zbMATH — the first resource for mathematics

The role of microscopic anisotropy in the macroscopic behavior of a phase boundary. (English) Zbl 0639.58038
A Landau-Ginzburg approach is implemented to study the influence microscopic anisotropy on the interfacial boundary between two phases. The continuum limit for a lattice spin system with anisotropic interactions \(J_ i\) and spacing \(a_ i\) leads to the system of equations \[ \tau (\partial \phi /\partial t)=\sum^{d}_{i=1}\xi^ 2_ i(\partial^ 2\phi /\partial x^ 2_ i)+(\phi -\phi^ 3)+2u, \]
\[ \partial u/\partial t+(\ell /2)(\partial \phi /\partial t)=K\Delta u. \] Here, \(\phi\) is an order parameter, u is the temperature, \(\tau\) is a relaxation time, \(\xi_ i\) are related to \(J_ i\) and \(a_ i\), \(\ell\) is the latent heat of fusion, and K is the diffusivity. Both equilibrium and non-equilibrium conditions are considered. A modified Gibbs-Thompson relation \[ \Delta su(r,\theta)=-[\sigma (\theta)+\sigma ''(\theta)]\kappa -\tau \nu \sigma (\theta)/\xi^ 2_ A(\theta)+{\mathcal O}(\xi^ 2) \] is obtained, where s is entropy density, \(\kappa\) is curvature, \(\nu\) is normal velocity of the interface, and \(\xi_ A(\theta)\) is a measure of the thickness of the interface.

58Z05 Applications of global analysis to the sciences
82B99 Equilibrium statistical mechanics
58J99 Partial differential equations on manifolds; differential operators
Full Text: DOI
[1] Wulff, G., Krystallograph. mineral., 34, 499, (1901)
[2] Hoffman, D.W.; Cahn, J.W., Surface sci., 31, 368, (1972)
[3] Cahn, J.W.; Hoffman, D.W., Acta metallurg., 22, 1205, (1974)
[4] Taylor, J.E., Bull. amer. math. soc., 84, 568, (1978)
[5] Taylor, J.E., ()
[6] {\scJ. E. Taylor and J. W. Cahn}, Catalog of saddle shaped surfaces in crystals, Acta Metallurg., in press.
[7] Cahn, J.W.; Taylor, J.E., A contribution to the theory of surface energy minimizing shapes, Scripta metallurg., 18, 1117, (1984)
[8] Arrow, J.E.; Taylor, J.E.; Zia, R.K.P., Equilibrium shapes of crystals in a gravitational field: crystals on a table, J. statist. phys., 33, 493, (1983)
[9] Herring, C., Diffusional viscosity of a polycrustalline solid, J. appl. phys., 21, 437, (1950)
[10] Herring, C., The use of classical macroscopic concepts in surface energy problems, (), 5
[11] Allen, S.; Cahn, J.W., A microscopic theory for autophase boundary motion and its application to antiphase domain coarseing, Acta metallurg., 27, 1085, (1979)
[12] Hartman, P., ()
[13] Chalmers, B., ()
[14] Cahn, J.W.; Hilliard, J.E., Free energy of a nonuniform system. I. interfacial free energy, J. chem. phys., 28, 258, (1957)
[15] Langer, J.S., Theory of condensation point, Ann. of phys., 41, 108, (1967)
[16] Caginalp, G., An analysis of a phase field model of a free boundary, Arch. rat. mech. anal., 92, 205, (1986) · Zbl 0608.35080
[17] Caginalp, G., Surface tension and supercoding in solidification theory, ()
[18] Caginalp, G., Phase field models of solidification: free boundary problems as systems of nonlinear parabolic differential equations, (), 107-121, Proceedings, Col. Int., Maubuisson-Carcaus, France, June 1984
[19] Caginalp, G., Solidification problems as systems of nonlinear differential equations, () · Zbl 0593.35099
[20] Caginalp, G.; Hastings, S., Properties of some ordinary differential equations related to free boundary problems, (1984), Univ. of Pittsburgh, preprint
[21] Caginalp, G.; McLeod, B., The interior transition layer for an ordinary differential equation arising from solidification theory, Quart. appl. math., 44, 155, (1986) · Zbl 0605.34022
[22] Caginalp, G.; Fife, P.C., Elliptic problems involving phase boundaries satisfying a curvature condition, (1985), Univ. of Arizona, preprint · Zbl 0645.35101
[23] Caginalp, G.; Fife, P.C., Elliptic problems with layers representing phase interfaces, (), in press · Zbl 0657.35060
[24] Lin, J.T.; Fix, G., Numerical solution of moving boundary problems using phase field methods, (1985), Carnegie-Mellon Univ, preprint
[25] Ashcroft, N.; Mermin, N., ()
[26] Fisher, M.E.; Caginalp, G., Wall and boundary free energies. I. ferromagnetic scalar spin systems, Commun. math. phys., 56, 11, (1977) · Zbl 1155.82308
[27] Caginalp, G.; Fisher, M.E., All and boundary free energies 2. general domains and complete boudaries, Commun. math. phys., 65, 247, (1979)
[28] Caginalp, G., The φ4 lattice field theory as an asymptotic expansion about the Ising limit, Ann. of phys., 124, 189, (1980)
[29] Caginalp, G., Thermodynamic properties of the φ4 lattice field theory near the Ising limit, Ann. of phys., 126, 500, (1980)
[30] Jasnow, D., Critical phenomena at interfaces, Rep. progr. phys., 47, 1059, (1984)
[31] Hohenberg, P.C.; Halpern, B.I., Theory of dynamic critical phenomena, Rev. mod. phys., 49, 435, (1977)
[32] Landau, L.; Lifshitz, E., ()
[33] Stanley, H.E., ()
[34] Gibbs, J.W.; Gibbs, J.W., (), 242, also
[35] J. Rice, “A Commentary on the Scientific Writings of J. W. Gibbs, “Thermodynamics” Vol. I (F. G. Donnan and A. Haas, Eds.), pp. 518-532.
[36] Mullins, W.W., The thermodynamics of crystal phases with curved interfaces: special case of interface isotropy and hydrostatic pressure, ()
[37] Mullins, W.W., Thermodynamic equilibrium of a crystal sphere in a fluid, J. chem. phys., 81, 1436, (1984)
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.