Book contents
- Frontmatter
- Contents
- Preface
- Acknowledgements
- Abbreviations
- 1 Diffusion and its measurement
- 2 Theory of NMR diffusion and flow measurements
- 3 PGSE measurements in simple porous systems
- 4 PGSE measurements in complex and exchanging systems
- 5 PGSE hardware
- 6 Setup and analysis of PGSE experiments
- 7 PGSE hardware and sample problems
- 8 Specialised PGSE and related techniques
- 9 NMR imaging studies of translational motion
- 10 B1 gradient methods
- 11 Applications
- Appendix
- Index
- References
3 - PGSE measurements in simple porous systems
Published online by Cambridge University Press: 06 August 2010
Book contents
- Frontmatter
- Contents
- Preface
- Acknowledgements
- Abbreviations
- 1 Diffusion and its measurement
- 2 Theory of NMR diffusion and flow measurements
- 3 PGSE measurements in simple porous systems
- 4 PGSE measurements in complex and exchanging systems
- 5 PGSE hardware
- 6 Setup and analysis of PGSE experiments
- 7 PGSE hardware and sample problems
- 8 Specialised PGSE and related techniques
- 9 NMR imaging studies of translational motion
- 10 B1 gradient methods
- 11 Applications
- Appendix
- Index
- References
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
- Information
- NMR Studies of Translational MotionPrinciples and Applications, pp. 120 - 146Publisher: Cambridge University PressPrint publication year: 2009
References
, , and , 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
, NMR Survey of the Reflected Brownian Motion. Rev. Mod. Phys. 79 (2007), 1077–136.CrossRefGoogle Scholar
, Time-Dependent Diffusion Coefficient as a Probe of Geometry. Concepts Magn. Reson. 23A (2004), 1–21.CrossRefGoogle Scholar
, 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
and , 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
and , Spin Echo NMR Diffusion Studies. In Annual Reports on NMR Spectroscopy, ed. . vol. 61. (London: Elsevier, 2007), pp. 51–131.Google Scholar
, Multiexponential Attenuation of the CPMG Spin Echoes Due to a Geometrical Confinement. J. Magn. Reson. 180 (2006), 118–26.CrossRefGoogle ScholarPubMed
, Self-Diffusion Measurements on Fluids Undergoing Restricted Diffusion. J. Phys. Chem. 67 (1963), 1365–7.CrossRefGoogle Scholar
and , Decay of Nuclear Magnetization by Bounded Diffusion in a Constant Field Gradient. J. Chem. Phys. 100 (1994), 5597–604.CrossRefGoogle Scholar
, , , , and , Spin Echoes in a Constant Gradient and in the Presence of Simple Restriction. J. Magn. Reson. A 113 (1995), 260–4.CrossRefGoogle Scholar
, , , , and , 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
and , Measurement of Translational Displacement Probabilities by NMR: An Indicator of Compartmentation. Magn. Reson. Med. 14 (1990), 435–44.CrossRefGoogle ScholarPubMed
, , , and , High-Resolution q-Space Imaging in Porous Structures. J. Magn. Reson. 90 (1990), 177–82.Google Scholar
and , PGSE NMR and Molecular Translational Motion in Porous Media. In NMR Probes and Molecular Dynamics, ed. . (Dordrecht: Kluwer, 1994), pp. 489–523.Google Scholar
and , 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
, Oil Reservoir Rocks Examined by MRI. In Encyclopedia of Nuclear Magnetic Resonance, ed. and . vol. 5. (New York: Wiley, 1996), pp. 3365–75.Google Scholar
, , and , 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
, , , and , Contribution of Rotational Diffusion to Pulsed Field Gradient Diffusion Measurements. J. Chem. Phys. 127 (2007), 114505-1–114505-8.CrossRefGoogle ScholarPubMed
