Rate of Convergence of Phase Field Equations in Strongly Heterogeneous Media towards their Homogenized Limit

TL;DR

This paper derives and rigorously justifies an upscaled phase field model for heterogeneous porous media, providing error estimates and enabling better simulation of interfacial phenomena in complex materials.

Contribution

It presents a general formal derivation and error analysis of upscaled phase field equations for heterogeneous porous media with polynomial free energies.

Findings

Error between microscopic and macroscopic solutions is of order ε^{1/2}.

Provides a rigorous justification for the upscaled model.

Enables improved modeling of interfacial transport in heterogeneous environments.

Abstract

We study phase field equations based on the diffuse-interface approximation of general homogeneous free energy densities showing different local minima of possible equilibrium configurations in perforated/porous domains. The study of such free energies in homogeneous environments found a broad interest over the last decades and hence is now widely accepted and applied in both science and engineering. Here, we focus on strongly heterogeneous materials with perforations such as porous media. To the best of our knowledge, we present a general formal derivation of upscaled phase field equations for arbitrary free energy densities and give a rigorous justification by error estimates for a broad class of polynomial free energies. The error between the effective macroscopic solution of the new upscaled formulation and the solution of the microscopic phase field problem is of order $\epsilon^1/2$ for a material given characteristic heterogeneity $\epsilon$. Our new, effective, and reliable macroscopic porous media formulation of general phase field equations opens new modelling directions and computational perspectives for interfacial transport in strongly heterogeneous environments.