  We explore the impact of the flexoelectric effect and Vegard effect (chemical
pressure) on the phase diagrams, long-range polar order and related physical
properties of the spherical ferroelectric nanoparticles using
Landau-Ginzburg-Devonshire phenomenological approach. The synergy of these
effects can lead to the remarkable changes of the nanoparticles' phase
diagrams. In particular, a commonly expected transition from ferroelectric to
paraelectric phase at some small critical size is absent; so that the critical
size loses its sense. Contrary, the stabilization of the ferroelectric phase
manifests itself by the enhancement of the transition temperature and
polarization with the particle size decrease. Ferroelectric phase reentrant
phenomenon was observed earlier in the tetragonal BaTiO3 nanospheres of radii
5-50 nm [Zhu et al., JAP 112, 064110 (2012)] and stayed unexplained up to now.
Our calculations have shown the physical mechanism of the exciting phenomenon
is the flexo-chemo-effect. Since the spontaneous flexoelectric coupling, as
well as ion vacancies, should exist in any nanostructured ferroelectrics,
obtained analytical results can be valid for many nanoferroelectrics, where
reentrant phases appearance can be forecasted.
