Parameter identification in phase field models for ferroelectrics

Abstract

1 Field equations The ferroelectric body under consideration occupies a region B with boundary ∂B. The balance equations for the mechanical stress σ and the electric displacement D are given by divσ = 0 and divD = 0 in B, respectively, where inertia terms, volume forces, and volume charges are neglected. The linearized strain tensor and the electric field are defined as ε = 1 2 (gradu + (gradu) T) and E = −gradϕ in B, respectively, where u is the mechanical displacement, and ϕ the electric potential. Using the spontaneous polarization P as the order parameter, the phase field potential H is given by H = 1 2 (ε − ε 0) : [(ε − ε 0)] − (ε − ε 0) : T E − 1 2 E · KE − P · E + κ s G ψ(P) + 1 2 κ i GG P 2 0 gradP 2 , where the last two terms are the phase separation energy H sep and the interface energy H int , respectively, and the remaining terms contribute to the electric enthalpy H ent (cf. [1]), i. e. H = H ent + H sep + H int. In this equation, , , and K are the elastic stiffness, the piezoelectric coupling constants, and the dielectric tensor. The spontaneous strain ε 0 and depend on P and are defined as ε 0 (P) = 3 2 ε 0 P 2 P 2 0 e ⊗ e − 1 3 1 , e = P P and kij (P) = P 3 P 3 0 b e i e j e k + b ⊥ (δ ij − e i e j)e k + b = 1 2 [(δ ki − e k e i)e j + (δ kj − e k e j)e i ]. Furthermore, P 0 and ε 0 are the equilibrium polarization and spontaneous strain of the poled phase with no external loads, and b , b ⊥ , and b = are the piezoelectric constants. As shown in [2], the parameters G and can be identified as the specific energy and width of a 180 • domain wall, respectively, if the calibration constants κ s and κ i are chosen appropriately. Finally, in 2d, the energy landscape of the phase separation potential is represented by the sixth oder polynomial ψ = 1 + a 1 (P 2 1 + P 2 2) + a 2 (P 4 1 + P 4 2) + a 3 P 2 1 P 2 2 + a 4 (P 6 1 + P 6 2). The constitutive equations can be derived from the potential by standard arguments of rational thermodynamics: σ = ∂H ∂ε = (ε − ε 0) − T E and D = − ∂H ∂E = (ε − ε 0) + KE + P. The time evolution of the order parameter P is given by a Ginzburg-Landau type equation. Due to the appearance of the gradient of P in the potential the variational derivative appears in the evolution equation ˙ P = −M δH δP = −M ∂H ∂P − div ∂H int ∂gradP = −M ∂H ent ∂P + κ sep G ∂ψ ∂P − κ int GG P 2 0 ∆P , where M is a mobility constant and ∆ the Laplace operator. For details of the evaluation of the partial derivatives the reader is referred to [1]. We briefly note that boundary conditions need to be specified for u, ϕ, and P or their respective gradients.