- Email: [email protected]
Characterization of pore geometry using correlations between magnetic field and internal gradient
Microporous and Mesoporous Materials xxx (2017) 1e4
Contents lists available at ScienceDirect
Microporous and Mesoporous Materials journal homepage: www.elsevier.com/locate/micromeso
Characterization of pore geometry using correlations between magnetic field and internal gradient Rhiannon T. Lewis, John Georg Seland* Department of Chemistry, University of Bergen, Allegaten 41, N-5007 Bergen, Norway
a r t i c l e i n f o
a b s t r a c t
Article history: Received 15 January 2017 Received in revised form 4 May 2017 Accepted 18 May 2017 Available online xxx
We have applied a novel multi-dimensional NMR experiment to characterize the pore geometry in liquid saturated porous systems with variation in material content and pore structure. Using the correlation between the internal gradient and the difference in magnetic susceptibility we have established a novel approach for determining the pore size distributions. The obtained pore size distributions correspond well with distributions obtained from measurements based on Decay due to Diffusion in the Internal Field, but seems to overestimate the size of pores larger than approximately 100 mm. © 2017 Elsevier Inc. All rights reserved.
Keywords: Internal magnetic field Internal gradients Pore size Porous materials Relaxation Magnetic susceptibility
1. Introduction In a liquid saturated porous system the rate constant of the Free Induction Decay (FID), T2 , is related to the line broadening, Dn, which is caused by the difference in magnetic susceptibility in the sample [1,2]
1
pT2
¼ Dn ¼
g
B Dc 2p 0 app
(1)
It is assumed that T2 is totally dominated by susceptibility differences. The apparent difference in magnetic susceptibility is given by Dcapp ¼ C Dc, where C is a dimensionless constant that takes into account effects from large local variations in the internal magnetic field [2,3]. As a first order approximation the spatial variation of the internal magnetic field in a single pore is linear, and it scales with B0 Dcapp and the size of the pore, a [4,5]
VBi ≡G0 z
B0 Dcapp a
* Corresponding author. E-mail address: [email protected] (J.G. Seland).
(2)
where G0 is the internal gradient, which is assumed to be constant within the pore. However, simulations have shown that the internal gradients are stronger close to the liquid-solid interface compared to the middle of the pore [2,5e7]. The characteristic spatial properties of the internal magnetic field can be used to determine important properties of the porous structure. Sun and Dunn [8] presented a modified Carr-PurcellMeiboom-Gill (CPMG) pulse sequence with varying echo spacing, which enables determination of a correlation between the transverse relaxation time constant (T2 ) and G0 . Another method is the Decay due to Diffusion in the Internal Field (DDIF) [9,10], where a distribution of pore sizes, a, can be determined based on higher order relaxation terms based on the theory of Brownstein and Tarr [11]. Both of these methods, the modified CPMG and DDIF, are based on relating the spatial variation in the magnetic field to the size of the pore. Using a different approach Mitchell [12] measured correlations between the longitudinal relaxation time constant (T1 ) and Dcapp in sand stones, and showed that a strong correlation indicated high degree of heterogeneity in mineralogy. Recently [13] we presented a novel multi-dimensional NMR experiment that enabled us to measure correlations between T1 Dcapp , G0 Dcapp , G0 T1 , and G0 a in one experiment. We showed that these correlations can be used to examine the heterogeneity of the porous system at different length scales. In the current paper we investigate the correlations that involves Dcapp in
http://dx.doi.org/10.1016/j.micromeso.2017.05.041 1387-1811/© 2017 Elsevier Inc. All rights reserved.
Please cite this article in press as: R.T. Lewis, J.G. Seland, Characterization of pore geometry using correlations between magnetic field and internal gradient, Microporous and Mesoporous Materials (2017), http://dx.doi.org/10.1016/j.micromeso.2017.05.041
2
R.T. Lewis, J.G. Seland / Microporous and Mesoporous Materials xxx (2017) 1e4
more detail and in particular we demonstrate that the G0 Dcapp correlation can be used as a novel approach to determine pore size distributions.
