The $(2 + \delta)$-dimensional theory of the electromechanics of lipid membranes: I. Electrostatics

The coupling of electric fields to the mechanics of lipid membranes gives rise to intriguing electromechanical behavior, as, for example, evidenced by the deformation of lipid vesicles in external electric fields. Electromechanical effects are relevant for many biological processes, such as the propagation of action potentials in axons and the activation of mechanically-gated ion channels. Currently, a theoretical framework describing the electromechanical behavior of arbitrarily curved and deforming lipid membranes does not exist. Purely mechanical models commonly treat lipid membranes as two-dimensional surfaces, ignoring their finite thickness. While holding analytical and numerical merit, this approach cannot describe the coupling of lipid membranes to electric fields and is thus unsuitable for electromechanical models. In a sequence of articles, we derive an \textit{effective} surface theory of the electromechanics of lipid membranes, named a $(2+\delta)$-dimensional theory, which has the advantages of surface descriptions while accounting for finite thickness effects. The present article proposes a new, generic dimension-reduction procedure relying on low-order spectral expansions. This procedure is applied to the electrostatics of lipid membranes to obtain a $(2+\delta)$-dimensional theory that captures potential differences across and electric fields within lipid membranes. This new model is tested on different geometries relevant for lipid membranes, showing good agreement with the corresponding three-dimensional electrostatics theory.