Effective macroscopic interfacial transport equations in strongly heterogeneous environments for general homogeneous free energies

Markus Schmuck,Serafim Kalliadasis,Grigorios A Pavliotis

doi:10.1016/j.aml.2014.03.011

Abstract

We study phase field equations in perforated domains for arbitrary free energies. These equations have found numerous applications in a wide spectrum of both science and engineering problems with homogeneous environments. Here, we focus on strongly heterogeneous materials with perforations such as porous media. To the best of our knowledge, we provide the first derivation of upscaled equations for general free energy densities. In view of the versatile applications of phase field equations, we expect that our study will lead to new modelling and computational perspectives for interfacial transport and phase transformations in strongly heterogeneous environments.

Highlights

We study phase field equations in perforated domains for arbitrary free energies
In view of the versatile applications of phase field equations, we expect that our study will lead to new modelling and computational perspectives for interfacial transport and phase transformations in strongly heterogeneous environments
Our starting point is the widely accepted diffuse-interface formulation [1] describing the dynamics of interfaces between different phases

Summary

Introduction: phase field formulation in heterogeneous media

Our starting point is the widely accepted diffuse-interface formulation [1] describing the dynamics of interfaces between different phases. This formulation captures different thermodynamic states of a system by a continuous macroscopic variable obtained from averaged microscopic degrees of freedom Such a macro variable represents a locally conserved order parameter, denoted as φ, which defines different phases as local equilibrium limiting values of a free energy associated with the system under consideration. Our main objective is the derivation of a systematic and reliable homogenized/upscaled phase field formulation valid for general energy densities (1) by passing to the limit ε → 0 in (4). We formally achieve this by asymptotic multiscale expansions [17,18].

Main results

Basic steps in the derivation using formal asymptotic expansions