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3 - PGSE measurements in simple porous systems

Published online by Cambridge University Press:  06 August 2010

William S. Price
William S. Price
Affiliation:
University of Western Sydney
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Summary

Introduction

In the previous chapter we considered the various methods for relating echo attenuation with diffusion in the case of free isotropic diffusion for a single diffusing species. It was observed that the echo signal attenuation was single exponential with respect to q2 and the correct value of the diffusion coefficient was determined irrespective of the measuring time (i.e., Δ). Due to the relatively long timescale of the diffusion measurement (i.e., Δ), gradient-based measurements are sensitive to the enclosing geometry (or pore) in which the diffusion occurs (i.e., restricted diffusion) and an appropriate model must be used to account for the effects of restricted diffusion when analysing the data. The effects of the restriction can be used to provide structural information for pores with characterisitc distances (a) in the range of 0.01–100 μm. Thus, gradient methods are especially suited to studying the physics of restricted diffusion and transport in porous materials.

Non-single-exponential decays can arise in a number of ways including multicomponent systems, anisotropic or restricted diffusion. These effects are the subject of the next two chapters (more complex models are studied in Chapter 4). The relevant analytical formulae for diffusion between planes and inside spheres are presented (diffusion in cylinders is presented in the following chapter). It is remarked that these are the commonly used models for benchmarking numerical approaches. We also mention that Grebenkov has recently presented a review of NMR studies of restricted Brownian motion.

Type
Chapter
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NMR Studies of Translational Motion
Principles and Applications
 , pp. 120 - 146
Publisher: Cambridge University Press
Print publication year: 2009

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References

Hagslätt, H., Jönsson, B., Nydén, M. and Söderman, O., Predictions of Pulsed Field Gradient NMR Echo-Decays for Molecules Diffusing in Various Restrictive Geometries. Simulations of Diffusion Propagators Based on a Finite Element Method. J. Magn. Reson. 161 (2003), 138–47.CrossRefGoogle ScholarPubMed
Grebenkov, D. S., NMR Survey of the Reflected Brownian Motion. Rev. Mod. Phys. 79 (2007), 1077–136.CrossRefGoogle Scholar
Sen, P. N., Time-Dependent Diffusion Coefficient as a Probe of Geometry. Concepts Magn. Reson. 23A (2004), 1–21.CrossRefGoogle Scholar
Tanner, J. E., Transient Diffusion in a System Partitioned by Permeable Barriers. Application to NMR Measurements with a Pulsed Field Gradient. J. Chem. Phys. 69 (1978), 1748–54.CrossRefGoogle Scholar
Zielinski, L. J. and Sen, P. N., Effects of Finite Width Pulses in the Pulsed Field Gradient Measurement of the Diffusion Coefficient in Connected Porous Media. J. Magn. Reson. 165 (2003), 153–61.CrossRefGoogle ScholarPubMed
Stallmach, F. and Galvosas, P., Spin Echo NMR Diffusion Studies. In Annual Reports on NMR Spectroscopy, ed. Webb, G. A.. vol. 61. (London: Elsevier, 2007), pp. 51–131.Google Scholar
