Abstract: Electric field plays an important role in ferroelectric phase transition. There have been numerous phase field formulations attempting to account for electrostatic interactions subject to different boundary conditions to study phase transitions and polarization switching in ferroelectric thin films. We develop new variational forms of the phase field electrostatic energy and the relaxation dynamics of the polarization vector that involve a hybrid representation in both real and Fourier variables. The new formulations avoid ambiguities appeared in earlier studies and lead to much more effective ways to perform variational studies and numerical simulations. An efficient numerical scheme for the variational phase field model based on variational forms of the electrostatic energy and the relaxation dynamics of the polarization vector is also developed. The spatial discretization combines the Fourier spectral method with the finite difference method to handle three-dimensional mixed boundary conditions. It allows for an efficient semi-implicit discretization for the time integration of the relaxation dynamics. This method avoids explicitly solving the electrostatic equilibrium equation (a Poisson equation) and eliminates the use of associated Lagrange multipliers. We present several numerical examples including phase transitions and polarization switching processes in a single domain to demonstrate the effectiveness of the proposed method.

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