The objective of this review is to provide an overview and a classification of the key literature on models of non-Fickian case II type diffusion. Several extensive review articles concerning non-Fickian diffusion exist in the literature; our objective is not to reproduce these worthy contributions. Rather, we focus on a limited number of, seemingly disparate, notable models and attempt to unify them using the language of thermodynamics and continuum mechanics. This attempted unification of selected models arising from various modeling communities serves to elucidate the key strengths and potential weaknesses of the models.
Issue Section:
Review Articles
References
1.
Fick
, A.
, 1855, “Über Diffusion
,” Ann. Phys. Chem.
, 94
, 58
–86
.2.
Alfrey
, T.
, Gurnee
, E. F.
, and Lloyd
, W. G.
, 1966, “Diffusion in Glassy Polymers
,” J. Polym. Sci. C
, 12
, 249
–261
.3.
De Kee
, D.
, Liu
, Q.
, and Hinestroza
, J.
, 2005, “Viscoelastic (Non-Fickian) Diffusion
,” Can. J. Chem. Eng.
, 83
, 913
–929
.4.
Vesely
, D.
, 2008, “Diffusion of Liquids in Polymers
,” Int. Mater. Rev.
, 53
(5
), 299
–315
.5.
Collins
, R.
, 1998, “Mathematical Modelling of Controlled Release From Implanted Drug-Impregnated Monoliths
,” PSTT
1
(6
), 269
–276
.6.
Govindjee
, S.
, and Simo
, J. C.
, 1993, “Coupled Stress-Diffusion: Case I
.” Int. J. Mech. Phys. Solids
, 41
(5
), pp.
863–887
.7.
Astarita
G.
, and Sarti
, G. C.
, 1978, “A Class of Mathematical Models for Sorption of Swelling Solvents in Glassy Polymers
,” Polym. Eng. Sci.
, 18
(5
), 388
–395
.8.
Thomas
, N. L.
, and Windle
, A. H.
, 1980, “A Deformation Model for Case II Diffusion
,” Polymer
, 21
(6
), 613
–619
.9.
Thomas
, N. L.
, and Windle
, A. H.
, 1981, “Diffusion Mechanics of the System PMMA-methanol
,” Polymer
, 22
(5
), 627
–639
.10.
Thomas
N. L.
, and Windle
, A. H.
, 1982, “A Theory of Case II Diffusion
,” Polymer
, 23
(4
), 529
–542
.11.
Cohen
, D. S.
, 1984, “Theoretical Models for Diffusion in Glassy Polymers
,” J. Polym. Sci.: Polym. Phys. Ed.
, 2
(22
), 1001
–1009
.12.
Durning
, C. J.
, 1985, “Differential Sorption in Viscoelastic Fluids
,” J. Polym. Sci.: Polym. Phys. Ed.
, 23
, 1831
–1855
.13.
Hui
, C.-Y.
, and Wu
, K.-C.
, 1987, “Case-II Diffusion in Polymers, I. Transient Swelling
,” J. Appl. Phys.
, 61
(11
), 5129
–5136
.14.
Hui
, C.-Y.
, and Wu
, K.-C.
, 1987, “Case-II Diffusion in Polymers, II. Steady-State Front Motion
,” J. Appl. Phys.
, 61
(11
), 5137
–5149
.15.
Cohen
, D. S.
, and White
, A. B.
, 1989, “Sharp Fronts due to Diffusion and Stress at the Glass Transition in Polymers
,”J. Polym. Sci., Part B: Polym. Phys.
, 27
, 1731
–1747
.16.
Jou
, D.
, Camacho
, J.
, and Grmela
, M.
, 1991, “On the Nonequilibrium Thermodynamics of Non-Fickian Diffusion
,” Macromolecules
, 24
, 3597
–3602
.17.
Kalospiros
, N. S.
, Ocone
, R.
, Astarita
, G.
, and Meldon
, J. H.
, 1991, “Analysis of Anomalous Diffusion and Relaxation in Solid Polymers
,” Ind. Eng. Chem. Res.
, 30
, 851
–864
.18.
Wu
, J. C.
, and Peppas
, N. A.
, 1993, “Modeling of Penetrant Diffusion in Glassy Polymers With an Integral Sorption Deborah Number
,” J. Polym. Sci., Part B: Polym. Phys.
, 31
, 1503
–1518
.19.
Rossi
, G.
, Pincus
, P. A.
, and de Gennes
, P. G.