Grebenkov, D. S., Multiexponential Attenuation of the CPMG Spin Echoes Due to a Geometrical Confinement. J. Magn. Reson. 180 (2006), 118–26.CrossRefGoogle ScholarPubMed
Woessner, D. E., Spin-Echo, N. M. R.Self-Diffusion Measurements on Fluids Undergoing Restricted Diffusion. J. Phys. Chem. 67 (1963), 1365–7.CrossRefGoogle Scholar
Swiet, T. M. and Sen, P. N., Decay of Nuclear Magnetization by Bounded Diffusion in a Constant Field Gradient. J. Chem. Phys. 100 (1994), 5597–604.CrossRefGoogle Scholar
Hürlimann, M. D, Helmer, K. G., Swiet, T. M., Sen, P. N., and Sotak, C. H., Spin Echoes in a Constant Gradient and in the Presence of Simple Restriction. J. Magn. Reson. A 113 (1995), 260–4.CrossRefGoogle Scholar
Helmer, K. G., Hürlimann, M. D., Swiet, T. M., Sen, P. N., and Sotak, C. H., Determination of Ratio of Surface Area to Pore Volume from Restricted Diffusion in a Constant Field Gradient. J. Magn. Reson. A 115 (1995), 257–69.CrossRefGoogle Scholar
Cory, D. G. and Garroway, A. N., Measurement of Translational Displacement Probabilities by NMR: An Indicator of Compartmentation. Magn. Reson. Med. 14 (1990), 435–44.CrossRefGoogle ScholarPubMed
Callaghan, P. T., MacGowan, D., Packer, K. J., and Zelaya, F. O., High-Resolution q-Space Imaging in Porous Structures. J. Magn. Reson. 90 (1990), 177–82.Google Scholar
Callaghan, P. T. and Coy, A., PGSE NMR and Molecular Translational Motion in Porous Media. In NMR Probes and Molecular Dynamics, ed. Tycko, R.. (Dordrecht: Kluwer, 1994), pp. 489–523.Google Scholar
Linse, P. and Söderman, O., The Validity of the Short-Gradient-Pulse Approximation in NMR studies of Restricted Diffusion. Simulations of Molecules Diffusing between Planes, in Cylinders and Spheres. J. Magn. Reson. A 116 (1995), 77–86.CrossRefGoogle Scholar
Packer, K. J., Oil Reservoir Rocks Examined by MRI. In Encyclopedia of Nuclear Magnetic Resonance, ed. Grant, D. M. and Harris, R. K.. vol. 5. (New York: Wiley, 1996), pp. 3365–75.Google Scholar
Callaghan, P. T., Eccles, C. D., and Xia, Y., NMR Microscopy of Dynamic Displacements: gradient or diffusion weighting factor, more commonly written as b-Space and q-Space Imaging. J. Phys. E: Sci. Instrum. 21 (1988), 820–2.CrossRefGoogle Scholar
Baldwin, A. J., Christodolou, J., Barker, P. D., and Dobson, C. M., Contribution of Rotational Diffusion to Pulsed Field Gradient Diffusion Measurements. J. Chem. Phys. 127 (2007), 114505-1–114505-8.CrossRefGoogle ScholarPubMed
Szafer, A., Zhong, J., and Gore, J. C., Theoretical Model for Water Diffusion in Tissues. Magn. Reson. Med. 33 (1995), 697–712.CrossRefGoogle ScholarPubMed
Balinov, B., Jönsson, B., Linse, P., and Söderman, O., The NMR Self-Diffusion Method Applied to Restricted Diffusion. Simulation of Echo Attenuation from Molecules in Spheres and Between Planes. J. Magn. Reson. A 104 (1993), 17–25 and J. Magn. Reson. A 108 (1994), 130.CrossRefGoogle Scholar
Sen, P. N. and Hürlimann, M. D., Analysis of Nuclear Magnetic Resonance Spin Echoes Using Simple Structure Factors. J. Chem. Phys. 101 (1994), 5423–30.CrossRefGoogle Scholar
Hürlimann, M. D., Swiet, T. M., and Sen, P. N., Comparison of Diffraction and Diffusion Measurements in Porous Media. J. Non-Cryst. Solids 182 (1995), 198–205.CrossRefGoogle Scholar