, , and , Theoretical Model for Water Diffusion in Tissues. Magn. Reson. Med. 33 (1995), 697–712.CrossRefGoogle ScholarPubMed
, , , and , 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
and , Analysis of Nuclear Magnetic Resonance Spin Echoes Using Simple Structure Factors. J. Chem. Phys. 101 (1994), 5423–30.CrossRefGoogle Scholar
, , and , Comparison of Diffraction and Diffusion Measurements in Porous Media. J. Non-Cryst. Solids 182 (1995), 198–205.CrossRefGoogle Scholar
, , and , Debye-Porod Law of Diffraction for Diffusion in Porous Media. Phys. Rev. B 51 (1995), 601–4.CrossRefGoogle ScholarPubMed
, , and , 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
, Diffusion in Porous Materials as Probed by Pulsed Gradient NMR Measurements. Physica A 241 (1997), 122–7.CrossRefGoogle Scholar
, , , and , 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
, Translational Motions as Studied by Nuclear Magnetic Resonance. In Molecular Motions in Liquids, ed. . (Dordrecht: Reidel, 1974), pp. 337–57.CrossRefGoogle Scholar
, , , , and , Diffraction-Like Effects in NMR Diffusion of Fluids in Porous Solids. Nature 351 (1991), 467–9.CrossRefGoogle Scholar
, , , and , Diffusion of Fluids in Porous Solids Probed by Pulsed Field Gradient Spin Echo NMR. J. Mol. Liquids 54 (1992), 239–51.CrossRefGoogle Scholar
and , NMR as a Generalized Incoherent Scattering Experiment. NMR Basic Princ. Progr. 30 (1994), 157–207.Google Scholar
, , , , and , Pulsed-Field-Gradient NMR Analogue of the Single-Slit Diffraction Pattern. J. Magn. Reson. A 122 (1996), 248–50.CrossRefGoogle Scholar
and , PFG NMR Studies of Anomalous Diffusion. In Diffusion in Condensed Matter, ed. and . (Berlin: Springer, 2006), pp. 417–59.Google Scholar
and , Fourier Transforms in NMR, Optical, and Mass Spectroscopy. A User's Handbook. (Amsterdam: Elsevier, 1990).Google Scholar
, Principles of Nuclear Magnetic Resonance Microscopy. (Oxford: Clarendon Press, 1991).Google Scholar
and , Introduction to Magnetic Resonance Imaging. Concepts Magn. Reson. 3 (1991), 145–59.CrossRefGoogle Scholar
, Contrast in NMR Imaging and Microscopy. Concepts Magn. Reson. 8 (1996), 205–25.3.0.CO;2-2>CrossRefGoogle Scholar
and , ‘Diffraction’ and Microscopy in Solids and Liquids by NMR. Phys. Rev. B 12 (1975), 3618–34.CrossRefGoogle Scholar
, NMR Imaging, NMR Diffraction and Applications of Pulsed Gradient Spin Echoes in Porous Media. Magn. Reson. Imaging 14 (1996), 701–9.CrossRefGoogle ScholarPubMed
, , and , NMR Diffraction and Spatial Statistics of Stationary Systems. Science 255 (1992), 714–17.CrossRefGoogle ScholarPubMed
and , Restricted Self-Diffusion of Protons in Colloidal Systems by the Pulsed-Gradient, Spin-Echo Method. J. Chem. Phys. 49 (1968), 1768–77.CrossRefGoogle Scholar
, Measurement of Membrane Permeability to Slowly Penetrating Molecules by a Pulse Gradient NMR Method. J. Magn. Reson. 21 (1976), 479–84.Google Scholar
, , and , 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
and , 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
, Spin-Echo Decay of Spins Diffusing in a Bounded Region. Phys. Rev. 151 (1966), 273–7.CrossRefGoogle Scholar
, Spin Echo of Spins Diffusing in a Bounded Medium. J. Chem. Phys. 60 (1974), 4508–11.CrossRefGoogle Scholar
, Diffusion in a Closed Sphere. In Annual Reports on NMR Spectroscopy, ed. . vol. 50. (London: Elsevier, 2003), pp. 201–16.CrossRefGoogle Scholar
and , Self-Diffusion Coefficient of Liquid Lithium. J. Chem. Phys. 48 (1968), 4938–45.CrossRefGoogle Scholar
and , NMR Studies of Complex Surfactant Systems. Prog. NMR Spectrosc. 26 (1994), 445–82.CrossRefGoogle Scholar