2. Experimental 2.1. Method The multi-dimensional pulse sequence presented in Ref. [13] consist of a DDIF experiment followed by a modified CPMG sequence where the echo spacing (tE ) is varied [8] by varying the number of p-pulses, and finally the detection of the FID signal: p=2 te p=2 td p=2 te ntE tFID . The corresponding reference pulse sequence necessary to subtract the contribution from T1 relaxation is: p=2 te p te p=2 td p=2 ntE tFID . By performing a series of experiments with varying values of td and tE , while keeping the time intervals te and ntE constant and detecting the FID signal, various correlations can be determined [13]. Our experiments are performed at relatively large values of Dcapp B0 and the dynamics of water molecules in a spin echo experiment is most likely somewhere in between the fast diffusion (FD) or localization (LOC) regime [4]. To evaluate this dynamics Mitchell et al. [14,15] have suggested to express the diffusion exponent in CPMG data in a general form
MðntE Þ k ¼ ebntE MðtE ¼ 0Þ
(3)
where b is a general decay constant due to diffusion in internal gradients and k describes a general echo time dependence of the diffusion exponent. For instance, in the FD regime k ¼ 3 and 1 Dg2 G2 . By measuring CPMG decays with varying echo spacb ¼ 12 0 ings and using the data collapse plot of Hurlimann [4], is then possible to find the optimal value of k that gives the most correct description of the data [14]. In this paper we focus in more detail on one of the correlations obtained from the multi-dimensional pulse sequence. Using the reference experiment and keeping tD short while varying values of tE , the rate constant b can be correlated with T2 , giving the attenuation
Z Z MðtE ; tFID Þ ¼
2.2. Experimental details All NMR experiments were performed at 25+ C on a Bruker Avance 500 spectrometer (B0 ¼ 11.7 T), using a commercial probe from Bruker Biospin (DIFF30). Three different types of watersaturated random packed compact glass beads (Duke Scientific) were analyzed: Bead diameter of 30 mm (Soda lime glass); Bead diameter of 100 mm (Soda lime glass); Bead diameters with distribution between 5 and 50 mm (Borosilicate glass). The samples were prepared by adding water to 5 mm NMR-tubes first and then adding the glass beads. A sample of water-saturated Berea sandstone was also analysed. The Berea plug had a diameter of 3.5 mm and a length of 20 mm, and was wrapped in parafilm to prevent water evaporating during the experiments. The motivation for the choice of these samples is to study two different mono-disperse bead sizes of the same material but that will produce two different pore geometries when closely packed. Furthermore, the packings of 5e50 mm glass beads produces a sample where the solid material is the same, i.e. a uniform distribution of paramagnetic substances, but where the pore geometry is more complex. Finally, the Berea sandstone plug is a sample where the material content, and in particular paramagnetic substances, can vary in a non-uniform way, and where the pore geometry is very complex. To determine the optimal value of k regular CPMG experiments with echo spacing varying between 0.1 and 30 ms were performed in all of the samples. The data was then plotted as a function of tEk , and the right condition for the data collapse was determined visually by varying k manually [15]. The encoding period te was 50 ms and td was varied logarithmically from 50 ms to 8 s in 32 steps for all four samples. The echo spacing, tE was varied logarithmically between 0.1 and 20.4 ms by keeping the total echo time (ntE ) constant at 20.4 ms, with a corresponding variation of the number of p pulses from 204 to 1, in 32 steps. Notably, G20 is not measured directly, but through DG20 , which can be normalized using the diffusion coefficient for water ðDH2 O ¼ 2:3 105 cm2 /s at 25+ C). In all the experiments the FID signal was acquired with a dwell time (dw) of 6.65 ms and with a total of 512 data points (td), so that tFID ¼ td,dw. The obtained data were analyzed as 2D Fredholm integral equations using software [16] based on the Schlumberger algorithm [17]. 3. Results and discussion
tFID k P b; T2 ebntE e T2 dbT2
(4)
where P(b; T2 ) is a 2D distribution function of b and T2 . It has been experimentally verified that a single exponential decay for the FIDsignal is a good approximation for a single pore size [2]. Furthermore, assuming the FD regime the distribution involving internal gradients, PðG0 ; T2 Þ, can be determined using k ¼ 3 and b ¼ 1 2 2 12 Dg G0 according to Eq. (3). Finally, using Eq. (1) the T2 -axis can be rescaled to give the distributions Pðb; Dcapp Þ or PðG0 ; Dcapp Þ. Assuming that Eq. (2) is valid for the PðDcapp ; G0 Þ distribution, one can determine the correlation between G0 and Dcapp , and use this correlation to determine a pore size distribution. This is done using the following procedure on the obtained data: For each point in pðG0 Þ the maximum value in PðDcapp ; G0 Þ can be determined and the corresponding value of Dcapp can be found. Thus, the G0 -axis can then be rescaled according to a ¼ Dcapp B0 =G0 , giving an axis that describes range of pore sizes, a, and a corresponding distribution pðaÞ can be obtained from pðG0 Þ.
