − τse tse K2 (T2 , nte ) = − exp − 2 T1 T2 nte × exp − , T2
(66)
and K1 (D, g) =
n 1 tδ o exp −tδ2 g2 D t∆ − . 2 3
(67)
As the timings are fixed and known, the first exponent in K2 is included readily in the optimization problem as a fixed scaling (amplitude) parameter. Generally, tse (1/T2 − 1/T1 ) vanishes for bulk liquids and is often ignored in porous materials, and the remaining (t∆ + tstore + τse ) /T1 factor is assumed to be constant and therefore also removed from the optimization problem by normalization of the 104
data [151]. The factor 1/2 in each kernel function is associated with the stimulated echo that returns only half the initial magnetization [91]. A few experiments generate 2D data with non-separable dimensions. Such experiments include the STRAFI D-T2 correlations discussed in Section 4.5.2. As both dimensions of the experiment depend on tse , the kernel function for the FFG-2SE experiment has the form 1 2 2 3 K0 (D, T2 ,tse , nte ) = exp − γ g Dtse 6 2tse + nte × exp − , T2,eff
(68)
and, similarly, the kernel function for the PFG-2SE (bench-top equivalent) experiment is [198] n tδ o K0 (D, T2 ,tδ ,tse , nte ) = exp −2γ 2 g2 Dtδ2 tse − 3 2tse + nte × exp − . T2
(69)
The kernel function may be simplified by substituting tδ with (tse − tc ) /2, where tc is a constant time difference between tδ and tse . However, this modification results in a non-rectangular data matrix that must be truncated or extrapolated with undesirable consequences [151]. It is preferable to construct the full kernel matrix K0 when possible [198, 240]. 6. Interpretation Relaxation times are often used in low-field measurements to infer some property of the sample; for example, changes in T2 have been related to the degree of cross-linking in polymeric materials [27]. However, a generic use of relaxation data (notably T2 decays) is the determination of total sample volume. In the case 105
of a CPMG decay, the exponential relaxation curve is fitted to obtain the signal intensity at the experimental time origin. In this way a signal intensity is determined (independent of relaxation) which is proportional to the number of spins and hence interpreted, for example, as a liquid volume. Quantification is achieved by comparison to a calibrated signal intensity. In this Section we consider some of the more sophisticated methods used to convert NMR signals into physical parameters describing a property of the sample. We describe pore size measurements (Section 6.1), surface interaction strength (Section 6.2), and exchange rates (Section 6.3) determined from relaxation times. Droplet size measurements are obtained using pulsed field gradient techniques (Section 6.4). The phenomenon of magnetic susceptibility gradients (Section 6.5) that are ubiquitous in heterogeneous samples is discussed. 6.1. Surface-to-volume ratio Distributions of relaxation time are a useful method of recording low-field data when the SNR is poor. Here we consider the interpretation of T1,2 distributions as measurements of surface-to-volume ratio. If a liquid is in contact with a solid surface then spins within molecules adsorbed on the surface will undergo enhanced relaxation due to reduced molecular mobility [241]. In the fast diffusion limit (where ρ2 `s /D0 1 as defined by Brownstein and Tarr [242], with ρ2 being the surface (transverse) relaxivity, `s being the length-scale of the confining geometry, and D0 being the bulk liquid diffusion coefficient) exchange is sufficiently rapid that surface and the bulk relaxing spins are well mixed, the observed relaxation time will be 1 1 1 = + , T1,2 T1,2,bulk T1,2,surf
(70)
106
where T1,2,bulk is the bulk liquid relaxation time and the relaxation time of molecules adsorbed on the surface is given by T1,2,surf =
1 V , ρ1,2 S
(71)
where ρ1,2 is the surface relaxivity term associated with longitudinal or transverse surface relaxation processes respectively, and S/V is the surface-to-volume ratio. For many samples, the bulk relaxation term in eq. (70) will be negligible compared to the surface relaxation term and can be ignored in the analysis so 1 S ' ρ1,2 . T1,2 V
(72)
In the context of porous media, S/V can be converted to pore size by assuming a pore geometry. For spherical pores, S/V ≡ 3/`s where `s is a characteristic length scale associated with the pore radius. Using eq. (72), distributions of relaxation are rescaled to pore size [243, 244]; an example is presented in Fig. 32. There are several notable limitations in the application of eq. (70) to real systems. The first is the assumption that the fast diffusion limit is valid at all times. This is generally true for small pores or weak surface relaxation. However, for liquid saturated large pores (or strong surface relaxation) the spins in the center of the pore will exhibit bulk-like relaxation. This situation is encountered in the intermediate or slow diffusion regimes [242]. Therefore, the surface-to-volume ratio term will apply only to the liquid layer containing spins influenced by the surface. This layer is considered to be 1 − 3 monolayers thick [245, 246]. Under these conditions, the observed relaxation rate becomes 1 S 1 = p ρ1,2 + (1 − p) , T1,2 V T1,2,bulk
(73)
where p is the fraction of spins in a pore that are influenced by the surface and S/V described the geometry of the surface-influenced layer. Generally p will be 107
unknown although it can be estimated from the peak areas in a T1,2 distribution if distinct relaxation time components associated with bulk and surface relaxation are present. The same concept applies in S/V measurements of granular suspensions. `s / µ m 10
−8
−7
−6
10
10
−5
10
P{log10(T2 )}
0.06 0.04 0.02 0 −3 10
10
−2
−1
10 T2 / s
0
10
1
10
Figure 32: Simulated example of a T2 relaxation time distribution (lower abscissa) converted to a distribution of pore radii `s (upper abscissa). The scaling has been achieved using ρ2 = 2 µm s−1 . In this example, the blue area corresponds to liquid influenced by surface interactions and the red area corresponds to bulk liquid. Assuming the pore liquid relaxes in the fast diffusion limit, then T2 maps directly to pore size. The small relaxation time component at T2 = 10−2 s is an artifact of the inversion process and is ignored in the analysis.
The effect of exchange between surface and bulk liquid is subtly different for T1 and T2 measurements. The T1 measurement averages over a longer time-scale than the T2 measurement. In a saturation or inversion recovery experiment, mixing occurs over the time τ1 which may vary from hundreds of microseconds up to many seconds so the effective time for exchange is long. In the case of a CPMG experiment, the relevant time-scale for mixing is the echo time te = 2τ2 which is typically less than a millisecond. Also, ρ1 ≤ ρ2 . Consequently, it is possible to envisage a system that is in the slow diffusion limit for a T2 measurement whilst 108
in the fast diffusion limit for a T1 measurement. Accordingly, T1 is considered a volume-averaged probe of pore structures whereas T2 is considered a local probe of pore surface structure. Further discussions on exchange will be given in Sections 6.3 and 7.3.1. A further complication of eq. (70) is the accurate determination of ρ1,2 . Theoretical expressions for the surface relaxivity term exist although attempts to calculate values have proven inadequate [152]. Ideally a calibration value of ρ1,2 would be obtained from a material of known S/V . However, even similar chemical surfaces have proven to result in very different relaxivities ensuring reference measurements are unreliable [247]. Direct estimates of ρ1,2 can be obtained by comparing the shortest relaxation time component observed in a sample (considered to be a measure of relaxation on the surface) to BET gas adsorption surface area measurements [248]. If S/V is very large or ρ1,2 is very small, the assumption that 1/T1,2,bulk is negligible in eq. (70) will not be valid. When converting T1,2 to S/V , the bulk relaxation time sets an upper limit on the surface-to-volume ratio (or pore size) that can be observed. Only pores of size `s < 6ρ1,2 T1,2,bulk will be observable. In the example presented in Fig. 32 this corresponds to a diameter of `s ∼ 10−5 m. Larger pores may be present but the liquid they contain will be indistinguishable from bulk. We note that diffusion measurements are also used to determine S/V in porous materials. At short observation times, when the mean displacement of the diffusing molecules is less than the pore size (practical values of t∆ lead to a lower pore size limit of `s ∼ 1 µm), the observed diffusion coefficient is related to the bulk
109
diffusion coefficient by [249] D 4 S (D0t∆ )1/2 . ≈ 1− √ D0 9 πV
(74)
At long observation times, such that diffusing molecules explore many connected pores, the tortuosity of the pore network is obtained. In diffusion NMR experiments it is common practice to define (dimensionless) tortuosity τtort as [250] D 1 lim ≡ . (75) t∆ →∞ D0 τtort The surface-to-volume ratio and tortuosity are therefore obtained readily as the √ initial slope and long-time plateau amplitude in a plot of D/D0 versus t∆ , respectively. The diffusion coefficients of the liquid in the porous material and in bulk are obtained using the APGSTE sequence (see Section 4.2) with a fixed observation time and variable gradient amplitude. The observation time is incremented in successive measurements. The “difftrain” pulse sequence [251] enables a rapid measure of surface-to-volume ratio as reviewed in [252]. Finally we draw attention to the method of determining surface-to-volume ratio distributions from the melting point depression of a confined frozen liquid. NMR cryoporometry [253] has been implemented at both low and high field, although the low-field measurements are considered more reliable as low Q probes (typical on benchtop hardware) are robust to thermal drift that is unavoidable during the experiment. The liquid-saturated porous material is cooled to well below the expected freezing point of the liquid and then warmed in situ so the melting transition can be observed by NMR. The confining geometry hinders the formation of regular crystal structures within the pores. Therefore, an irregular structure will form that has a lower melting point than the bulk crystal lattice. The temperature difference ∆Tm between the bulk melting point Tm (∞) (crystal of infinite size) 110
and the depressed melting point Tm (`c ) is proportional to the crystal size `c (and hence pore size `s when the crystal size is determined by geometric confinement) according to the Gibbs-Thomson equation for a small isolated crystal in its own liquid [254] ∆Tm (`c ) = Tm (∞) − Tm (`c ) =
4σsl Tm (∞) , `c ∆Hf ρs
(76)
where σsl is the surface energy at the crystal-liquid interface, ∆Hf is the bulk enthalpy of fusion (per gram of material), and ρs is the density of the solid. Consequently, the melting point depression of a confined liquid depends only on the properties of the liquid, its own solid, and the interface between these two states. Cryoporometry is independent of the chemistry of the pore surface; unlike relaxometry it does not require a term that depends on the adsorbate-adsorbent interaction. Instead the melting point depression constant CGT for a given liquid is obtained empirically based on measurements in controlled porosity materials (such as silica gels). The surface-to-volume ratio is determined simply as S ∆Tm = CGT . V
(77)
As the T2 of a solid is significantly shorter (usually by an order of magnitude or more) than the T2 of its liquid even when confined in pores, the quantification of the liquid fraction at any given temperature is usually straight forward. Complications arise when the solid is a plastic crystal (long T2 ); in this case robust solid/liquid phase separation is achieved by fitting relaxation time distributions to CPMG decays [247]. The range of pore sizes that can be probed is limited by the chosen liquid (through CGT ), the thermal stability of the sample, and the sensitivity of the temperature measurement. Practically, cryoporometry is suited to the study of small pores where `s < 10 µm. Cryoporometry has been applied to 111
numerous porous materials including cements, rocks (gas shales), coal, wood, and bones. A thorough review of the theory, experimental techniques, and applications is presented in [255, 256]. 6.2. Surface interaction strength Relaxation times T1,2 are sensitive to molecular motions occurring at the Larmor frequency. When molecular mobility is reduced, such as a liquid adsorbing on a pore surface, the frequency of molecular motion changes. T1 and T2 are sensitive to different types of motion: T1 is predominantly determined by molecular rotation whereas in heterogeneous systems T2 is often determined by translation motion. Therefore, adsorption on a surface influences T1 and T2 to different extents and the ratio T1 /T2 is useful as an indicator of the strength of surface adsorption. This ratio has an advantage over absolute relaxation time measurements in that the sensitivity to geometry is (to leading order) removed. A robust method for determining T1 /T2 is the 2D T1 -T2 correlation experiment; an example is presented in Fig. 33. The 2D plot provides a useful visual guide to unambiguously correlating T1 and T2 relaxation time components. To interpret surface relaxation, we represent a solid surface as a lattice of relaxation sinks; these sinks can take the form of active surface sites where molecules can adsorb at paramagnetic relaxation centers on the surface (such as metal impurities). Motion of a molecule adsorbed on a surface is governed by two time scales: the average surface diffusion correlation time τm and an average surface residence time τs . The diffusion correlation time describes the time for a molecule to move between relaxation sinks on the surface. The residence time describes the length of time an adsorbed molecule remains on the surface before desorbing back into the surrounding bulk liquid. Typically, τs τm , which means an adsorbed 112
1
10
a
0
T1 / s
10
−1
10
−2
10
T 1 /T 2 = 1.5
−3
10
1
10
b
0
T1 / s
10
−1
10
−2
10
T 1 /T 2 = 5
−3
10
−3
10
−2
10
−1
10 10 T2 / s
0
10
1
Figure 33: Simulated example T1 -T2 correlation plots of a liquid interacting (a) weakly and (b) strongly with the surface of a porous medium. The diagonal lines indicate T1 = T2 (dashed line) and T1 > T2 (solid line). Contour spacings are constant. In both plots a bulk liquid component is observed at long T1,2 times with T1 = T2 . When the liquid interacts with the surface, its relaxation times are reduced. The surface influences T1 and T2 to differing extents, resulting in T1 > T2 ; the stronger the interaction with the surface, the greater the ratio of T1 /T2 . For example, (a) weak surface interactions may provide T1 /T2 = 1.5 whereas (b) strong surface interactions may provide T1 /T2 = 5.
113
molecule undergoes some motion on the surface before desorbing. For a spin diffusing across a surface, the spectral density function is [257, 258] # " 2 1 + ω 2 τm . J (ω ) ∝ ln 2 (τm /τs )2 + ω 2 τm
(78)
If we consider first a “clean” surface (i.e., no paramagnetic relaxation sinks) [241] then the relaxation time ratio is T2,surf J (ω ) + 4J (2ω ) . =2 T1,surf 6J(0) + 5J (ω ) + 2J (2ω )
(79)
However, if the surface contains paramagnetic relaxation sinks, then the interaction of the unlike proton I and electron S spins gives rise to a modified relaxation time ratio [143, 257] T2,surf 3J (ωI ) + 7J (ωS ) , =2 T1,surf 8J(0) + 3J (ωI ) + 13J (ωS )
(80)
where ωS = 658.21ωI (i.e., ωS ωI ). At low field (within the fast diffusion limit [242]) we make the assumption that ωI 1/τs so that T1,2,surf T1,2,bulk and it then follows that the observed relaxation rate is approximately equal to the surface relaxation rate according to eq. (72). 6.3. Diffusive or chemical exchange Mass transport is often an important measurement in process engineering. NMR exchange time analysis can provide an estimate of mass transfer over distances of tens of microns, such as diffusion between neighboring pores in a liquid saturated porous medium. These exchange times are determined at low field using T2 -T2 experiments [143, 152, 153], wherein a pair of CPMG echo trains are separated by a z-storage interval of duration tstore , as detailed in Section 4.3.3. Here, we consider the analysis of these data. Off-diagonal exchange peaks (XP) are 114
observed in the T2 -T2 distribution when spins move between environments characterized by different T2 relaxation rates on the time-scale of the storage interval. The different T2 environments are characterized by the main peaks (MP) in the 2D distribution. Simulated data are presented in Fig. 34. It is important to realize that the positions of the peaks in the T2 -T2 distribution are determined by a combination of the absolute T2 relaxation time in each environment and the exchange rate. We therefore denote the observed relaxation as an apparent time T2,app . The same logic applies to longitudinal relaxation measurements in the presence of exchange, leading to T1,app . Technically, the observation of an apparent relaxation time occurs in 1D acquisitions as well (whenever exchange occurs) although this differentiation is rarely made in the literature. In 2D correlations the observation of apparent relaxation times is more obvious, particularly in T1 -T2 correlations where relaxation time components can have negative intensity in the presence of exchange. A full derivation of the effects of exchange on relaxation time components is given in [259]. Returning to the T2 -T2 plots in Fig. 34, the variation in amplitude of the offdiagonal peaks IXP as tstore changes is related to the rate of exchange. To determine the exchange rate, therefore, it is necessary to determine IXP over, say, 8 different storage times. IXP is calculated as the volume under the peaks in the 2D distribution. The T2 -T2 plot should be symmetric. However, the inversion process and real (i.e., noisy) data often leads to asymmetries in the distributions. The XP amplitude can be taken as the average volume under the two exchange peaks. As the tstore time is increased, the amount of T1 relaxation occurring likewise increases. This longitudinal relaxation results in a decrease in the overall intensity of the 2D plot. Therefore, it is often easier to consider the variation in IXP normalized by
115
10 10 10 10 10
T 2,app / s
10 10 10 10 10 10 10 10 10 10
1
a
0
MP
−1
MP −2 −3
1
b
0
XP
−1
XP
−2 −3
1
c
0 −1 −2 −3
−3
10
−2
−1
10 10 10 T 2,app / s
0
10
1
Figure 34: Simulated T2 -T2 maps showing two-site exchange. The main peaks (MP) and exchange peaks (XP) are labeled. The data were generated with an exchange time τex,a,b = 0.5 s and 2D plots are shown with storage times of (a) tstore = 1 ms, (b) 100 ms, and (c) 1 s. As the storage time increases, the relative intensity of the exchange peaks increases but the total intensity of the plot decreases due to T1 relaxation (visible in the 1D T2 projections). Contour spacings are constant across all plots. The spin system was simulated with Meq,a,b = 1, T1,a,b = 2 s, T2,a = 0.05 s, and T2,b = 0.5 s. The position and intensity of the peaks on the 2D plot do not match the absolute T2 relaxation times and magnetizations, due to exchange; apparent relaxation times and magnetizations are observed instead.
116
the total peak amplitude ITP . Several authors have proposed methods for fitting exchange rates [152, 153, 237]. If we consider two spin reservoirs (denoted by a and b) then it is straightforward to determine the Bloch equations describing relaxation under the condition of exchange as dMa Mb Meq,a − Ma Ma + + , =− dt τex,a τex,b T1,2,a
(81)
dMb Ma Meq,b − Mb Mb =− + + , dt τex,b τex,a T1,2,b
(82)
and
where Ma,b is the magnetization in reservoir a or b respectively at time t, Meq,a,b is the equilibrium magnetization in each reservoir as t → ∞, 1/τex,a,b describes the rate at which spins move from a → b and vice versa, and T1,2,a,b is the longitudinal or transverse relaxation time of each reservoir (independent of exchange). The equilibrium magnetization will depend on the type of relaxation; for longitudinal relaxation, Meq,a,b = Ma,b (0), i.e., the maximum (fully recovered) magnetization. For transverse relaxation, Meq,a,b = 0. The definition of exchange rate used here requires the condition of detailed balance such that τex,b Ma (0) = τex,a Mb (0); otherwise, spins will accumulate in one of the reservoirs. If Ma (0) = Mb (0) then exchange occurs at the same rate in either direction and we need only define a single exchange time as τex . It is convenient to express eq. (81) and (82) in matrix notation as [152] d(Meq,a −Ma ) − 1 − 1 dt = T1,2,a τex,a d(Meq,b −Mb ) 1 τex,a
dt
×
Meq,a − Ma
Meq,b − Mb 117
1
τex,b 1
1
− T − τex,b 1,2,b .
(83)
The simple but lengthy solutions to this equation are known [260] and have been reviewed in [237]. This analytic model provides a method of fitting exchange times that is reliable and robust for two-site exchange, assuming the relaxation times and relative spin populations of each reservoir can be determined independent of exchange [152, 154]. Of course, this is usually impractical. If the exchange time is similar to, or less than, the relaxation times such that T1,2,a τa,b ≤ T1,2,b or T1,2,a,b τa,b then the apparent relaxation times and magnetizations observed in the T2 -T2 experiment may be used in place of the absolute values in the fit; an example is given in Fig. 35. Otherwise, accurate exchange rates may not be readily determined and an alternative fitting method is required, which we discuss in the next paragraph. The two-site analytic model can be extended to multi-site exchange assuming the exchange pathways are independent [261]. The analytic solution for interconnected pathways in a three-site exchange model is discussed in [237]. An alternative determination of exchange time is achieved by simply fitting an exponential recovery curve to the plot of normalized XP amplitude against tstore [153]. An effective exchange time τex,eff is obtained from the equation tstore IXP (tstore ) IXP (∞) = 1 − exp − . ITP (tstore ) ITP (∞) τex,eff
(84)
As demonstrated in Fig. 35, τex,eff τex , specifically by a factor 2 in this example (a consequence of having equal spin populations by construction). Washburn and Callaghan suggested a 1/T1 correction factor be added to the exchange rate [153], although in this case the difference is negligible. Even when lacking knowledge regarding the spin populations in the absence of exchange (required to determine the mass balance governing exchange from a → b and b → a), the effective exchange rate is a useful parameter. This quick exponential fit can be used 118
0.5
IXP /ITP
0.4 0.3 0.2 0.1 0 0
0.5
1 tstore / s
1.5
2
Figure 35: Ratio of exchange peak to total peak intensity as a function of storage time (×) derived from simulated 2D T2 -T2 plots; see Fig. 34 for examples. The solid blue line indicates a best fit of the analytic solution to eq. (83) starting with initial guesses of Meq,a = 0.6, Meq,b = 1, T2,app,a = 0.045 s, T2,app,b = 0.265 s, and T1,a,b = 1 s, where the Meq and T2,app values were determined from Fig. 34(a). Despite the poor estimates of absolute relaxation time and magnetization (independent of exchange), the fit provides correct values of τex,a,b = 0.5 s and Meq,a,b = 1. Also shown is a best fit to the exponential recovery eq. (84) (red dashed line) which provides an effective exchange time of τex,eff = 0.25 s.
119
to compare effective exchange rates and is easily extended to multiple exchange pathways. Furthermore, a distribution of effective exchange times can be obtained readily using 1D Tikhonov regularization as described in Section 5.4.1. A distribution of exchange times can also be obtained from the analytic model in eq. (83) although this process is more complicated and requires many assumptions [261]. Another possible fit could be obtained using a sinc-like function of IXP (tstore ) as derived for two-site chemical exchange observed in 2D spectroscopy experiments [200]; to date, this model has not been applied to the T2 -T2 exchange experiment. Interpretation of exchange rates can be complicated by exchange processes other than pore-to-pore diffusion, notably diffusion between regions of different internal gradient strength (see Section 6.5 and [50]) or exchange between bulk liquid and surface adsorbed molecules in the slow-diffusion regime [242]. 6.4. Droplet sizing Pulsed field gradient measurements are used to determine distributions of droplet size in emulsions [22, 23]. Early bench-top diffusion experiments relied on incrementing the gradient pulse duration tδ or observation time t∆ to step through q-space. Relaxation or diffusion pre-conditioning was commonly used in such cases to eliminate the continuous liquid phase from the acquired signal. Van Duynhoven et al. advised that, given the capabilities of modern bench-top spectrometers, the experimentalist should increment gradient strength instead (the method described in Section 4.2), allowing simultaneous detection of oil and water diffusion coefficients without the concern of variable relaxation time contrast [262]. Now, as spectral resolution becomes routine on intermediate-field benchtop spectrometers, oil and water signals may be distinguished by chemical shift [263]. This method is more versatile as discrete signals are acquired from both 120
liquid phases simultaneously, enabling the investigation of complicated systems such as oil-in-water-in-oil multiple emulsions. Based on our discussion in Section 4.2 we know the signal acquired for a free liquid in a standard PGSTE experiment, Fig. 18(b), is [97] h b(g) tδ i ln = −D0 (γ tδ g)2 t∆ − , b(0) 3
(85)
assuming ideal rectangular gradient pulses. If the liquid is confined by a boundary the spins are unable to diffuse the expected distance through the gradient g. The signal attenuation is therefore less for confined liquid than free liquid. The signal attenuation for liquid confined within a spherical boundary of radius a (as applicable to an emulsion droplet) can be represented by three different models. • The Gaussian phase distribution (gpd) model [264–266] assumes the diffusing spin magnetization vector exhibits a phase that is distributed normally at all times. This approximate theoretical model describes the signal attenuation for diffusion bounded within a sphere as " ∞ b(g) 1 2tδ 2 = exp −2 (γ g) ∑ 2 2 2 2 b(0) m=1 βm (βm a − 2) βm D0 )# 2 + E (t∆ − tδ ) − 2E (t∆ ) − 2E (tδ ) + E (t∆ + tδ ) , − 2 (βm2 D0 )
(86)
where βm is given by the positive roots of J3/2 (β a) = β aJ5/2 (β a), Jn is the nth order Bessel function, and E(t) = exp −βm2 D0t .
• The short gradient pulse (sgp) model [267–269] provides an exact expres-
sion for signal attenuation under the assumption that tδ is negligibly small so that no diffusion occurs during the application of the gradient pulses. For spherical geometry, the signal decay is given by [γ tδ ga cos (γ tδ ga) − sin (γ tδ ga)]2 b(g) = 9 b(0) (γ tδ ga)6 121
∞
∞
2 (2n + 1)βnm βnm D0t∆ + 6∑ ∑ 2 exp − 2 a2 n=1 m=1 βnm − n − n γ tδ ga jn0 (γ tδ ga) 2 , × 2 − (γ t ga)2 βnm δ
(87)
where βnm are the roots of the equation jn0 (βnm ) = 0 (where jn is the nth order spherical Bessel function). Numerous versions of eq. (87) can be found in the literature; the correct form is given by Veeman [270]. • The third model is the block gradient pulse (bgp) approximation [263, 271]. This model is derived by solving the Bloch-Torrey equation [272] for diffusion in a single droplet using an eigenfunction expansion. The solution, applicable to a constant gradient, is then applied piecewise [273] to obtain a model for the entire PGSTE pulse sequence: b(g) v† G (t˜δ ; γ˜) D (t˜∆ − t˜δ ) G∗ (t˜δ ; γ˜) v = , b(0) v† D (t˜∆ + t˜δ ) v
(88)
where ∗ denotes the complex conjugate and D(t˜) = exp[−Λt˜] and G(t˜; γ˜) = exp{(−Λ− 2 δ where δ is the Krojγ˜Z)t˜}, with the elements of matrix Λ given by Λuv = βnm uv uv
necker delta; the elements of vector v are given by vu = δu1 . The matrix elements in Z have been determined analytically by Grebenkov [271, 274, 275] for a spherical droplet. The variable substitutions are t˜ = D0t/a2 and γ˜ = γ ga3 /D0 . Only 2 ) are required so the matrices G and D are limited the 50 largest eigenvalues (βnm
to dimensions of 50 × 50 elements and likewise the vector v has a length of 50 elements. In the limit that a → 0, G = D. In all three models, the only free parameter is a. These models can also be expanded to account for surface relaxation [270]. A droplet size distribution [265] is obtained using the regularization method outlined in Section 5.4.1. We note for completeness that when the continuous liquid phase or the droplet surface contains 122
paramagnetic ions, relaxation time measurements can be used to determine the droplet size distribution by analogy to the pore sizing measurement in Section 6.1 above. However, few samples are appropriate for study using this method; T2 relaxation is used for the probe droplet size distribution of water in crude oil emulsions when the crude oil contains metallic impurities [276]. 6.5. Internal gradients Magnetic susceptibility contrast ∆χ is an important topic for NMR applications to industrial processes. Many of the samples to be considered are heterogeneous, potentially causing gross distortions in the magnetic field B0 far greater than any inhomogeneities inherent in the permanent magnet design. As noted in Section 1, magnetic susceptibility induced gradients are often referred to as internal gradients because they are commonly encountered in porous materials (rocks, cements, catalysts). Internal gradients are known to be stronger at higher magnetic fields, although the exact scaling with field strength is complicated [50, 277]. Therefore, one method for minimizing these field inhomogeneities is the use of low-field magnets; this is the solution adopted in petrophysics, see Section 7.3.1. We have chosen to include this important discussion of internal gradients at a late stage in the review because this topic requires a clear understanding of the interplay between magnet, sample, and spin dynamics. Internal gradients influence almost all NMR measurements in heterogeneous materials, including imaging [278], diffusion [104, 279], and transverse relaxation [280]. T1 is insensitive to magnetic field gradients [281], although the maximum observable signal will be limited depending on the acquisition technique. Molecular diffusion occurring within a spatially variant magnetic field will result in a reduction in the T2∗ observed in a FID [282]; hence internal gradients cause line broadening in a spec123
trum. It is possible for the rf system dead time to exceed T2∗ , at which point no signal will be observed. Therefore, if very heterogeneous samples are to be studied, the NMR system must be carefully selected to provide a low magnetic field strength and fast rf receiver chain. When imaging porous materials, SPI-based methods are preferred because they are insensitive to relaxation time distortions [173], within the limit that a useful SNR is obtained. To leading order (for “weak” internal gradients, such that ∆χ B0 10 µT), the 3/2
maximum gradient strength gmax can be considered proportional to B0
[277], al-
though as B0 becomes large other factors (specifically pore size) limit gmax [240]. Nevertheless, this rule-of-thumb scaling indicates that internal gradients increase in magnitude rapidly as the magnetic field strength is increased. Even samples with a low magnetic susceptibility contrast can generate sufficient internal gradients so as to prevent quantitative measurements at intermediate field [50]. In much of the existing literature on NMR measurements in porous media, the influence of internal gradients has been overlooked, dismissed, or – worse – misinterpreted as a physical property of the sample. The classic example is the observation of a multi-modal T2 distribution in a rock known to have a mono-modal pore size distribution [50, 153]. Detailed understanding of internal gradients has largely originated from transverse relaxation measurements. This is a consequence of the well logging industry where T2 is the metric of choice and rocks (notably sandstones and shales) can exhibit a large susceptibility contrast [277]. In the presence of internal gradients, the effective relaxation time observed from a CPMG measurement is T2,eff , which is a combination of the actual relaxation time T2 and a diffusive component. The nature of the diffusion is determined by the relative significance of the experimental parameters, internal gradient strength, and
124
pore size. When one of these factors dominates over the other two, an asymptotic limit to the diffusion behavior is observed. The three asymptotic limits are: 1. The Short Time (ST) regime [283]. The archetypal regime where the diffusion exponent is determined by the echo time in the CPMG experiment. The ST regime is encountered in large pores with weak internal gradients. Dif√ fusion occurs over the length-scale `e ≈ D0te and the diffusion exponent varies as nte3 .
2. The Motional Averaging (MAV) regime [280]. Encountered in very small pores. The pore size `s defines the maximum diffusion distance and spins will explore an entire pore several times during te . The diffusion exponent varies with `2s and te . 3. The Localization (LOC) regime [284]. This regime corresponds to diffusion in very strong internal gradients. Spins dephase by more than 2π radians when diffusing over a distance `g = (D0 /γ g)1/3 . As the gradient is spatially variant, diffusing spins are not refocused properly by a spin echo. The diffusion exponent varies with te . The ratio `g /`s in this regime indicates the fraction of pore volume containing coherent spins. The relative importance of the length scales `e , `s , and `g determine the diffusion regime as illustrated in Fig. 36. At the asymptotic limits, analytic solutions exist for the diffusion exponent that contributes to the decay in the CPMG experiment. Outside the asymptotic limits, pre-asymptotic regimes exist. No exact analytic solutions exist in these ill-defined regimes, except that the diffusion exponent will ς
vary with nte where the power can have any value in the range 1 < ς < 3 [284]. The power ς effectively indicates the relative importance of the echo time on the degree of signal attenuation due to diffusion. When ς = 1 (asymptotic limit of 125
MAV and LOC regimes) the diffusion exponent is insensitive to te . However, when ς = 3 (asymptotic limit of ST regime) the diffusive decay is dependent on the chosen te . Therefore, in the ST regime reducing te reduces the amount of diffusive attenuation in the signal [277]. In real samples, the asymptotic limits of the MAV and LOC regimes (ς = 1) are not expected to be encountered: diffusion through connected pores will prevent the MAV asymptote from being reached and features of the LOC asymptote are not expected to be observed in 3D pore geometry due to very rapid magnetization decay [284]. 2
ST
10
MAV
`g /`s
0
10
−2
10
LOC −4
10
−4
10
−2
10
0
10
2
10
`e /`s
Figure 36: Schematic indicating the diffusion regimes ST, MAV, and LOC, as defined by the diffusion length-scales `e , `s , and `g , respectively. The dashed lines indicate equivalence of the length-scales. The boundaries between the asymptotic limits of the diffusion regimes (gray areas) are not well defined.
The use of low fields is intended to push the diffusion behavior into the ST regime [277]. In this asymptote the diffusion exponent becomes −
1 2 2 γ geff D0 nte3 , 12
(89)
which readers will recognize as the equation for free diffusion in a magnetic field gradient. Here, geff is an effective gradient strength determined as the average of all point gradients g(r) over the diffusion length-scale `e . It is important to note
126
that due to this averaging, the diffusion exponent is insensitive to local variations in g(r). The observed signal decay is given by b (nte ) 1 2 2 nte 3 exp − γ geff D0 nte . = exp − b(0) T2 12
(90)
By acquiring CPMG data with different echo times, it is possible to determine if the diffusive term is significant. The decay due to T2 will not vary with echo time. Therefore, if the CPMG decay becomes more rapid as te is increased, there must be a contribution from diffusion and hence the influence of the internal gradients is significant. In a porous material it is likely that a distribution of both T2 and geff will exist. Therefore, eq. (90) can be expressed as a 2D Fredholm integral equation b (nte ,te ) = b(0, 0)
Z∞ Z∞
d (log10 T2 ) d (log10 geff )
0 0
× K0 (nte ,te , T2 , geff ) f (T2 , geff ) + e (nte ,te ) ,
(91)
where the kernel function is given by the R.H.S. of eq. (90). A set of CPMG data acquired with a range of te can be inverted using the method outlined in Section 5.5.1, noting that the kernel is non-separable. A 2D distribution f (T2 , geff ) is obtained, from which the actual T2 distribution is extracted. In this way, the contributions from relaxation and diffusion are decoupled in the CPMG experiment [240]. Attempting to fit the CPMG decay described in eq. (90) with a simple relaxation kernel function of exp{−nte /T2,eff } will result in a distribution that is overly broad and containing unphysical components [50]. An illustration is given in Fig. 37. In this demonstration it is clear that even using a very short echo time 127
1
τ τ τ τ τ
a b/b(0)
0.8 0.6
≡ = = = =
0s 0.5 × 10 −4 1.5 × 10 −4 3.0 × 10 −4 6.0 × 10 −4
s s s s
0.4 0.2 0 0
0.2
0.4
0.6
0.8
1
nte / s 0.1
P[log10(T2,eff )]
b 0.08 0.06 0.04 0.02 0 −3 10
10
−2
−1
10
10
0
T2,eff / s
Figure 37: Simulation of internal gradient effects on measurements of transverse relaxation time: (a) CPMG decays were generated using eq. (90) with SNR = 100. The T2 distribution was represented by a log-normal distribution with mean 0.3 s and standard deviation 0.06 s; the geff distribution was represented by a log-normal distribution with mean 1 T m−1 and standard deviation 0.25 T m−1 . The true T2 decay (equivalent to τ → 0 s) is indicated by the dashed line. The solid lines show CPMG decays generated with both relaxation and diffusion exponents. As the half-echo time increases, the effect of diffusion is more significant, leading to faster signal decay. The T2,eff distributions obtained from the data in (a) are shown in (b), generated assuming the kernel function K0 = exp(−nte /T2 ). Again, the true T2 distribution is indicated by the dashed line. The other distributions are influenced by internal gradients resulting in a reduced mean effective relaxation time and unphysical components. The legend applies to both plots.
128
te = 100 µs (typically the shortest practical for benchtop instruments) the CPMG decay exhibits some diffusive attenuation. It is therefore very important to test for the presence of susceptibility induced gradients in any heterogeneous materials and interpret the data accordingly. In the pre-asymptotic diffusion regimes, the diffusion exponent is described by ς
−ante where a is an unphysical distribution containing the dimensionless terms
D0te /`2 and γ g`3 /D0 , each raised to some power, with ` being `e , `s , or `g . De-
scribing the diffusive exponent in this way allows the contributions of surface relaxation and diffusion in T2,eff to be decoupled in the pre-asymptotic regimes under the condition ς > 1. The correct scaling of ς is determined empirically using a the “data-collapse” protocol as detailed in [285]. This approach is useful when internal gradients are large and diffusion occurs in one of the pre-asymptotic regimes. 7. Industrial Applications In this Section we consider some of the most successful applications of lowfield NMR in industry with the specific examples of food manufacture (Section 7.1), the built environment (Section 7.2), and petroleum production (Section 7.3). There are, of course, many other examples of low-field NMR in industry to be found throughout the literature. For example, unilateral devices such as the NMRMOUSE have been applied extensively to the study of polymeric materials [27] and biomedical systems [286]. Furthermore, even within the topics we have considered here the examples are not exhaustive; wherever possible we have included citations to dedicated review articles. We present these applications to provide the reader with a link between the NMR hardware, experiments, analysis, and 129
interpretation in Sections 3 through 6 and the world of industrial process. 7.1. Food One of the most prevalent industrial applications of low-field bench-top magnets is process control in the food industry. NMR systems operating at ν0 = 20 MHz for 1 H are popular; these spectrometers have dominated the low-field market for the last two decades. The typical ν0 = 20 MHz spectrometer is configured to provide a few robust and reliable measurements at the press of a button: total signal intensity, relaxation time distributions, or diffusion measurements for droplet size distributions (as used in dairy product characterization, see Section 7.1.3). As such, these low-field instruments can be installed and operated readily in an industrial environment. Low-field NMR and MRI has been used as a characterization and quality control tool for a wide range of food products including bread [287–289] and biscuit dough [290, 291], potatoes [292–298] and potato starch [299–301], tomatoes [302, 303], apples [304–306], processed soybean protein [307], and powdered food products [308–310], to cite but a few of the more recent studies. Advanced imaging techniques, such as SPRITE (see Section 4.4.5), have been used to investigate oil and water in fried foods [311, 312]. Here we consider applications of low-field NMR in three areas of the food industry: meat, fish, and dairy products. 7.1.1. Meat A primary focus of low-field NMR in meat production and processing is the determination of the water holding capacity (WHC) – a metric of meat quality. A recent review details the importance of water in meat science [313]. Consequently, NMR provides a useful characterization of meat properties because the mobility 130
and distribution of water within muscle (myowater) affects the meat quality, including properties such as tenderness. Mammalian skeletal muscle consists of water (65 − 80 %), proteins (16 − 22 %), fats (1 − 13 %), carbohydrates(1 − 2 %) and less than 1 % other soluble materials. Water is held by microfibrils (protein chains in muscle fibers, see Fig. 38) and so WHC is mainly determined by microfibrillar shinkage during the onset of rigor mortis as the muscle converts to meat. In vivo, about 85 % of the water is associated with the myofibrils (intra-myofibrillar water). The majority of this intra-myofibrillar water is bound to, or trapped between, the myofibril filaments; this water may be mobilized post mortem due to changes in cellular structure. There will be some water associated with proteins that cannot be removed by external physical forces (mechanical processing, freezing, heating) although this protein-bound water will exchange with surrounding mobile water. The remaining water is located in the space between myofibrils (inter-myofibrillar water), between muscle fibers, and in or between the muscle fasciculi. The primary low-field measurement used in studies of muscle conversion to meat is the CPMG echo train, see Section 4.1.2. It is typical, as we shall see in the following discussion, for the transverse magnetization decay to contain only two or three components. Therefore, the data may be analyzed using a least squares fit with an appropriate sum of exponentials, as mentioned in Section 5.3. In the meat science literature it is common to see T2 relaxation time distributions, as described in Section 5.4. Conventional relaxation time interpretation relates T2 to the local environment of the water. Myowater typically provides two T2 relaxation time components: the intra-myofibrillar water exhibits T21 ∼ 30 − 50 ms whereas inter-myofibrillar water exhibits T22 ∼ 100 − 250 ms. The longer T2 of the inter-myofibrillar water reflects the greater mobility of this spin population.
131
Accordingly, an increase in the T21 relaxation time is interpreted as an increase in the intra-myofibrillar spacing, and an increase in the T22 relaxation time is interpreted as an increase in the inter-myofibrillar spacing. The associated spin populations (signal amplitude) are used to infer the fraction of intra- and intermyofibrillar water. Both T2 components exhibit a strong correlation with WHC [314], as well as animal age and muscle type [315]. The relaxation time component T21 is known to correlate to sensory attributes of the meat: juiciness and taste [316–318]. CPMG data acquired in the presence of a gradient confirmed that both water populations are mobile [319]. Diffusion mapping at intermediate field has been used to confirm the spatial distribution of water [320]. In some experiments, a short T2 < 10 ms relaxation time component has been observed and attributed to water strongly associated with proteins. Intra−myofibrillar water
Inter−myofibrillar water Nuclei
Single Inter−fascicular water muscle fiber Perimysium
Myofibril Sarcoplasm Sarcolemma Endomysium (between fibers) Fasciculus Extra−fascicular water
Epimysium (deep fascia)
Muscle
Figure 38: Illustration of the structuring within mammalian muscle, adapted from [313].
Early post mortem muscle is converted to meat by the process of rigor mortis: muscle fibers contract longitudinally and fuse, expelling intra-myofibrillar water. Water redistribution also occurs at a cellular level due to changes in ionic strength (salt content) [321–323] and hence osmotic transport across cell membranes. The shrinkage of muscle fibers alters the mobility of the water and is
132
monitored by changes in T2 . These processes are summarized in Fig. 39 alongside typical T2 results for porcine muscle (M. longissimus dorsi) [324]. Swelling of the myofibrils is observed as an increase in both the relaxation time and amplitude of the T21 component, Fig. 39 (upper graph). As contraction of the fibers overcomes the osmotic swelling, the amplitude of the T21 component decreases, whereas the amplitude of the T22 component increases, see Fig. 39 (lower graph). Other factors contribute to the fibril spacing (and hence water content), such as pH [314, 321, 323, 325], which results in lateral shrinkage of muscle fibers. The change in T2 behavior shown in Fig. 39 is interpreted as intra-myofibrillar water moving to the inter-myofibrillar space and then into gaps between the cells, fasciculi, and fibers as the cellular membranes destabilize [326, 327]. The shrinkage of muscle fibers reduces the WHC of meat. This process is countered by the proteolytic degradation of cytoskeletal proteins which prevents cell shrinkage; proteolysis causes the meat to become more tender. The swelling of the myofibrils during ageing results in an increase of intra-myofibrillar water and hence an improved WHC [318]. The formation of channels with diameter 20 − 50 µm allows free water to move to the surface of the meat, leading to “drip” during ageing [328]. The correlation between water environment and T2 has provided a facility for determining the factors that influence meat quality. For example, pork quality depends on antemortem stress and stunning method [329]. The presence of the porcine major gene Rendement Napole (RN) also affects WHC [330, 331]. WHC is improved by the addition of water-binding agents; conventional active waterbinding compounds include sodium chloride, sodium phosphate, sodium lactate, calcium lactate, lactic acid, and calcium chloride. However, T2 measurements
133
100
contraction rate T21 time T21 amplitude
48 99
98
46
60 44 40 42 20
40 97
80
38 0
4
8
12
16
20
contraction rate
50
T21 / s
T21 amplitude / %
100
0 24
Time / h Muscle fibers
cellular swelling
myofibrillar shrinkage and longitundinal contraction
disruption of cell membranes
Myofibril
4
80
3
60
2
40
1
Py-value T22 amplitude
0 0
4
8
12
16
20
Py-value
T22 amplitude / %
drip channel formation
20 0 24
Time / h
Figure 39: Illustration of the correlation between T2 , water environment, and structure in post mortem porcine muscle, adapted from [324]. The upper graph shows the variation in the short T21 transverse relaxation component in post mortem porcine muscle, along with the muscle fiber contraction rate [unspecified units]. The lower graph shows the variation in the long T22 transverse relaxation component in post mortem porcine muscle, along with an electrical impedance measurement (Py-value [unspecified units]). The impedance provides an indication of mobile water content.
134
have revealed that better WHC is achieved in sodium pyrophosphate and sodium bicarbonate enhanced meat [332]. The bicarbonate also resulted in less protein denaturing on cooking. Elsewhere, the water content of meat during cooking has been investigated by low-field NMR [317, 327, 333–335]. Storage conditions prior to cooking influence the meat quality as well: fast chilling provides an improved WHC [330, 336, 337]. Low-field NMR has been applied to studies of poultry meat where the use of sodium bicarbonate as a water-binding agent has been considered [338]. The stability and performance of mechanically recovered poultry meat in the presence of enzymes has been studied using T1 and T2 relaxation time analysis [339, 340]. Comprehensive reviews of the applications of low-field NMR in meat science are presented in [328, 341]. 7.1.2. Fish Low-field measurements of fish muscle are very similar to those of mammalian muscle discussed in the previous section. Again, two transverse relaxation time components are commonly observed in CPMG experiments with T21 ∼ 40−60 ms and T22 ∼ 150−400 ms. Like mammalian meat, the WHC of fish meat can be predicted based on relaxation time analysis [342, 343]. However, there are some differences to the interpretation. Fish have a high fat content, making separation of the oil and water signals difficult based on relaxation time analysis alone. Seasonal variations in salmon fat have been shown to give rise to very different relaxation times [344]. Fat and water separation is achieved efficiently using diffusion-weighting to remove the water signal, thereby allowing the fat content to be determined [345]. Appropriate pulse sequences for diffusion-weighting are discussed in Section 4.2. Two-dimensional diffusion-relaxation correlations, as 135
described in Section 4.3.2, have shown that the T22 relaxation time component in fish contains a significant signal contribution from fat [346]. The fat content of live salmon is determined in vivo using a NMR MOUSE (Section 3.1.2) where the inherent magnetic field gradient of the unilateral device provides automatic diffusion suppression of the water signal in the T22 component [347]. Low-field NMR has also been used as a non-invasive measurement of oyster mass and phenotype [348]. The convention of relating the short T21 component to restricted or bound water, and the long T22 component to mobile water/fat, in fish muscle is considered appropriate when the water content is high. However, relaxation time measurements of low water content systems, such as dried fish, need a different interpretation. Recently, models describing water adsorption on macromolecules and proton exchange have been introduced to explain the relaxation phenomena observed in low water content food products [349]. The texture [350] and quality [351] of frozen fish has been monitored using T2 relaxation time analysis. However, conflicting reports on the mobility of water determined during storage does raise some questions regarding the robustness of interpretation in these relaxation time measurements [352–354]. Salting and desalting also have a profound influence on the water content distribution in cod [355–359], salmon [356, 360, 361], and shrimp [362]. Some of these studies were supported by high-field 23 Na imaging [355, 360, 361] although given the advances in modern low-field spectrometers, sodium images could now be obtained with bench-top imaging systems. Low fields are advantageous for 23 Na measurements as the signal loss due to quadrupolar interactions is minimized. Exposure of fish fillets to high salinity brine was found to decrease the fraction of high mobility
136
water (long T2 ), whereas lower salinity brines promoted the uptake of water. Relaxation time analysis also revealed differences in muscle structure between wild and farmed fish, and reflected muscle structure differences according to the degree of antemortem stress. The effects of chilling method prior to storage on water content have also been studied [363, 364]. Detailed reviews of low-field NMR in fish processing are presented in [364, 365]; a table listing the published applications to fish and fish products is given in [364]. 7.1.3. Dairy Dairy products present an interesting challenge for low-field NMR. The longitudinal and transverse relaxation rates of the water fraction in milk and cheese are proportional to the protein content [366]. However, the mixture of oil (fat) and water often means that, whilst relaxation time analysis provides some information, interpretation of product structure based on T1 or T2 analysis alone can be uninformative. Diffusion is the ideal probe for these complicated mixtures of liquids and/or soft solids. Notwithstanding, there are plenty of examples in the literature of dairy product characterization by relaxation time analysis. The water and fat content of cream cheeses has been probed by T2 relaxation where a bi-exponential recovery is observed [367, 368]. Gel formation in cheese is similarly followed; a third (long) T2 component is observed when the cheese is mechanically stressed resulting in the release of (bulk) whey water [369]. The uptake of salt and the water content of Feta cheese during brining has been monitored with a combination of relaxation time analysis and MRI [370]. A combination of low-field relaxation time and high-field spectroscopy has been used to elucidate the process of protein digestion: the in vitro digestion of Parmigiano Reggiano cheese at different ages was observed [371]. The authors argued that the low-field NMR technique 137
would provide a route to on-line monitoring of digestive processes. Similarly, T1 relaxation time is a good indicator of albumen quality in hen eggs: another possible on-line monitoring application [372]. NMR spin relaxation in ice has been used to study crystal formation in ice cream [373] and probe the solid-to-fat ratio [374]. Relaxation time measurements also reveal fluid-phase (whey) separation in acidified milk drinks [375] and the liquid fat content in pumped milk [376]. −8
10 10 10 10
D / m2 s−1
10 10 10 10 10 10 10 10
Skimmed milk
Whole milk
Yogurt
Clotted cream
Cream cheese
Brie
Mozzarella
Monterey Jack
´ Pont l’Eveque
Cheddar
Raclette
Gruy`ere (1)
Emmentaler
Gruy`ere (2)
T´ete de Moine
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Figure 40: D-T2 correlations for a selection of dairy products, obtained at ν0 = 5 MHz using a fixed field gradient. In each plot the horizontal dashed line indicates the bulk diffusion coefficient of water at ambient conditions; the diagonal dashed line indicates the canonical “alkane line” corresponding to the D/T2 ratio of saturated hydrocarbons. The contour intervals correspond to 10 % steps up to the maximum intensity in each plot. Data previously presented in [141, 366].
