  Using the Landau-Ginzburg-Devonshire phenomenological approach, we perform
finite element modeling of the electric polarization, electric field, and
elastic stresses and strains in a core-shell nanoparticle, where the
ferroelectric core has a shape of prolate cylinder. Calculations reveal the
quadrupolar-type diffuse domain structure consisting of two oppositely oriented
diffuse axial domains located near the cylinder ends, which are separated by a
region with a zero-axial polarization; we have termed this "flexon" to
underline the flexoelectric nature of its axial polarization. Analytical
calculations and FEM results have proven that a change of the flexoelectric
coefficient sign leads to a reorientation of the flexon axial polarization; as
well as an anisotropy of the flexoelectric coupling critically influences the
flexon formation and related domain morphology. The flexon polarization forms a
drop-shaped region near the ends of the cylinder, with a distinct chiral
structure that is determined by the sign of the flexoelectric coupling
constant. Its rounded shape, combined with its distinct chiral properties and
the localization nature near the surface are reminiscent of a those of Chiral
Bobber structures in magnetism. In the azimuthal plane, the flexon displays the
polarization state of a meron. We show that this new type of chiral
polarization structure is stabilized by an anisotropic flexoelectric coupling.
It is important to note that the Lifshitz invariant describing the
flexoelectric effect, which couples the electric polarization and elastic
strain gradients, plays a determining role in the stabilization of these chiral
states. It thereby provides an energetic interaction that, similar to the
recently predicted ferroelectric Dyzaloshinskii-Moryia interaction, can lead to
the formation of chiral polarization states, and, by extension, ferroelectric
skyrmions.
