Multiscale modeling for ferroelectric materials: identification of the phase-field model's free energy for PZT from atomistic simulations

Within a knowledge-based multiscale simulation approach for ferroelectric materials, the atomic level can be linked to the mesoscale by transferring results from first-principles calculations into a phase-field model. A recently presented routine (Volker et al 2011 Contin. Mech. Thermodyn. 23 435-51) for adjusting the Helmholtz free energy coefficients to intrinsic and extrinsic ferroelectric material properties obtained by DFT calculations and atomistic simulations was subject to certain limitations: caused by too small available degrees of freedom, an independent adjustment of the spontaneous strains and piezoelectric coefficients was not possible, and the elastic properties could only be considered in cubic instead of tetragonal symmetry. In this work we overcome such restrictions by expanding the formulation of the free energy function, i.e. by motivating and introducing new higher-order terms that have not appeared in the literature before. Subsequently we present an improved version of the adjustment procedure for the free energy coefficients that is solely based on input parameters from first-principles calculations performed by Marton and Elsasser, as documented in Volker et al (2011 Contin. Mech. Thermodyn. 23 435-51). Full sets of adjusted free energy coefficients for PbTiO3 and tetragonal Pb(Zr, Ti)O-3 are presented, and the benefits of the newly introduced higher-order free energy terms are discussed.