A Diffuse Interface Model for Alloys with Multiple Components and Phases

A nonisothermal phase field model for alloys with multiple phases and components is derived. The model allows for arbitrary phase diagrams. We relate the model to classical sharp interface models by formally matched asymptotic expansions. In addition we discuss several examples and relate our model to the ones already existing.

[1]  Lee,et al.  Three-dimensional dendrite-tip morphology at low undercooling , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[2]  Daniel M. Anderson,et al.  Thin interface asymptotics for an energy/entropy approach to phase-field models with unequal conductivities , 2000 .

[3]  Harald Garcke,et al.  A mathematical model for grain growth in thin metallic films , 2000 .

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

[5]  Danan Fan,et al.  Diffuse-interface description of grain boundary motion , 1997 .

[6]  Long-Qing Chen.  Phase-Field Models for Microstructure Evolution , 2002 .

[7]  I. Steinbach,et al.  A phase field concept for multiphase systems , 1996 .

[8]  Harald Garcke,et al.  A singular limit for a system of degenerate Cahn-Hilliard equations , 2000, Advances in Differential Equations.

[9]  Geoffrey B. McFadden,et al.  Phase-field model for solidification of a eutectic alloy , 1996, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[10]  Harald Garcke,et al.  Anisotropy in multi-phase systems: a phase field approach , 1999 .

[11]  Barbara Stoth,et al.  A sharp interface limit of the phase field equations: one-dimensional and axisymmetric , 1996, European Journal of Applied Mathematics.

[12]  Chen,et al.  Computer simulation of the domain dynamics of a quenched system with a large number of nonconserved order parameters: The grain-growth kinetics. , 1994, Physical review. B, Condensed matter.

[13]  A. Karma,et al.  Multiscale random-walk algorithm for simulating interfacial pattern formation. , 2000, Physical review letters.

[14]  A. Karma.  Phase-field formulation for quantitative modeling of alloy solidification. , 2001, Physical review letters.

[15]  Harald Garcke,et al.  Modelling of microstructure formation and interface dynamics , 2003 .

[16]  Harald Garcke,et al.  A MultiPhase Field Concept: Numerical Simulations of Moving Phase Boundaries and Multiple Junctions , 1999, SIAM J. Appl. Math..

[17]  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.

[18]  Britta Nestler,et al.  A multi-phase-field model of eutectic and peritectic alloys: numerical simulation of growth structures , 2000 .

[19]  Gunduz Caginalp,et al.  An Analysis of Phase‐Field Alloys and Transition Layers , 1998 .

[20]  B. Chalmers.  Principles of Solidification , 1964 .

[21]  Britta Nestler,et al.  Phase-field model for solidification of a monotectic alloy with convection , 2000 .

[22]  Harald Garcke,et al.  On Fully Practical Finite Element Approximations of Degenerate Cahn-Hilliard Systems , 2001 .

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

[24]  A. A. Wheeler,et al.  Thermodynamically-consistent phase-field models for solidification , 1992 .

[25]  N. Goldenfeld,et al.  Adaptive Mesh Refinement Computation of Solidification Microstructures Using Dynamic Data Structures , 1998, cond-mat/9808216.

[26]  Morton E. Gurtin,et al.  Dynamic solid-solid transitions with phase characterized by an order parameter , 1994 .

[27]  J. Lowengrub,et al.  Quasi–incompressible Cahn–Hilliard fluids and topological transitions , 1998, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[28]  J. Waals.  The thermodynamic theory of capillarity under the hypothesis of a continuous variation of density , 1979 .

[29]  G. Caginalp,et al.  Stefan and Hele-Shaw type models as asymptotic limits of the phase-field equations. , 1989, Physical review. A, General physics.

[30]  A. Visintin.  Models of Phase Transitions , 1996 .

[31]  A. Karma,et al.  Phase-field method for computationally efficient modeling of solidification with arbitrary interface kinetics. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[32]  A. Karma,et al.  Quantitative phase-field modeling of dendritic growth in two and three dimensions , 1996 .

[33]  William L. George,et al.  A Parallel 3D Dendritic Growth Simulator Using the Phase-Field Method , 2002 .

[34]  Harald Garcke,et al.  On anisotropic order parameter models for multi-phase system and their sharp interface limits , 1998 .

[35]  P. Sternberg.  Vector-Valued Local Minimizers of Nonconvex Variational Problems , 1991 .

[36]  A. Karma,et al.  Phase-Field Simulation of Solidification , 2002 .

[37]  Robert Almgren,et al.  Second-Order Phase Field Asymptotics for Unequal Conductivities , 1999, SIAM J. Appl. Math..

[38]  Wei Bo-kang.  Phase-field simulations of solidification of Al-Cu binary alloys , 2004 .

[39]  Paul C. Fife,et al.  Thermodynamically consistent models of phase-field type for the kinetics of phase transitions , 1990 .

[40]  Halil Mete Soner,et al.  Convergence of the phase-field equations to the mullins-sekerka problem with kinetic undercooling , 1995 .

[41]  J. Langer.  Models of Pattern Formation in First-Order Phase Transitions , 1986 .

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