Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-29T00:19:34.075Z Has data issue: false hasContentIssue false

1 - Diffusion and its measurement

Published online by Cambridge University Press:  06 August 2010

William S. Price
William S. Price
Affiliation:
University of Western Sydney
Get access

Summary

Introduction

This chapter introduces the concept of diffusion and other associated forms of translational motion such as flow, together with their physical significance. Measurements of translational motion and their interpretation are necessarily tied to a mathematical framework. Consequently, a detailed coverage of the mathematics, including the partial differential equation known as the diffusion equation, is presented. Finally, the common techniques for measuring diffusion are discussed.

Types of translational motion – physical interpretation and significance

‘Diffusion’ is used in the scientific literature with imprecision and ambiguity as there are a number of types of diffusion. With respect to molecular motion, diffusion is used to denote self-diffusion, mutual diffusion and ‘distinct’ (not in the sense of individual to a species) diffusion coefficients. Confusion arises since, although related and having the same units (i.e., m2s−1), these phenomena are physically distinct. The confusion is exacerbated in the NMR literature with the term ‘spin-diffusion’ which is a distinct NMR cross relaxation – based phenomenon involving the random migration of magnetisation via mutual spin flips in neighbouring nuclei, even though it can be measured using techniques related to those outlined in this book. In this book ‘diffusion’ signifies self-diffusion, which will also be referred to as translational diffusion, although some consideration will be given to mutual diffusion since many of the alternative methods for measuring diffusion, especially those based on scattering, provide information on mutual diffusion which is often compared with the results of NMR measurements of translational diffusion.

Type
Chapter
Information
NMR Studies of Translational Motion
Principles and Applications
 , pp. 1 - 68
Publisher: Cambridge University Press
Print publication year: 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Tyrrell, H. J. V. and Harris, K. R., Diffusion in Liquids: A Theoretical and Experimental Study. (London: Butterworths, 1984).Google Scholar
Cussler, E. L., Diffusion Mass Transfer in Fluid Systems, 2nd edn. (Cambridge: Cambridge University Press, 1997).Google Scholar
Poling, B. E., Prausnitz, J. M., and O'Connell, J. P., The Properties of Gases and Liquids, 5th edn. (New York: McGraw Hill, 2001).Google Scholar
Bird, R. B., Lightfoot, E. N., and Stewart, W. E., Transport Phenomena, 2nd edn. Revised (New York: Wiley, 2007).Google Scholar
Hanna, S., Hess, W., and Klein, R., Self-Diffusion of Spherical Brownian Particles with Hard-Core Interaction. Physica A 111 (1982), 181–99.CrossRefGoogle Scholar
Kalk, A. and Berendsen, H. J. C., Proton Magnetic Relaxation and Spin Diffusion in Proteins. J. Magn. Reson. 24 (1976), 343–66.Google Scholar
Fatkullin, N., Theory of Stimulated Spin Echo in Polymer System. Sov. Phys. JETP 72 (1991), 563–69.Google Scholar
Zhang, W. and Cory, D. G., First Direct Measurement of the Spin Diffusion Rate in a Homogeneous Solid. Phys. Rev. Lett. 80 (1998), 1324–7.CrossRefGoogle Scholar
Komlosh, M. E. and Callaghan, P. T., Spin Diffusion in Semidilute Random Coil Polymers Studied by Pulsed Gradient Spin-Echo NMR. Macromolecules 33 (2000), 6824–7.CrossRefGoogle Scholar
Johnson, Jr. C. S. and Gabriel, D. A., Laser Light Scattering. (New York: Dover, 1994).Google Scholar
Chandrasekhar, S., Stochastic Problems in Physics and Astronomy. Rev. Mod. Phys. 15 (1943), 1–89.CrossRefGoogle Scholar
Kac, M., Random Walk and the Theory of Brownian Motion. Am. Math. Mon. 54 (1947), 369–91.CrossRefGoogle Scholar
Crank, J., The Mathematics of Diffusion, 2nd edn. (Oxford: Oxford University Press, 1975).Google Scholar
Cantor, C. R. and Schimmel, P. R., Biophysical Chemistry, Part II: Techniques for the Study of Biological Structure and Function. (New York: W. H. Freeman, 1980).Google Scholar
Lauffer, M. A., Motion in Biological Systems. (New York: Alan R. Liss, Inc., 1989).Google Scholar
Berg, H. C., Random Walks in Biology. (New Jersey: Princeton University Press, 1993).Google Scholar
Glaser, R., Biophysics, Revised (Berlin: Springer-Verlag, 2000).Google Scholar
Nicholson, C., Diffusion and Related Transport Mechanisms in Brain Tissue. Rep. Prog. Phys. 64 (2001), 815–84.CrossRefGoogle Scholar
Dill, K. A. and Bromberg, S., Molecular Driving Forces. (New York: Garland Science, 2003).Google Scholar
Truskey, G. A., Yuan, F., and Katz, D. F., Transport Phenomena in Biological Systems. (New York: Prentice Hall, 2003).Google Scholar
Green, P. F., Kinetics, Transport, and Structure in Hard and Soft Materials. (Baton Rouge: CRC Press, 2005).CrossRefGoogle Scholar
Barzykin, A. V., Seki, K., and Tachiya, M., Kinetics of Diffusion-Assisted Reactions in Microheterogeneous Systems. Adv. Colloid Interface Sci. 89–90 (2001), 47–140.CrossRefGoogle ScholarPubMed
Traytak, S. D. and Price, W. S., Exact Solution for Anisotropic Diffusion-Controlled Reactions with Partially Reflecting Conditions. J. Chem. Phys. 127 (2007), 184508-1–184508-8.CrossRefGoogle ScholarPubMed
Halsey, T. C., Diffusion-Limited Aggregation: A Model for Pattern Formation. Phys. Today 53 (2000), 36–41.CrossRefGoogle Scholar
