  In this work we numerically study the switching dynamics of a 2D cholesteric
emulsion droplet immersed in an isotropic fluid under an electric field, which
is either uniform or rotating with constant speed. The overall dynamics depend
strongly on the magnitude and on the direction (with respect to the cholesteric
axis) of the applied field, on the anchoring of the director at the droplet
surface and on the elasticity. If the surface anchoring is homeotropic and a
uniform field is parallel to the cholesteric axis, the director undergoes deep
elastic deformations and the droplet typically gets stuck into metastable
states which are rich in topological defects. When the surface anchoring is
tangential, the effects due to the electric field are overall less dramatic, as
a small number of topological defects form at equilibrium. The application of
the field perpendicular to the cholesteric axis usually has negligible effects
on the defect dynamics. The presence of a rotating electric field of varying
frequency fosters the rotation of the defects and of the droplet as well,
typically at lower speed than that of the field, due to the inertia of the
liquid crystal. If the surface anchoring is homeotropic a periodic motion is
found. Our results represent a first step to understand the dynamical response
of a cholesteric droplet under an electric field and its possible application
in designing novel liquid crystal-based devices.