Sen, P. N., Hürlimann, M. D., and Swiet, T. M., Debye-Porod Law of Diffraction for Diffusion in Porous Media. Phys. Rev. B 51 (1995), 601–4.CrossRefGoogle ScholarPubMed
Hürlimann, M. D., Schwartz, L. M., and Sen, P. N., Probability of Return to the Origin at Short Times: A Probe of Microstructure in Porous Media. Phys. Rev. B 51 (1995), 14936–40.CrossRefGoogle ScholarPubMed
Mitra, P. P., Diffusion in Porous Materials as Probed by Pulsed Gradient NMR Measurements. Physica A 241 (1997), 122–7.CrossRefGoogle Scholar
Mitra, P. P., Latour, L. L., Kleinberg, R. L., and Sotak, C. H., Pulsed-Field-Gradient NMR Measurements of Restricted Diffusion and the Return-to-the Origin Probability. J. Magn. Reson. A 114 (1995), 47–58.CrossRefGoogle Scholar
Hertz, H. G., Translational Motions as Studied by Nuclear Magnetic Resonance. In Molecular Motions in Liquids, ed. Lascombe, J.. (Dordrecht: Reidel, 1974), pp. 337–57.CrossRefGoogle Scholar
Callaghan, P. T., Coy, A., MacGowan, D., Packer, K. J., and Zelaya, F. O., Diffraction-Like Effects in NMR Diffusion of Fluids in Porous Solids. Nature 351 (1991), 467–9.CrossRefGoogle Scholar
Callaghan, P. T., Coy, A., MacGowan, D., and Packer, K. J., Diffusion of Fluids in Porous Solids Probed by Pulsed Field Gradient Spin Echo NMR. J. Mol. Liquids 54 (1992), 239–51.CrossRefGoogle Scholar
Cotts, R. M., Diffusion and Diffraction. Nature 351 (1991), 443–4.CrossRefGoogle Scholar
Fleischer, G. and Fujara, F., NMR as a Generalized Incoherent Scattering Experiment. NMR Basic Princ. Progr. 30 (1994), 157–207.Google Scholar
Appel, M., Fleischer, G., Geschke, D., Kärger, J., and Winkler, M., Pulsed-Field-Gradient NMR Analogue of the Single-Slit Diffraction Pattern. J. Magn. Reson. A 122 (1996), 248–50.CrossRefGoogle Scholar
Kärger, J. and Stallmach, F., PFG NMR Studies of Anomalous Diffusion. In Diffusion in Condensed Matter, ed. Heitjans, P. and Kärger, J.. (Berlin: Springer, 2006), pp. 417–59.Google Scholar
McQuarrie, D. A., Statistical Mechanics. (New York: Harper & Row, 1976).Google Scholar
Marshall, A. G. and Verdun, F. R., Fourier Transforms in NMR, Optical, and Mass Spectroscopy. A User's Handbook. (Amsterdam: Elsevier, 1990).Google Scholar
Callaghan, P. T., Principles of Nuclear Magnetic Resonance Microscopy. (Oxford: Clarendon Press, 1991).Google Scholar
Talagala, S. L. and Lowe, I. J., Introduction to Magnetic Resonance Imaging. Concepts Magn. Reson. 3 (1991), 145–59.CrossRefGoogle Scholar
Xia, Y., Contrast in NMR Imaging and Microscopy. Concepts Magn. Reson. 8 (1996), 205–25.3.0.CO;2-2>CrossRefGoogle Scholar
Mansfield, P. and Grannell, P. K., ‘Diffraction’ and Microscopy in Solids and Liquids by NMR. Phys. Rev. B 12 (1975), 3618–34.CrossRefGoogle Scholar
Callaghan, P. T., NMR Imaging, NMR Diffraction and Applications of Pulsed Gradient Spin Echoes in Porous Media. Magn. Reson. Imaging 14 (1996), 701–9.CrossRefGoogle ScholarPubMed
Barrall, G. A., Frydman, L., and Chingas, G. C., NMR Diffraction and Spatial Statistics of Stationary Systems. Science 255 (1992), 714–17.CrossRefGoogle ScholarPubMed
Tanner, J. E. and Stejskal, E. O., Restricted Self-Diffusion of Protons in Colloidal Systems by the Pulsed-Gradient, Spin-Echo Method. J. Chem. Phys. 49 (1968), 1768–77.CrossRefGoogle Scholar
