April  2011, 4(2): 311-350. doi: 10.3934/dcdss.2011.4.311

Anisotropic phase field equations of arbitrary order

1. 

Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260

2. 

Mathematics Department, University of Pittsburgh, Pittsburgh, PA 15260, United States

Received  April 2009 Revised  June 2009 Published  November 2010

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.
Citation: G. Caginalp, Emre Esenturk. Anisotropic phase field equations of arbitrary order. Discrete & Continuous Dynamical Systems - S, 2011, 4 (2) : 311-350. doi: 10.3934/dcdss.2011.4.311
References:
[1]

G. Lamé and B. P. Clapeyron, Memoire sur la solidification par refroiddissement d'un globe solide,, Ann. Chem. Physics, 47 (1831), 250. Google Scholar

[2]

J. Stefan, Uber einige probleme der theorie der warmeleitung,, S.-B Wien Akad. Mat. Natur, 98 (1889), 173. Google Scholar

[3]

L. A. Caffarelli, Continuity of the temperature in the Stefan problem,, Indiana Univ. Math. J., 28 (1979), 53. doi: 10.1512/iumj.1979.28.28004. Google Scholar

[4]

A. M. Meirmanov, On a classical solution of the multidimensional Stefan problem for quasi-linear parabolic equations,, Math. Sbornik, 112 (1980), 170. Google Scholar

[5]

J. W. Gibbs, "Collected Works,", Yale University Press, (1948). Google Scholar

[6]

B. Chalmers, "Principles of Solidification,", John Wiley & Sons, (1964). Google Scholar

[7]

X. Chen and F. Reitich, Local existence and uniqueness of solution of the Stefan problem,, J. Math. Anal. Appl., 162 (1992), 350. Google Scholar

[8]

E. Radkevitch, The Gibbs-Thomson correction and conditions for the solutions of modified Stefan Problem,, Sov. Math. Doklady, 43 (1991). Google Scholar

[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. doi: 10.1017/S0956792500000103. Google Scholar

[10]

J. Duchon and R. Robert, Evolution d'une interface par capillarite et diffusion de volume,, Ann. Inst. Henri Poincare, 1 (1984), 361. Google Scholar

[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. Google Scholar

[12]

C. Evans and J. Spruck, Motion by mean curvature,, J. Diff. Geom, 33 (1991), 635. Google Scholar

[13]

H. M. Soner, Motion of a set by the curvature of its boundary,, J. Diff. Geom, 101 (1993), 313. Google Scholar

[14]

O. A. Oleinik, A method of solution of the general Stefan problem,, Sov. Math. Dokl., 1 (1960), 1350. Google Scholar

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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. doi: 10.1016/0001-6160(79)90196-2. Google Scholar

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G. Caginalp and P. Fife, Higher order phase field models and detailed anisotropy,, Phys. Review B, 34 (1986), 4940. doi: 10.1103/PhysRevB.34.4940. Google Scholar

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G. Caginalp, A microscopic derivation of macroscopic sharp interface problems involving phase transitions,, J. of Statistical Physics, 59 (1990), 869. doi: 10.1007/BF01025855. Google Scholar

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G. Caginalp, "The Limiting Behavior of a Free Boundary in the Phase Field Model,", CMU Research Report, 82 (1982). Google Scholar

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G. Caginalp, An analysis of a phase field model of a free boundary,, Arch. Rational Mech. Anal., 92 (1986). Google Scholar

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G. Caginalp, Mathematical models of phase boundaries,, in, (1985). Google Scholar

[26]

G. Caginalp and E. Socolovsky, Efficient computation of a sharp interface by spreading via phase field methods,, Applied Math. Letters, 2 (1989), 117. doi: 10.1016/0893-9659(89)90002-5. Google Scholar

[27]

X. Chen, G. Caginalp and C. Eck, A rapidly converging phase field model,, Discrete and Continuous Dynamical Systems, 15 (2006), 1017. doi: 10.3934/dcds.2006.15.1017. Google Scholar

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G. Caginalp and X. Chen, Convergence of the phase field model to its sharp interface limits,, Euro. J. of Applied Mathematics, 9 (1998), 417. doi: 10.1017/S0956792598003520. Google Scholar

[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. doi: 10.1007/BF00386194. Google Scholar

[30]

B. Stoth, Convergence of Cahn-Hilliard equation to the Mullins-Sekerka problem in spherical symmetry,, J. of Differential Equations, 125 (1996), 154. doi: 10.1006/jdeq.1996.0028. Google Scholar

[31]

X. Chen, The Hele-Shaw Problem as area-preserving curve shortening motions,, Arch. Rat. Mech. Anal., 123 (1993), 117. doi: 10.1007/BF00695274. Google Scholar

[32]