Sander, L. M., Diffusion-Limited Aggregation: A Kinetic Critical Phenomenon?Contemp. Phys. 41 (2000), 203–18.CrossRefGoogle Scholar
Verkman, A. S., Solute and Macromolecular Diffusion in Cellular Aqueous Compartments. Trends Biochem. Sci. 27 (2002), 27–33.CrossRefGoogle ScholarPubMed
Albright, J. G. and Mills, R., A Study of Diffusion in the Ternary System, Labeled Urea-Urea-Water, at 25° by Measurements of the Intradiffusion Coefficients of Urea. J. Phys. Chem. 69 (1965), 3120–6.CrossRefGoogle Scholar
Glicksman, M. E., Diffusion in Solids: Field Theory, Solid-State Principles, and Applications. (New York: Wiley, 2000).Google 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
Mair, R. W., Cory, D. G., Peled, S., Tseng, C.-H., Patz, S., and Walsworth, R. L., Pulsed-Field-Gradient Measurements of Time-Dependent Gas Diffusion. J. Magn. Reson. 135 (1998), 478–86.CrossRefGoogle ScholarPubMed
Kuchel, P. W., Chapman, B. E., and Lennon, A. J., Diffusion of Hydrogen in Aqueous Solutions Containing Protein. Pulsed Field Gradient NMR Measurements. J. Magn. Reson. A 103 (1993), 329–31.CrossRefGoogle Scholar
Weingärtner, H., Self-Diffusion in Liquid Water. A Reassessment. Z. Phys. Chem. 132 (1982), 129–49.CrossRefGoogle Scholar
Mills, R., Self-Diffusion in Normal and Heavy Water in the Range 1–45°. J. Phys. Chem. 77 (1973), 685–8.CrossRefGoogle Scholar
Price, W. S. and Kuchel, P. W., Restricted Diffusion of Bicarbonate and Hypophosphite Ions Modulated by Transport in Suspensions of Red Blood Cells. J. Magn. Reson. 90 (1990), 100–10.Google Scholar
Price, W. S., Chapman, B. E., Cornell, B. A., and Kuchel, P. W.Translational, Diffusion of Glycine in Erythrocytes Measured at High Resolution with Pulsed Field Gradients. J. Magn. Reson. 83 (1989), 160–6.Google Scholar
Gmeiner, W. H., Hudalla, C. J., Soto, A. M., and Marky, L., Binding of Ethidium to DNA Measured Using a 2D Diffusion-Modulated Gradient COSY NMR Experiment. FEBS Lett. 465 (2000), 148–52.CrossRefGoogle ScholarPubMed
Adachi, K., Natsuisaka, M., and Tanioka, A., Measurements of Self-Diffusion Coefficients of Monensin in Chloroform Solution by PFG-NMR. J. Chem. Soc., Faraday Trans. 93 (1997), 3347–50.CrossRefGoogle Scholar
Kato, T., Kikuchi, K., and Achiba, Y., Measurement of the Self-Diffusion Coefficient of C60 in Benzene-embedding dimension6 Using 13C Pulsed-Gradient Spin-Echo. J. Phys. Chem. 97 (1993), 10251–3.CrossRefGoogle Scholar
Gaemers, S., Elsevier, C. J., and Bax, A., NMR of Biomolecules in Low Viscosity, Liquid CO2. Chem. Phys. Lett. 301 (1999), 138–44.CrossRefGoogle Scholar
Pan, H., Barany, G., and Woodward, C., Reduced BPTI is Collapsed. A Pulsed Field Gradient NMR Study of Unfolded and Partially Folded Bovine Pancreatic Trypsin Inhibitor. Protein Sci. 6 (1997), 1985–92.CrossRefGoogle ScholarPubMed
Mackay, J. P., Shaw, G. L., and King, G. F., Backbone Dynamics of the c-Jun Leucine Zipper: 15N NMR Relaxation Studies. Biochemistry 35 (1996), 4867–77.CrossRefGoogle ScholarPubMed
Lin, M. and Larive, C. K., Detection of Insulin Aggregates with Pulsed-Field Gradient Nuclear Magnetic Resonance Spectroscopy. Anal. Biochem. 229 (1995), 214–20.CrossRefGoogle ScholarPubMed
Price, W. S., Tsuchiya, F., and Arata, Y., Lysozyme Aggregation and Solution Properties Studied Using PGSE NMR Diffusion Measurements. J. Am. Chem. Soc. 121 (1999), 11503–12.CrossRefGoogle Scholar
Gibbs, S. J., Chu, A. S., Lightfoot, E. N., and Root, T. W., Ovalbumin Diffusion at Low Ionic Strength. J. Phys. Chem. 95 (1991), 467–71.CrossRefGoogle Scholar
Everhart, C. H. and Johnson, C. S., Jr., The Determination of Tracer Diffusion Coefficients for Proteins by Means of Pulsed Field Gradient NMR with Applications to Hemoglobin. J. Magn. Reson. 48 (1982), 466–74.Google Scholar
Callaghan, P. T. and Pinder, D. N., Influence of Polydispersity on Polymer Self-Diffusion Measurements by Pulsed Field Gradient Nuclear Magnetic Resonance. Macromolecules 18 (1985), 373–9.CrossRefGoogle Scholar
Furó, I. and Dvinskikh, S. V., NMR Methods Applied to Anisotropic Diffusion. Magn. Reson. Chem. 40 (2002), S3–14.CrossRefGoogle Scholar
Bihan, D., Diffusion, Molecular, Tissue Microdynamics and Microstructure. NMR Biomed. 8 (1995), 375–86.CrossRefGoogle ScholarPubMed
Hsu, E. W., Aiken, N. R., and Blackband, S. J., A Study of Diffusion Isotropy in Single Neurons by Using NMR Microscopy. Magn. Reson. Med. 37 (1997), 624–27.CrossRefGoogle ScholarPubMed
Ben-Avraham, D. and Havlin, S., Diffusion and Reactions in Fractals and Disordered Systems. (Cambridge: Cambridge University Press, 2000).CrossRefGoogle Scholar
Job, G. and Herrmann, F., Chemical Potential – A Quantity in Search of Recognition. Eur. J. Phys. 27 (2006), 353–71.CrossRefGoogle Scholar
Brady, J. B., Reference Frames and Diffusion Coefficients. Am. J. Sci. 275 (1975), 954–83.CrossRefGoogle Scholar
Weingärtner, H., The Molecular Description of Mutual Diffusion Processes in Liquid Mixtures. In Diffusion in Condensed Matter, ed. Heitjans, P. and Kärger, J. (Berlin: Springer, 2005), pp. 555–78.CrossRefGoogle Scholar
Friedman, H. L. and Mills, R., Velocity Cross Correlations in Binary Mixtures of Simple Fluids. J. Solution Chem. 10 (1981), 395–409.CrossRefGoogle Scholar
Friedman, H. L., Raineri, F. O., and Wood, M. D., Distinct Diffusion Coefficients, Probes of Ion-Ion Interactions. Chem. Scr. 29A (1989), 49–59.Google Scholar
Hawlicka, E., Self-Diffusion in Multicomponent Liquid Systems. Chem. Soc. Rev. 24 (1995), 367–77.CrossRefGoogle Scholar
Newling, B., Gas Flow Measurements by NMR. Prog. NMR Spectrosc. 52 (2008), 31–48.CrossRefGoogle Scholar