Andrasko, J., Measurement of Membrane Permeability to Slowly Penetrating Molecules by a Pulse Gradient NMR Method. J. Magn. Reson. 21 (1976), 479–84.Google Scholar
Price, W. S., Stilbs, P., and Söderman, O., Determination of Pore Space Shape and Size in Porous Systems Using NMR Diffusometry. Beyond the Short Gradient Pulse Approximation. J. Magn. Reson. 160 (2003), 139–43.CrossRefGoogle ScholarPubMed
Topgaard, D. and Söderman, O., Experimental Determination of Pore Shape Using q-Space NMR Microscopy in the Long Diffusion-Time Limit. Magn. Reson. Imaging 21 (2003), 69–76.CrossRefGoogle ScholarPubMed
Robertson, B., Spin-Echo Decay of Spins Diffusing in a Bounded Region. Phys. Rev. 151 (1966), 273–7.CrossRefGoogle Scholar
Neuman, C. H., Spin Echo of Spins Diffusing in a Bounded Medium. J. Chem. Phys. 60 (1974), 4508–11.CrossRefGoogle Scholar
Veeman, W. S., Diffusion in a Closed Sphere. In Annual Reports on NMR Spectroscopy, ed. Webb, G. A.. vol. 50. (London: Elsevier, 2003), pp. 201–16.CrossRefGoogle Scholar
Abramowitz, M. and Stegun, I. A., Handbook of Mathematical Functions. (New York: Dover, 1970).Google Scholar
Murday, J. S. and Cotts, R. M., Self-Diffusion Coefficient of Liquid Lithium. J. Chem. Phys. 48 (1968), 4938–45.CrossRefGoogle Scholar
Söderman, O. and Stilbs, P., NMR Studies of Complex Surfactant Systems. Prog. NMR Spectrosc. 26 (1994), 445–82.CrossRefGoogle Scholar
Söderman, O., Pulsed-Field-Gradient NMR Studies of Emulsions. Droplet Sizes and Concentrated Emulsions. Progr. Colloid Polym. Sci. 106 (1997), 34–41.CrossRefGoogle Scholar
Söderman, O., Stilbs, P., and Price, W. S., NMR Studies of Surfactants. Concepts Magn. Reson. 23A (2004), 121–35.CrossRefGoogle Scholar
Johns, M. L. and Hollingsworth, K. G., Characterisation of Emulsion Systems Using NMR and MRI. Prog. NMR Spectrosc. 50 (2007), 51–70.CrossRefGoogle Scholar
Packer, K. J. and Rees, C., Pulsed NMR Studies of Restricted Diffusion. 1. Droplet Size Distributions in Emulsions. J. Colloid Interface Sci. 40 (1972), 206–18.CrossRefGoogle Scholar
Callaghan, P., Jolley, K. W., and Humphrey, R. S., Diffusion of Fat and Water in Cheese as Studied by Pulsed Field Gradient Nuclear Magnetic Resonance. J. Colloid Interface Sci. 93 (1983), 521–9.CrossRefGoogle Scholar
Ambrosone, L., Murgia, S., Cinelli, G., Monduzzi, M., and Ceglie, A., Size Polydispersity Determination in Emulsion Systems by Free Diffusion Measurements via PFG-NMR. J. Phys. Chem. B 108 (2004), 18472–8.CrossRefGoogle Scholar
Ambrosone, L., Ceglie, A., Colafemmina, G., and Palazzo, G., A Novel Approach for Determining the Droplet Size Distribution in Emulsion Systems by Generating Function. J. Chem. Phys. 107 (1997), 10756–63.CrossRefGoogle Scholar
Ambrosone, L., Ceglie, A., Colafemmina, G., and Palazzo, G., General Methods for Determining the Droplet Size Distribution in Emulsion Systems. J. Chem. Phys. 110 (1999), 797–854.CrossRefGoogle Scholar
Enden, J. C., Waddington, D., Aalst, H., Kralingen, C. G., and Packer, K. J., Rapid Determination of Water Droplet Size Distributions by PFG-NMR. J. Colloid Interface Sci. 140 (1990), 105–13.CrossRefGoogle Scholar
Fourel, I., Guillement, J. P., and Botlan, D., Determination of Water Droplet Size Distributions by Low Resolution PFG-NMR. I. Liquid Emulsions. J. Colloid Interface Sci. 164 (1994), 48–53.CrossRefGoogle Scholar
