Article Contents
Article Contents

# Anisotropic phase field equations of arbitrary order

• We derive a set of higher order phase field equations using a microscopic interaction Hamiltonian with detailed anisotropy in the interactions of the form $a_{0}+\delta\sum_{n=1}^{N}{a_{n}\cos( 2n\theta) + b_{n}\sin( 2n\theta) }$ where $\theta$ is the angle with respect to a fixed axis, and $\delta$ is a parameter. The Hamiltonian is expanded using complex Fourier series, and leads to a free energy and phase field equation with arbitrarily high order derivatives in the spatial variable. Formal asymptotic analysis is performed on these phase field equation in terms of the interface thickness in order to obtain the interfacial conditions. One can capture $2N$-fold anisotropy by retaining at least $2N^{th}$ degree phase field equation. We derive, in the limit of small $\delta,$ the classical result $( T-T_{E} ) [s]_{E}=-\kappa {\sigma( \theta ) + \sigma^{''}( \theta) }$ where $T-T_{E}$ is the difference between the temperature at the interface and the equilibrium temperature between phases, $[s]_{E}$ is the entropy difference between phases, $\sigma$ is the surface tension and $\kappa$ is the curvature. If there is only one mode in the anisotropy [i.e., the sum contains only one term: $A_{n}\cos( 2n\theta)$] then this identity is exact (valid for any magnitude of $\delta$) if the surface tension is interpreted as the sharp interface limit of excess free energy obtained by the solution of the $2N^{th}$ degree differential equation. The techniques rely on rewriting the sums of derivatives using complex variables and combinatorial identities, and performing formal asymptotic analyses for differential equations of arbitrary order.
Mathematics Subject Classification: Primary 35R35; Secondary 35R50, 82C24.

 Citation:

•  [1] G. Lamé and B. P. Clapeyron, Memoire sur la solidification par refroiddissement d'un globe solide, Ann. Chem. Physics, 47 (1831), 250-256. [2] J. Stefan, Uber einige probleme der theorie der warmeleitung, S.-B Wien Akad. Mat. Natur, 98 (1889), 173-484. [3] L. A. Caffarelli, Continuity of the temperature in the Stefan problem, Indiana Univ. Math. J., 28 (1979), 53-70.doi: 10.1512/iumj.1979.28.28004. [4] A. M. Meirmanov, On a classical solution of the multidimensional Stefan problem for quasi-linear parabolic equations, Math. Sbornik, 112 (1980), 170-192. [5] J. W. Gibbs, "Collected Works," Yale University Press, New Haven, 1948. [6] B. Chalmers, "Principles of Solidification," John Wiley & Sons, Inc., 1964. [7] X. Chen and F. Reitich, Local existence and uniqueness of solution of the Stefan problem, J. Math. Anal. Appl., 162 (1992), 350-362. [8] E. Radkevitch, The Gibbs-Thomson correction and conditions for the solutions of modified Stefan Problem, Sov. Math. Doklady, 43 (1991), 1. [9] S. Luckhaus, Solutions for the two-phase Stefan problem with Gibbs-Thomson law for the melting temperature, Euro. J. Appl. Math, 1 (1990), 101-111.doi: 10.1017/S0956792500000103. [10] J. Duchon and R. Robert, Evolution d'une interface par capillarite et diffusion de volume, Ann. Inst. Henri Poincare, Analyse non lineaire, 1 (1984), 361-378. [11] Y. G. Chen, Y. Giga and S. Goto, Uniqueness and existence of viscosity solution of generalized mean curvature equations, J. Diff. Geom, 33 (1991), 749-786. [12] C. Evans and J. Spruck, Motion by mean curvature, J. Diff. Geom, 33 (1991), 635-681. [13] H. M. Soner, Motion of a set by the curvature of its boundary, J. Diff. Geom, 101 (1993), 313-372. [14] O. A. Oleinik, A method of solution of the general Stefan problem, Sov. Math. Dokl., 1 (1960), 1350-1354. [15] L. D. Landau and E. M. Lifshitz, "Statistical