Diffusion provides an alternative method of probing structure in dairy products. The ability to determine emulsion droplet size distributions, as described in Section 6.4, has been used to characterize cheese (protein-stabilized oil-in-water emulsions) [377]. M´etais and Mariette proposed the inclusion of a T1 -null to sup138
press the continuous oil phase for monitoring water-in-oil emulsions [378]. However, as noted in Section 4.2, modern bench-top spectrometers now have the capability to acquire diffusion coefficients for both liquid phases in a single experiment [262, 263]. Diffusion coefficients have also been used to probe the fat-water interface in emulsions [379] and the tortuosity of fat crystal networks, with the crystal S/V determined by relaxation time analysis [380]. The stray field of unilateral devices is useful for providing an inherent diffusion-weighting in T2,eff measurements, as demonstrated with the NMR-MOUSE as a device for in situ monitoring of dairy product quality [381]. Reviews of low-field NMR applications to food emulsions are given in [141, 382, 383]. Improved characterization of dairy products is achieved using 2D D-T2 correlations [366]; see Sections 4.3.2, 4.5.2, and 5.5.1 for further details. Enhanced separation of oil (fat) and water signals is achieved in the 2D plots; examples from a selection of dairy products are given in Fig. 40. Low fat content products (milk, yogurt) exhibit a strong signal at the bulk diffusion coefficient of water. As the fat content increases (cream, soft cheese) a signal associated with the oil (fat) emerges with a much slower diffusion coefficient. In all the examples, this fat component lies on or near the canonical “alkane” line showing the expected D-T2 response of saturated hydrocarbons; further details are given Section 7.3.1. In the solid-fat cheese, the mobility of the water is restricted and the diffusion coefficient reduced below the bulk water line. It is clear from Fig. 40 that the D-T2 correlation provides a robust and reliable method of characterizing the properties of dairy products. As an aside we note that this application of the D-T2 experiment has derived from the oil industry where salty dairy products (specifically cheese) provide a convenient proxy to the mixture of oil and brine found in a petroleum
139
reservoir. 7.2. The built environment Civil engineering and construction is a world-wide industry. Here we consider three areas where low-field NMR is providing increased understanding of construction materials: cement, wood, and the application of mobile NMR for conserving ancient buildings. 7.2.1. Cement Concrete, formed by mixing cement (or lime mortar) with aggregates has been used for thousands of years as a construction material. Portland cement is the most common form and consists of calcium silicates and metal oxides. When water is added, cement hydrates to form a structure of calcium silicate hydrate (CSH) gel [384]. The initial stages of hydration occur quickly, on the order of seconds to minutes. However, cement continues to react throughout its lifetime leading to structural changes that occur over many years. These slow processes can cause in-service failures, notably reinforcement corrosion that results from transport of sodium and chloride ions to embedded metal surfaces [385]. Despite the long history of cement as a building material, it is only recently that the structure of the CSH gel is being elucidated, partly due to the insights available from NMR studies. High-field spectroscopy has been used to explore the atomic arrangements in CSH and reviews of these measurements are to be found in [386, 387]. Fundamental insights into the arrangement of CSH structures have come from low-field NMR studies of the hydration water [388]. Better understanding of the nanoscale structure and water transport will enable the design of materials with greater strength and endurance in hostile environments, as well as enabling the prediction 140
of long-term properties and behavior of cements and concrete in the built environment. There is also a continual drive to develop improved building materials that are low in cost and have a reduced carbon footprint compared to conventional solutions. For example, global concrete production generated approximately 536 million tonnes of CO2 during 2010 [389]. Calcium silicate hydrate is a complicated material whose gel morphology depends on the available water for hydration (the water-to-cement ratio w/c), the cement formulation (initial composition plus additives), and environmental factors (temperature, humidity). Furthermore, the morphology alters with degree of hydration (time) and all these factors combine to make CSH gel a difficult material to study. Disordered on the nano-scale, CSH is formed through a dissolutionprecipitation reaction of tricalcium silicate (Alite) and dicalcium silicate (Belite) in water. CSH structures exist over a hierarchy of length-scales from nanometer thick sheets of Ca(OH)2 up to macroscopic cracks formed through chemical shrinkage and surface drying. To predict the performance of cement and concrete it is necessary to understand water transport over all these length-scales. Most low-field laboratory NMR experiments have been conducted on white (architectural) cement. This formulation is similar to Portland cement except for a significantly lower concentration of paramagnetic metal oxides and is therefore preferable for NMR analysis. The interpretation holds for Portland but the analysis (notably of T2 distributions) is more robust in white cement where internal gradients are negligible. Numerous interpretations of T1 and T2 relaxation time measurements have been proposed. Schreiner et al. introduced a three-phase model of 1 H relaxation that has been applied widely to studies of cement [390]. Bound water chemically combined in the hydration solids has a long T1 > 100 ms
141
and short T2 ∼ 10 µs. Water in the gel pores of the CSH structure exhibits short T1 and T2 on the order of T1,2 = 0.5 − 1 ms. Water in large capillary pores formed by chemical shrinkage exhibits long T1,2 ∼ 5 − 10 ms. Three exponential components fitted to a combination of FID and CPMG data correspond to the model of Schreiner et al. as noted in [391]. The component amplitudes also correspond to the fractions of water expected in each environment based on Neville’s theory of cement hydration [384]. More complicated five-phase [392] and seven-phase models [393] have been proposed that introduce concepts of water layers on pore surfaces. 5
P[log10(T2 )]
4
w/c = 0.48
3
w/c = 0.40
2
1
w/c = 0.32 0 −6
10
−5
10
10
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10
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Figure 41: T2 relaxation time distributions of cement pastes prepared at different w/c ratios after 10 days hydration. Data previously presented in [394].
A new interpretation of relaxometry data in hydrating cement was presented recently [394], supported by x-ray diffraction measurements and based on an earlier model [395]. A combination of solid (quadrature) echo and CPMG decays (see Sections 4.1.2 and 4.5) revealed five discrete relaxation time components as shown in Fig. 41. The quadrature echoes provided a short Gaussianlike and exponential-like T2 component [396]. The Gaussian-like component had T2 ∼ 10 µs and was attributed to chemically combined water in the crystalline 142
calcium hydroxide and ettringite (hexacalcium aluminate trisulfate hydrate) structures. The liquid-like signals (i.e., Lorentzian) were deconvolved from CPMG data to provide T2 ∼ 100 µs (water in CSH interlayers), T2 ∼ 400 µs (water in CSH gel porosity), T2 ∼ 1 ms (water in interhydrate spaces), and T2 ∼ 10 ms (water in capillary porosity). All the water in the sample was observed and the rate of signal loss with drying was as expected [397]. These measurements were implemented on a Bruker Minispec bench-top system operating at ν0 ≈ 7.5 MHz and highlight the capability of modern spectrometers to provide robust and reliable NMR analysis of cements. Based on surface relaxivity estimates in earlier work [152], the authors suggested characteristic length-scales of 0.85 nm (CSH interlayer spacing), 2.5 nm (gel pore size), and 8 nm (interhydrate pore size) for the different structures. The NMR signal intensity is rescaled to hydrate density, allowing the mass (and volume) fractions of the different cement phases to be determined [397]. The high temporal resolution of the NMR data acquisition permits monitoring of the structural changes during hydration [394], as demonstrated in Fig. 42 for white cement pastes prepared with different initial water-to-cement ratios. The evolution of porosity in hydrating cement has been elucidated through 2D T1 -T2 correlations [59, 143] and field-cycling experiments to probe changes in the specific surface area of the CSH structure [398, 399]. Exchange of water between CSH interlayers and gel pores has been explored using 2D T2 -T2 analysis (see Section 6.3) [152, 400]. Relaxometry has also been used to monitor the hydration of cement in the presence of additives [401–403] including light-weight aggregates [404] and controlled water release polymers [405]. The behavior of lime-pozzolan mixtures have been considered [406], as well as the hydration of
143
cement-wood composites [407]. Elsewhere, drying of cement has been monitored [408] and the mobility of water in cement has been assessed with diffusion measurements [409]. w/c = 0.32
Mass fraction
1
w/c = 0.40
w/c = 0.48
0.8 0.6 0.4 0.2 0
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Figure 42: Compositions of cement pastes by (top) mass and (bottom) volume as a function of hydration for pastes prepared at different w/c ratios, adapted from [394]. The compositions were determined from NMR signal intensities, except for the void volumes which were determined from chemical shrinkage experiments.
Beyond relaxation time analysis, the gel pores of size < 100 nm can be characterized using cryoporometry (see Section 6.1) [410]. Water distribution [411] and fractures [412, 413] are assessed using robust broadline imaging techniques such as SPRITE (see Section 4.4.5). The ingress of surface coatings has been monitored using STRAFI [391, 414] (see Section 4.5) and SPRITE [415]. A review of MRI studies of cements is presented in [416]. So far we have considered the monitoring of hydration kinetics and elucidation of nano-scale structuring in cements through bench-top analyses. However, there is interest in monitoring the long term in situ evolution of cements and concretes 144
using portable NMR devices. There have been several solutions to this challenge. Rf sensors may be embedded in the poured concrete and then interrogated when a mobile magnet is located nearby. The concept of embedded sensors was introduced in [417] and discussed in [230, 418]. Alternatively, single-sided sensors (see Section 3.1.2) may be used, circumventing the requirement of embedding a sensor during construction. The use of the NMR-MOUSE as a mobile device for monitoring hydration and drying was demonstrated in [230]. Elsewhere, the dedicated Surface-GARField magnet was developed to examine the water content in reinforced concrete structures in the 50 mm of material above the reinforcement bars [44]. The Surface-GARField, based on the concept of a linear Halbach array, provides a plane of uniform magnetic field that is stepped through the concrete. Two-dimensional T1 -T2 relaxation time correlations are obtained from water in the sensitive plane to enable improved interpretation of the concrete pore structure [59, 419]. Here, we have focused on the use of low-field NMR for studying microstructure and hydration in cementitious materials. We note that similar studies have been published for gypsum plaster, a light-weight alternative to cement used primarily as a surface coating for interior wall and ceiling and sold commercially as pre-fabricated “plasterboard”. The manufacture of dry plaster powder is energy intensive and there is a drive to optimize the drying process. The hydration of plaster powder results in the formation of a porous solid, the properties of which depend on the pore structure. Low-field NMR has been used to monitor the structural changes that occur during the hydration process [230, 420, 421]. As gypsum plaster does not generally contain paramagnetic impurities, the hydration process and structure-transport relations have been studied in detail at an intermediate
145
magnetic field strength [147, 422–424]. 7.2.2. Wood The requirement for low cost, renewable construction materials is driving a global interest in sustainable forestry [425]. Wood may be used directly as timber or reformed into modern composite materials such as plywood. Beyond its application as a building material, wood is also used in paper production and as an energy source. The water mobility and cellular structure of wood have been probed by laboratory NMR using relaxation time [426–428] and diffusion measurements [429, 430]. Transport of water between cells has been monitored using 2D T2 -T2 exchange measurements [431]. The tree rings in Scots pine sapwood have been observed using MRI [432]. The process of water adsorption [433] and desorption [434] in sawn wood, of relevance to the production of timber, has been investigated. Adsorption and effectiveness of wood treatments have been examined using the NMR-MOUSE [435]. Elsewhere, wood composites have been studied [436, 437]. The forestry industry has a particular interest in monitoring the transport, uptake, and loss of water in living trees. Portable NMR devices offer the opportunity to non-destructively probe in situ the water distribution in trees. These measurements can be used to infer the health of the tree and the quality of the wood. Monitoring water flow in plants with MRI was pioneered in the laboratory by van As and co-workers [438]; a recent review of the subject is presented in [439]. Specialized mobile magnets have been constructed to examine trees in orchards [41, 64, 440, 441]. In the most recent example an open access imaging magnet (operating at ν0 ≈ 1 MHz for 1 H and capable of investigating tree trunks up to 100 mm in diameter) dubbed the “Tree Hugger” has been used to verify the spi146
ral uptake of water in the sapwood of a bird cherry tree due to the phenomenon known as grain rotation [41]. Low-field imaging methods are discussed in Section 4.4.4. Such large bore, low-field portable magnets will be valuable in future studies of environmental influences on tree growth and health. Portable magnets are also used to study the condition of fruit prior to harvest [442]. 7.2.3. Conservation An aspect of the built environment where mobile NMR is providing a significant contribution is in the conservation of ancient buildings and monuments. Damage to ancient stone, concrete, brick, and mortar can come from a number of sources: water condensation and evaporation leading to erosion or salt dissolution, biological attack, and pollution (notably from motor vehicles in urban environments). Hydrophobic treatments provide an obvious solution to the problems of weathering by water invasion. We have mentioned already the use of MRI (STRAFI and SPRITE) for monitoring coating ingress into cement-based materials (see Section 7.2.1) [391, 414, 415]. Elsewhere, the more relevant challenge of monitoring surface coating efficacy on stone has been addressed [443–449]. Relaxation time analysis is useful for determining the effect of these treatments. For example, Sharma et al. showed that stone strengtheners fill the largest pores in stone and reduce the mobile water content [450]. Mobile devices like the NMR-MOUSE are enabling the measurement of pore size distributions in ancient stone structures [56], plus in situ determination of hydrophobic treatment efficacy [451, 452]. The NMR-MOUSE has also been used to assess the moisture content in plasters and mortars behind ancient frescoes (wall paintings) [453–456]. The presence of sub-surface water influences the choice of preservation technique applied to these paintings and mobile NMR is one of the 147
few non-destructive methods capable of mapping moisture content in these rare samples. The NMR-MOUSE has been used to observe capillary uptake of ground water into painted plaster before [457] and after treatment [458, 459]. A recent development – the Profile NMR-MOUSE capable of high-resolution 1D profiles into a remote sample [61] – is providing new insights into the structure of ancient wall paintings. The Profile NMR-MOUSE is capable of resolving multiple layers of paint and mortar, as well as providing a moisture content measurement. The Profile NMR-MOUSE has been applied to stones and mortars [68, 449] as well as a study of mosaic tesseræ [68]. Coatings such as paint can have negative consequences for the underlying stone or cement-based material. Some paints prevent evaporation of water, exacerbating damage from environmental effects and salt crystallization [460]. Salt redistribution is driven by water migration, notably surface drying.
23 Na
MRI
provides a method of monitoring salt transport in building materials [461, 462]. Similarly, salt crystallization which can result in damage to the pore structure [463] has been monitored [464], and the presence of salt has been observed to increase water adsorption in carbonate rocks [465]. Where possible, controlled humidity is expected to reduce damage to ancient building materials caused by salt migration. NMR, both at low field and high field, has been used in a range of other conservation projects including the study of paper in historic documents [57] and fired-clay pottery [466, 467]. A complete review is presented in [58]. 7.3. Petrophysics Low-field NMR plays two important roles in oil and gas production: welllogging tools provide access to “reservoir-scale” measurements of the formation 148
fluids, whereas bench-top instruments are used to examine fluids and cored plugs of rock recovered from the reservoir. Interpretations of well logs are supported by laboratory-scale measurements. Bench-top NMR systems provide fluid characterization, (see Section 7.3.3), rock properties (see Section 7.3.2), and a platform for trialling new recovery methods prior to single-well pilots in the reservoir [142, 468, 469]. 7.3.1. Well-logging The most successful commercial application of ex situ NMR has been welllogging for the petroleum industry [45, 73]. A brief history of the development and design of well-logging tools was presented in Section 3.1.3 [74, 470, 471]. We now address the measurement capability of these instruments. Well-logging tools are used in three situations: 1. Exploration. A new well (typically uncased) will be logged to determine the formation type, fluid content, and physical properties (porosity, permeability) as a function of well depth. These parameters will be used to determine whether production is commercially feasible and provide estimates of production volumes. Rock cores extracted during drilling will be measured in the laboratory to provide additional information not available from downhole analysis and support the interpretation of the well logs. The log will be acquired while the tool-string is in motion. 2. Piloting. A relatively new concept, the single-well in situ enhanced oil recovery (EOR) pilot provides a rapid, cost-efficient method of trialling new recovery methods in a reservoir formation. Oil recovery factors measured by NMR have been shown to agree with equivalent estimates obtained from
149
coherent dielectric logs and the conventional single-well chemical tracer test. The tool will be stationary while the logging data are acquired. 3. Monitoring / surveillance. A cased well is used to monitor the transport of oil or gas through a formation to obtain estimates of total production volumes and lifespan of the reservoir. The tool may be stationary while the logs are acquired. The archetypal data acquisition for NMR well-logging is a transverse relaxation decay obtained using a modified CPMG pulse sequence [77, 190, 192, 472]. These data may be acquired when the tool is stationary in a cased well, as the tool is retracted along an open or cased hole, or in logging while drilling (LWD) operations. The CPMG data are inverted to form a T2 distribution using methods similar to those described in Section 5.4.1, as introduced to petrophysics by Brown et al. [473]. A T2 distribution has the potential to provide a wealth of information regarding both the fluid and the rock matrix environment. An example T2 distribution for a water saturated sandstone is presented in Fig. 43. The total integral area under the distribution is proportional to the porosity of the formation; this is a particularly useful measurement as the NMR-derived porosity is independent of rock lithology and Archie parameters [474] that influence other logging tools. It is usual to analyze T2 distributions by positioning so-called T2 cut-offs across the distribution to segregate the regions of interest, as illustrated in Fig. 43. The area under each segment is obtained by integration; in sandstones the volume of clay-bound [475], capillary-bound, and mobile water is determined. The mobile water fraction corresponds to the porosity available for transport of fluids (water, oil, gas) and is referred to as the “producible porosity” [476]. However, care must be taken when analyzing logs (or core-plugs) obtained near the well bore 150
1 0.8 0.6
0.01
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P{log10(T2 )}
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clay capillary 0.02 bound bound water water
P
P{log10(T2 )}
`s
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10 T2 / s
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Figure 43: Simulation of a typical T2 probability distribution (blue line, left ordinate) obtained for a water-saturated sandstone. The water is present in three environments: (left) clay-bound, (center) capillary-bound (trapped in small pores) and (right) “free” (present in large pores). These environments are delimited by T2 cut-offs (vertical black lines). The large pores are assumed to be well connected and control fluid transport; hence the area under the portion of the distribution corresponding to “free” water is considered a measure of the producible fluid volume. As described in Section 6.1, the T2 can be rescaled to pore size `s if an appropriate value of the surface relaxivity
ρ2 is known. The T2 cumulative probability (green line, right ordinate) shows the sum over the distribution; the total liquid signal (here, normalized to unity) is equivalent to the total porosity of the formation.
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as drilling fluid infiltration will adversely influence relaxation time interpretations [477–479]. In clastic formations, notably well-sorted sands, a strong correlation exists between pore body size and permeability (determined by pore throat size) [480, 481]. It is therefore possible to infer a hydraulic permeability from
κ = Cs φ 4 (T2LM )2 ,
(92)
where φ is porosity, Cs is an empirical constant [482], the logarithmic mean T2 is defined as log10 (T2LM ) = hlog10 (T2 )i =
∑ [Pi log10 (T2i )] i
∑ Pi
i
,
(93)
and Pi represents the probability (amplitude) of the ith T2 component. Alternatively, the T2 cut-off corresponding to the producible porosity may be used instead (with an appropriately modified constant Cs ) as this better represents the earlier concept of relating permeability to the ratio of free to bound water [483– 485]. This comparison between T2 and permeability is dimensionally consistent through the scaling of ρ2 T2 having units of length, i.e., a pore body size according to eq. 71 (see Section 6.1), whilst permeability has units of length squared [248]. Surface relaxation in rocks occurs predominantly through adsorption on crystalline defects in the exposed mineral surface [486]. Additional relaxation will occur in clastic formations if paramagnetic metal species (e.g., iron or manganese oxides) are present, usually associated with clays [487]. Consequently, the constant cs must incorporate some empirical estimate of the surface relaxivity ρ2 . Independent estimates of ρ2 are difficult to obtain in sandstones: clays prevent quantitative determination of S/V using BET gas adsorption [248] and influence 152
diffusion measurements [488]. Instead, ρ2 may be estimated based on mercury intrusion porosimetry (MIP) [489–491] or capillary pressure measurements (centrifuge desaturation) [492, 493], again assuming a fixed relationship between pore throat size (as determined by MIP) and pore body size (as determined by NMR) [473, 475, 476, 494]. Recently, an alternative determination of ρ2 was proposed for sandstones based on independent MRI measurements of grain size [495]. Under favorable circumstances the T2 distribution can be used to determine the hydrocarbon content of the reservoir. A T2 cut-off is used to separate signal associated with the aqueous (brine) and hydrocarbon (oil, gas) fluid-phases. In light oil / gas reservoirs, the T2 of the hydrocarbons is typically longer than that of capillary-bound water. However, there will be overlap between the hydrocarbon and mobile water signal. Heavier oils will have shorter relaxation times (see Section 7.3.3 for more details) and may be distinguishable from the mobile water [142, 469]; wettability of the rock surface will play an important role in determining the actual relaxation times of the fluid-phases. Real wireline well logs are complicated to interpret, containing the results from many different tools mounted on the tool-string; examples are to be found in [45, 46]. Here we present an idealized NMR well-log in Fig. 44 to show the type of information typically obtained. The T2 cut-off would be determined to provide producible porosity and oil saturations consistent with the coherent measurements obtained from other logs such as gamma-ray, resistivity, and dielectric. The T2 distributions are stacked as a function of well depth, and this implication of spatial dimension has resulted in NMR well-logging being considered an imaging protocol; tools such as Schlumberger’s MRScanner1 can also provide information at different depths into the formation, adding an extra spatial dimension.
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Porosity Perm T2 distribution p.u. mD ms 30 0 50 0 0.3 3000 A
Well depth
B
C
Figure 44: Simulated well-log response for a T2 measurement of a sandstone reservoir containing brine (water) and a moderately heavy oil, η ∼ 100 cP. Individual T2 distributions are stacked on the right as a function of well depth. A T2 cut-off (red vertical line) has been determined to provide differentiation between oil (short T2 ) and mobile brine (long T2 ). Capillary-bound water is present, indicated by the very short (low intensity) T2 components. A permeability estimate is included (center) based on the T2 cut-off. On the right, the total porosity of the formation is shown (dashed line) along with the producible oil (black area) and mobile water (blue area). The formation is divided into three zones: (A) a low permeability clay-based cap-rock, (B) a high permeability oil reservoir, and (C) a moderate permeability sandstone below the free water level. In reality, such NMR data would be interpreted alongside numerous other coherent well logs with support from laboratory core analysis.
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Isolated pores
−1
0
1
Coupled pores
−1
10 10 10 ρ2 T2 / µm
0
10 10 10 ρ2 T2 / µm
1
Figure 45: Simulation of the effect of pore-to-pore coupling on T2 distributions obtained in carbonate formations, adapted from [496]. Diffusion of water occurs between intra-granular microporosity (red shading) and inter-granular macroporosity (unshaded). In all cases, the micropores constitute 17 porosity units ( p.u.) and the macropores constitute 14 p.u. porosity. The five cases (top to bottom) represent different surface relaxivities of ρ2 = 1.5 µm s−1 , 7.5 µm s−1 , 37.5 µm s−1 , 187.5 µm s−1 , and 937.5 µm s−1 . In the isolated pores (left) the short T2 component associated with the microporosity remains constant whereas the long T2 component associated with the macroporosity broadens as ρ2 increases and relaxation shifts from the fast diffusion limit to the slow-diffusion limit. In the coupled pore system (right) exchange causes the long T2 component to shift until the microporosity and macroporosity are indistinguishable. As the surface relaxivity increases, the effect of exchange becomes less significant and is eventually dominated by surface relaxation.
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Historically, many NMR studies of rocks have focused on sandstones due to their microstructural simplicity. However, understanding of fluid transport in carbonate reservoirs is of significant commercial interest. Unlike sandstones, it is possible to determine an accurate surface relaxivity in clay-free carbonate formations by comparing the initial relaxation rate to a measure of surface area from BET gas adsorption [248]. Carbonate formations have variable surface relaxivities, but the values are usually small and within the range 1 µm s−1 < ρ2 < 3 µm s−1 [248]. When the surface relaxivity of a carbonate formation is unknown, the industry standard “unit-normalized” value of ρ1 = 1 µm s−1 is used to obtain an estimate of pore body size. Attempts to infer permeability from the T2,LM can succeed in carbonates using a modified scaling of [248]
κ = Cc φ 2 (ρ2 T2LM )2 ,
(94)
when formations contain well-defined crystal or grain structures. This assumption breaks down in mud dominated formations where the pore body size and pore throat size are poorly correlated. Carbonates offer two additional challenges for NMR relaxation time analysis. 1. Diffusive coupling between microporosity and macroporosity [496] on the time-scale of the measurement removes the sensitivity of T2 to pore size; an apparent T2 is observed instead, as defined in Section 6.3, which will not rescale to pore body diameter. An illustration of the effect of poreto-pore coupling on the T2 distribution is given in Fig. 45. When surface relaxation is weak (ρ2 D0 /`s ) then diffusion and hence exchange will dominate; conversely if surface relaxation is strong (ρ2 D0 /`s ) then the 156
true T2 distribution will be observed. Unfortunately, carbonate formations tend to exhibit weak surface relaxivity so exchange dominates over relaxation. Fleury et al. have used the T2 -T2 exchange analysis to quantify the mass transport in benchtop measurements of carbonate plugs [157]. 2. The presence of very large macropores (`s > 10 µm) in a carbonate formation with weak surface relaxivity also results in a loss of sensitivity to pore size in the T2 measurement. The T2,bulk relaxation rate term dominates over the surface relaxation ρ2 S/V term in eq. (70). If we assume ρ2 = 1 µm s−1 and T2,bulk = 2 s for water, then surface relaxation is indistinguishable from bulk relaxation in pores of size `s > 10 µm. The standard T2 distribution is not the only measurement available with welllogging tools. When T2 is insufficient for an accurate fluid-phase discrimination, a T1 -D-T2 correlation may be obtained instead [60, 139, 196, 197, 497, 498]. Under ideal conditions these data can be acquired while the tool is in motion, or though practically these correlations are usually measured when the tool is stationary for piloting and monitoring operations. The D-T2 portion of the measurement is described in Section 4.5.2; the T1 dimension is appended by varying the recovery delay between scans. The addition of the diffusion dimension provides clear discrimination between oil, gas, and water in most formations. An example D-T2 plot is given in Fig. 46. The solid lines in this plot correspond to the bulk fluid responses expected for each fluid-phase. When measuring the fluids in a reservoir formation, the influence of the pore geometry will modify the D-T2 response. Very small pores will cause a reduction in the apparent diffusion coefficient as described by Tanner and Stejskal for diffusing molecules in a confined volume [96]. However, whilst the diffusion measurement is sensitive to confinement in rocks 157
[149] the diffusion coefficient cannot be reliably scaled to pore size due to connectivity between neighboring pores [499]. Furthermore, water bound on mineral surfaces or clays will exhibit very slow diffusion. Notwithstanding, Zielinski et al. demonstrated that it is possible to generate an approximate restricted diffusion line for water [500] "
D (T2 ) = D0 1 − D 0
`ρ `∆ + (`∆ /`H )2 `ρ `∆ + (`∆ /`H )2 + D 0
#
,
where ρeff T2 is a proxy for geometric restriction, `∆ =
(95) √ D0t∆ , D 0 = 1 − D∞ /D0
(with D∞ denoting the observed diffusion coefficient as t∆ → ∞), and 1 4 `ρ = √ . 9 π ρeff T2,surf
(96)
The length `H represents the scale of macroscopic heterogeneities in the formation with `H `∆ so the term (`∆ /`H )2 → 0 in eq. (95). In mixed wet rocks
ρeff =
Sg,w ρ2 , Sg,w + Sg,o
(97)
where Sg,w represents the grain area contacted by water and Sg,o is the grain area contacted by oil. In a water-wet rock, ρeff ≡ ρ2 for all aqueous fluid-phases. We note, therefore, that fitting eq. (95) to a bench-top D-T2 correlation obtained from a water-saturated water-wet plug provides an alternative method of estimating ρ2 . Oils and oil-based muds (OBM) tend to lie near the canonical “alkane line” which is determined from the expected D/T2 ratio that depends on viscosity; see Section 7.3.3. D-T2 plots are therefore useful for identifying the occurrence of mud invasion. The maximum oil viscosity that can be probed with NMR diffusometry is η ∼ 100 cP. Live oils with a high gas-to-oil ratio (GOR) will shift up from this alkane line. Undissolved gases (as found in gas reservoirs) have very high diffusion coefficients and reside above the water line in D-T2 plots. The 158
concept of eq. (95) is applied to hydrocarbons as well as water, but large spreads in D and T2 due to variable viscosity complicate the generation of a modified alkane line [501]. Therefore, whilst the D-T2 plot provides excellent fluid-phase sensitivity in most environments, careful interpretation of these 2D plots is still required.
−7
10
−8
gas
D / m2 s−1
10
−9
water
10
−10
oil
10
−11
10
−3
10
10
−2
10
−1
0
10
T2 / s
Figure 46: Representation of a D-T2 correlation plot for various fluid-phases trapped in a microporous rock formation. The dotted lines indicate the expected D-T2 behavior for gas (red, bulk diffusion), water (blue, bulk diffusion), and oil (green, alkane line). The solid lines indicate the modified D-T2 behavior [500, 501] to account for restricted diffusion effects. The exact positions of these lines will be temperature and pressure dependent. In this example, a mixed-viscosity oil is shown: the light fractions provide a signal component that lies on the expected D-T2 line; heavy fractions (viscosity η > 100 cP, including asphaltenes) diffuse too slowly to be measured and result in a tail on the oil distribution. Brine, confined in small pores, lies on the water (blue) restricted diffusion line. Similarly, a small quantity of gas provides a peak on the gas (red) diffusion line. In reality it is unusual to observe all three fluid-phases in a single measurement. It is important to note that in this example, the diffusion dimension (projection, right) provides improved fluid-phase discrimination over the T2 dimension (projection, top).
159
Another use of T2 analysis is the in situ estimation of wettability. The wettability is the tendency of a fluid to preferentially adsorb on a solid surface in the presence of other immiscible fluid-phases [502]. In the laboratory, wettability is usually characterized by either the Amott [503] or United States Bureau of Mines (USBM) indexes [504] which provide an indication of whether a mineral surface is preferentially wet by water, or by oil, or is mixed wet. NMR measurements of wettability provide a rapid alternative. The connection between relaxation and wettability was recognized by Brown and Fatt [505]. The details of extending laboratory measurements of wettability to well-logging tools have been discussed by Looyestijn [506, 507]. To infer wettability, the T2 relaxation time distribution of the oil measured downhole is compared to a bench-top T2 analysis of produced oil measured at downhole pressure and temperature. If the T2 of the oil in the reservoir is shorter than that of the produced oil, the in situ oil is undergoing surface relaxation and is therefore wetting the mineral surface, and the rock is assumed to be mixed-wet [508]. If no change in the T2 distribution is observed between the downhole and produced oil, the rock is inferred to be water-wet. Of course, a good separation of the downhole oil and water signals is required to make an unambiguous determination of the rock wettability and the D-T2 experiment again offers the advantage of improved fluid-phase discrimination [501, 509]. An improved indicator of wettability is achieved using the T1 /T2 ratio [14, 144] as discussed in Section 6.2. Presently, there is interest in determining wettability in unconventional shale reservoirs [501, 510, 511]. Reviews of wettability determination via NMR measurements are published in [512, 513].
160
7.3.2. Laboratory core analysis All of the experiments available downhole are also available in the laboratory for studies of core-plugs. Plugs may be extracted from cored material and, if handled with care, will contain the connate fluids. However, it is more usual to clean cored material and resaturate with reservoir fluids for analysis. Laboratory relaxation and diffusion experiments are used to improve interpretation of well logs [468, 469, 476, 514]. General discussions on NMR core analysis have been presented in [515–517], with articles focusing on unconventional gas hydrates [518] and shales [519]. One significant facility of laboratory core-plug studies is the spatial resolution of petrophysical parameters. In a recent review we considered the use of MRI in core analysis [14]. During the 1990s many experimentalists attempted to image fluid distribution in core-plugs using intermediate-field biomedical “small animal” magnets operating between B0 = 1 − 3 T. However, the internal gradients generated within the pores at these field strengths prevented quantification of the fluid volumes, as demonstrated by Straley et al. [475]. Consequently, MRI lost favor amongst the core analysis community. Recently, with the advent of modern bench-top spectrometers offering improved SNR at low field, there has been a resurgence in 1D profiling of core plugs. Two areas of particular interest have evolved from these low field MRI studies. 1. Rapid capillary pressure measurements. 2. Monitoring of forced displacement of oil. We now consider examples of these applications. Capillary pressure is an important petrophysical parameter: fluid transport and saturation states are affected by capillary-driven processes [520]. Capillary pres161
sure is generated by the contact of wetting and non-wetting fluids at a solid interface. “Capillary pressure curves” describe the variation in capillary pressure as a function of fluid saturation state and are used in core analysis to calculate petrophysical quantities such as irreducible water saturation, free water level, and residual oil saturation in the reservoir. Oil recovery predictions and simulations rely on capillary pressure measurements and therefore accurate determination of capillary pressure curves has significant economic implications. A conventional capillary pressure curve is obtained using a centrifuge, where a core-plug is spun at a range of rotation speeds until the liquid distribution in the rock reaches equilibrium. The volume of liquid ejected is recorded as a function of centrifugal force [493, 521]. Assuming the capillary pressure at the outlet face of the plug is zero, the capillary pressure as a function of radius rc is ! 2 1 2 Pc (rc ) = ∆ρϖc2 rc2 , − rc1 2
(98)
where the angular velocity of the centrifuge is ϖc , ∆ρ is the density difference between the resident and displacing fluids, and the radii rc1,2 are associated with the inlet and outlet faces of the plug respectively. In the conventional centrifuge method a range of capillary pressures will exist within the length of the plug. However, only the average volume of displaced liquid will be observed. The average saturation S¯ is assumed to be a sum over all capillary pressures so 1 S¯ = rc2 − rc1
PcZ(rc1 ) 0
S {Pc (rc )} dPc (rc ) . ∆ρϖc2 rc
(99)
The Hassler-Brunner integral for average saturation is obtained by rearranging eq. (99) as ¯ c (rc1 ) = cos2 SP
PcZ(rc1 ) 1 rc1 cos−1 dPc (rc ) 2 rc2 0
162
S {Pc (rc )} × r h i h i . Pc (rc ) rc1 2 1 −1 1 − Pc (r ) sin 2 cos rc2
(100)
c1
Unfortunately there is a recognized flaw in this method to calculate capillary pressure: as the average saturation is measured, eq. (100) is only valid when rc1 ≈ rc2 (i.e., in short plugs). Other approximate solutions are also known to have limitations [522]. MRI profiles of fluid saturation acquired after centrifugation offer a solution this problem. Rather than measure average saturation for the entire plug, eq. (100) is applied to each pixel in the profile where the assumption that rc1 ≈ rc2 is valid. Each pixel (radial position) is associated with a different centrifugal acceleration even though the centrifuge is running at a constant speed. Therefore, under ideal conditions it is possible to determine the capillary pressure at all fluid saturations S = 0 → 1 in a single experiment, although practically two or three centrifugation stages are often required to construct a full capillary pressure curve. SPRITE (see Section 4.4.5) is an ideal profiling pulse sequence for these capillary pressure curve measurements [523–527], providing robust and quantitative saturation profiles [528]; an example is given in Fig. 47. The addition of a spatially resolved T2 dimension [167, 184] (see Section 4.4.4) provides correlations between capillary pressure and pore size, as well as providing a route to measuring oil-brine capillary pressure curves using a T2 cut-off for fluid-phase discrimination [14]. Relaxation time T2 mapping [167, 184] is also useful for monitoring forced displacement of oil by brine or enhanced oil recovery (EOR) agents, such as polymers. A T2 map allows the simultaneous detection of oil and aqueous fluid-phases in a core-plug utilizing a T2 cut-off as described in Section 7.3.1. The data produced from a laboratory T2 mapping pulse sequence, whether through frequency 163
Local Sw
1 0.8
a r c1
r c2
0.6 0.4 0.2 0 10
12
14
16
18
Pc / kPa
r c / cm 160 140 120 100 80 60 40 20 0 0
b
0.2
0.4
0.6
0.8
1
Sw
Figure 47: Capillary pressure curve measurement of brine draining from a Berea sandstone plug, obtained by SPRITE profiling. Data previously presented in [523]. MRI profiles (a) are obtained before (4) and after (◦) centrifugation. The vertical dashed line indicate the inlet and outlet faces of the plug; fluid-flow occurred from left to right. The ratio of the intensities in these two profiles, obtained on a pixel-by-pixel basis, provides the water saturation Sw as a function of position rc2 − rc1 , which gives Pc (rc ) from eq. (98). Hence a capillary pressure curve (b) is obtained for a wide range of saturation states in a single experiment.
164
[167] or phase encoding [184], is equivalent to a well-log as demonstrated in Fig. 44 except that the vertical resolution is on the order of millimeters rather than tens of centimeters. Mitchell et al. have used T2 maps to monitor oil displaced by brine [529], brine and alkaline surfactant [468], and a combination of brine, polymer, and alkaline surfactant [142, 469]. In the surfactant studies, excellent agreement was observed between the laboratory recovery and single-well in situ EOR pilots. Elsewhere, low-field NMR has been used to study emulsion formation for enhanced recovery of heavy oils [530]. It is also possible to obtain spatially resolved D-T2 maps for improved fluid-phase discrimination [531] and spatially resolved T1 -T2 correlations for wettability mapping [14]. Finally, we note that another important core analysis measurement available in the laboratory is the flow propagator. The propagator provides an indication of the average displacement and dispersion of liquid flow at intermediate to high interstitial velocities [115], plus the fraction of stagnant fluid in the pores [131, 132]. To date, very few low-field propagator measurements have been published [86]. However, significant efforts have been made to understand and interpret flow propagators obtained for rocks at intermediate field [115–117, 130], including fast [133, 252] and quantitative measurements [134]. At present, flow propagators are used primarily to support simulations of fluid transport [135, 136, 532, 533]. 7.3.3. Hydrocarbon characterization We have touched already on the idea of fluid-phase identification through relaxation time and diffusion coefficient measurements (see Section 7.3.1). Separation of oil and water signals in bulk liquid samples is straightforward and often possible from 1D relaxation time distributions [534]. Here we consider in more detail the quantification of crude oil components using diffusion [535] or relax165
ation [536] as a proxy for chemical spectroscopy at low field. Let us consider a mixture of alkanes as a proxy for a real crude oil, where the mth alkane has a length (number of carbon atoms) Nn > 2. An approximate scaling for the diffusion coefficient of the nth alkane is derived from the Einstein-Stokes relation such that [535] D = X N¯ −Ξ Nn−ξ .
(101)
At ambient conditions, mixtures of alkanes will be described by eq. (101) with the empirically determined pre-factor X = 3.5 × 10−7 m2 s−1 and exponents Ξ ≈ 1.73 and ξ ≈ 0.7. If a distribution of diffusion coefficients has been determined, a given component with diffusion coefficient Dn will be associated with an alkane of chain length 1 Ξ+ξ
D
1 ξ
E Ξ+Ξ ξ
−1
Nn = X D Dn ξ , (102) D E where D 1/ξ is the (1/ξ )th moment of the measured distribution f (log10 D) such that D 1E Z 1 D ξ ≡ d log10 D f (log10 D) D ξ .
(103)
A distribution of alkane chain lengths is obtained from the distribution of diffusion coefficients simply as f (log10 N) = ξ f (log10 D) .
(104)
An approximate scaling law also exists for the longitudinal and transverse relaxation rates. Assuming the viscosity η < 100 cP such that T1,n ' T2,n (based on the assumption that frequencies of motion in light oils are > 100 MHz so that T1 and T2 are frequency-independent at low field) then [536] T2,n = Y N¯ −ϒ Nn−υ .
(105) 166
In this case the empirical pre-factor Y = 672 s and exponents ϒ = 1.25 and υ = 1.24 [536]. Again, a distribution of relaxation times will have components T2,n associated with alkanes of chain length ϒ 1 ϒ+υ 1 −1 Nn = Y ϒ+υ T2υ T2,nυ ,
(106)
where Z 1 1 T2υ = d log10 T2 f (log10 T2 ) T2υ ,
(107)
and the alkane chain length distribution is given by f (log10 N) = υ f (log10 T2 ) .
(108)
The pre-factors and exponents in eq. (101) and (105) are pressure- and temperaturedependent as described in [537]. Although crude oils tend to contain a complicated mixture of organic molecules (unsaturated, branched, or cyclic components) these scaling laws have been shown to apply to light crude oils. A summary is presented in [538]. We note that the viscosity dependences of D and T2 are the origin of the bulk D/T2 line in Fig. 46. Elsewhere, Chen et al. found T2 correlated better to specific gravity than viscosity in heavy oils [539], and in other work multivariant analysis was used to provide a reliable oil viscosity prediction based on T2 relaxation times across a wide range of molecular weights [540]. The existence of either very light (N < 2) or heavy components (such as asphaltenes and waxes) will complicate the analysis [535]. A low-field NMR study of asphaltene content and aggregation is presented in [541]. Relaxation and diffusion experiments have also been applied to heavy oils [542, 543] and emulsified crude oils to determine stability, sedimentation, and coalescence in the emulsion [542, 544, 545]. We conclude by noting that although our discussion here has focused on characterization 167
of light crude oils, there are many examples in the literature of low-field NMR applied to unconventional heavy crudes classified as bitumen [546, 547] and pitch [548–550], and even solid hydrocarbons in the form of coal [551–554]. 8. Future Directions In this final Section of the review we look forward to some possible future developments in permanent magnet technology and low-field data analysis. 8.1. Hardware Recent developments in permanent magnet technology have countered many of the perceived limitations of low-field NMR. Improved pole-piece and gradient coil design, plus the inclusion of shim coils, have provided significant increase in field homogeneity. Advances in probe rf electronics and signal detection have improved the achievable SNR. The advent of fully digital spectrometers now means low-field systems are capable of reproducing most of the experiments employed at high field. At the moment, there is a trend for the production of permanent magnet benchtop instruments with intermediate field strengths in the range B0 = 1 − 2 T for spectroscopy and biomedical imaging. These magnets are still limited in some cases by thermal drifts which result in large changes in the B0 field (on the order ∼ 100 ppm) over several hours. Even with the best thermal regulation it is difficult to provide a temperature stability corresponding to < 1 ppm frequency drift over a long time. The almost inevitable inclusion of a B0 shim and lock channel, available on the latest generation of spectrometers, will increase the versatility of these bench-top instruments.
