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

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.