Abshagen, J., Schulz, A., and Pfister, G., The Couette-Taylor Flow: A Paradigmatic System for Instabilities, Pattern Formation and Routes to Chaos. In Nonlinear Physics of Complex Systems. ed. Parisi, J., Müller, S. C., and Zimmermann, W., Lect. Notes Phys. vol. 476 (Berlin: Springer, 1997), pp. 63–72.Google Scholar
Bénard, H., Les tourbillons celluaries dans une nappe liquide transportant de la chaleur par convection en regime permanent. Ann. Chim. Phys. 23 (1901), 62–144.Google Scholar
,Lord Rayleigh, On Convection Currents on a Horizontal Layer of Fluid When the Higher Temperature is on the Under Side. Philos. Mag. 32 (1916), 529–46.CrossRefGoogle Scholar
Koch, D. L. and Brady, J. F., A Non-Local Description of Advection-Diffusion with Application to Dispersion in Porous Media. J. Fluid. Mech. 180 (1987), 387–403.CrossRefGoogle Scholar
Callaghan, P. T., Codd, S. L., and Seymour, J. D., Spatial Coherence Phenomena Arising from Translational Spin Motion in Gradient Spin Echo Experiments. Concepts Magn. Reson. 11 (1999), 181–202.3.0.CO;2-T>CrossRefGoogle Scholar
Callaghan, P. T., Some Perspectives on Dispersion and the Use of Ensemble-Averaged PGSE NMR. Magn. Reson. Imaging 23 (2005), 133–7.CrossRefGoogle ScholarPubMed
Turq, P., Barthel, J., and Chemla, M., Transport, Relaxation and Kinetic Processes in Electrolyte Solutions. (Berlin: Springer-Verlag, 1992).CrossRefGoogle Scholar
Bockris, J. O. and Reddy, A. K. N., Modern Electrochemistry, 2nd edn. (New York: Plenum Press, 1998).Google Scholar
Holz, M., Field-Assisted Diffusion Studied by Electrophoretic NMR. In Diffusion in Condensed Matter, ed. Kärger, J. and Heitjans, P.. (Berlin: Springer, 2005), pp. 717–42.CrossRefGoogle Scholar
Fatkullin, N., Theory of Diffusive Damping of Spin-Echo Signal in a Medium with Random Obstacles. Sov. Phys. JETP 71 (1990), 1141–4.Google Scholar
Fulinski, A., On Marian Smoluchowski's Life and Contribution to Physics. Acta Physiol. Pol. B 29 (1998), 1523–37.Google Scholar
Valiullin, R. and Skirda, V., Time Dependent Self-Diffusion Coefficient of Molecules in Porous Media. J. Chem. Phys. 114 (2001), 452–58.CrossRefGoogle Scholar
Berezhkovskii, A. M. and Sutmann, G., Time and Length Scales for Diffusion in Liquids. Phys. Rev. E 65 (2002), 060201-1–060201-4.CrossRefGoogle ScholarPubMed
Kubo, R., The Fluctuation-Dissipation Theorem. Rep. Prog. Phys. 29 (1966), 255–84.CrossRefGoogle Scholar
Stepišnik, J., Analysis of NMR Self-Diffusion Measurements by a Density Matrix-Calculation. Physica B & C 104 (1981), 350–64.CrossRefGoogle Scholar
Stepišnik, J., Time-Dependent Self-Diffusion by NMR Spin-Echo. Physica B 183 (1993), 343–50.CrossRefGoogle Scholar
Traytak, S. D. and Kudryavtsev, A. G., On the Coagulation Equation for a Passive Admixture in a Stochastic Medium. Physica A 260 (1998), 381–90.CrossRefGoogle Scholar
Gardiner, C. W., Handbook of Stochastic Methods: For Physics, Chemistry and the Natural Sciences, 2nd edn. (Berlin: Springer-Verlag, 1996).Google Scholar
Callaghan, P. T. and Stepišnik, J., Generalized Analysis of Motion Using Magnetic Field Gradients. Adv. Magn. Opt. Reson. 19 (1996), 325–88.CrossRefGoogle Scholar
Avramov, I., Viscosity in Disordered Media. J. Non-Cryst. Solids 351 (2005), 3163–73.CrossRefGoogle Scholar
Sutherland, W., A Dynamical Theory of Diffusion for Nonelectrolytes and the Molecular Mass of Albumin. Philos. Mag. S.6, 9 (1905), 781–85.CrossRefGoogle Scholar
Einstein, A., Investigations on the Theory of Brownian Movement. (New York: Dover, 1956).Google Scholar
Bearman, R. J., Statistical Mechanical Theory of the Diffusion Coefficients in Binary Liquid Solutions. J. Chem. Phys. 32 (1960), 1308–13.CrossRefGoogle Scholar
Tanford, C., Physical Chemistry of Macromolecules. (New York: Wiley, 1961).Google Scholar
Edward, J. T., Molecular Volume and the Stokes–Einstein Equation. J. Chem. Educ. 47 (1970), 261–70.CrossRefGoogle Scholar
Kruus, P., Liquids and Solutions Structure and Dynamics. (New York: Marcel Dekker, 1977).Google Scholar
Renn, J., Einstein's Invention of Brownian Motion. Ann. Physik. 14 (2005), 23–7.CrossRefGoogle Scholar
Marqusee, J. A. and Deutch, J. M., Concentration Dependence of the Self-Diffusion Coefficient. J. Chem. Phys. 73 (1980), 5396–7.CrossRefGoogle Scholar
MacDowell, L. G., Garzón, B., Calero, S., and Lago, S., Dynamical Properties and Transport Coefficients of Kihara Linear Fluids. J. Chem. Phys. 106 (1997), 4753–67.CrossRefGoogle Scholar
Branca, C., Magazù, S., Maisano, G., Migliardo, P., and Tettamanti, E., Anomalous Translational Diffusive Processes in Hydrogen-Bonded Systems Investigated by Ultrasonic Technique, Raman Scattering and NMR. Physica B 291 (2000), 180–9.CrossRefGoogle Scholar
Balucani, U., Vallauri, R., and Gaskell, T., Generalized Stokes–Einstein Relation. Ber. Bunsenges. Phys. Chem. 94 (1990), 261–4.CrossRefGoogle Scholar
Teller, D. C., Swanson, E., and Haën, C., The Translational Friction Coefficient of Proteins. Methods Enzymol. 61 (1979), 104–24.CrossRefGoogle ScholarPubMed
Zhou, H.-X., Calculation of Translational Friction and Intrinsic Viscosity. II. Application to Globular Proteins. Biophys. J. 69 (1995), 2298–303.CrossRefGoogle ScholarPubMed
Torre, J. García de la, Huertas, M. L., and Carrasco, B., Calculation of Hydrodynamic Properties of Globular Proteins from their Atomic-Level Structure. Biophys. J. 78 (2000), 719–30.CrossRefGoogle Scholar
Torre, J. García de la, Hydration from Hydrodynamics. General Considerations and Applications of Bead Modelling to Globular Proteins. Biophys. Chem. 93 (2001), 159–70.CrossRefGoogle Scholar