Physics (Part 1)," 3rd edition, Pergamon, New York, 1980. [16] P. C. Hohenberg and B. I. Halperin, Theory of dynamics in critical phenomena, Rev. Mod. Phys., 49 (1977), 435-480.doi: 10.1103/RevModPhys.49.435. [17] J. W. Cahn and J. H. Hilliard, Free energy of a non-uniform system I, Interfacial free energy, J. of Chem. Physics, 28 (1957), 258-267.doi: 10.1063/1.1744102. [18] S. M. Allen and J. W. Cahn, A microscopic theory of antiphase boundary motion and its application to antiphase domain coarsening, Acta. Metal. Mater., 27 (1979), 1084-1095.doi: 10.1016/0001-6160(79)90196-2. [19] J. Langer, Theory of condensation point, Annals of Physics, 41 (1967), 108-157.doi: 10.1016/0003-4916(67)90200-X. [20] G. Caginalp, The role of microscopic physics in the macroscopic behavior of a phase boundary, Annals of Physics, 172 (1986), 136-155.doi: 10.1016/0003-4916(86)90022-9. [21] G. Caginalp and P. Fife, Higher order phase field models and detailed anisotropy, Phys. Review B, 34 (1986), 4940-4943.doi: 10.1103/PhysRevB.34.4940. [22] G. Caginalp, A microscopic derivation of macroscopic sharp interface problems involving phase transitions, J. of Statistical Physics, 59 (1990), 869-884.doi: 10.1007/BF01025855. [23] G. Caginalp, "The Limiting Behavior of a Free Boundary in the Phase Field Model," CMU Research Report, 82, 1982. [24] G. Caginalp, An analysis of a phase field model of a free boundary, Arch. Rational Mech. Anal., 92 (1986), 205. [25] G. Caginalp, Mathematical models of phase boundaries, in "Material Instabilities in Continuum Mechanics: Related Mathematical Problems" (J. M. Ball, ed.), Lecture at Heriot-Watt University, (1985). [26] G. Caginalp and E. Socolovsky, Efficient computation of a sharp interface by spreading via phase field methods, Applied Math. Letters, 2 (1989), 117-120.doi: 10.1016/0893-9659(89)90002-5. [27] X. Chen, G. Caginalp and C. Eck, A rapidly converging phase field model, Discrete and Continuous Dynamical Systems, 15 (2006), 1017-1034.doi: 10.3934/dcds.2006.15.1017. [28] G. Caginalp and X. Chen, Convergence of the phase field model to its sharp interface limits, Euro. J. of Applied Mathematics, 9 (1998), 417-445.doi: 10.1017/S0956792598003520. [29] H. M. Soner, Convergence of the phase field equation to the Mullins-Sekerka problem with kinetic undercooling, Arch. Rational. Mech. Anal., 131 (1995), 139-197.doi: 10.1007/BF00386194. [30] B. Stoth, Convergence of Cahn-Hilliard equation to the Mullins-Sekerka problem in spherical symmetry, J. of Differential Equations, 125 (1996), 154-183.doi: 10.1006/jdeq.1996.0028. [31] X. Chen, The Hele-Shaw Problem as area-preserving curve shortening motions, Arch. Rat. Mech. Anal., 123 (1993), 117-151.doi: 10.1007/BF00695274. [32] X. Chen, Spectrums of the Allen-Cahn, Cahn-Hilliard, and phase field equations for generic interfaces, Comm. Partial Differential Equations, 19 (1994), 1371-1395.doi: 10.1080/03605309408821057. [33] S. Gatti, M. Grasselli and V. Pata, Exponential attractors for a conserved phase-field system, Physica D, 189 (2004), 31-48. [34] M. Grasselli and V. Pata, Attractor for a conserved phase-field system with hyperbolic heat conduction, Mathematical Methods in the Applied Sciences, 27 (2004), 1917-1934.doi: 10.1002/mma.533. [35] A. Miranville and R. Quintanilla, A generalization of the Caginalp phase-field system based on the Cattaneo Law, Nonlinear Analysis, 71 (2009), 2278-2290.doi: 10.1016/j.na.2009.01.061. [36] G. Schimperna and U. Stefanelli, A quasi-stationary phase field model with micro-movements, Applied Mathematics and Optimization, 50 (2004), 67-86. [37] L. Cherfils, S. Gatti and A. Miranville, Existence of global solutions to the Caginalp