Physical models of diffusion for polymer solutions, gels and solids

Diffusion in polymer solutions and gels has been studied by various techniques such as gravimetry, membrane permeation, fluorescence and radioactive labeling. These studies have led to a better knowledge on polymer morphology, transport phenomena, polymer melt and controlled release of drugs from polymer carriers. Various theoretical descriptions of the diffusion processes have been proposed. The theoretical models are based on different physical concepts such as obstruction effects, free volume effects and hydrodynamic interactions. With the availability of pulsed field gradient NMR techniques and other modern experimental methods, the study of diffusion has become much easier and data on diffusion in polymers have become more available. This review article summarizes the different physical models and theories of diffusion and their uses in describing the diffusion in polymer solutions, gels and even solids. Comparisons of the models and theories are made in an attempt to illustrate the applicability of the physical concepts. Examples in the literature are used to illustrate the application and applicability of the models in the treatment of diffusion data in various systems. q 1999 Elsevier Science Ltd. All rights reserved.

[1]  Hisahiko Watanabe,et al.  DIFFUSION MEASUREMENT OF TRITIUM LABELED POLYSTYRENE IN POLYMER BULK , 1979 .

[2]  B. Roux,et al.  A New Physical Model for the Diffusion of Solvents and Solute Probes in Polymer Solutions , 1996 .

[3]  V. Klepko,et al.  Self-diffusion of water in gelatin gels: 1. Macroscopic measurements by tracer technique , 1993 .

[4]  B. Amsden Solute diffusion in hydrogels. , 1998 .

[5]  John F. Brady,et al.  Hindered transport of spherical macromolecules in fibrous membranes and gels , 1989 .

[6]  J. Piau,et al.  Rheology for polymer melt processing , 1996 .

[7]  T. Higuchi MECHANISM OF SUSTAINED-ACTION MEDICATION. THEORETICAL ANALYSIS OF RATE OF RELEASE OF SOLID DRUGS DISPERSED IN SOLID MATRICES. , 1963, Journal of pharmaceutical sciences.

[8]  X. Zhu,et al.  Self-Diffusion of Oligo- and Poly(ethylene glycol)s in Poly(vinyl alcohol) Aqueous Solutions As Studied by Pulsed-Gradient NMR Spectroscopy , 1998 .

[9]  K. Foster,et al.  Self-diffusion of water in ionic and nonionic microemulsions , 1985 .

[10]  T. Barbari,et al.  Diffusion of methyl ethyl ketone in polyisobutylene: Comparison of spectroscopic and gravimetric techniques , 1997 .

[11]  E. V. Meerwall Pulsed and steady field gradient NMR diffusion measurements in polymers , 1985 .

[12]  J. S. Vrentas,et al.  Solvent self-diffusion in rubbery polymer-solvent systems , 1994 .

[13]  Edward L Cussler,et al.  Diffusion: Mass Transfer in Fluid Systems , 1984 .

[14]  P. Stilbs,et al.  Self‐diffusion of small molecules in cellulose gels using FT‐pulsed field gradient NMR , 1984 .

[15]  Kerman Tube leakage during electrophoresis retards reptating DNA in unmodified and hydroxyethylated agarose gels. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[16]  X. Zhu,et al.  Study of the self-diffusion of poly(ethylene glycol)s in poly(vinyl alcohol) aqueous systems , 1999 .

[17]  H. Höltje Principles of medicinal chemistry, von W. O. Foye (Ed.), 857 S., Preis $28,50, Lea & Febiger, Philadelphia 1974 , 1975 .

[18]  N. Peppas,et al.  Solute diffusion in swollen membranes. IV. Theories for moderately swollen networks , 1985 .

[19]  M. Daoud,et al.  Solutions of Flexible Polymers. Neutron Experiments and Interpretation , 1975 .

[20]  J. L. Duda,et al.  Diffusion in polymer—solvent systems. I. Reexamination of the free‐volume theory , 1977 .

[21]  N A Peppas,et al.  Water, solute and protein diffusion in physiologically responsive hydrogels of poly (methacrylic acid-g-ethylene glycol). , 1996, Biomaterials.