Peña, A. A. and Hirasaki, G. J., Enhanced Characterizations of Oil Field Emulsions via NMR Diffusion and Transverse Relaxation Experiments. Adv. Colloid Interface Sci. 105 (2003), 103–50.CrossRefGoogle Scholar
Kuchel, P. W., Eykyn, T. R., and Regan, D. G., Measurement of Compartment Size in q-Space Experiments: Fourier Transform of the Second Derivative. Magn. Reson. Med. 52 (2004), 907–12.CrossRefGoogle ScholarPubMed
Price, W. S. and Söderman, O., Some ‘Reflections’ on the Effects of Finite Gradient Pulse Lengths in PGSE NMR Experiments in Restricted Systems. Isr. J. Chem. 43 (2003), 25–32.CrossRefGoogle Scholar
Coy, A. and Callaghan, P. T., Pulsed Gradient Spin Echo Nuclear Magnetic Resonance for Molecules Diffusing Between Partially Reflecting Rectangular Barriers. J. Chem. Phys. 101 (1994), 4599–609.CrossRefGoogle Scholar
Sheltraw, D. and Kenkre, V. M., The Memory-Function Technique for the Calculation of Pulsed-Gradient NMR Signals in Confined Geometries. J. Magn. Reson. A 122 (1996), 126–36.CrossRefGoogle Scholar
Lori, N. F., Conturo, T. E., and Bihan, D., Definition of Displacement Probability and Diffusion Time in q-Space Magnetic Resonance Measurements That Use Finite-Duration Diffusion-Encoding Gradients. J. Magn. Reson. 165 (2003), 185–95.CrossRefGoogle ScholarPubMed
Helmer, K. G., Meiler, M. R., Sotak, C. H., and Petrucelli, J. D., Comparison of the Return-to-the-Origin Probability and the Apparent Diffusion Coefficient of Water as Indicators of Necrosis in RIF-1 Tumors. Magn. Reson. Med. 49 (2003), 468–78.CrossRefGoogle ScholarPubMed
Callaghan, P. T., A Simple Matrix Formalism for Spin Echo Analysis of Restricted Diffusion under Generalized Gradient Waveforms. J. Magn. Reson. 129 (1997), 74–84.CrossRefGoogle ScholarPubMed
Malmborg, C., Sjöbeck, M., Brockstedt, S., Englund, E., Söderman, O., and Topgaard, D., Mapping the Intracellular Fraction of Water by Varying the Gradient Pulse Length in q-Space Diffusion MRI. J. Magn. Reson. 180 (2006), 280–5.CrossRefGoogle ScholarPubMed
Blees, M. H., The Effect of Finite Duration of Gradient Pulses on the Pulsed-Field-Gradient NMR Method for Studying Restricted Diffusion. J. Magn. Reson. A 109 (1994), 203–9.CrossRefGoogle Scholar
Mitra, P. P. and Halperin, B. I., Effects of Finite Gradient Pulse Widths in Pulsed Field Gradient Diffusion Measurements. J. Magn. Reson. A 113 (1995), 94–101.CrossRefGoogle Scholar
Wang, L. Z., Caprihan, A., and Fukushima, E., The Narrow-Pulse Criterion for Pulsed-Gradient Spin-Echo Diffusion Measurements. J. Magn. Reson. A 117 (1995), 209–19.CrossRefGoogle Scholar
Mair, R. W., Sen, P. N., Hürlimann, M. D., Patz, S., Cory, D. G., and Walsworth, R. L., The Narrow Pulse Approximation and Long Length Scale Determination in Xenon Gas Diffusion NMR Studies of Model Porous Media. J. Magn. Reson. 156 (2002), 202–12.CrossRefGoogle ScholarPubMed
Malmborg, C., Topgaard, D., and Söderman, O., NMR Diffusometry and the Short Gradient Pulse Limit Approximation. J. Magn. Reson. 169 (2004), 85–91.CrossRefGoogle ScholarPubMed
Kenkre, V. M., Fukushima, E., and Sheltraw, D., Simple Solution of the Torrey–Bloch Equations in the NMR Study of Molecular Diffusion. J. Magn. Reson. 128 (1997), 62–9.CrossRefGoogle Scholar
