The peculiarities of the specific heat and dielectric permittivity related to the grain size distribution in ferroelectric nanomaterials

We have performed calculations of the size effect in the temperature dependence of the BaTiO3 nanograin ceramic specific heat and dielectric permittivity. We took into account the distribution of grain sizes that exists in any real nanomaterial. This distribution led to a distribution of the temperatures of the size driven transition from the ferroelectric to the paraelectric phase because of the relation between the temperature and the size. We calculated the transition temperature distribution function on the basis of the size distribution function. This function allows us to calculate the temperature dependence of any physical quantity for a nanomaterial. As examples, we calculated the specific heat and dielectric permittivity for nanograin ferroelectric ceramics. The results demonstrate the strong influence of the size distribution on the observed properties and especially on the values of the critical size and temperature extracted from experiment. We carried out a comparison of the theory with the measured specific heat and dielectric permittivity for BaTiO3 nanomaterial. The theory developed described the experimental data fairly well. The possibility of extracting size distribution function parameters as well as real values of critical parameters from experimental data is discussed.