×

Diffuse interfaces with sharp corners and facets: Phase field models with strongly anisotropic surfaces. (English) Zbl 0930.35201

Summary: We provide the general outline of an analysis of the motion of diffuse interfaces in the order-parameter (phase field) formulation which includes nondifferentiable and nonconvex gradient energy terms. Nondifferentiability leads to equilibrium and motion equations that are apparently undefined and nonconvexity leads to equations that are apparently ill-posed. The problem of nondifferentiability is resolved by using nonlocal variations to move entire facets or line segments with orientations having such facets or line segments in the underlying Wulff shape. The problem of ill-posedness is resolved by using varifolds (infinitesimally corrugated diffuse interfaces) constructed from the edges and corners in the underlying Wulff shape, or equivalently by using the convexification of the gradient energy term and then reinterpreting the solutions as varifolds. This is justified on a variational basis. We conclude that after an initial transient, level sets move by weighted mean curvature, in agreement with the sharp interface limit. We provide equations for tracking the shocks that develop as edges and corners in level sets.

MSC:

35R35 Free boundary problems for PDEs
58E50 Applications of variational problems in infinite-dimensional spaces to the sciences
35Q55 NLS equations (nonlinear Schrödinger equations)
35R25 Ill-posed problems for PDEs
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bouchitte, G., Singular perturbations of variational problems arising from a two-phase transition model, Appl. Math. Optim., 21, 289-314 (1990) · Zbl 0695.49003
[2] Barroso, A. C.; Fonseca, I., Anisotropic singular perturbations — the vectorial case, (Proc. Roy. Soc. Edin. Ser. A, 124 (1994)), 527-571 · Zbl 0804.49013
[3] R. Goglione and M. Paolini, Numerical simulations of crystalline motion by mean curvature with Allen-Cahn relaxation, preprint.; R. Goglione and M. Paolini, Numerical simulations of crystalline motion by mean curvature with Allen-Cahn relaxation, preprint. · Zbl 0864.35125
[4] Cahn, J. W.; Allen, S. M., A microscopic theory of domain wall motion and its experimental verification in FeAl alloy domain growth kinetics, J. de Physique (Colloq. 7), 38, 51-55 (1977)
[5] Allen, S. M.; Cahn, J. W., A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metall., 27, 1085-1095 (1979)
[6] Cahn, J. W., On spinodal decomposition, Acta Metall., 9, 795-801 (1961)
[7] Hilliard, J. E., Spinodal decomposition, (Phase Transformations (1970), ASM: ASM Cleveland), 497-560, Ch. 12
[8] Rayleigh, Lord, (Scientific Papers, Vol. 3 (1902), Cambridge University Press: Cambridge University Press Cambridge), 513, (reprinted by: Dover, New York, 1964) · JFM 33.0046.01
[9] Rowlinson, J. S., J. Stat. Phys., 20, 197-244 (1979), translated from the German into English by · Zbl 1245.82006
[10] Taylor, J. E.; Cahn, J.; Handwerker, C., Geometric models of crystal growth, Acta Metall. Mater., 40, 1443-1474 (1992)
[11] Kobayashi, R., Modeling and numerical simulations of dendritic crystal growth, Physica D, 63, 410-423 (1993) · Zbl 0797.35175
[12] Owen, N. C.; Sternberg, P., Nonconvex variational problems with anisotropic perturbations, Nonlinear Anal., 78, 705-719 (1991) · Zbl 0748.49034
[13] Elliott, C. M.; Schätzle, R., The limit of the fully anisotropic double-obstacle Allen-Cahn equation in the non-smooth case, SIAM J. Math. Anal., 28, 274-303 (1997) · Zbl 0870.35128
[14] Taylor, J. E.; Cahn, J. W., Linking anisotropic sharp and diffuse surface motion laws via gradient flows, J. Stat. Phys., 77, 183-197 (1994) · Zbl 0844.35044
[15] 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, 2016-2024 (1993) · Zbl 0791.35159
[16] Taylor, J. E., Book review of Wulff Construction, A Global Shape from Local Interaction, (Dobrushin, R.; Kotecky, R.; Shlosman, S., Bull Amer. Math. Soc., 31 (1994)), 291-296