Axelrod, S. and Sen, P. N., Nuclear Magnetic Resonance Spin Echoes for Restricted Diffusion in an Inhomogeneous Field: Methods and Asymptotic Regimes. J. Chem. Phys. 114 (2001), 6878–95.CrossRefGoogle Scholar
Kenkre, V. M. and Sevilla, F. J., Analytic Considerations in the Theory of NMR Microscopy. Physica A 371 (2006), 139–43.CrossRefGoogle Scholar
Sevilla, F. J. and Kenkre, V. M., Theory of the Spin Echo Signal in NMR Microscopy: Analytic Solutions of a Generalized Torrey–Bloch Equation. J. Phys. Condens. Matter 19 (2007), 065113-1–065113-14.CrossRefGoogle Scholar
Wayne, R. C. and Cotts, R. M., Nuclear-Magnetic-Resonance Study of Self-Diffusion in a Bounded Medium. Phys. Rev. 151 (1966), 264–72.CrossRefGoogle Scholar
Zientara, G. P. and Freed, J. H., Spin-Echoes for Diffusion in Bounded, Heterogeneous Media: A Numerical Study. J. Chem. Phys. 72 (1980), 1285–92.CrossRefGoogle Scholar
Daragan, V. A. and Il'ina, E. E., Pulsed Field Gradient NMR for the Study of the Structure of Membrane Systems. Chem. Phys. 158 (1991), 105–11.CrossRefGoogle Scholar
Novikov, E. G., Dusschoten, D., and As, H., Modeling of Self-Diffusion and Relaxation Time NMR in Multi-Compartment Systems. J. Magn. Reson. 135 (1998), 522–8.CrossRefGoogle ScholarPubMed
Zielinski, L. J. and Sen, P. N., Relaxation of Nuclear Magnetization in a Nonuniform Magnetic Field Gradient and in a Restricted Geometry. J. Magn. Reson. 147 (2000), 95–103.CrossRefGoogle Scholar
Sen, P. N., André, A., and Axelrod, S., Spin Echoes of Nuclear Magnetization Diffusing in a Constant Magnetic Field Gradient and in a Restricted Geometry. J. Chem. Phys. 111 (1999), 6548–55.CrossRefGoogle Scholar
Hwang, S. N., Chin, C.-L., Wehrli, F. W., and Hackney, D. B., An Image-Based Finite Difference Model for Simulating Restricted Diffusion. Magn. Reson. Med. 50 (2003), 373–82.CrossRefGoogle ScholarPubMed
Leibig, M., Random Walks and NMR Measurements in Porous Media. J. Phys. A: Math. Gen. 26 (1993), 3349–67.CrossRefGoogle Scholar
Callaghan, P. T., Coy, A., Halpin, T. P. J., MacGowan, D., Packer, K. J., and Zelaya, F. O., Diffusion in Porous Systems and the Influence of Pore Morphology in Pulsed Field Gradient Spin-Echo Nuclear Magnetic Resonance Studies. J. Chem. Phys. 97 (1992), 651–62.CrossRefGoogle Scholar
Celebre, G., Coppola, L., and Raineri, G. A., Water Self-Diffusion in Lyotropic Systems by Simulation of Pulsed Field Gradient Spin Echo Nuclear Magnetic Resonance Experiments. J. Chem. Phys. 97 (1992), 7781–5.CrossRefGoogle Scholar
Mitra, P. P., Sen, P. N., and Schwartz, L. M., Short-Time Behaviour of the Diffusion Coefficient as a Geometrical Probe of Porous Media. Phys. Rev. B 47 (1993), 8565–74.CrossRefGoogle ScholarPubMed
Lennon, A. J. and Kuchel, P. W., Enhancement of the ‘Diffraction-Like’ Effect in NMR Diffusion Experiments. J. Magn. Reson. A 111 (1994), 208–11.CrossRefGoogle Scholar
Sen, P. N., Schwartz, L. M., Mitra, P. P., and Halperin, B. I., Surface Relaxation and the Long-Time Diffusion Coefficient in Porous Media: Periodic Geometries. Phys. Rev. B 49 (1994), 215–25.CrossRefGoogle ScholarPubMed
Håkansson, B., Jönsson, B., Linse, P., and Söderman, O., The Influence of a Nonconstant Magnetic-Field Gradient on PFG NMR Diffusion Experiments. A Brownian-Dynamics Computer Simulation Study. J. Magn. Reson. 124 (1997), 343–51.CrossRefGoogle Scholar