, 1995, “A Phenomenological Description of Case-II Diffusion in Polymeric Materials
,” Europhys. Lett.
, 32
(5
), 391
.20.
Friedman
, A.
, and Rossi
, G.
, 1997, “Phenomenological Continuum Equations to Describe Case II Diffusion in Polymeric Materials
,” Macromolecules
, 30
153
–154
.21.
El Afif
, A.
, 2000, “Diffusion de la matirière dans les Polymeres
,” Ph.D. thesis, Ecole Polytechnique de Montréal, Montreal, Quebec, Canada.22.
Qian
, T.
, and Taylor
, P. L.
, 2000, “From the Thomas–Windle Model to a Phenomenological Description of Case-II Diffusion in Polymers
,” Polymer
, 41
(19
), 7159
–7163
.23.
El Afif
, A.
, and Grmela
, M.
, 2002, “Non-Fickian Mass Transport in Polymers
,” J. Rheol.
, 46
(3
), 591
–628
.24.
Vijalapura
, P. K.
, and Govindjee
, S.
, 2003, “Numerical Simulation of Coupled-Stress Case II Diffusion in One Dimension
,” J. Polym. Sci., Part B: Polym. Phys.
, 41
, 2091
–2108
.25.
Vijalapura
, P. K.
, and Govindjee
, S.
, 2005, “An Adaptive Hybrid Time-Stepping Scheme for Highly Non-linear Strongly Coupled Problems
,” Int. J. Numer. Methods Eng.
, 64
, 819
–848
.26.
Riviére
, B.
, and Shaw
, S.
, 2006, “Discontinuous Galerkin Finite Element Approximation of Nonlinear Non-Fickian Diffusion in Viscoelastic Polymers
,” SIAM (Soc. Ind. Appl. Math) J. Numer. Anal.
, 44
(6
), 2650
–2670
.27.
Aifantis
, E. C.
, 1980, “On the Problem of Diffusion in Solids
,” Acta Mech.
, 37
265
–296
.28.
Grmela
, M.
, and Öttinger
, H. C.
, 1997, “Dynamics and Thermodynamics of Complex Fluids. I. Development of a General Formalism
,” Phys. Rev. E
, 56
(6
) 6620
–6632
.29.
Öttinger
, H. C.
, and Grmela
, M.
, 1997, “Dynamics and Thermodynamics of Complex Fluids. II. Illustrations of a General Formalism
,” Phys. Rev. E
, 56
(6
), 6633
–6655
.30.
Vijalapura
, P. K.
, Strain
, J.
, and Govindjee
, S.
, 2005, “Fractional Step Methods for Index-1 Differential-Algebraic Equations
,” J. Comput. Phys.
203
, 305
–320
.31.
Ö
ttinger
, H. C.
, 2005, Beyond Equilibrium Thermodynamics
, Wiley
, New York
.32.
Grmela
, M.
, 2004, “Geometry of Mesoscopic Dynamics and Thermodynamics
,” J. Non-Newtonian Fluid Mech.
, 120
, 137
–147
.33.
Hütter
, M.
, and Tervoort
, T. A.
, 2009, “Coarse Graining in Elasto-Viscoplasticity: Bridging the Gap From Microscopic Fluctuations to Dissipation
,” Adv. Appl. Mech.
, 42
, 253
–317
.34.
Hütter
, M.
, and Tervoort
, T. A.
, 2008, “Continuum Damage Mechanics: Combining Thermodynamics With a Thoughtful Characterization of the Microstructure
,” Acta Mech.
, 201
(1–4
), 297
–312
.35.
Truesdell
, C.
, and Noll
, W.
, 2007, The Non-Linear Field Theories of Mechanics
, 3rd ed., Springer
, New York
.36.
Gurtin
, M. E.
, 1981, An Introduction to Continuum Mechanics
, Academic Press
, Orlando, FL
.37.
Ogden
, R. W.
, 1982, Nonlinear Elastic Deformations
, Ellis Horwood
, West Susex, UK
.38.
Ciarlet
, P. G.
, 1988, Mathematical Elasticity, Volume I: Three-Dimensional Elasticity
, North-Holland Publishing Company
, Amsterdam
.39.
Crank
, J.
, 1970, The Mathematics of Diffusion
, Oxford University Press, Oxford
, UK
.40.
Crank
, J.
, 1953, “A Theoretical Investigation of the Influence of Molecular Relaxation and Internal Stress on Diffusion in Polymers
,” J. Polym. Sci.
, 6
(2
), 1831
–1855
.41.
