Models for phase separation and their mathematics

Electronic Journal of Differential Equations

Volumn 2000, Issue , 2000, Pages

  Fif, Paul C ^a   Fife, Paul C ^a  

^a UNIVERSITY OF UTAH   (United States)

Author keywords

Cahn hilliard equation; Gradient flows; Phase separation; Phase field equations

Indexed keywords

EID: 54749104300     PISSN: 10726691     EISSN: 10726691     Source Type: Journal    
DOI: None     Document Type: Article

References (83)

1. N. D. Alikakos, P. W. Bates, and G. Fusco, Slow motion for the Cahn-Hilliard equation in one space dimension, J. Differential Equations, 90, 8f-f35 (f99f).
2. P. W. Bates and P. C. Fife, Spectral comparison principles related to coarsening for the Cahn-Hilliard and phase-field equations, Physica D 4-3, 335-348 (1990).
3. P. W. Bates and P. C. Fife, Dynamics of nucleation for the Cahn-Hilliard equation, SIAM J. Appl. Math. 53, 990-1008 (1993).
4. G. Caginalp, An analysis of a phase field model of a free boundary, Arch. Rat. Mech. Anal. 92, 205-245 (1986).
5. G. Caginalp, The role of microscopic anisotropy in the macroscopic behavior of a phase boundary, Ann. Phys. 172, 136-155 (1986).
6. G. Caginalp and P. C. Fife, Phase field methods for interfacial boundaries, Phys. Rev. B, 33, 7792-7794 (1986).
7. G. Caginalp and P. C. Fife, Higher order phase field models and detailed anisotropy, Phys. Rev. B, 34, 4940-4943.
8. G. Caginalp and P. C. Fife, Dynamics of layered interfaces arising from phase boundaries, SIAM J. Appl. Math. 48, 506-518 (1988).
9. J. W. Cahn, Theory of crystal growth and interface motion in crystalline materials, Acta Metallurgica 48, 554-562 (1960).
10. J. W. Cahn, On spinodal decomposition, Acta Metallurgica 9, 795-801 (1961).
11. J. W. Cahn, On spinodal decomposition in cubic crystals, Acta Metallurgica 10, 179-183 (1962).
12. J. W. Cahn, Phase separation by spinodal decomposition in isotropic systems, J. Chem. Physics 42, 93-99 (1965).
13. J. W. Cahn, Spinodal decomposition, Trans. Metallurg. Soc. of AIME 242 , 166-180 (1968).
14. J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system I. Interfacial free energy, J. Chem. Phys. 28, 258-267 (1958).
15. J. Carr, M. E. Gurtin, and M. Slemrod, Structured phase transitions on a finite interval, Arch. Rat. Mech. Anal. 86, 317-351 (1984).
16. J. B. Collins and H. Levine, Diffuse interface model of diffusion-limited crystal growth, Phys. Rev. B 31, 6119-6122 (1985). Erratum loc. cit. 33, 2020.
17. E. A. Coutsias and J. C. Neu, The aging of nuclei in a binary mixture, Physica D 12, 295-302 (1984).
18. W. A. Curtin and N. W. Ashcroft, Weighted-density-functional theory of inhomogeneous liquids and the freezing transition, Phys. Rev. A 32, 29092919 (1985).
19. J. C. Eilbeck, P. C. Fife, J. E. Furter, M. Grinfeld, and M. Mimura, Stationary states associated with phase separation in a pure material. I-The large latent heat case, Phys. Rev. Lett. A 139, 42-46 (1988).
20. J. C. Eilbeck, J. E. Furter, and M. Grinfeld, On a stationary state characterization of transition from spinodal decomposition to nucleation behaviour in the Cahn-Hilliard model of phase separation, Phys. Lett. A 135, 272 (1989).
21. C. M. Elliott, The Cahn-Hilliard model for the kinetics of phase separation, in Mathematical Models for Phase Change Problems, J. F. Rodrigues, éd., pp. 35-73, Int. Ser. of Numerical Math. Vol. 88, Birkhüser Verlag, Basel (1989).
22. C. M. Elliott and D. A. French, Numerical studies of the Cahn-Hilliard equation for phase separation, IM A Jour. Appl. Math 38, 97-128 (1987).
23. C. M. Elliott and S. Zheng, On the Cahn-Hilliard equation, Arch. Rat. Mech. Anal. 96, 339-357 (1986).
24. C. M. Elliott and S. Zheng, Global existence and stability of solutions to the phase field equations, in: Free Boundary Problems, K-H. Huffman and J. Sprekels, eds., Int. Ser. of Numerical Math. Vol. 95, Birkhüser Verlag, Basel (1990).
25. P. C. Fife and G. S. Gill, The phase-field description of mushy zones, Physica D 35, 267-275 (1989).
26. P. C. Fife and G. S. Gill, Phase-transition mechanisms for the phase-field model under internal heating, Phys. Rev. A 43, 843-851 (1991).
