Thermodynamically Consistent Modeling and Stable ALE Approximations of Reactive Semi-Permeable Interfaces

TL;DR

This paper develops a thermodynamically consistent model and stable numerical scheme for reactive semi-permeable interfaces, capturing complex biological processes involving coupled surface reactions, transport, and deformation.

Contribution

It introduces a unified continuum framework with an energy-based derivation and a finite element ALE scheme that ensures conservation and stability, verified through numerical experiments.

Findings

The scheme accurately captures complex interfacial dynamics.

Numerical experiments confirm convergence and conservation.

Applications demonstrate biological relevance in drug delivery and lipid metabolism.

Abstract

Reactive, semi-permeable interfaces play important roles in key biological processes such as targeted drug delivery, lipid metabolism, and signal transduction. These systems involve coupled surface reactions, transmembrane transport, and interfacial deformation, often triggered by local biochemical signals. The strong mechanochemical couplings complicate the modeling of such interfacial dynamics. We propose a thermodynamically consistent continuum framework that integrates bulk fluid motion, interfacial dynamics, surface chemistry, and selective solute exchange, derived via an energy variation approach to ensure mass conservation and energy dissipation. To efficiently solve the resulting coupled system, we develop a finite element scheme within an Arbitrary Lagrangian-Eulerian (ALE) framework, incorporating the Barrett-Garcke-Nurnberg (BGN) strategy to maintain mesh regularity and preserve conservation laws. Numerical experiments verify the convergence and conservation properties of the scheme and demonstrate its ability in capturing complex interfacial dynamics. Two biologically inspired examples showcase the model's versatility: cholesterol efflux via the ABCG1 pathway, involving multistage interfacial reactions and HDL uptake; and a self-propelled droplet system with reaction-activated permeability, mimicking drug release in pathological environments. This work provides a unified computational platform for studying strongly coupled biochemical and mechanical interactions at interfaces, offering new insights into reactive transport processes in both biological and industrial contexts.