Effective Macroscopic Interfacial Transport Equations in Strongly Heterogeneous Environments for General Homogeneous Free Energies

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.


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 macrovariable 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.
Diffuse interface formulations show a high versatility which is further extended due to increasing computational power. This leads continuously to new and increasingly complex scientific and engineering applications such as more realistic descriptions for the computation of transport in porous media [2] which represents a high-dimensional multiscale problem with many numerical challenges [3]. Our main result here is the systematic and general derivation of effective macroscopic equations which reliable account for multiple phases invading strongly heterogeneous environments such as porous materials.
The physical basis of the diffuse interface formulation relies on the following class of abstract energy densities The free energy density F defines equilibrium phases φ i , i = 1, 2, . . . , M as M ∈ N local minima and the gradient term λ 2 |∇φ| 2 penalizes the interfacial area between these equilibrium phases. In thermodynamic contexts, F represents the (Helmholtz) free energy density F (φ) := U − T S , where U is the internal energy, T is the temperature, and S is the entropy. Popular examples include the energy of regular solutions (also known as the Flory-Huggins energy [4]). The regular solution theory plays a crucial role in many important applications such as ionic melts [5], water sorption in porous solids [6], and micellization in binary surfactant mixtures [7]. Also wetting phenomena, often studied using classical sharp-interface approximations, e.g. [8], are also described by phase-field equations [9,10,11] which have been extended to include electric fields (so-called electrowetting, e.g. [12]). This energy-functionals based framework has also been applied in image processing such as inpainting, see e.g. [13].
Periodic covering by cells Y In a previous study [14], we focused on a specific form of the homogeneous free energy density and we recently extended it towards Stokes flow [15]. Here, we provide an upscaling for H −1 -gradient flows of arbitrary free energy densities based on a Taylor expansion of the free energy density at the effective upscaled solution. Before we can state our main result, we formulate the basic setting to study general interfacial dynamics.
[1]] under homogeneous Neumann boundary conditions, i.e., g = 0, and free energy densities F . (b) Heterogeneous/perforated domains Ω . Our main focus concentrates on (1) in a perforated domain Ω ⊂ R d instead of a homogeneous Ω ⊂ R d . The parameter = L > 0 is called heterogeneity where represents the characteristic pore size and L is the macroscopic length of the porous medium, see Figure  1. Herewith, we can define the porous medium by a reference pore/cell Y : For simplicity, we set 1 = 2 = · · · = d = 1. The pore (Ω ) and the solid phase (B ) are defined by where the subsets Y 1 , Y 2 ⊂ Y are such that Ω is a connected set. The set Y 1 ⊂ Y represents the pore phase (e.g. liquid or gas phase in wetting problems), see Figure 1. Herewith, we can can rewrite (2) as the following microscopic porous media problem with the boundary (∇ n φ := n · ∇φ = 0 on ∂Ω T , ∇ n ∆φ = 0 on ∂Ω T ) and initial (φ (x, 0) = ψ(x) on Ω ) conditions. 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 → 0 in (4). We formally achieve this by asymptotic multiscale expansions [17,18].
The main results are stated in Section 2 and subsequently justified in Section 3.

Main results
Before we state our main result of effective macroscopic phase field equations (including the Cahn-Hilliard equation) which is valid for arbitrary energy densities (1), we introduce the following scale separation property of the chemical potential.
where φ 0 (x) is the upscaled/slow variable solving the upscaled phase field equation.
Remark 2.2. Definition 2.1 accounts for the problem specific separation between the large (macroscopic) scale x with slow processes and the small (microscopic) scale y with fast processes.
In the homogenization/upscaling of nonlinear equations, Definition 2.1 appears naturally in the sense that it leads to the same class of equations on the macroscale as on the microscale and that it guarantees that resulting cell problems are well-posed [15,19,20]. These considerations together with the splitting strategy [14,15], which decouples 4-th order problems (4) into two 2-nd order equations, allow us to state our Main Result: (Upscaled Cahn-Hilliard equations) Suppose that ψ(x) ∈ H 2 E (Ω). For scale separated chemical potentials µ 0 = ∇ φ E(φ 0 ) (Definition 2.1), the microscopic porous media formulation (4) can be effectively and reliably approximated by the following macroscopic problem, with boundary (∇ n φ 0 = n · ∇φ 0 = 0 on ∂Ω T , ∇ n ∆φ 0 = 0 on ∂Ω T ) and initial (φ 0 (x, 0) = ψ(x) in Ω) conditions, where θ 1 := |Y 1 | |Y | is the porosity and the porous media correction tensorsD : The corrector functions ξ k φ ∈ H 1 per (Y 1 ) and ξ k w ∈ L 2 (Ω; H 1 per (Y 1 )) for 1 ≤ k ≤ d solve in the distributional sense the following reference cell problems ξ k w : The expression ∇ φ E(φ) denotes the Fréchet derivative of E with respect to φ. The upscaled equations show the mathematically and physically convincing feature that they preserve the structure from the microscopic formulation except for the effective correction tensors (6). For a rigorous error quantification, we refer the interested reader to [21]. Remark 2.3. i) For an isotropic mobility, i.e.,M := mÎ whereÎ is the identity matrix, we have ξ k w = ξ k φ . In this case, one can find computational results in Ref. [22] for the cell problems.

Basic steps in the derivation using formal asymptotic expansions
We introduce the micro-scale x =: y ∈ Y and apply the standard multiscale property for spatial differentiation ∂f (x) y) is an arbitrary function depending on two variables x ∈ Ω, y ∈ Y . The Laplace operators ∆ and div M ∇ then can be written as follows, such that we can identify A := −2 A 0 + −1 A 1 + A 2 = ∆ and correspondingly B = div M ∇ . We account for the multiscale nature of strongly heterogeneous environments [19,17,18] by the following ansatz ξ ≈ ξ 0 (x, y, t) + ξ 1 (x, y, t) + 2 ξ 2 (x, y, t) + . . . , for ξ ∈ {w, φ} .
Before we can insert (9) into the microscopic formulation (4), we need to approximate the derivative of the nonlinear homogeneous free energy f := F by a Taylor expansion of the form where φ 0 denotes the leading-order term in (9). Substituting (9) and (10) into (4), which we split into two second order problems as suggested in [14], and using (8) provides the following sequence of problems O( −1 ) : The first problem (11) immediately implies that the leading-orders φ 0 and w 0 are independent of the microscale y [19,18]. This suggests the following ansatz for w 1 and φ 1 , i.e., Inserting (14) into (12) 2 provides an equation for the correctors ξ k w and ξ k v . The resulting equation for ξ k v can be immediately written for 1 ≤ k ≤ d as, for ξ k φ (y) Y -periodic with M Y 1 (ξ k φ ) = 0 and associated boundary condition n · ∇ξ k φ + e k = 0 on ∂Y 1 w ∩ ∂Y 2 w .