168
Improvements to portable magnets are likely to continue much as they have, with experiments previously considered impossible on unilateral devices now being implemented routinely. As with the bench-top systems, advances in magnet pole-piece geometry and mechanical shimming techniques will provide larger homogeneous volumes or regions of uniform gradient, improving signal quality and reducing acquisition times. An area that might develop is the implementation of “lab-on-a-chip” low-field microfluidic devices for process monitoring and control; such systems are currently being demonstrated in high-field magnets [555, 556]. As the world’s supply of liquid helium becomes increasingly scarce, driving up the cost of operating older superconducting magnets, more research laboratories will turn to low-field NMR as the technology of choice. Improvements in commercial bench-top platforms are driven by the consumer, so as experimentalists attempt to implement more and more complicated acquisitions and protocols, we shall see further hardware developments. 8.2. Experiments It is difficult to envision future directions for NMR experiments except to say that they will depend on hardware improvements discussed already. Experiments considered routine in high-field NMR (such as nD-spectroscopy correlations, velocity mapping) will probably become feasible at low field, although the underlying pulse sequences are not expected to change significantly. Rather than modify the underlying experiments, it is likely that sophisticated processing and analysis techniques will be introduced; some examples are discussed below in Section 8.3. One area that is likely to see an advance in the next few years is intermediatefield spectroscopy. The advent of compact, bench-top instruments operating in the range ν0 = 45 − 60 MHz and capable of providing useful spectroscopic resolution 169
will enable monitoring of flow and reaction systems in a commercial production environment [4]. These instruments have been used for reaction monitoring [66, 557–563]. In order to advance the applications of these instruments we should look to experimental methods of signal and sensitivity enhancement. Spectroscopic sensitivity can be improved by careful nucleus selection. For example, 13 C nuclei exhibit a much wider chemical shift range than 1 H. However, 13 C
is only 1.07 % naturally abundant and has a gyromagnetic ratio four times
smaller than that of 1 H. Therefore, a NMR system operating at ν0 = 45 MHz for 1H
will need to operate at ν0 ≈ 11 MHz for 13 C. Overall, natural abundance 13 C
measurements are 5870 times less sensitive than equivalent 1 H measurements. It will be necessary, therefore, to adopt signal enhancement techniques to probe carbon nuclei at low field. Current options include isotopic enrichment to increase the number of resonant carbon nuclei in a sample, or polarization transfer techniques such as proton-decoupled 13 C distortionless enhancement by polarization transfer (DEPT) [564]. An alternative to switching nucleus is the enhancement of the 1 H signal. Highfield solutions, such as cryo-probes, are unlikely to be introduced at low field as this would remove the advantage of cryogen-free operation. However, enhancement of spectral lines through para-hydrogen induced polarization (PHIP) enhancement may be an important technique in the low-field spectroscopy arena. Para-hydrogen signal enhancement techniques are reviewed in [565, 566]. This method of signal enhancement is particularly interesting for reaction monitoring as specific spectral lines can be enhanced, allowing the observation of intermediaries or low conversion products [567].
170
8.3. Analysis It is expected that the next significant advances in low-field NMR will occur at the data processing stage. Rather than improving the hardware or experiment, the data currently available will be analyzed in a more intelligent manner. We describe two such examples here. Bayesian analysis uses prior knowledge of the system and a model of the data to predict the most likely fit parameters from undersampled or noisy data. Compressed sensing (CS) is a modification to the Fourier transform that takes advantage of fact that NMR data, when transformed to some other domain (e.g., following a total variation or wavelet transform), is typically sparse. Significantly undersampled data can be reconstructed, leading to an improvement in acquisition time. 8.3.1. Bayesian inference Bayes’ theorem [568] relates the probability of a hypothesis (set of fitting parameters) to a set of measured data. Formally, this is expressed by the posterior probability distribution P(ϑ |ι ), which describes the probability of a hypothesis
ϑ given the measurements ι . The posterior probability distribution is calculated from P(ϑ |ι ) =
P(ι |ϑ )P(ϑ ) , P(ι )
(109)
where P(ι |ϑ ) is the likelihood of observing the measured data given that the hypothesis ϑ is true, and P(ϑ ) is the prior which describes our a priori knowledge of the system. The denominator in eq. 109, P(ι ), is called the evidence and describes the probability of measuring any set of data ι . In applications to parameter estimation, the evidence is a constant that acts only to normalize the probability distribution and can be ignored. 171
NMR data
Model data (likelihood)
Posterior probability
Prior knowledge
Most likely parameter
Figure 48: Illustration of inputs for Bayesian analysis of NMR data. The measured NMR data (e.g., FID, k-space acquisition), a model of the expected data for a given set of fitting parameters (e.g., signal intensity, mean and standard deviation of a distribution), and prior knowledge about the system (e.g., model function, range of fitting parameters) are combined to provide the posterior probability distribution. This process is repeated for a selection of parameter sets. The parameter set with the highest probability of generating the measured data is determined.
The inputs required for Bayesian inference are 1. the measured NMR data ι , 2. a model of the expected data for a given parameter set ϑ , 3. prior knowledge including the model function and range of the parameter set to be tested. The protocol is illustrated in Fig. 48. Effectively, the Bayesian inference approach compares a range of parameter sets to the data and determines which parameter set is most likely to give rise to the measurement. The input parameter set may contain a single term (e.g., signal intensity) or multiple terms (e.g., mean and standard deviation of a distribution). If each parameter set provides unique model data and the model is a good approximation for the real system, a significant amount of information can be inferred from very few data. Bayesian analysis has been used previously in NMR applications to extract chemical spectra from noisy data [569], flow velocities from under-sampled data [570], and bubble size distributions in bubble columns [571, 572]. Recently, 172
Bayesian analysis has been ported to low-field NMR for the determination of grain size distributions in reservoir rocks [495] and particle sizing with Earth’s field NMR [219]. The ability of the inference approach to overcome poor SNR and provide robust results from very few data makes it an ideal analysis approach for future low field process control applications. 8.3.2. Compressed sensing Compressed sensing (CS) provides a protocol for undersampling and reconstructing NMR data acquired in the inverse Fourier domain. So far, CS has be applied to MRI [183, 573], velocity mapping [574], spectroscopy [575], and flow propagators [576]. The concepts of CS have even been extended to multidimensional relaxation time measurements [577], although the application to numerical inversions of Fredholm integral equations is not currently robust for routine application. CS is readily applied to MRI when incomplete data are acquired in k-space using a pseudo-random sampling scheme. If we have prior knowledge that the image is sparse in some transform domain, e.g., following the discrete cosine, wavelet, or curvelet transform, then the theory of CS allows us to reconstruct the complete image from a subsample of the full data. If the data (in this case k-space points) are undersampled incoherently (meaning the sampling scheme is not sparse in the transform domain [578]) then the image is obtained using an appropriate non-linear reconstruction algorithm. As the CS protocol returns only the signal and not the noise in the data, the noise-like artifacts associated with the sampling scheme are removed. The CS reconstruction replaces the FT processing stage. Here we present an example CS reconstruction method suitable for use with undersampled MRI data after [574, 579]. This CS method is derived from basis 173
pursuit [580, 581]. The image reconstruction uses the `1 -norm to characterize the sparsity of the reconstructed image. First the nD k-space data are stacked into a vector b. The image to be reconstructed is likewise stacked as a vector f and this is transformed into the sparse domain by operator Ψ. The k-space data vector b is obtained from the image by the undersampled Fourier transform F . The reconstruction is achieved by solving the constrained optimization problem min kΨfk1 , s.t. kF f − bk2 ≤ e,
(110)
where e is a threshold that is set equal to the expected noise level and the `1 norm is a surrogate for sparsity. Therefore, minimizing the objective in eq. (110) provides an image that is consistent with the acquired k-space data but has the most sparse representation achievable in the transform domain. Equation (110) is appealing because it is a convex optimization problem and therefore readily solved using a number of existing algorithms. Equation (110) can either be solved directly or in its unconstrained Lagrangian form arg min kF f − bk2 + λ1 kΨfk1 .
(111)
f
using, for example, a projected conjugate gradients algorithm [581], iterative thresholding [582], interior point methods [583], or Bregman iterations [584]. It is common to include the spatial finite differences (total variation) transform with another sparse transform [585], such as a wavelet transform. The optimization problem then becomes min (kΨa fk1 + λ2 kΨb fk1 ) , s.t. kF f − bk2 ≤ e,
(112) 174
where Ψa is the wavelet operator, Ψb is the spatial finite-differences operator, and the constant λ2 trades sparsity in each of the two domains. As with eq. (110), eq. (112) may be solved in an unconstrained Lagrangian form arg min kF f − bk2 + λa kΨa fk1 + λb kΨb fk1 . f
(113)
Methods for reconstructing phase sensitive k-space data are described in [574]. CS reconstruction provides a particular improvement to phase encoded MRI acquisitions due to the nature of the sampling scheme (one wavenumber per measurement) where any k-space co-ordinate can be accessed readily. SPI-based acquisitions provide an obvious example of an imaging method wherein k-space wave-numbers may be sampled incoherently [183, 573]. As the image dimensionality increases (i.e., 2D or 3D images) then k-space generally becomes more sparse, allowing additional undersampling. Three-dimensional images are reconstructed without noticeable error from data comprising just a few percent of the full k-space raster [586]. Therefore, CS allows significant improvements in acquisition time which will be important for imaging at low field where multiple scans are required to provide a useful SNR. 9. Summary This is an exciting time to be working with low-field NMR instrumentation. The recent availability of fully digital spectrometers capable of performing complicated experiments on permanent magnets has greatly increased the range of applications accessible using bench-top systems. In particular, the recent introduction of intermediate-field spectroscopy with permanent magnets has enhanced the capabilities of NMR as an on-line tool suitable for implementation in industrial environments. 175
In this review we have presented a guide to obtaining the best results from modern low-field spectrometers. We started by considering the most appropriate permanent magnet designs for a given application and moved on to cover the following topics. • Radio frequency pulse behavior, including a discussion of phase transients and the effect of rf pulse errors on spin dynamics. • Digital filtering – a recent addition to low-field NMR but a technique that can provide significant noise reductions in the time domain if implemented correctly. • Strategies for maximizing the SNR. • Pre-processing stages to improve data analysis and interpretation. • Time-domain fitting of spectroscopy data that is more robust than a conventional FT at intermediate field. • Time-domain fitting of relaxation and diffusion data, including 1D and 2D distributions – now the most common method for analyzing low-field data. The literature pertaining to the uses of low-field NMR in industrial process monitoring is vast and saturated with niche applications. Rather than attempt to encompass all these experiments, we presented instead a summary of the most common methods of acquiring and interpreting data. We provided a link between the NMR magnet and the industrial environment with three examples: food production, the built environment, and the petroleum industry. Many advancements in low-field NMR have been driven by the requirements of petrophysical well-logging and 176
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255
Appendix A. Nomenclature a
Sphere (droplet) radius
ad
Amplitude component
A
Variable substitution (current)
A
Gradient ramp function
α
Smoothing parameter
b; b; B Signal (data) function/scalar; vector; matrix beq
Equilibrium signal
B0
Magnitude of static magnetic field
B1
Magnitude of applied (rf) magnetic field
∆B0
Variation in static magnetic field
∇B0
Magnetic field gradient
Brel
Relaxation magnetic field
Bpol
Polarizing magnetic field
Bacq
Acquisition magnetic field
β
Bessel function roots
c
Scalar
c
Vector of fitting parameters
C
Capacitance
256
CGT
Gibbs-Thomson coefficient
Cs,c
Permeability scaling constant (sandstone, carbonate)
d
Model order
D
Maximum model order
D
bgp parameter matrix
D
Diffusion coefficient
δ
Width parameter
δuv
Kronecker delta
∆χ
Magnetic susceptibility contrast
e, e
Experimental error (noise) function/scalar; vector
E
Variable substitution (exponent)
E
Sum of squared errors
E
Efficiency factor
ε
Gradient ramp duration
f ; f; F Distribution function; vector; matrix ˆf; Fˆ
Fitted distribution vector; matrix
fd
Frequency component
fpb
Passband frequency
fsw
Sweep width
F
Fourier transform operator
257
φ
Porosity
φrot
Magnetic field phase rotation
ϕ
Phase shift
ΦGCV
GCV score
g; g
Magnetic field gradient magnitude; vector
g0
Background magnetic field gradient magnitude
geff
Effective internal magnetic field gradient magnitude
G
bgp parameter matrix
γ
Gyromagnetic ratio
H
Smoothing function
∆Hf
Enthalpy of fusion
H
Hankel matrix
η
Viscosity
i
Complex current
I
Physical current
I
Identity matrix
ι
Experimental data (Bayes’ theorem)
ℑ
Imaginary component √ Imaginary unit −1
j jn
nth order spherical Bessel function
258
J
Spectral density
Jn
nth order Bessel function
k; k
Wavenumber magnitude; vector
∆k
Resolution in k-space
K; K
Kernel function; matrix
κ
Permeability
`c
Crystal size
`∆
Diffusion distance during t∆
`e
Diffusion path length over time te
`g
Dephasing length scale
`H
Length-scale of macroscopic heterogeneities
`ρ
Relaxivity length-scale
`s
Characteristic pore size
ld
Eigenvalues
L
Inductance
L
Sample length
L
Smoothing operator
λ
Weighting parameter
λ
bgp coefficient matrix
m
Index
259
M
Array size
M
Magnetization
n
Index
ns
Number of saturation pulses
n0
Offset from origin
N
Index
Ns
Number of scans
ν0
Resonant frequency:
∆ν
Frequency bandwidth of slice
O
Array size
O
Solution matrix
p
Surface spin population
Pc
Capillary pressure
P
Probability
q; q Magnetization wave number; vector Q
Quality factor
Q
Second derivative matrix
rd
Rate component
rc
Radius of rotation in centrifuge
r
Position vector
260
R
Displacement vector
R
Resistance
R
Residuals
ρ
Surface relaxivity
ρs
Solid density
∆ρ
Density difference
S
Saturation state
S
Singular values (diagonal) matrix
S/V
Surface-to-volume ratio
σsl
Surface energy
ς
Power
t; t
Experimental time scalar; vector
tc
Constant time
tDE
Dead time
tdw
Dwell time
tδ
Gradient pulse duration
tδ1,2
Delay before / after gradient pulse
t∆
Observation time
te
Echo time (CPMG)
tGD
Group delay
261
tp
Phase gradient duration
tr
Read gradient duration
tRD
Recycle delay between scans
ts
Slice gradient duration
tse
Echo time (diffusion)
tstore
Storage time of spins on z-axis
tω
Pulse duration
t90
90◦ pulse duration
t180
180◦ pulse duration
T1
Longitudinal relaxation time
T1ρ
Longitudinal relaxation in the rotating frame
T2
Transverse relaxation time
T2∗
Spin dephasing time
T2,eff
Effective transverse relaxation time
T1,2,bulk
Bulk relaxation times
T1,2,surf
Surface relaxation time
T21
Short T2 component
T22
Long T2 component
T2LM
Log-mean T2
Tm
Melting temperature
262
∆Tm
Melting point depression
T
Transpose
†
Conjugate transpose
τex
Exchange time
τm
Surface diffusion correlation time
τs
Surface residence time
τse
Half echo time (diffusion)
τtort
Tortuosity
τ1
Longitudinal recovery time
τ2
Half echo time (CPMG)
∆t
Time increment
θ
Phase angle
u; u
Fitting parameter scalar; vector
U
SVD matrix
υ
Empirical exponent
ϒ
Empirical exponent
v
Complex voltage
v
bgp Kronecker matrix
V
Physical voltage
V
SVD matrix
263
ω0
Larmor frequency
ωd
Driving (forcing) frequency
ωI
Proton Larmor frequency
ωL
Probe (tuned) frequency
ωr
Free ringing frequency
ωS
Electron Larmor frequency
∆ω
Frequency offset
W
Precision matrix
ϖc
Angular velocity of centrifuge
x, y, z
Cartesian coordinates
∆x, ∆y, ∆z Spatial resolution X
Empirical pre-factor
ξ
Empirical exponent
Ξ
Empirical exponent
∆ys
Slice thickness in image space
Y
Empirical pre-factor
Ψ
Sparse domain transform operator
Z
bgp parameter matrix
ζ
Relaxation shape parameter
Appendix B. Abbreviations AIC
Akaike information criterion
264
bgp
Block gradient pulse
CMR
Combinable magnetic resonance tool
CPMG
Carr-Purcell Meiboom-Gill
CS
Compressed sensing
CSH
Calcium silicate hydrate
CUFF
Cut-open, uniform, force free
DOSY
Diffusion ordered spectroscopy
FFC
Fast field cycling
FID
Free induction decay
FOV
Field of view
fMRI
Functional MRI
FT
Fourier transform
FWHM
Full width at half maximum
GARField
Gradient at right angles to the field
GCV
Generalized cross validation
GdCo
Gadolinium cobalt
gpd
Gaussian phase distribution
HTLS
Hankel total least squares
HSVD
Hankel singular value decomposition
LCR
Inductor capacitor resistor
265
LP
Linear prediction
LWD
Logging while drilling
MOUSE
Mobile universal surface explorer
MRI
Magnetic resonance imaging
MRIL
Magnetic resonance imaging logging
MRScanner Magnetic resonance scanner tool MSME
Multi-slice multi-echo (pulse sequence)
NeFeB
Neodymium iron boron
NML
Nuclear magnetic logging tool
NMT
Nuclear magnetism tools
NMR
Nuclear magnetic resonance
p.u.
Porosity units
PV
Pore volume
QPP
Quadratic programming problem
RARE
Rapid acquisition with relaxation enhancement
rf
Radio frequency
RN
Rendement Napole gene
SESPI
Spin echo single point imaging
sgp
Short gradient pulse
SmCo
Samarian cobalt
266
SNR
Signal-to-noise ratio
SPI
Single point imaging
SPRITE
Single point ramped imaging with T1 enhancement
STRAFI
Stray-field imaging
SQUID
Superconducting quantum interference device
SVD
Singular value decomposition
USBM
United States Bureau of Mines
w/c
Water-to-cement ratio
WHC
Water holding capacity
1D, 2D, 3D
One, two, three dimensional
267
Earth’s Abstract Graphical field
ultra low field
intermediate field
low field
40 µT
50 mT
0.2 T
0.5 T
2 kHz
2 MHz
7.5 MHz
20 MHz
well−logging
quality control
1T
1.5 T
45 MHz
60 MHz
process monitoring
SQUIDs petrophysics
relaxation
clinical diagnosis
imaging
relaxation
spectroscopy
B0 ν0
!1 " H
*Highlights
! ! ! ! !
We review low-field NMR technology for industrial process monitoring. Designs for bench-top and portable magnets are considered. Radio frequency performance and system optimization are discussed. We review experimental techniques, data acquisition, and interpretation. Example applications are given for food, construction, and petroleum industries.
S0079-6565(13)00088-5 http://dx.doi.org/10.1016/j.pnmrs.2013.09.001 JPNMRS 1376
To appear in:
Progress in Nuclear Magnetic Resonance Spectroscopy
Received Date: Accepted Date:
17 July 2013 19 September 2013
Please cite this article as: J. Mitchell, L.F. Gladden, T.C. Chandrasekera, E.J. Fordham, Low-Field Permanent Magnets for Industrial Process and Quality Control, Progress in Nuclear Magnetic Resonance Spectroscopy (2013), doi: http://dx.doi.org/10.1016/j.pnmrs.2013.09.001
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Low-Field Permanent Magnets for Industrial Process and Quality Control J. Mitchella,b , L.F. Gladdena,∗, T.C. Chandrasekeraa , E.J. Fordhamb a Department
of Chemical Engineering and Biotechnology, University of Cambridge, Pembroke Street, Cambridge CB2 3RA, United Kingdom. b Schlumberger Gould Research, High Cross, Madingley Road, Cambridge CB3 0EL, United Kingdom.
Keywords: Low-field, Permanent magnets, Processing, Quality control, Industry PACS: 07.57.Pt, 76.60.Pc, 76.60.-k, 82.56.-b Contents 1
Introduction
2
Which Field Strength?
11
2.1
NMR magnets and industrial environments . . . . . . . . . . . .
12
2.2
A rough guide to selecting a magnet . . . . . . . . . . . . . . . .
14
3
5
Low-Field Spectrometers
20
3.1
Permanent magnet configurations . . . . . . . . . . . . . . . . . .
20
3.1.1
Bench-top magnets . . . . . . . . . . . . . . . . . . . . .
20
3.1.2
Ex situ NMR . . . . . . . . . . . . . . . . . . . . . . . .
22
3.1.3
Well-logging tools . . . . . . . . . . . . . . . . . . . . .
25
Radio frequency pulses . . . . . . . . . . . . . . . . . . . . . . .
27
3.2
∗ Corresponding
author. Tel: +44 (0)1223 334762 Email address: [email protected] (L.F. Gladden)
Preprint submitted to Progress in Nuclear Magnetic Resonance Spectroscopy September 26, 2013
4
3.2.1
Elementary circuit theory . . . . . . . . . . . . . . . . . .
29
3.2.2
Spin dynamics during pulse transients . . . . . . . . . . .
32
3.3
Digital filtering . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
3.4
Maximizing signal-to-noise ratio . . . . . . . . . . . . . . . . . .
46
Pulse Sequences and Experiments
50
4.1
Relaxation time measurements . . . . . . . . . . . . . . . . . . .
50
4.1.1
Longitudinal relaxation . . . . . . . . . . . . . . . . . . .
50
4.1.2
Transverse relaxation . . . . . . . . . . . . . . . . . . . .
51
4.2
Diffusion and flow . . . . . . . . . . . . . . . . . . . . . . . . .
54
4.3
Multi-dimensional correlations . . . . . . . . . . . . . . . . . . .
58
4.3.1
Relaxation time correlations . . . . . . . . . . . . . . . .
59
4.3.2
Diffusion-relaxation correlations . . . . . . . . . . . . . .
60
4.3.3
Exchange rate measurements . . . . . . . . . . . . . . . .
62
Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
62
4.4.1
Frequency encoding . . . . . . . . . . . . . . . . . . . .
63
4.4.2
Phase encoding . . . . . . . . . . . . . . . . . . . . . . .
65
4.4.3
Slice selection . . . . . . . . . . . . . . . . . . . . . . .
65
4.4.4
Multi-dimensional imaging . . . . . . . . . . . . . . . . .
67
4.4.5
Pure phase-encoded imaging . . . . . . . . . . . . . . . .
69
Stray field experiments . . . . . . . . . . . . . . . . . . . . . . .
74
4.5.1
Relaxation time measurements . . . . . . . . . . . . . . .
74
4.5.2
Diffusion measurements . . . . . . . . . . . . . . . . . .
78
4.5.3
Stray field imaging . . . . . . . . . . . . . . . . . . . . .
79
4.6
Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . .
80
4.7
Fast field cycling . . . . . . . . . . . . . . . . . . . . . . . . . .
81
4.4
4.5
2
4.8 5
7
83
Data Analysis
84
5.1
Pre-processing for optimum results . . . . . . . . . . . . . . . . .
85
5.1.1
Zeroth-order phase correction . . . . . . . . . . . . . . .
85
5.1.2
Data compression . . . . . . . . . . . . . . . . . . . . . .
87
5.1.3
First-order phase correction . . . . . . . . . . . . . . . .
88
5.2
Time-domain fitting for spectroscopy . . . . . . . . . . . . . . . .
89
5.3
Time-domain fitting for relaxation and diffusion . . . . . . . . . .
94
5.4
Distributions of relaxation and diffusion (1D) . . . . . . . . . . .
95
5.4.1
Data fitting . . . . . . . . . . . . . . . . . . . . . . . . .
95
5.4.2
Kernel functions . . . . . . . . . . . . . . . . . . . . . .
98
5.4.3
Limitations and considerations . . . . . . . . . . . . . . .
99
5.5
6
Ultra-low field and Earth’s field . . . . . . . . . . . . . . . . . . .
Distributions of relaxation and diffusion (2D) . . . . . . . . . . . 101 5.5.1
Data fitting . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.5.2
Kernel functions . . . . . . . . . . . . . . . . . . . . . . 103
Interpretation
105
6.1
Surface-to-volume ratio . . . . . . . . . . . . . . . . . . . . . . . 106
6.2
Surface interaction strength . . . . . . . . . . . . . . . . . . . . . 112
6.3
Diffusive or chemical exchange . . . . . . . . . . . . . . . . . . . 114
6.4
Droplet sizing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
6.5
Internal gradients . . . . . . . . . . . . . . . . . . . . . . . . . . 123
Industrial Applications 7.1
129
Food . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 7.1.1
Meat . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 3
7.2
7.3
8
9
7.1.2
Fish . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
7.1.3
Dairy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
The built environment . . . . . . . . . . . . . . . . . . . . . . . . 140 7.2.1
Cement . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
7.2.2
Wood . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
7.2.3
Conservation . . . . . . . . . . . . . . . . . . . . . . . . 147
Petrophysics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 7.3.1
Well-logging . . . . . . . . . . . . . . . . . . . . . . . . 149
7.3.2
Laboratory core analysis . . . . . . . . . . . . . . . . . . 161
7.3.3
Hydrocarbon characterization . . . . . . . . . . . . . . . 165
Future Directions
168
8.1
Hardware . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
8.2
Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
8.3
Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 8.3.1
Bayesian inference . . . . . . . . . . . . . . . . . . . . . 171
8.3.2
Compressed sensing . . . . . . . . . . . . . . . . . . . . 173
Summary
175
10 Acknowledgments
177
Appendix A
Nomenclature
256
Appendix B
Abbreviations
264
4
1. Introduction Nuclear magnetic resonance (NMR) is now widely used as a process and quality control tool in a numerous industries including food [1], pharmaceuticals [2, 3], and chemicals manufacturing [4]. Increasingly, a diversity of NMR measurements is exploited. For example, whilst magnetic resonance imaging (MRI) has traditionally been associated with clinical medicine, presenting a non-destructive diagnosis tool and a powerful technique for in situ biomedical investigations [5–8], it is now well established in applications to materials science and is gaining popularity for probing dynamics in optically opaque media [9–14]. High-resolution NMR spectroscopy provides a method of structural determination for complicated molecules such as proteins [15, 16], and enables studies of molecular interactions including enzyme activity [17]. Solid state spectroscopy is useful for materials characterization, such as determining the structures of catalyst metals and supports [18–21]. Finally, NMR measurements of molecular diffusion or advection are possible with pulsed field gradient (PFG) techniques, enabling the determination of droplet size distributions in emulsions [22, 23], dissolution of pharmaceutical tablets [24], degree of polymerization [25–27], or structure-transport relations in porous media [14, 28–30]. In this review we focus on the technology associated with low-field NMR. We present the current state-of-the-art in low-field NMR hardware and experiments, considering general magnet designs, radio frequency (rf) performance, data processing and interpretation. We provide guidance on obtaining the optimum results from these instruments, along with an introduction for those new to low-field NMR. The applications of low-field NMR are now many and diverse. Furthermore, niche applications have spawned unique magnet designs to accom5
modate the extremes of operating environment or sample geometry. Trying to capture all the applications, methods, and hardware encompassed by low-field NMR would be a daunting task and likely of little interest to researchers or industrialists working in specific subject areas. Instead we discuss only a few applications, to highlight uses of the hardware and experiments in an industrial environment. For details on more particular methods and applications, we provide citations to specialized review articles. Let us start by defining “low field”. The terms low field, intermediate field, and high field do not carry any strict meaning. We choose to define low field as the range of magnetic field strengths corresponding to B0 = 10 mT to 1 T; for 1 H,
this range is equivalent to Larmor frequencies of ν0 = 425 kHz to 42.5 MHz
and corresponds to typical bench-top and portable NMR devices based on permanent magnet technology. A summary of the most common experiments and applications within the remit of low-field NMR is given in Fig. 1. Below our low field range exists “ultra-low field” NMR, corresponding to B0 1 mT. The ultralow field [31] applications often involve superconducting quantum interference devices (SQUIDs) [32], required to detect the very weak NMR signals encountered at such small polarizing magnetic fields. Earth’s field NMR [33] is a special case of ultra-low field technology wherein the NMR signal is detected using a conventional rf tuned circuit but polarized to provide sufficient magnetization for detection. The Earth’s magnetic field varies depending on location; it is typically in the range B0 ≈ 30 − 40 µT, corresponding to a 1 H frequency of ν0 ≈ 2 kHz. Above the low field range we define intermediate field (B0 = 1 − 3 T) associated primarily with clinical medicine and full-body MRI scanners. Further still we define high field (B0 > 3 T) as the domain of high-resolution spectroscopy.
6
Earth’s field
ultra low field
intermediate field
low field
40 µT
50 mT
0.2 T
0.5 T
2 kHz
2 MHz
7.5 MHz
20 MHz
well−logging
quality control
1T
1.5 T
45 MHz
60 MHz
B0 ν0
!1 " H
process monitoring
SQUIDs petrophysics
relaxation
clinical diagnosis
imaging
relaxation
spectroscopy
Figure 1: Summary of magnetic field strengths – primary applications and measurements using permanent magnets. The range of B0 encompassed by the term “low field” is considered to extend from B0 = 10 mT (ν0 = 425 kHz) to B0 = 1 T (ν0 = 42.5 MHz). Permanent magnets can provide stronger field strengths up to B0 = 2.1 T (ν0 = 90 MHz), considered to fall in the “intermediate range”. Ultra-low field strengths encompass the range B0 < 1 mT; NMR measurements at “ultralow field” are typically conducted using SQUIDs, although Earth’s field NMR (a special case of low-field NMR where B0 ≈ 30 − 40 µT, ν0 ≈ 2 kHz) is achieved using a polarizing coil to generate a detectable signal. A wide range of applications utilize regular permanent magnet technology, from oil reservoir well-logging to process (chemical reaction) monitoring. Relaxation, diffusion, and imaging are possible across the entire range of low-field NMR. Petrophysical and quality control applications rely mainly on relaxation time analysis. Spectroscopy is possible only at intermediate field over small sample volumes where a homogeneous magnetic field is achievable with permanent magnets. Images of a selection of low-field magnets designed for different purposes are shown at the top (left to right): a Magritek Earth’s field NMR system (B0 ∼ 40 µT), an Oxford Instruments bench-top rock core analyzer (B0 = 50 mT), a Siemens C-shaped whole body MRI system (B0 = 0.35 T), a Bruker bench-top analyzer (B0 = 0.5 T), and a Magritek bench-top spectroscopy magnet (B0 = 1 T). Permanent magnet designs are often tailored to specific applications and can be open-access or single-sided.
7
The maximum field strength that can be currently achieved with a bench-top permanent magnet is around B0 ∼ 1.5 T, corresponding to ν0 ∼ 60 MHz. This field strength is sufficient for basic spectroscopy, and we are seeing a resurgence in spectroscopic measurements at such modest field strengths for in-line process monitoring. Spectroscopic resolution is achievable with intermediate-field permanent magnets on small samples (5 mm diameter tubes), though most low-field instrumentation provides a magnetic field that is insufficiently homogeneous for 1H
spectroscopy. Within NMR research and application in general, there remains
a drive for stronger magnets to enable improved signal-to-noise ratio (SNR) and spectroscopic resolution. Commercial spectroscopy systems are available up to about B0 ≈ 23.5 T (ν0 = 1 GHz for 1 H) and full-body MRI systems up to B0 = 7 T
(ν0 = 300 MHz for 1 H) and even higher for clinical research (combined MRI and
spectroscopy) [8, 34]. However, with the rising cost of liquid helium required for superconducting magnets, plus the safety and economic factors of installing such systems in industrial environments, there is a persistent interest in low-field technology. Low-field bench-top magnets (typically B0 ≈ 0.5 T, ν0 ≈ 20 MHz for 1 H) are popular in the food and agriculture industry for product screening and quality control [35]. Bench-top magnets operating at lower frequencies (ν0 = 2 MHz for 1 H)
are preferred for studies of petrophysical rock and environmental soil sam-
ples [14]. Bench-top MRI systems are available for biomedical studies of rodents [36]. Full-body MRI scanners based on permanent magnet technology are also available [37]; these C-shaped magnets offer the advantages of (1) open access to the subject, (2) improved patient experience (being less claustrophobic than an enclosed superconducting magnet), (3) treatment during scans, and (4) reduced safety hazards. Smaller C-shaped magnets are used to observe tissue damage in
8
body extremities such as hands and feet [38–40]. A similar magnet design was implemented recently for examining in situ water transport in trees [41]. Portable low-field NMR systems are available for specialized applications, such as monitoring moisture transport in skin [42], analysis of car tyres [43], measurements of cement or soils [44], and oil well-logging [45, 46]; the last example is perhaps the most significant commercial application of low-field NMR. Portable magnets are often single-sided (unilateral) devices that project an inhomogeneous B0 field, allowing the user to “see” a short distance into a sample. A popular commercial implementation is the hand-held Mobile Universal Surface Explorer (MOUSE) [47]. The so-called stray field of unilateral devices can also be used to acquire one-dimensional (1D) profiles – a technique known as stray field imaging (STRAFI) [48]. A further advantage is that the very large magnetic field gradient associated with the B0 field of unilateral devices can provide a pixel resolution over small samples that is difficult to achieve with conventional MRI methods. A specialized magnet design with a Gradient at Right Angles to the Field (GARField) [49] optimizes the STRAFI technique with shaped magnet pole pieces; this system is ideal for studying drying and curing of surface coatings. Profiling with a NMR-MOUSE is achieved by moving the magnet, and hence the resonant volume, through the sample. Well-logging tools operate on a similar principle, acquiring signal from a defined volume within the rock formation but near the well bore [45]. NMR instruments are complicated systems often capable of performing many different measurements, such as relaxation time analysis, diffusion, imaging, and spectroscopy. Such diverse instrumentation requires a skilled operator and this necessity has, to some extent, limited the commercial application of NMR in in-
9
dustrial environments. The use of low-field NMR for process monitoring has led to the development of automated spectrometers, capable of self-calibration and designed to perform a single type of experiment. These simplified systems are adequate for basic quality control, but will not necessarily provide optimum performance. For the skilled experimentalist, there are many factors to consider when installing and operating a low-field NMR system. First and foremost is the magnet: in Section 2 we consider the advantages and disadvantages of low-field NMR hardware compared to superconducting magnets. We also offer generic guidelines for selecting an appropriate magnetic field for a particular application in Section 2.2. Common hardware (magnet and rf probe) designs are considered in Section 3, including mobile NMR devices. Detailed aspects of the rf pulse performance and signal reception are also discussed. In Section 4 we summarize the experiments that are most commonly conducted at low field. Longitudinal T1 and transverse T2 relaxation times, or selfdiffusion coefficient D, are used as proxies for chemical sensitivity when spectroscopic resolution is unavailable. Multi-dimensional relaxation and diffusion experiments are popular because they provide more information than their onedimensional counterparts, notably detection of diffusive or chemical exchange. Careful data analysis is required at low field, especially when the SNR is poor; data processing techniques are considered in Section 5 where we focus on time domain analysis. We note that even in spectroscopic measurements, time-domain fitting can provide more robust identification of chemical species than traditional Fourier transform spectroscopy at low field. Many of the industrial samples investigated using low-field NMR fall within the remit of “porous media”. Heterogeneous, fluid-saturated materials are studied
10
using low-field NMR techniques as the magnetic susceptibility contrast between the solid and fluid in such systems results in pore-scale magnetic field gradients (so-called “internal gradients”) which distort the NMR signal. These internal gradients are known to scale with magnetic field strength, and so the use of a low-field spectrometer is one method for reducing sample-dependent magnetic field inhomogeneities. Porous materials that have been studied include cellular structures in biomedicine and food production, cement-based construction materials, reservoir rocks in petrophysics, and catalysts in chemical engineering. Interpretation of the data obtained from such systems can be complicated and we offer advice for converting NMR parameters into sample properties in Section 6. We consider three industrial applications of low-field NMR in Section 7: food, construction, and petroleum. Finally, in Section 8, we consider the changing world of low-field NMR and look to possible future developments. The majority of low-field experiments interrogate 1 H (proton) spins; the high natural abundance and large gyromagnetic ratio of the hydrogen nucleus provides a detectable magnetization even with low-field instrumentation. However, heteronuclear NMR is possible at low field; other detectable nuclei include 13 C,
23 Na,
and/or 31 P, although their lower natural abundance and gyromagnetic ratios
make detection difficult except at the upper end of the low field range. Unless otherwise stated, all techniques described herein pertain to 1 H detection. 2. Which Field Strength? There has been a continual drive in NMR and MRI for the manufacture of stronger magnets. A higher magnetic field strength provides better SNR, enabling enhanced spectroscopic resolution and improved image resolution, or shorter ac11
quisition times. In MRI, the construction of high field imaging magnets enables in situ spectroscopy, providing localized studies of tissue chemistry. However, very strong magnetic fields are not always desirable in an industrial context. In this Section we consider the advantages and disadvantages of high and low-field magnet technology, and offer suggestions for selecting an appropriate low-field magnet for industrial applications. 2.1. NMR magnets and industrial environments There are several disadvantages to the installation of a superconducting magnet in an industrial environment: • The stray magnetic field presents a serious safety hazard. • High economic cost (an average system is around $500, 000). • Supply and containment of cryogens (liquid helium and nitrogen). Cryogenic cooling presents an additional operating cost and hazard. The worldwide supply of liquid helium is limited and non-renewable although modern magnet designs require only small volumes of cryogenic liquids, have low boil-off rates, and include mechanical refrigeration to reduce or even eliminate helium losses. Beyond the practical limitations of installing a high-field magnet in an industrial environment, strong magnetic fields are inappropriate for the study of fluids in many heterogeneous (porous) materials such as oil-field reservoir rocks, cementbased building materials, and some biological systems, e.g., lungs. The magnetic susceptibility contrast between the solid and fluid phases results in large, porescale magnetic field gradients (so-called “internal gradients”) that distort measure12
ments of relaxation time, diffusion, and chemical spectra, or can prevent detection of a NMR signal entirely due to rapid dephasing of the spin ensemble [50]. Low-field permanent magnets can provide a suitable compromise between magnetic field strength and experimental versatility for installation in industrial environments and the investigation of heterogeneous materials. Several advantages of low-field instruments over their high-field superconducting counterparts are: • No cryogenic cooling. • Lower economic cost (a typical bench-top system is around $100, 000). • Reduced safety considerations (negligible stray field). • Versatility of design including the possibility of open-access and singlesided magnet arrangements. A particular disadvantage of low-field permanent magnets is the field stability. Thermal fluctuations in the magnet poles will cause the B0 field to vary and hence cause a shift in the Larmor frequency. To improve temperature stability, permanent magnets are typically thermostatted to super-ambient temperatures such as 35 − 40◦ C, depending on the local climate. The presence of a large thermal mass, such as an iron yoke, improves the temperature regulation. Magnetic materials with opposite temperature coefficients may be combined to reduce field drift when the temperature fluctuates. Small temperature drifts are unavoidable and methods exist to compensate for such inevitable fluctuations [51]. For example, the Larmor frequency may be measured and corrected between scans. Alternatively, a frequency lock can be installed including a B0 shim coil, to automatically 13
and continuously maintain the correct Larmor frequency. A low cost solution to improving temperature performance is to surround the magnet with insulation, such as a layer of expanded polystyrene or similar material; this simple approach is particularly effective where air-conditioning units produce circulating currents of cold air. Small shifts in frequency can be accommodated in relaxation or diffusion experiments when the signal is acquired in the time domain and summed over all frequencies. However, frequency shifts can be much more problematic during imaging or spectroscopy measurements. Excellent thermal regulation is essential on permanent magnet systems designed for such experiments. 2.2. A rough guide to selecting a magnet Selecting an appropriate low-field magnet for a given application will be a compromise between numerous factors: • Sample size. • Magnetic field strength. • Magnetic field homogeneity. • Experimental capability. • Logistical limitations (size, weight, portability). • Cost. Many process monitoring and quality control tasks rely on basic relaxation time analysis of bulk liquid or soft-solid samples. For such industrial applications, a standard bench-top magnet is appropriate. These systems have a typical field strength of B0 = 0.5 T (ν0 ≈ 20 MHz) and rf probe bore diameters in the range 14
10 − 20 mm. Stronger permanent magnets can be manufactured by the addition of more magnetic material, although this will increase the cost, size, and weight of the instrument. For a given magnet size there is an inverse relationship between pole gap (which determines the rf probe size and hence sample volume) and magnetic field strength. In addition to the magnet, a basic relaxometry system requires only a spectrometer (including acquisition computer) and rf amplifier. These days, fully digital spectrometers are available, capable of direct synthesis and acquisition of rf signals up to about ν0 = 40 MHz, thereby reducing the complexity of the hardware. Such digital spectrometers were originally based on the technology (chip sets) manufactured by the mobile telecommunications industry. For heterogeneous materials or larger sample volumes it is necessary to adopt a lower field strength such as B0 ∼ 0.24 T (ν0 ≈ 10 MHz). The pole gap can be increased to accommodate rf probe diameters of 50 mm within the confines of a bench-top system. Studies of extremely heterogenenous samples will require field strengths of B0 < 100 mT (ν0 < 4 MHz) in order to extract quantitative signal intensities. Low-field NMR systems are appropriate for studies of large sample volumes. At low frequencies, solenoid rf probes with small bore diameters can be difficult to tune due to their low inductance. A larger bore typically leads to an increased inductance which is easier to tune. For more complicated NMR experiments such as diffusion, flow, and droplet sizing, a 1D magnetic field gradient coil is required. Most commercial benchtop magnet designs can accommodate a single-axis gradient without modification, although the maximum bore diameter, and hence rf probe bore diameter, will be reduced. The addition of even a 1D gradient set increases the complexity of the extraneous hardware, requiring a high-powered gradient (audio) amplifier and
15
sometimes a gradient plate cooling system (typically a circulating water chiller). The strength of the gradient produced will depend on the design of the gradient coils; the orientation (gradient axis) may also influence the gradient strength. Again, depending on design, it may be possible to obtain a stronger gradient by changing the orientation. This approach is useful for bulk diffusion studies. A 1D gradient is most usefully aligned parallel to the long axis of the rf probe bore, allowing flow measurements and 1D profiling to be implemented, though this orientation may not provide the maximum achievable gradient strength. Full three-dimensional (3D) imaging obviously requires a 3D gradient set plus three audio amplifiers. Considering each audio amplifier will have a typical cost of $50, 000, the change from a basic relaxometry spectrometer (no gradients) to a 3D imaging magnet effectively doubles the cost of the entire system. Most bench-top magnet designs can be upgraded to accommodate a 3D gradient set; the potential value of 3D gradients should be considered carefully. Obviously, applications in MRI for clinical diagnosis, biomedicine, and pharmaceuticals (e.g., imaging of drug tablet dissolution [24]) demand this capability. Very low field applications, such as petrophysics, may not benefit: the SNR per voxel will be inherently poor, necessitating long acquisition times (many repeat scans) and low image resolution [14]. For such applications a 1D gradient coil for diffusion and profiling is sufficient. Spectroscopy systems have the very different requirement of excellent magnetic field homogeneity. Useful 1 H spectra can be recorded at field strengths of B0 = 1 − 1.5 T. The limitation in permanent magnet technology is the volume over which the magnetic field can be made sufficiently homogeneous for subparts per million (ppm) resolution. In current bench-top designs the sample vol-
16
ume is restricted to a 5 mm diameter tube. A particularly important consideration when purchasing a bench-top spectroscopy system is thermal stability; the results are susceptible to slight drifts in magnetic field (or shim) caused by temperature fluctuations. As with other bench-top systems, a 1D gradient set can be advantageous for chemical identification through diffusion ordered spectroscopy (DOSY) measurements, as well as enabling the usual diffusion-based applications such as droplet sizing. Spectroscopic resolution may also be obtained on more conventional magnet designs operating at lower magnetic field strengths by using rf micro-coils to examine very small sample volumes [52], or even on unilateral devices with careful design of the magnetic field profile [53–55]. Another factor to consider is portability. Numerous designs for portable NMR devices can be found in the literature although only a few, such as the NMRMOUSE [27, 47, 56], are commercially available. Of course the concept of portability is open to interpretation: early “portable” NMR systems were large and heavy, having to be transported on a vehicle. Most bench-top NMR systems can be made portable under the same criteria, although locating or transporting a suitable power supply, notably for gradient amplifiers, can be a limitation. Here we refer to portable systems as magnets conceptualized specifically to be taken to the sample: the NMR-MOUSE is effectively a hand-held device that is used outside the laboratory for studies of art and cultural heritage [57, 58], the “Surface GARField” magnetic is used to study concrete in the built environment [44, 59], and the “tree-hugger” magnet is designed to examine water transport in trees [41]. Oil well-logging tools are another example of specialized mobile NMR instrumentation [45, 46]. These unilateral portable systems acquire signal from an external (stray) magnetic field. Such stray fields are known to be characterized by large
17
and spatially variant magnetic field gradients. The signal is therefore detected from a defined sensitive volume: a plane or volume over which the magnetic field is constant. The shape and position of the sensitive volume will be determined by the design of the magnet and also the interaction of the B0 and B1 fields; further discussion is given in Section 3. Therefore, the sensitive volume can be adjusted by altering the rf excitation and detection frequency; this approach is used in some well-logging tool designs [60], see Section 3.1.3. Unilateral NMR devices offer the possibility of high-resolution 1D profiling by moving the sensitive volume through the sample. In additional, both the Surface GARField and Profile NMRMOUSE [61] provide spatial (depth) resolution in surface studies [62]. However, unilateral devices typically provide poor SNR compared to conventional benchtop systems due to the small resonant volume within a sample and hence the poor rf coil filling factor. For laboratory applications, the presence of a large magnetic field gradient can cause other complications: if a sample is enclosed by fluorinated plastic or composite materials, commonly used due to their low 1 H content, the resonance condition for 19 F will be met somewhere in the magnetic field. Therefore, signal will be obtained simultaneously from the required 1 H sensitive volume and also a different volume sensitive to 19 F. Unilateral NMR devices are best used in remote locations, away from sources of rf interference (noise). To summarize, there are three simple questions that should be answered before purchasing a low-field permanent magnet: • Where will the NMR device be located? For most laboratory or industrial process control applications, a bench-top instrument will be appropriate. The lack of a stray magnetic field on these instruments makes them suitable for location in industrial environments without the safety concerns associ18
ated with superconducting magnets. If the system is to be used outside a laboratory or factory, or if the sample is an unusual size or shape, then a unilateral (mobile) NMR device should be considered. • What magnetic field strength is required? The magnetic field strength will be determined largely by the sample. Heterogeneous (porous) materials containing large quantities of paramagnetic or ferromagnetic impurities (e.g., rocks, catalysts) will benefit from very low field strengths (B0 < 100 mT) to minimize internal gradients and allow quantitative signal intensities to be obtained. Otherwise, higher field strengths are preferable (improved SNR) although a compromise between sample volume and B0 will exist. Field strengths of B0 > 1.5 T can be achieved with permanent magnets but only by significantly increasing the volume of magnetic material. • Which experiments are to be conducted? Most NMR experiments (relaxation, diffusion, imaging) can be conducted at any field strength; the exception is spectroscopy which requires an intermediate-strength homogeneous magnetic field (B0 ∼ 1 T). Basic relaxation time (T1 and T2 ) distributions can be acquired on any bench-top system. Diffusion-based experiments and profiling requires a 1D gradient. Imaging will require 2D or 3D gradients. If a system is to be used only for a single, defined application – as is typical in an industrial environment – the minimum required specification should be chosen to simplify operation. For research applications it is desirable to over-specify the instrument capabilities to provide versatility for future projects.