[22]  H. Fricke,et al.  A Mathematical Treatment of the Electric Conductivity and Capacity of Disperse Systems I. The Electric Conductivity of a Suspension of Homogeneous Spheroids , 1924 .

[23]  S. W. Kim,et al.  Macromolecular diffusion through collagen membranes , 1984 .

[24]  P. Meares,et al.  The diffusion of electrolytes in a cation-exchange resin membrane I. Theoretical , 1955, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[25]  P. Stilbs,et al.  Associative thickeners: NMR self-diffusion and rheology studies of aqueous solutions of hydrophobically modified poly(oxyethylene) polymers , 1993 .

[26]  P. Russo,et al.  Rotational and Translational Diffusion of a Rodlike Virus in Random Coil Polymer Solutions , 1997 .

[27]  P. Russo,et al.  Diffusion of Dextran in Aqueous (Hydroxypropyl)cellulose , 1994 .

[28]  J. Koenig,et al.  Study of cyclic sorption-desorption into poly(methyl methacrylate) rods using NMR imaging , 1992 .

[29]  G. Phillies Range of validity of the hydrodynamic scaling model , 1992 .

[30]  G. Phillies,et al.  Transport in intermediate and high molecular weight hydroxypropylcellulose/water solutions , 1993 .

[31]  P. Russo,et al.  Interactions between polystyrene latex spheres and a semiflexible polymer, hydroxypropylcellulose☆ , 1988 .

[32]  W. Foye,et al.  Principles of medicinal chemistry , 1974 .

[33]  Robert I. Cukier,et al.  Diffusion of Brownian spheres in semidilute polymer solutions , 1984 .

[34]  K. Jansons,et al.  Flow and diffusion through random suspensions of aggregated rods: application to proteoglycan solutions , 1990 .

[35]  P. Gennes Scaling Concepts in Polymer Physics , 1979 .

[36]  H. Kramers Brownian motion in a field of force and the diffusion model of chemical reactions , 1940 .

[37]  L. Johansson,et al.  Diffusion and interaction in gels and solution. 5. Nonionic micellar systems , 1993 .

[38]  V. Klepko,et al.  Liquid flow through polymer gels , 1996 .

[39]  R. Phillips,et al.  Measurement of protein diffusion in Dextran solutions by holographic interferometry , 1995 .

[40]  K. Freed,et al.  On dynamic scaling theories of polymer solutions at nonzero concentrations , 1980 .

[41]  K. Zick,et al.  Stray field magnetic resonance imaging of the diffusion of acetone into poly(vinyl chloride) , 1994 .

[42]  J. Viovy Reptation theories of electrophoresis , 1996, Molecular biotechnology.

[43]  X. Zhu,et al.  Pulsed-gradient spin-echo NMR measurements of the diffusion coefficients of ketones in poly(methyl methacrylate) , 1992 .

[44]  J. Petit,et al.  Solute Probe Diffusion in Aqueous Solutions of Poly(vinyl alcohol) As Studied by Pulsed-Gradient Spin-Echo NMR Spectroscopy , 1996 .

[45]  T. Lodge,et al.  Monomer diffusion and the kinetics of methyl methacrylate radical polymerization at intermediate to high conversion , 1994 .

[46]  Marmonier,et al.  Reptation and tube renewal in entangled polymer solutions. , 1985, Physical Review Letters.

[47]  P. Callaghan,et al.  NMR Investigations of Self-Diffusion and Shear Thinning for Semidilute Polymer Solutions near the Demixing Transition , 1997 .

[48]  S. A. Stern,et al.  Tests of a ``free-volume'' model of gas permeation through polymer membranes. II. Pure Ar, SF6, CF4, and C2H2F2 in polyethylene , 1986 .

[49]  N. Peppas Hydrogels in Medicine and Pharmacy , 1987 .

[50]  H. Yasuda,et al.  Permeability of Solutes through Hydrated Polymer Membranes Part I. Diffusion of Sodium Chloride , 1968 .

[51]  J. S. Vrentas,et al.  A new equation relating self-diffusion and mutual diffusion coefficients in polymer-solvent systems , 1979 .

[52]  Matthew Tirrell,et al.  Polymer Self-Diffusion in Entangled Systems , 1984 .

[53]  Michael E. Moseley,et al.  Molecular motion of small molecules in cellulose gels studied by NMR , 1981 .