X. Chen, Spectrums of the Allen-Cahn, Cahn-Hilliard, and phase field equations for generic interfaces,, Comm. Partial Differential Equations, 19 (1994), 1371. doi: 10.1080/03605309408821057. Google Scholar

[33]

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[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. doi: 10.1002/mma.533. Google Scholar

[35]

A. Miranville and R. Quintanilla, A generalization of the Caginalp phase-field system based on the Cattaneo Law,, Nonlinear Analysis, 71 (2009), 2278. doi: 10.1016/j.na.2009.01.061. Google Scholar

[36]

G. Schimperna and U. Stefanelli, A quasi-stationary phase field model with micro-movements,, Applied Mathematics and Optimization, 50 (2004), 67. Google Scholar

[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. doi: 10.1016/j.jmaa.2008.01.077. Google Scholar

[38]

N. Kenmochi and K. Shirakawa, Stability for steady state patterns in phase field dynamics associated with total variation energies,, Nonlinear Analysis, 53 (2003), 425. Google Scholar

[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. Google Scholar

[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. doi: 10.1007/s00030-008-7029-9. Google Scholar

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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. doi: 10.1016/0022-0248(89)90142-5. Google Scholar

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E. R. Rubinstein and M. E. Glicksman, Dentritic growth kinetics and structure,, J. of Crystal Growth, 112 (1991), 84. doi: 10.1016/0022-0248(91)90914-Q. Google Scholar

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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. doi: 10.1103/PhysRevA.46.1038. Google Scholar

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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. doi: 10.1016/S1359-6454(01)00306-8. Google Scholar

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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. doi: 10.1063/1.449884. Google Scholar

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[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. doi: 10.1103/PhysRevB.73.024116. Google Scholar

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show all references

References:
[1]

G. Lamé and B. P. Clapeyron, Memoire sur la solidification par refroiddissement d'un globe solide,, Ann. Chem. Physics, 47 (1831), 250. Google Scholar

[2]

J. Stefan, Uber einige probleme der theorie der warmeleitung,, S.-B Wien Akad. Mat. Natur, 98 (1889), 173. Google Scholar

[3]

L. A. Caffarelli, Continuity of the temperature in the Stefan problem,, Indiana Univ. Math. J., 28 (1979), 53. doi: 10.1512/iumj.1979.28.28004. Google Scholar

[4]

A. M. Meirmanov, On a classical solution of the multidimensional Stefan problem for quasi-linear parabolic equations,, Math. Sbornik, 112 (1980), 170. Google Scholar

[5]

J. W. Gibbs, "Collected Works,", Yale University Press, (1948). Google Scholar

[6]

B. Chalmers, "Principles of Solidification,", John Wiley & Sons, (1964). Google Scholar

[7]

X. Chen and F. Reitich, Local existence and uniqueness of solution of the Stefan problem,, J. Math. Anal. Appl., 162 (1992), 350. Google Scholar

[8]

E. Radkevitch, The Gibbs-Thomson correction and conditions for the solutions of modified Stefan Problem,, Sov. Math. Doklady, 43 (1991). Google Scholar

[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. doi: 10.1017/S0956792500000103. Google Scholar

[10]

J. Duchon and R. Robert, Evolution d'une interface par capillarite et diffusion de volume,, Ann. Inst. Henri Poincare, 1 (1984), 361. Google Scholar

[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. Google Scholar

[12]

C. Evans and J. Spruck, Motion by mean curvature,, J. Diff. Geom, 33 (1991), 635. Google Scholar

[13]

H. M. Soner, Motion of a set by the curvature of its boundary,, J. Diff. Geom, 101 (1993), 313. Google Scholar

[14]

O. A. Oleinik, A method of solution of the general Stefan problem,, Sov. Math. Dokl., 1 (1960), 1350. Google Scholar

[15]

L. D. Landau and E. M. Lifshitz, "Statistical Physics (Part 1),", 3rd edition, (1980). Google Scholar

[16]

P. C. Hohenberg and B. I. Halperin, Theory of dynamics in critical phenomena,, Rev. Mod. Phys., 49 (1977), 435. doi: 10.1103/RevModPhys.49.435. Google Scholar

[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. doi: 10.1063/1.1744102. Google Scholar

[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. doi: 10.1016/0001-6160(79)90196-2. Google Scholar

[19]

J. Langer, Theory of condensation point,, Annals of Physics, 41 (1967), 108. doi: 10.1016/0003-4916(67)90200-X. Google Scholar

[20]

G. Caginalp, The role of microscopic physics in the macroscopic behavior of a phase boundary,, Annals of Physics, 172 (1986), 136. doi: 10.1016/0003-4916(86)90022-9. Google Scholar

[21]

G. Caginalp and P. Fife, Higher order phase field models and detailed anisotropy,, Phys. Review B, 34 (1986), 4940. doi: 10.1103/PhysRevB.34.4940. Google Scholar