19
3. Low-Field Spectrometers In this Section we consider the hardware behind low-field NMR experiments with permanent magnets. First, we examine the most common arrangements of magnets and rf probes currently used in commercial NMR spectrometers. Then we consider the optimization of the tuned rf circuit and the recent phenomenon of digital filtering in low-field spectrometers. Finally we discuss noise sources and hardware modifications to maximize the SNR achievable in low-field experiments. 3.1. Permanent magnet configurations 3.1.1. Bench-top magnets There are two designs that currently dominate commercial bench-top magnets for use in laboratory environments. A simple and robust design consists of two magnetic pole pieces (parallel plates) mounted in an iron yoke, see Fig. 2(a). The iron yoke minimizes the stray field. The amount and type of magnetic material used in the pole pieces, plus the pole gap, determines the field strength B0 between the poles. Typically the pole pieces are formed from neodymium iron boron (NdFeB). An alternate permanent magnetic material used in some designs is samarian-cobalt (SmCo) which has an inverse temperature coefficient compared to NdFeB. Therefore, magnet designs incorporating both SmCo and NdFeB magnetic material can provide a magnetic field that is insensitive to temperature fluctuations; a similar effect is achieved with an alloy of SmCo and GdCo. Temperature regulation in the parallel plate magnet is achieved by heating the iron yoke to superambient temperatures, usually 35 − 40◦ C. The B0 homogeneity may be improved by the inclusion of shim coils; gradient coils can also be added to enable diffusion and imaging experiments. The direction of the B0 field across 20
the pole gap allows the rf antenna to be constructed as an inductive solenoid, unlike in superconducting magnets where the rf antenna is a birdcage or saddle coil arrangement. Solenoid rf coils are simpler to design and tune, and provide improved B1 homogeneity compared to the alternatives. The B1 homogeneity can be further improved by adjusting the loop spacing at either end of the solenoid to compensate for the finite length of the inductor. The iron yoke and rf probe housing act as effective screens to external noise sources. The parallel plate magnet design shown in Fig. 2(a) can be converted to an open access (so-called C-shaped magnet) by the removal of one end face of the iron yoke. The parallel plate design is therefore straightforward to construct and versatile in implementation. An alternative magnet design that has become popular for bench-top NMR is the cylindrical Halbach permanent magnet [63], see Fig. 2(b). A series of polarized magnetic blocks are placed in a circular pattern, where the direction of polarization of each block is rotated slightly relative to its immediate neighbors. In the limit that an infinite number of blocks are used, a uniform B0 field is generated across the ring of magnets with zero stray field. Identical rings are stacked to form a cylindrical magnet. Obviously the practical constraints of building these Halbach magnets ensure that the B0 field is not perfectly homogeneous and there will be some small stray field. Notwithstanding, careful construction can provide a very good approximation to the ideal design and the Halbach magnet can provide improved B0 homogeneity compared to the parallel plate magnet design in Fig. 2(a). Constructing a good Halbach magnet does rely on the selection of nearidentical magnetic blocks [63]; heterogeneities in the B0 field of Halbach magnets are considered harder to correct by shimming than in their parallel plate counterparts, and temperature regulation is more difficult due to the lack of an iron
21
yoke. Gradient coils and the rf coil are located inside the Halbach magnet ring. Again, the alignment of the B0 field across the long axis of the magnet, as shown in Fig. 2(b), enables the use of a solenoid rf coil. Halbach magnets can be designed for open access by placing a second Halbach cylinder of magnets around the first: by rotating the outer cylinder, the magnetic field of the inner cylinder can be negated, allowing the magnets to be separated with minimum resistance. This open-access Halbach concept is embodied in the NMR-CUFF [64], used to monitor water transport in plant stems. Both the parallel plate magnet and the Halbach magnet can be constructed with a wide range of magnetic field strengths for most laboratory applications. Parallel plate and Halbach magnets are readily available at field strengths ranging from B0 = 50 mT (ν0 ≈ 2 MHz) for rock core analysis [14] to B0 = 1 T (ν0 ≈ 42.5 MHz) for spectroscopy [65, 66]. 3.1.2. Ex situ NMR Portable magnet designs are required for in situ measurements of materials in industrial environments. The NMR-MOUSE is one of the simplest and most robust NMR devices available [67]. It was designed to allow NMR surface measurements to be conducted on arbitrarily large samples [56, 61, 68]. The original NMR-MOUSE was designed around a bar magnet and surface rf antenna. More complicated versions of the MOUSE have been constructed over the years with increasing numbers of permanent magnetic components: the two-pole MOUSE (also called the U-shaped MOUSE) is illustrated in Fig. 3 where the B0 field is generated by two anti-parallel polarized permanent magnets mounted on an iron yoke to increase the magnetic flux on the upper surface [43]. Improved B0 homogeneity can be achieved using multiple magnets (four or more) [69], or by affixing 22
a B1
N
S y z
B0 b
B0
x z
Figure 2: Schematics of standard bench-top (laboratory) permanent magnet configurations: (a) parallel plate magnet and (b) cylindrical Halbach magnet. In (a) the B0 magnetic field is generated on the z-axis between the pole pieces (light gray). Shim and gradient plates (medium / dark gray) can be included. The sample is placed in a solenoid rf coil (blue) which generates a B1 field in the y-axis. In (b) the magnetic pole pieces are arranged in a rotating pattern (indicated by arrows) to form a cylindrical Halbach magnet with a B0 field in the z-axis. Shim and gradient coils (dark gray) can be included. The sample is placed in a solenoid rf coil (blue) which generates a B1 field in the y-axis. Note the reference axes are different in (a) and (b).
23
shaped pole pieces molded in soft iron to the pole faces [70, 71]. The rf antenna, usually of a 2D surface design (e.g., printed circuit board) is located between the permanent magnet pole pieces. Signal is detected from a sensitive volume in the stray field of the magnet where the inhomogeneous B0 and B1 fields are perpendicular [62]. This sensitive volume is located a short distance (∼ 10 mm) above the face of the magnet. Hence the MOUSE can be used to “see inside” a planar object. As the strength of the B0 field decays away from the surface of the magnets, the exact height of the sensitive volume on the y-axis is determined by the chosen Larmor frequency. Each position in the B0 field will also have an associated magnetic field gradient that can be used for diffusion measurements. The NMR-MOUSE concept is particularly useful for studying molecular reorientations in soft solids such as polymeric rubber [72]. sensitive volume
B0 B1 N
rf antenna S
y z iron yoke
Figure 3: Schematic of the two-pole NMR Mobile Universal Surface Explorer (MOUSE). In this basic configuration, two permanent magnet pole pieces are mounted on an iron yoke. The B0 field is approximately parallel to the z-axis above the center of the magnet. A surface rf antenna generates a B1 field approximately parallel to the y-axis above the center of the magnet. Signal is obtained from a sensitive volume defined by the exact shapes of the B0 and B1 fields. More complicated MOUSE designs have been presented with larger numbers of pole pieces to provide better definition of the sensitive volume.
24
3.1.3. Well-logging tools The history of NMR well logging dates back almost to the inception of the NMR experiment: a patent was filed in 1952 by Russell Varian on the use of the Earth’s field for NMR logging. The Nuclear Magnetic Logging (NML) tool was developed by Chevron and Schlumberger, although it did not achieve great commercial success. However, continued investment in that technology led to the development of successful logging tools by Schlumberger and NUMAR (now Halliburton). Schlumberger launched its series of Nuclear Magnetism Tools (NMT) in 1978; these were superseded in the 1990’s by the Combinable Magnetic Resonance (CMR)1 tool. This design had the important capability of being combined with a multitude of other logging tools on a single tool string (as indicated by its name). The CMR tool contains permanent magnets arranged in such a way as to generate a defined “sweet spot” of known volume from which the NMR signal is detected, as illustrated in Fig. 4(a). The resonant frequency of the sweet spot is nominally ν0 = 2 MHz. An alternative design, adopted by NUMAR, is a gradienttool implementation where the magnetic field decays continuously away from the pole face of the magnets; this design is akin to the NMR-MOUSE concept. Different resonant “shells” are excited by varying the Larmor frequency, each with a unique position and associated magnetic field gradient strength. Halliburton now operate the Magnetic Resonance Imaging Logging (MRIL)2 tool [73] – here, “imaging” refers to the (vertical) spatial resolution achieved as the tool moves up the well-bore measuring continuously as well as to the ability to acquire signal at different depths into the formation. The MRIL tool operates at a nominal reso1 Mark
of Schlumberger. is a registered trade mark of Halliburton.
2 MRIL
25
nance frequency of ν0 = 1 MHz. Schlumberger offer a similar tool design called the Magnetic Resonance Scanner (MRScanner)1 , illustrated in Fig. 4(b), which operates at ν0 = 1.4 MHz or less, depending on the resonant shell selected [46]. Logging tool designs are reviewed in [74]. a
well bore bowspring
S
N
S
N
sensitive volume rf antenna
b
N
S
well bore bowspring
multiple resonant shells rf antenna
Figure 4: Cross-section schematics of Schlumberger’s magnetic resonance logging tool sensor: (a) the Combinable Magnetic Resonance (CMR) tool [45] and (b) the Magnetic Resonance Scanner (MRScanner) [46]. The CMR tool (a) has magnet pole pieces designed to provide a defined “sweet spot” of known volume. The MRScanner (b) has a decaying magnetic field gradient away from the tool, so varying the resonance frequency of the rf field enables detection of signal from different resonant “shells”. Each resonant shell has an associated magnetic field gradient that is used for diffusion measurements. In both tools the bowspring keeps the face of the tool in contact with the well bore.
The industry standard NMR logging measurement is a T2 relaxation time distribution and all logging tools offer this capability. The MRIL and MRScanner 26
tools offer the ability to measure diffusion coefficients utilizing the magnetic field gradient that is approximately constant across the resonant volume [46, 75]. CMR, on the other hand, is not appropriate for diffusion measurements because the field gradient varies across the resonant volume; however, because the sweet spot volume is defined precisely, the CMR provides calibrated measures of liquid volume and hence porosity in the vicinity of the well bore [45]. MRScanner is the tool of choice in monitoring operations with cased wells to determine long-term changes in a reservoir, whereas CMR is the tool of choice in piloting and exploration wells as it can be run simultaneously with other logging tools. Logging tool designs are modified to enhance the sensitivity and reliably of acquired data, and hence improve the interpretation. One area of ongoing development is the implementation of logging-while-drilling (LWD), in which an NMR logging tool is mounted on the drill string to provide immediate feedback on formation and enable informed decisions for directional drilling. 3.2. Radio frequency pulses Theoretical descriptions of spin physics are often based on ideal rf behaviour, where the spin ensemble is assumed to experience an applied oscillating field of the amplitude, duration, and frequency specified. However, this situation is never encountered in reality and consideration must be given to non-ideal rf transmission. This is especially important at low field where a rf pulse may contain only a few cycles. The Larmor (precession) frequency of the spins is defined by the magnet according to ωL = −γ B0 . The rf probe is tuned so its impedance is entirely resistive at the angular frequency ω0 ; tuning such that ωL = ω0 is usual but not necessarily accurately achieved. Furthermore, the probe (as for any linear resonant system) 27
will have a free-ringing frequency ωr that differs slightly from ω0 . This is of practical consequence where rf pulses have a significant period of free ringing, after the rf drive from the transmitter amplifier is removed. Ideally, the condition
ωr = ωL would be satisfied; however the tuning is unlikely to be either accurate or stable enough for this to be achieved in practice. The experimentalist defines a forcing (driving) frequency ωd to be produced by the spectrometer and amplified prior to transmission to the probe. Depending on the fidelity of the rf amplifier stage, the excitation pulse may or may not be a good representation of an ideal sine wave constrained within the envelope of the rf pulse. The forcing frequency is typically set to ωd = ωL , although in broadline samples spins will precess at a range of frequencies and excitation will never be uniform across the spin ensemble; the CPMG experiment is particularly sensitive to off-resonance spin dynamics [76, 77]. Furthermore, in imaging and spectroscopy, slice- or chemically-selective soft rf pulses are applied off-resonance, with the forced frequency chosen specifically so that ωd 6= ωL . In these cases, one assumes that the quality factor Q (a measure of the width of the resonance curve) of the probe is sufficiently low that the excitation pulse is transmitted without significant distortion. Theoretically, then, the condition ωr ' ωd ' ωL would always be met. Clearly this is not practical, or necessarily desirable. It is important to realize that a mismatch between the forcing frequency and the free-ringing frequency of the probe may lead to unexpected transient responses, which in turn leads to complicated spin dynamics. The duration of the transients is approximately 2Q/ω0 ; hence, at high field (typically Q ∼ 100 but also very large ω0 ) the transient may be shortlived compared to the duration of the rf pulse and if so can be ignored. However,
28
at very low fields (a smaller Q ∼ 10 but also much smaller ω0 ) the transient can be a significant fraction of the rf pulse. It is therefore important to understand the origin and influence of pulse transients in low-field experiments. We now present the theoretical expressions governing pulse transients in order to highlight the interplay between the forcing, free-ringing, and resonance frequencies. In Section 3.2.2 we provide a visual demonstration of theoretical pulse transients in low-field experiments. 3.2.1. Elementary circuit theory Transient effects in NMR excitation pulses can result in phase errors in the spin dynamics [78, 79]. These transients occur if: 1. The forcing frequency is very different from the free ringing frequency of the probe (ωd 6= ωr ). 2. Pulse switching (on/off) does not occur at zero-crossings of the carrier wave. 3. Pulse switching is not coherent with Larmor periods. Let us consider a simple rf probe tuned so that the impedance is real at the resonance frequency, i.e., ω0 = ωL . Conventional rf probes are constructed around the tuned series-parallel circuit illustrated in Fig. 5(a). Discussion of rf resonance circuits (tank circuits) appropriate for low-field probes is given in [80]. More complicated rf circuit designs are considered in [81]. Analysis of the pulse transients is conducted by representing our rf probe as an equivalent series LCR (inductor-capacitor-resistor) circuit, see Fig. 5(b). This series circuit representation is possible for any linear system, and has sufficient generality in its impedance to cover any actual single-resonance probe constructed from linear passive circuit elements. 29
LP
a CS
CP RP
b LS
CS
RS
Figure 5: Basic schematics for (a) series-parallel LCR circuit on which the designs for conventional low-field solenoid rf probes are based and (b) the equivalent series LCR circuit. The subscripts S and P indicate series or parallel components, respectively.
The time-varying current in the rf probe I(t) is proportional to the magnetic field B1 generated within the coil, and will be represented as the imaginary part of a complex current I(t) = ℑ{i(t)}, following the arbitrary convention in [78]. Given a complex driving voltage v(t) (such that the physical voltage V (t) = ℑ{v(t)}), the current obeys the differential equation dv di d2 i LC 2 + RC + i(t) = C , dt dt dt
(1)
and the driving voltage is harmonic with a driving (angular) frequency ωd , an abrupt start at t = 0, and initial phase angle θ : V (t) = 0
for t < 0
(2)
= V0 exp { j (ωdt + θ )} for t ≥ 0. The series circuit differential equation is conveniently represented as d2 i dv di + ∆ω + ω02 i(t) = ω02C , 2 dt dt dt
(3)
where ω02 = 1/LC and ω0 /Q = R/L. At the angular frequency ω0 , the circuit impedance is purely real (resistive), and ω0 /Q is a measure of the half-width of the circuit resonance written in terms of the circuit quality factor. It is shown in 30
elementary texts that the free-ringing frequency ωr , arising from solutions of the homogeneous equation d2 i ω0 di + + ω02 i(t) = 0, dt 2 Q dt
(4)
differs from the real-impedance frequency ω0 according to
ω02 − ωr2 =
ω02 . 4Q2
(5)
The steady-state solution to the forced problem, eq. (1) given eq. (2), is found by standard complex impedance methods, and given by is (t) = I0 exp { j (ωdt + θ − ϕ )} ,
(6)
with the current amplitude I0 =
V0 cos ϕ , R
(7)
and the phase lag ϕ from the forcing voltage given by ! Q ωd2 − ω02 . tan ϕ = ωd ω0
(8)
It can be shown [78] that the transient response that occurs when the driving current is switched on is ih (t) = I0 e−ω0t/Q [(A − 1) exp {− j(ωrt + −θ + ϕ )} −A exp {+ j(ωrt + θ − ϕ )}] ,
(9)
where 1 A= 2
ωd + ωr ω0 ω0 exp {+ jϕ } −j . +j ωr 2ωr Q 2ωr Q cos ϕ
(10)
The last term in A (neglected in [78]) describes the impulsive part of the forcing function. When the rf pulse is applied, if other than at a zero-crossing, there will 31
be an impulse response corresponding to the step change in driving voltage. The current in the rf coil will be i(t) = is (t) + ih (t) over the time interval 0 ≤ t ≤ tω , where tω is the duration of the rf pulse: the time for which the forcing frequency is applied. There will be a similar transient response when the driving current is turned off. If the rf pulse has a duration of tω then the transient response after t = tω is obtained from eq. (10) by the simple substitutions t → t 0 +tω and θ → θ 0 − ωdtω . The current in the coil after the forcing frequency has been removed will be −ih (t) for t > tω . Barbara et al. [79] extended the theory to include the response of the tuning and matching capacitors in the rf probe. They also demonstrated that such theoretical descriptions of phase transients do not precisely match experimental observations; they attributed the differences to treating the rf amplifier as an ideal non-reactive voltage source in their calculations. As we will discuss in the next section, the finite response time of the rf amplifier can reduce both the severity of phase transients and the sensitivity to voltage phase when the pulse is turned on or off. 3.2.2. Spin dynamics during pulse transients A steady-state current of amplitude I0 applied to the rf probe results in a steady-state magnetic field of magnitude B1 . If we define B1 as being on the y-axis (rotating frame) then the spins experience a rotation about the y-axis of angle γ B1tω , where the spin precession frequency around the applied magnetic field is ω1 = −γ B1 . However, the transient response of the rf pulse will generate a time-varying magnetic field in the xy-plane, and hence the spins will experience a time-varying axis of precession. The full effect of a non-ideal rf pulse on the spin 32
dynamics is therefore complicated, and most readily determined by simulation [79]. Here, we present three examples of rf pulses under different conditions. In all cases the driving frequency ωd (supplied by the spectrometer) is 4π × 103 rad s−1
higher than ωL and the tuned frequency of the probe ω0 is 4π × 103 rad s−1 lower than ωL . Practically, it is easy to set both the probe tuning and driving frequencies closer to the Larmor frequency than ±4π × 103 rad s−1 , but these values were chosen to provide a clear visual demonstration of the effects of such experimental imperfections on the spin dynamics. The first two examples are relevant for very low-field experiments with ν0 = 2 MHz. In Fig. 6, the performance of a moderate Q = 15 rf probe is simulated. The current in the probe, Fig. 6(a), is dominated by the slow rise and fall times (> 10 µs) of the transient responses when the rf pulse starts (t = 0 µs) and stops (t = 14.62 µs). The current is set to zero whenever the corresponding driving voltage amplitude V0 < 0.2 V to simulate crossed diodes to ground in the LCR circuit. The B1 field generated by the rf pulse is shown in Fig. 6(b); again, the slow transient response of the probe is evident in the magnitude of the magnetic field. The angle of phase rotation of the B1 vector relative to the y-axis is also shown in Fig. 6(b). There is a significant deviation in φrot during the transient intervals; these notable phase rotations have little impact on the spin dynamics, shown in Fig. 6(c), due to the corresponding low magnitude of the B1 field. The transient behavior is sensitive to errors in the probe tuning frequency ωp of ∼ 10 kHz relative to the Larmor frequency. A small yet continual drift in the B1 phase angle is observed throughout the rf pulse, caused by the offset between the driving and the Larmor frequencies. This change in phase angle shifts the
33
magnetization vector away from the x-axis. Although small, such phase errors will accumulate in multi-pulse sequences, notably CPMG. The steady-state behavior is sensitive to errors in the forcing frequency ωd of ∼ 1 kHz relative to the Larmor frequency. It is therefore extremely important, especially in CPMG-like experiments, to set the driving frequency correctly. An example of a CPMG echo train using an imperfect pulse as shown in Fig. 6, is shown in Fig. 7, where the effects of the imperfect rf pulses are clearly visible. A “zig-zag” phenomenon is present as a beating oscillatory pattern in the components of the magnetization vector. This simplistic simulation neglects two relevant physical properties of the experiment: (1) spin relaxation that leads to complications in CPMG experiments due to off-resonance spin dynamics, and (2) the transient response of the rf amplifier. Depending on the response time of the rf amplifier, the transmitted pulse may be subject to significant pulse transients; logically, trying to generate a large, instantaneous voltage (e.g., θ = π rad, equivalent to a step from zero to maximum voltage) will provide a transient response that will not be present if the rf pulse starts at a zero-crossing in the waveform. To some extent, the damping effects of finite amplifier response time appear to minimize the influence of start-up condition on the rf pulse. Paradoxically, impulse response effects may be more pronounced with high-specification amplifiers than with slow-response devices. It is found empirically that improved CPMG echo train performance is obtained with either coherent pulsing (fixed θ ) or the use of shaped (e.g., trapezoidal) rf pulses to control the rise and fall times. The performance of very low-field rf probes (ν0 < 5 MHz) can be improved using more sophisticated rf circuit designs. For example, Oxford Instruments’ GeoSpec instruments, shown in Fig. 1, include “Q-Sense technology” based on
34
a
4
={i(t)} / A
2 0 −2 −4 0
5
10
15
20
25
30
time / µs b 0.5 B1 / mT
0.5
0.3 0 0.2 −0.5
0.1 0 0
5
10
15
20
φrot / rad
1
0.4
−1 30
25
time / µs
M(t)/M(0)
c
1 0.8
Mx My Mz
0.6 0.4 0.2 0 0
5
10
15
20
25
30
time / µs
Figure 6: Simulated transient response of a very low-field rf probe with modest Q = 15. In (a) the probe current (solid blue line) is calculated with a driving frequency offset of +2 kHz and a tuning frequency offset of −2 kHz relative to the Larmor frequency of 2 MHz (red dotted line), and assuming a 500 W rf amplifier driving into a matched 50 Ω load. The corresponding magnetic field (b) is obtained for an ideal solenoid of length L = 10 cm and N = 10 turns. The spin dynamics (c) indicate that the chosen pulse length of tω = 14.62 µs provides a 90◦ rotation (Mz → Mx ). The solid black line in each plot indicates the end of the driving current at time tω = 14.62 µs.
the works of Hoult [82]. A combination of passive circuit components provides a low Q ' 3 probe with the SNR of a higher Q circuit. In the general LCR series circuit, Q is reduced by increasing R since Q = ω0 L/R. A low Q probe has a much faster transient response, as demonstrated by the simulation in Fig. 8. The transient response now lasts only 2 µs, hence providing a significant reduction in the physical probe dead time after the end of the rf pulse. The overall spin dynamics in Fig. 8(c) are unchanged, with the obvious exception that the rotation of the magnetization vector is almost complete when the forcing current stops. As a rule-of-thumb, the transient response lasts approximately Q cycles of
35
1
M(t)/M(0)
0.8 0.6
Mx My Mz
0.4 0.2 0 −0.2 −0.4 0
50
100
150
200
250
300
350
400
450
500
number of echoes
Figure 7: Simulation of magnetization at spin echo maxima in a standard CPMG pulse sequence with rf pulses of the quality shown in Fig. 6. Imperfect refocusing results in a regular oscillation in the components of the magnetization vector. This simulation ignores spin relaxation, other offresonance effects, and phase cycling schemes that are usually employed to minimize such artifacts.
the rf frequency. Hence the influence of the rf probe quality factor decreases as the Larmor frequency increases. The third rf pulse simulation, Fig. 9, shows the response of a rf pulse on a ν0 = 20 MHz system with a modest Q = 30 probe. Despite the higher Q compared to the previous examples, the transient response only accounts for a small proportion of the total rf pulse, with rise and fall times of < 2 µs. A higher Larmor frequency also reduces the sensitivity to offsets in the tuning and driving frequencies, as reflected in the final state of the magnetization vector in Fig. 9(c); the phase shift in the xy-plane is notably smaller than in the earlier examples (Fig. 6 and 8). Incidentally, precise setting of the tuning frequency becomes more important as the Q is increased as an rf pulse applied away from ω0 will suffer significant electrical loses, thereby requiring longer rf pulses to achieve the desired tip angle. 3.3. Digital filtering Modern low-field spectrometers now have the same digital filtering capabilities as their high-field counterparts. Digital filtering is important for low-field experiments as careful design and implementation of filters can provide a significant reduction in amount of noise observed in a time-domain signal. In spectroscopy 36
a
4
={i(t)} / A
2 0 −2 −4 0
5
10
15
20
25
30
time / µs b 0.5 B1 / mT
0.5
0.3 0 0.2 −0.5
0.1 0 0
5
10
15
20
φrot / rad
1
0.4
−1 30
25
time / µs
M(t)/M(0)
c
1 0.8
Mx My Mz
0.6 0.4 0.2 0 0
5
10
15
20
25
30
time / µs
Figure 8: Simulated transient response of a very low-field rf probe with low Q = 3, consistent with Oxford Instruments’ Q-Sense technology. In (a) the probe current (solid blue line) is determined with a driving frequency offset of +2 kHz and a tuning frequency offset of −2 kHz relative to the Larmor frequency of 2 MHz (red dotted line), and assuming a 500 W rf amplifier driving into a matched 50 Ω load. The corresponding magnetic field (b) is obtained for an ideal solenoid of length L = 10 cm and N = 10 turns. The spin dynamics (c) indicate that the chosen pulse length of tω = 14.62 µs provides a 90◦ rotation (Mz → Mx ). The solid black line in each plot indicates the end of the driving current at time tω = 14.62 µs.
and imaging, filters are useful for removing baseline artifacts, plus aliased signals and noise. In the case of low-field experiments, 1 H observation benefits from 19 F signal suppression achieved with careful filtering. A detailed description of digital filter design and operation has been presented by Moskau [83]. Here, we focus on the significance of the digital filter stage on data analysis. Our aim is to apply a filter that discriminates efficiently between the signal and the noise whilst leaving the signal unchanged. In general, this means any filter can only be designed to suppress noise outside the frequency range associated with the signal. Separa-
37
a
4
={i(t)} / A
2 0 −2 −4 0
2
4
6
8
10
12
time / µs b
1 0.5 0.4
0
0.2 0 0
−0.5 2
4
6
8
φrot / rad
B1 / mT
0.6
−1 12
10
time / µs
M(t)/M(0)
c
1 0.8
Mx My Mz
0.6 0.4 0.2 0 0
2
4
6
8
10
12
time / µs
Figure 9: Simulated transient response of a low-field rf probe with Q = 30. The probe current (a) is determined with a driving frequency offset of +2 kHz and a tuning frequency offset of −2 kHz relative to the Larmor frequency of 20 MHz, and assuming a 300 W rf amplifier driving into a matched 50 Ω load. The corresponding magnetic field (b) is obtained for an ideal solenoid of length L = 10 cm and N = 20 turns. The spin dynamics (c) indicate that the chosen pulse length of tω = 8.99 µs provides a 90◦ rotation (Mz → Mx ). The solid black line in each plot indicates the end of the driving current at time tω = 8.992 µs.
tion of signal and noise acquired at the same frequency is achieved through data processing, which will be discussed in Section 5. The process by which an analogue signal detected in the rf probe is converted into a digital signal is represented in Fig. 10. The analogue signal is initially amplified and filtered using a fast electronic (analogue) low-pass filter to remove frequency components and harmonics above the Nyquist limit (2ω0 ) of the resonance frequency. This filtering stage is necessary to prevent aliasing in the analogue to digital converter (ADC). The digital signal is then mixed with sine and cosine waveforms to provide a complex signal. Digital multipliers do not intro-
38
input
analogue filter digital digital multipliers filters
ADC preamplifier
complex signal
Figure 10: Schematic of NMR signal detection. The analogue signal passes through a preamplifier and an analogue low-pass filter before being converted to a digital signal in the ADC. Digital multipliers are then used to obtain a quadrature signal which is filtered digitally. The output is a complex digital signal.
duce gain or phase errors into the signal, unlike analogue multipliers [84]. The digital filtering is then applied before the final (complex) NMR signal is stored. 40
passband
gain / dB
0 −40 −80
stopband attenuation
transition band
−120
stopband −160 0
100
200
300
400
500
frequency / kHz
Figure 11: Schematic of a digital filter function, indicating the parameters used to define the filter performance. Frequency and gain scales are shown as an example of typical ranges. The filter shown has a total bandwidth of fsw = 1 MHz (hence a dwell time tdw = 1 µs) but the passband has a width of fpb = 400 kHz, so any frequency components outside the range ν0 ± fpb /2 will be suppressed. The passband and stopband ripple has been exaggerated for clarity.
Digital NMR filters are usually formed as a chain of digital filters that start with very high sampling rates and apply successive levels of data decimation over several stages. They are usually of the finite impulse response (FIR) class (also known as causal or nonrecursive filters) [83]. The exact form of the digital filter will depend on the receiver hardware design. However, the entire digital filter 39
chain can be described by one overall set of characteristics, mostly determined by the final filter stage, which in some instruments is user-programmable. Most digital filters will be designed to operate as low-pass filters operating on the detected, or heterodyned, signal centered on the Larmor frequency. An example of an oversampling filter is given in Fig. 11, showing the performance of the filter at frequencies > ν0 ; an antisymmetric filter function exists for frequencies < ν0 . Only signal components at frequencies within the passband, i.e., ν0 ± fpb /2, will be retained by the filter. Any signal or noise in the stopband will be suppressed; an input attenuated by 120 dB (10−6 in power) is considered to have zero amplitude. The transition band describes the manner in which the filter changes from “pass” to “stop” and has important consequences for the NMR experimentalist. On either side of the transition band, there will be some ripple in the gain; passband ripples are significant as these will distort the NMR signal. a 0
−120
gain / dB
b 0
−120
c 0
−120
frequency
Figure 12: Examples of filter designs appropriate for (a) noise suppression at very low field, (b) imaging, and (c) spectroscopy. In each case, the dotted vertical lines indicate the transition band. Exact frequency ranges will depend on sample line width and experimental parameters.
An ideal filter function will have minimal passband ripple, be close to 0 dB 40
gain over the entire passband, and have a narrow transition band. However, any filter design will always be a compromise between quality (shape) in the frequency domain and response time in the time domain. Filter selection should be determined primarily by the application. The impulse response of the filter is characterized by the group delay tGD , the time required for a signal to pass through the filter. Any signal recorded after filtering will be time-shifted by tGD ; we consider the implications of this time shift on the NMR experiment in Figs. 13 and 14. There is also a corresponding phase shift, but this is easily removed at the processing stage. The group delay is important as it defines the rise time of the filter. Data passing through the filter should not be stored until after the rise time (or dead time) of the filter tDE . As a rule-of-thumb, tDE ≈ 2tGD . The group delay is determined from the Fourier reciprocal of the filter function, and hence a high quality filter (narrow transition band) will have a long group delay and vice versa. The rf resonator itself has the properties of a linear filter, measured conventionally by the quality factor Q, introduced in Section 3.2, and so by extension we note that the rf probe has a physical group delay tGD = 2Q/ω0 that should be included in the experimental timing. This becomes more significant at low Larmor frequencies. Examples of digital filter functions for different applications are given in Fig. 12. A noise suppression filter, Fig. 12(a), suitable for very low-field applications, will have a narrow passband (determined by the spectral line width of the sample or magnet heterogeneity) and a “lazy” transition band, leading to a short impulse response time. A short response time and group delay is important when studying broadline samples or acquiring CPMG echo trains with short echo spacings. An imaging filter, in contrast, needs a flat passband to prevent image distortion, see Fig. 12(b). The transition band should be narrow (preventing aliasing of the
41
image) which will lead to a long response time. Most imaging sequences, particularly in medical applications, do not require very short acquisition times and so a long filter dead time can be accommodated. Filters for spectroscopic applications tend to be of intermediate quality, see Fig. 12(c). Again, the passband must be flat over the frequency range of interest, and the transition band should be as short as possible within the limits of the dead time imposed by the FID acquisition. transmitter output, V0 / kV
a probe group delay (transmit), 2Q/ω0
probe current, I0 / A or rf field, B1 / T
b probe dead time
magnetization, Mxy / A m−1
c probe group delay (receive), 2Q/ω0
d
e
probe dead time
induced voltage, VM / V
filter group delay digitized signal, S / arb. units
filter dead time
time / µs
total group delay
Figure 13: Timing diagram for a FID experiment. The spectrometer provides the rf transmission pulse in (a) and the experimentalist expects to acquire the signal shown by the dotted line. However, the actual recorded signal (e) is delayed. The timebase of the observed FID must be shifted by a time equal to the total group delay of the system to match the timebase of the transmitted rf pulse in (a). The intermediate stages (b – d) are discussed in the text.
Correct assimilation of the digital filter into the data acquisition process is extremely important. When implementing or designing a pulse sequence it is necessary to be aware of the time delays that are incurred by the filtering stages and also the rf probe. A schematic showing the various delays in a FID experiment 42
transmitter output, V0 / kV
te
a τ
τ
probe dead time
b
probe current, I0 / A or rf field, B1 / T
τ
probe group delay (transmit), 2Q/ω0
magnetization, Mxy / A m−1
c probe group delay (receive), 2Q/ω0
d probe group delay (total), 4Q/ω0
e
filter group delay
filter dead time
induced voltage, VM / V
digitized signal, S / arb. units
total group delay
time / µs
Figure 14: Timing diagram for a spin echo experiment. The spectrometer provides the rf transmission pulses (a) and the experimentalist expects to acquire the echo shown by the dotted line. However, the actual recorded echo (e) is delayed. The timebase of the observed echo must be shifted by a time equal to the total group delay of the system to match the timebase of the transmitted rf pulses in (a). The intermediate stages (b – d) are discussed in the text.
is given in Fig. 13. The spectrometer issues a 90◦ rf pulse at time t = 0, Fig. 13(a); the pulse has a duration of t90 . The resonant circuit of the rf probe does not respond immediately. The B1 field experienced by the spins is time shifted by the physical group delay of the probe, equal to 2Q/ω0 , Fig. 13(b). It follows that the probe cannot be used to detect the evolution of the magnetization Mxy until the rf pulse has decayed, as described in Section 3.2.2, leading to a dead time associated with the probe, Fig. 13(c). Once the probe is able to detect the magnetization, there is a further group delay of 2Q/ω0 on detection, Fig. 13(d). The observed signal will then be time shifted by the group delay of the digital filter tGD . Furthermore, the dead time of the filter prevents the initial response of the signal from being 43
detected, and so data are collected only once the filter has settled, Fig. 13(e). On the spectrometer time-base (relative to the issued rf pulse), the zero time origin of the data is actually t = t90 + 4Q/ω0 + tGD . Therefore, if the data are to be fitted to determine the total signal amplitude S0 , the time base of the acquired data must be adjusted accordingly. Failure to do so will result in a significant over-estimation of the total signal. The dead time of the probe and filter result in phase errors in a direct FT of the signal; methods for extracting an undistorted spectrum from time-shifted data will be discussed in Section 5.2. A similar situation arises in the case of spin echoes. A schematic showing the various delays in a spin echo experiment is given in Fig. 14. The spin echo forms between a pair of 180◦ rf pulses, each of duration t180 , Fig. 14(a). The detected signal is delayed relative to the issued rf pulses by 4Q/ω0 which is the total group delay of the probe on transmit and receive, Fig. 14(b,c,d). If the half-echo time
τ is sufficiently long, the dead time of the probe can be neglected. However, this time may become significant if the rf pulse separation is small and the ringdown duration of the rf pulses is long. Similarly, the dead time of the filter can be neglected for fast filters and sufficiently long echo times. The actual echo center will be displaced on the spectrometer time base by the total group delay of the system tGD + 4Q/ω0 , Fig. 14(e). It is important to know the exact temporal placement of the echo in order to acquire the data correctly. If the acquisition command is not delayed relative to the rf transmission time base, the echo will be off-center in the acquisition window. Such an error in timing is more critical when only the echo amplitude is recorded (single point acquisition): the echo maximum will not be observed, thereby leading to a significant error in the measurement. Similarly, problems can arise with very short echo times where the acquisition
44
window overlaps with the next rf pulse. In such circumstances it is necessary to design fast digital filters to suit the experimental parameters. On the spectrometer time base, the echo center occurs at a time t180 /2 + τ2 + tGD + 4Q/ω0 after the start of the 180◦ refocusing pulse. amplitude / arb. units
3 2 1 0 0
5 signal / arb. units
a
4
0.2 0.4 0.6 0.8 time / ms
1
c
4 3 2 1 0 0
0.2 0.4 0.6 0.8 time / ms
1
b
0.8 0.6 0.4 0.2 0 −500 −250 0 250 frequency / kHz
amplitude / arb. units
signal / arb. units
5
1
1
500
d
0.8 0.6 0.4 0.2 0 −500 −250 0 250 frequency / kHz
500
Figure 15: Simulation of digital filtering. The FID signal from a liquid sample (a), acquired with a dwell time tdw = 1 µs, is inhomogeneously broadened by heterogeneities in the magnetic field; SNR ≈ 50. The FT of the signal (b) clearly shows that the passband of the filter covers the full ±500 kHz range. If a carefully chosen digital filter is used for the data acquisition instead (c), then the noise observed in the time domain is reduced, whilst the SNR, defined in the frequency domain (d), remains constant. This filter has a passband approximately corresponding to ±20 kHz, and so the noise is suppressed outside this frequency band.
The difference between timing in the FID, Fig. 13, and spin echo experiments, Fig. 14, is subtle and easy to overlook. In the FID acquisition, there is no need to account for the group delay in the pulse sequence. The data are acquired immediately following the probe and filter dead times. However, the time base of the data must be adjusted at the processing stage to account for the group delays 45
in the system. In the spin echo experiment, the acquisition must be shifted by the total group delay of the system relative to the expected position of the echo center. The observed echo can then be considered to have reached a maximum midway between the two 180◦ rf pulses as issued on the spectrometer time base and no further adjustment of the time base is necessary during processing. An example highlighting the significance of careful digital filter design in lowfield experiments is presented in Fig. 15. The FT of an FID provides a spectrum in which noise has been acquired across the full frequency range, Fig. 15(a,b). The frequency range is determined by the dwell time tdw used in the acquisition. However, if a digital filter is applied, it is possible to suppress much of the noise in the spectral window without adversely affecting the shape of on-resonance components, Fig. 15(d). This technique is especially useful at low field where the entire 1 H proton signal is expected to be on, or close to, the Larmor frequency. Furthermore, the digital filter is useful in suppressing signal contamination from 19 F,
often present in the rf probe body or sample container. Although the SNR
of the data acquired at the Larmor frequency is unchanged, the digital filter has suppressed noise that would otherwise be integrated into the time-domain signal, Fig. 15(c). The noise has therefore been reduced in the time domain, allowing a more robust interpretation of the time domain data. 3.4. Maximizing signal-to-noise ratio Low-field NMR is considered to have an inherently poor SNR compared to experiments conducted at intermediate and high field. The rule-of-thumb that SNR ∝ ω0 demonstrates why low-field NMR is associated with poor SNR, although Hoult has argued that the exact theoretical SNR owes as much to the performance of the rf probe (both in transmission and reception due to the principle 46
of reciprocity) as to the static field strength [85]. It is important to realize that SNR is a property of the measurement and not of the instrument. There are a few straightforward steps that can be taken when designing a lowfield experiment to maximize the available SNR. Wherever possible, the strongest static magnetic field should be used to maximize the inherent SNR of the experiment. There is one exception to this rule: heterogeneous materials (such as porous media, dense composite materials, or any systems with gas/liquid/solid interfaces) will be prone to the formation of internal gradients due to the magnetic susceptibility contrast in the sample [50]. These magnetic field gradients scale with B0 , and so a higher field strength will result in a larger distortion in the static magnetic field. Strong internal gradients can lead to very rapid signal decay within the dead-time of the rf probe, preventing the detection of any signal. Therefore, depending on the properties of the sample, improved SNR may be achieved at lower field strengths. Capacitive coupling between the probe and the sample can lead to significant changes in the probe tuning, particularly if the sample is electrically conductive, such as a high salinity brine. Low Q circuits will be robust to small changes in sample composition (conductivity) during the course of an experiment. Maximizing the coil filling factor, i.e., the fraction of the sensitive volume of the rf probe occupied by the sample, is also important. For bench-top solenoid coils, the filling factor is easily calculated and can be close to unity (volume of sample is equal to volume of probe). However, due to the cylindrical geometry of solenoid rf coils, small reductions in sample radius result in a significant reduction in filling factor. We note that it is advisable to leave some space between rf coil and sample as local B1 “hot spots” will be present in close proximity to the wire. Filling factor
47
is more difficult to estimate for surface coils used on single-sided NMR devices; the filling factor will always be less than
1 2
because the sample exists only on one
side of the rf probe. Elimination of external rf sources is significant for optimizing the SNR. In bench-top magnet systems, as illustrated in Fig. 2, the outer frame of the magnet (combined with the rf probe housing) acts to screen the antenna from external sources. However, this screening can do little to reduce noise sources conducted directly into the system along cables. Unfortunately, gradient power cables and temperature sensors (thermocouples or PRT’s) are obvious transmission routes both for rf noise and coherent interference. White, Gaussian noise is not a particular problem for the experimentalist: the SNR can be improved by increasing the √ number of repeat scans Ns (where SNR ∝ Ns ) and signal may be extracted or fitted in the presence of white noise by careful data analysis as discussed in Section 5. Interference that is coherent with the receiver will sum at the same rate as the signal and is not reduced by signal averaging. Furthermore, robust separation of signal and interference can be difficult to achieve at the analysis stage. It is important, therefore, to remove sources of interference whenever possible. In the laboratory environment these sources can be identified using a network analyzer tuned to the Larmor frequency of the magnet and a simple pick-up coil (a loop of copper wire is sufficient). Switch-mode power supplies, now common in electrical devices, are particularly egregious sources of rf interference. It is usual to ground the NMR equipment using the laboratory Earth line, although in some cases this may be a source of interference. If the building ground is particularly noisy, the NMR equipment may be powered through an isolation transformer and grounded independently using a local Earth. Similarly, laboratory Ethernet connections can
48
be interference sources and computers should be isolated with wireless or optical Ethernet connections. Furthermore, all coaxial cables used on the NMR system should be shielded, with the shield grounded only at one end (preferably as close to the magnet housing as possible); this mode of grounding prevents the formation of earth loops. A main route of interference into the NMR system is via the gradient connections; at modest frequencies (e.g., B0 > 20 MHz) it is possible to filter the gradient lines without adversely affecting gradient amplifier performance, although at lower frequencies this is not a practical solution. The other common routes of noise injection are temperature sensors either on the sample or in the gradient unit. For superambient measurements, electronic sensors could be replaced with temperature-sensitive fiber-optic Bragg gratings that are interrogated remotely and are pure electrical insulators. With more traditional devices it is important to keep the temperature sensor away from the rf coil (preferably outside the probe housing) to limit rf interference unless considerable efforts are made to effectively ground the sensor. Grounding of fluid flow lines during NMR measurements is also important as discussed in [86]. The issue of noise and interference sources is quite different for single-sided NMR instruments where the rf probe is unshielded. Devices such as the NMRMOUSE suffer from poor SNR in “noisy” laboratory environments and it may be necessary to construct a Faraday cage around the instrument and sample. However, when using single-sided devices in remote locations (construction sites or oil wells) the sample itself will act as a rf shield and SNR performance can be significantly better than expected based on laboratory tests. Again, the ideas of minimizing or eliminating local sources of rf interference apply. Overall, it is possible to optimize low-field hardware to provide a usable SNR
49
within a practical acquisition time. Careful data processing can allow information to be extracted even when the SNR is poor and so for many industrial process or quality control applications, the perceived SNR performance should not be considered a reason to avoid low-field instrumentation. 4. Pulse Sequences and Experiments 4.1. Relaxation time measurements Relaxation time analysis is the most common technique used in industrial applications of low-field NMR. Distributions of relaxation time are used to characterize fluid content, fluid type, and pore size. Here, we consider the basic pulse sequences for determining longitudinal T1 and transverse T2 relaxation times. These relaxation time measurements can be combined, or performed in conjunction with diffusion measurements, to provide multi-dimensional relaxation time and diffusion-relaxation correlations. Additionally, relaxation time measurements are incorporated within MRI protocols to provide relaxation weighted images. 4.1.1. Longitudinal relaxation Recovery of the spin ensemble to equilibrium is governed by the exponential time constant T1 . Accordingly, the observed signal (magnetization) b at time τ1 is described by b (τ1 ) τ1 = 1 − c exp − . beq (∞) T1
(11)
The change in magnetization process is typically monitored using either the inversion recovery or saturation recovery method [87]; eq. (11) applies to both methods. Pulse sequence schematics for both inversion and saturation methods are shown in Fig. 16. Inversion recovery (c = 2) begins with the spin ensemble 50
aligned along −z, whereas saturation recovery (c = 1) begins with the spin ensemble fully saturated, i.e., zero net magnetization, on the x-y plane. Inversion recovery provides twice the dynamic range in the data compared to saturation recovery but relies on the application of a good 180◦ inversion rf pulse. Furthermore, the spin ensemble must be at equilibrium on +z prior to the inversion pulse. To guarantee this condition, the recycle delay tRD between scans must be long and satisfy tRD ≡ 5 × T1 . Saturation of the spin ensemble is easier to achieve when B0 and B1 are non-uniform, or in broadline samples. The saturation recovery experiment does not require a long tRD as effective saturation is achieved regardless of the initial state of the spin ensemble. The saturation recovery experiment is therefore much faster than the inversion recovery experiment when long T1 times are observed (i.e., when T1 > 1 s). A rapid double-shot T1 measurement is described in [88]; at present this pulse sequence has not been implemented at low field due to the reliance on small tip angle rf pulses and hence reduced SNR. 4.1.2. Transverse relaxation The observed NMR signal amplitude following a single excitation pulse (nominally 90◦ ) decays over time as the spin ensemble loses phase coherence in the x-y plane due to local magnetic field fluctuations. These fluctuations arise from dipolar interactions, heterogeneities in the background magnetic field, and other terms in the nuclear spin Hamiltonian. The observed free induction decay (FID) [89] is governed by the exponential time constant T2∗ , as [90] 1 1 ≈ + γ ∆B0 , ∗ T2 T2
(12)
51
τ1
a
τ1
b ns time
Figure 16: Pulse sequence schematic for T1 relaxation time measurements via (a) inversion recovery and (b) saturation recovery [87]. In all cases, the thin and thick vertical bars represent 90◦ and 180◦ rf pulses, respectively. The saturation comb of 90◦ pulses is repeated nc times. Homospoil gradients (gray trapezoids) may be included between the rf pulses to eliminate unwanted coherence pathways. The initial magnetization b at time τ1 = 0 is set to (a) b(0) = −beq or (b) b(0) = 0. The spin ensemble is allowed to recover on the z-axis for a time τ1 before being interrogated by a 90◦ excitation pulse. The signal amplitude immediately after the excitation pulse is determined by fitting the FID. At intermediate field, the FID can be Fourier transformed to obtain a chemical spectrum. The experiment is repeated for a range of τ1 times to determine the T1 recovery time.
where ∆B0 is a characteristic measure of magnetic field variation. T2∗ is inversely related to the full-width half-maximum (FWHM) of the spectral line, such that 1 + γ ∆B0 ≈ πFWHM. T2
(13)
When using intermediate-field spectroscopy magnets, ∆B0 ≈ 0 in bulk liquids and
so T2∗ → T2 , where T2 is the transverse relaxation time determined by spin inter-
actions. However, when using low-field permanent magnets with poor B0 homogeneity, ∆B0 is large and the FID will be characterized by T2∗ T2 . A similar situation arises when measuring heterogeneous samples (such as liquids confined in porous media) because the magnetic susceptibility contrast between the solid and liquid distorts the static magnetic field. To measure T2 relaxation in the presence of magnetic field inhomogeneities, the spin ensemble is refocused in the x-y plane to form a spin echo [91] by a 180◦ rf 52
pulse applied a time τ2 after the initial excitation pulse. The spin echo, comprising two FIDs back-to-back, has a maximum at time te = 2τ2 after the excitation pulse. By repeatedly applying 180◦ rf pulses, separated in time by te , a train of n echoes is formed [92]. The method of forming this echo train has been refined into the standard Carr-Purcell Meiboom-Gill (CPMG) experiment [93]. The envelope of the echo maxima decays exponentially with T2 as nte b (t) . = exp − b (0) T2
(14)
τ2
n
time
Figure 17: Schematic of the standard CPMG pulse sequence [92, 93] for T2 relaxation time measurements. In all cases, the thin and thick vertical bars represent 90◦ and 180◦ rf pulses, respectively. The echo time te = 2τ2 ; the echo maximum forms at a time τ2 after the 180◦ pulse. The repeated application of 180◦ refocusing rf pulses generates a train of n echoes. The maximum signal intensity of each echo is acquired at time nte to provide an entire echo train in a single scan. At intermediate field, the FID (latter half of final echo) is acquired after n 180◦ rf pulses and Fourier transformed to provide a chemical spectrum. The experiment is repeated to obtain a spectrum for a range of n echoes.