The optimal value of k was determined to be 2.00, 2.05, 1.95 and 1.70 for the 30 mm, 100 mm, 5e50 mm glass beads, and Berea sandstone sample, respectively [13]. Thus, as expected the dynamics of the molecules during the modified CPMG part of the sequence is in between the FD and LOC regimes. In the further analysis we chose to evaluate the obtained data according to the general decay constant, b, and the corresponding optimal value of k, but also assuming the FD regime (k ¼ 3) where G0 can be determined. We are then able to compare the plots based on b and G0 , evaluate the effect of not being in the FD regime, and thus carefully use the G0 data to make physical interpretations of the obtained data. Both the Pða; bÞ and Pða; G0 Þ plots presented in Ref. [13] showed a clear tendency of a stronger and more negative correlation between a and b (and G0 ) when going from the more homogeneous pore geometry of the packings of mono-disperse glass beads, via the packings of multi-disperse glass beads, to the more heterogeneous pore structure of the Berea sandstone. Therefore, Pða; bÞ or
Please cite this article in press as: R.T. Lewis, J.G. Seland, Characterization of pore geometry using correlations between magnetic field and internal gradient, Microporous and Mesoporous Materials (2017), http://dx.doi.org/10.1016/j.micromeso.2017.05.041
R.T. Lewis, J.G. Seland / Microporous and Mesoporous Materials xxx (2017) 1e4
Fig. 1. PðDcapp ; bÞ for distilled water confined in random packings of 30 mm (a), 100 mm (b) and 5e50 mm (c) glass beads, and in Berea sandstone (d).
Pða; G0 Þ correlation plots can be used as a measure of pore size heterogeneity. This is also an experimental verification of the approximation given in Eq. (2). Recently, pore size -G0 correlations have been measured also at low field strength [18] with similar conclusions. The PðDcapp ; bÞ plots obtained in the different samples are presented in Fig. 1, and show a clear positive correlation between these two parameters in the more heterogenous samples. Similar to
Fig. 2. PðDcapp ; G0 Þ for distilled water confined in random packings of 30 mm (a), 100 mm (b) and 5e50 mm (c) glass beads, and in Berea sandstone (d). The dotted lines corresponds to the maximum value in the distributions, used for determining the correlation between Dcapp and G0 , and the rescaling of the G0 -axis to a pore size distribution, as presented below in Fig. 3.
3
what was observed for the Pða; bÞ data, the correlation is less obvious, or even non-existing, in the more homogeneous samples. The corresponding PðDcapp ; G0 Þ plots (k ¼ 3) obtained in the different samples are presented in Fig. 2. As in the PðDcapp ; bÞ plots, Dcapp has a positive correlation with G0 , even though the dynamic behaviour of the molecules is not in the FD-regime. Interestingly, in the PðDcapp ; G0 Þ plots the positive correlation is pronounced also in the more homogeneous samples. We have no clear explanation for this difference between PðDcapp ; bÞ and PðDcapp ; G0 Þ, but it is most likely connected to the fact that our data is not obtained in the FD regime. Experiments performed at lower field strength, satisfying the criteria of the FD regime, should be performed on the same samples in order to investigate this effect. The positive correlation between Dcapp and G0 is expected from Eq. (2), and using the approach described above under section 2.1 it can be used as a novel method for determining a pore size distribution pðaÞ. The results from this procedure is presented in Fig. 3. The results are compared to pore size distributions obtained using a DDIF analysis on the same set of multidimensional data [13]. The average pore size obtained for the 30 mm and 100 mm glass bead samples using DDIF is 17 mm and 44 mm, respectively. This correspond well with the results obtained by Barker et al. [19] who studied the structure and dynamics of powders, using a computersimulation approach. They found that in packings of mono-sized spheres the mean diameter of a pore is 0.45 times the size of the sphere, which should result in expected values of 14 mm and 45 mm for our samples. For the 30 mm glass bead sample there is a good correspondence between the pore size distribution obtained using DDIF and the Dcapp - G0 correlation. However, for the 100 mm glass bead sample the pore size distribution obtained using the Dcapp G0 correlation is shifted to higher values (average value is 70 mm) compared to the DDIF results. For the 5e50 mm glass beads, there is again a good correspondence between the two methods, with an average pore size of approximately 30 mm, which is reasonable for this sample. In the Berea sandstone sample there is again a mismatch between the two different methods, with the pore size distribution obtained using the Dcapp - G0 correlation is shifted to
Fig. 3. Pore size distributions obtained from Dcapp - G0 correlations in random packings of 30 mm (a), 100 mm (b) and 5e50 mm (c) glass beads, and in Berea sandstone (d). The corresponding pore size distributions obtained from DDIF experiments (dashed lines) are shown for comparison.