Maxwell
, J. C.
, 1867, “On the Dynamical Theory of Gases
,” Philos. Trans. R. Soc. London
, 157
, 49
–88
.42.
Peshkov
, V.
, 1944, “Second Sound in Helium I
,” J. Phys. (USSR)
, 8
, 381
.43.
Peshkov
, V.
, 1946, International Conference on Fundamental Particles and Low Temperatures Report
, Cambridge, UK.44.
Pellam
, J. R.
, 1948, “Investigations of Pulsed Second Sound in Liquid Helium
,” Phys. Rev.
, 75
(8
), 1183
–1194
.45.
Ackermann
, C. C.
, Bertram
, B.
, Fairbank
, H. A.
, and Guyer
, R. A.
, 1966, “Second Sound in Solid Helium
,” Phys. Rev.
, 18
, 789
–791
.46.
Ackermann
, C. C.
, and Overton
, W. C.
, 1969, “Second Sound in Solid Helium-3
,” Phys. Review. Lett.
22
(15
), 764
–767
.47.
Jackson
, H. E.
, Walker
, C. T.
, and McNelly
, T. F.
, 1970, “Second Sound in NaF
,” Phys. Rev.
, 25
(1
), 26
–28
.48.
Narayanamurti
, V.
, and Dynes
, R. C.
, 1972, “Observation of Second Sound in Bismuth
,” Phys. Rev.
, 28
, 1461
–1464
.49.
Cattaneo
, C.
, 1948, “Atti del Seminario Matematico e Fisico della Universita di Modena
,” Atti Semin. Mat. Fis. Univ. Modena
, 3
, 3
–21
.50.
Vernotte
, P.
, 1958, “Les paradoxes de la théorie Continue de léquation de la chaleur
,” C. R. Acad. Sci.
, 246
, 3154
–3155
.51.
Wilson
, R. K.
, and Aifantis
, E. C.
, 1982, “On the Theory of Stress-Assisted Diffusion, I
,” Acta Mech.
, 45
, 273
–296
.52.
Unger
, D. J.
, and Aifantis
, E. C.
, 1983, “On the Theory of Stress-Assisted Diffusion, II
,” Acta Mech.
47
, 117
–151
.53.
Ferry
, J. D.
, 1980, Viscoelastic Properties of Polymers
, 3rd ed., Wiley
, New York
.54.
Grest
, G. S.
, and Cohen
, M. H.
, 1981, “Liquids, Glasses, and the Glass Transition: A Free-volume Approach
,” Advances in Chemical Physics
, Vol. 48
, I.
Prigogine
and S. A.
Rice
, eds., Wiley
, New York
, pp. 455
–525
.55.
Hong
, W.
, Zhao
, X.
, Zhou
, J.
, and Suo
, Z.
, 2008, “A Theory of Coupled Diffusion and Large Deformation in Polymeric Gels
,” J. Mech. Phys. Solids
, 56
, 1779
–1793
.56.
Hong
, W.
, Zhao
, X.
, and Suo
, Z.
, 2010, “Large Deformation and Electrochemistry of Polyelectrolyte Gels
,” J. Mech. Phys. Solids
, 58
, 558
–577
.57.
For more information, please see http://imechanica.org/node/2487http://imechanica.org/node/2487, 2008.
58.
Simo
, J. C.
, 1998, “Topics on the Numerical Analysis and Simulation of Plasticity
,” Handbook of Numerical Analysis
, Vol. VI
, P. G.
Ciarlet
and J. L.
Lions
, eds., North-Holland
, Amsterdam
, pp. 183
–499
.59.
Cottrell
, A. H.
, 1948, “Effect of Solute Atoms on the Behavior of Dislocations
,” Report of a Conference on Strength of Solids, The Physical Society, London
, H. H.
Wills
, ed., University of Bristol
, Bristol, UK
,pp. 30
–36
.60.
Zohdi
, T. I.
, and Wriggers
, P.
, 1999, “On the Effects of Microstress on Macroscopic Diffusion Processes
,” Acta Mech.
, 136
, 91
–107
.61.
Thomas
, N.
, and Windle
, A. H.
, 1977, “Discontinuous Shape Changes Associated With Case II Transport of Methanol in Thin Sheets of PMMA
,” Polymer
, 18
(11
), 1195.62.
Thomas
, N.
, and Windle
, A. H.
, 1978, “Transport of Methanol in Poly(methylmethacrylate)
,” Polymer
, 19
, 255
–265
.63.
Durning
, C. J.
