  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.