Please cite this article in press as: R.T. Lewis, J.G. Seland, Characterization of pore geometry using correlations between magnetic field and internal gradient, Microporous and Mesoporous Materials (2017), http://dx.doi.org/10.1016/j.micromeso.2017.05.041
4
R.T. Lewis, J.G. Seland / Microporous and Mesoporous Materials xxx (2017) 1e4
higher values compared to the DDIF results. Thus, in the part of the samples with relatively large pores the method based on the Dcapp - G0 correlation overestimates the pore sizes. This could be explained by the fact that the total echo time (ntE ) in the modified CPMG part of the experiment was 20.4 ms, giving a diffusion length of 10 mm during this time interval. An assumption is Eq. (2) is that the gradient is constant across the pore. If the pore size is large compared to the diffusion length, the spins will not experience an average (constant) gradient, and will therefore be more sensitive to local gradients that varies on a scale which is considerably smaller than the pore size. This means that molecules located in the middle of a large pore will experience a much weaker gradient than the average value across the pore, leading to an underestimation of the value of G0 and a corresponding overestimation of the pore size. Another explanation can be that the dynamics of the molecules is not in the FD regime, leading to an error in the measured G0 . It should also be mentioned that the signal attenuation during the modified CPMG part of the sequence did not go all the way to the noise level, which could also lead to an underestimation of G0 . All these possible effects should be investigated by performing experiments at lower magnetic field strengths, and making sure a sufficient G0 -attenuation. 4. Conclusion We have shown that the correlation between Dcapp and G0 can be used for a novel approach of determining pore size distributions in liquid saturated porous systems. The obtained pore size distributions correspond well with corresponding distributions obtained using the well established DDIF technique for pore sizes up to approximately 100 mm, but seems to overestimate the size of larger
pores. This effect should be studied in more detail in future studies, in particular at lower magnetic field strengths.
Acknowledgments We acknowledge financial support from the Research Council of Norway (Project no. 210448). We also thank Uni-Research CIPR for providing us with the Berea sandstone material.
References [1] J. Karger, H. Pfeifer, S. Rudtsch, J. Mag. Reson 85 (2) (1989) 381e387. [2] Q. Chen, A.E. Marble, B.G. Colpitts, B.J. Balcom, J. Mag. Reson 175 (2) (2005) 300e308. [3] J. Mitchell, T.C. Chandrasekera, L.F. Gladden, J. Chem. Phys. 132 (24) (2010) 244705. [4] M.D. Hurlimann, J. Mag. Reson 131 (2) (1998) 232e240. [5] P.N. Sen, S. Axelrod, J. Appl. Phys. 86 (8) (1999) 4548e5454. [6] B. Audoly, P.N. Sen, S. Ryu, Y.Q. Song, J. Mag. Reson 164 (1) (2003) 154e159. [7] L.M. Burcaw, P.T. Callaghan, J. Mag. Reson 216 (2012) 144e151. [8] B.Q. Sun, K.J. Dunn, Phys. Rev. E 65 (5) (2002) 051309. [9] Y.-Q. Song, Concepts Mag. Reson 18A (2) (2003) 97e110. [10] N.V. Lisitza, Y.Q. Song, J. Chem. Phys. 114 (2001) 9120. [11] K.R. Brownstein, C.E. Tarr, Phys. Rev. A 19 (6) (1979) 2446e2453. [12] J. Mitchell, J. Mag. Reson 240 (2014) 52e60. [13] R.T. Lewis, J.G. Seland, J. Mag. Reson 263 (2016) 19e32. [14] J. Mitchell, T.C. Chandrasekera, J. Chem. Phys. 141 (22) (2014) 224201e224217. [15] J. Mitchell, T.C. Chandrasekera, L.F. Gladden, J. Chem. Phys. 139 (7) (2013) 074205. [16] P.T. Callaghan, Provided with Courtesy of Paul Callaghan, 2002. [17] L. Venkataramanan, Y.Q. Song, M.D. Hurlimann, IEEE Trans. Sign. Process 50 (5) (2002) 1017e1026. [18] Y. Zhang, L. Xiao, G. Liao, B. Bluemich, J. Mag. Reson 267 (2016) 37e42. [19] G.C. Barker, A. Mehta, Phys. Rev. A 45 (6) (1992) 3435e3446.