[17] Hoffman, D. W.; Cahn, J. W., A vector thermodynamics for anisotropic surfaces I. Fundamentals and application to plane surface junctions, Surf. Sci., 31, 368-388 (1972)
[18] Cahn, J. W.; Hoffman, D. W., A vector thermodynamics for anisotropic surfaces II. Curved and faceted surfaces, Acta Metall., 22, 1205-1214 (1974)
[19] Arbel, E.; Cahn, J. W., On invariances in surface thermodynamic properties and their applications to low symmetry crystals, Surf. Sci., 51, 305-309 (1975)
[20] Taylor, J. E., Geometric crytal growth in 3D via faceted interfaces, (Taylor, J. E., Computational Crystal Growers Workshop. Computational Crystal Growers Workshop, Selected Lectures in Mathematics (1992), American Mathematical Society: American Mathematical Society Providence, RI), 111-113, plus video 20:25-26:00
[21] Taylor, J. E., Surface motion due to crystalline surface energy gradient flows, (Peters, A. K., Elliptic and Parabolic Methods in Geometry (1996)), 145-162, Wellesley · Zbl 0915.49024
[22] Taylor, J. E., Mean curvature and weighted mean curvature, Acta Metall, Mater., 40, 1475-1485 (1992)
[23] Carr, J.; Pego, R., Metastable patterns in solutions of \(u_1 = ϵ^2u_{ xx } \) − \(f(u)\), CPAM, 42, 523-576 (1989) · Zbl 0685.35054
[24] Bronsard, L.; Kohn, R. V., Motion by mean curvature as the singular limit of Ginzburg-Landau dynamics, J. Diff. Eqs., 90, 211-237 (1991) · Zbl 0735.35072
[25] Wheeler, A. A.; McFadden, G. B., A ξ vector formulation of anisotropic phase field models: 3D asymptotics, European J. Appl. Math., 7, 367-381 (1996) · Zbl 0909.35160
[26] Kobayashi, R., A numerical approach to three-dimensional dendritic solidification, Experimental Math., 3, 59-81 (1994) · Zbl 0811.65126
[27] Taylor, J. E., Motion of curves by crystalline curvature, including triple junctions and boundary points, Differential Geometry, (Proc. Symp. Pure Math., 51 (1993)), 417-438, (part 1) · Zbl 0823.49028
[28] Almgren, F.; Taylor, J. E., Flat flow is motion by crystalline curvature for curves with crystalline energies, J. Diff. Geom., 42, 1-22 (1995) · Zbl 0867.58020
[29] Almgren, F. J.; Taylor, J. E.; Wang, L., A variational approach to motion by weighted mean curvature, (Taylor, J. E., Computational Crystal Growers Workshop. Computational Crystal Growers Workshop, Selected Lectures in Mathematics (1992), American Mathematical Society: American Mathematical Society Providence, RI), 9-12
[30] Taylor, J. E., Complete catalog of minimizing embedded crystalline cones, (Proc. Symp. Math., 44 (1986)), 379-403
[31] Cahn, J. W.; Taylor, J. E., Catalog of saddle shaped surfaces in crystals, Acta Metall., 34, 1-12 (1986)
[32] Cahn, J. W.; Taylor, J. E., A contribution to the theory of surface energy minimizing surfaces, Scripta Metall., 18, 1117-1120 (1984)
[33] Kikuchi, R.; Cahn, J. W., Theory of domain walls in ordered structures. II. Pair approximation for nonzero temperatures, J. Phys. Chem. Solids, 23, 137-151 (1962)
[34] Cahn, J. W., Interfacial free energy and interfacial stress: The case of an internal interface in a solid, Acta Metall., 37, 773-776 (1989)
[35] Braun, R. J.; Cahn, J. W.; Hagedorn, J.; McFadden, G. B.; Wheeler, A. A., Anisotropic interfaces and ordering in fcc alloys: A multiple-order-parameter continuum theory, (Chen, L.-Q.; etal., Mathematics of Microstructure Evolution (1996), TMS/SIAM), 225-244
[36] R.J. Braun, J.W. Cahn, G.B. McFadden and A.A. Wheeler, Anisotropy of interfaces in an ordered alloy: A multiple-order parameter model, preprint.; R.J. Braun, J.W. Cahn, G.B. McFadden and A.A. Wheeler, Anisotropy of interfaces in an ordered alloy: A multiple-order parameter model, preprint. · Zbl 1067.74513
[37] Cahn, J. W.; Hilliard, J. E., Free energy of a nonuniform system. I, J. Chem. Phys., 28, 258-267 (1958) · Zbl 1431.35066
[38] Carter, C.; Roosen, A.; Cahn, J.; Taylor, J. E., Shape evolution by surface diffusion and surface attachment limited kinetics on completely faceted surfaces, Acta Metall. Mater., 43, 4309-4323 (1995)
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.