A Stabilized Diffuse-Interface Electroporation Model with a Semi-Analytical Spectral Electrolyte Solver

We develop a diffuse-interface continuum model for membrane electroporation that couples a phase field for pore geometry to a quasi-static electrolyte potential and a spatially varying leaky-dielectric model for the transmembrane voltage. The main contribution is a stabilized time-integration strategy for transmembrane voltage $V_m$: the stiff leakage term is treated implicitly while the electrolyte-to-membrane ionic current is lagged, yielding a closed-form update that removes the restriction imposed by the fast dielectric relaxation time. The electrolyte potential is computed efficiently using a semi-analytical spectral Laplace solver: a 2D DCT in the membrane plane reduces the 3D problem to independent 1D ODEs in $z$, solved in closed form and reconstructed by an inverse transform. The coupled method is robust under grid refinement, reproduces the sharp-interface critical-radius bifurcation, and captures electric-field focusing through conductive pores. We also demonstrate stochastic pore nucleation by adding thermal noise to the phase-field dynamics, enabling fully emergent electroporation events without prescribing initial defects.