Flexoelectric Fluid Membranes in External Electric Fields

This manuscript presents a geometrical framework for flexoelectric membranes based on stress tensors which mimic the response of a flexoelectric fluid membrane to an external electric field. This framework is used to study numerically the morphology of spherically confined flexoelectric fluid membrane vesicles in an external uniform electric field. The confinement induces membrane deformations, which lead to its polarization and interactions with the external field. Without such electric fields, the equilibrium shapes of the vesicle were categorized in a geometrical phase diagram as a function of reduced volume and the scaled area in the past [1, 2]. When the area of the flexoelectric fluid membrane is a bit larger than the area of the confining sphere, an axisymmetric invagination can be found with a simple numerical integration scheme. A non-vanishing electric field induces an additional elongation of the confined vesicle, which is either perpendicular or parallel depending on the sign of the electric field parameter. Higher values of surface area or the electric field parameter reduce the symmetry of the system resulting in more complex folding. To find equilibrium configurations as a function of volume, area and coupling with the electric field, two numerical methods were employed. Despite some rather crude approximations such as assuming a constant electric field, interesting shape transformations and symmetry breaking are found. Moreover, the resulting shapes indicate that transition lines are shifted in the presence of an electric field, which leads to the transition shapes reminiscent of the main protagonist in the video game "Pac-Man". In these shapes, the invagination of the membrane is not axisymmetric but deforms into a large elongated slit reminiscent of shapes that can be found with the area-difference-elasticity (ADE) model for confined membranes without an electric field [3]. Self-contacts, as observed in this work, can potentially lead to a shape transition, from a spherical to a toroidal vesicle topology via membrane fusion. It turns out that the spherical topology is preferred for typical values of the material parameters when the electric field vanishes. Flexoelectricity could potentially facilitate topology changes. The obtained folding patterns could be of interest to biophysical and technological applications alike.