Please cite this article in press as: R.T. Lewis, J.G. Seland, Characterization of pore geometry using correlations between magnetic field and internal gradient, Microporous and Mesoporous Materials (2017), http://dx.doi.org/10.1016/j.micromeso.2017.05.041
Contents lists available at ScienceDirect
Microporous and Mesoporous Materials journal homepage: www.elsevier.com/locate/micromeso
Characterization of pore geometry using correlations between magnetic field and internal gradient Rhiannon T. Lewis, John Georg Seland* Department of Chemistry, University of Bergen, Allegaten 41, N-5007 Bergen, Norway
a r t i c l e i n f o
a b s t r a c t
Article history: Received 15 January 2017 Received in revised form 4 May 2017 Accepted 18 May 2017 Available online xxx
We have applied a novel multi-dimensional NMR experiment to characterize the pore geometry in liquid saturated porous systems with variation in material content and pore structure. Using the correlation between the internal gradient and the difference in magnetic susceptibility we have established a novel approach for determining the pore size distributions. The obtained pore size distributions correspond well with distributions obtained from measurements based on Decay due to Diffusion in the Internal Field, but seems to overestimate the size of pores larger than approximately 100 mm. © 2017 Elsevier Inc. All rights reserved.
Keywords: Internal magnetic field Internal gradients Pore size Porous materials Relaxation Magnetic susceptibility
1. Introduction In a liquid saturated porous system the rate constant of the Free Induction Decay (FID), T2 , is related to the line broadening, Dn, which is caused by the difference in magnetic susceptibility in the sample [1,2]
1
pT2
¼ Dn ¼
g
B Dc 2p 0 app
(1)
It is assumed that T2 is totally dominated by susceptibility differences. The apparent difference in magnetic susceptibility is given by Dcapp ¼ C Dc, where C is a dimensionless constant that takes into account effects from large local variations in the internal magnetic field [2,3]. As a first order approximation the spatial variation of the internal magnetic field in a single pore is linear, and it scales with B0 Dcapp and the size of the pore, a [4,5]
VBi ≡G0 z
B0 Dcapp a
* Corresponding author. E-mail address: [email protected] (J.G. Seland).
(2)
where G0 is the internal gradient, which is assumed to be constant within the pore. However, simulations have shown that the internal gradients are stronger close to the liquid-solid interface compared to the middle of the pore [2,5e7]. The characteristic spatial properties of the internal magnetic field can be used to determine important properties of the porous structure. Sun and Dunn [8] presented a modified Carr-PurcellMeiboom-Gill (CPMG) pulse sequence with varying echo spacing, which enables determination of a correlation between the transverse relaxation time constant (T2 ) and G0 . Another method is the Decay due to Diffusion in the Internal Field (DDIF) [9,10], where a distribution of pore sizes, a, can be determined based on higher order relaxation terms based on the theory of Brownstein and Tarr [11]. Both of these methods, the modified CPMG and DDIF, are based on relating the spatial variation in the magnetic field to the size of the pore. Using a different approach Mitchell [12] measured correlations between the longitudinal relaxation time constant (T1 ) and Dcapp in sand stones, and showed that a strong correlation indicated high degree of heterogeneity in mineralogy. Recently [13] we presented a novel multi-dimensional NMR experiment that enabled us to measure correlations between T1 Dcapp , G0 Dcapp , G0 T1 , and G0 a in one experiment. We showed that these correlations can be used to examine the heterogeneity of the porous system at different length scales. In the current paper we investigate the correlations that involves Dcapp in
http://dx.doi.org/10.1016/j.micromeso.2017.05.041 1387-1811/© 2017 Elsevier Inc. All rights reserved.
Please cite this article in press as: R.T. Lewis, J.G. Seland, Characterization of pore geometry using correlations between magnetic field and internal gradient, Microporous and Mesoporous Materials (2017), http://dx.doi.org/10.1016/j.micromeso.2017.05.041
2
R.T. Lewis, J.G. Seland / Microporous and Mesoporous Materials xxx (2017) 1e4
more detail and in particular we demonstrate that the G0 Dcapp correlation can be used as a novel approach to determine pore size distributions.