[22]

G. Caginalp, A microscopic derivation of macroscopic sharp interface problems involving phase transitions,, J. of Statistical Physics, 59 (1990), 869. doi: 10.1007/BF01025855. Google Scholar

[23]

G. Caginalp, "The Limiting Behavior of a Free Boundary in the Phase Field Model,", CMU Research Report, 82 (1982). Google Scholar

[24]

G. Caginalp, An analysis of a phase field model of a free boundary,, Arch. Rational Mech. Anal., 92 (1986). Google Scholar

[25]

G. Caginalp, Mathematical models of phase boundaries,, in, (1985). Google Scholar

[26]

G. Caginalp and E. Socolovsky, Efficient computation of a sharp interface by spreading via phase field methods,, Applied Math. Letters, 2 (1989), 117. doi: 10.1016/0893-9659(89)90002-5. Google Scholar

[27]

X. Chen, G. Caginalp and C. Eck, A rapidly converging phase field model,, Discrete and Continuous Dynamical Systems, 15 (2006), 1017. doi: 10.3934/dcds.2006.15.1017. Google Scholar

[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. doi: 10.1017/S0956792598003520. Google Scholar

[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. doi: 10.1007/BF00386194. Google Scholar

[30]

B. Stoth, Convergence of Cahn-Hilliard equation to the Mullins-Sekerka problem in spherical symmetry,, J. of Differential Equations, 125 (1996), 154. doi: 10.1006/jdeq.1996.0028. Google Scholar

[31]

X. Chen, The Hele-Shaw Problem as area-preserving curve shortening motions,, Arch. Rat. Mech. Anal., 123 (1993), 117. doi: 10.1007/BF00695274. Google Scholar

[32]

X. Chen, Spectrums of the Allen-Cahn, Cahn-Hilliard, and phase field equations for generic interfaces,, Comm. Partial Differential Equations, 19 (1994), 1371. doi: 10.1080/03605309408821057. Google Scholar

[33]

S. Gatti, M. Grasselli and V. Pata, Exponential attractors for a conserved phase-field system,, Physica D, 189 (2004), 31. Google Scholar

[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. doi: 10.1002/mma.533. Google Scholar

[35]

A. Miranville and R. Quintanilla, A generalization of the Caginalp phase-field system based on the Cattaneo Law,, Nonlinear Analysis, 71 (2009), 2278. doi: 10.1016/j.na.2009.01.061. Google Scholar

[36]

G. Schimperna and U. Stefanelli, A quasi-stationary phase field model with micro-movements,, Applied Mathematics and Optimization, 50 (2004), 67. Google Scholar

[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. doi: 10.1016/j.jmaa.2008.01.077. Google Scholar

[38]

N. Kenmochi and K. Shirakawa, Stability for steady state patterns in phase field dynamics associated with total variation energies,, Nonlinear Analysis, 53 (2003), 425. Google Scholar

[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. Google Scholar

[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. doi: 10.1007/s00030-008-7029-9. Google Scholar

[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. doi: 10.1016/0022-0248(89)90142-5. Google Scholar

[42]

E. R. Rubinstein and M. E. Glicksman, Dentritic growth kinetics and structure,, J. of Crystal Growth, 112 (1991), 84. doi: 10.1016/0022-0248(91)90914-Q. Google Scholar

[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. doi: 10.1103/PhysRevA.46.1038. Google Scholar

[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. doi: 10.1016/S1359-6454(01)00306-8. Google Scholar

[45]

R. E. Napolitano, S. Liu and R. Trivedi, Experimental measurement of anisotropy in the interfacial free energy,, Interface Science, 10 (2002), 217. doi: 10.1023/A:1015884415896. Google Scholar

[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. doi: 10.1063/1.449884. Google Scholar

[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. doi: 10.1103/PhysRevLett.85.4751. Google Scholar

[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. doi: 10.1103/PhysRevLett.86.5530. Google Scholar

[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). doi: 10.1063/1.1563248. Google Scholar

[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. doi: 10.1103/PhysRevB.73.024116. Google Scholar

[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. doi: 10.1103/PhysRevB.78.144112. Google Scholar

[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. doi: 10.1063/1.2149859. Google Scholar

[53]

J. R. Morris and X. Y. Song, Anisotropic free energy of the Lennard-Jones crystal-melt interface,, J. Chem. Physics, 119 (2003), 3920. doi: 10.1063/1.1591725. Google Scholar

[54]

G. Wulff, Zur frage der geschwindigkeit des wachstums und der auflö der kristallflä zeitschr,, Zeitschr. F. Kristallog., 34 (1901), 449. Google Scholar

[55]

C. Herring, , in, (1952). Google Scholar

[56]

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