, Pulsed-Field-Gradient NMR Studies of Emulsions. Droplet Sizes and Concentrated Emulsions. Progr. Colloid Polym. Sci. 106 (1997), 34–41.CrossRefGoogle Scholar
, , and , NMR Studies of Surfactants. Concepts Magn. Reson. 23A (2004), 121–35.CrossRefGoogle Scholar
and , Characterisation of Emulsion Systems Using NMR and MRI. Prog. NMR Spectrosc. 50 (2007), 51–70.CrossRefGoogle Scholar
and , Pulsed NMR Studies of Restricted Diffusion. 1. Droplet Size Distributions in Emulsions. J. Colloid Interface Sci. 40 (1972), 206–18.CrossRefGoogle Scholar
, , and , 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
, , , , and , Size Polydispersity Determination in Emulsion Systems by Free Diffusion Measurements via PFG-NMR. J. Phys. Chem. B 108 (2004), 18472–8.CrossRefGoogle Scholar
, , , and , A Novel Approach for Determining the Droplet Size Distribution in Emulsion Systems by Generating Function. J. Chem. Phys. 107 (1997), 10756–63.CrossRefGoogle Scholar
, , , and , General Methods for Determining the Droplet Size Distribution in Emulsion Systems. J. Chem. Phys. 110 (1999), 797–854.CrossRefGoogle Scholar
, , , , and , Rapid Determination of Water Droplet Size Distributions by PFG-NMR. J. Colloid Interface Sci. 140 (1990), 105–13.CrossRefGoogle Scholar
, , and , Determination of Water Droplet Size Distributions by Low Resolution PFG-NMR. I. Liquid Emulsions. J. Colloid Interface Sci. 164 (1994), 48–53.CrossRefGoogle Scholar
and , Enhanced Characterizations of Oil Field Emulsions via NMR Diffusion and Transverse Relaxation Experiments. Adv. Colloid Interface Sci. 105 (2003), 103–50.CrossRefGoogle Scholar
, , and , Measurement of Compartment Size in q-Space Experiments: Fourier Transform of the Second Derivative. Magn. Reson. Med. 52 (2004), 907–12.CrossRefGoogle ScholarPubMed
and , 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
and , Pulsed Gradient Spin Echo Nuclear Magnetic Resonance for Molecules Diffusing Between Partially Reflecting Rectangular Barriers. J. Chem. Phys. 101 (1994), 4599–609.CrossRefGoogle Scholar
and , 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
, , and , 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
, , , and , 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
, A Simple Matrix Formalism for Spin Echo Analysis of Restricted Diffusion under Generalized Gradient Waveforms. J. Magn. Reson. 129 (1997), 74–84.CrossRefGoogle ScholarPubMed
, , , , , and , 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
, 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
and , Effects of Finite Gradient Pulse Widths in Pulsed Field Gradient Diffusion Measurements. J. Magn. Reson. A 113 (1995), 94–101.CrossRefGoogle Scholar
, , and , The Narrow-Pulse Criterion for Pulsed-Gradient Spin-Echo Diffusion Measurements. J. Magn. Reson. A 117 (1995), 209–19.CrossRefGoogle Scholar
, , , , , and , 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
, , and , NMR Diffusometry and the Short Gradient Pulse Limit Approximation. J. Magn. Reson. 169 (2004), 85–91.CrossRefGoogle ScholarPubMed
, , and , Simple Solution of the Torrey–Bloch Equations in the NMR Study of Molecular Diffusion. J. Magn. Reson. 128 (1997), 62–9.CrossRefGoogle Scholar
and , 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
and , Analytic Considerations in the Theory of NMR Microscopy. Physica A 371 (2006), 139–43.CrossRefGoogle Scholar
and , 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
and , Nuclear-Magnetic-Resonance Study of Self-Diffusion in a Bounded Medium. Phys. Rev. 151 (1966), 264–72.CrossRefGoogle Scholar