[54]  P. E. Rouse A Theory of the Linear Viscoelastic Properties of Dilute Solutions of Coiling Polymers , 1953 .

[55]  J. S. Vrentas,et al.  Solvent self‐diffusion in crosslinked polymers , 1991 .

[56]  Hyuk Yu,et al.  Self-diffusion of polystyrenes by forced Rayleigh scattering , 1984 .

[57]  R. Landel,et al.  The Temperature Dependence of Relaxation Mechanisms in Amorphous Polymers and Other Glass-Forming Liquids , 1955 .

[58]  C. H. Wang,et al.  Holographic grating relaxation studies of probe diffusion in amorphous polymers , 1995 .

[59]  George D. J. Phillies,et al.  Universal scaling equation for self-diffusion by macromolecules in solution , 1986 .

[60]  Diffusion of latex spheres through solutions of hydroxypropylcellulose in water , 1988 .

[61]  Balinov,et al.  NMR Self-Diffusion Studies of the Water and the Oil Components in a W/O/W Emulsion , 1997, Journal of colloid and interface science.

[62]  Noreen L. Thomas,et al.  Transport of methanol in poly(methyl methacrylate) , 1978 .

[63]  James M. Caruthers,et al.  Self- and mutual-diffusion coefficients in the dodecane/polystyrene system , 1994 .

[64]  W. Brown,et al.  Transport from an aqueous phase through cellulosic gels and membranes , 1981 .

[65]  R. Lostritto,et al.  Diffusion of benzocaine in poly(ethylene-vinyl acetate) membranes: Effects of vehicle ethanol concentration and membrane vinyl acetate content , 1996 .

[66]  Charles S. Johnson,et al.  Probe Diffusion in Polyacrylamide Gels as Observed by Means of Holographic Relaxation Methods: Search for a Universal Equation , 1990 .

[67]  E. Kramer,et al.  Diffusion of styrene-acrylonitrile copolymers , 1995, Journal of Materials Science.

[68]  J. L. Duda,et al.  Diffusion in polymer-solvent systems. II. A predictive theory for the dependence of diffusion coefficients on temperature, concentration, and molecular weight , 1977 .

[69]  J. L. Duda,et al.  Self-diffusion in polymer-solvent-solvent systems , 1984 .

[70]  Stephen Z. D. Cheng,et al.  SELF-DIFFUSION OF POLY(ETHYLENE OXIDE) FRACTIONS AND ITS INFLUENCE ON THE CRYSTALLINE TEXTURE , 1991 .

[71]  J. Koenig,et al.  NMR imaging of diffusion processes in polymers: measurement of the spatial dependence of solvent mobility in partially swollen PMMA rods , 1990 .

[72]  S. Pickup,et al.  Self-diffusion of toluene in polystyrene solutions , 1989 .

[73]  David Turnbull,et al.  Molecular Transport in Liquids and Glasses , 1959 .

[74]  D. Hu,et al.  Kinetics of water swelling and development of porous structure in ionic poly(acrylonitrile-acrylamide-acrylic acid) hydrogels , 1996 .

[75]  John Crank,et al.  The Mathematics Of Diffusion , 1956 .

[76]  L. Johansson,et al.  Diffusion and interaction in gels and solutions. 2. Experimental results on the obstruction effect , 1991 .

[77]  S. A. Stern,et al.  The diffusion of CO2, CH4, C2H4, and C3H8 in polyethylene at elevated pressures , 1983 .

[78]  Self-diffusion and probe diffusion in dilute and semidilute aqueous solutions of dextran , 1991 .

[79]  G. Fleischer Self diffusion in melts of polystyrene and polyethylene measured by pulsed field gradient NMR , 1983 .

[80]  T. Lodge,et al.  Tracer diffusion of linear polystyrenes in dilute, semidilute, and concentrated poly(vinyl methyl ether) solutions , 1989 .

[81]  G. Phillies,et al.  Macroparticle diffusion in dextran solutions , 1989 .

[82]  G. Zacchi,et al.  Diffusion measurement in gels , 1994 .

[83]  J. L. Duda,et al.  Predicting polymer/solvent diffusion coefficients using free-volume theory , 1992 .