Aragon, S. and Hahn, D. K., Precise Boundary Element Computation of Protein Transport Properties: Diffusion Tensors, Specific Volume, and Hydration. Biophys. J. 91 (2006), 1591–603.CrossRefGoogle ScholarPubMed
Halle, B. and Davidovic, M., Biomolecular Hydration: From Water Dynamics to Hydrodynamics. Proc. Natl. Acad. Sci. U.S.A. 100 (2003), 12135–40.CrossRefGoogle ScholarPubMed
Byron, O., Construction of Hydrodynamic Bead Models from High-Resolution X-Ray Crystallographic or Nuclear Magnetic Resonance Data. Biophys. J. 72 (1997), 408–15.CrossRefGoogle ScholarPubMed
Allison, S., Chen, C., and Stigter, D., The Length Dependence of Translational Diffusion, Free Solution Electrophoretic Mobility, and Electrophoretic Tether Force of Rigid Rod-Like Model Duplex DNA. Biophys. J. 81 (2001), 2558–68.CrossRefGoogle ScholarPubMed
Davies, J. A. and Griffiths, P. C., A Phenomenological Approach to Separating the Effects of Obstruction and Binding for the Diffusion of Small Molecules in Polymer Solutions. Macromolecules 36 (2003), 950–2.CrossRefGoogle Scholar
Sharma, M. and Yashonath, S., Breakdown of the Stokes–Einstein Relationship: Role of Interactions in the Size Dependence of Self-Diffusivity. J. Phys. Chem. B 110 (2006), 17207–11.CrossRefGoogle ScholarPubMed
Gierer, V. A. and Wirtz, K., Molekulare Theorie der Mikroreibung. Z. Naturforsch. 8a (1953), 532–8.Google Scholar
Chen, H.-C. and Chen, S.-H., Diffusion of Crown Ethers in Alcohols. J. Phys. Chem. 88 (1984), 5118–21.CrossRefGoogle Scholar
Balucani, U., Nowotny, G., and Kahl, G., The Generalized Stokes–Einstein Relation for Liquid Sodium. J. Phys. Condens. Matter 9 (1997), 3371–6.CrossRefGoogle Scholar
Lamanna, R., Delmelle, M., and Cannistraro, S., Solvent Stokes–Einstein Violation in Aqueous Protein Solutions. Phys. Rev. E 49 (1994), 5878–80.CrossRefGoogle ScholarPubMed
Brilliantov, N. V. and Krapivsky, P. L., Stokes Laws for Ions in Solutions with Ion-Induced Inhomogeneity. J. Phys. Chem. 95 (1991), 6055–7.CrossRefGoogle Scholar
Doi, M. and Edwards, S. F., The Theory of Polymer Dynamics. (Oxford: Clarendon Press, 1986).Google Scholar
Koenig, S. H., Brownian Motion of an Ellipsoid. A Correction to Perrin's Results. Biopolymers 14 (1975), 2421–3.CrossRefGoogle Scholar
Happel, J. and Brenner, H., Low Reynolds Number Hydrodynamics − With Special Applications to Particulate Matter. (Dordrecht: Kluwer, 1983).Google Scholar
Probstein, R. F., Physicochemical Hydrodynamics, 2nd edn. (New York: Wiley, 2003).Google Scholar
Tirado, M. M. and Torre, J. García de la, Translational Friction Coefficients of Rigid, Symmetric Top Macromolecules. Application to Circular Cylinders. J. Chem. Phys. 71 (1979), 2581–7.CrossRefGoogle Scholar
Tirado, M. M., Martínez, C. López, and Torre, J. García de la, Comparison of Theories for the Translational and Rotational Diffusion Coefficients of Rod-Like Macromolecules. Application to Short DNA Fragments. J. Chem. Phys. 81 (1984), 2047–52.CrossRefGoogle Scholar
Hansen, S., Translational Friction Coefficients for Cylinders of Arbitrary Axial Ratios Estimated by Monte Carlo Simulation. J. Chem. Phys. 121 (2004), 9111–15.CrossRefGoogle ScholarPubMed
Torre, J. García de la and Bloomfield, V. A., Hydrodynamic Properties of Complex, Rigid, Biomolecules: Theory and Applications. Q. Rev. Biophys. 14 (1981), 81–139.CrossRefGoogle Scholar
Bloomfield, V., Dalton, W. O., and Holde, K. E., Frictional Coefficients of Multisubunit Structures. I. Theory. Biopolymers 5 (1967), 135–48.CrossRefGoogle ScholarPubMed
Torre, J. García de la, Hydrodynamic Properties of Macromolecular Assemblies. In Dynamic Properties of Biomolecular Assemblies, ed. Harding, S. E. and Rowe, A. J.. (Cambridge: Royal Society of Chemistry, 1989), pp. 3–31.Google Scholar
Harding, S. E., On the Hydrodynamic Analysis of Macromolecular Conformation. Biophys. Chem. 55 (1995), 69–93.CrossRefGoogle ScholarPubMed
Robert, C. H., Estimating Friction Coefficients of Mixed Globular/Chain Molecules, such as Protein/DNA Complexes. Biophys. J. 69 (1995), 840–8.CrossRefGoogle ScholarPubMed
Zhou, H.-X., Calculation of Translational Friction and Intrinsic Viscosity. I. General Formulation for Arbitrarily Shaped Particles. Biophys. J. 69 (1995), 2286–97.CrossRefGoogle ScholarPubMed
Harding, S. E., The Intrinsic Viscosity of Biological Macromolecules. Progress in Measurement, Interpretation and Application to Structure in Dilute Solution. Prog. Biophys. Mol. Biol. 68 (1997), 207–62.CrossRefGoogle ScholarPubMed
Torre, J. García de la and Carrasco, B., Intrinsic Viscosity and Rotational Diffusion of Bead Models for Rigid Macromolecules and Bioparticles. Eur. Biophys. J. 27 (1998), 549–57.CrossRefGoogle Scholar
Carrasco, B. and Torre, J. García de la, Hydrodynamic Properties of Rigid Particles: Comparison of Different Modeling and Computational Procedures. Biophys. J. 76 (1999), 3044–57.CrossRefGoogle ScholarPubMed
Torre, J. García de la, Harding, S. E., and Carrasco, B., Calculation of NMR Relaxation, Covolume, Scattering-Related Properties of Bead Molecules Using the SOLPRO Computer Program. Eur. Biophys. J. 28 (1999), 119–32.CrossRefGoogle Scholar
Torre, J. García de la, Ortega, A., Sanchez, H. E. Perez, and Cifre, J. G. Hernandez, MULTIHYDRO and MONTEHYDRO: Conformational Search and Monte Carlo Calculation of Solution Properties of Rigid or Flexible Bead Models. Biophys. Chem. 116 (2005), 121–8.CrossRefGoogle Scholar
