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We study phase field equations based on the diffuse-interface approximationof general homogeneous free energy densities showing different local minima ofpossible equilibrium configurations in perforated/porous domains. The study ofsuch free energies in homogeneous environments found a broad interest over thelast decades and hence is now widely accepted and applied in both science andengineering. Here, we focus on strongly heterogeneous materials withperforations such as porous media. To the best of our knowledge, we present ageneral formal derivation of upscaled phase field equations for arbitrary freeenergy densities and give a rigorous justification by error estimates for abroad class of polynomial free energies. The error between the effectivemacroscopic solution of the new upscaled formulation and the solution of themicroscopic phase field problem is of order $\epsilon^1/2$ for a material givencharacteristic heterogeneity $\epsilon$. Our new, effective, and reliablemacroscopic porous media formulation of general phase field equations opens newmodelling directions and computational perspectives for interfacial transportin strongly heterogeneous environments.

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