It is important to note that the CPMG sequence does not refocus J-coupling interactions, and so at intermediate field the signal decay will be modified when studying chemical species with multiple functionality. The J-coupling interaction still exists at low field, but in general the B0 and B1 fields are sufficiently inhomogeneous as to mask scalar coupling processes. At ultra-low fields, J-coupling is used for chemical identification in lieu of conventional spectroscopy [94].
53
4.2. Diffusion and flow Pulsed field gradient (PFG) diffusion measurements are used to determine coefficients of self-diffusion, restricted diffusion, and advection (flow) [95–98]. A magnetic field gradient g is applied for a time tδ to provide the spins with a phase shift proportional to their position r(0). After an observation time t∆ , a second gradient pulse of equal area is applied to remove the phase shift. If the spins have not moved, their net phase shift is zero. However, if the spins have moved to a new position r(t∆ ), then the spins have a net phase shift equal to
ϕ = γ g · [r(t∆ ) − r(0)] .
(15)
This phase shift can be simplified to ϕ = 2πq · R where q = γ tδ g/2π is the magnetization wave vector and R = r(t∆ ) − r(0) is the displacement vector. The signal decay due to diffusion is then b(g) = exp −4π2 q2 Dt∆ , b(0)
(16)
where q is the magnitude of q and D is the diffusion coefficient. It is usual to apply the gradient along a single axis so we need only consider the gradient magnitude g. The diffusion coefficient is readily determined by repeating the measurement for a range of g. If all the timings remain constant, the only contribution to the normalized signal decay is diffusion. There are exceptions to this condition, notably when g is fixed by the hardware as discussed in Section 4.5.2. The basic principles of PFG diffusion measurements are the same regardless of the acquisition method. However, the details of q in eq. (16) will depend on the exact form of the experiment. For example, the pulsed gradient spin echo (PGSE) experiment [95] with arbitrary gradient pulse shape gives n h io tδ b(g) 2 2 2 + A (ε ,tδ ) , = exp −γ g D tδ t∆ − b(0) 3 54
(17)
τse g
a tδ
t∆
τse g
b tδ
t∆ time
τse tδ1
c tδ /2
tδ2
g
t∆
Figure 18: Basic diffusion pulse sequence schematics for (a) PGSE, (b) PGSTE, and (c) APGSTE. Thin and thick vertical lines indicate 90◦ and 180◦ rf pulses, respectively. Timings are specified following the notation of Tanner [97]: in each case, the spin echo time is τse and the observation time (defined as the time between leading gradient edges) is t∆ . Trapezoidal gradient pulses are shown with effective duration tδ and amplitude g. The gradient amplitude is incremented in successive experiments. Homospoil gradients are indicated by gray trapezoids. The delay between the rf pulse and leading gradient edge is tδ1 , and the delay between the trailing gradient edge and the rf pulse is tδ2 . At intermediate field the measured FID may be Fourier transformed to provide a spectrum, as in the DOSY experiment.
55
where A (ε ,tδ ) is an approximate correction term in the Stejskal-Tanner equation that depends only on the shape of the gradient pulse, following the notation of Price and Kuchel [99]. For example, a trapezoidal pulse requires
ε 3 tδ ε 2 A (ε ,tδ ) = − , 30 6
(18)
where ε is the duration of the gradient ramp (assuming the rise and fall rates are equal). Other correction terms for a variety of pulse shapes are given by Price [100, 101], and exact exponents in the Stejskal-Tanner equation for a selection of pulse shapes are given by Sinnaeve [102]. It can be helpful to use shaped gradient pulses (e.g., trapezoids or half-sine pulses) to provide robust and reproducible gradients. If the ramp time is short compared to the pulse duration (i.e., approximate rectangular gradient pulses), such that ε 2 tδ , we may assume A (ε ,tδ ) → 0. The basic PGSE pulse sequence is shown in Fig. 18(a). The maximum amount of signal obtained in the PGSE sequence will depend on the ratio of T2 to t∆ . In heterogeneous materials, T1 > T2 (even at low field [103]) and so it is desirable to use a pulsed gradient stimulated echo (PGSTE) experiment instead [97], see Fig. 18(b); eqs. (17) and (18) still apply to this pulse sequence. A stimulated echo [91] stores the spin ensemble on the z-axis during the observation time, so the signal decays only with T1 , enabling longer observation times to be employed. It is worth noting that we have neglected the relaxation terms in eq. (17) on the assumption that all pulse timings are constant and so these fixed pre-factors are encompassed in the signal normalization b(g)/b(0), and this assumption is sufficient when determining diffusion coefficients. However, if quantitative signal intensities are required, terms describing T2 relaxation (during spin echoes) and T1 relaxation (during stimulated echoes) need consideration; the exact form of these terms will depend on the pulse sequence. It is also necessary to note that a 56
stimulated echo returns only half the initial signal amplitude, whereas a spin echo returns the entire signal amplitude. As well as the applied gradient pulses, distortions in the B0 field will contribute to diffusive attenuation. In the presence of a constant background gradient g0 , the signal attenuation observed in a PGSTE experiment (assuming rectangular gradient pulses) will be [104] n b(g) tδ 2 g = exp −γ 2 D tδ2 t∆ − b(0) 3 !2 2tδ2 2 + tδ 2τset∆ − − tδ (tδ1 + tδ2 ) − tδ1 + tδ2 gg0 3 τse 2 o 2 g0 . + τse t∆ − 3
(19)
Cotts et al. [104] devised several alternating pulsed gradient stimulated echo (APGSTE) sequences to reduce the influence of background gradients in diffusion experiments. The popular version is the Condition I “13-interval” APGSTE pulse sequence illustrated in Fig. 18(c). Here, the gradient pulses are split into positive and negative amplitude lobes around a 180◦ refocusing pulse. Each lobe has an area gtδ /2, so that the spins experience a total applied gradient of gtδ (as in the simpler PGSE and PGSTE sequences) due to the refocusing pulse. Assuming rectangular pulses, this 13-interval APGSTE experiment provides a signal that varies as 2 t tδ 2 b(g) 2 = exp −γ D δ 4t∆ − 2τse − g b(0) 4 3 4 3 2 , + τsetδ (tδ1 − tδ2 ) gg0 + τse g0 3
(20)
such that the cross term in gg0 vanishes when tδ1 = tδ2 . This condition on the pulse sequence timing is the ideal mode of operation and should only be waived 57
when very short τse times are required. Assuming tδ g τse g0 , the overall signal attenuation is dominated by the applied gradient pulses and not the internal gradients. Improved compensation for background gradients can be achieved using the “17-interval” APGSTE pulse sequence [104] or higher order versions [105]. These pulse sequences are designed to remove the cross term in gg0 regardless of the ratio of tδ1 to tδ2 , and provide additional reduction of the term in g20 relative to that in g2 . However, the increased complexity of implementation and greater relaxation attenuation associated with these long pulse sequences leads to them not being widely used. The diffusion pulse sequences in Fig. 18 are also used to acquire a probability distribution of advective displacement (the so-called “flow propagator”) [98], averaged over all possible starting positions of the spins in the volume of interest. The only modification required to the aforementioned pulse sequences is that the gradient amplitude is ramped between ±g(max) in successive scans. Flow propagators have been used to assess pre-asymptotic Stokes’ flow in bead packs [106–117], catalysts and chemical reactors [98, 118, 119], biofilms [120–122], foams [123, 124], and rocks [86, 115–117, 125–136]. However, to date the implementation of the propagator experiment on low-field hardware has been limited [86]. 4.3. Multi-dimensional correlations Two-dimensional (2D) correlations are becoming increasingly popular at low field. The 2D correlations of relaxation time or diffusion coefficient can provide robust chemical identification when spectroscopic resolution is not available. A notable example is the separation of oil and water signals in well-logging using D58
T2 correlations as will be discussed in Section 7.3.1. Other applications, including monitoring of hydration in cement-based materials (Section 7.2.1), are to be found in the literature. Higher order correlations, such as three-dimensional T1 -D-T2 correlations, are possible, although the experiment duration and complexity of data processing have thus far limited the use of these experiments. Diffusive or chemical exchange processes can be monitored by including a defined “storage time” between similar encoding intervals, such as the D-D or T2 -T2 experiments. We note that at intermediate field, all these experiments can be acquired with chemical resolution. However, doing so greatly increases the acquisition time. Examples of chemically resolved correlations can be found in the literature (e.g., [137]) although they are rare. Here, we consider only the standard low-field implementations without chemical resolution. 4.3.1. Relaxation time correlations T1 -T2 correlations have been applied to rocks [138–140] and to dairy products [141] where the additional information in the 2D distribution simplifies discrimination of the oil and aqueous signals [142]. T1 -T2 correlations also provide the relaxation time ratio T1 /T2 which is related to the strength of surface interaction between the imbibed liquid and a solid pore matrix [143–145]. The ratio T1 /T2 is therefore useful as an indicator of wettability [144]. The relaxation behavior observed in the 2D data array is exponential in a linear combination of terms in T1 and T2 . Therefore, the NMR data have the form nte b (nte , τ1 ) τ1 = exp − 1 − c exp − , b(0, 0) T2 T1
(21)
where the constant c again describes the scaling of the initial magnetization perturbation as in eq. (11) with c = 1 for saturation recovery and c = 2 for inversion 59
recovery. τ1
τ2
a n τ1
τ2
b ns n
time
Figure 19: Pulse sequence schematic for T1 -T2 relaxation time measurements via (a) inversion recovery [141] and (b) saturation recovery [146, 147]. In all cases, the thin and thick vertical bars represent 90◦ and 180◦ rf pulses, respectively. The saturation comb of 90◦ pulses is repeated nc times. Homospoil gradients may be included between the pulses in the T1 encoding portion to eliminate unwanted coherence pathways. The initial magnetization M at time t = 0 is set to (a) b(0) = −b(max) or (b) b(0) = 0. The spin ensemble is allowed to recover on the z-axis for a time
τ1 before being interrogated by a CPMG echo train. The experiment is repeated for a range of τ1 times to encode T1 , followed in each case by n echoes separated in time by te = 2τ2 to encode T2 .
4.3.2. Diffusion-relaxation correlations A combined PGSE and CPMG experiment was introduced for diffusion imaging [148]. At low field the PGSTE sequence is preferable for bulk liquid samples, with the APGSTE sequence being used predominantly in studies of porous media. In such applications to porous media, the 13-interval Cotts sequence is combined with the standard CPMG experiment [107, 134] as shown in Fig. 20. An additional z-storage delay of duration tstore may be incorporated between the diffusion encoding and acquisition sections of the experiment to provide eddy current stabilization prior to the CPMG sequence [149] or ensure the echo time in the CPMG sequence is constant for all echoes [134]. The quality of the diffusion data is improved by including dummy gradient pulses for eddy current stabilization at the start of the sequence [150]. 60
τse
tδ /2
te
g
t∆
ts
n time
Figure 20: A schematic of the APGSTE-CPMG pulse sequence [107, 134], based on the “13interval” sequence of Cotts et al. combined with the standard CPMG echo train. Thin and thick vertical lines indicate 90◦ and 180◦ rf pulses, respectively. Timings on the diffusion encoding portion are specified following the notation of Tanner [97]: the echo time is tse = 2τse and the observation time (defined as the time between leading gradient edges) is t∆ . Trapezoidal gradient pulses are shown with duration tδ and amplitude g; the gradient amplitude is incremented in successive experiments. Homospoil gradients are indicated by gray trapezoids. Data are acquired only in the CPMG each train comprising n echoes separated in time by te . An extra z-storage (stimulated echo) delay of tstore is included prior to the CPMG sequence.
Of course a CPMG echo train may be appended to any of the diffusion sequences illustrated in Fig. 18. A summary of the pulse sequence combinations appropriate for D-T2 experiments has been presented in [151]. The 2D data obtained from the APGSTE-CPMG experiment shown in Fig. 20 will be of the form nte + 4τse b (nte , g) = exp − b(0, 0) T2 n h io tδ 2 2 × exp −g D tδ t∆ − (22) + A (ε ,tδ ) , 3 with the approximate correction term for arbitrary gradient ramps given by A (ε ,tδ ), see Section 4.2. An additional (fixed) time is present in the T2 exponent to account for transverse relaxation during the spin echoes in the diffusion encoding portion of the sequence. An additional T1 relaxation term t∆ − 2τse + tstore exp − , T1 must also be included if the signal intensity is to be quantitative. 61
(23)
4.3.3. Exchange rate measurements Diffusive or chemical exchange processes can be monitored using T2 -T2 experiments [152–154], wherein a pair of CPMG echo trains are separated by a z-storage interval of duration tstore . A similar exchange measurement can be obtained with the D-D experiment [155, 156], although to the best of our knowledge this pulse sequence has not been implemented at low field. The T2 -T2 pulse sequence, as applied at low field to cements [143, 152] and rocks [50, 157], is illustrated in Fig. 21. Data are acquired only in the second CPMG train. We use (1)
the convention that the first (direct) dimension is T2
and the second (indirect)
(2)
dimension is T2 . The experimental data have the form ( ) ( ) b (nte , mte ) nte mte = exp − (2) exp − (1) . b(0, 0) T T 2
(24)
2
The total signal amplitude is reduced due to T1 relaxation during the storage interval, so a factor exp {−tstore /T1 } needs to be included when comparing T2 -T2 data acquired with different storage intervals. Methods for determining exchange rates from these data are discussed in Section 6.3. 4.4. Imaging Spatial position is encoded in the reciprocal k-space signal b(k) and the image ˆ amplitude b(r) is obtained from the FT of b(k); k-space is therefore the Fourier inverse of real space. The Larmor frequency ω (r) of the spins at position r is related to the applied magnetic field gradient vector g (in the laboratory reference frame) by
ω (r) = − (γ B0 + γ g · r) .
(25)
62
te
tstore
m
n
time
Figure 21: A schematic of the T2 -T2 exchange experiment pulse sequence. This sequence consists of two CPMG echo trains separated by a z-storage delay (stimulated echo) of duration tstore . The first CPMG echo train has a total duration mte ; the number of echoes m is incremented over sequential experiments to construct a 2D data array. The second CPMG train has a total duration of nte . The usual implementation consists of a single-scan acquisition of the echo amplitudes in the second CPMG train, although chemical resolution can be retained by incrementing n over separate experiments and recording the final half-echo. To determine an exchange rate, the storage time tstore is incremented in separate experiments.
When k-space data are acquired on resonance in the rotating frame, the voxel intensity is obtained by the 3D FT of b(k) such that ˆ b(r) =
ZZZ
b(k) exp [+i2πk · r] dk.
(26)
Complete introductions to k-space and imaging are to be found in [8, 90, 158]. 4.4.1. Frequency encoding In medical MRI it is typical to acquire a single dimension in multi-dimensional imaging using a frequency-encoded gradient echo [8]. As the amplitude of a gradient echo depends on T2∗ , it is desirable when imaging materials to include a coincident frequency-encoded spin echo (amplitude depends on T2 ); this modification requires only the addition of a refocusing rf pulse [90]. A frequency-encoded spin echo provides a single line of k-space points. Therefore, frequency encoding has the advantage of providing a profile in a single scan. It is important to realise that the shape of a frequency-encoded profile is convolved with the spectral lineshape. Therefore the profile resolution is determined by T2∗ . 63
A frequency-encoded spin-echo profile is obtained by applying a magnetic field “read” gradient during data acquisition such that excited nuclear spins precess with a frequency determined by their position. If the magnetic field gradient is applied for a time tr , then the range of wavenumbers spanned is given by ±k(max) = ±
γ gtr , 4π
(27)
where kmax is the maximum deviation from k(0), the central k-space position corresponding to ω0 . Each point in k-space will have a width defined by ∆k =
γ gtdw . 2π
(28)
The field of view (FOV) of the profile is then determined by the reciprocal relation FOV = 1/∆k such that the spatial resolution is ∆z = FOV/m where m is the number of complex data points acquired. When using digital filters, the practical FOV will be defined by the filter passband fpb rather than the reciprocal dwell time. Frequency-encoded profiles must be acquired with sufficient bandwidth to exceed the frequency range ∆ν0 =
γ gL . 2π
(29)
A frequency-encoded profile can be measured only in a single direction in any one experiment or acquisition interval. Images can be generated by acquiring read profiles in multiple directions and then reconstructing with a back-projection algorithm [158]. In modern MRI, it is far more common to use FT imaging reconstruction by combining frequency, phase, and slice techniques [8, 90]. Complicated k-space trajectories may be defined using time-varying read gradients, although such acquisition techniques have not yet been applied at low field. Rather, a single frequency-encoded echo is acquired in the direction requiring the highest 64
spatial resolution; frequency encoding is used since it provides the most efficient sampling of a large number of k-space points (pixels). 4.4.2. Phase encoding To select a k-space position using phase encoding, a “phase” gradient of variable amplitude g is applied for a time tp . T2∗ blurring is negated when using phase encoding because the acquisition time (i.e., time after initial excitation) of each datum is fixed. The k-space point sampled at a particular gradient strength is determined by k=
γ gtp . 2π
(30)
By adjusting the gradient amplitude between ±g(max) over m subsequent scans, a line of k-space is traversed. The corresponding FOV is FOV =
mπ
γ g(max)tp
,
(31)
and the pixel size is ∆z =
π
γ g(max)tp
.
(32)
The phase-encoded gradient can be ramped in multiple directions (x, y, or z) in subsequent experiments, so that k-space may be traversed in one, two, or three dimensions. When all the image dimensions are acquired in this manner, the technique is referred to as a “pure phase-encode method”. Details of pure phaseencoding experiments will be given in Section 4.4.5. 4.4.3. Slice selection A spatial slice is selected by applying a shaped rf pulse in the presence of a gradient [159]. Slice selection differs from frequency and phase encoding in that 65
the obtained signal intensity is directly proportional to the spin density in the slice. A profile may therefore be obtained by moving the location y0 of the resonant slice sequentially through the sample; no further data processing is required. The width of the slice ∆ys is determined by the frequency bandwidth ∆ν of the rf pulse and the gradient amplitude g as ∆ys =
2π∆ν , γg
(33)
where the rf pulse bandwidth is a property of the pulse shape. The position of the slice is determined by the frequency offset, γ gy0 /2π. The inverse relationship between frequency and time means a narrow slice width is achieved only with a long rf pulse duration, often 100 µs when using a modest gradient amplitude. For low flip angles, the slice shape is approximately the Fourier inverse of the rf pulse shape, so an ideal boxcar slice is obtained from a sinc pulse with an infinite number of oscillations. The practical quality of the slice will be influenced by the truncation of the rf pulse width to a realistic duration, coupled with the ability of the spectrometer and rf amplifier to reproduce the required shape. It is often easier to define a Gaussian rf pulse and accept that the slice shape will also be a Gaussian; the simplified waveform provides better reproducibility and control for the experimentalist. The slice-selection gradient of amplitude g is applied for a time ts , ideally equal to the duration of the rf pulse. During this time, the excited portion of the spin ensemble will acquire a phase shift. It is therefore necessary to apply a rephasing gradient lobe of approximate area gts /2 after the slice-selection gradient to remove the phase shift. The exact area of the rephase gradient lobe relative to the slice-selection gradient is dependent on the details of the rf pulse shape [8].
66
Slice selection can be inappropriate when imaging short T2∗ materials due to the long rf pulse durations. 4.4.4. Multi-dimensional imaging The primary application of MRI has been in medicine, where anatomical images of humans and rodents are used for clinical diagnosis and biomedical studies, respectively. The concept of applying MRI to medical diagnosis was suggested by Damadian [160]. The practical implementation of MRI was demonstrated by Mansfield and Grannell [161], and Lauterbur [162]. The first full-body MRI scanners were constructed nearly a decade later [163]. There has since been a drive in magnet technology to attain higher field strengths for improved SNR (leading to improved image resolution) and spectroscopic resolution for studies of tissue chemistry. Imaging of materials (soft solids, porous media, solids) presents different challenges due to the short relaxation times associated with these samples. The earliest example of MRI of materials was presented by Gummerson et al. [164] who profiled 1D spontaneous ingress of water into stones and cement based materials using a low-field magnet. High resolution 3D imaging is possible with the spin-warp pulse sequence [165, 166]. The standard implementation utilizes a frequency-encoded echo in the read dimension, combined with two phase-encoded gradients to achieve full 3D k-space sampling. There is inherent relaxation contrast in the images generated by the spin warp method, although the timing of each k-space acquisition is constant, so the final image has a uniform weighting within the limitations of the frequencyencoded spin echo acquisition (i.e., T2∗ ). The spin warp sequence does not provide the temporal resolution required in medical imaging, although it is suitable for materials which do not change with time. The basic spin warp pulse sequence is 67
rf
te read(z)
phase(y)
phase(x)
time
Figure 22: 3D spin warp imaging sequence utilizing one read gradient and two phase-encode dimensions. The thin and thick vertical lines represent 90◦ and 180◦ rf pulses, respectively. The spin echo is centered at time te after the initial rf excitation pulse. The phase-encoded gradients are incremented individually in successive experiments.
shown in Fig. 22. The acquisition of a full isotropic image can take several hours using the spin warp sequence, and therefore rapid imaging techniques are often chosen for studies of materials. Quantitative MRI is achievable if the relaxation weighting associated with each voxel is determined. Majors et al. demonstrated a 1D pulsed gradient version of the multi-echo imaging sequence [167], illustrated in Fig. 23. A series of spin echoes, each acquired in the presence of a read gradient, provides a CPMG decay for each pixel in the profile. These decay curves are fitted to obtain b(0), the projected signal at zero time, and a T2 distribution. The multi-slice multi-echo (MSME) sequence detailed in [8] provides a method for obtaining relaxation time maps in 2 or 3D. The phase-encoded gradient is ramped over subsequent scans to generate an echo train that decays with a mixture of T1 and T2 for each voxel. Relaxation time maps are again obtained by fitting the decay curves. A pure T2 relaxation time measurement is achieved by adding spoiler gradients to shift the stimulated echo contribution out of the acquisition window; note that this modifi68
cation reduces the acquired signal intensity. MSME offers quantitative 3D imaging using conventional frequency, phase, and slice encoding techniques. τ2 rf
n read(z)
g
tr
time
Figure 23: Pulse sequence schematic for multi-echo, spatially resolved T2 mapping [167]. The thin and thick vertical lines represent 90◦ and 180◦ rf pulses, respectively. A series of n CPMG-like echoes are acquired in the presence of a read gradient of amplitude g and duration tr . A T2 decay is associated with each k-space point (and hence pixel after Fourier transformation). The read gradient can be applied on any one axis; multiple directions must be scanned in separate experiments. Acquiring nD T2 maps requires the addition of phase-encoded gradients. The addition of bipolar phase-encoded gradients extends the multi-echo sequence to 2D. A third dimension is added by making the rf pulses slice-selective in the MSME sequence [8].
An alternative method of acquiring quantitative images is pre-conditioning of the spin ensemble [168, 169]. Rf pulses applied prior to the main imaging sequence are used to encode relaxation time or diffusion. The image acquisition is repeated for different pre-conditioning states, so that the voxel intensity changes as a function of T1 , T2 or D over a series of images. If the relaxation rate is slow relative to the image acquisition time, this approach may be used to generate quantitative images from fast imaging protocols such as the rapid acquisition with relaxation enhancement (RARE) sequence [170, 171]. 4.4.5. Pure phase-encoded imaging Another method of obtaining quantitative images is single point imaging (SPI) [172]. In SPI, each k-space point is acquired at a fixed time tp after a rf excitation pulse in the presence of a phase-encoded gradient; see Fig. 24(a). Accordingly, 69
the T2∗ relaxation weighting is constant across k-space and therefore does not distort the profile. If the acquisition time tp of each k-space point is short compared to T2∗ , then the resultant profile is considered quantitative. As such, SPI is well suited to the measurement of broadline samples. SPI is time consuming because each kspace point is acquired in a separate scan. Accordingly, SPI has been improved in the form of single point ramped imaging with T1 enhancement (SPRITE) [173] which uses repeated applications of small tip angle rf pulses to acquire a complete k-space raster in a single scan; see Fig. 24(b). This pulse sequence relies on short T1 relaxation times to continually restore a fraction of the spin ensemble to the z-axis, thus providing signal over a long acquisition window. SPRITE offers a significant reduction in experiment duration compared to SPI, although the required high gradient duty cycle does mean some commercial hardware struggles with the necessary gradient waveforms, which can last for several seconds. Similarly, commercial hardware may fail to deliver suitably short rf probe response times (probe dead time); ideally, the time between rf excitation and detection should be tp ∼ 10 µs. As SPI and SPRITE data are acquired in the presence of a phase-encoded gradient, the frequency bandwidth limit given in eq. (29) applies. SPRITE is a robust and reliable method for imaging a wide range of materials [13, 173] although application to very low magnetic field strengths (B0 < 0.1 T) is restricted by the inherent poor SNR arising from the the small tip angle rf pulses and wide frequency bandwidth. SPRITE is inherently insensitive to sample chemistry due to the lack of relaxation time weighting in the images. Methods for introducing chemical sensitivity into SPRITE through pre-conditioning have been explored for separation of oil and water signals [174], although these techniques are not straightforward to implement.
70
rf
a
tp phase(x, y, z)
time tp rf
n
b phase(x, y, z)
Figure 24: Pulse sequence schematics for (a) SPI and (b) SPRITE. The rf excitation in (a) is via a 90◦ pulse, and in (b), via a small tip angle pulse. The usual implementation involves the acquisition of a single point at time tp after the rf pulse. Multiple points may be acquired and adjusted to a constant FOV using a z-chirp transform. The phase-encoded gradients are ramped independently between ±g(max) in either 1, 2, or 3D. For (b) SPRITE, the gradients are incremented on each loop to sample n k-space positions in a single scan.
The pure phase-encoded nature of SPRITE lends itself readily to k-space sampling schemes other than rectilinear grids. For example, SPRITE data may therefore be acquired with spiral (2D) or conical (3D) sampling schemes [175]. The majority of the intensity in an image is acquired at low k positions. Therefore, under-sampling of high k positions reduces both the image fidelity and acquisition time but does not reduce the quantitative nature of the image. In materials imaging there is a bias towards achieving quantification, as opposed to high spatial resolution, which favors this type of undersampling. The SNR of SPRITE is improved by sampling the origin of k-space first, as encompassed in the 1D centric (double half-k) SPRITE acquisition, where the phase-encoded gradients are ramped from g = 0 to +g(max) in one scan, and g = 0 to −g(max) in the subsequent scan. When using complicated sampling schemes (such as 3D cones) the origin of k-space may be sampled multiple times, provid-
71
ing a more accurate total image intensity. Furthermore, multiple points may be sampled after each rf excitation in both SPI and SPRITE, as indicated in Fig. 24. However, as each point has a different associated tp , a series of images – each with a different field of view determined by tp according to eq. (31) – is obtained. The fields of view are standardized using the z-chirp transform [176–178]. Noise observed in the time domain signal can also be reduced by customizing the digital filter bandwidth used to sample each k-space location [179], although this method can be difficult to implement depending on the available hardware. a
τ2
rf
n
phase(x, y, z)
tp
b
τ2
τ20
tp
n−1
rf
g phase(x, y, z)
time
Figure 25: Pulse sequence schematics for (a) the original SESPI sequence and (b) the modified SESPI sequence. The thin and thick vertical bars represent 90o and 180o rf pulses, respectively. The phase-encoded gradients are ramped independently in either 1, 2, or 3 dimensions. In (a) the bipolar phase-encoded gradients may be incremented on each echo to sample a line of k-space in a single scan (turbo SESPI), or incremented on sequential scans to provide a T2 map. The modified SESPI sequence (b) only provides a T2 map.
An alternative to SPI is the spin echo single point imaging (SESPI) sequence 72
[180, 181], shown in Fig. 25(a). Bipolar phase-encoded gradients of amplitude ranging over ±g(max) are applied for a time tp around each echo. Only the echo amplitude is acquired in the SESPI experiment to provide a single k-space point. As the signal is not acquired in the presence of a gradient, there is no FOV limitation on the minimum frequency bandwidth that can be used. A narrow-band digital filter provides an additional noise reduction over SPI. The advantage of this SESPI sequence is the simplicity of the spin dynamics: the spin ensemble is coherent during the rf transmit and receive intervals. However, imaging artifacts will occur if the positive and negative gradient lobes are not matched precisely. Like SPI, the phase-encoded gradients may be applied on any axis to generate a 2 or 3D image. If the phase-encoded gradients are incremented on every echo, a line of k-space is traversed in a single scan; this acquisition mode is referred to as turbo-SPI [180, 182]. Alternatively, the phase-encoded gradients may be incremented on each scan to provide a CPMG-like decay, and hence T2 distribution, for each voxel [181]; 2D T2 mapping by this method is discussed in [183]. A modification to the SESPI sequence for short T2 samples is described in [184]. The modified SESPI sequence, Fig. 25(b), has only a single phase-encoded gradient of variable amplitude ranging over ±g(max) and duration tp between the leading excitation and refocusing rf pulses. The position of the gradient pulse allows the subsequent echo amplitudes in the CPMG-like echo train to be acquired with a reduced echo time te0 = 2τ20 . However, the spin dynamics are more complicated than in the standard SESPI sequence. To overcome the phase shifts that accumulate by repeated reversal of the incoherent spin ensemble, the xy-16 phase cycle [185] is applied to the 180o rf pulses along the echo train. Consequently, only the decay described by every 16th echo is a good approximation of the T2
73
relaxation behavior. The effective echo time for the determination of accurate T2 relaxation rates therefore becomes te0 = 32τ20 if accurate T2 relaxation rates are to be determined. Additionally, composite rf pulses can be used to improve the quality of the echo shape [184, 186]. These long rf pulses further limit the minimum echo time achievable. Notwithstanding, the direct implementation, as shown in Fig. 25(b), provides a reasonable T2 decay when fitting all the echo amplitudes. The greater SNR achieved with SESPI sequences (compared to SPI and SPRITE) makes them ideal for implementation at low field. 4.5. Stray field experiments Stray field measurements of relaxation time, diffusion, and image profiles are similar to those conducted in homogeneous field, bench-top instruments. However, there are a few important differences to the pulse sequences and analysis that need to be considered when using stray field devices. Of general significance we note that 1. All rf pulses are inherently slice-selective in the presence of the field gradient, resulting in a SNR reduction due to the poor rf probe filling factor. 2.
19 F
signal contamination will occur in 1 H experiments if fluorinated com-
ponents are used for sample containment or as part of the magnet assembly. Here we describe pulse sequences appropriate for use in stray fields; data analysis will be discussed in Section 5. 4.5.1. Relaxation time measurements When the CPMG pulse sequence, as illustrated in Fig. 17, is applied in the presence of an inhomogeneous magnetic field the accuracy of the T2 measurement is influenced by off-resonance spin dynamics [76, 77, 187–189]. This is 74
because the application of the rf pulses required to generate a CPMG echo in a grossly inhomogeneous magnetic field results in multiple coherence pathways that contribute to the observed signal. For off-resonance spins, the magnetization following the excitation 90◦ pulse will contain contributions from spins with a phase shift that is approximately proportional to their offset frequency ∆ω0 . The phase shift is removed by allowing the spins to precess for a time equal to [190] cos (ω1t90 ) − 1 2t90 →− , ω1 sin (ω1t90 ) π
(34)
assuming all spins experience the nominal rf field such that ω1t90 = π/2. Consequently, the time between the leading 90◦ and 180◦ pulses in a CPMG sequence applied in a grossly inhomogeneous field should be τ2 − 2t90 /π; the timing be-
tween subsequent 180◦ pulses remains at 2τ2 . Depending on the quality of the B1
field, the exact time shift can be slightly different from the theoretical prediction and should be calibrated empirically. H¨urlimann demonstrated that the modified timing results in a 1.2 dB (26 %) improvement in SNR. The performance of the CPMG sequence in the stray field is also improved with the use of composite pulses [186, 191]. Another important consideration when using CPMG experiments on a stray field system is the influence of diffusion. For static (or very slowly diffusing) solids, the CPMG sequence provides an accurate T2 measurement [192]. However, most samples measured in stray field experiments are liquid or liquid-like, so an effective transverse relaxation time T2,eff is observed. Diffusion in a weak (background) gradient of amplitude g0 during a CPMG experiment results in a signal nte 1 2 2 b (nte ) 3 = exp − exp − γ g0 Dnte . b(0) T2 12 75
(35)
However, in the presence of a strong gradient off-resonance effects ensure that n coherence pathways contribute to the nth echo amplitude; Goelman and Prammer showed that in this case the contribution from diffusion in the nth echo will be [192] n
1 n 2 2 3 ∑ exp − 2 3 + n − 1 γ g0Dte . 1
(36)
It is clear, therefore, that determining the expected signal decay over an entire CPMG echo train is extremely complicated. H¨urlimann and Griffin provided a generalized form of eq. (36), viz [77]: Z∞ nte b (nte ) = exp − dg0 f (g0 ) b(0) T2 −∞ 1 2 2 3 × exp − E γ g0 Dnte , 12
(37)
where f (g0 ) is the distribution of background gradients and E is an “efficiency factor” that depends on the off-resonance effects observed empirically as a reduction in the signal intensity. On stray field instruments the influence of diffusion on the observed magnetization decay is significant, especially for bulk samples of rapidly diffusing species such as water. Under these circumstances, accurate T2 relaxation time measurements are difficult, if not impossible. There are alternatives to the CPMG sequence for stray field measurements. A similar multi-echo measurement of relaxation time is achieved with a quadrature echo train using the Ostroff-Waugh pulse sequence [193]. The quadrature echo sequence, illustrated in Fig. 26, is based on the “solid echo” 90◦x −90◦y −echo pulse sequence that forms an echo by refocusing dipolar interactions [194]. In a stray field experiment, any two rf pulses will generate an echo due to the distribution of B0 and B1 . The quadrature echo can provide improved signal compared to CPMG 76
τ2
n
time
Figure 26: Schematic of the quadrature echo (or solid echo) pulse sequence for T2 relaxation time measurements in short T2∗ systems. The pulse sequence contains only 90◦ rf pulses, denoted by the thin vertical bars. The echo time te = 2τ2 ; the echo maximum forms a time τ2 after the 90◦ pulse. The repeated application of 90◦ refocusing rf pulses generates a train of n echoes. When studying solids (or other short T2∗ materials), a FID (latter half of the nth echo) is acquired and Fourier transformed to provide a chemical spectrum; the process is repeated for a range of n. When using the quadrature echo pulse sequence to overcome B0 inhomogeneities (e.g., stray-field measurements), where chemical resolution is not available, a few data are acquired at the top of each echo (or encompassing each echo, depending on line width) to determine the maximum signal intensity. Following this protocol, an entire echo train is interrogated in a single scan.
in very strong gradients as long as τ2 ≤ T2,eff . Like the CPMG sequence, a shorter
τ2 provides a slower effective relaxation decay; however, the signal intensity in the quadrature echo train is determined by a mixture of T2 and T1ρ , where T1ρ is the longitudinal relaxation time in the rotating frame (i.e., relaxation in the B1 field) [195]. Therefore, quantification of signal amplitudes obtained from quadrature echo sequences can be unreliable. In general the T2∗ associated with the measurement of materials using unilateral NMR devices will be very short regardless of sample. As an echo comprises two FID decays, back-to-back, narrow echoes are observed. It is therefore important to calculate the rf timing accurately in these experiments. Failure to account for, say, filter group delays (see Section 3.3), can result in the echo being located outside of the acquisition window.
77
4.5.2. Diffusion measurements The diffusion pulse sequences described in Section 4.2 utilize fixed experimental timings and variable gradient amplitude. In stray field instruments, the inherent magnetic field gradient ∇B0 is used to measure diffusion. We refer to these pulse sequences as fixed field gradient (FFG) experiments. To provide signal attenuation, the spin echo time τse is incremented, thereby increasing both the observation time t∆ and the gradient duration tδ . However, the relaxation weighting also increases and this needs to be considered in the data analysis. Carr and Purcell noted that spin echoes acquired in the presence of a weak gradient can be used to determine a diffusion coefficient [92] according to eq. (35). However, as indicated by eq. (37), the contribution from strong, spatially variant gradients (as found in many STRAFI devices) complicates the interpretation of such diffusion data. Rather, it is useful to consider stray field diffusion experiments in terms of the D-T2 correlation described in Section 4.3.2 in order to separate the contributions from diffusion and relaxation [139, 196, 197]. By acquiring a CPMG echo train (fixed echo time) after the diffusion encoding (variable echo time), the effects of relaxation and diffusion are decoupled. In practice, a double spin echo is acquired in the so-called “FFG-2SE” experiment to provide motion compensation; the pulse sequence is illustrated in Fig. 27. A comparable PFG-2SE diffusion experiment, appropriate for use on bench-top instruments, is available [198]. The only significant difference is found in the CPMG acquisition: T2,eff is measured in the presence of the FFG, whilst T2 is observed in the PFG case. Data acquired using the FFG-2SE experiment will have the form b (tse , nte ) = b(0, 0)
Z∞
−∞
1 3 dg0 f (g0 ) exp − γ 2 g20 Dtse 6
78
τse
te g0
tδ
t∆
tδ + te /2
n
time
Figure 27: Pulse sequence schematics for the stray field diffusion-relaxation correlation FFG2SE experiment. A background (fixed) gradient of amplitude g0 is present at all times; the initial echo interval tse ≡ t∆ (tse /2 ≡ tδ ) is incremented in successive scans to encode diffusion. The spin echoes are narrowed by the presence of the gradient and the CPMG echo train measurements T2,eff .
2tse + nte , × exp − T2,eff
(38)
assuming a background gradient distribution, where it is clear that both the relaxation and diffusion contributions depend on the initial echo time separation, tse . Therefore, as we shall see in Section 5, the analysis of FFG D-T2,eff correlation experiments differs from that used for PFG D-T2 correlations. 4.5.3. Stray field imaging Stray field imaging (STRAFI) [199] is achieved in one of two ways. If the field gradient is uniform across the sensitive volume, acquired echoes are considered to be frequency-encoded, see Section 4.4.1, and a profile is obtained by Fourier transform. This is the method of choice for studying planar samples in the GARField magnet [49]. Alternatively, if the sensitive volume is well defined, a profile may be obtained by moving either the sample or magnet in order to step the sensitive volume through the region of interest. A single data point (or projected total signal from a CPMG-like decay) is obtained at each spatial location and so a profile is constructed directly from the data; no FT processing stage is required. This SPI-like acquisition is the method used with the Profile MOUSE [61] and Surface GARField [44]. In both cases it is usual to acquire an echo train, thereby providing the T2,eff relaxation time as a function of spatial location. STRAFI is 79
ideally suited to 1D profiles, and can achieve a higher spatial resolution across a small sample than is readily achievable with standard low-field pulsed gradient imaging technology. To perform 2D or 3D imaging with STRAFI requires the sample to be rotated relative to the direction of the magnetic field gradient [48]. However, as the primary use for STRAFI in industrial process design has been the investigation of thin films and coatings, a 1D profile is usually sufficient [62]. 4.6. Spectroscopy In homogeneous magnetic fields, the Fourier transform (FT) of the FID provides a chemical spectrum. Individual nuclei within a molecule experience distortions in the background magnetic field created by their local electronic environment. This chemical shielding results in resonant spins at different positions within a molecule precessing at subtly different frequencies. The spectral line associated with each unique local environment is therefore observed to be shifted relative to the spectrometer reference frequency. For example, the 1 H spectrum of ethanol will contain lines associated with the CH2 , CH3 , and OH functional groups. The size of the chemical shift (in Hz) is determined by the magnet field strength. Furthermore, each of the spectral lines will be split into multiplets due to the J-coupling (indirect dipolar coupling, or scalar coupling) interaction between NMR active nuclei. For example, the CH3 line of ethanol is split into a triplet due to the presence of the neighbouring hydrogen nuclei. Unlike chemical shifts, the J-coupling interaction is independent of the background magnetic field strength. Many excellent texts are available on the subject of chemical spectroscopy; see for example [4, 200, 201]. In non-uniform magnetic fields, 1 H spectra are typically dominated by the line broadening caused by the ∆B0 term in eq. (12) and so spectroscopy of large samples is not practical with current permanent mag80
net technology. Permanent intermediate-field magnets capable of spectral resolution over small liquid samples (diameter ∼ 5 mm) are available [4], based on the standard designs for bench-top instruments shown in Fig. 2. Spectroscopic resolution has also been achieved with specialized single-sided magnets based on the MOUSE concept [53–55]. The quality of chemical spectra acquired with permanent magnet technology will suffer from poor SNR compared to similar experiments conducted at high field. It is recommended that time-domain fitting techniques (Section 5.2), rather than a conventional FT analysis, be applied to the data. 4.7. Fast field cycling Relaxation times are dependent on magnetic field strength and so information on processes that influence spin relaxation, notably molecular dynamics, can be accessed by variable field measurements. Fast field cycling (FFC) technology provides a method of determining the frequency dependence of the longitudinal relaxation time, T1 (ω0 ); a plot of T1 versus Larmor frequency is known as a T1 dispersion measurement. Commercial FFC hardware is based on electromagnet technology where a B0 coil is used to vary the fixed field between, typically, B0 ∼ 10 µT and 1 T. The minimum field will depend on the local environment and can be improved (reduced) with magnetic shielding. Acceptable SNR is achieved by polarization and detection of the spin ensemble near the upper limit of the magnetic field strength, corresponding to 1 H frequencies of ν0 ∼ 10 MHz. A typical FFC T1 relaxation time pulse sequence is illustrated in Fig. 28. The observed relaxation decay is described by τ1 b (τ1 ) = beq (∞) + b (0) − beq (∞) exp − , T1 81
(39)
where beq is the equilibrium magnetization in the Brel field as τ1 → ∞, b(0) is the magnetization at the start of the relaxation time (end of the polarization time), and
τ1 is the variable relaxation time in the Brel field. In practice, eq. (39) should be modified slightly to account for the finite switching times of the field as discussed in [202]. rf time τ1 Bpol B0
Bacq Brel
Figure 28: Schematic of a standard fast field cycling pulse sequence. The B0 field strength is varied during the course of the experiment: initially, the spin ensemble is polarized at Bpol . Then the field is shifted and the spins are allowed to relax towards equilibrium in Brel . After time τ1 , the field is shifted again to Bacq , at which point an rf excitation pulse is applied and a FID recorded. The experiment is repeated for different values of τ1 to obtain the T1 relaxation rate in the Brel field. The polarization Bpol and acquisition Brel are the same in each experiment. Brel is varied over different experiments to obtain the T1 (ω0 ) dispersion curve.