Arfken, G. and Weber, H. J., Mathematical Methods for Physicists, 4th edn. (New York: Academic Press, 1995).Google Scholar
Metzler, R., Glöcke, W. G., and Nonnenmacher, T. F., Fractional Model Equation for Anomalous Diffusion. Physica A 211 (1994), 13–24.CrossRefGoogle Scholar
Arkhincheev, V. E., Anomalous Diffusion and Charge Relaxation on Comb Model: Exact Solutions. Physica A 280 (2000), 304–14.CrossRefGoogle Scholar
Metzler, R., The Restaurant at the End of the Random Walk: Recent Developments in the Description of Anomalous Transport by Fractional Dynamics. J. Phys. A Math. Gen. 37 (2004), R161–208.CrossRefGoogle Scholar
Egelstaff, P. A., An Introduction to the Liquid State, 2nd edn. (Oxford: Oxford Science Publications, 1994).Google Scholar
Kärger, J. and Heink, W., The Propagator Representation of Molecular Transport in Microporous Crystallites. J. Magn. Reson. 51 (1983), 1–7.Google Scholar
Roach, G. F., Green's Functions, 2nd edn. (Cambridge: Cambridge, 1982).Google Scholar
Özisik, M. N., Boundary Value Problems of Heat Conduction. (New York: Dover, 1989).Google Scholar
Barton, G., Elements of Green's Functions and Propagation Potentials, Diffusion and Waves. (Oxford: Oxford University Press, 1995).Google Scholar
Duffy, D. G., Green's Functions With Applications. (Boca Raton: CRC, 2001).CrossRefGoogle Scholar
Callaghan, P. T., Principles of Nuclear Magnetic Resonance Microscopy. (Oxford: Clarendon Press, 1991).Google Scholar
Kampen, N. G., Stochastic Processes in Physics and Chemistry, 3rd edn. (Amsterdam: North Holland, 2001).Google Scholar
Kärger, J., Pfeifer, H., and Heink, W., Principles and Applications of Self-Diffusion Measurements by Nuclear Magnetic Resonance. Adv. Magn. Reson. 12 (1988), 1–89.CrossRefGoogle Scholar
Hoskins, R. F., Delta Functions: An Introduction to Generalised Functions. (New York: Albion/Horwood Publishing House, 1999).Google Scholar
Duffy, D. G., Advanced Engineering Mathematics With MATLAB, 2nd edn. (Boca Raton: CRC, 2003).Google Scholar
Moon, P. and Spencer, D. E., Field Theory Handbook, 2nd edn. (Berlin: Springer-Verlag, 1971).CrossRefGoogle Scholar
Sneddon, I. N., Mixed Boundary Value Problems in Potential Theory. (Amsterdam: North-Holland Publishing Company, 1966).Google Scholar
Brownstein, K. R. and Tarr, C. E., Importance of Classical Diffusion in NMR Studies of Water in Biological Cells. Phys. Rev. A 19 (1979), 2446–53.CrossRefGoogle Scholar
Callaghan, P. T., Pulsed Gradient Spin Echo NMR for Planar, Cylindrical and Spherical Pores under Conditions of Wall Relaxation. J. Magn. Reson. A 113 (1995), 53–59.CrossRefGoogle Scholar
Barzykin, A. V., Price, W. S., Hayamizu, K., and Tachiya, M., Pulsed Field Gradient NMR of Diffusive Transport Through a Spherical Interface into an External Medium Containing a Relaxation Agent. J. Magn. Reson. A 114 (1995), 39–46.CrossRefGoogle Scholar
Price, W. S., Barzykin, A. V., Hayamizu, K., and Tachiya, M., A Model for Diffusive Transport Through a Spherical Interface Probed by Pulsed-Field Gradient NMR. Biophys. J. 74 (1998), 2259–71.CrossRefGoogle ScholarPubMed
Buchanan, G. R., Finite Element Analysis. (New York: McGraw-Hill, 1995).Google Scholar
Carslaw, H. S. and Jaeger, J. C., Conduction of Heat in Solids, 2nd edn. (Oxford: Oxford University Press, 1959).Google Scholar
Zwillinger, D., Handbook of Differential Equations, 3rd edn. (Boston: Academic Press, 1998).Google Scholar
Duffy, D. G., Transform Methods for Solving Partial Differential Equations, 2nd edn. (Boca Raton: CRC Press, 2004).CrossRefGoogle Scholar
Bergman, D. J. and Dunn, K.-J., Self-Diffusion in a Periodic Porous Medium With Interface Absorption. Phys. Rev. E 51 (1995), 3401–16.CrossRefGoogle Scholar
Bergman, D. J., Dunn, K.-J., Schwartz, L. M., and Mitra, P. P., Self-Diffusion in a Periodic Porous Medium: A Comparison of Different Approaches. Phys. Rev. E 51 (1995), 3393–400.CrossRefGoogle Scholar
Dunn, K.-J. and Bergman, D. J., Self Diffusion of Nuclear Spins in a Porous Medium with a Periodic Microstructure. J. Chem. Phys. 102 (1995), 3041–54.CrossRefGoogle Scholar
Zayed, A. I., Handbook of Generalized Function Transformations. (Boca Raton: CRC, 1996).Google Scholar
Onsager, L., Reciprocal Relations in Irreversible Processes. I. Phys. Rev. 37 (1931), 405–26.CrossRefGoogle Scholar
Onsager, L., Reciprocal Relations in Irreversible Processes. II. Phys. Rev. 38 (1931), 2265–79.CrossRefGoogle Scholar
Groot, S. R. and Mazur, P., Non-Equilibrium Thermodynamics. (New York: Dover, 1984).Google Scholar
Traytak, S. D., Barzykin, A. V., and Tachiya, M., Effect of Anisotropic Diffusion and External Electric Field on the Rate of Diffusion-Controlled Reactions. J. Chem. Phys. 120 (2004), 10111–17.CrossRefGoogle ScholarPubMed
Basser, P. J., Mattiello, J., and Bihan, D., MR Diffusion Tensor Spectroscopy and Imaging. Biophys. J. 66 (1994), 259–67.CrossRefGoogle Scholar
Gelderen, P., DesPres, D., Zijl, P. C. M., and Moonen, C. T. W., Evaluation of Restricted Diffusion in Cylinders. Phosphocreatine in Rabbit Leg Muscle. J. Magn. Reson. B 103 (1994), 255–60.CrossRefGoogle ScholarPubMed
Swiet, T. M. and Mitra, P. P., Possible Systematic Errors in Single-Shot Measurements of the Trace of the Diffusion Tensor. J. Magn. Reson. B 111 (1996), 15–22.CrossRefGoogle ScholarPubMed
Wang, M. C. and Uhlenbeck, G. E., On the Theory of the Brownian Motion II. Rev. Mod. Phys. 17 (1945), 323–42.CrossRefGoogle Scholar
Stejskal, E. O., Use of Spin Echoes in a Pulsed Magnetic-Field Gradient to Study Anisotropic Restricted Diffusion and Flow. J. Chem. Phys. 43 (1965), 3597–603.CrossRefGoogle Scholar