2. Experimental 2.1. Method The multi-dimensional pulse sequence presented in Ref. [13] consist of a DDIF experiment followed by a modified CPMG sequence where the echo spacing (tE ) is varied [8] by varying the number of p-pulses, and finally the detection of the FID signal: p=2 te p=2 td p=2 te ntE tFID . The corresponding reference pulse sequence necessary to subtract the contribution from T1 relaxation is: p=2 te p te p=2 td p=2 ntE tFID . By performing a series of experiments with varying values of td and tE , while keeping the time intervals te and ntE constant and detecting the FID signal, various correlations can be determined [13]. Our experiments are performed at relatively large values of Dcapp B0 and the dynamics of water molecules in a spin echo experiment is most likely somewhere in between the fast diffusion (FD) or localization (LOC) regime [4]. To evaluate this dynamics Mitchell et al. [14,15] have suggested to express the diffusion exponent in CPMG data in a general form
MðntE Þ k ¼ ebntE MðtE ¼ 0Þ
(3)
where b is a general decay constant due to diffusion in internal gradients and k describes a general echo time dependence of the diffusion exponent. For instance, in the FD regime k ¼ 3 and 1 Dg2 G2 . By measuring CPMG decays with varying echo spacb ¼ 12 0 ings and using the data collapse plot of Hurlimann [4], is then possible to find the optimal value of k that gives the most correct description of the data [14]. In this paper we focus in more detail on one of the correlations obtained from the multi-dimensional pulse sequence. Using the reference experiment and keeping tD short while varying values of tE , the rate constant b can be correlated with T2 , giving the attenuation
Z Z MðtE ; tFID Þ ¼
2.2. Experimental details All NMR experiments were performed at 25+ C on a Bruker Avance 500 spectrometer (B0 ¼ 11.7 T), using a commercial probe from Bruker Biospin (DIFF30). Three different types of watersaturated random packed compact glass beads (Duke Scientific) were analyzed: Bead diameter of 30 mm (Soda lime glass); Bead diameter of 100 mm (Soda lime glass); Bead diameters with distribution between 5 and 50 mm (Borosilicate glass). The samples were prepared by adding water to 5 mm NMR-tubes first and then adding the glass beads. A sample of water-saturated Berea sandstone was also analysed. The Berea plug had a diameter of 3.5 mm and a length of 20 mm, and was wrapped in parafilm to prevent water evaporating during the experiments. The motivation for the choice of these samples is to study two different mono-disperse bead sizes of the same material but that will produce two different pore geometries when closely packed. Furthermore, the packings of 5e50 mm glass beads produces a sample where the solid material is the same, i.e. a uniform distribution of paramagnetic substances, but where the pore geometry is more complex. Finally, the Berea sandstone plug is a sample where the material content, and in particular paramagnetic substances, can vary in a non-uniform way, and where the pore geometry is very complex. To determine the optimal value of k regular CPMG experiments with echo spacing varying between 0.1 and 30 ms were performed in all of the samples. The data was then plotted as a function of tEk , and the right condition for the data collapse was determined visually by varying k manually [15]. The encoding period te was 50 ms and td was varied logarithmically from 50 ms to 8 s in 32 steps for all four samples. The echo spacing, tE was varied logarithmically between 0.1 and 20.4 ms by keeping the total echo time (ntE ) constant at 20.4 ms, with a corresponding variation of the number of p pulses from 204 to 1, in 32 steps. Notably, G20 is not measured directly, but through DG20 , which can be normalized using the diffusion coefficient for water ðDH2 O ¼ 2:3 105 cm2 /s at 25+ C). In all the experiments the FID signal was acquired with a dwell time (dw) of 6.65 ms and with a total of 512 data points (td), so that tFID ¼ td,dw. The obtained data were analyzed as 2D Fredholm integral equations using software [16] based on the Schlumberger algorithm [17]. 3. Results and discussion
tFID k P b; T2 ebntE e T2 dbT2
(4)
where P(b; T2 ) is a 2D distribution function of b and T2 . It has been experimentally verified that a single exponential decay for the FIDsignal is a good approximation for a single pore size [2]. Furthermore, assuming the FD regime the distribution involving internal gradients, PðG0 ; T2 Þ, can be determined using k ¼ 3 and b ¼ 1 2 2 12 Dg G0 according to Eq. (3). Finally, using Eq. (1) the T2 -axis can be rescaled to give the distributions Pðb; Dcapp Þ or PðG0 ; Dcapp Þ. Assuming that Eq. (2) is valid for the PðDcapp ; G0 Þ distribution, one can determine the correlation between G0 and Dcapp , and use this correlation to determine a pore size distribution. This is done using the following procedure on the obtained data: For each point in pðG0 Þ the maximum value in PðDcapp ; G0 Þ can be determined and the corresponding value of Dcapp can be found. Thus, the G0 -axis can then be rescaled according to a ¼ Dcapp B0 =G0 , giving an axis that describes range of pore sizes, a, and a corresponding distribution pðaÞ can be obtained from pðG0 Þ.