Field cycling is used to study the relaxation effects of paramagnetic species and molecular dynamics. As the T1 relaxation time is sensitive to molecular motions at the Larmor frequency, a hindrance of motion through adsorption results in a change in the relaxation rate; therefore, field cycling is of particular use for studying liquids in porous materials [203]. We have chosen to mention FFC here as a special case of low-field relaxation experiment, although, being a specialized technique requiring unique hardware, it is not widely used in industrial process control. Comprehensive reviews on FFC and its applications are available elsewhere [202–205]. 82
4.8. Ultra-low field and Earth’s field Another specialized area is ultra low-field NMR using superconducting quantum interference devices (SQUIDs) to detect signals in microtesla fields. The extreme sensitivity of SQUIDs enables the detection of NMR signals in such weak magnetic fields. In much of the literature, SQUIDs are referred to simply as “lowfield” instruments. The potential of SQUIDs as pre-clinical instruments has been explored and a review of the applications to medical MRI is presented in [32]. The properties of SQUIDs suggest they would be appropriate for use in industrial environments, but the requirement for cryogenic cooling counters one of the advantages of NMR with permanent magnets. To date, the use of SQUIDs for process control has been limited. Using the definition of ultra-low field in Section 1, this range also includes Earth’s field NMR, where signal is detected directly in the local magnetic environment [33]. The magnitude of the Earth’s magnetic field varies depending on latitude, being stronger at the poles than at the equator, and will have a value on the order of a few tens of microtesla. An extreme yet common use of Earth’s field NMR is the detection of groundwater aquifers [206]. A more practical approach using conventional rf probe technology requires that the spin ensemble first be polarized in a stronger field (typically a few millitesla) generated using a polarizing coil [207]. Earth’s field NMR has the advantage that the local field is (ideally) very homogeneous, although in practice it is difficult to experience the Earth’s field in a laboratory environment, especially when other NMR magnets are nearby [207]. Earth’s field NMR has achieved success in two distinct arenas: (1) applications in remote locations [208–210] and (2) J-coupling spectroscopy [94, 211–213]. The very weak, yet uniform, magnetic field yields narrow spectral lines but chemical
83
shifts (being field dependent) are negligible. However, as J-coupling interactions are independent of field strength, line splitting is still observed, allowing structural information to be obtained [33, 94]. J-coupling spectroscopy is of course well known at zero-field [214–216], and at near-zero-field whereby the application of small magnetic fields restores the loss of nuclear species information encountered at zero-field [217]. Low-resolution imaging is also a possibility with Earth’s field NMR [218]. Despite the obvious advantages of NMR without a permanent magnet, Earth’s field NMR has not achieved great success in industrial applications; for example, attempts at well-logging in the Earth’s field did not develop into a commercial service. The poor SNR achievable, even with pre-polarization, has thus far limited the use of Earth’s field NMR in industrial process control, and it is largely viewed as a novelty instead of a serious replacement for high-field magnets: Magritek sell a commercial Earth’s field NMR system primarily as an educational demonstration of NMR [207]. Notwithstanding, recent advances in data processing (such as Bayesian analysis, see Section 8.3.1) are enabling useful experiments on Earth’s field instruments [219] and may provide a route to industrial applications. 5. Data Analysis Early low-field spectrometers were crude analogue systems. The data quality did not warrant intensive processing and so many experimentalists often resorted to studying only the magnitude of the NMR signal. Nowadays, low-field spectrometers are complicated digital systems, fully capable of reproducing both the experiments and data quality of their high-field counterparts. Careful pulse sequence timing and appropriate use of digital filtering, as discussed in Section 3, 84
will yield time domain data that deserves equal care in processing to obtain the most from the experiments. Here we describe various stages of data processing and fitting necessary for reliable interpretation of low-field results. 5.1. Pre-processing for optimum results There are several stages of initial processing that can be applied to low-field NMR data prior to fitting. The exact choice of pre-processing method will depend on the data acquired. We list possible pre-processing stages and note where they become important. 5.1.1. Zeroth-order phase correction All NMR data contain a zeroth -order phase shift. Ideally the phase of the receiver would match that of on-resonance spins, so the NMR data are acquired entirely in the real channel. However, it is common for the spectrometer acquisition chain to introduce small (constant) phase shifts. These phase shifts can be attributed, for example, to filters [83] and lengths of cable, and manifest as a portion of the signal rotated into the quadrature channel. The total phase shift (angle)
θ (in radians) is simply the angle of the data in the complex plane. The phasecorrected signal is therefore obtained by multiplying the acquired signal with the phase rotation exp {− jθ }; this correction can be applied to any data in the time domain. The phase correction has the effect of rotating the signal into the real channel. In the case of FIDs or CPMG decays, the phase angle is obtained from an average of the first few data points. An example of a zeroth -order phase rotation applied to CPMG data is demonstrated in Fig. 29(a) and (c). In imaging, the phase angle must be obtained at k = 0 m−1 to ensure the correct zeroth -order phase shift is applied. There are other examples where the data are phase-sensitive, such as 85
inversion recovery data where the zeroth order phase shift must be obtained from the data acquired with the longest recovery time and applied consistently to all the other data; otherwise the correct inversion will not be observed. signal / arb. units
a 0.5 0 −0.5
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Figure 29: Simulation of zeroth -order phase correction applied to CPMG data. The recorded echo amplitudes (a) include a zeroth -order phase shift, so some of the signal is observed on the quadrature channel. If the magnitudes of the data are used (b), the noise adds to form a nonzero baseline (dashed horizontal line). However, if the data are phase rotated (c), the real channel contains all the signal and has a baseline with zero mean (dashed horizontal line). The (real) CPMG data may be compressed using window sums (d) to reduce the computational requirements of curve fitting.
Historically, it has been convenient on low-field instruments to store the magnitude of the complex data. When qualitative images are sufficient, magnitude data may be used. For quantification it is necessary to apply a zeroth -order phase correction. The importance of phase correcting relaxation data is highlighted by the simulation in Fig. 29: the magnitude of a CPMG decay (b) contains a non-zero baseline generated by summing over the noise. Fitting the magnitude data will result in an apparent long T2 component due to the baseline offset. By applying a 86
phase correction and fitting the real channel in Fig. 29(c) the mean of the baseline is now zero (assuming Gaussian white noise). A fit to the phase-corrected data will provide an accurate T2 relaxation time. 5.1.2. Data compression We have taken the opportunity to include an example of window averaging (data compression) on the simulated CPMG data in fig. 29. Window averaging is a useful technique when data are noisy and reduces the computation required for fitting large quantities of data. The phase-corrected real channel of the data, Fig. 29(c), is selected and divided into bins whose sizes increase logarithmically [198]. The data points in each bin are averaged to provide the compressed data shown in Fig. 29(d). In this example, an input of n = 1024 echoes is compressed to n˜ = 16 elements (where ˜ represents a compressed value). It is often necessary to adjust the width of the initial bins to accommodate an integer number of data. The SNR of the compressed data varies from point to point due to the increasing widths of the bins. The compressed data are accompanied by a weighting vector w that describes the variation in SNR across the data; the weighting vector is required to fit the data. We note that if the echoes are narrow (such that the entire envelope of each echo is observed), as in STRAFI experiments, then the intensity of each echo needs to be determined as part of the pre-processing stage. Depending on the SNR, simply determining the value of the highest signal intensity datum within an echo may not be sufficient; similarly, if the echo is narrow and the dwell time comparatively long, there may not be a single datum corresponding to the peak intensity of the echo. In such cases, it is better to fit each echo to determine the intensity. If the echoes happen to be described well by a simple function, e.g., a 87
Gaussian, then a direct fit to each echo is practical. Alternatively, the exact functional (or analytic) form of the echo could be derived to provide an appropriate fitting function [77, 220]. When the echo shape is complicated, or when SNR is low, direct fitting of each echo becomes impractical. Under these circumstances a generic echo shape is defined by summing over the first few echoes in an echo train. The relative intensity of each echo in the train is then determined by vector division of the echo with the generic echo shape. This straight-forward method will fail if the position of each echo within the acquisition window changes along the echo train; such timing errors can be eliminated through careful pulse sequence design. 5.1.3. First-order phase correction A first-order phase shift observed in the frequency domain corresponds to a time shift in the time domain data. First-order phase shifts manifest when data are Fourier transformed. The most common source of a first-order phase shift in NMR data occurs when the initial data point is not acquired at time t = 0 s. An extreme example is presented in Fig. 30: (a) a frequency-encoded spin echo has been acquired in the presence of a read gradient and stored with the timebase running sequentially from t = 0 to 1 ms. The FT of these data is illustrated in Fig. 30(b) where a first-order phase shift (oscillation) is apparent throughout the top-hat profile. If the data are time shifted, Fig. 30(c), so that t = 0 ms corresponds to the center of k-space, the FT in Fig. 30(d) is a top-hat entirely in the real channel (no first-order phase shift). Less extreme first-order phase shifts are encountered commonly in frequency-encoded spin echoes as it is difficult to obtain a datum corresponding to k = 0 m−1 . Such small phase shifts can be mediated by interpolating the data to provide an approximation for the k = 0 m−1 datum. A 88
similar problem arises in spectroscopy when a FID does not start at t = 0 s. Simply assuming that the initial data point corresponds to zero time will introduce a first-order phase shift in the resulting spectrum and prevent quantitative determination of peak areas. It is preferable to project the data back to zero time using concepts of linear prediction, which we will discuss in Section 5.2. In the case of relaxation time measurements, first-order phase shifts can be neglected as the data are fitted directly on the correct time base. 0.4
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Figure 30: Simulation of the Fourier transform of a frequency-encoded spin echo for a top-hat profile. The echo (a) is stored with the initial data point at t = 0. When the FT is applied (b), a phase shift is observed in the profile. However, if the echo is time shifted (c) so that the echo maximum corresponds to t = 0, the FT (d) provides a profile with the correct phase. The echo is simulated with SNR = 100; in each case the blue line indicates the real channel and the red line indicates the quadrature channel.
5.2. Time-domain fitting for spectroscopy Liquid phase spectroscopy relies on the Fourier transform of FIDs into chemical spectra. It is typical in high-field experiments for the SNR to be good and 89
probe plus filter dead times to be short. Small zeroth - and first-order phase corrections, plus baseline corrections, are applied to the spectrum after the FT stage. However, intermediate-field spectroscopy systems have relatively poor SNR and probe dead times may be 100 µs or more, resulting in significant spectral phase distortions that are difficult to correct. Examples of these situations are presented in Fig. 31(a – f). An intermediate-field measurement (with “poor” SNR) is equivalent to Fig. 31(a,b). If a FT is applied to recorded data including a significant dead time in the FID, Fig. 31(c), then the spectrum is distorted Fig. 31(d); this distortion manifests as oscillations in the baseline. However, if the initial data are simply discarded, Fig. 31(e), the spectrum in Fig. 31(f) contains a significant firstorder phase shift that is difficult to correct at a post-processing stage. The areas under the peaks are no longer quantitative. An alternative to the FT is a direct fit to the data in the time domain [221]. An example FID and spectrum reconstructed from the fit parameters are shown in Fig. 31(g,h). We proceed to describe one method of time domain fitting for spectroscopy data. Time-domain fitting is based on the concepts of linear prediction (LP), used to project FID decays forward and backward to overcome baseline and phase artifacts in the corresponding spectrum. Numerous LP methods are available in the literature; a review covering several LP methods is given in [222]. Although several possible methods exist, Hankel total least squares (HTLS) [223] provides a robust, computationally efficient solution that is appropriate for intermediatefield data, which we will summarize here. This non-iterative method is based on the Hankel singular value decomposition (HSVD) [224] method but provides enhanced performance through an additional filtering stage. Ideally, only the signal contained within the data is fitted and not the noise. The FID is modeled as a sum
90
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Figure 31: Demonstration of time domain fitting of simulated intermediate-field spectroscopy data, adapted from [222]. The entire FID (a) would ideally be acquired (SNR ≈ 100) and Fourier transformed to provide a real spectrum (b). In practice there is a dead time on the FID acquisition (c); here tDE = 100 µs. This dead time results in baseline distortions in the spectrum (d). The FID can be time shifted (e) so the initial zero amplitude points are ignored, but this introduces a firstorder phase shift in the spectrum (f) and results in incorrect peak intensities. A better approach is to fit the data in (c) in the time domain using the concepts of linear prediction to determine the frequency, T2∗ , and intensity of each component, from which a FID (g) and spectrum (h) may be reconstructed. In each plot the blue line indicates the real channel and the red line indicates the quadrature channel.
91
of sinusoids, damped exponentially, according to D
bn =
∑ {ad exp [ j (ϕ0 + ϕd )]} exp [(−rd + j2π fd )tn] ,
(40)
d=1
for the number of data n = 0 . . . N − 1. D is the total number of sinusoidal components (model order), fd are the component frequencies, rd are the component decay rates (effectively 1/T2∗ ), ad are the component amplitudes, and the component phases are described by (ϕ0 + ϕd ), for d = 1 . . . D. Typically, ϕd is zero and the phases of all components can be described simply by the zeroth order phase shift ϕ0 . The data are uniformly sampled at times t = (n + n0 ) ∆t, n = 0 . . . N − 1,
(41)
where ∆t is the constant sampling interval (equivalent to dwell time) and n0 ∆t describes the time-shift of the first datum relative to the origin (the dead time). This model enforces the assumption that all the spectral lines are Lorentzian. This assumption is generally valid for liquid and liquid-like samples generally measured in spectroscopy; if broadline components (solids or macromolecular species) are known to be present, multiple Lorentizan peaks will be fitted; these can be summed by the experimentalist. If the spectral lines are known to be Gaussian or Voigt (sum of Lorentizan and Gaussian) peaks, an alternative fitting approach should be adopted where a different model may be assumed. A review of time-domain fitting methods has been presented in [225]. The choice of model order D is a crucial step in time domain fitting. The model order is selected using a model information criterion, such as the Akaike information criterion (AIC) [226] which is appropriate for FIDs acquired with a large number of data points, although alternatives can be found in the literature [227]. In SVD-based analyses, the model order corresponds to the number of 92
significant singular values (those singular values corresponding to signal rather than noise) in the SVD of the Hankel data matrix b0 b1 b2 · · · bO−1 b1 b2 · HM×O = b2 , · .. .. .. . . . bM−1 · · · bN−1
(42)
consisting of the measured data bn , n = 0 . . . N −1, with dimensions M, O > D and
satisfying M + O = N + 1. Note that H contains complex data. The SVD yields H = USV† ,
(43)
where † denotes a conjugate transpose and S is a diagonal matrix containing the singular values in descending order. Regularization is achieved by truncating the SVD matrices to D significant values, as determined by the chosen information criterion. If N is large, the SVD stage may be achieved efficiently with a truncated eigenvalue decomposition of H† H [223, 228]. ˆ is obtained (where ˆ denotes an estimated or fitted paraNext, the solution O meter) that satisfies + U− K O ≈ UK ,
(44)
− where U+ K and UK are obtained from UK by omitting either the first or last row of
ˆ is obtained in a total least squares sense, then the SVD matrix, respectively. If O ˆ = −V12 V−1 where the matrices V12 and V−1 are determined from the SVD of O 22 22 − + the augmented data matrix UK , UK as described in [223]. The eigenvalues lˆd of ˆ are O
! lˆd = exp −ˆrd + j2π fˆd ∆t , d = 1 . . . D, 93
(45)
from which the estimates of frequency fˆd and decay rates rˆd are obtained for the D components. Finally, the corresponding amplitude terms aˆd exp [ jϕ0 ] are obtained from a least squares fit of the damped sinusoids to the data. The estimated fre∗ , and amplitudes aˆ can be used to reconstruct quencies fˆd , decay rates rˆd = 1/T2,d d
a spectrum if required, noting that aˆd are the equal to the areas under the spectral peaks, not the heights. A simulated result is given in Fig. 31. 5.3. Time-domain fitting for relaxation and diffusion When relaxation time or diffusion data are expected to contain only a few components (between 1 and 3) then the most reliable way to the fit these data is with a straightforward least-squares fit to the expected exponential function. Appropriate functions are given throughout Sections 4 and 5 with a summary given in Table 1 (located in Section 5.4.2). An alternative approach that has been used for relaxation time analysis is an exponential “stretch” fit [229]. For example, the fitting function for CPMG data would be modified from eq. (14) to ( ) b(t) nte ζ = exp − , b(0) T2
(46)
where ζ defines the shape of the relaxation curve relative to a perfect exponential (ζ = 1). This fit is useful for monitoring changes in a sample property, particularly when the data are non-exponential, such as in the presence of a stray field gradient [230]. If the relaxation time or diffusion data contain numerous components, it is better to fit a distribution of component amplitudes. This situation is likely to occur in studies of porous materials or emulsion droplets. We discuss distribution fitting methods next.
94
5.4. Distributions of relaxation and diffusion (1D) 5.4.1. Data fitting To generate a distribution of relaxation time or diffusion coefficients it is necessary to solve the Fredholm integral equation that models the expected relaxation behavior. In general, the NMR signal is described by the 1st kind Fredholm integral equation h(t) = h(0)
Z∞
K0 (u,t) p (u) du,
(47)
0
where h(t) is the noiseless signal acquired at an experimental time t, p is the (probability) density distribution governed by the time constants in u, and K0 is the model function (kernel) describing the signal. However, it is typical to solve for u on the log10 scale in the presence of noise e(t) so the acquired data are modeled by b(t) = b(0)
Z∞
K0 (u,t) f (u) d (log10 u) + e (t) .
(48)
−∞
The numerical inversion provides the distribution f (u) rather than the actual density distribution p(u). It is typical in NMR acquisitions for the data to contain insufficient information to determine a unique solution to the distribution in the presence of noise. Consequently, the numerical inversion is an ill-posed problem and so the determination of f (u) is difficult. A more stable result can be obtained by fitting the normalized signal b(t)/b(0), rather than scaling the expected form to match the data. To provide quantitative results, the integral area under the fitted distribution fˆ(u) is rescaled to equal b(0) at the end of the fitting process. Practically, b(0) is rarely measured, and so the datum with maximum signal cor-
95
responding to b(t → 0), is used for scaling instead. The rescaled distribution area
is then b(t → 0) fˆ(u) ≈ b(0), equal to the signal amplitude at the time origin. R
In 1D, the statistical approach of Wilson [231] for solving this first-kind Fredholm integral equation is adopted. In reality, our data consist of N discrete points. We achieve a stable inversion in the presence of noise using regularization, whereby we assume the distribution f (u) is always smooth and positive, and the integral over log10 u is constrained. Accordingly, we wish to determine a functional form of f (u) that minimizes the residuals 2 Z N R = ∑ bn − K0 (u,t) f (u) d (log10 u) n=1
+ α
Z
(H f ) (u) d (log10 u) ,
(49)
where H is the smoothing function and α is a constant (smoothing parameter) that defines the trade-off between smoothness and consistency with the data. Equations (48) and (49) are restated in vector-matrix notation as b = K0 f + e,
(50)
where lower case letters represent vectors, and upper case letters represent matrices. We solve for f using regularization by choosing a solution ˆf ≥ 0 that minimizes the residuals matrix (E + α L) where L is the smoothing operator and the weighted sum of squared errors is E = (g − Kf)† W (g − Kf) ,
(51)
where W is a diagonal precision matrix, required if the data have different weights (e.g., when using window sums of echoes as described in Section 5.1.2). Otherwise, W becomes the identity matrix I and can therefore be ignored in subsequent 96
calculations. An appropriate smoothing function for 1D regularization is the integral over the square of the second derivative of f (u). This choice of smoothing term forces the end-points of the distribution to be zero. As demonstrated by Wilson [231], the corresponding smoothing operator L ≡ f† Qf where Q is symmetric, with dimensions equal to the length of f, and having the form 6 −4 1 0 ··· −4 6 −4 1 · · · 1 Q= 1 −4 6 −4 · · · , 3 [δ (log10 u)] 0 1 −4 6 · · · .. .. .. .. . . . . . . .
(52)
residuals matrix then becomes [231] E + α L = −2b† WKf + f† K† WK + α Q f.
(53)
where δ (log10 u) is the constant width of each interval in the log10 u scale. The
Therefore, a solution ˆf is found that satisfies the quadratic programming problem (QPP)
ˆfα ≡ arg min 1 f† K† WK + α Q f − b† WKf, (54) f≥0 2 for a given value of α ; ˆfα can be determined using an appropriate medium-scale QPP solver. It is also necessary to optimize the value of α to provide a suitable balance between smoothness and accuracy in the solution. Generalized cross validation (GCV) [232] is considered an appropriate method, wherein a GCV score ΦGCV (α ) is calculated such that the minimum score corresponds to the optimum smoothing parameter αopt . An initial guess for α is obtained from Tr K† WK , α1 = Tr [Q] 97
(55)
where Tr[· · ·] denotes the trace of a matrix. An exact GCV score for the constrained minimization of eq. (54) is given in [233]; however, this calculation is computationally inefficient and an approximate GCV score may be obtained instead, based on the unconstrained minimization of eq. (54), as ! ! 1 ˆfα † W b − Kˆfα b − K n ΦGCV (α ) = h i2 , −1 1 1 − n Tr K (K† WK + α Q) K† W
(56)
where n is the number of data. An appropriate update formula for the estimate of
α is α←
h ! −2 † i † K W −Tr K K WK + α Q f† K−1 f
ΦGCV (α ) .
(57)
5.4.2. Kernel functions The kernel functions describe the expected form of the data and are therefore experiment-specific. For NMR experiments in general, the kernel function will be an exponential decay of the form K0 (u,t) = exp {−ut} ,
(58)
where u might be a relaxation rate or diffusion coefficient. A list of kernel functions for the 1D experiments described in Section 4 is given in Table 1. The kernel matrix K0 is generated by inputing all combinations of u and t into the appropriate function. In this way, the kernel matrix provides the expected (normalized) signal amplitude for any set of measurement times and fit parameters. The fitting process outlined in Section 5.4.1 effectively compares these expected values to the actual data in order to determine the most likely set of (smooth) component amplitudes that recreates the measured signal.
98
Experiment
Kernel function
Fitted
Section
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parameter
number
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4.1.1
T1
4.1.1
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4.1.2
D
4.2
n o t Inversion recovery 1 − 2 exp − T1 o n t Saturation recovery 1 − exp − T1 n o CPMG exp − Tt2 n o ! PFG diffusion exp − (γ gtδ )2 D t∆ − t3δ
Table 1: Example kernel functions for 1D relaxation and diffusion experiments. The Section in which experimental details are given is noted in each case.
5.4.3. Limitations and considerations It is important to note that when obtaining a distribution of relaxation time or diffusion coefficients following the method presented here (or related inversion techniques), the result is sensitive to the mathematics of the method. For example, the appropriate degree of smoothing is determined primarily by the SNR of the data and noisy data are smoothed more than “clean” data. However, a very high SNR > 1000 can result in under-smoothing and instabilities in the solution; that is, the fit becomes sensitive to non-idealities in the spin dynamics (imperfect rf pulses, motion/diffusion artifacts) and unphysical peaks will be introduced into the distribution. An ideal SNR is ∼ 100 [143] and white noise may be added to the raw data to provide a smooth but stable solution. To compare the shapes of distributions, it is necessary to ensure that the raw data have been acquired with a consistent SNR. It is better practice to compare peak positions and areas than shape. At low SNR, the noise may provide discrete peaks in a distribution. It is therefore sensible to ignore low intensity peaks (e.g., < 10 % of the maximum intensity) when interpreting relaxation time or diffusion coefficient distributions 99
unless there is a physical reason to expect such a small component to be present. Another factor influencing the resulting distribution is the choice of fitting parameters, specifically the range and number of relaxation time or diffusion coefficients being fitted. Common artifacts are peaks of significant area appearing at the extremities of the fitting range. These peaks arise due to non-exponential behavior in the first or last points of the raw data (such as a baseline offset). To determine whether these components are artifacts, the fit must be repeated with a wider fitting range; artifact peaks will move and remain at the end of the range; these peaks should be ignored in the interpretation. A stable and reliable inversion will be obtained with reasonable SNR and a sufficient number of data points. For the types of dataset encountered in low-field NMR, a useful rule of thumb is to acquire 32 points, appropriately spaced on the time axis [234]. For example, if fitting a sum of exponentials, the temporal separation of the data points should increase logarithmically. In general, fewer data points will give a poor fit, whilst additional points will not improve the quality of the distribution but will slow the computation. For experiments where every point is acquired in a separate scan, such as conventional T1 or D measurement, 32 experiments must be performed. For single-shot CPMG where 1000’s of echoes may be accrued rapidly, it is advisable to window sum the echo train (see Section 5.1.2) in order to reduce the number of data to 32; this pre-processing stage reduces the computational requirements of the QPP without influencing the result. Care should be taken not to over-interpret the data obtained from these fitting procedures (such as assigning significance to every low intensity component [235]) and it is important to understand the limitations of the numerical inversion algorithms. The same artifacts and limitations will apply to the mathematics of
100
2D inversions that we discuss next. 5.5. Distributions of relaxation and diffusion (2D) 5.5.1. Data fitting The process of fitting 2D distributions of relaxation time or diffusion coefficient is conceptually identical to that employed in Section 5.4.1 for 1D data, although a slightly different mathematical approach is required to handle the larger matrices. Full details of current 2D fitting methods are given in a recent review [236] and a discussion of errors in 2D inversions is given in [237]. The 2D data are described by b (t1 ,t2 ) = b(0, 0)
Z∞ Z∞
d (log10 u1 ) d (log10 u2 )
0 0
× K0 (u1 ,t1 , u2 ,t2 ) f (u1 , u2 ) + e (t1 ,t2 ) ,
(59)
where the subscripts 1, 2 denote the first and second dimensions of the experiment, respectively. The 2D data are stacked to form a vector (conversely the fitted distribution is a vector rearranged to form a 2D matrix) so the 2D Fredholm integral equation (59) is expressible in vector-matrix notation according to eq. (50). However, the corresponding K0 matrix will be very large unless the data are compressed (e.g., using window sums) and the solution size limited. The complete K0 matrix can be generated on a 32-bit desktop PC if both the data matrix and solution matrix are reduced to 32 × 32 elements. A precision matrix will be required to account for the data compression and so the optimization problem becomes
2
1/2 ˆf = arg min (60)
W0 (g − K0 f) , f≥0
where k· · ·k indicates the `2 (Euclidean) norm of the vector. 101
A computationally efficient method of solving the 2D inverse problem utilizes the Kronecker product structure of the kernel matrix. If the two dimensions of the experiment are separable (i.e., do not share a common time base) then we may write K0 = K2 ⊗ K1 . The much smaller K1,2 matrices can be computed and handled separately, so there is no need to generate the full K0 matrix. Now the 2D Fredholm integral equation is expressed in matrix notation as B = K1 FKT2 + E,
(61)
where T denotes a matrix transpose. It is usual to compress only the first dimension of the experiment (CPMG echo train) and so we only need include a W1 precision matrix associated with K1 . The optimization problem is
2
1/2 ! Fˆ = arg min W1 G − K1 FKT2 , F≥0
(62)
where k· · ·k now indicates the Frobenius norm of the matrix.
A common solution for reducing the size of the minimization problem in 1/2
eq. (60) is to project the data onto a truncated singular value basis of the W0 K0 matrix. This reduction is achieved by determining the significant singular values 1/2
of the kernel matrix. A singular value decomposition of W0 K0 is performed, equivalent to eq. (43) and detailed in [236]. The number of significant singular values is selected based on the SNR of the data. If the kernel matrix exhibits Kronecker product structure, the SVD of the smaller K1,2 matrices is performed instead, as described in [238]. Following truncation of the SVD, the reduced data ˜ 0. (stacked as a vector) are b˜ and the reduced kernel matrix is K We solve the reduced optimization problem using Tikhonov regularization by introducing a smoothing parameter α , forming the QPP ˆfα ≡ arg min 1 f† K ˜ †K ˜ 0 + α I 0 f − b† K ˜ 0 f. 0 f≥0 2 102
(63)
The reader will notice the similarity between this QPP and eq. (54), except here our smoothing matrix is just the identity matrix I0 . This large-scale QPP is solved using the method of Butler, Reeds, and Dawson [239]. For a given α , we make the substitution ˜ † c, f=K 0
(64)
where c is a vector of fitting parameters such that c=
˜ 0f b˜ − K . α
(65)
The fitting vector is determined by an unconstrained minimization [238, 239] from which the stacked vector ˆf is interpolated. Again, non-negativity is enforced in ˆ As with the the solution. The 2D distribution is obtained by reshaping ˆf → F. 1D inversion, numerous methods are available for determining the optimum value of α . The GCV method has been found to provide a robust, reliable, and fast estimate of αopt ; formulas for the initial estimate of α , GCV score, and α update appropriate for 2D optimization are given in [236]. 5.5.2. Kernel functions The kernel functions that describe 2D data can be grouped into two categories: separable and non-separable. For convenience, we first consider separable kernels. Most 2D relaxation and diffusion measurements fall into this category. The complete kernel functions are obtained by multiplying the 1D functions that describe the expected form of the data in each dimension of the experiment. A list of separable 2D relaxation kernel functions is given in Table 2. The complete kernel matrix K0 provides the expected (normalized) signal amplitude for every combination of the measurement time t1 ,t2 and fitting parameter u1 , u2 . 103
Experiment
Direct kernel Indirect kernel
function K2 n o t T1 -T2 (inversion) exp − T2 n o t T1 -T2 (saturation) exp − T2 n o T2 -T2 exp − Tt2
function K1 n o t 1 − 2 exp − T1 n o t 1 − exp − T1 n o exp − Tt2
Section number 4.3.1 4.3.1 4.3.3
Table 2: Example kernel functions for separable 2D relaxation correlation and exchange experiments. The Section in which experimental details are given is noted in each case.
The separable kernel functions for diffusion-relaxation experiments are a little more complicated to define due to the inherent relaxation that occurs during the diffusion encoding portion of the pulse sequence. If the diffusion timings remain constant, the kernel function is still separable. For example, the APGSTE-CPMG experiment, as shown in Fig. 20, has separable kernel functions (assuming rectangular gradient pulses) of 1 t∆ + tstore − τse tse K2 (T2 , nte ) = − exp − 2 T1 T2 nte × exp − , T2
(66)
and K1 (D, g) =
n 1 tδ o exp −tδ2 g2 D t∆ − . 2 3
(67)
As the timings are fixed and known, the first exponent in K2 is included readily in the optimization problem as a fixed scaling (amplitude) parameter. Generally, tse (1/T2 − 1/T1 ) vanishes for bulk liquids and is often ignored in porous materials, and the remaining (t∆ + tstore + τse ) /T1 factor is assumed to be constant and therefore also removed from the optimization problem by normalization of the 104
data [151]. The factor 1/2 in each kernel function is associated with the stimulated echo that returns only half the initial magnetization [91]. A few experiments generate 2D data with non-separable dimensions. Such experiments include the STRAFI D-T2 correlations discussed in Section 4.5.2. As both dimensions of the experiment depend on tse , the kernel function for the FFG-2SE experiment has the form 1 2 2 3 K0 (D, T2 ,tse , nte ) = exp − γ g Dtse 6 2tse + nte × exp − , T2,eff
(68)
and, similarly, the kernel function for the PFG-2SE (bench-top equivalent) experiment is [198] n tδ o K0 (D, T2 ,tδ ,tse , nte ) = exp −2γ 2 g2 Dtδ2 tse − 3 2tse + nte × exp − . T2
(69)
The kernel function may be simplified by substituting tδ with (tse − tc ) /2, where tc is a constant time difference between tδ and tse . However, this modification results in a non-rectangular data matrix that must be truncated or extrapolated with undesirable consequences [151]. It is preferable to construct the full kernel matrix K0 when possible [198, 240]. 6. Interpretation Relaxation times are often used in low-field measurements to infer some property of the sample; for example, changes in T2 have been related to the degree of cross-linking in polymeric materials [27]. However, a generic use of relaxation data (notably T2 decays) is the determination of total sample volume. In the case 105
of a CPMG decay, the exponential relaxation curve is fitted to obtain the signal intensity at the experimental time origin. In this way a signal intensity is determined (independent of relaxation) which is proportional to the number of spins and hence interpreted, for example, as a liquid volume. Quantification is achieved by comparison to a calibrated signal intensity. In this Section we consider some of the more sophisticated methods used to convert NMR signals into physical parameters describing a property of the sample. We describe pore size measurements (Section 6.1), surface interaction strength (Section 6.2), and exchange rates (Section 6.3) determined from relaxation times. Droplet size measurements are obtained using pulsed field gradient techniques (Section 6.4). The phenomenon of magnetic susceptibility gradients (Section 6.5) that are ubiquitous in heterogeneous samples is discussed. 6.1. Surface-to-volume ratio Distributions of relaxation time are a useful method of recording low-field data when the SNR is poor. Here we consider the interpretation of T1,2 distributions as measurements of surface-to-volume ratio. If a liquid is in contact with a solid surface then spins within molecules adsorbed on the surface will undergo enhanced relaxation due to reduced molecular mobility [241]. In the fast diffusion limit (where ρ2 `s /D0 1 as defined by Brownstein and Tarr [242], with ρ2 being the surface (transverse) relaxivity, `s being the length-scale of the confining geometry, and D0 being the bulk liquid diffusion coefficient) exchange is sufficiently rapid that surface and the bulk relaxing spins are well mixed, the observed relaxation time will be 1 1 1 = + , T1,2 T1,2,bulk T1,2,surf
(70)
106
where T1,2,bulk is the bulk liquid relaxation time and the relaxation time of molecules adsorbed on the surface is given by T1,2,surf =
1 V , ρ1,2 S
(71)
where ρ1,2 is the surface relaxivity term associated with longitudinal or transverse surface relaxation processes respectively, and S/V is the surface-to-volume ratio. For many samples, the bulk relaxation term in eq. (70) will be negligible compared to the surface relaxation term and can be ignored in the analysis so 1 S ' ρ1,2 . T1,2 V
(72)
In the context of porous media, S/V can be converted to pore size by assuming a pore geometry. For spherical pores, S/V ≡ 3/`s where `s is a characteristic length scale associated with the pore radius. Using eq. (72), distributions of relaxation are rescaled to pore size [243, 244]; an example is presented in Fig. 32. There are several notable limitations in the application of eq. (70) to real systems. The first is the assumption that the fast diffusion limit is valid at all times. This is generally true for small pores or weak surface relaxation. However, for liquid saturated large pores (or strong surface relaxation) the spins in the center of the pore will exhibit bulk-like relaxation. This situation is encountered in the intermediate or slow diffusion regimes [242]. Therefore, the surface-to-volume ratio term will apply only to the liquid layer containing spins influenced by the surface. This layer is considered to be 1 − 3 monolayers thick [245, 246]. Under these conditions, the observed relaxation rate becomes 1 S 1 = p ρ1,2 + (1 − p) , T1,2 V T1,2,bulk
(73)
where p is the fraction of spins in a pore that are influenced by the surface and S/V described the geometry of the surface-influenced layer. Generally p will be 107
unknown although it can be estimated from the peak areas in a T1,2 distribution if distinct relaxation time components associated with bulk and surface relaxation are present. The same concept applies in S/V measurements of granular suspensions. `s / µ m 10
−8
−7
−6
10
10
−5
10
P{log10(T2 )}
0.06 0.04 0.02 0 −3 10
10
−2
−1
10 T2 / s
0
10
1
10
Figure 32: Simulated example of a T2 relaxation time distribution (lower abscissa) converted to a distribution of pore radii `s (upper abscissa). The scaling has been achieved using ρ2 = 2 µm s−1 . In this example, the blue area corresponds to liquid influenced by surface interactions and the red area corresponds to bulk liquid. Assuming the pore liquid relaxes in the fast diffusion limit, then T2 maps directly to pore size. The small relaxation time component at T2 = 10−2 s is an artifact of the inversion process and is ignored in the analysis.
The effect of exchange between surface and bulk liquid is subtly different for T1 and T2 measurements. The T1 measurement averages over a longer time-scale than the T2 measurement. In a saturation or inversion recovery experiment, mixing occurs over the time τ1 which may vary from hundreds of microseconds up to many seconds so the effective time for exchange is long. In the case of a CPMG experiment, the relevant time-scale for mixing is the echo time te = 2τ2 which is typically less than a millisecond. Also, ρ1 ≤ ρ2 . Consequently, it is possible to envisage a system that is in the slow diffusion limit for a T2 measurement whilst 108
in the fast diffusion limit for a T1 measurement. Accordingly, T1 is considered a volume-averaged probe of pore structures whereas T2 is considered a local probe of pore surface structure. Further discussions on exchange will be given in Sections 6.3 and 7.3.1. A further complication of eq. (70) is the accurate determination of ρ1,2 . Theoretical expressions for the surface relaxivity term exist although attempts to calculate values have proven inadequate [152]. Ideally a calibration value of ρ1,2 would be obtained from a material of known S/V . However, even similar chemical surfaces have proven to result in very different relaxivities ensuring reference measurements are unreliable [247]. Direct estimates of ρ1,2 can be obtained by comparing the shortest relaxation time component observed in a sample (considered to be a measure of relaxation on the surface) to BET gas adsorption surface area measurements [248]. If S/V is very large or ρ1,2 is very small, the assumption that 1/T1,2,bulk is negligible in eq. (70) will not be valid. When converting T1,2 to S/V , the bulk relaxation time sets an upper limit on the surface-to-volume ratio (or pore size) that can be observed. Only pores of size `s < 6ρ1,2 T1,2,bulk will be observable. In the example presented in Fig. 32 this corresponds to a diameter of `s ∼ 10−5 m. Larger pores may be present but the liquid they contain will be indistinguishable from bulk. We note that diffusion measurements are also used to determine S/V in porous materials. At short observation times, when the mean displacement of the diffusing molecules is less than the pore size (practical values of t∆ lead to a lower pore size limit of `s ∼ 1 µm), the observed diffusion coefficient is related to the bulk
109
diffusion coefficient by [249] D 4 S (D0t∆ )1/2 . ≈ 1− √ D0 9 πV
(74)
At long observation times, such that diffusing molecules explore many connected pores, the tortuosity of the pore network is obtained. In diffusion NMR experiments it is common practice to define (dimensionless) tortuosity τtort as [250] D 1 lim ≡ . (75) t∆ →∞ D0 τtort The surface-to-volume ratio and tortuosity are therefore obtained readily as the √ initial slope and long-time plateau amplitude in a plot of D/D0 versus t∆ , respectively. The diffusion coefficients of the liquid in the porous material and in bulk are obtained using the APGSTE sequence (see Section 4.2) with a fixed observation time and variable gradient amplitude. The observation time is incremented in successive measurements. The “difftrain” pulse sequence [251] enables a rapid measure of surface-to-volume ratio as reviewed in [252]. Finally we draw attention to the method of determining surface-to-volume ratio distributions from the melting point depression of a confined frozen liquid. NMR cryoporometry [253] has been implemented at both low and high field, although the low-field measurements are considered more reliable as low Q probes (typical on benchtop hardware) are robust to thermal drift that is unavoidable during the experiment. The liquid-saturated porous material is cooled to well below the expected freezing point of the liquid and then warmed in situ so the melting transition can be observed by NMR. The confining geometry hinders the formation of regular crystal structures within the pores. Therefore, an irregular structure will form that has a lower melting point than the bulk crystal lattice. The temperature difference ∆Tm between the bulk melting point Tm (∞) (crystal of infinite size) 110
and the depressed melting point Tm (`c ) is proportional to the crystal size `c (and hence pore size `s when the crystal size is determined by geometric confinement) according to the Gibbs-Thomson equation for a small isolated crystal in its own liquid [254] ∆Tm (`c ) = Tm (∞) − Tm (`c ) =
4σsl Tm (∞) , `c ∆Hf ρs
(76)
where σsl is the surface energy at the crystal-liquid interface, ∆Hf is the bulk enthalpy of fusion (per gram of material), and ρs is the density of the solid. Consequently, the melting point depression of a confined liquid depends only on the properties of the liquid, its own solid, and the interface between these two states. Cryoporometry is independent of the chemistry of the pore surface; unlike relaxometry it does not require a term that depends on the adsorbate-adsorbent interaction. Instead the melting point depression constant CGT for a given liquid is obtained empirically based on measurements in controlled porosity materials (such as silica gels). The surface-to-volume ratio is determined simply as S ∆Tm = CGT . V
(77)
As the T2 of a solid is significantly shorter (usually by an order of magnitude or more) than the T2 of its liquid even when confined in pores, the quantification of the liquid fraction at any given temperature is usually straight forward. Complications arise when the solid is a plastic crystal (long T2 ); in this case robust solid/liquid phase separation is achieved by fitting relaxation time distributions to CPMG decays [247]. The range of pore sizes that can be probed is limited by the chosen liquid (through CGT ), the thermal stability of the sample, and the sensitivity of the temperature measurement. Practically, cryoporometry is suited to the study of small pores where `s < 10 µm. Cryoporometry has been applied to 111
numerous porous materials including cements, rocks (gas shales), coal, wood, and bones. A thorough review of the theory, experimental techniques, and applications is presented in [255, 256]. 6.2. Surface interaction strength Relaxation times T1,2 are sensitive to molecular motions occurring at the Larmor frequency. When molecular mobility is reduced, such as a liquid adsorbing on a pore surface, the frequency of molecular motion changes. T1 and T2 are sensitive to different types of motion: T1 is predominantly determined by molecular rotation whereas in heterogeneous systems T2 is often determined by translation motion. Therefore, adsorption on a surface influences T1 and T2 to different extents and the ratio T1 /T2 is useful as an indicator of the strength of surface adsorption. This ratio has an advantage over absolute relaxation time measurements in that the sensitivity to geometry is (to leading order) removed. A robust method for determining T1 /T2 is the 2D T1 -T2 correlation experiment; an example is presented in Fig. 33. The 2D plot provides a useful visual guide to unambiguously correlating T1 and T2 relaxation time components. To interpret surface relaxation, we represent a solid surface as a lattice of relaxation sinks; these sinks can take the form of active surface sites where molecules can adsorb at paramagnetic relaxation centers on the surface (such as metal impurities). Motion of a molecule adsorbed on a surface is governed by two time scales: the average surface diffusion correlation time τm and an average surface residence time τs . The diffusion correlation time describes the time for a molecule to move between relaxation sinks on the surface. The residence time describes the length of time an adsorbed molecule remains on the surface before desorbing back into the surrounding bulk liquid. Typically, τs τm , which means an adsorbed 112
1
10
a
0
T1 / s
10
−1
10
−2
10
T 1 /T 2 = 1.5
−3
10
1
10
b
0
T1 / s
10
−1
10
−2
10
T 1 /T 2 = 5
−3
10
−3
10
−2
10
−1
10 10 T2 / s
0
10
1
Figure 33: Simulated example T1 -T2 correlation plots of a liquid interacting (a) weakly and (b) strongly with the surface of a porous medium. The diagonal lines indicate T1 = T2 (dashed line) and T1 > T2 (solid line). Contour spacings are constant. In both plots a bulk liquid component is observed at long T1,2 times with T1 = T2 . When the liquid interacts with the surface, its relaxation times are reduced. The surface influences T1 and T2 to differing extents, resulting in T1 > T2 ; the stronger the interaction with the surface, the greater the ratio of T1 /T2 . For example, (a) weak surface interactions may provide T1 /T2 = 1.5 whereas (b) strong surface interactions may provide T1 /T2 = 5.
113
molecule undergoes some motion on the surface before desorbing. For a spin diffusing across a surface, the spectral density function is [257, 258] # " 2 1 + ω 2 τm . J (ω ) ∝ ln 2 (τm /τs )2 + ω 2 τm
(78)
If we consider first a “clean” surface (i.e., no paramagnetic relaxation sinks) [241] then the relaxation time ratio is T2,surf J (ω ) + 4J (2ω ) . =2 T1,surf 6J(0) + 5J (ω ) + 2J (2ω )
(79)
However, if the surface contains paramagnetic relaxation sinks, then the interaction of the unlike proton I and electron S spins gives rise to a modified relaxation time ratio [143, 257] T2,surf 3J (ωI ) + 7J (ωS ) , =2 T1,surf 8J(0) + 3J (ωI ) + 13J (ωS )
(80)
where ωS = 658.21ωI (i.e., ωS ωI ). At low field (within the fast diffusion limit [242]) we make the assumption that ωI 1/τs so that T1,2,surf T1,2,bulk and it then follows that the observed relaxation rate is approximately equal to the surface relaxation rate according to eq. (72). 6.3. Diffusive or chemical exchange Mass transport is often an important measurement in process engineering. NMR exchange time analysis can provide an estimate of mass transfer over distances of tens of microns, such as diffusion between neighboring pores in a liquid saturated porous medium. These exchange times are determined at low field using T2 -T2 experiments [143, 152, 153], wherein a pair of CPMG echo trains are separated by a z-storage interval of duration tstore , as detailed in Section 4.3.3. Here, we consider the analysis of these data. Off-diagonal exchange peaks (XP) are 114
observed in the T2 -T2 distribution when spins move between environments characterized by different T2 relaxation rates on the time-scale of the storage interval. The different T2 environments are characterized by the main peaks (MP) in the 2D distribution. Simulated data are presented in Fig. 34. It is important to realize that the positions of the peaks in the T2 -T2 distribution are determined by a combination of the absolute T2 relaxation time in each environment and the exchange rate. We therefore denote the observed relaxation as an apparent time T2,app . The same logic applies to longitudinal relaxation measurements in the presence of exchange, leading to T1,app . Technically, the observation of an apparent relaxation time occurs in 1D acquisitions as well (whenever exchange occurs) although this differentiation is rarely made in the literature. In 2D correlations the observation of apparent relaxation times is more obvious, particularly in T1 -T2 correlations where relaxation time components can have negative intensity in the presence of exchange. A full derivation of the effects of exchange on relaxation time components is given in [259]. Returning to the T2 -T2 plots in Fig. 34, the variation in amplitude of the offdiagonal peaks IXP as tstore changes is related to the rate of exchange. To determine the exchange rate, therefore, it is necessary to determine IXP over, say, 8 different storage times. IXP is calculated as the volume under the peaks in the 2D distribution. The T2 -T2 plot should be symmetric. However, the inversion process and real (i.e., noisy) data often leads to asymmetries in the distributions. The XP amplitude can be taken as the average volume under the two exchange peaks. As the tstore time is increased, the amount of T1 relaxation occurring likewise increases. This longitudinal relaxation results in a decrease in the overall intensity of the 2D plot. Therefore, it is often easier to consider the variation in IXP normalized by
115
10 10 10 10 10
T 2,app / s
10 10 10 10 10 10 10 10 10 10
1
a
0
MP
−1
MP −2 −3
1
b
0
XP
−1
XP
−2 −3
1
c
0 −1 −2 −3
−3
10
−2
−1
10 10 10 T 2,app / s
0
10
1
Figure 34: Simulated T2 -T2 maps showing two-site exchange. The main peaks (MP) and exchange peaks (XP) are labeled. The data were generated with an exchange time τex,a,b = 0.5 s and 2D plots are shown with storage times of (a) tstore = 1 ms, (b) 100 ms, and (c) 1 s. As the storage time increases, the relative intensity of the exchange peaks increases but the total intensity of the plot decreases due to T1 relaxation (visible in the 1D T2 projections). Contour spacings are constant across all plots. The spin system was simulated with Meq,a,b = 1, T1,a,b = 2 s, T2,a = 0.05 s, and T2,b = 0.5 s. The position and intensity of the peaks on the 2D plot do not match the absolute T2 relaxation times and magnetizations, due to exchange; apparent relaxation times and magnetizations are observed instead.
116
the total peak amplitude ITP . Several authors have proposed methods for fitting exchange rates [152, 153, 237]. If we consider two spin reservoirs (denoted by a and b) then it is straightforward to determine the Bloch equations describing relaxation under the condition of exchange as dMa Mb Meq,a − Ma Ma + + , =− dt τex,a τex,b T1,2,a
(81)
dMb Ma Meq,b − Mb Mb =− + + , dt τex,b τex,a T1,2,b
(82)
and
where Ma,b is the magnetization in reservoir a or b respectively at time t, Meq,a,b is the equilibrium magnetization in each reservoir as t → ∞, 1/τex,a,b describes the rate at which spins move from a → b and vice versa, and T1,2,a,b is the longitudinal or transverse relaxation time of each reservoir (independent of exchange). The equilibrium magnetization will depend on the type of relaxation; for longitudinal relaxation, Meq,a,b = Ma,b (0), i.e., the maximum (fully recovered) magnetization. For transverse relaxation, Meq,a,b = 0. The definition of exchange rate used here requires the condition of detailed balance such that τex,b Ma (0) = τex,a Mb (0); otherwise, spins will accumulate in one of the reservoirs. If Ma (0) = Mb (0) then exchange occurs at the same rate in either direction and we need only define a single exchange time as τex . It is convenient to express eq. (81) and (82) in matrix notation as [152] d(Meq,a −Ma ) − 1 − 1 dt = T1,2,a τex,a d(Meq,b −Mb ) 1 τex,a
dt
×
Meq,a − Ma
Meq,b − Mb 117
1
τex,b 1
1
− T − τex,b 1,2,b .
