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diffusion correlations in porous media
Magnetic Resonance Imaging 21 (2003) 381–383
2D relaxation/diffusion correlations in porous media S. Godefroy, P. T. Callaghan* MacDiarmid Institute of Advanced Materials and Nanotechnology, SCPS, Victoria University of Wellington, Wellington, New Zealand
Abstract 2D correlations between NMR relaxation and/or diffusion have been used to investigate water and oil dynamics in food and micro-emulsion systems. In the case of Mozzarella and Gouda cheese samples, a significant change in D/T2 correlation is appearing with cheese aging. In the case of a water/toluene micro-emulsion, some evidence for coalescence effects is suggested by D/D exchange spectra. © 2003 Elsevier Inc. All rights reserved. Keywords: NMR; 2D correlation experiments; Food material; Micro-emulsions; Chemical shift; 2-D Laplace inversion
1. Introduction Two dimensional (2-D) spin-relaxation and diffusion experiments aim to correlate the molecular dynamics and interactions. For example, T1 vs. T2 correlation experiments allow one to identify molecular species and to study dynamics, their importance relying mostly on the ratio between T1 and T2. For complex fluids, semisolids, or fluids in porous media where there may be strong dipole-dipole interactions between spins, T2 can be much shorter than T1. D vs. T2 correlations allow molecular motion to be distinguished from molecular interactions of the system of spins. D vs. D correlation experiments provide evidence on the exchange between compartments with different diffusion behaviors. Because of the heterogeneity of the materials under study in this work, both the diffusion and relaxation signals can have multi-exponential behavior. Therefore the multi-exponential 2-D data were analyzed by using two-dimensional Laplace Inversion [1]. These techniques were used to study water and oil in cheese as a function of aging. We also studied D vs. D diffusion exchange during the stabilization time of micro-emulsions by droplet coalescence.
2. Experiments Two kinds of samples were used to illustrate the 2-D NMR correlation experiments in porous material, using high resolution NMR, namely cheese and micro-emulsions. * Corresponding author. Tel.: 64-4-4635237; fax: 64-4-4635945. E-mail address: [email protected] (P. Callaghan). 0730-725X/03/$ – see front matter © 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0730-725X(03)00144-9
Dry-salted Gouda-style and Mozzarella-style cheese were manufactured by Fonterra Research Center (Palmerston North, New Zealand) especially for this project. The block cheeses were vacuum-packed and stored at 5°C until their examination by NMR spectroscopy. The samples plugged for the NMR experiments were 20 mm long and in a 10 mm diameter glass open-ended NMR tube. Tight polytetrafluoroethylene plugs were then positioned on both sides of the sample to prevent loose of moisture by evaporation. NMR experiments were performed at 5 and 40°C at ages between 1 to 74 days, to investigate the water and oil characteristics of the cheese sample. The toluene/brine/SDS micro-emulsions, with butanol as a cosurfactant, were prepared with the percentages 47.25/46.8/ 1.99/3.96, respectively [5]. The variation in the brine salinity allows transition from a two-phase micro-emulsion with water droplets in oil or oil droplets in water to a three-phase, one with bicontinuous water and oil phases. We studied in particular a micro-emulsion of salinity S (percentage of NaCl in weight) of 6.45 (bicontinuous water and oil phases), during the phase of equilibration of the micro-emulsion. The NMR experiments were performed on an AVANCE 300 MHz Bruker spectrometer. The spin-lattice vs. spinspin relaxation correlation sequence consists of a combination of an inversion-recovery and a CPMG sequence. The experiments were performed by varying independently the recovery time (t1) and the number of pulses (described by t2 as the duration of the CPMG). The 2-D signal decay as a function of t1 and t2 is given in the equation [1], for a multi-exponential behavior. M共t 1,t 2兲 ⫽
冘A共T ,T 兲e ⫺ Tt
1
1
2
1
e⫺
t2 T2
(1)
382
S. Godefroy, P.T. Callaghan / Magnetic Resonance Imaging 21 (2003) 381–383
Fig. 2. D vs. T2 distribution maps of water for the Mozzarella sample (left) and the Gouda sample (right) measured at 40°C, at the age of 74 days. The black cross correspond to the position of the peaks at the age of 1 day.