The optimal value of k was determined to be 2.00, 2.05, 1.95 and 1.70 for the 30 mm, 100 mm, 5e50 mm glass beads, and Berea sandstone sample, respectively [13]. Thus, as expected the dynamics of the molecules during the modified CPMG part of the sequence is in between the FD and LOC regimes. In the further analysis we chose to evaluate the obtained data according to the general decay constant, b, and the corresponding optimal value of k, but also assuming the FD regime (k ¼ 3) where G0 can be determined. We are then able to compare the plots based on b and G0 , evaluate the effect of not being in the FD regime, and thus carefully use the G0 data to make physical interpretations of the obtained data. Both the Pða; bÞ and Pða; G0 Þ plots presented in Ref. [13] showed a clear tendency of a stronger and more negative correlation between a and b (and G0 ) when going from the more homogeneous pore geometry of the packings of mono-disperse glass beads, via the packings of multi-disperse glass beads, to the more heterogeneous pore structure of the Berea sandstone. Therefore, Pða; bÞ or
Please cite this article in press as: R.T. Lewis, J.G. Seland, Characterization of pore geometry using correlations between magnetic field and internal gradient, Microporous and Mesoporous Materials (2017), http://dx.doi.org/10.1016/j.micromeso.2017.05.041
R.T. Lewis, J.G. Seland / Microporous and Mesoporous Materials xxx (2017) 1e4
Fig. 1. PðDcapp ; bÞ for distilled water confined in random packings of 30 mm (a), 100 mm (b) and 5e50 mm (c) glass beads, and in Berea sandstone (d).
Pða; G0 Þ correlation plots can be used as a measure of pore size heterogeneity. This is also an experimental verification of the approximation given in Eq. (2). Recently, pore size -G0 correlations have been measured also at low field strength [18] with similar conclusions. The PðDcapp ; bÞ plots obtained in the different samples are presented in Fig. 1, and show a clear positive correlation between these two parameters in the more heterogenous samples. Similar to
Fig. 2. PðDcapp ; G0 Þ for distilled water confined in random packings of 30 mm (a), 100 mm (b) and 5e50 mm (c) glass beads, and in Berea sandstone (d). The dotted lines corresponds to the maximum value in the distributions, used for determining the correlation between Dcapp and G0 , and the rescaling of the G0 -axis to a pore size distribution, as presented below in Fig. 3.
3
what was observed for the Pða; bÞ data, the correlation is less obvious, or even non-existing, in the more homogeneous samples. The corresponding PðDcapp ; G0 Þ plots (k ¼ 3) obtained in the different samples are presented in Fig. 2. As in the PðDcapp ; bÞ plots, Dcapp has a positive correlation with G0 , even though the dynamic behaviour of the molecules is not in the FD-regime. Interestingly, in the PðDcapp ; G0 Þ plots the positive correlation is pronounced also in the more homogeneous samples. We have no clear explanation for this difference between PðDcapp ; bÞ and PðDcapp ; G0 Þ, but it is most likely connected to the fact that our data is not obtained in the FD regime. Experiments performed at lower field strength, satisfying the criteria of the FD regime, should be performed on the same samples in order to investigate this effect. The positive correlation between Dcapp and G0 is expected from Eq. (2), and using the approach described above under section 2.1 it can be used as a novel method for determining a pore size distribution pðaÞ. The results from this procedure is presented in Fig. 3. The results are compared to pore size distributions obtained using a DDIF analysis on the same set of multidimensional data [13]. The average pore size obtained for the 30 mm and 100 mm glass bead samples using DDIF is 17 mm and 44 mm, respectively. This correspond well with the results obtained by Barker et al. [19] who studied the structure and dynamics of powders, using a computersimulation approach. They found that in packings of mono-sized spheres the mean diameter of a pore is 0.45 times the size of the sphere, which should result in expected values of 14 mm and 45 mm for our samples. For the 30 mm glass bead sample there is a good correspondence between the pore size distribution obtained using DDIF and the Dcapp - G0 correlation. However, for the 100 mm glass bead sample the pore size distribution obtained using the Dcapp G0 correlation is shifted to higher values (average value is 70 mm) compared to the DDIF results. For the 5e50 mm glass beads, there is again a good correspondence between the two methods, with an average pore size of approximately 30 mm, which is reasonable for this sample. In the Berea sandstone sample there is again a mismatch between the two different methods, with the pore size distribution obtained using the Dcapp - G0 correlation is shifted to
Fig. 3. Pore size distributions obtained from Dcapp - G0 correlations in random packings of 30 mm (a), 100 mm (b) and 5e50 mm (c) glass beads, and in Berea sandstone (d). The corresponding pore size distributions obtained from DDIF experiments (dashed lines) are shown for comparison.