(83)
The simple but lengthy solutions to this equation are known [260] and have been reviewed in [237]. This analytic model provides a method of fitting exchange times that is reliable and robust for two-site exchange, assuming the relaxation times and relative spin populations of each reservoir can be determined independent of exchange [152, 154]. Of course, this is usually impractical. If the exchange time is similar to, or less than, the relaxation times such that T1,2,a τa,b ≤ T1,2,b or T1,2,a,b τa,b then the apparent relaxation times and magnetizations observed in the T2 -T2 experiment may be used in place of the absolute values in the fit; an example is given in Fig. 35. Otherwise, accurate exchange rates may not be readily determined and an alternative fitting method is required, which we discuss in the next paragraph. The two-site analytic model can be extended to multi-site exchange assuming the exchange pathways are independent [261]. The analytic solution for interconnected pathways in a three-site exchange model is discussed in [237]. An alternative determination of exchange time is achieved by simply fitting an exponential recovery curve to the plot of normalized XP amplitude against tstore [153]. An effective exchange time τex,eff is obtained from the equation tstore IXP (tstore ) IXP (∞) = 1 − exp − . ITP (tstore ) ITP (∞) τex,eff
(84)
As demonstrated in Fig. 35, τex,eff τex , specifically by a factor 2 in this example (a consequence of having equal spin populations by construction). Washburn and Callaghan suggested a 1/T1 correction factor be added to the exchange rate [153], although in this case the difference is negligible. Even when lacking knowledge regarding the spin populations in the absence of exchange (required to determine the mass balance governing exchange from a → b and b → a), the effective exchange rate is a useful parameter. This quick exponential fit can be used 118
0.5
IXP /ITP
0.4 0.3 0.2 0.1 0 0
0.5
1 tstore / s
1.5
2
Figure 35: Ratio of exchange peak to total peak intensity as a function of storage time (×) derived from simulated 2D T2 -T2 plots; see Fig. 34 for examples. The solid blue line indicates a best fit of the analytic solution to eq. (83) starting with initial guesses of Meq,a = 0.6, Meq,b = 1, T2,app,a = 0.045 s, T2,app,b = 0.265 s, and T1,a,b = 1 s, where the Meq and T2,app values were determined from Fig. 34(a). Despite the poor estimates of absolute relaxation time and magnetization (independent of exchange), the fit provides correct values of τex,a,b = 0.5 s and Meq,a,b = 1. Also shown is a best fit to the exponential recovery eq. (84) (red dashed line) which provides an effective exchange time of τex,eff = 0.25 s.
119
to compare effective exchange rates and is easily extended to multiple exchange pathways. Furthermore, a distribution of effective exchange times can be obtained readily using 1D Tikhonov regularization as described in Section 5.4.1. A distribution of exchange times can also be obtained from the analytic model in eq. (83) although this process is more complicated and requires many assumptions [261]. Another possible fit could be obtained using a sinc-like function of IXP (tstore ) as derived for two-site chemical exchange observed in 2D spectroscopy experiments [200]; to date, this model has not been applied to the T2 -T2 exchange experiment. Interpretation of exchange rates can be complicated by exchange processes other than pore-to-pore diffusion, notably diffusion between regions of different internal gradient strength (see Section 6.5 and [50]) or exchange between bulk liquid and surface adsorbed molecules in the slow-diffusion regime [242]. 6.4. Droplet sizing Pulsed field gradient measurements are used to determine distributions of droplet size in emulsions [22, 23]. Early bench-top diffusion experiments relied on incrementing the gradient pulse duration tδ or observation time t∆ to step through q-space. Relaxation or diffusion pre-conditioning was commonly used in such cases to eliminate the continuous liquid phase from the acquired signal. Van Duynhoven et al. advised that, given the capabilities of modern bench-top spectrometers, the experimentalist should increment gradient strength instead (the method described in Section 4.2), allowing simultaneous detection of oil and water diffusion coefficients without the concern of variable relaxation time contrast [262]. Now, as spectral resolution becomes routine on intermediate-field benchtop spectrometers, oil and water signals may be distinguished by chemical shift [263]. This method is more versatile as discrete signals are acquired from both 120
liquid phases simultaneously, enabling the investigation of complicated systems such as oil-in-water-in-oil multiple emulsions. Based on our discussion in Section 4.2 we know the signal acquired for a free liquid in a standard PGSTE experiment, Fig. 18(b), is [97] h b(g) tδ i ln = −D0 (γ tδ g)2 t∆ − , b(0) 3
(85)
assuming ideal rectangular gradient pulses. If the liquid is confined by a boundary the spins are unable to diffuse the expected distance through the gradient g. The signal attenuation is therefore less for confined liquid than free liquid. The signal attenuation for liquid confined within a spherical boundary of radius a (as applicable to an emulsion droplet) can be represented by three different models. • The Gaussian phase distribution (gpd) model [264–266] assumes the diffusing spin magnetization vector exhibits a phase that is distributed normally at all times. This approximate theoretical model describes the signal attenuation for diffusion bounded within a sphere as " ∞ b(g) 1 2tδ 2 = exp −2 (γ g) ∑ 2 2 2 2 b(0) m=1 βm (βm a − 2) βm D0 )# 2 + E (t∆ − tδ ) − 2E (t∆ ) − 2E (tδ ) + E (t∆ + tδ ) , − 2 (βm2 D0 )
(86)
where βm is given by the positive roots of J3/2 (β a) = β aJ5/2 (β a), Jn is the nth order Bessel function, and E(t) = exp −βm2 D0t .
• The short gradient pulse (sgp) model [267–269] provides an exact expres-
sion for signal attenuation under the assumption that tδ is negligibly small so that no diffusion occurs during the application of the gradient pulses. For spherical geometry, the signal decay is given by [γ tδ ga cos (γ tδ ga) − sin (γ tδ ga)]2 b(g) = 9 b(0) (γ tδ ga)6 121
∞
∞
2 (2n + 1)βnm βnm D0t∆ + 6∑ ∑ 2 exp − 2 a2 n=1 m=1 βnm − n − n γ tδ ga jn0 (γ tδ ga) 2 , × 2 − (γ t ga)2 βnm δ
(87)
where βnm are the roots of the equation jn0 (βnm ) = 0 (where jn is the nth order spherical Bessel function). Numerous versions of eq. (87) can be found in the literature; the correct form is given by Veeman [270]. • The third model is the block gradient pulse (bgp) approximation [263, 271]. This model is derived by solving the Bloch-Torrey equation [272] for diffusion in a single droplet using an eigenfunction expansion. The solution, applicable to a constant gradient, is then applied piecewise [273] to obtain a model for the entire PGSTE pulse sequence: b(g) v† G (t˜δ ; γ˜) D (t˜∆ − t˜δ ) G∗ (t˜δ ; γ˜) v = , b(0) v† D (t˜∆ + t˜δ ) v
(88)
where ∗ denotes the complex conjugate and D(t˜) = exp[−Λt˜] and G(t˜; γ˜) = exp{(−Λ− 2 δ where δ is the Krojγ˜Z)t˜}, with the elements of matrix Λ given by Λuv = βnm uv uv
necker delta; the elements of vector v are given by vu = δu1 . The matrix elements in Z have been determined analytically by Grebenkov [271, 274, 275] for a spherical droplet. The variable substitutions are t˜ = D0t/a2 and γ˜ = γ ga3 /D0 . Only 2 ) are required so the matrices G and D are limited the 50 largest eigenvalues (βnm
to dimensions of 50 × 50 elements and likewise the vector v has a length of 50 elements. In the limit that a → 0, G = D. In all three models, the only free parameter is a. These models can also be expanded to account for surface relaxation [270]. A droplet size distribution [265] is obtained using the regularization method outlined in Section 5.4.1. We note for completeness that when the continuous liquid phase or the droplet surface contains 122
paramagnetic ions, relaxation time measurements can be used to determine the droplet size distribution by analogy to the pore sizing measurement in Section 6.1 above. However, few samples are appropriate for study using this method; T2 relaxation is used for the probe droplet size distribution of water in crude oil emulsions when the crude oil contains metallic impurities [276]. 6.5. Internal gradients Magnetic susceptibility contrast ∆χ is an important topic for NMR applications to industrial processes. Many of the samples to be considered are heterogeneous, potentially causing gross distortions in the magnetic field B0 far greater than any inhomogeneities inherent in the permanent magnet design. As noted in Section 1, magnetic susceptibility induced gradients are often referred to as internal gradients because they are commonly encountered in porous materials (rocks, cements, catalysts). Internal gradients are known to be stronger at higher magnetic fields, although the exact scaling with field strength is complicated [50, 277]. Therefore, one method for minimizing these field inhomogeneities is the use of low-field magnets; this is the solution adopted in petrophysics, see Section 7.3.1. We have chosen to include this important discussion of internal gradients at a late stage in the review because this topic requires a clear understanding of the interplay between magnet, sample, and spin dynamics. Internal gradients influence almost all NMR measurements in heterogeneous materials, including imaging [278], diffusion [104, 279], and transverse relaxation [280]. T1 is insensitive to magnetic field gradients [281], although the maximum observable signal will be limited depending on the acquisition technique. Molecular diffusion occurring within a spatially variant magnetic field will result in a reduction in the T2∗ observed in a FID [282]; hence internal gradients cause line broadening in a spec123
trum. It is possible for the rf system dead time to exceed T2∗ , at which point no signal will be observed. Therefore, if very heterogeneous samples are to be studied, the NMR system must be carefully selected to provide a low magnetic field strength and fast rf receiver chain. When imaging porous materials, SPI-based methods are preferred because they are insensitive to relaxation time distortions [173], within the limit that a useful SNR is obtained. To leading order (for “weak” internal gradients, such that ∆χ B0 10 µT), the 3/2
maximum gradient strength gmax can be considered proportional to B0
[277], al-
though as B0 becomes large other factors (specifically pore size) limit gmax [240]. Nevertheless, this rule-of-thumb scaling indicates that internal gradients increase in magnitude rapidly as the magnetic field strength is increased. Even samples with a low magnetic susceptibility contrast can generate sufficient internal gradients so as to prevent quantitative measurements at intermediate field [50]. In much of the existing literature on NMR measurements in porous media, the influence of internal gradients has been overlooked, dismissed, or – worse – misinterpreted as a physical property of the sample. The classic example is the observation of a multi-modal T2 distribution in a rock known to have a mono-modal pore size distribution [50, 153]. Detailed understanding of internal gradients has largely originated from transverse relaxation measurements. This is a consequence of the well logging industry where T2 is the metric of choice and rocks (notably sandstones and shales) can exhibit a large susceptibility contrast [277]. In the presence of internal gradients, the effective relaxation time observed from a CPMG measurement is T2,eff , which is a combination of the actual relaxation time T2 and a diffusive component. The nature of the diffusion is determined by the relative significance of the experimental parameters, internal gradient strength, and
124
pore size. When one of these factors dominates over the other two, an asymptotic limit to the diffusion behavior is observed. The three asymptotic limits are: 1. The Short Time (ST) regime [283]. The archetypal regime where the diffusion exponent is determined by the echo time in the CPMG experiment. The ST regime is encountered in large pores with weak internal gradients. Dif√ fusion occurs over the length-scale `e ≈ D0te and the diffusion exponent varies as nte3 .
2. The Motional Averaging (MAV) regime [280]. Encountered in very small pores. The pore size `s defines the maximum diffusion distance and spins will explore an entire pore several times during te . The diffusion exponent varies with `2s and te . 3. The Localization (LOC) regime [284]. This regime corresponds to diffusion in very strong internal gradients. Spins dephase by more than 2π radians when diffusing over a distance `g = (D0 /γ g)1/3 . As the gradient is spatially variant, diffusing spins are not refocused properly by a spin echo. The diffusion exponent varies with te . The ratio `g /`s in this regime indicates the fraction of pore volume containing coherent spins. The relative importance of the length scales `e , `s , and `g determine the diffusion regime as illustrated in Fig. 36. At the asymptotic limits, analytic solutions exist for the diffusion exponent that contributes to the decay in the CPMG experiment. Outside the asymptotic limits, pre-asymptotic regimes exist. No exact analytic solutions exist in these ill-defined regimes, except that the diffusion exponent will ς
vary with nte where the power can have any value in the range 1 < ς < 3 [284]. The power ς effectively indicates the relative importance of the echo time on the degree of signal attenuation due to diffusion. When ς = 1 (asymptotic limit of 125
MAV and LOC regimes) the diffusion exponent is insensitive to te . However, when ς = 3 (asymptotic limit of ST regime) the diffusive decay is dependent on the chosen te . Therefore, in the ST regime reducing te reduces the amount of diffusive attenuation in the signal [277]. In real samples, the asymptotic limits of the MAV and LOC regimes (ς = 1) are not expected to be encountered: diffusion through connected pores will prevent the MAV asymptote from being reached and features of the LOC asymptote are not expected to be observed in 3D pore geometry due to very rapid magnetization decay [284]. 2
ST
10
MAV
`g /`s
0
10
−2
10
LOC −4
10
−4
10
−2
10
0
10
2
10
`e /`s
Figure 36: Schematic indicating the diffusion regimes ST, MAV, and LOC, as defined by the diffusion length-scales `e , `s , and `g , respectively. The dashed lines indicate equivalence of the length-scales. The boundaries between the asymptotic limits of the diffusion regimes (gray areas) are not well defined.
The use of low fields is intended to push the diffusion behavior into the ST regime [277]. In this asymptote the diffusion exponent becomes −
1 2 2 γ geff D0 nte3 , 12
(89)
which readers will recognize as the equation for free diffusion in a magnetic field gradient. Here, geff is an effective gradient strength determined as the average of all point gradients g(r) over the diffusion length-scale `e . It is important to note
126
that due to this averaging, the diffusion exponent is insensitive to local variations in g(r). The observed signal decay is given by b (nte ) 1 2 2 nte 3 exp − γ geff D0 nte . = exp − b(0) T2 12
(90)
By acquiring CPMG data with different echo times, it is possible to determine if the diffusive term is significant. The decay due to T2 will not vary with echo time. Therefore, if the CPMG decay becomes more rapid as te is increased, there must be a contribution from diffusion and hence the influence of the internal gradients is significant. In a porous material it is likely that a distribution of both T2 and geff will exist. Therefore, eq. (90) can be expressed as a 2D Fredholm integral equation b (nte ,te ) = b(0, 0)
Z∞ Z∞
d (log10 T2 ) d (log10 geff )
0 0
× K0 (nte ,te , T2 , geff ) f (T2 , geff ) + e (nte ,te ) ,
(91)
where the kernel function is given by the R.H.S. of eq. (90). A set of CPMG data acquired with a range of te can be inverted using the method outlined in Section 5.5.1, noting that the kernel is non-separable. A 2D distribution f (T2 , geff ) is obtained, from which the actual T2 distribution is extracted. In this way, the contributions from relaxation and diffusion are decoupled in the CPMG experiment [240]. Attempting to fit the CPMG decay described in eq. (90) with a simple relaxation kernel function of exp{−nte /T2,eff } will result in a distribution that is overly broad and containing unphysical components [50]. An illustration is given in Fig. 37. In this demonstration it is clear that even using a very short echo time 127
1
τ τ τ τ τ
a b/b(0)
0.8 0.6
≡ = = = =
0s 0.5 × 10 −4 1.5 × 10 −4 3.0 × 10 −4 6.0 × 10 −4
s s s s
0.4 0.2 0 0
0.2
0.4
0.6
0.8
1
nte / s 0.1
P[log10(T2,eff )]
b 0.08 0.06 0.04 0.02 0 −3 10
10
−2
−1
10
10
0
T2,eff / s
Figure 37: Simulation of internal gradient effects on measurements of transverse relaxation time: (a) CPMG decays were generated using eq. (90) with SNR = 100. The T2 distribution was represented by a log-normal distribution with mean 0.3 s and standard deviation 0.06 s; the geff distribution was represented by a log-normal distribution with mean 1 T m−1 and standard deviation 0.25 T m−1 . The true T2 decay (equivalent to τ → 0 s) is indicated by the dashed line. The solid lines show CPMG decays generated with both relaxation and diffusion exponents. As the half-echo time increases, the effect of diffusion is more significant, leading to faster signal decay. The T2,eff distributions obtained from the data in (a) are shown in (b), generated assuming the kernel function K0 = exp(−nte /T2 ). Again, the true T2 distribution is indicated by the dashed line. The other distributions are influenced by internal gradients resulting in a reduced mean effective relaxation time and unphysical components. The legend applies to both plots.
128
te = 100 µs (typically the shortest practical for benchtop instruments) the CPMG decay exhibits some diffusive attenuation. It is therefore very important to test for the presence of susceptibility induced gradients in any heterogeneous materials and interpret the data accordingly. In the pre-asymptotic diffusion regimes, the diffusion exponent is described by ς
−ante where a is an unphysical distribution containing the dimensionless terms
D0te /`2 and γ g`3 /D0 , each raised to some power, with ` being `e , `s , or `g . De-
scribing the diffusive exponent in this way allows the contributions of surface relaxation and diffusion in T2,eff to be decoupled in the pre-asymptotic regimes under the condition ς > 1. The correct scaling of ς is determined empirically using a the “data-collapse” protocol as detailed in [285]. This approach is useful when internal gradients are large and diffusion occurs in one of the pre-asymptotic regimes. 7. Industrial Applications In this Section we consider some of the most successful applications of lowfield NMR in industry with the specific examples of food manufacture (Section 7.1), the built environment (Section 7.2), and petroleum production (Section 7.3). There are, of course, many other examples of low-field NMR in industry to be found throughout the literature. For example, unilateral devices such as the NMRMOUSE have been applied extensively to the study of polymeric materials [27] and biomedical systems [286]. Furthermore, even within the topics we have considered here the examples are not exhaustive; wherever possible we have included citations to dedicated review articles. We present these applications to provide the reader with a link between the NMR hardware, experiments, analysis, and 129
interpretation in Sections 3 through 6 and the world of industrial process. 7.1. Food One of the most prevalent industrial applications of low-field bench-top magnets is process control in the food industry. NMR systems operating at ν0 = 20 MHz for 1 H are popular; these spectrometers have dominated the low-field market for the last two decades. The typical ν0 = 20 MHz spectrometer is configured to provide a few robust and reliable measurements at the press of a button: total signal intensity, relaxation time distributions, or diffusion measurements for droplet size distributions (as used in dairy product characterization, see Section 7.1.3). As such, these low-field instruments can be installed and operated readily in an industrial environment. Low-field NMR and MRI has been used as a characterization and quality control tool for a wide range of food products including bread [287–289] and biscuit dough [290, 291], potatoes [292–298] and potato starch [299–301], tomatoes [302, 303], apples [304–306], processed soybean protein [307], and powdered food products [308–310], to cite but a few of the more recent studies. Advanced imaging techniques, such as SPRITE (see Section 4.4.5), have been used to investigate oil and water in fried foods [311, 312]. Here we consider applications of low-field NMR in three areas of the food industry: meat, fish, and dairy products. 7.1.1. Meat A primary focus of low-field NMR in meat production and processing is the determination of the water holding capacity (WHC) – a metric of meat quality. A recent review details the importance of water in meat science [313]. Consequently, NMR provides a useful characterization of meat properties because the mobility 130
and distribution of water within muscle (myowater) affects the meat quality, including properties such as tenderness. Mammalian skeletal muscle consists of water (65 − 80 %), proteins (16 − 22 %), fats (1 − 13 %), carbohydrates(1 − 2 %) and less than 1 % other soluble materials. Water is held by microfibrils (protein chains in muscle fibers, see Fig. 38) and so WHC is mainly determined by microfibrillar shinkage during the onset of rigor mortis as the muscle converts to meat. In vivo, about 85 % of the water is associated with the myofibrils (intra-myofibrillar water). The majority of this intra-myofibrillar water is bound to, or trapped between, the myofibril filaments; this water may be mobilized post mortem due to changes in cellular structure. There will be some water associated with proteins that cannot be removed by external physical forces (mechanical processing, freezing, heating) although this protein-bound water will exchange with surrounding mobile water. The remaining water is located in the space between myofibrils (inter-myofibrillar water), between muscle fibers, and in or between the muscle fasciculi. The primary low-field measurement used in studies of muscle conversion to meat is the CPMG echo train, see Section 4.1.2. It is typical, as we shall see in the following discussion, for the transverse magnetization decay to contain only two or three components. Therefore, the data may be analyzed using a least squares fit with an appropriate sum of exponentials, as mentioned in Section 5.3. In the meat science literature it is common to see T2 relaxation time distributions, as described in Section 5.4. Conventional relaxation time interpretation relates T2 to the local environment of the water. Myowater typically provides two T2 relaxation time components: the intra-myofibrillar water exhibits T21 ∼ 30 − 50 ms whereas inter-myofibrillar water exhibits T22 ∼ 100 − 250 ms. The longer T2 of the inter-myofibrillar water reflects the greater mobility of this spin population.
131
Accordingly, an increase in the T21 relaxation time is interpreted as an increase in the intra-myofibrillar spacing, and an increase in the T22 relaxation time is interpreted as an increase in the inter-myofibrillar spacing. The associated spin populations (signal amplitude) are used to infer the fraction of intra- and intermyofibrillar water. Both T2 components exhibit a strong correlation with WHC [314], as well as animal age and muscle type [315]. The relaxation time component T21 is known to correlate to sensory attributes of the meat: juiciness and taste [316–318]. CPMG data acquired in the presence of a gradient confirmed that both water populations are mobile [319]. Diffusion mapping at intermediate field has been used to confirm the spatial distribution of water [320]. In some experiments, a short T2 < 10 ms relaxation time component has been observed and attributed to water strongly associated with proteins. Intra−myofibrillar water
Inter−myofibrillar water Nuclei
Single Inter−fascicular water muscle fiber Perimysium
Myofibril Sarcoplasm Sarcolemma Endomysium (between fibers) Fasciculus Extra−fascicular water
Epimysium (deep fascia)
Muscle
Figure 38: Illustration of the structuring within mammalian muscle, adapted from [313].
Early post mortem muscle is converted to meat by the process of rigor mortis: muscle fibers contract longitudinally and fuse, expelling intra-myofibrillar water. Water redistribution also occurs at a cellular level due to changes in ionic strength (salt content) [321–323] and hence osmotic transport across cell membranes. The shrinkage of muscle fibers alters the mobility of the water and is
132
monitored by changes in T2 . These processes are summarized in Fig. 39 alongside typical T2 results for porcine muscle (M. longissimus dorsi) [324]. Swelling of the myofibrils is observed as an increase in both the relaxation time and amplitude of the T21 component, Fig. 39 (upper graph). As contraction of the fibers overcomes the osmotic swelling, the amplitude of the T21 component decreases, whereas the amplitude of the T22 component increases, see Fig. 39 (lower graph). Other factors contribute to the fibril spacing (and hence water content), such as pH [314, 321, 323, 325], which results in lateral shrinkage of muscle fibers. The change in T2 behavior shown in Fig. 39 is interpreted as intra-myofibrillar water moving to the inter-myofibrillar space and then into gaps between the cells, fasciculi, and fibers as the cellular membranes destabilize [326, 327]. The shrinkage of muscle fibers reduces the WHC of meat. This process is countered by the proteolytic degradation of cytoskeletal proteins which prevents cell shrinkage; proteolysis causes the meat to become more tender. The swelling of the myofibrils during ageing results in an increase of intra-myofibrillar water and hence an improved WHC [318]. The formation of channels with diameter 20 − 50 µm allows free water to move to the surface of the meat, leading to “drip” during ageing [328]. The correlation between water environment and T2 has provided a facility for determining the factors that influence meat quality. For example, pork quality depends on antemortem stress and stunning method [329]. The presence of the porcine major gene Rendement Napole (RN) also affects WHC [330, 331]. WHC is improved by the addition of water-binding agents; conventional active waterbinding compounds include sodium chloride, sodium phosphate, sodium lactate, calcium lactate, lactic acid, and calcium chloride. However, T2 measurements
133
100
contraction rate T21 time T21 amplitude
48 99
98
46
60 44 40 42 20
40 97
80
38 0
4
8
12
16
20
contraction rate
50
T21 / s
T21 amplitude / %
100
0 24
Time / h Muscle fibers
cellular swelling
myofibrillar shrinkage and longitundinal contraction
disruption of cell membranes
Myofibril
4
80
3
60
2
40
1
Py-value T22 amplitude
0 0
4
8
12
16
20
Py-value
T22 amplitude / %
drip channel formation
20 0 24
Time / h
Figure 39: Illustration of the correlation between T2 , water environment, and structure in post mortem porcine muscle, adapted from [324]. The upper graph shows the variation in the short T21 transverse relaxation component in post mortem porcine muscle, along with the muscle fiber contraction rate [unspecified units]. The lower graph shows the variation in the long T22 transverse relaxation component in post mortem porcine muscle, along with an electrical impedance measurement (Py-value [unspecified units]). The impedance provides an indication of mobile water content.
134
have revealed that better WHC is achieved in sodium pyrophosphate and sodium bicarbonate enhanced meat [332]. The bicarbonate also resulted in less protein denaturing on cooking. Elsewhere, the water content of meat during cooking has been investigated by low-field NMR [317, 327, 333–335]. Storage conditions prior to cooking influence the meat quality as well: fast chilling provides an improved WHC [330, 336, 337]. Low-field NMR has been applied to studies of poultry meat where the use of sodium bicarbonate as a water-binding agent has been considered [338]. The stability and performance of mechanically recovered poultry meat in the presence of enzymes has been studied using T1 and T2 relaxation time analysis [339, 340]. Comprehensive reviews of the applications of low-field NMR in meat science are presented in [328, 341]. 7.1.2. Fish Low-field measurements of fish muscle are very similar to those of mammalian muscle discussed in the previous section. Again, two transverse relaxation time components are commonly observed in CPMG experiments with T21 ∼ 40−60 ms and T22 ∼ 150−400 ms. Like mammalian meat, the WHC of fish meat can be predicted based on relaxation time analysis [342, 343]. However, there are some differences to the interpretation. Fish have a high fat content, making separation of the oil and water signals difficult based on relaxation time analysis alone. Seasonal variations in salmon fat have been shown to give rise to very different relaxation times [344]. Fat and water separation is achieved efficiently using diffusion-weighting to remove the water signal, thereby allowing the fat content to be determined [345]. Appropriate pulse sequences for diffusion-weighting are discussed in Section 4.2. Two-dimensional diffusion-relaxation correlations, as 135
described in Section 4.3.2, have shown that the T22 relaxation time component in fish contains a significant signal contribution from fat [346]. The fat content of live salmon is determined in vivo using a NMR MOUSE (Section 3.1.2) where the inherent magnetic field gradient of the unilateral device provides automatic diffusion suppression of the water signal in the T22 component [347]. Low-field NMR has also been used as a non-invasive measurement of oyster mass and phenotype [348]. The convention of relating the short T21 component to restricted or bound water, and the long T22 component to mobile water/fat, in fish muscle is considered appropriate when the water content is high. However, relaxation time measurements of low water content systems, such as dried fish, need a different interpretation. Recently, models describing water adsorption on macromolecules and proton exchange have been introduced to explain the relaxation phenomena observed in low water content food products [349]. The texture [350] and quality [351] of frozen fish has been monitored using T2 relaxation time analysis. However, conflicting reports on the mobility of water determined during storage does raise some questions regarding the robustness of interpretation in these relaxation time measurements [352–354]. Salting and desalting also have a profound influence on the water content distribution in cod [355–359], salmon [356, 360, 361], and shrimp [362]. Some of these studies were supported by high-field 23 Na imaging [355, 360, 361] although given the advances in modern low-field spectrometers, sodium images could now be obtained with bench-top imaging systems. Low fields are advantageous for 23 Na measurements as the signal loss due to quadrupolar interactions is minimized. Exposure of fish fillets to high salinity brine was found to decrease the fraction of high mobility
136
water (long T2 ), whereas lower salinity brines promoted the uptake of water. Relaxation time analysis also revealed differences in muscle structure between wild and farmed fish, and reflected muscle structure differences according to the degree of antemortem stress. The effects of chilling method prior to storage on water content have also been studied [363, 364]. Detailed reviews of low-field NMR in fish processing are presented in [364, 365]; a table listing the published applications to fish and fish products is given in [364]. 7.1.3. Dairy Dairy products present an interesting challenge for low-field NMR. The longitudinal and transverse relaxation rates of the water fraction in milk and cheese are proportional to the protein content [366]. However, the mixture of oil (fat) and water often means that, whilst relaxation time analysis provides some information, interpretation of product structure based on T1 or T2 analysis alone can be uninformative. Diffusion is the ideal probe for these complicated mixtures of liquids and/or soft solids. Notwithstanding, there are plenty of examples in the literature of dairy product characterization by relaxation time analysis. The water and fat content of cream cheeses has been probed by T2 relaxation where a bi-exponential recovery is observed [367, 368]. Gel formation in cheese is similarly followed; a third (long) T2 component is observed when the cheese is mechanically stressed resulting in the release of (bulk) whey water [369]. The uptake of salt and the water content of Feta cheese during brining has been monitored with a combination of relaxation time analysis and MRI [370]. A combination of low-field relaxation time and high-field spectroscopy has been used to elucidate the process of protein digestion: the in vitro digestion of Parmigiano Reggiano cheese at different ages was observed [371]. The authors argued that the low-field NMR technique 137
would provide a route to on-line monitoring of digestive processes. Similarly, T1 relaxation time is a good indicator of albumen quality in hen eggs: another possible on-line monitoring application [372]. NMR spin relaxation in ice has been used to study crystal formation in ice cream [373] and probe the solid-to-fat ratio [374]. Relaxation time measurements also reveal fluid-phase (whey) separation in acidified milk drinks [375] and the liquid fat content in pumped milk [376]. −8
10 10 10 10
D / m2 s−1
10 10 10 10 10 10 10 10
Skimmed milk
Whole milk
Yogurt
Clotted cream
Cream cheese
Brie
Mozzarella
Monterey Jack
´ Pont l’Eveque
Cheddar
Raclette
Gruy`ere (1)
Emmentaler
Gruy`ere (2)
T´ete de Moine
−9
−10
−11
−8
−9
−10
−11
−8
−9
−10
−11
10
−3
10
−2
10
−1
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10 10
−3
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−2
−1
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10 10
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−3
10
−2
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−1
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0
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−3
10 10
−2
10
10
−1
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0
10
1
T2 / s
Figure 40: D-T2 correlations for a selection of dairy products, obtained at ν0 = 5 MHz using a fixed field gradient. In each plot the horizontal dashed line indicates the bulk diffusion coefficient of water at ambient conditions; the diagonal dashed line indicates the canonical “alkane line” corresponding to the D/T2 ratio of saturated hydrocarbons. The contour intervals correspond to 10 % steps up to the maximum intensity in each plot. Data previously presented in [141, 366].
Diffusion provides an alternative method of probing structure in dairy products. The ability to determine emulsion droplet size distributions, as described in Section 6.4, has been used to characterize cheese (protein-stabilized oil-in-water emulsions) [377]. M´etais and Mariette proposed the inclusion of a T1 -null to sup138
press the continuous oil phase for monitoring water-in-oil emulsions [378]. However, as noted in Section 4.2, modern bench-top spectrometers now have the capability to acquire diffusion coefficients for both liquid phases in a single experiment [262, 263]. Diffusion coefficients have also been used to probe the fat-water interface in emulsions [379] and the tortuosity of fat crystal networks, with the crystal S/V determined by relaxation time analysis [380]. The stray field of unilateral devices is useful for providing an inherent diffusion-weighting in T2,eff measurements, as demonstrated with the NMR-MOUSE as a device for in situ monitoring of dairy product quality [381]. Reviews of low-field NMR applications to food emulsions are given in [141, 382, 383]. Improved characterization of dairy products is achieved using 2D D-T2 correlations [366]; see Sections 4.3.2, 4.5.2, and 5.5.1 for further details. Enhanced separation of oil (fat) and water signals is achieved in the 2D plots; examples from a selection of dairy products are given in Fig. 40. Low fat content products (milk, yogurt) exhibit a strong signal at the bulk diffusion coefficient of water. As the fat content increases (cream, soft cheese) a signal associated with the oil (fat) emerges with a much slower diffusion coefficient. In all the examples, this fat component lies on or near the canonical “alkane” line showing the expected D-T2 response of saturated hydrocarbons; further details are given Section 7.3.1. In the solid-fat cheese, the mobility of the water is restricted and the diffusion coefficient reduced below the bulk water line. It is clear from Fig. 40 that the D-T2 correlation provides a robust and reliable method of characterizing the properties of dairy products. As an aside we note that this application of the D-T2 experiment has derived from the oil industry where salty dairy products (specifically cheese) provide a convenient proxy to the mixture of oil and brine found in a petroleum
139
reservoir. 7.2. The built environment Civil engineering and construction is a world-wide industry. Here we consider three areas where low-field NMR is providing increased understanding of construction materials: cement, wood, and the application of mobile NMR for conserving ancient buildings. 7.2.1. Cement Concrete, formed by mixing cement (or lime mortar) with aggregates has been used for thousands of years as a construction material. Portland cement is the most common form and consists of calcium silicates and metal oxides. When water is added, cement hydrates to form a structure of calcium silicate hydrate (CSH) gel [384]. The initial stages of hydration occur quickly, on the order of seconds to minutes. However, cement continues to react throughout its lifetime leading to structural changes that occur over many years. These slow processes can cause in-service failures, notably reinforcement corrosion that results from transport of sodium and chloride ions to embedded metal surfaces [385]. Despite the long history of cement as a building material, it is only recently that the structure of the CSH gel is being elucidated, partly due to the insights available from NMR studies. High-field spectroscopy has been used to explore the atomic arrangements in CSH and reviews of these measurements are to be found in [386, 387]. Fundamental insights into the arrangement of CSH structures have come from low-field NMR studies of the hydration water [388]. Better understanding of the nanoscale structure and water transport will enable the design of materials with greater strength and endurance in hostile environments, as well as enabling the prediction 140
of long-term properties and behavior of cements and concrete in the built environment. There is also a continual drive to develop improved building materials that are low in cost and have a reduced carbon footprint compared to conventional solutions. For example, global concrete production generated approximately 536 million tonnes of CO2 during 2010 [389]. Calcium silicate hydrate is a complicated material whose gel morphology depends on the available water for hydration (the water-to-cement ratio w/c), the cement formulation (initial composition plus additives), and environmental factors (temperature, humidity). Furthermore, the morphology alters with degree of hydration (time) and all these factors combine to make CSH gel a difficult material to study. Disordered on the nano-scale, CSH is formed through a dissolutionprecipitation reaction of tricalcium silicate (Alite) and dicalcium silicate (Belite) in water. CSH structures exist over a hierarchy of length-scales from nanometer thick sheets of Ca(OH)2 up to macroscopic cracks formed through chemical shrinkage and surface drying. To predict the performance of cement and concrete it is necessary to understand water transport over all these length-scales. Most low-field laboratory NMR experiments have been conducted on white (architectural) cement. This formulation is similar to Portland cement except for a significantly lower concentration of paramagnetic metal oxides and is therefore preferable for NMR analysis. The interpretation holds for Portland but the analysis (notably of T2 distributions) is more robust in white cement where internal gradients are negligible. Numerous interpretations of T1 and T2 relaxation time measurements have been proposed. Schreiner et al. introduced a three-phase model of 1 H relaxation that has been applied widely to studies of cement [390]. Bound water chemically combined in the hydration solids has a long T1 > 100 ms
141
and short T2 ∼ 10 µs. Water in the gel pores of the CSH structure exhibits short T1 and T2 on the order of T1,2 = 0.5 − 1 ms. Water in large capillary pores formed by chemical shrinkage exhibits long T1,2 ∼ 5 − 10 ms. Three exponential components fitted to a combination of FID and CPMG data correspond to the model of Schreiner et al. as noted in [391]. The component amplitudes also correspond to the fractions of water expected in each environment based on Neville’s theory of cement hydration [384]. More complicated five-phase [392] and seven-phase models [393] have been proposed that introduce concepts of water layers on pore surfaces. 5
P[log10(T2 )]
4
w/c = 0.48
3
w/c = 0.40
2
1
w/c = 0.32 0 −6
10
−5
10
10
−4
−3
10
−2
10
10
−1
T2 / s
Figure 41: T2 relaxation time distributions of cement pastes prepared at different w/c ratios after 10 days hydration. Data previously presented in [394].
A new interpretation of relaxometry data in hydrating cement was presented recently [394], supported by x-ray diffraction measurements and based on an earlier model [395]. A combination of solid (quadrature) echo and CPMG decays (see Sections 4.1.2 and 4.5) revealed five discrete relaxation time components as shown in Fig. 41. The quadrature echoes provided a short Gaussianlike and exponential-like T2 component [396]. The Gaussian-like component had T2 ∼ 10 µs and was attributed to chemically combined water in the crystalline 142
calcium hydroxide and ettringite (hexacalcium aluminate trisulfate hydrate) structures. The liquid-like signals (i.e., Lorentzian) were deconvolved from CPMG data to provide T2 ∼ 100 µs (water in CSH interlayers), T2 ∼ 400 µs (water in CSH gel porosity), T2 ∼ 1 ms (water in interhydrate spaces), and T2 ∼ 10 ms (water in capillary porosity). All the water in the sample was observed and the rate of signal loss with drying was as expected [397]. These measurements were implemented on a Bruker Minispec bench-top system operating at ν0 ≈ 7.5 MHz and highlight the capability of modern spectrometers to provide robust and reliable NMR analysis of cements. Based on surface relaxivity estimates in earlier work [152], the authors suggested characteristic length-scales of 0.85 nm (CSH interlayer spacing), 2.5 nm (gel pore size), and 8 nm (interhydrate pore size) for the different structures. The NMR signal intensity is rescaled to hydrate density, allowing the mass (and volume) fractions of the different cement phases to be determined [397]. The high temporal resolution of the NMR data acquisition permits monitoring of the structural changes during hydration [394], as demonstrated in Fig. 42 for white cement pastes prepared with different initial water-to-cement ratios. The evolution of porosity in hydrating cement has been elucidated through 2D T1 -T2 correlations [59, 143] and field-cycling experiments to probe changes in the specific surface area of the CSH structure [398, 399]. Exchange of water between CSH interlayers and gel pores has been explored using 2D T2 -T2 analysis (see Section 6.3) [152, 400]. Relaxometry has also been used to monitor the hydration of cement in the presence of additives [401–403] including light-weight aggregates [404] and controlled water release polymers [405]. The behavior of lime-pozzolan mixtures have been considered [406], as well as the hydration of
143
cement-wood composites [407]. Elsewhere, drying of cement has been monitored [408] and the mobility of water in cement has been assessed with diffusion measurements [409]. w/c = 0.32
Mass fraction
1
w/c = 0.40
w/c = 0.48
0.8 0.6 0.4 0.2 0
Volume fraction
1 0.8 0.6 0.4 0.2 0 0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
Degree of hydration free water
gel pore water
CSH
ettringite
Ca(OH) 2
unreacted cement
void
Figure 42: Compositions of cement pastes by (top) mass and (bottom) volume as a function of hydration for pastes prepared at different w/c ratios, adapted from [394]. The compositions were determined from NMR signal intensities, except for the void volumes which were determined from chemical shrinkage experiments.
Beyond relaxation time analysis, the gel pores of size < 100 nm can be characterized using cryoporometry (see Section 6.1) [410]. Water distribution [411] and fractures [412, 413] are assessed using robust broadline imaging techniques such as SPRITE (see Section 4.4.5). The ingress of surface coatings has been monitored using STRAFI [391, 414] (see Section 4.5) and SPRITE [415]. A review of MRI studies of cements is presented in [416]. So far we have considered the monitoring of hydration kinetics and elucidation of nano-scale structuring in cements through bench-top analyses. However, there is interest in monitoring the long term in situ evolution of cements and concretes 144
using portable NMR devices. There have been several solutions to this challenge. Rf sensors may be embedded in the poured concrete and then interrogated when a mobile magnet is located nearby. The concept of embedded sensors was introduced in [417] and discussed in [230, 418]. Alternatively, single-sided sensors (see Section 3.1.2) may be used, circumventing the requirement of embedding a sensor during construction. The use of the NMR-MOUSE as a mobile device for monitoring hydration and drying was demonstrated in [230]. Elsewhere, the dedicated Surface-GARField magnet was developed to examine the water content in reinforced concrete structures in the 50 mm of material above the reinforcement bars [44]. The Surface-GARField, based on the concept of a linear Halbach array, provides a plane of uniform magnetic field that is stepped through the concrete. Two-dimensional T1 -T2 relaxation time correlations are obtained from water in the sensitive plane to enable improved interpretation of the concrete pore structure [59, 419]. Here, we have focused on the use of low-field NMR for studying microstructure and hydration in cementitious materials. We note that similar studies have been published for gypsum plaster, a light-weight alternative to cement used primarily as a surface coating for interior wall and ceiling and sold commercially as pre-fabricated “plasterboard”. The manufacture of dry plaster powder is energy intensive and there is a drive to optimize the drying process. The hydration of plaster powder results in the formation of a porous solid, the properties of which depend on the pore structure. Low-field NMR has been used to monitor the structural changes that occur during the hydration process [230, 420, 421]. As gypsum plaster does not generally contain paramagnetic impurities, the hydration process and structure-transport relations have been studied in detail at an intermediate
145
magnetic field strength [147, 422–424]. 7.2.2. Wood The requirement for low cost, renewable construction materials is driving a global interest in sustainable forestry [425]. Wood may be used directly as timber or reformed into modern composite materials such as plywood. Beyond its application as a building material, wood is also used in paper production and as an energy source. The water mobility and cellular structure of wood have been probed by laboratory NMR using relaxation time [426–428] and diffusion measurements [429, 430]. Transport of water between cells has been monitored using 2D T2 -T2 exchange measurements [431]. The tree rings in Scots pine sapwood have been observed using MRI [432]. The process of water adsorption [433] and desorption [434] in sawn wood, of relevance to the production of timber, has been investigated. Adsorption and effectiveness of wood treatments have been examined using the NMR-MOUSE [435]. Elsewhere, wood composites have been studied [436, 437]. The forestry industry has a particular interest in monitoring the transport, uptake, and loss of water in living trees. Portable NMR devices offer the opportunity to non-destructively probe in situ the water distribution in trees. These measurements can be used to infer the health of the tree and the quality of the wood. Monitoring water flow in plants with MRI was pioneered in the laboratory by van As and co-workers [438]; a recent review of the subject is presented in [439]. Specialized mobile magnets have been constructed to examine trees in orchards [41, 64, 440, 441]. In the most recent example an open access imaging magnet (operating at ν0 ≈ 1 MHz for 1 H and capable of investigating tree trunks up to 100 mm in diameter) dubbed the “Tree Hugger” has been used to verify the spi146
ral uptake of water in the sapwood of a bird cherry tree due to the phenomenon known as grain rotation [41]. Low-field imaging methods are discussed in Section 4.4.4. Such large bore, low-field portable magnets will be valuable in future studies of environmental influences on tree growth and health. Portable magnets are also used to study the condition of fruit prior to harvest [442]. 7.2.3. Conservation An aspect of the built environment where mobile NMR is providing a significant contribution is in the conservation of ancient buildings and monuments. Damage to ancient stone, concrete, brick, and mortar can come from a number of sources: water condensation and evaporation leading to erosion or salt dissolution, biological attack, and pollution (notably from motor vehicles in urban environments). Hydrophobic treatments provide an obvious solution to the problems of weathering by water invasion. We have mentioned already the use of MRI (STRAFI and SPRITE) for monitoring coating ingress into cement-based materials (see Section 7.2.1) [391, 414, 415]. Elsewhere, the more relevant challenge of monitoring surface coating efficacy on stone has been addressed [443–449]. Relaxation time analysis is useful for determining the effect of these treatments. For example, Sharma et al. showed that stone strengtheners fill the largest pores in stone and reduce the mobile water content [450]. Mobile devices like the NMR-MOUSE are enabling the measurement of pore size distributions in ancient stone structures [56], plus in situ determination of hydrophobic treatment efficacy [451, 452]. The NMR-MOUSE has also been used to assess the moisture content in plasters and mortars behind ancient frescoes (wall paintings) [453–456]. The presence of sub-surface water influences the choice of preservation technique applied to these paintings and mobile NMR is one of the 147
few non-destructive methods capable of mapping moisture content in these rare samples. The NMR-MOUSE has been used to observe capillary uptake of ground water into painted plaster before [457] and after treatment [458, 459]. A recent development – the Profile NMR-MOUSE capable of high-resolution 1D profiles into a remote sample [61] – is providing new insights into the structure of ancient wall paintings. The Profile NMR-MOUSE is capable of resolving multiple layers of paint and mortar, as well as providing a moisture content measurement. The Profile NMR-MOUSE has been applied to stones and mortars [68, 449] as well as a study of mosaic tesseræ [68]. Coatings such as paint can have negative consequences for the underlying stone or cement-based material. Some paints prevent evaporation of water, exacerbating damage from environmental effects and salt crystallization [460]. Salt redistribution is driven by water migration, notably surface drying.