and , Spin-Echoes for Diffusion in Bounded, Heterogeneous Media: A Numerical Study. J. Chem. Phys. 72 (1980), 1285–92.CrossRefGoogle Scholar
and , Pulsed Field Gradient NMR for the Study of the Structure of Membrane Systems. Chem. Phys. 158 (1991), 105–11.CrossRefGoogle Scholar
, , and , Modeling of Self-Diffusion and Relaxation Time NMR in Multi-Compartment Systems. J. Magn. Reson. 135 (1998), 522–8.CrossRefGoogle ScholarPubMed
and , Relaxation of Nuclear Magnetization in a Nonuniform Magnetic Field Gradient and in a Restricted Geometry. J. Magn. Reson. 147 (2000), 95–103.CrossRefGoogle Scholar
, , and , 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
, , , and , An Image-Based Finite Difference Model for Simulating Restricted Diffusion. Magn. Reson. Med. 50 (2003), 373–82.CrossRefGoogle ScholarPubMed
, Random Walks and NMR Measurements in Porous Media. J. Phys. A: Math. Gen. 26 (1993), 3349–67.CrossRefGoogle Scholar
, , , , , and , 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
, , and , 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
, , and , Short-Time Behaviour of the Diffusion Coefficient as a Geometrical Probe of Porous Media. Phys. Rev. B 47 (1993), 8565–74.CrossRefGoogle ScholarPubMed
and , Enhancement of the ‘Diffraction-Like’ Effect in NMR Diffusion Experiments. J. Magn. Reson. A 111 (1994), 208–11.CrossRefGoogle Scholar
, , , and , Surface Relaxation and the Long-Time Diffusion Coefficient in Porous Media: Periodic Geometries. Phys. Rev. B 49 (1994), 215–25.CrossRefGoogle ScholarPubMed
, , , and , 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
, , and , Computer Simulation of the Spin-Echo Spatial Distribution in the Case of Restricted Self-Diffusion. J. Magn. Reson. 148 (2001), 257–66.CrossRefGoogle ScholarPubMed
, , , and , 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
, , and , Restricted Diffusion in a Model Acinar Labyrinth by NMR: Theoretical and Numerical Results. J. Magn. Reson. 184 (2007), 143–56.CrossRefGoogle Scholar
, 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
, , and , A Multiple-Narrow-Pulse Approximation for Restricted Diffusion in a Time-Varying Field Gradient. J. Magn. Reson. A 118 (1996), 94–102.CrossRefGoogle Scholar
and , 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
, , , , and , 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
and , 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
and , Effects of Restricted Diffusion on MR Signal Formation. J. Magn. Reson. 157 (2002), 92–105.CrossRefGoogle ScholarPubMed
, Exact Solution of the Torrey–Bloch Equation for a Spin Echo in Restricted Geometries. Phys. Rev. B 58 (1998), 14171–4.CrossRefGoogle Scholar
, Theory of Spin Echo in Restricted Geometries Under a Step-Wise Gradient Pulse Sequence. J. Magn. Reson. 139 (1999), 342–53.CrossRefGoogle Scholar
, On Quantum Theory of Transport Phenomena. Prog. Theor. Phys. 20 (1958), 948–59.CrossRefGoogle Scholar
and , Lattice Boltzmann Description of Magnetization in Porous Media. Phys. Rev. B 62 (2000), 3674–88.CrossRefGoogle Scholar
, Multiple Correlation Function Approach: Rigorous Results for Simple Geometries. Diffusion Fundamentals 5 (2007), 1.1–1.34.Google Scholar
and , Neural Networks Used to Interpret Pulsed-Gradient Restricted-Diffusion Data. J. Magn. Reson. A 107 (1994), 229–35.CrossRefGoogle Scholar
and , Permeability Coefficients from NMR q-Space Data: Models with Unevenly Spaced Semi-Permeable Parallel Membranes. J. Magn. Reson. 139 (1999), 258–72.CrossRefGoogle ScholarPubMed