, and Tabor
, M.
, 1986, “Mutual Diffusion in Concentrated Polymer Solutions Under a Small Driving Force
,” Macromolecules
, 19
, 2220
–2232
.64.
Fu
, T. Z.
, and Durning
, C. J.
, 1993, “Numerical Simulation of Case II Transport
,” AIChE J.
, 39
(6
), 1030
–1044
.65.
Cohen
, D. S.
, White
, A. B.
, and Witelski
, P. T.
, 1995, “Shock Formation in a Multidimensional Viscoelastic Diffusive System
,” SIAM J. Appl. Math.
, 55
(2
), 348
–368
.66.
Simo
, J. C.
, and Hughes
, T. J. R.
, 1998, “Computational Inelasticity
,” Interdisciplinary Applied Mathematics
, Vol. 7
, J. E.
Marsden
, S.
Wiggins
, and L.
Sirovich
, eds., Springer
, Berlin
.67.
Lustig
, S. R.
, Caruthers
, J. M.
, and Peppas
, N. A.
, 1992, “Continuum Thermodynamics and Transport Theory of Case II Transport
,” Chem. Eng. Sci.
, 47
, 3037
–3057
.68.
Flory
, P. J.
, 1941, “Thermodynamics of High Polymer Solutions
,” J. Chem. Phys.
, 9
(8
), 660
–661
.69.
Huggins
, M. L.
, 1941, “Solutions of Long Chain Compounds
,” J. Chem. Phys.
, 9
, 440
.70.
Hildebrand
, J. H.
, 1947, “The Entropy of Solution of Molecules of Different Size
,” J. Chem. Phys.
, 15
(5
) 225
–228
.71.
Malvern
, L. E.
, 1969, Introduction to the Mechanics of a Continuous Medium
, Prentice-Hall
, Englewood Cliffs, NJ
.72.
Coleman
, B. D.
, and Noll
, W.
, 1963, “The Thermodynamics of Elastic Materials With Heat Conduction and Viscosity
,” Arch. Ration. Mech. Anal.
, 13
, 167
–178
.73.
74.
Bird
, R. B.
, Armstrong
, R. C.
, and Hassager
, O.
, 1987, Dynamics of Polymeric Fluids, Vol. 1: Fluid Mechanics
, 2nd ed., Wiley
, New York
.75.
Bird
, R. B.
, Armstrong
, R. C.
, and Hassager
, O.
, 1987, Dynamics of Polymeric Fluids, Vol. 2: Kinetic Theory
, 2nd ed., John Wiley
, New York
.76.
El
Afif
, A.
, Grmela
, M.
, and Lebon
, G.
, 1999, “Rheology and Diffusion in Simple and Complex Fluids
,” J. Non-Newtonian Fluid Mech.
, 86
, 253
–275
.77.
Doi
, M.
, and Onuki
, A.
, 1992, “Dynamic Coupling Between Stress and Composition in Polymer Solutions and Blends
,” J. Phys. II
, 2
1631
–1656
.78.
Manero
, O.
, and Rodríguez
, R. F.
, 1999, “A Thermodynamic Description of Coupled Flow and Diffusion in a Viscoelastic Binary Mixture
,” J. Non-Equilib. Thermodyn.
, 24
(2
), 177
–195
.79.
Holzapfel
, G. A.
, 2001, Nonlinear Solid Mechanics: A Continuum Approach for Engineering
, Wiley
, New York
.80.
Carreau
, P. J.
, and Grmela
, M.
, 1991, “Conformation Tensor Rheological Models
,” Rheological Modelling: Thermodynamical and Statistical Approaches
(Lecture Notes in Physics, Vol. 381
), Springer
, Berlin
, pp. 126
–157
.81.
Marsden
, J. E.
, and Hughes
, T. J. R.
, 1994, Mathematical Foundations of Elasticity
, Dover
, New York
.82.
Beris
, A. N.
, and Edwards
, B. J.
, 1994, Thermodynamics of Flowing Systems: With Internal Microstructure
, (Oxford Engineering Science Series, Vol. 36
), Oxford University Press
, Oxford, UK
.83.
Zohdi
, T. I.
, Hutter
, K.
, and Wriggers
, P.
, 1999, “A Technique to Describe the Macroscopic Pressure Dependence of Diffusive Properties of Solid Materials Containing Heterogeneities
,” Comput. Mater. Sci.
, 15
, 69
–88
.Copyright © 2011
by American Society of Mechanical Engineers
You do not currently have access to this content.