[84]  S. Prager,et al.  Dynamics of Entangled Polymer Liquids: Do Linear Chains Reptate? , 2007 .

[85]  J. Ferry,et al.  Self-diffusion in solutions of polystyrene in tetrahydrofuran: comparison of concentration dependences of the diffusion coefficients of polymer, solvent, and a ternary probe component , 1985 .

[86]  Balaji Narasimhan,et al.  On the importance of chain reptation in models of dissolution of glassy polymers , 1996 .

[87]  N. Peppas,et al.  Solute diffusion in swollen membranes. Part I. A new theory , 1983 .

[88]  C. L. Bell,et al.  Biomedical membranes from hydrogels and interpolymer complexes , 1995 .

[89]  Y. Uemura,et al.  Associative behavior and diffusion coefficients of hydrophobically modified poly(N,N-dimethylacrylamides) , 1995 .

[90]  T. Lodge,et al.  Tracer diffusion of linear polystyrenes in poly(vinyl methyl ether) gels , 1992 .

[91]  A. Gent,et al.  Diffusion of polymer molecules into polymer networks: Effect of stresses and contraints , 1991 .

[92]  J. L. Duda,et al.  Free‐volume theories for self‐diffusion in polymer–solvent systems. II. Predictive capabilities , 1985 .

[93]  J. Kirkwood,et al.  Errata: The Intrinsic Viscosities and Diffusion Constants of Flexible Macromolecules in Solution , 1948 .

[94]  U. Gedde,et al.  Diffusion of small-molecule penetrants in semicrystalline polymers , 1996 .

[95]  G. Phillies,et al.  Chain and sphere diffusion in polyisobutylene-trichloromethane: a reanalysis , 1992 .

[96]  T. Lodge,et al.  Diffusion of Spheres in Entangled Polymer Solutions: A Return to Stokes-Einstein\ Behavior , 1994 .

[97]  J. Wang,et al.  Theory of the Self-diffusion of Water in Protein Solutions. A New Method for Studying the Hydration and Shape of Protein Molecules , 1954 .

[98]  J. D. Wells,et al.  On the transport of compact particles through solutions of chain-polymers , 1973, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[99]  N. Peppas,et al.  Solute diffusion in swollen membranes. Part II. Influence of crosslinking on diffusive properties , 1984 .

[100]  T. Hanai Theory of the dielectric dispersion due to the interfacial polarization and its application to emulsions , 1960 .

[101]  S. L. Wright,et al.  Solute diffusion in poly(vinyl alcohol)/poly(acrylic acid) interpenetrating networks , 1996 .

[102]  Wim E. Hennink,et al.  Controlled release of proteins from dextran hydrogels , 1996 .

[103]  H. Eyring Viscosity, Plasticity, and Diffusion as Examples of Absolute Reaction Rates , 1936 .

[104]  Noreen L. Thomas,et al.  A theory of case II diffusion , 1982 .

[105]  M. Tirrell,et al.  On the theory of self‐diffusion in a polymer gel , 1984 .

[106]  Tadashi Inoue,et al.  Self diffusion of polymers in the concentrated regime. Part 2. Self diffusion and tracer-diffusion coefficient and viscosity of concentrated solutions of linear polystyrenes in dibutyl phthalate , 1989 .

[107]  P. G. de Gennes,et al.  Dynamics of Entangled Polymer Solutions. I. The Rouse Model , 1976 .

[108]  J. S. Vrentas,et al.  Solvent self-diffusion in glassy polymer-solvent systems , 1994 .

[109]  H. Wennerström,et al.  Self-diffusion of small molecules in colloidal systems , 1986 .

[110]  Alan H. Muhr,et al.  Diffusion in gels , 1982 .

[111]  John Crank,et al.  Diffusion in polymers , 1968 .

[112]  P. Gennes Dynamics of Entangled Polymer Solutions. II. Inclusion of Hydrodynamic Interactions , 1976 .

[113]  R. Zwanzig,et al.  Diffusion in a rough potential. , 1988, Proceedings of the National Academy of Sciences of the United States of America.

[114]  P. Gennes Reptation of a Polymer Chain in the Presence of Fixed Obstacles , 1971 .

