Phase-field dynamics with transfer of materials: The Cahn–Hilliard equation with reaction rate dependent dynamic boundary conditions

Abstract

The Cahn–Hilliard equation is one of the most common models to describe phase separation processes of a mixture of two materials. For a better description of short-range interactions between the material and the boundary, various dynamic boundary conditions for the Cahn–Hilliard equation have been proposed and investigated in recent times. Of particular interests are the model by Goldstein et al. [Phys. D 240 (2011) 754–766] and the model by Liu and Wu [Arch. Ration. Mech. Anal. 233 (2019) 167–247]. Both of these models satisfy similar physical properties but differ greatly in their mass conservation behaviour. In this paper we introduce a new model which interpolates between these previous models, and investigate analytical properties such as the existence of unique solutions and convergence to the previous models mentioned above in both the weak and the strong sense. For the strong convergences we also establish rates in terms of the interpolation parameter, which are supported by numerical simulations obtained from a fully discrete, unconditionally stable and convergent finite element scheme for the new interpolation model.

A Review on the Cahn-Hilliard Equation: Classical Results and Recent Advances in Dynamic Boundary Conditions

“…is stable with respect to perturbations of the kinetic rate and constructed a robust family exponential attractors for L ∈ [0, 1] (see [83,Theorem 6.5]). We note that the results obtained in [83,[117][118][119] for those extended models are only valid for regular potentials including (2.3). However, singular potentials like the logarithmic potential (2.2) or the obstacle potential (2.4) are not admissible.…”

“…We note that in the formulations (3.10), (3.21) and (3.32), some "strong" relations were imposed on the phase-field function or the chemical potential via the trace operator, namely, in terms of some nonhomogeneous Dirichlet boundary conditions. To provide a more general description of the interactions between the materials in the bulk and the materials on the boundary, extended models with certain relaxed coupling relations between the bulk and boundary variables (e.g., via the Robin type boundary conditions) were introduced and analyzed in [83,117,118]. For an extended model with nonlocal effects, we refer to [119].…”

“…Similar relaxations can be imposed on the chemical potentials, which account for possible reactions between the materials in Ω and on ∂Ω. In [118] for some constants β = 0 and L > 0. The Robin type boundary condition indicates that the mass flux ∂ n µ is driven by differences in the chemical potentials and the constant L −1 can be interpreted as the reaction rate.…”

“…The Robin type boundary condition indicates that the mass flux ∂ n µ is driven by differences in the chemical potentials and the constant L −1 can be interpreted as the reaction rate. Taking β = 1, this boundary condition also allows people to establish a connection between models (3.21) and (3.32), via the formal limits L → 0 + and L → +∞, which correspond to the limit cases of an instantaneous reaction (L → 0 + ) and a vanishing reaction rate L → +∞, see [118] for detailed discussions. Thus, problem (3. )…”

Bulk-Boundary eigenvalues for Bilaplacian problems

“…In recent years, several types of dynamic boundary conditions have been proposed in order to account for the interactions of the material with the solid wall. In [41] the authors introduced a model which can be derived as a gradient flow for a Ginzburg-Landau type energy that contains both a bulk and a boundary contribution (compare to our (1.1)), which gives a third-order dynamic boundary condition…”