27. G. Fix, Phase field methods for free boundary problems, in: Free Boundary Problems, A. Fasano and M. Primicerio, eds., 580-589, Pitman, London (1983).
28. I. Fonseca and L. Tartar, The gradient theory of phase transitions for systems with two potential wells, Proc. Roy. Soc. Edinburgh A 111, 89-102 (1989).
29. C. Grant, Spinodal decomposition for the Cahn-Hilliard equation, Commun, in Partial Differential Equations 18, 453-490 (1993); see also: The dynamics of phase separation for the Cahn-Hilliard equation, PhD Thesis, Univ. of Utah, 1991.
30. J. D. Gunton and M. Droz, Introduction to the Theory of Metastable and Unstable States, Lecture Notes in Physics No. 183, Springer-Verlag, Berlin (1983).
31. M. E. Gurtin, On a theory of phase transitions with interfacial energy, Arch. Rat. Mech. Anal. 87, 187-212 (1985).
32. M. E. Gurtin, On the two-phase Stefan problem with interfacial energy and entropy, Arch. Rat. Mech. Anal. 96, 199-241 (1986).
33. M. E. Gurtin, On phase transitions with bulk, interfacial, and boundary energy, Arch. Rat. Mech. Anal. 96, 243-264 (1986).
34. M. E. Gurtin and H. Matano, On the structure of the equilibrium phase transition within the gradient theory of fluids, Quart. Appl. Math. 46 (1988), 301-317.
35. B. I. Halperin, P. C. Hohenberg, and S-K. Ma, Renormalization-group methods for critical dynamics: I. Recursion relations and effects of energy conservation, Phys. Rev. B 10, 139-153 (1974).
36. P. C. Hohenberg and B. I. Halperin, Theory of dynamic critical phenomena, Rev. Modern Physics 49, 435-485 (1977).
37. R. V. Kohn and P. Sternberg, Local minimizers and singular perturbation, Proc. Roy. Soc. Edin. 3, 69-84 (1989).
38. L. D. Landau and E. M. Lifshitz, Statistical Physics, 2nd Ed., Pergamon Press, Oxford (1969)
39. J. S. Langer, Theory of the condensation point, Annals of Physics 4 -, 108157 (1967).
40. J. S. Langer, Theory of spinodal decomposition in alloys. Ann. Physics 65 , 53-86 (1971).
41. J. S. Langer, Models of pattern formation in first-order phase transitions, pp. 164-186 in Directions in Condensed Matter Physics, World Science Publishers (1986).
42. Yu. S. Lipatov and V., V. Shilov, Spinodal decomposition in polymeric systems, Russ. Chem. Revs. 53 , 697-710 (1984).
43. S. Luckhaus and L. Modica, The Gibbs-Thompson relation within the gradient theory of phase transitions, Arch. Rat. Mech. Anal. 107 , 71-83 (1989).
44. L. Modica, The gradient theory of phase transitions and the minimal interface criterion, Arch. Rat. Mech. Anal. 98 , 123-142 (1987).
45. L. Modica, Gradient theory of phase transitions with boundary contact energy, Ann. Inst. H. Poincaré, Anal. Nonlinéare 5, 453-486 (1987).
46. B. Nicolaenko and B. Scheurer, Low-dimensional behavior of the patternforming Cahn-Hilliard equation, Trends in the Theory and Practice of Nonlinear Analysis, V. Lakshmikantham, ed., 323-336, Eisevier (1985).
47. B. Nicolaenko, B. Scheurer, and R. Temam, Inertial manifolds for the CahnHilliard model of phase transition, in: Ordinary and Partial Differential Equations, Proc. Dundee Conf. (1986), B. Sleeman and R. J. Jarvis, eds., Pitman Res. Notes in Math. Vol. 157 , Longman, N.Y. (1987).
48. B. Nicolaenko, B. Scheurer, and R. Temam, Some global dynamical properties of a class of pattern-forming equations, Comm. Par. Diff. Eqns. 14, 245-297 (1989).
49. A. Novick-Cohen, The nonlinear Cahn-Hilliard equation: transition from spinodal decomposition to nucleation behavior, J. Stat. Physics 38, 707-723 (1985).
50. A. Novick-Cohen and R. L. Pego, Stable patterns in a viscous diffusion equation, Trans. Amer. Math. Soc. 324, 331-351 (1991).
51. A. Novick-Cohen and L. A. Segel, Nonlinear aspects of the Cahn-Hilliard equation, Physica lOD, 277-298 (1984).
52. R. L. Pego, Front migration in the nonlinear Cahn-Hilliard equation, Proc. Roy. Soc. London A 422, 261-278 (1989).
53. O. Penrose, Statistical mechanics and the kinetics of phase separation, in: Material Instabilities in Continuum Mechanics and Related Mathematical Problems, J. M. Ball, ed., 373-394, Clarendon Press, Oxford (1988).
54. O. Penrose and P. C. Fife, Thermodynamically consistent models of phasefield type for the kinetics of phase transitions, Physica D 4-3, 44-62 (1990).