[115]  G. Mey,et al.  Structural information on hyaluronic acid solutions as studied by probe diffusion experiments , 1994 .

[116]  J. Noolandi,et al.  A new theory of entanglements and dynamics in dense polymer systems , 1988 .

[117]  N. Peppas,et al.  Transport of ionizable drugs and proteins in crosslinked poly(acrylic acid) and poly(acrylic acid-co-2-hydroxyethyl methacrylate) hydrogels. II. Diffusion and release studies , 1997 .

[118]  Hyuk Yu,et al.  Polymer diffusion in linear matrixes: polystyrene in toluene , 1986 .

[119]  J. Torkelson,et al.  Small-molecule probe diffusion in polymer solutions: Studies by Taylor dispersion and phosphorescence quenching , 1996 .

[120]  Nikolaos A. Peppas,et al.  Solute diffusion in swollen membranes. IX: Scaling laws for solute diffusion in gels , 1988 .

[121]  Perkins,et al.  Self-diffusion of an entangled DNA molecule by reptation. , 1995, Physical review letters.

[122]  A. Langdon,et al.  Self-diffusion studies of gel hydration and the obstruction effect , 1971 .

[123]  G. Phillies Quantitative prediction of .alpha. in the scaling law for self-diffusion , 1988 .

[124]  Noolandi,et al.  New view of entanglements in dense polymer systems. , 1987, Physical review letters.

[125]  G. Phillies Dynamics of polymers in concentrated solutions: the universal scaling equation derived , 1987 .

[126]  J. Joanny,et al.  Dynamics of Semiflexible Polymers in Solution , 1980 .

[127]  M. Tirrell,et al.  Hydrodynamic screening and particle dynamics in porous media, semidilute polymer solutions and polymer gels , 1986 .

[128]  Balaji Narasimhan,et al.  Disentanglement and reptation during dissolution of rubbery polymers , 1996 .

[129]  R. Waggoner,et al.  Dependence of the Solvent Diffusion Coefficient on Concentration in Polymer Solutions , 1993 .

[130]  L. Johansson,et al.  Diffusion and interaction in gels and solutions. 4. Hard sphere Brownian dynamics simulations , 1993 .

[131]  G. Phillies The hydrodynamic scaling model for polymer self-diffusion , 1989 .

[132]  J. Labský,et al.  Measurements of diffusion processes in HEMA-DEGMA hydrogels by ESR imaging , 1995 .

[133]  J. L. Duda,et al.  Penetrant transport in polyethylene‐polystyrene semi‐interpenetrating polymer networks , 1997 .

[134]  S. Matsukawa,et al.  Study of Self-Diffusion of Molecules in a Polymer Gel by Pulsed-Gradient Spin-Echo 1H NMR. 2. Intermolecular Hydrogen-Bond Interaction between the Probe Polymer and Network Polymer in N,N-Dimethylacrylamide−Acrylic Acid Copolymer Gel Systems , 1997 .

[135]  T. Cosgrove,et al.  The critical overlap concentration measured by pulsed field gradient nuclear magnetic resonance techniques , 1994 .

[136]  L. Léger,et al.  Reptation in entangled polymer solutions by forced Rayleigh light scattering , 1981 .

[137]  H. Fujita Diffusion in polymer-diluent systems , 1961 .

[138]  J. L. Duda,et al.  Free‐volume theories for self‐diffusion in polymer–solvent systems. I. Conceptual differences in theories , 1985 .

[139]  N. Peppas,et al.  EFFECT OF THE MORPHOLOGY OF HYDROPHILIC POLYMERIC MATRICES ON THE DIFFUSION AND RELEASE OF WATER SOLUBLE DRUGS , 1981 .

[140]  M. Sefton,et al.  Permeability of a heparin-polyvinyl alcohol hydrogel to thrombin and antithrombin III. , 1988, Journal of biomedical materials research.

[141]  B. Lindman,et al.  Water self-diffusion in nonionic surfactant solutions. Hydration and obstruction effects , 1983 .

[142]  W. Graessley,et al.  Measurement of self-diffusion coefficient in polymer melts by a small-angle neutron scattering method , 1983 .

[143]  Noreen L. Thomas,et al.  Diffusion mechanics of the system PMMA-methanol , 1981 .