Fig. 1. (a) D vs. T2 correlation sequence, as a combination of CPMG sequence and Pulsed Gradient Spin Echo sequence with stimulated echoes. The CPMG duration and the strength G of the gradients are varied independently. (b) Diffusion exchange sequence as a combination of two PGSE sequences separated by an exchange time excG1 and G2 are varied independently.
The D vs. T2 correlation sequence, as shown on the Fig. 1 -a, consists of a combination of a Pulsed Gradient Spin Echo (PGSE) sequence with stimulated echoes and a CPMG sequence. To get the 2-D signal, we varied the number of echoes and the strength G of the gradient independently. The resulting signal is described by the equation (2). M共t,q 2兲 ⫽
冘A共T ,D兲e ⫺ 2
t ⫺q2D⌬ T2e
(2)
The D vs. D sequence (Fig. 1-b) consists of two PGSE sequences with stimulated echoes, separated by an exchange time where the gradient strengths of the two different parts are varied independently [2]. The resulting multiexponential signal as a function of q ⫽ (␥G␦) is given by: M共q 12,q 22兲 ⫽
冘A共D ,D 兲e 1
2
⫺q12D1⌬ ⫺q22D2⌬
e
(3)
As the 2-D signals are expected to be multi-exponential because of the heterogeneity of the systems, their analysis requires a 2-D Laplace inversion. The 1-D Laplace inversion, comprises a non-negative least square fit with the addition of a smoothing term and is well described in the literature [3,4]. We developed a 2-D Laplace inversion from the 1-D version according to the algorithm of Song et al. [1]. The major problem arises from the large size of the matrices to compute. The solution consists in reducing the size of the matrices of the system by a Singular Value Decompositionalgorithm, and transforming the data into a 1D vector on which a simple Laplace Inversion is applied. For the cheese samples, the water and oil signals were differentiated by their chemical shifts, separated by 3.3 ppm.
(right) and Mozzarella sample (left), as measured at 40°C at the age of 74 days. The black crosses on the plots correspond to the position of the peaks at the age of 1 day. As a function of the age, we observe an exchange in intensity for the peaks at low diffusion coefficient with a peak at higher D and very short T2 (for Mozzarella) and two different peaks at shorter T2 with a faster diffusion. One explanation of this phenomenon could be the migration of water molecules from pools between fat globules to absorption sites on proteins. Fig. 3 displays preliminary diffusion exchange (D vs. D) experiments of water in micro-emulsion with a brine salinity S ⫽ 6.45 (3-phase micro-emulsion), during the period of equilibration to a water and oil bi-continuous phase. The diffusion exchange correlations were measured for exchange times of 10 ms (left) and 100 ms (right). For very short exchange time, one observes a range of diffusion coefficient that could be explain by the range of droplet size of the emulsion. At longer exchange time, we observe two symmetrical peaks that could come from fluctuations, ordroplet coalescence, as those preliminary experiments were done during the period of equilibration of the micro-emulsion. 4. Conclusion The D/T2 correlation spectra shown here for water and oil in cheese indicate the potential for 2D Laplace Inversion techniques to elucidate molecular dynamics in complex soft materials such as food. A specific window on fluctuations is observed by D/D exchange spectroscopy. We demonstrate the potential of the method in a preliminary investigation using micro-emulsion approaching a bicontinuous phase.
3. Results and discussion Fig. 2 shows the Diffusion vs. spin-spin relaxation correlation distribution maps of water for the Gouda samples
Fig. 3. Preliminary diffusion exchange D vs. D experiments of water in the micro-emulsion with S⫽6.45, for an exchange time exc of 10 ms (left) and 100 ms (right).
S. Godefroy, P.T. Callaghan / Magnetic Resonance Imaging 21 (2003) 381–383
References [1] Song Y-Q, Venkataramanan L, Hurlimann MD, Flaumm M, Frulla P, Straley C. T1-T2 correlation spectra obtained using a fast two-dimensional laplace inversion. J Magn Reson 2002;154:261. [2] Callaghan PT, Manz B. Velocity exchange spectroscopy. J Magn Reson 1994;106:260.
383
[3] Provencher SW. CONTIN: a general purpose constrained regularization program for inverting noisy linear algebraic and integral equations. Comput Phys Commun 1982;7:229. [4] Borgia GC, Brown RJS, Fantazzini P. Uniform-penalty inversion of multiexponential decay data. 132, 65–77, 1998. [5] Clarkson MT, Beaglehole D, Callaghan PT. Molecular diffusion in a microemulsion. Phys Rev Let 1985;54(15).
