Effect of surface tension and depolarization field on ferroelectric   nanomaterials properties
  The theory of size effects of the properties of nanocrystalline ferroelectric ceramic or nanoparticle powder allowing for surface tension and depolarization field is proposed. Surface tension was included into free energy functional and surface energy was expressed via surface tension coefficient. The latter was shown to be dependent on temperature due to its relation to dielectric permittivity of the nanoparticles. The depolarization field effect was calculated in the model taking into account the space charge layer on the surface, this space-charge being able to compensate depolarization field in the bulk material. Euler-Lagrange Equation for inhomogeneous polarization of nanomaterial with boundary condition where extrapolation length was shown to be temperature dependent quantity was solved analytically both in paraelectric and ferroelectric phase of size driven phase transition. This phase transition critical temperature dependence on the particle size was calculated. Temperature and size dependence of nanomaterials polarization and dielectric susceptibility was obtained. The possibility to calculate these and other properties by minimization of conventional free energy in the form of different power polarization series, but with the coefficients which depend on particles size, temperature, contribution of depolarization field and surface tension coefficient was demonstrated. These latter effects were shown to influence essentially the nanomaterial properties. The comparison with available experimental data is performed. 
M. D. Glinchuk, A. N. Morozovskaya