Abramowitz, M., and Stegun, I. A., Handbook of Mathematical Functions. (New York: Dover, 1970).Google 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
Kac, M., Can One Hear the Shape of a Drum?Am. Math. Mon. 73 (1966), 1–23.CrossRefGoogle Scholar
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
Schwartz, L. M., Hürlimann, M. D., Dunn, K.-J., Mitra, P. P., and Bergman, D. J., Restricted Diffusion and the Return to the Origin Probability at Intermediate and Long Times. Phys. Rev. E 55 (1997), 4225–34.CrossRefGoogle Scholar
Neuman, C. H., Spin Echo of Spins Diffusing in a Bounded Medium. J. Chem. Phys. 60 (1974), 4508–11.CrossRefGoogle Scholar
Lenk, R., Fluctuations, Diffusion and Spin Relaxation. (Amsterdam: Elsevier, 1986).Google Scholar
Gradshteyn, I. S. and Ryzhik, I. M., Table of Integrals, Series, and Products, 7th edn. (New York: Academic Press, 2007).Google Scholar
Woessner, D. E., Spin-Echo, N. M. R.Self-Diffusion Measurements on Fluids Undergoing Restricted Diffusion. J. Phys. Chem. 67 (1963), 1365–67.CrossRefGoogle Scholar
Chandler, D., Introduction to Modern Statistical Mechanics, (Oxford: Oxford University Press, 1987).Google Scholar
Basser, P. J., Relationships Between Diffusion Tensor and q-Space MRI. Magn. Reson. Med. 47 (2002), 392–7.CrossRefGoogle ScholarPubMed
Topgaard, D., Probing Biological Tissue Microstructure with Magnetic Resonance Diffusion Techniques. Curr. Opin. Colloid Interface Sci. 11 (2006), 7–12.CrossRefGoogle Scholar
Basser, P. J., Inferring Microstructural Features and the Physiological State of Tissues from Diffusion-Weighted Images. NMR Biomed. 8 (1995), 333–44.CrossRefGoogle ScholarPubMed
Minati, L. and Węglarz, W. P., Physical Foundations, Models, and Methods of Diffusion Magnetic Resonance Imaging of the Brain: A Review. Concepts Magn. Reson. 30A (2007), 278–307.CrossRefGoogle Scholar
Callaghan, P. T. and Komlosh, M. E., Locally Anisotropic Motion in a Macroscopically Isotropic System: Displacement Correlations Measured Using Double Pulsed Gradient Spin-Echo NMR. Magn. Reson. Chem. 40 (2002), S15–19.CrossRefGoogle Scholar
Price, W. S., Pulsed Field Gradient NMR as a Tool for Studying Translational Diffusion, Part I. Basic Theory. Concepts Magn. Reson. 9 (1997), 299–336.3.0.CO;2-U>CrossRefGoogle Scholar
Sen, P. N., Time-Dependent Diffusion Coefficient as a Probe of Geometry. Concepts Magn. Reson. 23A (2004), 1–21.CrossRefGoogle Scholar
Kimmich, R., Strange Kinetics, Porous Media, and NMR. Chem. Phys. 284 (2002), 253–85.CrossRefGoogle Scholar
Ardelean, I. and Kimmich, R., Principles and Unconventional Aspects of NMR Diffusometry. In Annual Reports on NMR Spectroscopy, ed. Webb, G. A.. vol. 49. (London: Elsevier, 2003), pp. 43–115.Google Scholar
Vojta, G. and Renner, U., Diffusion in Fractals. In Diffusion in Condensed Matter, ed. Kärger, J. and Heitjans, P.. (Berlin: Springer-Verlag, 2005), pp. 793–811.Google Scholar
Mitra, P. P. and Sen, P. N., Effects of Surface Relaxation on NMR Pulsed Field Gradient Experiments in Porous Media. Physica A 186 (1992), 109–14.CrossRefGoogle Scholar
Mitra, P. P., Sen, P. N., Schwartz, L. M., and Doussal, P., Diffusion Propagator as a Probe of the Structure of Porous Media. Phys. Rev. Lett. 68 (1992), 3555–8.CrossRefGoogle ScholarPubMed
Mitra, P. P. and Sen, P. N., Effects of Microgeometry and Surface Relaxation on NMR Pulsed-Field-Gradient Experiments: Simple Pore Geometries. Phys. Rev. B 45 (1992), 143–56.CrossRefGoogle ScholarPubMed
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
Latour, L. L., Mitra, P. P., Kleinberg, R. L., and Sotak, C. H., Time-Dependent Diffusion Coefficient of Fluids in Porous Media as a Probe of Surface-to-Volume Ratio. J. Magn. Reson. A 101 (1993), 342–6.CrossRefGoogle Scholar
Sen, P. N., Time-Dependent Diffusion Coefficient as a Probe of Permeability of the Pore Wall. J. Chem. Phys. 119 (2003), 9871–6.CrossRefGoogle Scholar
Candela, D. and Wong, P.-Z., Using NMR to Measure Fractal Dimensions. Phys. Rev. Lett. 90 (2003), 039601-1.CrossRefGoogle ScholarPubMed
Banavar, J. R. and Schwartz, L. M., Probing Porous Media with Nuclear Magnetic Resonance. In Molecular Dynamics in Restricted Geometries, ed. Klafter, J. and Drake, J. M.. (New York: Wiley, 1989), pp. 273–309.Google Scholar
Sing, K. S. W., Everett, D. H., Haul, R. A. W., Moscou, L., Pierotti, R. A., Rocquérol, J., and Siemieniewska, T., Reporting Physisorption Data for Gas/Solid Systems with Special Reference to the Determination of Surface Area and Porosity (Recommendations 1984) Commission on Colloid and Surface Chemistry Including Catalysis. Pure Appl. Chem. 57 (1985), 603–19.CrossRefGoogle Scholar
Zimmerman, S. B. and Minton, A. P., Macromolecular Crowding: Biochemical, Biophysical, and Physiological Consequences. Annu. Rev. Biophys. Biomol. Struct. 22 (1993), 27–65.CrossRefGoogle ScholarPubMed
Bernadó, P., Torre, J. García de la, and Pons, M., Macromolecular Crowding in Biological Systems: Hydrodynamics and NMR Methods. J. Mol. Recognit. 17 (2004), 397–407.CrossRefGoogle ScholarPubMed
Qian, J. and Sen, P. N., Time Dependent Diffusion in a Disordered Medium with Partially Absorbing Walls: A Perturbative Approach. J. Chem. Phys. 125 (2006), 194508-1–194508-6.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
Bear, J., Dynamics of Fluids in Porous Media. (New York: Dover, 1988).Google Scholar
Dullien, F. A. L., Porous Media: Fluid Transport and Pore Structure, 2nd edn. (New York: Academic Press, 1992).Google Scholar