2D relaxation/diffusion correlations in porous media S. Godefroy, P. T. Callaghan* MacDiarmid Institute of Advanced Materials and Nanotechnology, SCPS, Victoria University of Wellington, Wellington, New Zealand
Abstract 2D correlations between NMR relaxation and/or diffusion have been used to investigate water and oil dynamics in food and micro-emulsion systems. In the case of Mozzarella and Gouda cheese samples, a significant change in D/T2 correlation is appearing with cheese aging. In the case of a water/toluene micro-emulsion, some evidence for coalescence effects is suggested by D/D exchange spectra. © 2003 Elsevier Inc. All rights reserved. Keywords: NMR; 2D correlation experiments; Food material; Micro-emulsions; Chemical shift; 2-D Laplace inversion
1. Introduction Two dimensional (2-D) spin-relaxation and diffusion experiments aim to correlate the molecular dynamics and interactions. For example, T1 vs. T2 correlation experiments allow one to identify molecular species and to study dynamics, their importance relying mostly on the ratio between T1 and T2. For complex fluids, semisolids, or fluids in porous media where there may be strong dipole-dipole interactions between spins, T2 can be much shorter than T1. D vs. T2 correlations allow molecular motion to be distinguished from molecular interactions of the system of spins. D vs. D correlation experiments provide evidence on the exchange between compartments with different diffusion behaviors. Because of the heterogeneity of the materials under study in this work, both the diffusion and relaxation signals can have multi-exponential behavior. Therefore the multi-exponential 2-D data were analyzed by using two-dimensional Laplace Inversion [1]. These techniques were used to study water and oil in cheese as a function of aging. We also studied D vs. D diffusion exchange during the stabilization time of micro-emulsions by droplet coalescence.
2. Experiments Two kinds of samples were used to illustrate the 2-D NMR correlation experiments in porous material, using high resolution NMR, namely cheese and micro-emulsions. * Corresponding author. Tel.: 64-4-4635237; fax: 64-4-4635945. E-mail address: [email protected] (P. Callaghan). 0730-725X/03/$ – see front matter © 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0730-725X(03)00144-9
Dry-salted Gouda-style and Mozzarella-style cheese were manufactured by Fonterra Research Center (Palmerston North, New Zealand) especially for this project. The block cheeses were vacuum-packed and stored at 5°C until their examination by NMR spectroscopy. The samples plugged for the NMR experiments were 20 mm long and in a 10 mm diameter glass open-ended NMR tube. Tight polytetrafluoroethylene plugs were then positioned on both sides of the sample to prevent loose of moisture by evaporation. NMR experiments were performed at 5 and 40°C at ages between 1 to 74 days, to investigate the water and oil characteristics of the cheese sample. The toluene/brine/SDS micro-emulsions, with butanol as a cosurfactant, were prepared with the percentages 47.25/46.8/ 1.99/3.96, respectively [5]. The variation in the brine salinity allows transition from a two-phase micro-emulsion with water droplets in oil or oil droplets in water to a three-phase, one with bicontinuous water and oil phases. We studied in particular a micro-emulsion of salinity S (percentage of NaCl in weight) of 6.45 (bicontinuous water and oil phases), during the phase of equilibration of the micro-emulsion. The NMR experiments were performed on an AVANCE 300 MHz Bruker spectrometer. The spin-lattice vs. spinspin relaxation correlation sequence consists of a combination of an inversion-recovery and a CPMG sequence. The experiments were performed by varying independently the recovery time (t1) and the number of pulses (described by t2 as the duration of the CPMG). The 2-D signal decay as a function of t1 and t2 is given in the equation [1], for a multi-exponential behavior. M共t 1,t 2兲 ⫽
冘A共T ,T 兲e ⫺ Tt
1
1
2
1
e⫺
t2 T2
(1)
382
S. Godefroy, P.T. Callaghan / Magnetic Resonance Imaging 21 (2003) 381–383
Fig. 2. D vs. T2 distribution maps of water for the Mozzarella sample (left) and the Gouda sample (right) measured at 40°C, at the age of 74 days. The black cross correspond to the position of the peaks at the age of 1 day.
Fig. 1. (a) D vs. T2 correlation sequence, as a combination of CPMG sequence and Pulsed Gradient Spin Echo sequence with stimulated echoes. The CPMG duration and the strength G of the gradients are varied independently. (b) Diffusion exchange sequence as a combination of two PGSE sequences separated by an exchange time excG1 and G2 are varied independently.
