Triple-junction motion for an Allen-Cahn/Cahn-Hilliard system

Abstract Long time asymptotics are developed here for an Allen–Cahn/Cahn–Hilliard system derived recently by Cahn and Novick-Cohen [J.W. Cahn, A. Novick-Cohen, J. Statist. Phys. 76 (1994) 877–909] as a diffuse interface model for simultaneous order–disorder and phase separation. Proximity to a deep quench limit is assumed, and spatial scales are chosen to model Krzanowski instabilities in which droplets of a minor disordered phase bounded by interphase boundaries (IPBs) of high curvature coagulate along a slowly curved antiphase boundaries (APBs) separating two ordered variants. The limiting motion couples motion by mean curvature of the APBs with motion by minus the surface Laplacian of the IPBs on the same timescale. Quasi-static surface diffusion of the chemical potential occurs along APBs. The framework here yields both sharp interface and diffuse interface modeling of sintering of small grains and thermal grain boundary grooving in polycrystalline films.

[1]  Michael F. Ashby,et al.  A first report on sintering diagrams , 1973 .

[2]  John W. Cahn,et al.  Linking anisotropic sharp and diffuse surface motion laws via gradient flows , 1994 .

[3]  Harald Garcke,et al.  A multi-phase Mullins–Sekerka system: matched asymptotic expansions and an implicit time discretisation for the geometric evolution problem , 1998, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[4]  Morton E. Gurtin,et al.  On the motion of a phase interface by surface diffusion , 1990 .

[5]  Long-Qing Chen,et al.  Computer simulation of structural transformations during precipitation of an ordered intermetallic phase , 1991 .

[6]  J. Rubinstein,et al.  Minimizers and gradient flows for singularly perturbed bi-stable potentials with a Dirichlet condition , 1990, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[7]  Lorenzo Giacomelli,et al.  Existence for an Allen-Cahn/Cahn-Hilliard system with degenerate mobility , 1999 .

[8]  Peter W. Bates,et al.  Slow motion for the Cahn-Hilliard equation in one space dimension , 1991 .

[9]  T. Young.  III. An essay on the cohesion of fluids , 1805, Philosophical Transactions of the Royal Society of London.

[10]  J. Schneibel,et al.  The sintering of two particles by surface and grain boundary diffusion -- A two-dimensional numerical study , 1995 .

[11]  J. Schneibel,et al.  Sintering of regular two-dimensional arrays of particles surface and grain boundary diffusion , 1994 .

[12]  J. Keller,et al.  Fast reaction, slow diffusion, and curve shortening , 1989 .

[13]  Charles M. Elliott,et al.  `A generalised diffusion equation for phase separation of a multi-component mixture with interfacial free energy' , 1991 .

[14]  J. Taylor,et al.  Shape evolution by surface diffusion and surface attachment limited kinetics on completely faceted surfaces , 1995 .

[15]  Irena Pawlow,et al.  A mathematical model of dynamics of non-isothermal phase separation , 1992 .

[16]  NONEQUILIBRIUM PATTERN FORMATION INVOLVING BOTH CONSERVED AND NONCONSERVED ORDER PARAMETERS AND EFFECT OF LONG-RANGE INTERACTIONS , 1993 .

[17]  Solute-drag effects at migrating diffuse interfaces—I. Theoretical analysis and application to apbs in FeAl alloys , 1986 .

[18]  Paul C. Fife,et al.  Dynamics of Layered Interfaces Arising from Phase Boundaries , 1988 .

[19]  Danielle Hilhorst,et al.  Finite-dimensional exponential attractor for a model for order-disorder and phase separation , 1994 .

[20]  Charles M. Elliott,et al.  The Cahn–Hilliard equation with a concentration dependent mobility: motion by minus the Laplacian of the mean curvature , 1996, European Journal of Applied Mathematics.

[21]  L. Bronsard,et al.  On three-phase boundary motion and the singular limit of a vector-valued Ginzburg-Landau equation , 1993 .

[22]  Lebowitz,et al.  Exact macroscopic description of phase segregation in model alloys with long range interactions. , 1996, Physical review letters.

[23]  Barbara Stoth,et al.  Convergence of the Cahn-Hilliard Equation to the Mullins-Sekerka Problem in Spherical Symmetry , 1996 .

[24]  David J. Srolovitz,et al.  The Thermodynamics and Kinetics of film agglomeration , 1995 .

[25]  Samuel M. Allen,et al.  Mechanisms of phase transformations within the miscibility gap of Fe-rich Fe-Al alloys , 1976 .

[26]  David J. Eyre,et al.  Systems of Cahn-Hilliard Equations , 1993, SIAM J. Appl. Math..

[27]  W. Mullins,et al.  Theory of the decay of a surface scratch to flatness , 1962 .

[28]  Robert L. Pego,et al.  Front migration in the nonlinear Cahn-Hilliard equation , 1989, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[29]  J. Cahn,et al.  Evolution equations for phase separation and ordering in binary alloys , 1994 .

[30]  Samuel A. Safran,et al.  Capillary instabilities in thin films. II. Kinetics , 1986 .

[31]  Solute-drag effects at migrating diffuse interfaces—II. Experimental investigation of APB migration kinetics and comparison to theory , 1986 .

[32]  Joel Lebowitz,et al.  Phase Segregation Dynamics in Particle Systems with Long Range Interactions II: Interface Motion , 1997, SIAM J. Appl. Math..

[33]  J. E. Hilliard,et al.  Free Energy of a Nonuniform System. I. Interfacial Free Energy , 1958 .

[34]  P. Fife.  Dynamics of Internal Layers and Diffusive Interfaces , 1988 .

[35]  F. Génin,et al.  Capillary instabilities in thin films: A model of thermal pitting at grain boundary vertices , 1992 .

[36]  W. Mullins.  Theory of Thermal Grooving , 1957 .

[37]  Harald Garcke,et al.  Diffusional phase transitions in multicomponent systems with a concentration dependent mobility matrix , 1997 .

[38]  John W. Cahn,et al.  Spinodal decomposition in ternary systems , 1971 .

[39]  Capillarity driven motion of solid film wedges , 1997 .