The anomalies of the properties of nanomaterials related to the distribution of the grain sizes

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