phase field system with dynamic boundary conditions and singular potentials, J. Math. Anal. Appl., 343 (2008), 557-566.doi: 10.1016/j.jmaa.2008.01.077. [38] N. Kenmochi and K. Shirakawa, Stability for steady state patterns in phase field dynamics associated with total variation energies, Nonlinear Analysis, Theory, Methods and Applications, 53 (2003), 425-440. [39] C. G. Gal and M. Grasselli, On the asymptotic behavior of Caginalp systems with dynamic boundary conditions, Communications on Pure and Applied Analysis, 9 (2009), 689-710. [40] C. G. Gal, M. Grasselli and A. Miranville, Robust exponential attractors for singularly perturbed equation with dynamical boundary conditions, NoDEA Nonlinear Differential Equations and Applications, 15 (2008), 535-556.doi: 10.1007/s00030-008-7029-9. [41] M. E. Glicksman and N. Singh, Effects of crystal-melt interfacial energy anisotropy on dentritic morphology and growth kinetics, J. of Crystal Growth, 98 (1989), 277-284.doi: 10.1016/0022-0248(89)90142-5. [42] E. R. Rubinstein and M. E. Glicksman, Dentritic growth kinetics and structure, J. of Crystal Growth, 112 (1991), 84-96.doi: 10.1016/0022-0248(91)90914-Q. [43] M. Muschol, D. Liu and H. Z. Cummins, Surface tension measurements of succinonitrile and pivalic acid: Comparison with microscopic solvability theory, Phys. Rev. A, 46 (1992), 1038-1050.doi: 10.1103/PhysRevA.46.1038. [44] S. Liu, R. E. Napolitano and R. Trivedi, Measurement of anisotropy of crystal-melt interfacial energy for a binary Al-Cu alloy, Acta. Mater., 49 (2001), 42710-4276.doi: 10.1016/S1359-6454(01)00306-8. [45] R. E. Napolitano, S. Liu and R. Trivedi, Experimental measurement of anisotropy in the interfacial free energy, Interface Science, 10 (2002), 217-232.doi: 10.1023/A:1015884415896. [46] J. Q. Broughton and G. H. Gilmer, Molecular dynamics investigation of crystal-fluid interface. VI Excess surface free energies of crystal liquid systems, J. of Chem. Physics, 84 (1986), 5759-5768.doi: 10.1063/1.449884. [47] R. L. Davidcheck and B. B. Laird, Direct calculation of the hard-sphere crystal-melt interfacial free energy, Phys. Rev. Lett., 85 (2000), 4751-4754.doi: 10.1103/PhysRevLett.85.4751. [48] J. J. Hoyt, M. Asta and A. Karma, Method for computing the anisotropy of the solid-liquid interfacial free energy, Phys. Rev. Lett., 86 (2001), 5530-5533.doi: 10.1103/PhysRevLett.86.5530. [49] R. L. Davidcheck and B. B. Laird, Direct calculation of interfacial free energies for continuous potentials: An application to Lennard-Jones systems, J. of Chem. Physics, 118 (2003), 7651.doi: 10.1063/1.1563248. [50] D. Y. Sun and et.al., Crystal-melt interfacial free energies in hcp metals: A molecular dynamics study of Mg, Phys. Rev. B, 73 (2008), 24116-24127.doi: 10.1103/PhysRevB.73.024116. [51] M. Amini and B. B. Laird, Crystal-melt interfacial free energy of binary hard-sphere from capillary fluctuation anisotropy, Phys. Rev. B, 78 (2008), 144112-144119.doi: 10.1103/PhysRevB.78.144112. [52] X. Feng and B. B. Laird, Calculation of the crystal-melt interfacial free energy of succinonitrile from molecular simulation, J. Chem. Physics, 124 (2006), 44707-44711.doi: 10.1063/1.2149859. [53] J. R. Morris and X. Y. Song, Anisotropic free energy of the Lennard-Jones crystal-melt interface, J. Chem. Physics, 119 (2003), 3920-3925.doi: 10.1063/1.1591725. [54] G. Wulff, Zur frage der geschwindigkeit des wachstums und der auflö der kristallflä zeitschr, Zeitschr. F. Kristallog., 34 (1901), 449-530. [55] C. Herring, in "Structure and Properties of Solid Surfaces" (R. Gomer), University of Chicago Press, Chicago, (1952), 5. [56] D. W. Hoffman and J. W. Cahn, A vector thermodynamics for