Duh, A., Mohorič, A., and Stepišnik, J., Computer Simulation of the Spin-Echo Spatial Distribution in the Case of Restricted Self-Diffusion. J. Magn. Reson. 148 (2001), 257–66.CrossRefGoogle ScholarPubMed
Valckenborg, R. M. E., Huinink, H. P., Sande, J. J., and Kopinga, K., Random Walk Simulations of NMR Dephasing Effects Due to Uniform Magnetic Field Gradients in a Pore. Phys. Rev. E 65 (2002), 021306-1–021306-8.CrossRefGoogle Scholar
Grebenkov, D. S., Guillot, G., and Sapoval, B., Restricted Diffusion in a Model Acinar Labyrinth by NMR: Theoretical and Numerical Results. J. Magn. Reson. 184 (2007), 143–56.CrossRefGoogle Scholar
Grebenkov, D. S., Nuclear Magnetic Resonance Restricted Diffusion Between Parallel Planes in a Cosine Magnetic Field: An Exactly Solvable Model. J. Chem. Phys. 126 (2007), 104706-1–104706-15.CrossRefGoogle Scholar
Caprihan, A., Wang, L. Z., and Fukushima, E., A Multiple-Narrow-Pulse Approximation for Restricted Diffusion in a Time-Varying Field Gradient. J. Magn. Reson. A 118 (1996), 94–102.CrossRefGoogle Scholar
Codd, S. L. and Callaghan, P. T., Spin Echo Analysis of Restricted Diffusion Under Generalized Gradient Waveforms: Planar, Cylindrical, and Spherical Pores with Wall Relaxivity. J. Magn. Reson. 137 (1999), 358–72.CrossRefGoogle ScholarPubMed
Hayamizu, K., Akiba, E., Bando, T., Aihara, Y., and Price, W. S., NMR Studies on Poly(ethylene oxide)-Based Polymer Electrolytes with Different Cross-Linking Doped with LiN(SO2CF3)2. Restricted Diffusion of the Polymer and Lithium Ion and Time-Dependent Diffusion of the Anion. Macromolecules 36 (2003), 2785–92.CrossRefGoogle Scholar
Ryland, B. N. and Callaghan, P. T., Spin Echo Analysis of Restricted Diffusion under Generalized Gradient Waveforms for Spherical Pores with Relaxivity and Interconnections. Isr. J. Chem. 43 (2003), 1–7.CrossRefGoogle Scholar
Sukstanskii, A. L. and Yablonskiy, D. A., Effects of Restricted Diffusion on MR Signal Formation. J. Magn. Reson. 157 (2002), 92–105.CrossRefGoogle ScholarPubMed
Barzykin, A. V., Exact Solution of the Torrey–Bloch Equation for a Spin Echo in Restricted Geometries. Phys. Rev. B 58 (1998), 14171–4.CrossRefGoogle Scholar
Barzykin, A. V., Theory of Spin Echo in Restricted Geometries Under a Step-Wise Gradient Pulse Sequence. J. Magn. Reson. 139 (1999), 342–53.CrossRefGoogle Scholar
Nakajima, S., On Quantum Theory of Transport Phenomena. Prog. Theor. Phys. 20 (1958), 948–59.CrossRefGoogle Scholar
Zwanzig, R., Nonequilibrium Statistical Mechanics. (Oxford: Oxford, 2001).Google Scholar
Guyer, R. A. and McCall, K. R., Lattice Boltzmann Description of Magnetization in Porous Media. Phys. Rev. B 62 (2000), 3674–88.CrossRefGoogle Scholar
Grebenkov, D. S., Multiple Correlation Function Approach: Rigorous Results for Simple Geometries. Diffusion Fundamentals 5 (2007), 1.1–1.34.Google Scholar
Lennon, A. J. and Kuchel, P. W., Neural Networks Used to Interpret Pulsed-Gradient Restricted-Diffusion Data. J. Magn. Reson. A 107 (1994), 229–35.CrossRefGoogle Scholar
Kuchel, P. W. and Durrant, C. J., Permeability Coefficients from NMR q-Space Data: Models with Unevenly Spaced Semi-Permeable Parallel Membranes. J. Magn. Reson. 139 (1999), 258–72.CrossRefGoogle ScholarPubMed

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