(2+δ)-dimensional theory of the electromechanics of lipid membranes. III. Constitutive models.

Abstract

This article is the final part of a three-part series that develops a self-consistent theoretical framework describing the electromechanics of arbitrarily curved lipid membranes at the continuum scale. Owing to their small thickness, lipid membranes are commonly modeled as two-dimensional surfaces. However, this approach breaks down when considering their electromechanical behavior as it requires accounting for their finite thickness. To address this, we developed a new dimension reduction procedure in part I [Y. A. D. Omar et al. Phys. Rev. E 109, 054401 (2024)10.1103/PhysRevE.109.054401] to derive effective surface theories that explicitly capture the finite thickness of lipid membranes. We applied this method to dimensionally reduce Gauss' law and the electromechanical balance laws and referred to the resulting theory as (2+δ)-dimensional, where δ indicates the membrane thickness. However, the (2+δ)-dimensional balance laws for thin bodies derived in part II [Y. A. D. Omar et al., Phys. Rev. E 112, 024406 (2025)10.1103/75tt-k2f5] are general, and specific constitutive material models must be incorporated to specialize them to lipid membranes. In this work, we devise appropriate three-dimensional constitutive models that capture the material behavior of lipid membranes, which flow along their in-plane directions like viscous fluids but bend out-of-plane like elastic solids. The viscous material behavior is recovered by considering a three-dimensional Newtonian fluid model, leading to the same viscous stresses as strictly two-dimensional models of lipid membranes. The elastic resistance to bending is recovered by imposing a free energy penalty on local volume changes. While this material model does give rise to the characteristic bending resistance of lipid membranes, it differs in its higher-order curvature terms from the two-dimensional Canham-Helfrich-Evans theory. Furthermore, since lipid membranes only exhibit small midsurface stretch, they are often considered midsurface-area incompressible. In this work, this is captured by introducing reactive stresses that give rise to an effective surface tension. Finally, we use the viscous, elastic, and reactive stresses to derive the equations of motion and boundary conditions describing the electromechanics of lipid membranes. We conclude this article by providing the equations of motion and coupling and boundary conditions for a charged lipid membrane embedded in an electrolyte solution.