55. T. V. Ramakrishnan and M. Yussouff, First-principle order-parameter theory of freezing, Phys. Rev. B 19, 2775-2794 (1979).
56. S. Sachdev and D. R. Nelson, Order in metallic glasses and icosahedral crystals, Phys. Rev. B 32, 4592-4606 (1985).
57. V. P. Skripov and A. V. Skripov, Spinodal decomposition (phase transitions via unstable states). Soviet Physics Uspekhi 22, 389-410 (1979).
58. J. D. van der Waals, The thermodynamic theory of capillarity under the hypothesis of a continuous variation of density (translation of Dutch title): Konink. Akad. Weten. Amsterdam (sec. 1), Vol. 1, No. 8 (1893).
59. English trans.: J. S. Rowlinson, J. Stat. Phys. 20, 197-244 (1979).
60. S. Zheng, Asymptotic behavior of the solution to the Cahn-Hilliard equation, Appl. Anal. 23, 165-184 (1986).
61. J. W. Cahn and J. E. Taylor, Surface motion by surface diffusion, Acta metall. mater. 42, 1045-1063 (1994).
62. M. E. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance, Physica D 92, 178-192 (1996).
63. N. Alikakos, L. Bronsard, and G. Fusco, Slow motion in the gradient theory of phase transitions via energy and spectrum, Calculus of Variations 6, 39-66 (1998).
64. N. Alikakos, G. Fusco, The spectrum of the Cahn-Hilliard operator for generic interface in higher space dimensions, Indiana Mathematics Journal 42, 637-674 (1993).
65. N. Alikakos, G. Fusco, Slow dynamics for the Cahn-Hilliard equation in higher space dimensions. Part I: Spectral estimates, Comm. in P.D.E. 19, 1397-1447 (1994).
66. N. Alikakos, G. Fusco, Slow dynamics for the Cahn-Hilliard equation in higher space dimensions: the motion of bubbles, Arch. Rat. Mech. Anal. 141, 1-61 (1998).
67. N. Alikakos, G. Fusco, and M. Kowalczyk, Finite-dimensional dynamics and interfaces intersecting the boundary: equilibria and quasi-invariant manifolds, Indiana Mathematics Journal 45, 1119-1155 (1996).
68. P. Bates and J. Xun, Metastable patterns for the Cahn-Hilliard equation: Part I, J. Differential Equations 111, 421-457 (1995).
69. P. Bates and J. Xun, Metastable patterns for the Cahn-Hilliard equation: Part II, J. Differential Equations 117, 165-216 (1995).
70. Xinfu Chen, Global asymptotic limit of solutions of the Cahn-Hilliard equation, J. Differential Geometry 44, 262-311 (1996).
71. K. Elder, T. Rogers, and R. Desai, Early stages of spinodal decomposition for the Cahn-Hilliard-Cook model of phase separation. Phys. Rev. B 38, 4725-4739 (1988).
72. D. Eyre, Coarsening dynamics for solutions of the Cahn-Hilliard equation in one dimension, preprint.
73. D. Eyre, Systems of Cahn-Hilliard equations, SIAM J. Appl. Math. 53, 1686-1712 (1993).
74. D. Eyre, Cascades of Spinodal Decomposition in the Ternary Cahn-Hilliard Equations, to appea
75. P. C. Fife, Pattern formation in gradient systems, to appear as a chapter in Handbook for Dynamical Systems. Ill: Toward Applications, B. Fiedler and N. Kopell, eds., Eisevier.
76. S. Maier-Paape and T. Wanner, Spinodal decomposition for the CahnHilliard equation in higher dimensions. Part I: Probability and wavelength estimate, Comm. Math. Physics 195, 435-464 (1998).
77. S. Maier-Paape and T. Wanner, Spinodal decomposition for the CahnHilliard equation in higher dimensions: Nonlinear dynamics, Arch. Rat. Mech. Anal. 151, 187-219 (2000).
78. E. Sander and T. Wanner, Monte Carlo simulations for spinodal decomposition, J. Stat. Phys. 95, 925-948 (1999).
79. E. Sander and T. Wanner, Unexpectedly linear behavior for the CahnHilliard equation, SIAM J. Appl. Math. 60, 2182-2202 (2000).
80. P. Bates and G. Fusco, Equilibria with many nuclei for the Cahn-Hilliard equation, J. Differential Equations 160, 283-356 (2000).
81. C. Charach and P. Fife, On thermodynamically consistent schemes for phase field equations, Open Systems and Information Dynamics 5, 99-123 (1998).
82. E. Fried and M. E. Gurtin, Continuum theory of thermally induced phase transitions based on an order parameter. Physica D 68, 326-343 (1993).
83. A. P. Umantsev, Thermodynamic stability of phases and transition kinetics under adiabatic conditions. J. Chem. PHys. 96, 189-200 (1993).