A new methodology to obtain accurate models for ferroelectrics with application to BaTiO3

A new methodology based on semi-infinite optimization is proposed to obtain accurate yet simple phenomenological models for ferroelectric single crystals. The phenomenological models for ferroelectrics start with a Taylor series expansion of the governing thermodynamic potential, the elastic Gibbs function, in terms of the independent variables. The coefficients of the appropriately truncated series are determined, based on the experimental properties of the crystal. However, there is to date no method to determine the coefficients for an accurate correlation to the experimental measurements. To this end, a semi-infinite optimization problem is formulated, aiming at minimizing the error between the analytical model and experiments in terms of permittivity coefficients and spontaneous polarization. A model in the cubic and the tetragonal phases for barium titanate (BaTiO₃) single crystals for a particular choice of experimental measurements is used to demonstrate the workability of the proposed methodology. The resulting optimization problem has an infinity of inequality constraints. The optimal solution to the proposed semi-infinite optimization problem when used in the model, accurately predicts the ferroelectric properties of BaTiO₃ single crystals such as phase transitions, spontaneous polarization, permittivity, the range of temperature in the cubic and the tetragonal phase. The proposed methodology is not limited by the complexity of the phenomenological model, or the choice of the experimental measurements. Furthermore, the proposed methodology can be generalized to model ferroelastic materials