Latour, L. L., Kleinberg, R. L., Mitra, P. P., and Sotak, C. H., Pore-Size Distributions and Tortuosity in Heterogeneous Porous Media. J. Magn. Reson. A 112 (1995), 83–91.CrossRefGoogle Scholar
Heil, S. R. and Holz, M., Electrical Transport in a Disordered Medium: NMR Measurement of Diffusivity and Electrical Mobility of Ionic Charge Carriers. J. Magn. Reson. 135 (1998), 17–22.CrossRefGoogle Scholar
Hizi, U. and Bergman, D. J., Molecular Diffusion in Periodic Porous Media. J. Appl. Phys. 87 (2000), 1704–11.CrossRefGoogle Scholar
Hrabe, J., Hrabtová, S., and Segeth, K., A Model of Effective Diffusion and Tortuosity in the Extracellular Space of the Brain. Biophys. J. 87 (2004), 1606–17.CrossRefGoogle Scholar
Jönsson, B., Wennerström, H., Nilsson, P. G., and Linse, P., Self-Diffusion of Small Molecules in Colloidal Systems. Colloid Polym. Sci. 264 (1986), 77–88.CrossRefGoogle Scholar
Jóhannesson, H. and Halle, B., Solvent Diffusion in Ordered Macrofluids: A Stochastic Simulation Study of the Obstruction Effect. J. Chem. Phys. 104 (1996), 6807–17.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
Mitra, P. P., Diffusion in Porous Materials as Probed by Pulsed Gradient NMR Measurements. Physica A 241 (1997), 122–7.CrossRefGoogle Scholar
Hürlimann, M. D., Helmer, K. G., Latour, L. L., and Sotak, C. H., Restricted Diffusion in Sedimentary Rocks. Determination of Surface-Area-to-Volume Ratio and Surface Relaxivity. J. Magn. Reson. A 111 (1994), 169–78.CrossRefGoogle Scholar
Schlesinger, M. F., Zaslavsky, G. M., and Klafter, J., Strange Kinetics. Nature 363 (1993), 31–7.CrossRefGoogle Scholar
Bizzari, A. R. and Cannistraro, S., Molecular Dynamics Simulation Evidence of Anomalous Diffusion of Protein Hydration Water. Phys. Rev. E 53 (1996), R3040–3.CrossRefGoogle Scholar
Klemm, A., Metzler, R., and Kimmich, R., Diffusion on Random-Site Percolation Clusters: Theory and NMR Microscopy Experiments with Model objects. Phys. Rev. E 65 (2002), 021112-1–021112-11.CrossRefGoogle ScholarPubMed
Özarslan, E., Basser, P. J., Shepherd, T. M., Thelwall, P. E., Vemuri, B. C., and Blackband, S. J., Observation of Anomalous Diffusion in Excised Tissue by Characterizing the Diffusion-time Dependence of the MR Signal. J. Magn. Reson. 183 (2006), 315–23.CrossRefGoogle ScholarPubMed
Paul, W., Anomalous Diffusion in Polymer Melts. Chem. Phys. 284 (2003), 59–66.CrossRefGoogle Scholar
Bychuk, O. V. and O'Shaugnessy, B., Anomalous Surface Diffusion: A Numerical Study. J. Chem. Phys. 101 (1994), 772–80.CrossRefGoogle Scholar
Schnell, S. and Turner, T. E., Reaction Kinetics in Intracellular Environments with Macromolecular Crowding: Simulations and Rate Laws. Prog. Biophys. Molec. Biol. 85 (2004), 235–60.CrossRefGoogle ScholarPubMed
Kimmich, R., NMR: Tomography, Diffusometry, Relaxometry. (Berlin: Springer-Verlag, 1997).CrossRefGoogle Scholar
Banavar, J. R., Lipsicas, M., and Willemsen, J. F., Determination of the Random-Walk Dimension of Fractals by Means of NMR. Phys. Rev. B 32 (1985), 6066.CrossRefGoogle ScholarPubMed
Hentschel, H. G. E. and Procaccia, I., Relative Diffusion in Turbulent Media: The Fractal Dimension of Clouds. Phys. Rev. A 29 (1984), 1461–70.CrossRefGoogle Scholar
O'Shaugnessy, B. and Procaccia, I., Analytical Solutions for Diffusion on Fractal Objects. Phys. Rev. Lett. 54 (1985), 455–8.CrossRefGoogle Scholar
Damion, R. A. and Packer, K. J., Predictions for Pulsed-Field-Gradient NMR Experiments of Diffusion in Fractal Spaces. Proc. R. Soc. London A 453 (1997), 205–11.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
Crick, F., Diffusion in Embryogenesis. Nature 225 (1970), 420–2.CrossRefGoogle ScholarPubMed
Cooper, R. L., Chang, D. B., Young, A. C., Martin, C. J., and Ancker-Johnson, B., Restricted Diffusion in Biophysical Systems: Experiment. Biophys. J. 14 (1974), 161–77.CrossRefGoogle ScholarPubMed
Dudko, O. K., Berezhkovskii, A. M., and Weiss, G. H., Diffusion in the Presence of Periodically Spaced Permeable Membranes. J. Chem. Phys. 121 (2004), 11283–8.CrossRefGoogle ScholarPubMed
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
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
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
Packer, K. J., Stapf, S., Tessier, J. J., and Damion, R. A., The Characterisation of Fluid Transport in Porous Solids by Means of Pulsed Magnetic Field Gradient NMR. Magn. Reson. Imaging 16 (1998), 463–9.CrossRefGoogle ScholarPubMed
Stapf, S., Packer, K. J., Graham, R. G., Thovert, J.-F., and Adler, P. M., Spatial Correlations and Dispersion for Fluid Transport Through Packed Glass Beads Studied by Pulsed Field-Gradient NMR. Phys. Rev. E 58 (1998), 6206–21.CrossRefGoogle Scholar
Graham, R. G., Holmes, W. M., Panfilis, C., and Packer, K. J., Characterisation of Locally Anisotropic Structures Within Isotropic Porous Solids Using 2-D Pulsed Field Gradient NMR. Chem. Phys. Lett. 332 (2000), 319–23.CrossRefGoogle Scholar
Rayleigh, J. W., On the Influence of Obstacles Arranged in Rectangular Order Upon the Properties of a Medium. Philos. Mag. 34 (1892), 481–502.CrossRefGoogle Scholar
Lekkerkerker, H. N. W. and Dhont, J. K. G., On the Calculation of the Self-Diffusion Coefficient of Interacting Brownian Particles. J. Chem. Phys. 80 (1984), 5790–92.CrossRefGoogle Scholar
Muramatsu, N. and Minton, A. P., Tracer Diffusion of Globular Proteins in Concentrated Protein Solutions. Proc. Natl. Acad. Sci. U.S.A. 85 (1988), 2984–8.CrossRefGoogle ScholarPubMed
