×

On anisotropic order parameter models for multi-phase systems and their sharp interface limits. (English) Zbl 0936.82010

Summary: For a general class of diffuse anisotropic multi-phase order parameter (or phase-field) models we use formally matched asymptotic expansions to determine the asymptotic limit when a small parameter related to the thickness of the interface tends to zero. In the case of anisotropic Allen-Cahn systems we obtain in the limit that the interface moves by anisotropic mean curvature flow. At triple junctions a force balance holds which in the anisotropic case includes shear forces (Herring torque terms) acting normal to the interface. We further identify the singular limit of anisotropic Cahn-Hilliard systems.
In a forthcoming paper [H. Garcke, B. Nestler and B. Stoth, An anisotropic multiphase-field concept: numerical simulations of moving phase boundaries and multiple junctions (preprint)] we will present numerical simulations based on the anisotropic order parameter model which show that the model recaptures many features observed in anisotropic multi-phase systems.

MSC:

82B24 Interface problems; diffusion-limited aggregation arising in equilibrium statistical mechanics
35K99 Parabolic equations and parabolic systems
58E50 Applications of variational problems in infinite-dimensional spaces to the sciences
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Alikakos, N.; Bates, P.; Chen, X., Convergence of the Caha-Hilliard equation to the Hele-Shaw model, Arch. Rat. Mech. Anal., 128, 165-205 (1994) · Zbl 0828.35105
[2] Alt, H. W.; Pawlow, I., A mathematical model of dynamics of non-isothermal phase separation, Physica D, 59, 389-416 (1992) · Zbl 0763.58031
[3] Barroso, A. C.; Fonseca, I., Anisotropic singular perturbations-the vectorial case, (Proc. Roy. Soc. Edinburgh A, 124 (1994)), 527-571 · Zbl 0804.49013
[4] Bronsard, L.; Garcke, H.; Stoth, B., A multi-phase Mullins-Sekerka system: Matched asymptotic expansions and an implicit time discretization for the geometric evolution problem, (preprint no. 472 (1996), SFB 256 Universität Bonn) · Zbl 0924.35199
[5] Bronsard, L.; Gui, C.; Schatzman, M., A three layered minimizer in \(R^2\) for a variational problem with a symmetric three-well potential, Commun. Pure Appl. Math., 47, 677-715 (1996) · Zbl 0855.35035
[6] Bronsard, L.; Reitich, F., On three-phase boundary motion and the singular limit of a vector-valued Ginzburg-Landau equation, Arch. Rat. Mech. Anal., 124, 355-379 (1993) · Zbl 0785.76085
[7] Caginalp, G., An analysis of a phase field model of a free boundary, Arch Rat. Mech. Anal., 92, 205-245 (1986) · Zbl 0608.35080
[8] Caginalp, G.; Fife, P. C., Dynamics of layered interfaces arising from phase boundaries, SIAM J. Appl. Math., 48, 506-518 (1988)
[9] Cahn, J. W.; Hilliard, J. E., Free energy of a nonuniform system I. Interfacial free energy, J. Chem. Phys., 28, 258-267 (1958) · Zbl 1431.35066
[10] Cahn, J. W.; Novick-Cohen, A., Evolution equations for phase separation and ordering in binary alloys, J. Stat. Phys., 76, 877-909 (1994) · Zbl 0840.35110
[11] De Fontaine, D., An analysis of clustering and ordering in multicomponent solid solutions - I. Stability criteria, J. Phys. Chem. Solids, 33, 287-310 (1972)
[12] De Mottoni, P.; Schatzman, M., Geometrical evolution of developed interfaces, Trans. Am. Math. Soc., 347, 1533-1589 (1995) · Zbl 0840.35010
[13] C.M. Elliott, H. Garcke, Diffusional phase transitions in multicomponent systems with a concentration dependent mobility matrix, Physica D, to appear.; C.M. Elliott, H. Garcke, Diffusional phase transitions in multicomponent systems with a concentration dependent mobility matrix, Physica D, to appear. · Zbl 1194.35225
[14] C.M. Elliott, S. Luckhaus, A generalized diffusion equation for phase separation of a multi-component mixture with interfacial free energy, preprint no. 195, SFB256 University Bonn.; C.M. Elliott, S. Luckhaus, A generalized diffusion equation for phase separation of a multi-component mixture with interfacial free energy, preprint no. 195, SFB256 University Bonn.
[15] Elliott, C. M.; Schätzle, R., The limit of the anisotropic double-obstacle Allen-Cahn equation, (Proc. Roy. Soc. Edinburgh A, 126 (1996)), 1217-1234 · Zbl 0865.35073
[16] H. Garcke, B. Nestler, B. Stoth, An anisotropic multiphase-field concept: numerical simulations of moving phase boundaries and multiple junctions, preprint.; H. Garcke, B. Nestler, B. Stoth, An anisotropic multiphase-field concept: numerical simulations of moving phase boundaries and multiple junctions, preprint. · Zbl 0942.35095