[144]  D. L. Tipton,et al.  Dye diffusion in isotropic and liquid-crystalline aqueous (hydroxypropyl)cellulose , 1993 .

[145]  M. Antonietti,et al.  Self-diffusion of polystyrene chains in networks , 1985 .

[146]  D. N. Pinder,et al.  Influence of multiple length scales on the behavior of polymer self-diffusion in the semidilute regime , 1984 .

[147]  J. Hoffman,et al.  Membrane Physiology , 1987, Springer US.

[148]  D. Kee,et al.  Free volume model and diffusion of organic solvents in natural rubber , 1995 .

[149]  U. Alon,et al.  GEL ELECTROPHORESIS AND DIFFUSION OF RING-SHAPED DNA , 1997, cond-mat/9702154.

[150]  P. Stilbs,et al.  Self-diffusion of poly(ethylene oxide) in aqueous dextran solutions measured using FT-pulsed field gradient n.m.r. , 1983 .

[151]  J. MacElroy,et al.  Hindered diffusion of gases in Leaky membranes using the dusty gas model , 1985 .

[152]  N. Peppas,et al.  Modelling of water transport and solute release in physiologically sensitive gels , 1993 .

[153]  B. Zimm Dynamics of Polymer Molecules in Dilute Solution: Viscoelasticity, Flow Birefringence and Dielectric Loss , 1956 .

[154]  Noreen L. Thomas,et al.  Discontinuous shape changes associated with Case II transport of methanol in thin sheets of PMMA , 1977 .

[155]  Jungmoon Sung,et al.  Temperature dependence of probe diffusion in polymer matrix , 1993 .

[156]  P. Griffiths,et al.  PGSE-NMR studies of solvent diffusion in poly(N-isopropylacrylamide) colloidal microgels , 1995 .

[157]  E. V. Meerwall Self-diffusion in polymer systems, measured with field-gradient spin echo NMR methods , 1984 .

[158]  Nikolaos A. Peppas,et al.  Crystal dissolution-controlled release systems: I. Physical characteristics and modeling analysis , 1997 .

[159]  M. Antonietti,et al.  Diffusion of labeled macromolecules in molten polystyrenes studied by a holographic grating technique , 1984 .

[160]  T. Laurent,et al.  Diffusion of Linear Polymers in Hyaluronate Solutions , 1975 .

[161]  M. J. Turner,et al.  Self-diffusion and spin-spin relaxation in blends of linear and cyclic polydimethylsiloxane melts , 1996 .

[162]  Noreen L. Thomas,et al.  Case II swelling of PMMA sheet in methanol , 1978 .

[163]  V. Calhoun,et al.  The use of magnetic resonance imaging to track controlled drug release and transport in the brain. , 1993, Magnetic resonance imaging.

[164]  S. Matsukawa,et al.  A Study of Self-Diffusion of Molecules in Polymer Gel by Pulsed-Gradient Spin−Echo 1H NMR , 1996 .

[165]  P. Stilbs,et al.  Fourier transform pulsed-gradient spin-echo studies of molecular diffusion , 1987 .

[166]  N. Peppas,et al.  Controlled release from poly ( vinyl alcohol ) gels prepared by freezing-thawing processes , 1992 .

[167]  T. Laurent,et al.  ON THE INTERACTION BETWEEN POLYSACCHARIDES AND OTHER MACROMOLECULES. II. THE TRANSPORT OF GLOBULAR PARTICLES THROUGH HYALURONIC ACID SOLUTIONS. , 1963, Biochimica et biophysica acta.

[168]  Y. Uemura,et al.  Self-Diffusion Coefficients of Hydrophobic Ethoxylated Urethane Associating Polymers Using Pulsed-Gradient Spin-Echo Nuclear Magnetic Resonance , 1995 .

[169]  L. Johansson,et al.  Diffusion and interaction in gels and solutions. 3. Theoretical results on the obstruction effect , 1991 .

[170]  F. Rondelez,et al.  Sedimentation of large colloidal particles through semidilute polymer solutions , 1978 .

[171]  S. Edwards,et al.  Polymer viscosity in concentrated solutions , 1974 .

[172]  Lebowitz,et al.  Classical transport in modulated structures. , 1985, Physical review letters.