Please cite this article in press as: R.T. Lewis, J.G. Seland, Characterization of pore geometry using correlations between magnetic field and internal gradient, Microporous and Mesoporous Materials (2017), http://dx.doi.org/10.1016/j.micromeso.2017.05.041
4
R.T. Lewis, J.G. Seland / Microporous and Mesoporous Materials xxx (2017) 1e4
higher values compared to the DDIF results. Thus, in the part of the samples with relatively large pores the method based on the Dcapp - G0 correlation overestimates the pore sizes. This could be explained by the fact that the total echo time (ntE ) in the modified CPMG part of the experiment was 20.4 ms, giving a diffusion length of 10 mm during this time interval. An assumption is Eq. (2) is that the gradient is constant across the pore. If the pore size is large compared to the diffusion length, the spins will not experience an average (constant) gradient, and will therefore be more sensitive to local gradients that varies on a scale which is considerably smaller than the pore size. This means that molecules located in the middle of a large pore will experience a much weaker gradient than the average value across the pore, leading to an underestimation of the value of G0 and a corresponding overestimation of the pore size. Another explanation can be that the dynamics of the molecules is not in the FD regime, leading to an error in the measured G0 . It should also be mentioned that the signal attenuation during the modified CPMG part of the sequence did not go all the way to the noise level, which could also lead to an underestimation of G0 . All these possible effects should be investigated by performing experiments at lower magnetic field strengths, and making sure a sufficient G0 -attenuation. 4. Conclusion We have shown that the correlation between Dcapp and G0 can be used for a novel approach of determining pore size distributions in liquid saturated porous systems. The obtained pore size distributions correspond well with corresponding distributions obtained using the well established DDIF technique for pore sizes up to approximately 100 mm, but seems to overestimate the size of larger
pores. This effect should be studied in more detail in future studies, in particular at lower magnetic field strengths.
Acknowledgments We acknowledge financial support from the Research Council of Norway (Project no. 210448). We also thank Uni-Research CIPR for providing us with the Berea sandstone material.
References [1] J. Karger, H. Pfeifer, S. Rudtsch, J. Mag. Reson 85 (2) (1989) 381e387. [2] Q. Chen, A.E. Marble, B.G. Colpitts, B.J. Balcom, J. Mag. Reson 175 (2) (2005) 300e308. [3] J. Mitchell, T.C. Chandrasekera, L.F. Gladden, J. Chem. Phys. 132 (24) (2010) 244705. [4] M.D. Hurlimann, J. Mag. Reson 131 (2) (1998) 232e240. [5] P.N. Sen, S. Axelrod, J. Appl. Phys. 86 (8) (1999) 4548e5454. [6] B. Audoly, P.N. Sen, S. Ryu, Y.Q. Song, J. Mag. Reson 164 (1) (2003) 154e159. [7] L.M. Burcaw, P.T. Callaghan, J. Mag. Reson 216 (2012) 144e151. [8] B.Q. Sun, K.J. Dunn, Phys. Rev. E 65 (5) (2002) 051309. [9] Y.-Q. Song, Concepts Mag. Reson 18A (2) (2003) 97e110. [10] N.V. Lisitza, Y.Q. Song, J. Chem. Phys. 114 (2001) 9120. [11] K.R. Brownstein, C.E. Tarr, Phys. Rev. A 19 (6) (1979) 2446e2453. [12] J. Mitchell, J. Mag. Reson 240 (2014) 52e60. [13] R.T. Lewis, J.G. Seland, J. Mag. Reson 263 (2016) 19e32. [14] J. Mitchell, T.C. Chandrasekera, J. Chem. Phys. 141 (22) (2014) 224201e224217. [15] J. Mitchell, T.C. Chandrasekera, L.F. Gladden, J. Chem. Phys. 139 (7) (2013) 074205. [16] P.T. Callaghan, Provided with Courtesy of Paul Callaghan, 2002. [17] L. Venkataramanan, Y.Q. Song, M.D. Hurlimann, IEEE Trans. Sign. Process 50 (5) (2002) 1017e1026. [18] Y. Zhang, L. Xiao, G. Liao, B. Bluemich, J. Mag. Reson 267 (2016) 37e42. [19] G.C. Barker, A. Mehta, Phys. Rev. A 45 (6) (1992) 3435e3446.
Please cite this article in press as: R.T. Lewis, J.G. Seland, Characterization of pore geometry using correlations between magnetic field and internal gradient, Microporous and Mesoporous Materials (2017), http://dx.doi.org/10.1016/j.micromeso.2017.05.041