[17] H. Garcke, A. Novick-Cohen, A singular limit for a system of degenerate Cahn-Hilliard equations, preprint, SFB256 University Bonn.; H. Garcke, A. Novick-Cohen, A singular limit for a system of degenerate Cahn-Hilliard equations, preprint, SFB256 University Bonn. · Zbl 0988.35019
[18] Gurtin, M. E., Thermodynamics of Evolving Phase Boundaries in the Plane (1993), Clarendon Press: Clarendon Press Oxford · Zbl 0787.73004
[19] 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)
[20] Herring, C., Surface tension as a motivation for sintering, (Kingston, W. E., The Physics of Powder Metallurgy, Chapter 8 (1951), McGraw-Hill: McGraw-Hill New York)
[21] Hoffmann, D. W.; Cahn, J. W., A vector thermodynamics for anisotropic surfaces - I. Fundamentals and applications to plane surface junctions, Sur. Sci., 31, 368-388 (1972)
[22] Kobayashi, R., Modelling and numerical simulation of dendritic crystal growth, Physica D, 63, 410-423 (1993) · Zbl 0797.35175
[23] Landau, L. D.; Ginzburg, V. I., On the theory of superconductivity, (ter Haar, D., Collected papers of L.D. Landau (1965), Pergamon: Pergamon Oxford), 626-633, English transl.
[24] Landau, L. D.; Lifschitz, E. M., Lehrbuch der theoretischen Physik, (Statistische Physik Teil I (1979), Akademie Verlag: Akademie Verlag Berlin), Band 5 · Zbl 0201.30501
[25] Langer, J. S., Physica D, 63, 410-423 (1993)
[26] McFadden, G. B.; Wheeler, A. A.; Braun, R. J.; Coriell, S. R.; Sekerka, R. F., Phase field models for anisotropic interfaces, Phys. Rev. E, 48, 3, 2016-2024 (1993) · Zbl 0791.35159
[27] Morral, J. E.; Cahn, J. W., Spinodal decomposition in ternary alloys, Acta Metallurgica, 19, 1037-1045 (1971)
[28] B. Nestler, M. Seeßelberg, J. Tiaden, Work in progress.; B. Nestler, M. Seeßelberg, J. Tiaden, Work in progress.
[29] Penrose, O.; Fife, P. C., Thermodynamically consistent models of phase-field type for the kinetics of phase transition, Physica D, 43, 44-62 (1990) · Zbl 0709.76001
[30] Schmitz, G. J.; Nestler, B.; Seeßelberg, M., YBCO Melt-processing development by numerical simulation, J. Low Temp. Phys., 105, 5/6, 1451 (1996)
[31] Seeßelberg, M.; Tiaden, J.; Schmitz, G. J.; Steinbach, I., Peritectic and eutectic solidification: simulations of the microstructure using the multi-phase-field method, (Proceedings of Solidification Processing. Proceedings of Solidification Processing, Sheffield (1997))
[32] Soner, H. M., Convergence of the phase-field equations to the Mullins-Sekerka problem with kinetic undercooling, Arch. Rat. Mech. Anal., 131, 139-197 (1995) · Zbl 0829.73010
[33] Steinbach, I.; Pezolla, F.; Nestler, B.; Seeßelberg, M.; Prieler, R.; Schmitz, G. J.; Rezende, J. L.L., A phase-field concept for multiphase systems, Physica D, 94, 135-147 (1996) · Zbl 0885.35148
[34] Sternberg, P., Vector-valued local minimizers of nonconvex variational problems, Rocky Mt. J. Math., 21, 2, 799-807 (1991) · Zbl 0737.49009
[35] Stoth, B., A sharp interface limit of the phase field equations: one-dimensional and axisymmetric, Eur. J. Appl. Math., 7, 603-634 (1996) · Zbl 0876.35133
[36] Tiaden, J.; Nestler, B.; Diepers, H. J.; Steinbach, I., The multiphase-field model with an integrated concept for modelling solute diffusion, Physica D, 115, 73-86 (1998) · Zbl 0956.74038
[37] Rowlinson, J. S., J. Stat. Phys., 20, 197-244 (1979), English transl. (with commentary) · Zbl 1245.82006
[38] Wang, S.-L.; Sekerka, R. F.; Wheeler, A. A.; Murray, B. T.; Coriell, S. R.; Braun, R. J.; McFadden, G. B., Thermodynamically-consistent phase-field models for solidification, Physica D, 69, 189-200 (1993) · Zbl 0791.35159
[39] Wheeler, A. A.; McFadden, G. B., A ξ-vector formulation of anisotropic phase-field models: 3D asymptotics, Eur. J. Appl. Math., 7, 367-381 (1996) · Zbl 0909.35160
[40] A.A. Wheeler, G.B. McFadden, On the notion of a ξ-vector and a stress tensor for a general class of anisotropic diffuse interface models, preprint.; A.A. Wheeler, G.B. McFadden, On the notion of a ξ-vector and a stress tensor for a general class of anisotropic diffuse interface models, preprint. · Zbl 0894.60101
[41] Wheeler, A. A.; McFadden, G. B.; Boettinger, W. J., Phase-field model for solidification of a eutectic alloy, (Proc. Roy. Soc. Lond. A, 452 (1996)), 495-525
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.