On a nonequilibrium thermodynamics of capillarity and phase
Author:
Morton E. Gurtin
Journal:
Quart. Appl. Math. 47 (1989), 129-145
MSC:
Primary 80A15; Secondary 35K55, 76R50
DOI:
https://doi.org/10.1090/qam/987902
MathSciNet review:
987902
Full-text PDF Free Access
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Additional Information
J. W. Gibbs, On the equilibrium of heterogeneous substances, Trans. Connecticut Acad. 3, 108β248 (1876), 343β524 (1878). Reprinted in: The Scientific Papers of J. Willard Gibbs, Vol. 1, Dover, New York, 1961
J. D. van der Waals, The thermodynamic theory of capillarity under the hypothesis of a continuous variation of density (in Dutch), Verhandel. Konink. Akad. Weten. Amsterdam (Sec. 1) Vol. 1, No. 8 (1893)
D. J. Korteweg, Sur la forme que prennent les Γ©quations des mouvement des fluides si lβon tientcomple des forces capillaires par des variations de densitΓ©, Arch. Neerl. Sci. Exactes Nat. Ser. 11, 6, 1β24 (1901)
J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys. 28, 258β267 (1958)
L. D. Landau and E. M. Lifschitz, Electrodynamics of Continuous Bodies, Pergamon, New York, 1960
- Bernard D. Coleman and Walter Noll, The thermodynamics of elastic materials with heat conduction and viscosity, Arch. Rational Mech. Anal. 13 (1963), 167β178. MR 153153, DOI https://doi.org/10.1007/BF01262690
J. W. Cahn, On spinodal decomposition, Acta Metallurgica 9, 795β801 (1961)
J. W. Cahn, On spinodal decomposition in cubic crystals, Acta Metallurgica 10, 179β183 (1962)
J. W. Cahn and J. E. Hilliard, Spinodal decomposition: a reprise, Acta Metallurgica 19, 151β161 (1971)
D. W. Hoffman and J. W. Cahn, A vector thermodynamics for anisotropic surfaces-1. Fundamentals and applications to plane surface junctions, Surface Sci. 31, 368β388 (1972)
J. W. Cahn and D. W. Hoffman, A vector thermodynamics for anisotropic surfaces-2. Curved and faceted surfaces, Acta Metallurgica 22, 1205β1214 (1974)
B. Widom, Structure and thermodynamics of interfaces, in Statistical Mechanics and Statistical Methods in Theory and Application, Plenum, New York, 1977
- J. Fernandez-Diaz and W. O. Williams, A generalized Stefan condition, Z. Angew. Math. Phys. 30 (1979), no. 5, 749β755 (English, with German summary). MR 553277, DOI https://doi.org/10.1007/BF01590684
- Morton E. Gurtin, Toward a nonequilibrium thermodynamics of two-phase materials, Arch. Rational Mech. Anal. 100 (1988), no. 3, 275β312. MR 918798, DOI https://doi.org/10.1007/BF00251518
- J. E. Dunn and J. Serrin, On the thermomechanics of interstitial working, Arch. Rational Mech. Anal. 88 (1985), no. 2, 95β133. MR 775366, DOI https://doi.org/10.1007/BF00250907
- Basil Nicolaenko and Bruno Scheurer, Low-dimensional behavior of the pattern formation Cahn-Hilliard equation, Trends in the theory and practice of nonlinear analysis (Arlington, Tex., 1984) North-Holland Math. Stud., vol. 110, North-Holland, Amsterdam, 1985, pp. 323β336. MR 817507, DOI https://doi.org/10.1016/S0304-0208%2808%2972727-0
- Amy Novick-Cohen, Energy methods for the Cahn-Hilliard equation, Quart. Appl. Math. 46 (1988), no. 4, 681β690. MR 973383, DOI https://doi.org/10.1090/S0033-569X-1988-0973383-6
J. A. Serrin, Continuum model for chemical mixture dynamics, Developments in Mechanics 13, Appl. Mech. Div., Am. Soc. Mech. Eng., 1985
- Wolf von Wahl, On the Cahn-Hilliard equation $uβ+\Delta ^2u-\Delta f(u)=0$, Delft Progr. Rep. 10 (1985), no. 4, 291β310. Mathematics and mathematical engineering (Delft, 1985). MR 839466
- Charles M. Elliott and Zheng Songmu, On the Cahn-Hilliard equation, Arch. Rational Mech. Anal. 96 (1986), no. 4, 339β357. MR 855754, DOI https://doi.org/10.1007/BF00251803
- Morton E. Gurtin, On the two-phase Stefan problem with interfacial energy and entropy, Arch. Rational Mech. Anal. 96 (1986), no. 3, 199β241. MR 855304, DOI https://doi.org/10.1007/BF00251907
