The $(2+\delta)$-dimensional theory of the electromechanics of lipid membranes: II. Balance laws

This article is the second of a three-part series that derives a self-consistent theoretical framework of the electromechanics of arbitrarily curved lipid membranes. Existing continuum theories commonly treat lipid membranes as strictly two-dimensional surfaces. While this approach is successful in many purely mechanical applications, strict surface theories fail to capture the electric potential drop across lipid membranes, the effects of surface charges, and electric fields within the membrane. Consequently, they do not accurately resolve Maxwell stresses in the interior of the membrane and its proximity. Furthermore, surface theories are generally unable to capture the effects of distinct velocities and tractions at the interfaces between lipid membranes and their surrounding bulk fluids. To address these shortcomings, we apply a recently proposed dimension reduction method to the three-dimensional, electromechanical balance laws. This approach allows us to derive an effective surface theory without taking the limit of vanishing thickness, thus incorporating effects arising from the finite thickness of lipid membranes. We refer to this effective surface theory as $(2 + \delta)$-dimensional, where $\delta$ indicates the thickness. The resulting $(2 + \delta)$-dimensional equations of motion satisfy velocity and traction continuity conditions at the membrane-bulk interfaces, capture the effects of Maxwell stresses, and can directly incorporate three-dimensional constitutive models.