anisotropic surfaces, Surface Science, 31 (1972), 368-388.doi: 10.1016/0039-6028(72)90268-3. [57] J. W. Cahn and D. W. Hoffman, Vector thermodynamics for anisotropic surfaces 2. Curved and faceted surfaces, Acta. Metall., 22 (1974), 1205.doi: 10.1016/0001-6160(74)90134-5. [58] R. Kobayashi, Modelling and numerical simulations of dentritic crystal growth, Physica D, 63 (1993), 410-423.doi: 10.1016/0167-2789(93)90120-P. [59] J. E. Taylor, Mean curvature and weighted mean curvature, Acta. Metall. Mater., 40 (1992), 1475-1485.doi: 10.1016/0956-7151(92)90091-R. [60] A. A. Wheeler and G. B. McFadden, On the notion of $\xi$-vector and stress tensor for a general class of anisotropic diffuse interface models, Proc. R. Soc. London Ser. A, 453 (1997), 1611-1630.doi: 10.1098/rspa.1997.0086. [61] B. Nestler and A. A. Wheeler, Anisotropic multi-phase field model: Interfaces and junctions, Phys. Rev. E., 57 (1998), 2602-2609.doi: 10.1103/PhysRevE.57.2602. [62] A. D. J. Haymet and D. W. Oxtoby, A molecular theory of the solid-liquid interface, J. of Chem. Physics, 74 (1981), 2559-2565.doi: 10.1063/1.441326. [63] D. W. Oxtoby and A. D. J. Haymet, A molecular theory of solid-liquid interface. II. Study of bcc crystal-melt interfaces, J. of Chem. Physics, 76 (1982), 6262-6272.doi: 10.1063/1.443029. [64] W. H. Shih, Z.Q. Wang, X. C. Zeng and D. Stroud, Ginzburg-Landau theory for the solid-liquid interface of bcc elements, Phys. Rev. A, 35 (1987), 2611-2618.doi: 10.1103/PhysRevA.35.2611. [65] W. A. Curtin, Density functional theory of crystal melt interfaces, Phys. Rev. B, 39 (1989), 6775-6791.doi: 10.1103/PhysRevB.39.6775. [66] K.-A. Wu, A. Karma, J. J. Hoyt and M. Asta, Ginzburg-Landau theory of crystalline anisotropy for bcc liquid interfaces, Phys. Rev. B, 73 (2006), 94101-94107.doi: 10.1103/PhysRevB.73.094101. [67] S. Majaniemi and N. Provatas, Deriving surface energy anisotropy for phenomological phase-field models of solidification, Phys. Rev. E, 79 (2009), 11607-11618.doi: 10.1103/PhysRevE.79.011607. [68] L. Dobrushin, Roman Kotecký and S. Shlosman, "Wulff Construction: A Global Shape from Local Interactions," American Mathematical Society, 1992. [69] Markos A. Katsoulakis and Panagiotis E.Sounganidis, Generalized motion by mean curvature as macroscopic limit of stochastic ising models with long range interactions and Glauber dynamics, Communications in Mathematical Physics, 169 (1995), 61-97.doi: 10.1007/BF02101597. [70] Markos A. Katsoulakis and Panagiotis E.Sounganidis, Stochastic ising models and anistropic front propagation, 87 (1997), 63-90. [71] Herbert Spohn, Interface motion in models with stochastic dynamics, Journal of Statistical Physics, 71 (1993), 1081-1132.doi: 10.1007/BF01049962. [72] Giambattista Giacomin and Joel Lebowitz, Phase segregation dynamics in particle sytems with long range interactions II, SIAM Journal of Applied Mathematics, 58 (1998), 1707-1729.doi: 10.1137/S0036139996313046. [73] T. A. Abinandan and F. Haider, An extended Cahn Hilliard model for interfaces with cubic anisotropy, Philosophical Magazine, 81 (2001), 2457-2479.doi: 10.1080/01418610110038420. [74] S. M. Wise, J. S. Kim and J. S. Lowengrub, Solving the regularized, strongly anisotropic Cahn-Hilliard equation by an adaptive nonlinear multigrid method, J. Comput. Phys., 226 (2007), 414-446.doi: 10.1016/j.jcp.2007.04.020. [75] S. Torabi, J. S. Lowengrub, A. Voigt and S. M. Wise, A new phase-field model for strongly anisotropic systems, Proc. R. Soc. A, 465 (2009), 1337-1359,doi: 10.1098/rspa.2008.0385. [76] B. J. Spencer, Asymptotic solution for the equilibrium crystal shape with small corner energy regularization, Physical Review, 69 (2004), 2557-2567.