Modeling, shape analysis and computation of the equilibrium pore shape near a PEM-PEM intersection

TL;DR

This paper models the equilibrium shape of a pore interface in an elastomer using PDEs, variational methods, and shape derivatives, leading to a modified Young-Laplace equation and computational shape analysis.

Contribution

It introduces a coupled PDE model and variational approach for pore shape analysis, deriving a modified Young-Laplace equation and computational methods for equilibrium shapes.

Findings

Existence and uniqueness of electrical potential established.

Shape derivatives and differentiability of energy functional analyzed.

Modified Young-Laplace equation derived and compared with classical form.

Abstract

In this paper we study the equilibrium shape of an interface that represents the lateral boundary of a pore channel embedded in an elastomer. The model consists of a system of PDEs, comprising a linear elasticity equation for displacements within the elastomer and a nonlinear Poisson equation for the electric potential within the channel (filled with protons and water). To determine the equilibrium interface, a variational approach is employed. We analyze: i) the existence and uniqueness of the electrical potential, ii) the shape derivatives of state variables and iii) the shape differentiability of the corresponding energy and the corresponding Euler-Lagrange equation. The latter leads to a modified Young-Laplace equation on the interface. This modified equation is compared with the classical Young-Laplace equation by computing several equilibrium shapes, using a fixed point algorithm.