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**Thermodynamically consistent models of phase-field type for the kinetics of phase transitions.**
*(English)*
Zbl 0709.76001

Summary: A general framework is given for the phenomenological kinetics of phase transitions in which not only the order parameter but also the temperature may vary in time and space. Instead of a Ginzburg-Landau free energy functional, as used in formulating the Cahn-Hilliard equation, we use the analogous entropy functional. Model entropy functionals, and the kinetic equations resulting from them, are constructed for various cases: phase transitions with and without a critical point, and (in the former case) with or without a latent heat. The class considered is general enough to include the entropy functionals for the Ising model in mean- field approximation, the van der Waals fluid, and a simplified version of the density-functional theory of freezing. A case without critical point, for which the energy is conserved but the order parameter is not, provides a thermodynamically consistent derivation of the phase-field equations studied by e.g., G. Caginalp [Arch. Ration. Mech. Anal. 92, 205-245 (1986; Zbl 0608.35080)], and also leads in a natural way to the Lyapunov functional given by J. S. Langer [Models of pattern formation in first-order phase transitions, in: Directions in Condensed Matter Physics, 165-186 (1986)] for these equations; but the treatment also suggests that a modified version of the phase-field equations might provide a more realistic model of freezing.

### Keywords:

phenomenological kinetics of phase transitions; Ginzburg-Landau free energy functional; Cahn-Hilliard equation; entropy functional; van der Waals fluid; Lyapunov functional; phase-field equations### Citations:

Zbl 0608.35080
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\textit{O. Penrose} and \textit{P. C. Fife}, Physica D 43, No. 1, 44--62 (1990; Zbl 0709.76001)

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### References:

[1] | Bates, P.; Fife, P.C., Spectral comparison principles for the Cahn-Hilliard and phase-field equations, and the coarsening effect, Physica D, 43, (1990), in press · Zbl 0706.58074 |

[2] | Caginalp, G., Surface tension and supercooling in solidification theory, () |

[3] | Caginalp, G., An analysis of a phase field model of a free boundary, Arch. rat. mech. anal., 92, 205-245, (1986) · Zbl 0608.35080 |

[4] | Caginalp, G.; Fife, P.C., Higher-order phase field models and detailed anisotropy, Phys. rev. B, 34, 4940-4943, (1986) |

[5] | Caginalp, G.; Fife, P.C., Dynamics of layered interfaces arising from phase boundaries, SIAM J. appl. math., 48, 506-518, (1988) |

[6] | Cahn, J.W., Theory of crystal growth and interface motion in crystalline materials, Acta metall., 8, 554-562, (1960) |

[7] | Cahn, J.W., On spinodal decomposition, Acta metall., 9, 795-801, (1961) |

[8] | Cahn, J.W.; Hilliard, J.E., Free energy of a nonuniform system I. interfacial free energy, J. chem. phys., 28, 258-267, (1958) |

[9] | Collins, J.B.; Levine, H., Diffuse interface model of diffusion-limited crystal growth, Phys. rev. B, 31, 6119-6122, (1985) |

[10] | Fife, P.C., Mathematical aspects of reacting and diffusing systems, () · Zbl 0403.92004 |

[11] | Fife, P.C.; Gill, C.S., The phase-field description of mushy zones, Physica D, 35, 267-275, (1989) |

[12] | Fix, G.J., Phase field methods for free boundary problems, (), 580-589 · Zbl 0505.65061 |

[13] | Fonseca, I.; Tartar, L., The gradient theory of phase transitions for systems with two potential wells, (), 89-102 · Zbl 0676.49005 |

[14] | Halperin, B.I.; Hohenberg, P.C.; Ma, S.-K., Renormalization group methods for critical dynamics I. recursion relations and effects of energy conservation, Phys. rev. B, 10, 139-153, (1974) |

[15] | Hohenberg, P.C.; Halperin, B.I., Theory of dynamic critical phenomena, Rev. mod. phys., 49, 435-479, (1977) |

[16] | Landau, L.D.; Ginzburg, V.L., K teorii sverkhrovodimosti, (), 546-568, English translation: On the theory of superconductivity |

[17] | Landau, L.D.; Khalatnikov, I.M., Ob anomal’nom pogloshchenii zvuka vblizi tochek fazovo perekhoda vtorovo roda, (), 96, 626-633, (1954), English translation: On the anomalous absorption of sound near a second order transition point |

[18] | Langer, J.S., Models of pattern formation in first-order phase transitions, (), 165-186 |

[19] | Lebowitz, J.L.; Percus, J.K., Statistical thermodynamics of nonuniform fluids, J. math. phys., 4, 116-123, (1963) · Zbl 0112.22203 |

[20] | Modica, L., Gradient theory of phase transitions and minimal interface criterion, Arch. rat. mech. anal., 98, 123-142, (1987) · Zbl 0616.76004 |

[21] | Oleinik, O.A., A method of solution of the general Stefan problem, Sov. math. dokl., 1, 1350-1354, (1960) · Zbl 0131.09202 |

[22] | Ornstein, L.S.; Zernike, F., Konink. akad. weten. Amsterdam, 17, 793, (1914) |

[23] | Owen, N., Nonconvex variational problems with general singular perturbations, Trans. AMS, 310, 393-404, (1989) · Zbl 0718.34075 |

[24] | Rechtman, R.; Penrose, O., Continuity of temperature and derivation of the Gibbs canonical distribution in classical statistical mechanics, J. stat. phys., 19, 359-366, (1978) |

[25] | Ramakrishnan, T.V.; Yussouff, M., First-principles order-parameter theory of freezing, Phys. rev. B, 19, 2775-2794, (1979) |

[26] | Ruelle, D., Statistical mechanics, (1969), Benjamin New York · Zbl 0169.57502 |

[27] | Sachdev, S.; Nelson, D.R., Order in metallic glasses and icosahedral crystals, Phys. rev. B, 32, 4592-4606, (1985) |

[28] | Stell, G., Cluster expansions for classical systems in equilibrium, (), II-171-II-266 |

[29] | van der Waals, J.D.; Rowlinson, J.S., The thermodynamic theory of capillarity under the hypothesis of a continuous variation of density, Konink. akad. weten. Amsterdam, J. stat. phys., 20, No. 8, 197-244, (1979), English translation: (with commentary) · Zbl 1245.82006 |

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