- Amy Novick-Cohen and Lee A. Segel, Nonlinear aspects of the Cahn-Hilliard equation, Phys. D 10 (1984), no. 3, 277β298. MR 763473, DOI https://doi.org/10.1016/0167-2789%2884%2990180-5
- Zheng Songmu, Asymptotic behavior of solution to the Cahn-Hillard equation, Appl. Anal. 23 (1986), no. 3, 165β184. MR 870486, DOI https://doi.org/10.1080/00036818608839639
- Charles M. Elliott and Donald A. French, Numerical studies of the Cahn-Hilliard equation for phase separation, IMA J. Appl. Math. 38 (1987), no. 2, 97β128. MR 983721, DOI https://doi.org/10.1093/imamat/38.2.97
J. W. Gibbs, On the equilibrium of heterogeneous substances, Trans. Connecticut Acad. 3, 108β248 (1876), 343β524 (1878). Reprinted in: The Scientific Papers of J. Willard Gibbs, Vol. 1, Dover, New York, 1961
J. D. van der Waals, The thermodynamic theory of capillarity under the hypothesis of a continuous variation of density (in Dutch), Verhandel. Konink. Akad. Weten. Amsterdam (Sec. 1) Vol. 1, No. 8 (1893)
D. J. Korteweg, Sur la forme que prennent les Γ©quations des mouvement des fluides si lβon tientcomple des forces capillaires par des variations de densitΓ©, Arch. Neerl. Sci. Exactes Nat. Ser. 11, 6, 1β24 (1901)
J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys. 28, 258β267 (1958)
L. D. Landau and E. M. Lifschitz, Electrodynamics of Continuous Bodies, Pergamon, New York, 1960
B. D. Coleman and W. Noll, The thermodynamics of elastic materials with heat conduction and viscosity, Arch. Rat. Mech. Anal. 13, 167β178 (1963)
J. W. Cahn, On spinodal decomposition, Acta Metallurgica 9, 795β801 (1961)
J. W. Cahn, On spinodal decomposition in cubic crystals, Acta Metallurgica 10, 179β183 (1962)
J. W. Cahn and J. E. Hilliard, Spinodal decomposition: a reprise, Acta Metallurgica 19, 151β161 (1971)
D. W. Hoffman and J. W. Cahn, A vector thermodynamics for anisotropic surfaces-1. Fundamentals and applications to plane surface junctions, Surface Sci. 31, 368β388 (1972)
J. W. Cahn and D. W. Hoffman, A vector thermodynamics for anisotropic surfaces-2. Curved and faceted surfaces, Acta Metallurgica 22, 1205β1214 (1974)
B. Widom, Structure and thermodynamics of interfaces, in Statistical Mechanics and Statistical Methods in Theory and Application, Plenum, New York, 1977
J. Fernandez-Diaz and W. O. Williams, A generalized Stefan condition, Zeit. Angew. Math. Phys. 30, 749β755 (1979)
M. E. Gurtin, Toward a nonequilibrium thermodynamics of two phase materials, Arch. Rat. Mech. Anal. 100, 275β312 (1988)
J. E. Dunn and J. Serrin, On the thermomechanics of interstitial working, Arch. Rat. Mech. Anal. 88, 95β133 (1985)
B. Nicolaenko and B. Scheurer, Low-dimensional behavior of the pattern formation Cahn-Hilliard equation, in Trends in the Theory and Practice of Non-Linear Analysis, V. Lakschmikantham (ed.), Elsevier Science, New York, 1985
A. Novick-Cohen, Energy methods for the Cahn-Hilliard equation, IMA Preprint 157, Institute for Mathematics and its Applications, University of Minnesota, 1985
J. A. Serrin, Continuum model for chemical mixture dynamics, Developments in Mechanics 13, Appl. Mech. Div., Am. Soc. Mech. Eng., 1985
W. von Wahl, On the Cahn-Hilliard equation $uβ + {\Delta ^2}u - \Delta f\left ( u \right ) = 0$, Delft Progress Report 10, 291β310, Delft University, The Netherlands, 1985
C. M. Elliott and Z. Songmu, On the Cahn-Hilliard equation, Arch. Rat. Mech. Anal. 96, 339β357 (1986)
M. E. Gurtin, On the two-phase Stefan problem with interfacial energy and entropy, Arch. Rat. Mech. Anal. 96, 199β241 (1986)
A. Novick-Cohen and L. A. Segel, Nonlinear aspects of the Cahn-Hilliard equation, Physica (D) 10, 277β298 (1986)
Z. Songmu, Asymptotic behavior of the solution to the Cahn-Hilliard equation, Technical Report 25, Center for Applied Mathematics, Purdue University, 1986
C. M. Elliott and D. A. French, Numerical studies of the Cahn-Hilliard equation for phase separation, IMA J. Appl. Math. 38, 97β128 (1987)
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Article copyright:
© Copyright 1989
American Mathematical Society
