The phase field technique for modeling multiphase materials

Abstract

This paper reviews methods and applications of the phase field technique, one of the fastest growing areas in computational materials science. The phase field method is used as a theory and computational tool for predictions of the evolution of arbitrarily shaped morphologies and complex microstructures in materials. In this method, the interface between two phases (e.g. solid and liquid) is treated as a region of finite width having a gradual variation of different physical quantities, i.e. it is a diffuse interface model. An auxiliary variable, the phase field or order parameter φ( x), is introduced, which distinguishes one phase from the other. Interfaces are identified by the variation of the phase field. We begin with presenting the physical background of the phase field method and give a detailed thermodynamical derivation of the phase field equations. We demonstrate how equilibrium and non-equilibrium physical phenomena at the phase interface are incorporated into the phase field methods. Then we address in detail dendritic and directional solidification of pure and multicomponent alloys, effects of natural convection and forced flow, grain growth, nucleation, solid-solid phase transformation and highlight other applications of the phase field methods. In particular, we review the novel phase field crystal model, which combines atomistic length scales with diffusive time scales. We also discuss aspects of quantitative phase field modeling such as thin interface asymptotic analysis and coupling to thermodynamic databases. The phase field methods result in a set of partial differential equations, whose solutions require time-consuming large-scale computations and often limit the applicability of the method. Subsequently, we review numerical approaches to solve the phase field equations and present a finite difference discretization of the anisotropic Laplacian operator.

Nonlinear phase-field model for electrode-electrolyte interface evolution

“…Phase-field method is a versatile mesoscale computational approach that has been successfully applied to modeling temporal and spatial microstructure evolution of materials undergoing a wide variety of processes such as phase transformations, deformation, and particle coarsening [1][2][3][4][5]. However, all existing phase-field models are based on linear kinetics, i.e., the rate of change of a phase field is assumed to be linearly proportional to the thermodynamic driving force, which is, in principle, only valid for systems close to equilibrium.…”

A Gibbs formulation for reactive materials with phase change

“…All components including different phases of the same molecular material must have a complete EQB potential. Unlike classical phase field theory, that uses an order parameter or differences in molecular density to switch the constitutive description of the phases, [1], [2], [3], [4], the Gibbs formulation uses the mass fraction of the components, and equilibrium EOS descriptions of the mixtures are derived in a straightforward way. Recently we have applied this model t to both molecular premixed materials like RDX, and reaction diffusion problems like thermitic reactions in condensed phases, where energy is released at a molecular interface, and is hence controlled by both mass diffusion and heat conduction.…”

Phase Field Model for Solidification with Boundary Interface Interaction

“…This model was further elaborated in Caginalp [2], Fix [6] and Langer [11]. See also the recent reviews [13,17]. In the above mentioned works, however, the interaction of external boundary and interface was neglected.…”