The D vs. T2 correlation sequence, as shown on the Fig. 1 -a, consists of a combination of a Pulsed Gradient Spin Echo (PGSE) sequence with stimulated echoes and a CPMG sequence. To get the 2-D signal, we varied the number of echoes and the strength G of the gradient independently. The resulting signal is described by the equation (2). M共t,q 2兲 ⫽
冘A共T ,D兲e ⫺ 2
t ⫺q2D⌬ T2e
(2)
The D vs. D sequence (Fig. 1-b) consists of two PGSE sequences with stimulated echoes, separated by an exchange time where the gradient strengths of the two different parts are varied independently [2]. The resulting multiexponential signal as a function of q ⫽ (␥G␦) is given by: M共q 12,q 22兲 ⫽
冘A共D ,D 兲e 1
2
⫺q12D1⌬ ⫺q22D2⌬
e
(3)
As the 2-D signals are expected to be multi-exponential because of the heterogeneity of the systems, their analysis requires a 2-D Laplace inversion. The 1-D Laplace inversion, comprises a non-negative least square fit with the addition of a smoothing term and is well described in the literature [3,4]. We developed a 2-D Laplace inversion from the 1-D version according to the algorithm of Song et al. [1]. The major problem arises from the large size of the matrices to compute. The solution consists in reducing the size of the matrices of the system by a Singular Value Decompositionalgorithm, and transforming the data into a 1D vector on which a simple Laplace Inversion is applied. For the cheese samples, the water and oil signals were differentiated by their chemical shifts, separated by 3.3 ppm.
(right) and Mozzarella sample (left), as measured at 40°C at the age of 74 days. The black crosses on the plots correspond to the position of the peaks at the age of 1 day. As a function of the age, we observe an exchange in intensity for the peaks at low diffusion coefficient with a peak at higher D and very short T2 (for Mozzarella) and two different peaks at shorter T2 with a faster diffusion. One explanation of this phenomenon could be the migration of water molecules from pools between fat globules to absorption sites on proteins. Fig. 3 displays preliminary diffusion exchange (D vs. D) experiments of water in micro-emulsion with a brine salinity S ⫽ 6.45 (3-phase micro-emulsion), during the period of equilibration to a water and oil bi-continuous phase. The diffusion exchange correlations were measured for exchange times of 10 ms (left) and 100 ms (right). For very short exchange time, one observes a range of diffusion coefficient that could be explain by the range of droplet size of the emulsion. At longer exchange time, we observe two symmetrical peaks that could come from fluctuations, ordroplet coalescence, as those preliminary experiments were done during the period of equilibration of the micro-emulsion. 4. Conclusion The D/T2 correlation spectra shown here for water and oil in cheese indicate the potential for 2D Laplace Inversion techniques to elucidate molecular dynamics in complex soft materials such as food. A specific window on fluctuations is observed by D/D exchange spectroscopy. We demonstrate the potential of the method in a preliminary investigation using micro-emulsion approaching a bicontinuous phase.
3. Results and discussion Fig. 2 shows the Diffusion vs. spin-spin relaxation correlation distribution maps of water for the Gouda samples
Fig. 3. Preliminary diffusion exchange D vs. D experiments of water in the micro-emulsion with S⫽6.45, for an exchange time exc of 10 ms (left) and 100 ms (right).
S. Godefroy, P.T. Callaghan / Magnetic Resonance Imaging 21 (2003) 381–383
References [1] Song Y-Q, Venkataramanan L, Hurlimann MD, Flaumm M, Frulla P, Straley C. T1-T2 correlation spectra obtained using a fast two-dimensional laplace inversion. J Magn Reson 2002;154:261. [2] Callaghan PT, Manz B. Velocity exchange spectroscopy. J Magn Reson 1994;106:260.
383
[3] Provencher SW. CONTIN: a general purpose constrained regularization program for inverting noisy linear algebraic and integral equations. Comput Phys Commun 1982;7:229. [4] Borgia GC, Brown RJS, Fantazzini P. Uniform-penalty inversion of multiexponential decay data. 132, 65–77, 1998. [5] Clarkson MT, Beaglehole D, Callaghan PT. Molecular diffusion in a microemulsion. Phys Rev Let 1985;54(15).