23 Na
MRI
provides a method of monitoring salt transport in building materials [461, 462]. Similarly, salt crystallization which can result in damage to the pore structure [463] has been monitored [464], and the presence of salt has been observed to increase water adsorption in carbonate rocks [465]. Where possible, controlled humidity is expected to reduce damage to ancient building materials caused by salt migration. NMR, both at low field and high field, has been used in a range of other conservation projects including the study of paper in historic documents [57] and fired-clay pottery [466, 467]. A complete review is presented in [58]. 7.3. Petrophysics Low-field NMR plays two important roles in oil and gas production: welllogging tools provide access to “reservoir-scale” measurements of the formation 148
fluids, whereas bench-top instruments are used to examine fluids and cored plugs of rock recovered from the reservoir. Interpretations of well logs are supported by laboratory-scale measurements. Bench-top NMR systems provide fluid characterization, (see Section 7.3.3), rock properties (see Section 7.3.2), and a platform for trialling new recovery methods prior to single-well pilots in the reservoir [142, 468, 469]. 7.3.1. Well-logging The most successful commercial application of ex situ NMR has been welllogging for the petroleum industry [45, 73]. A brief history of the development and design of well-logging tools was presented in Section 3.1.3 [74, 470, 471]. We now address the measurement capability of these instruments. Well-logging tools are used in three situations: 1. Exploration. A new well (typically uncased) will be logged to determine the formation type, fluid content, and physical properties (porosity, permeability) as a function of well depth. These parameters will be used to determine whether production is commercially feasible and provide estimates of production volumes. Rock cores extracted during drilling will be measured in the laboratory to provide additional information not available from downhole analysis and support the interpretation of the well logs. The log will be acquired while the tool-string is in motion. 2. Piloting. A relatively new concept, the single-well in situ enhanced oil recovery (EOR) pilot provides a rapid, cost-efficient method of trialling new recovery methods in a reservoir formation. Oil recovery factors measured by NMR have been shown to agree with equivalent estimates obtained from
149
coherent dielectric logs and the conventional single-well chemical tracer test. The tool will be stationary while the logging data are acquired. 3. Monitoring / surveillance. A cased well is used to monitor the transport of oil or gas through a formation to obtain estimates of total production volumes and lifespan of the reservoir. The tool may be stationary while the logs are acquired. The archetypal data acquisition for NMR well-logging is a transverse relaxation decay obtained using a modified CPMG pulse sequence [77, 190, 192, 472]. These data may be acquired when the tool is stationary in a cased well, as the tool is retracted along an open or cased hole, or in logging while drilling (LWD) operations. The CPMG data are inverted to form a T2 distribution using methods similar to those described in Section 5.4.1, as introduced to petrophysics by Brown et al. [473]. A T2 distribution has the potential to provide a wealth of information regarding both the fluid and the rock matrix environment. An example T2 distribution for a water saturated sandstone is presented in Fig. 43. The total integral area under the distribution is proportional to the porosity of the formation; this is a particularly useful measurement as the NMR-derived porosity is independent of rock lithology and Archie parameters [474] that influence other logging tools. It is usual to analyze T2 distributions by positioning so-called T2 cut-offs across the distribution to segregate the regions of interest, as illustrated in Fig. 43. The area under each segment is obtained by integration; in sandstones the volume of clay-bound [475], capillary-bound, and mobile water is determined. The mobile water fraction corresponds to the porosity available for transport of fluids (water, oil, gas) and is referred to as the “producible porosity” [476]. However, care must be taken when analyzing logs (or core-plugs) obtained near the well bore 150
1 0.8 0.6
0.01
0.4
P{log10(T2 )}
producible fluids
clay capillary 0.02 bound bound water water
P
P{log10(T2 )}
`s
0.2 0 −3 10
10
−2
−1
10 T2 / s
10
0
0
Figure 43: Simulation of a typical T2 probability distribution (blue line, left ordinate) obtained for a water-saturated sandstone. The water is present in three environments: (left) clay-bound, (center) capillary-bound (trapped in small pores) and (right) “free” (present in large pores). These environments are delimited by T2 cut-offs (vertical black lines). The large pores are assumed to be well connected and control fluid transport; hence the area under the portion of the distribution corresponding to “free” water is considered a measure of the producible fluid volume. As described in Section 6.1, the T2 can be rescaled to pore size `s if an appropriate value of the surface relaxivity
ρ2 is known. The T2 cumulative probability (green line, right ordinate) shows the sum over the distribution; the total liquid signal (here, normalized to unity) is equivalent to the total porosity of the formation.
151
as drilling fluid infiltration will adversely influence relaxation time interpretations [477–479]. In clastic formations, notably well-sorted sands, a strong correlation exists between pore body size and permeability (determined by pore throat size) [480, 481]. It is therefore possible to infer a hydraulic permeability from
κ = Cs φ 4 (T2LM )2 ,
(92)
where φ is porosity, Cs is an empirical constant [482], the logarithmic mean T2 is defined as log10 (T2LM ) = hlog10 (T2 )i =
∑ [Pi log10 (T2i )] i
∑ Pi
i
,
(93)
and Pi represents the probability (amplitude) of the ith T2 component. Alternatively, the T2 cut-off corresponding to the producible porosity may be used instead (with an appropriately modified constant Cs ) as this better represents the earlier concept of relating permeability to the ratio of free to bound water [483– 485]. This comparison between T2 and permeability is dimensionally consistent through the scaling of ρ2 T2 having units of length, i.e., a pore body size according to eq. 71 (see Section 6.1), whilst permeability has units of length squared [248]. Surface relaxation in rocks occurs predominantly through adsorption on crystalline defects in the exposed mineral surface [486]. Additional relaxation will occur in clastic formations if paramagnetic metal species (e.g., iron or manganese oxides) are present, usually associated with clays [487]. Consequently, the constant cs must incorporate some empirical estimate of the surface relaxivity ρ2 . Independent estimates of ρ2 are difficult to obtain in sandstones: clays prevent quantitative determination of S/V using BET gas adsorption [248] and influence 152
diffusion measurements [488]. Instead, ρ2 may be estimated based on mercury intrusion porosimetry (MIP) [489–491] or capillary pressure measurements (centrifuge desaturation) [492, 493], again assuming a fixed relationship between pore throat size (as determined by MIP) and pore body size (as determined by NMR) [473, 475, 476, 494]. Recently, an alternative determination of ρ2 was proposed for sandstones based on independent MRI measurements of grain size [495]. Under favorable circumstances the T2 distribution can be used to determine the hydrocarbon content of the reservoir. A T2 cut-off is used to separate signal associated with the aqueous (brine) and hydrocarbon (oil, gas) fluid-phases. In light oil / gas reservoirs, the T2 of the hydrocarbons is typically longer than that of capillary-bound water. However, there will be overlap between the hydrocarbon and mobile water signal. Heavier oils will have shorter relaxation times (see Section 7.3.3 for more details) and may be distinguishable from the mobile water [142, 469]; wettability of the rock surface will play an important role in determining the actual relaxation times of the fluid-phases. Real wireline well logs are complicated to interpret, containing the results from many different tools mounted on the tool-string; examples are to be found in [45, 46]. Here we present an idealized NMR well-log in Fig. 44 to show the type of information typically obtained. The T2 cut-off would be determined to provide producible porosity and oil saturations consistent with the coherent measurements obtained from other logs such as gamma-ray, resistivity, and dielectric. The T2 distributions are stacked as a function of well depth, and this implication of spatial dimension has resulted in NMR well-logging being considered an imaging protocol; tools such as Schlumberger’s MRScanner1 can also provide information at different depths into the formation, adding an extra spatial dimension.
153
Porosity Perm T2 distribution p.u. mD ms 30 0 50 0 0.3 3000 A
Well depth
B
C
Figure 44: Simulated well-log response for a T2 measurement of a sandstone reservoir containing brine (water) and a moderately heavy oil, η ∼ 100 cP. Individual T2 distributions are stacked on the right as a function of well depth. A T2 cut-off (red vertical line) has been determined to provide differentiation between oil (short T2 ) and mobile brine (long T2 ). Capillary-bound water is present, indicated by the very short (low intensity) T2 components. A permeability estimate is included (center) based on the T2 cut-off. On the right, the total porosity of the formation is shown (dashed line) along with the producible oil (black area) and mobile water (blue area). The formation is divided into three zones: (A) a low permeability clay-based cap-rock, (B) a high permeability oil reservoir, and (C) a moderate permeability sandstone below the free water level. In reality, such NMR data would be interpreted alongside numerous other coherent well logs with support from laboratory core analysis.
154
Isolated pores
−1
0
1
Coupled pores
−1
10 10 10 ρ2 T2 / µm
0
10 10 10 ρ2 T2 / µm
1
Figure 45: Simulation of the effect of pore-to-pore coupling on T2 distributions obtained in carbonate formations, adapted from [496]. Diffusion of water occurs between intra-granular microporosity (red shading) and inter-granular macroporosity (unshaded). In all cases, the micropores constitute 17 porosity units ( p.u.) and the macropores constitute 14 p.u. porosity. The five cases (top to bottom) represent different surface relaxivities of ρ2 = 1.5 µm s−1 , 7.5 µm s−1 , 37.5 µm s−1 , 187.5 µm s−1 , and 937.5 µm s−1 . In the isolated pores (left) the short T2 component associated with the microporosity remains constant whereas the long T2 component associated with the macroporosity broadens as ρ2 increases and relaxation shifts from the fast diffusion limit to the slow-diffusion limit. In the coupled pore system (right) exchange causes the long T2 component to shift until the microporosity and macroporosity are indistinguishable. As the surface relaxivity increases, the effect of exchange becomes less significant and is eventually dominated by surface relaxation.
155
Historically, many NMR studies of rocks have focused on sandstones due to their microstructural simplicity. However, understanding of fluid transport in carbonate reservoirs is of significant commercial interest. Unlike sandstones, it is possible to determine an accurate surface relaxivity in clay-free carbonate formations by comparing the initial relaxation rate to a measure of surface area from BET gas adsorption [248]. Carbonate formations have variable surface relaxivities, but the values are usually small and within the range 1 µm s−1 < ρ2 < 3 µm s−1 [248]. When the surface relaxivity of a carbonate formation is unknown, the industry standard “unit-normalized” value of ρ1 = 1 µm s−1 is used to obtain an estimate of pore body size. Attempts to infer permeability from the T2,LM can succeed in carbonates using a modified scaling of [248]
κ = Cc φ 2 (ρ2 T2LM )2 ,
(94)
when formations contain well-defined crystal or grain structures. This assumption breaks down in mud dominated formations where the pore body size and pore throat size are poorly correlated. Carbonates offer two additional challenges for NMR relaxation time analysis. 1. Diffusive coupling between microporosity and macroporosity [496] on the time-scale of the measurement removes the sensitivity of T2 to pore size; an apparent T2 is observed instead, as defined in Section 6.3, which will not rescale to pore body diameter. An illustration of the effect of poreto-pore coupling on the T2 distribution is given in Fig. 45. When surface relaxation is weak (ρ2 D0 /`s ) then diffusion and hence exchange will dominate; conversely if surface relaxation is strong (ρ2 D0 /`s ) then the 156
true T2 distribution will be observed. Unfortunately, carbonate formations tend to exhibit weak surface relaxivity so exchange dominates over relaxation. Fleury et al. have used the T2 -T2 exchange analysis to quantify the mass transport in benchtop measurements of carbonate plugs [157]. 2. The presence of very large macropores (`s > 10 µm) in a carbonate formation with weak surface relaxivity also results in a loss of sensitivity to pore size in the T2 measurement. The T2,bulk relaxation rate term dominates over the surface relaxation ρ2 S/V term in eq. (70). If we assume ρ2 = 1 µm s−1 and T2,bulk = 2 s for water, then surface relaxation is indistinguishable from bulk relaxation in pores of size `s > 10 µm. The standard T2 distribution is not the only measurement available with welllogging tools. When T2 is insufficient for an accurate fluid-phase discrimination, a T1 -D-T2 correlation may be obtained instead [60, 139, 196, 197, 497, 498]. Under ideal conditions these data can be acquired while the tool is in motion, or though practically these correlations are usually measured when the tool is stationary for piloting and monitoring operations. The D-T2 portion of the measurement is described in Section 4.5.2; the T1 dimension is appended by varying the recovery delay between scans. The addition of the diffusion dimension provides clear discrimination between oil, gas, and water in most formations. An example D-T2 plot is given in Fig. 46. The solid lines in this plot correspond to the bulk fluid responses expected for each fluid-phase. When measuring the fluids in a reservoir formation, the influence of the pore geometry will modify the D-T2 response. Very small pores will cause a reduction in the apparent diffusion coefficient as described by Tanner and Stejskal for diffusing molecules in a confined volume [96]. However, whilst the diffusion measurement is sensitive to confinement in rocks 157
[149] the diffusion coefficient cannot be reliably scaled to pore size due to connectivity between neighboring pores [499]. Furthermore, water bound on mineral surfaces or clays will exhibit very slow diffusion. Notwithstanding, Zielinski et al. demonstrated that it is possible to generate an approximate restricted diffusion line for water [500] "
D (T2 ) = D0 1 − D 0
`ρ `∆ + (`∆ /`H )2 `ρ `∆ + (`∆ /`H )2 + D 0
#
,
where ρeff T2 is a proxy for geometric restriction, `∆ =
(95) √ D0t∆ , D 0 = 1 − D∞ /D0
(with D∞ denoting the observed diffusion coefficient as t∆ → ∞), and 1 4 `ρ = √ . 9 π ρeff T2,surf
(96)
The length `H represents the scale of macroscopic heterogeneities in the formation with `H `∆ so the term (`∆ /`H )2 → 0 in eq. (95). In mixed wet rocks
ρeff =
Sg,w ρ2 , Sg,w + Sg,o
(97)
where Sg,w represents the grain area contacted by water and Sg,o is the grain area contacted by oil. In a water-wet rock, ρeff ≡ ρ2 for all aqueous fluid-phases. We note, therefore, that fitting eq. (95) to a bench-top D-T2 correlation obtained from a water-saturated water-wet plug provides an alternative method of estimating ρ2 . Oils and oil-based muds (OBM) tend to lie near the canonical “alkane line” which is determined from the expected D/T2 ratio that depends on viscosity; see Section 7.3.3. D-T2 plots are therefore useful for identifying the occurrence of mud invasion. The maximum oil viscosity that can be probed with NMR diffusometry is η ∼ 100 cP. Live oils with a high gas-to-oil ratio (GOR) will shift up from this alkane line. Undissolved gases (as found in gas reservoirs) have very high diffusion coefficients and reside above the water line in D-T2 plots. The 158
concept of eq. (95) is applied to hydrocarbons as well as water, but large spreads in D and T2 due to variable viscosity complicate the generation of a modified alkane line [501]. Therefore, whilst the D-T2 plot provides excellent fluid-phase sensitivity in most environments, careful interpretation of these 2D plots is still required.
−7
10
−8
gas
D / m2 s−1
10
−9
water
10
−10
oil
10
−11
10
−3
10
10
−2
10
−1
0
10
T2 / s
Figure 46: Representation of a D-T2 correlation plot for various fluid-phases trapped in a microporous rock formation. The dotted lines indicate the expected D-T2 behavior for gas (red, bulk diffusion), water (blue, bulk diffusion), and oil (green, alkane line). The solid lines indicate the modified D-T2 behavior [500, 501] to account for restricted diffusion effects. The exact positions of these lines will be temperature and pressure dependent. In this example, a mixed-viscosity oil is shown: the light fractions provide a signal component that lies on the expected D-T2 line; heavy fractions (viscosity η > 100 cP, including asphaltenes) diffuse too slowly to be measured and result in a tail on the oil distribution. Brine, confined in small pores, lies on the water (blue) restricted diffusion line. Similarly, a small quantity of gas provides a peak on the gas (red) diffusion line. In reality it is unusual to observe all three fluid-phases in a single measurement. It is important to note that in this example, the diffusion dimension (projection, right) provides improved fluid-phase discrimination over the T2 dimension (projection, top).
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Another use of T2 analysis is the in situ estimation of wettability. The wettability is the tendency of a fluid to preferentially adsorb on a solid surface in the presence of other immiscible fluid-phases [502]. In the laboratory, wettability is usually characterized by either the Amott [503] or United States Bureau of Mines (USBM) indexes [504] which provide an indication of whether a mineral surface is preferentially wet by water, or by oil, or is mixed wet. NMR measurements of wettability provide a rapid alternative. The connection between relaxation and wettability was recognized by Brown and Fatt [505]. The details of extending laboratory measurements of wettability to well-logging tools have been discussed by Looyestijn [506, 507]. To infer wettability, the T2 relaxation time distribution of the oil measured downhole is compared to a bench-top T2 analysis of produced oil measured at downhole pressure and temperature. If the T2 of the oil in the reservoir is shorter than that of the produced oil, the in situ oil is undergoing surface relaxation and is therefore wetting the mineral surface, and the rock is assumed to be mixed-wet [508]. If no change in the T2 distribution is observed between the downhole and produced oil, the rock is inferred to be water-wet. Of course, a good separation of the downhole oil and water signals is required to make an unambiguous determination of the rock wettability and the D-T2 experiment again offers the advantage of improved fluid-phase discrimination [501, 509]. An improved indicator of wettability is achieved using the T1 /T2 ratio [14, 144] as discussed in Section 6.2. Presently, there is interest in determining wettability in unconventional shale reservoirs [501, 510, 511]. Reviews of wettability determination via NMR measurements are published in [512, 513].
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7.3.2. Laboratory core analysis All of the experiments available downhole are also available in the laboratory for studies of core-plugs. Plugs may be extracted from cored material and, if handled with care, will contain the connate fluids. However, it is more usual to clean cored material and resaturate with reservoir fluids for analysis. Laboratory relaxation and diffusion experiments are used to improve interpretation of well logs [468, 469, 476, 514]. General discussions on NMR core analysis have been presented in [515–517], with articles focusing on unconventional gas hydrates [518] and shales [519]. One significant facility of laboratory core-plug studies is the spatial resolution of petrophysical parameters. In a recent review we considered the use of MRI in core analysis [14]. During the 1990s many experimentalists attempted to image fluid distribution in core-plugs using intermediate-field biomedical “small animal” magnets operating between B0 = 1 − 3 T. However, the internal gradients generated within the pores at these field strengths prevented quantification of the fluid volumes, as demonstrated by Straley et al. [475]. Consequently, MRI lost favor amongst the core analysis community. Recently, with the advent of modern bench-top spectrometers offering improved SNR at low field, there has been a resurgence in 1D profiling of core plugs. Two areas of particular interest have evolved from these low field MRI studies. 1. Rapid capillary pressure measurements. 2. Monitoring of forced displacement of oil. We now consider examples of these applications. Capillary pressure is an important petrophysical parameter: fluid transport and saturation states are affected by capillary-driven processes [520]. Capillary pres161
sure is generated by the contact of wetting and non-wetting fluids at a solid interface. “Capillary pressure curves” describe the variation in capillary pressure as a function of fluid saturation state and are used in core analysis to calculate petrophysical quantities such as irreducible water saturation, free water level, and residual oil saturation in the reservoir. Oil recovery predictions and simulations rely on capillary pressure measurements and therefore accurate determination of capillary pressure curves has significant economic implications. A conventional capillary pressure curve is obtained using a centrifuge, where a core-plug is spun at a range of rotation speeds until the liquid distribution in the rock reaches equilibrium. The volume of liquid ejected is recorded as a function of centrifugal force [493, 521]. Assuming the capillary pressure at the outlet face of the plug is zero, the capillary pressure as a function of radius rc is ! 2 1 2 Pc (rc ) = ∆ρϖc2 rc2 , − rc1 2
(98)
where the angular velocity of the centrifuge is ϖc , ∆ρ is the density difference between the resident and displacing fluids, and the radii rc1,2 are associated with the inlet and outlet faces of the plug respectively. In the conventional centrifuge method a range of capillary pressures will exist within the length of the plug. However, only the average volume of displaced liquid will be observed. The average saturation S¯ is assumed to be a sum over all capillary pressures so 1 S¯ = rc2 − rc1
PcZ(rc1 ) 0
S {Pc (rc )} dPc (rc ) . ∆ρϖc2 rc
(99)
The Hassler-Brunner integral for average saturation is obtained by rearranging eq. (99) as ¯ c (rc1 ) = cos2 SP
PcZ(rc1 ) 1 rc1 cos−1 dPc (rc ) 2 rc2 0
162
S {Pc (rc )} × r h i h i . Pc (rc ) rc1 2 1 −1 1 − Pc (r ) sin 2 cos rc2
(100)
c1
Unfortunately there is a recognized flaw in this method to calculate capillary pressure: as the average saturation is measured, eq. (100) is only valid when rc1 ≈ rc2 (i.e., in short plugs). Other approximate solutions are also known to have limitations [522]. MRI profiles of fluid saturation acquired after centrifugation offer a solution this problem. Rather than measure average saturation for the entire plug, eq. (100) is applied to each pixel in the profile where the assumption that rc1 ≈ rc2 is valid. Each pixel (radial position) is associated with a different centrifugal acceleration even though the centrifuge is running at a constant speed. Therefore, under ideal conditions it is possible to determine the capillary pressure at all fluid saturations S = 0 → 1 in a single experiment, although practically two or three centrifugation stages are often required to construct a full capillary pressure curve. SPRITE (see Section 4.4.5) is an ideal profiling pulse sequence for these capillary pressure curve measurements [523–527], providing robust and quantitative saturation profiles [528]; an example is given in Fig. 47. The addition of a spatially resolved T2 dimension [167, 184] (see Section 4.4.4) provides correlations between capillary pressure and pore size, as well as providing a route to measuring oil-brine capillary pressure curves using a T2 cut-off for fluid-phase discrimination [14]. Relaxation time T2 mapping [167, 184] is also useful for monitoring forced displacement of oil by brine or enhanced oil recovery (EOR) agents, such as polymers. A T2 map allows the simultaneous detection of oil and aqueous fluid-phases in a core-plug utilizing a T2 cut-off as described in Section 7.3.1. The data produced from a laboratory T2 mapping pulse sequence, whether through frequency 163
Local Sw
1 0.8
a r c1
r c2
0.6 0.4 0.2 0 10
12
14
16
18
Pc / kPa
r c / cm 160 140 120 100 80 60 40 20 0 0
b
0.2
0.4
0.6
0.8
1
Sw
Figure 47: Capillary pressure curve measurement of brine draining from a Berea sandstone plug, obtained by SPRITE profiling. Data previously presented in [523]. MRI profiles (a) are obtained before (4) and after (◦) centrifugation. The vertical dashed line indicate the inlet and outlet faces of the plug; fluid-flow occurred from left to right. The ratio of the intensities in these two profiles, obtained on a pixel-by-pixel basis, provides the water saturation Sw as a function of position rc2 − rc1 , which gives Pc (rc ) from eq. (98). Hence a capillary pressure curve (b) is obtained for a wide range of saturation states in a single experiment.
164
[167] or phase encoding [184], is equivalent to a well-log as demonstrated in Fig. 44 except that the vertical resolution is on the order of millimeters rather than tens of centimeters. Mitchell et al. have used T2 maps to monitor oil displaced by brine [529], brine and alkaline surfactant [468], and a combination of brine, polymer, and alkaline surfactant [142, 469]. In the surfactant studies, excellent agreement was observed between the laboratory recovery and single-well in situ EOR pilots. Elsewhere, low-field NMR has been used to study emulsion formation for enhanced recovery of heavy oils [530]. It is also possible to obtain spatially resolved D-T2 maps for improved fluid-phase discrimination [531] and spatially resolved T1 -T2 correlations for wettability mapping [14]. Finally, we note that another important core analysis measurement available in the laboratory is the flow propagator. The propagator provides an indication of the average displacement and dispersion of liquid flow at intermediate to high interstitial velocities [115], plus the fraction of stagnant fluid in the pores [131, 132]. To date, very few low-field propagator measurements have been published [86]. However, significant efforts have been made to understand and interpret flow propagators obtained for rocks at intermediate field [115–117, 130], including fast [133, 252] and quantitative measurements [134]. At present, flow propagators are used primarily to support simulations of fluid transport [135, 136, 532, 533]. 7.3.3. Hydrocarbon characterization We have touched already on the idea of fluid-phase identification through relaxation time and diffusion coefficient measurements (see Section 7.3.1). Separation of oil and water signals in bulk liquid samples is straightforward and often possible from 1D relaxation time distributions [534]. Here we consider in more detail the quantification of crude oil components using diffusion [535] or relax165
ation [536] as a proxy for chemical spectroscopy at low field. Let us consider a mixture of alkanes as a proxy for a real crude oil, where the mth alkane has a length (number of carbon atoms) Nn > 2. An approximate scaling for the diffusion coefficient of the nth alkane is derived from the Einstein-Stokes relation such that [535] D = X N¯ −Ξ Nn−ξ .
(101)
At ambient conditions, mixtures of alkanes will be described by eq. (101) with the empirically determined pre-factor X = 3.5 × 10−7 m2 s−1 and exponents Ξ ≈ 1.73 and ξ ≈ 0.7. If a distribution of diffusion coefficients has been determined, a given component with diffusion coefficient Dn will be associated with an alkane of chain length 1 Ξ+ξ
D
1 ξ
E Ξ+Ξ ξ
−1
Nn = X D Dn ξ , (102) D E where D 1/ξ is the (1/ξ )th moment of the measured distribution f (log10 D) such that D 1E Z 1 D ξ ≡ d log10 D f (log10 D) D ξ .
(103)
A distribution of alkane chain lengths is obtained from the distribution of diffusion coefficients simply as f (log10 N) = ξ f (log10 D) .
(104)
An approximate scaling law also exists for the longitudinal and transverse relaxation rates. Assuming the viscosity η < 100 cP such that T1,n ' T2,n (based on the assumption that frequencies of motion in light oils are > 100 MHz so that T1 and T2 are frequency-independent at low field) then [536] T2,n = Y N¯ −ϒ Nn−υ .
(105) 166
In this case the empirical pre-factor Y = 672 s and exponents ϒ = 1.25 and υ = 1.24 [536]. Again, a distribution of relaxation times will have components T2,n associated with alkanes of chain length ϒ 1 ϒ+υ 1 −1 Nn = Y ϒ+υ T2υ T2,nυ ,
(106)
where Z 1 1 T2υ = d log10 T2 f (log10 T2 ) T2υ ,
(107)
and the alkane chain length distribution is given by f (log10 N) = υ f (log10 T2 ) .
(108)
The pre-factors and exponents in eq. (101) and (105) are pressure- and temperaturedependent as described in [537]. Although crude oils tend to contain a complicated mixture of organic molecules (unsaturated, branched, or cyclic components) these scaling laws have been shown to apply to light crude oils. A summary is presented in [538]. We note that the viscosity dependences of D and T2 are the origin of the bulk D/T2 line in Fig. 46. Elsewhere, Chen et al. found T2 correlated better to specific gravity than viscosity in heavy oils [539], and in other work multivariant analysis was used to provide a reliable oil viscosity prediction based on T2 relaxation times across a wide range of molecular weights [540]. The existence of either very light (N < 2) or heavy components (such as asphaltenes and waxes) will complicate the analysis [535]. A low-field NMR study of asphaltene content and aggregation is presented in [541]. Relaxation and diffusion experiments have also been applied to heavy oils [542, 543] and emulsified crude oils to determine stability, sedimentation, and coalescence in the emulsion [542, 544, 545]. We conclude by noting that although our discussion here has focused on characterization 167
of light crude oils, there are many examples in the literature of low-field NMR applied to unconventional heavy crudes classified as bitumen [546, 547] and pitch [548–550], and even solid hydrocarbons in the form of coal [551–554]. 8. Future Directions In this final Section of the review we look forward to some possible future developments in permanent magnet technology and low-field data analysis. 8.1. Hardware Recent developments in permanent magnet technology have countered many of the perceived limitations of low-field NMR. Improved pole-piece and gradient coil design, plus the inclusion of shim coils, have provided significant increase in field homogeneity. Advances in probe rf electronics and signal detection have improved the achievable SNR. The advent of fully digital spectrometers now means low-field systems are capable of reproducing most of the experiments employed at high field. At the moment, there is a trend for the production of permanent magnet benchtop instruments with intermediate field strengths in the range B0 = 1 − 2 T for spectroscopy and biomedical imaging. These magnets are still limited in some cases by thermal drifts which result in large changes in the B0 field (on the order ∼ 100 ppm) over several hours. Even with the best thermal regulation it is difficult to provide a temperature stability corresponding to < 1 ppm frequency drift over a long time. The almost inevitable inclusion of a B0 shim and lock channel, available on the latest generation of spectrometers, will increase the versatility of these bench-top instruments.
168
Improvements to portable magnets are likely to continue much as they have, with experiments previously considered impossible on unilateral devices now being implemented routinely. As with the bench-top systems, advances in magnet pole-piece geometry and mechanical shimming techniques will provide larger homogeneous volumes or regions of uniform gradient, improving signal quality and reducing acquisition times. An area that might develop is the implementation of “lab-on-a-chip” low-field microfluidic devices for process monitoring and control; such systems are currently being demonstrated in high-field magnets [555, 556]. As the world’s supply of liquid helium becomes increasingly scarce, driving up the cost of operating older superconducting magnets, more research laboratories will turn to low-field NMR as the technology of choice. Improvements in commercial bench-top platforms are driven by the consumer, so as experimentalists attempt to implement more and more complicated acquisitions and protocols, we shall see further hardware developments. 8.2. Experiments It is difficult to envision future directions for NMR experiments except to say that they will depend on hardware improvements discussed already. Experiments considered routine in high-field NMR (such as nD-spectroscopy correlations, velocity mapping) will probably become feasible at low field, although the underlying pulse sequences are not expected to change significantly. Rather than modify the underlying experiments, it is likely that sophisticated processing and analysis techniques will be introduced; some examples are discussed below in Section 8.3. One area that is likely to see an advance in the next few years is intermediatefield spectroscopy. The advent of compact, bench-top instruments operating in the range ν0 = 45 − 60 MHz and capable of providing useful spectroscopic resolution 169
will enable monitoring of flow and reaction systems in a commercial production environment [4]. These instruments have been used for reaction monitoring [66, 557–563]. In order to advance the applications of these instruments we should look to experimental methods of signal and sensitivity enhancement. Spectroscopic sensitivity can be improved by careful nucleus selection. For example, 13 C nuclei exhibit a much wider chemical shift range than 1 H. However, 13 C
is only 1.07 % naturally abundant and has a gyromagnetic ratio four times
smaller than that of 1 H. Therefore, a NMR system operating at ν0 = 45 MHz for 1H
will need to operate at ν0 ≈ 11 MHz for 13 C. Overall, natural abundance 13 C
measurements are 5870 times less sensitive than equivalent 1 H measurements. It will be necessary, therefore, to adopt signal enhancement techniques to probe carbon nuclei at low field. Current options include isotopic enrichment to increase the number of resonant carbon nuclei in a sample, or polarization transfer techniques such as proton-decoupled 13 C distortionless enhancement by polarization transfer (DEPT) [564]. An alternative to switching nucleus is the enhancement of the 1 H signal. Highfield solutions, such as cryo-probes, are unlikely to be introduced at low field as this would remove the advantage of cryogen-free operation. However, enhancement of spectral lines through para-hydrogen induced polarization (PHIP) enhancement may be an important technique in the low-field spectroscopy arena. Para-hydrogen signal enhancement techniques are reviewed in [565, 566]. This method of signal enhancement is particularly interesting for reaction monitoring as specific spectral lines can be enhanced, allowing the observation of intermediaries or low conversion products [567].
170
8.3. Analysis It is expected that the next significant advances in low-field NMR will occur at the data processing stage. Rather than improving the hardware or experiment, the data currently available will be analyzed in a more intelligent manner. We describe two such examples here. Bayesian analysis uses prior knowledge of the system and a model of the data to predict the most likely fit parameters from undersampled or noisy data. Compressed sensing (CS) is a modification to the Fourier transform that takes advantage of fact that NMR data, when transformed to some other domain (e.g., following a total variation or wavelet transform), is typically sparse. Significantly undersampled data can be reconstructed, leading to an improvement in acquisition time. 8.3.1. Bayesian inference Bayes’ theorem [568] relates the probability of a hypothesis (set of fitting parameters) to a set of measured data. Formally, this is expressed by the posterior probability distribution P(ϑ |ι ), which describes the probability of a hypothesis
ϑ given the measurements ι . The posterior probability distribution is calculated from P(ϑ |ι ) =
P(ι |ϑ )P(ϑ ) , P(ι )
(109)
where P(ι |ϑ ) is the likelihood of observing the measured data given that the hypothesis ϑ is true, and P(ϑ ) is the prior which describes our a priori knowledge of the system. The denominator in eq. 109, P(ι ), is called the evidence and describes the probability of measuring any set of data ι . In applications to parameter estimation, the evidence is a constant that acts only to normalize the probability distribution and can be ignored. 171
NMR data
Model data (likelihood)
Posterior probability
Prior knowledge
Most likely parameter
Figure 48: Illustration of inputs for Bayesian analysis of NMR data. The measured NMR data (e.g., FID, k-space acquisition), a model of the expected data for a given set of fitting parameters (e.g., signal intensity, mean and standard deviation of a distribution), and prior knowledge about the system (e.g., model function, range of fitting parameters) are combined to provide the posterior probability distribution. This process is repeated for a selection of parameter sets. The parameter set with the highest probability of generating the measured data is determined.
The inputs required for Bayesian inference are 1. the measured NMR data ι , 2. a model of the expected data for a given parameter set ϑ , 3. prior knowledge including the model function and range of the parameter set to be tested. The protocol is illustrated in Fig. 48. Effectively, the Bayesian inference approach compares a range of parameter sets to the data and determines which parameter set is most likely to give rise to the measurement. The input parameter set may contain a single term (e.g., signal intensity) or multiple terms (e.g., mean and standard deviation of a distribution). If each parameter set provides unique model data and the model is a good approximation for the real system, a significant amount of information can be inferred from very few data. Bayesian analysis has been used previously in NMR applications to extract chemical spectra from noisy data [569], flow velocities from under-sampled data [570], and bubble size distributions in bubble columns [571, 572]. Recently, 172
Bayesian analysis has been ported to low-field NMR for the determination of grain size distributions in reservoir rocks [495] and particle sizing with Earth’s field NMR [219]. The ability of the inference approach to overcome poor SNR and provide robust results from very few data makes it an ideal analysis approach for future low field process control applications. 8.3.2. Compressed sensing Compressed sensing (CS) provides a protocol for undersampling and reconstructing NMR data acquired in the inverse Fourier domain. So far, CS has be applied to MRI [183, 573], velocity mapping [574], spectroscopy [575], and flow propagators [576]. The concepts of CS have even been extended to multidimensional relaxation time measurements [577], although the application to numerical inversions of Fredholm integral equations is not currently robust for routine application. CS is readily applied to MRI when incomplete data are acquired in k-space using a pseudo-random sampling scheme. If we have prior knowledge that the image is sparse in some transform domain, e.g., following the discrete cosine, wavelet, or curvelet transform, then the theory of CS allows us to reconstruct the complete image from a subsample of the full data. If the data (in this case k-space points) are undersampled incoherently (meaning the sampling scheme is not sparse in the transform domain [578]) then the image is obtained using an appropriate non-linear reconstruction algorithm. As the CS protocol returns only the signal and not the noise in the data, the noise-like artifacts associated with the sampling scheme are removed. The CS reconstruction replaces the FT processing stage. Here we present an example CS reconstruction method suitable for use with undersampled MRI data after [574, 579]. This CS method is derived from basis 173
pursuit [580, 581]. The image reconstruction uses the `1 -norm to characterize the sparsity of the reconstructed image. First the nD k-space data are stacked into a vector b. The image to be reconstructed is likewise stacked as a vector f and this is transformed into the sparse domain by operator Ψ. The k-space data vector b is obtained from the image by the undersampled Fourier transform F . The reconstruction is achieved by solving the constrained optimization problem min kΨfk1 , s.t. kF f − bk2 ≤ e,
(110)
where e is a threshold that is set equal to the expected noise level and the `1 norm is a surrogate for sparsity. Therefore, minimizing the objective in eq. (110) provides an image that is consistent with the acquired k-space data but has the most sparse representation achievable in the transform domain. Equation (110) is appealing because it is a convex optimization problem and therefore readily solved using a number of existing algorithms. Equation (110) can either be solved directly or in its unconstrained Lagrangian form arg min kF f − bk2 + λ1 kΨfk1 .
(111)
f
using, for example, a projected conjugate gradients algorithm [581], iterative thresholding [582], interior point methods [583], or Bregman iterations [584]. It is common to include the spatial finite differences (total variation) transform with another sparse transform [585], such as a wavelet transform. The optimization problem then becomes min (kΨa fk1 + λ2 kΨb fk1 ) , s.t. kF f − bk2 ≤ e,
(112) 174
where Ψa is the wavelet operator, Ψb is the spatial finite-differences operator, and the constant λ2 trades sparsity in each of the two domains. As with eq. (110), eq. (112) may be solved in an unconstrained Lagrangian form arg min kF f − bk2 + λa kΨa fk1 + λb kΨb fk1 . f
(113)
Methods for reconstructing phase sensitive k-space data are described in [574]. CS reconstruction provides a particular improvement to phase encoded MRI acquisitions due to the nature of the sampling scheme (one wavenumber per measurement) where any k-space co-ordinate can be accessed readily. SPI-based acquisitions provide an obvious example of an imaging method wherein k-space wave-numbers may be sampled incoherently [183, 573]. As the image dimensionality increases (i.e., 2D or 3D images) then k-space generally becomes more sparse, allowing additional undersampling. Three-dimensional images are reconstructed without noticeable error from data comprising just a few percent of the full k-space raster [586]. Therefore, CS allows significant improvements in acquisition time which will be important for imaging at low field where multiple scans are required to provide a useful SNR. 9. Summary This is an exciting time to be working with low-field NMR instrumentation. The recent availability of fully digital spectrometers capable of performing complicated experiments on permanent magnets has greatly increased the range of applications accessible using bench-top systems. In particular, the recent introduction of intermediate-field spectroscopy with permanent magnets has enhanced the capabilities of NMR as an on-line tool suitable for implementation in industrial environments. 175
In this review we have presented a guide to obtaining the best results from modern low-field spectrometers. We started by considering the most appropriate permanent magnet designs for a given application and moved on to cover the following topics. • Radio frequency pulse behavior, including a discussion of phase transients and the effect of rf pulse errors on spin dynamics. • Digital filtering – a recent addition to low-field NMR but a technique that can provide significant noise reductions in the time domain if implemented correctly. • Strategies for maximizing the SNR. • Pre-processing stages to improve data analysis and interpretation. • Time-domain fitting of spectroscopy data that is more robust than a conventional FT at intermediate field. • Time-domain fitting of relaxation and diffusion data, including 1D and 2D distributions – now the most common method for analyzing low-field data. The literature pertaining to the uses of low-field NMR in industrial process monitoring is vast and saturated with niche applications. Rather than attempt to encompass all these experiments, we presented instead a summary of the most common methods of acquiring and interpreting data. We provided a link between the NMR magnet and the industrial environment with three examples: food production, the built environment, and the petroleum industry. Many advancements in low-field NMR have been driven by the requirements of petrophysical well-logging and 176
core analysis in the laboratory, and we shall likely see many more techniques emerging from this industry. As the capabilities of low-field NMR expand we shall see even more applications emerging, and it is probable that many will be driven by the introduction of intelligent data processing techniques such as Bayesian analysis and Compressed Sensing. NMR will become increasingly useful as an industrial tool for process monitoring and quality control through continued investment in low-field and intermediate-field permanent magnet technology. 10. Acknowledgments For financial support JM thanks Schlumberger Gould Research. LFG wishes to acknowledge funding through the Engineering and Physical Sciences Research Council (EPSRC) platform grant EP/F047991/1. References [1] L. Mannina, A. P. Sobolev, S. Viel, Liquid state 1 H high field NMR in food analysis, Prog. Nucl. Magn. Reson. Spectrosc. 66 (2012) 1–39. [2] B. J. Stockman, NMR spectroscopy as a tool for structure-based drug design, Prog. Nucl. Magn. Reson. Spectrosc. 33 (1998) 109–151. [3] J. W. Peng, J. Moore, N. Abdul-Manan, NMR experiments for lead generation in drug discovery, Prog. Nucl. Magn. Reson. Spectrosc. 44 (2004) 225–256. [4] F. Dalitz, M. Cudaj, M. Maiwald, G. Guthausen, Process and reaction
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255
Appendix A. Nomenclature a
Sphere (droplet) radius
ad
Amplitude component
A
Variable substitution (current)
A
Gradient ramp function
α
Smoothing parameter
b; b; B Signal (data) function/scalar; vector; matrix beq
Equilibrium signal
B0
Magnitude of static magnetic field
B1
Magnitude of applied (rf) magnetic field
∆B0
Variation in static magnetic field
∇B0
Magnetic field gradient
Brel
Relaxation magnetic field
Bpol
Polarizing magnetic field
Bacq
Acquisition magnetic field
β
Bessel function roots
c
Scalar
c
Vector of fitting parameters
C
Capacitance
256
CGT
Gibbs-Thomson coefficient
Cs,c
Permeability scaling constant (sandstone, carbonate)
d
Model order
D
Maximum model order
D
bgp parameter matrix
D
Diffusion coefficient
δ
Width parameter
δuv
Kronecker delta
∆χ
Magnetic susceptibility contrast
e, e
Experimental error (noise) function/scalar; vector
E
Variable substitution (exponent)
E
Sum of squared errors
E
Efficiency factor
ε
Gradient ramp duration
f ; f; F Distribution function; vector; matrix ˆf; Fˆ
Fitted distribution vector; matrix
fd
Frequency component
fpb
Passband frequency
fsw
Sweep width
F
Fourier transform operator
257
φ
Porosity
φrot
Magnetic field phase rotation
ϕ
Phase shift
ΦGCV
GCV score
g; g
Magnetic field gradient magnitude; vector
g0
Background magnetic field gradient magnitude
geff
Effective internal magnetic field gradient magnitude
G
bgp parameter matrix
γ
Gyromagnetic ratio
H
Smoothing function
∆Hf
Enthalpy of fusion
H
Hankel matrix
η
Viscosity
i
Complex current
I
Physical current
I
Identity matrix
ι
Experimental data (Bayes’ theorem)
ℑ
Imaginary component √ Imaginary unit −1
j jn
nth order spherical Bessel function
258
J
Spectral density
Jn
nth order Bessel function
k; k
Wavenumber magnitude; vector
∆k
Resolution in k-space
K; K
Kernel function; matrix
κ
Permeability
`c
Crystal size
`∆
Diffusion distance during t∆
`e
Diffusion path length over time te
`g
Dephasing length scale
`H
Length-scale of macroscopic heterogeneities
`ρ
Relaxivity length-scale
`s
Characteristic pore size
ld
Eigenvalues
L
Inductance
L
Sample length
L
Smoothing operator
λ
Weighting parameter
λ
bgp coefficient matrix
m
Index
259
M
Array size
M
Magnetization
n
Index
ns
Number of saturation pulses
n0
Offset from origin
N
Index
Ns
Number of scans
ν0
Resonant frequency:
∆ν
Frequency bandwidth of slice
O
Array size
O
Solution matrix
p
Surface spin population
Pc
Capillary pressure
P
Probability
q; q Magnetization wave number; vector Q
Quality factor
Q
Second derivative matrix
rd
Rate component
rc
Radius of rotation in centrifuge
r
Position vector
260
R
Displacement vector
R
Resistance
R
Residuals
ρ
Surface relaxivity
ρs
Solid density
∆ρ
Density difference
S
Saturation state
S
Singular values (diagonal) matrix
S/V
Surface-to-volume ratio
σsl
Surface energy
ς
Power
t; t
Experimental time scalar; vector
tc
Constant time
tDE
Dead time
tdw
Dwell time
tδ
Gradient pulse duration
tδ1,2
Delay before / after gradient pulse
t∆
Observation time
te
Echo time (CPMG)
tGD
Group delay
261
tp
Phase gradient duration
tr
Read gradient duration
tRD
Recycle delay between scans
ts
Slice gradient duration
tse
Echo time (diffusion)
tstore
Storage time of spins on z-axis
tω
Pulse duration
t90
90◦ pulse duration
t180
180◦ pulse duration
T1
Longitudinal relaxation time
T1ρ
Longitudinal relaxation in the rotating frame
T2
Transverse relaxation time
T2∗
Spin dephasing time
T2,eff
Effective transverse relaxation time
T1,2,bulk
Bulk relaxation times
T1,2,surf
Surface relaxation time
T21
Short T2 component
T22
Long T2 component
T2LM
Log-mean T2
Tm
Melting temperature
262
∆Tm
Melting point depression
T
Transpose
†
Conjugate transpose
τex
Exchange time
τm
Surface diffusion correlation time
τs
Surface residence time
τse
Half echo time (diffusion)
τtort
Tortuosity
τ1
Longitudinal recovery time
τ2
Half echo time (CPMG)
∆t
Time increment
θ
Phase angle
u; u
Fitting parameter scalar; vector
U
SVD matrix
υ
Empirical exponent
ϒ
Empirical exponent
v
Complex voltage
v
bgp Kronecker matrix
V
Physical voltage
V
SVD matrix
263
ω0
Larmor frequency
ωd
Driving (forcing) frequency
ωI
Proton Larmor frequency
ωL
Probe (tuned) frequency
ωr
Free ringing frequency
ωS
Electron Larmor frequency
∆ω
Frequency offset
W
Precision matrix
ϖc
Angular velocity of centrifuge
x, y, z
Cartesian coordinates
∆x, ∆y, ∆z Spatial resolution X
Empirical pre-factor
ξ
Empirical exponent
Ξ
Empirical exponent
∆ys
Slice thickness in image space
Y
Empirical pre-factor
Ψ
Sparse domain transform operator
Z
bgp parameter matrix
ζ
Relaxation shape parameter
Appendix B. Abbreviations AIC
Akaike information criterion
264
bgp
Block gradient pulse
CMR
Combinable magnetic resonance tool
CPMG
Carr-Purcell Meiboom-Gill
CS
Compressed sensing
CSH
Calcium silicate hydrate
CUFF
Cut-open, uniform, force free
DOSY
Diffusion ordered spectroscopy
FFC
Fast field cycling
FID
Free induction decay
FOV
Field of view
fMRI
Functional MRI
FT
Fourier transform
FWHM
Full width at half maximum
GARField
Gradient at right angles to the field
GCV
Generalized cross validation
GdCo
Gadolinium cobalt
gpd
Gaussian phase distribution
HTLS
Hankel total least squares
HSVD
Hankel singular value decomposition
LCR
Inductor capacitor resistor
265
LP
Linear prediction
LWD
Logging while drilling
MOUSE
Mobile universal surface explorer
MRI
Magnetic resonance imaging
MRIL
Magnetic resonance imaging logging
MRScanner Magnetic resonance scanner tool MSME
Multi-slice multi-echo (pulse sequence)
NeFeB
Neodymium iron boron
NML
Nuclear magnetic logging tool
NMT
Nuclear magnetism tools
NMR
Nuclear magnetic resonance
p.u.
Porosity units
PV
Pore volume
QPP
Quadratic programming problem
RARE
Rapid acquisition with relaxation enhancement
rf
Radio frequency
RN
Rendement Napole gene
SESPI
Spin echo single point imaging
sgp
Short gradient pulse
SmCo
Samarian cobalt
266
SNR
Signal-to-noise ratio
SPI
Single point imaging
SPRITE
Single point ramped imaging with T1 enhancement
STRAFI
Stray-field imaging
SQUID
Superconducting quantum interference device
SVD
Singular value decomposition
USBM
United States Bureau of Mines
w/c
Water-to-cement ratio
WHC
Water holding capacity
1D, 2D, 3D
One, two, three dimensional
267
Earth’s Abstract Graphical field
ultra low field
intermediate field
low field
40 µT
50 mT
0.2 T
0.5 T
2 kHz
2 MHz
7.5 MHz
20 MHz
well−logging
quality control
1T
1.5 T
45 MHz
60 MHz
process monitoring
SQUIDs petrophysics
relaxation
clinical diagnosis
imaging
relaxation
spectroscopy
B0 ν0
!1 " H
*Highlights
! ! ! ! !
We review low-field NMR technology for industrial process monitoring. Designs for bench-top and portable magnets are considered. Radio frequency performance and system optimization are discussed. We review experimental techniques, data acquisition, and interpretation. Example applications are given for food, construction, and petroleum industries.