Blees, M. H. and Leyte, J. C., The Effective Translational Self-Diffusion Coefficient of Small Molecules in Colloidal Crystals of Spherical Particles. J. Colloid Interface Sci. 166 (1994), 118–27.CrossRefGoogle Scholar
Han, J. and Herzfeld, J., Macromolecular Diffusion in Crowded Solutions. Biophys. J. 65 (1993), 1155–61.CrossRefGoogle ScholarPubMed
Bezrukov, O. Z., Lukyanov, A. E., Pozdyshev, V. K., and Struts, A. V., Analytical Calculations of Self-Diffusion Coefficients in a Medium with Obstacles. Physica Scr. 64 (2001), 382–5.CrossRefGoogle Scholar
Fujita, H., Diffusion in Polymer-Diluent Systems. Adv. Polym. Sci. 3 (1961), 1–47.Google Scholar
Masaro, L. and Zhu, X. X., Physical Models of Diffusion for Polymer Solutions, Gels and Solids. Progress in Polymer Science 24 (1999), 731–75.CrossRefGoogle Scholar
Tokuyama, M. and Oppenheim, I., Dynamics of Hard-Sphere Suspensions. Phys. Rev. E 50 (1994), R16–19.CrossRefGoogle ScholarPubMed
Bell, G. M., Self-Diffusion of Ions in the Electric Fields of Spherical Particles. Trans. Faraday Soc. 60 (1964), 1752–9.CrossRefGoogle Scholar
Schipper, F. J. M. and Leyte, J. C., Mass Transport in Polyelectrolyte Solutions. J. Phys. Condens. Matter 11 (1999), 1409–21.CrossRefGoogle Scholar
Darwish, M. I. M., Maarel, J. R. C., and Zitha, P. L. J., Ionic Transport in Polyelectrolyte Gels: Model and NMR Investigations. Macromolecules 37 (2005), 2307–12.CrossRefGoogle Scholar
Amsden, B., An Obstruction-Scaling Model for Diffusion in Homogeneous Hydrogels. Macromolecules 32 (1999), 874–9.CrossRefGoogle Scholar
Wang, J. H., Theory of the Self-Diffusion of Water in Protein Solutions. A New Method for Studying the Hydration and Shape of Protein Molecules. J. Am. Chem. Soc. 76 (1954), 4755–63.CrossRefGoogle Scholar
Clark, M. E., Burnell, E. E., Chapman, N. R., and Hinke, J. A. M., Water in Barnacle Muscle. Biophys. J. 39 (1982), 289–99.CrossRefGoogle ScholarPubMed
Teraoka, I., Polymer Solutions: An Introduction to Physical Properties. (New York: Wiley, 2002).CrossRefGoogle Scholar
Rouse, P. E., Jr., A Theory of the Linear Viscoelastic Properties of Dilute Solutions of Coiling Polymers. J. Chem. Phys. 21 (1953), 1272–80.CrossRefGoogle Scholar
Gennes, P. G., Reptation of a Polymer Chain in the Presence of Fixed Obstacles. J. Chem. Phys. 55 (1971), 572–9.CrossRefGoogle Scholar
Terentjev, E. M., Callaghan, P. T., and Warner, M., Pulsed Gradient Spin–Echo Nuclear Magnetic Resonance of Confined Brownian Particles. J. Chem. Phys. 102 (1995), 4619–24.CrossRefGoogle Scholar
Seymour, J. D. and Callaghan, P. T., Generalized Approach to NMR Analysis of Flow and Dispersion in Porous Media. AIChE J. 43 (1997), 2096–111.CrossRefGoogle Scholar
Callaghan, P. T. and Khrapitchev, A. A., Time-Dependent Velocities in Porous Media Dispersive Flow. Magn. Reson. Imaging 19 (2001), 301–5.CrossRefGoogle ScholarPubMed
Broeck, C., Taylor Dispersion Revisited. Physica A 168 (1990), 677–96.CrossRefGoogle Scholar
Crist, B., Polymer Self-Diffusion Measurements by Small-Angle Neutron Scattering. J. Non-Cryst. Solids 131–133 (1991), 709–14.CrossRefGoogle Scholar
Tang, X.-C., Song, X.-W., Shen, P.-Z., and Jia, D. Z., Capacity Intermittent Titration Technique (CITT): A Novel Technique for Determination of Li+ Solid Diffusion Coefficient of LiMn2O4. Electrochim. Acta. 50 (2007), 5581–7.CrossRefGoogle Scholar
Visser, A. J. W. G. and Hink, M. A., New Perspectives of Fluorescence Correlation Spectroscopy. J. Fluoresc. 9 (1999), 81–7.CrossRefGoogle Scholar
Braeckmans, K., Peeters, L., Sanders, N. N., Smedt, S. C., and DeMeester, J., Three-Dimensional Fluorescence Recovery after Photobleaching with the Confocal Scanning Laser Microscope. Biophys. J. 85 (2003), 2240–52.CrossRefGoogle ScholarPubMed
Lyon, S. B. and Phillipe, L. T. E., Direct Measurements of Ionic Diffusion in Protective Organic Coatings. Trans. Inst. Met. Finish. 84 (2006), 23–7.CrossRefGoogle Scholar
Minton, A. P., Analytical Centrifugation with Preparative Ultracentrifuges. Anal. Biochem. 176 (1989), 209–16.CrossRefGoogle ScholarPubMed
Lang, E. W. and Lüdemann, H.-D., Density Dependence of Rotational and Translational Molecular Dynamics in Liquids Studies by High Pressure NMR. Prog. NMR Spectrosc. 25 (1993), 507–633.CrossRefGoogle Scholar
McConnell, J., Nuclear Magnetic Relaxation in Liquids. (Cambridge: Cambridge University Press, 1987).Google Scholar
Debye, P., Polar Molecules. (New York: Dover Publications, 1945).Google Scholar
Bloembergen, N., Purcell, E. M., and Pound, R. V., Relaxation Effects in Nuclear Magnetic Resonance Absorption. Phys. Rev. 73 (1948), 679–712.CrossRefGoogle Scholar
Hennel, J. W. and Klinowski, J., Fundamentals of Nuclear Magnetic Resonance. (Essex: Longman Scientific & Technical, 1993).Google Scholar
Price, W. S., Kuchel, P. W., and Cornell, B. A., Microviscosity of Human Erythrocytes Studied with Hypophosphite and 31P- NMR. Biophys. Chem. 33 (1989), 205–15.CrossRefGoogle ScholarPubMed
Price, W. S., Perng, B.-C., Tsai, C.-L., and Hwang, L.-P., Microviscosity of Human Erythrocytes Studied using Hypophosphite Two-Spin Order Relaxation. Biophys. J. 61 (1992), 621–30.CrossRefGoogle ScholarPubMed
Woessner, D. E., Brownian Motion and Correlation Times. In Encyclopedia of Nuclear Magnetic Resonance, ed. Grant, D. M. and Harris, R. K.. vol. 2. (New York: Wiley, 1996), pp. 1068–84.Google Scholar
Woessner, D. E., Relaxation Effects of Chemical Exchange. In Encyclopedia of Nuclear Magnetic Resonance, ed. Grant, D. M. and Harris, R. K.. vol. 6. (New York: Wiley, 1996), pp. 4018–28.Google Scholar

Save book to Kindle

To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×