Phase Field Method for Inhomogeneous Modulus Systems

One of the advantages of the phase-field method (PFM) is its ability to incorporate elastic interactions that dominate solid-state processes including phase transformations and plastic deformation. As mechanical equilibrium is attained much faster than chemical equilibrium, the former should be used as a constraint explicitly in deriving the governing equations of time-evolution of PFM order parameters. Current models for elastically anisotropic and inhomogeneous media in the literature do not impose such a constraint in their governing equations. In particular, they ignore the dependence of the total strain on the order parameters while evaluating the variational derivative of the elastic energy (VDEE). There is no mathematical proof to support this treatment and the fundamental thermodynamic consistency of such a models could be challenged. In this work, we present a rigorous and physically transparent formulation of PFM governing equations for elastically anisotropic and inhomogeneous media in which the mechanical equilibrium is explicitly used as a constraint. From our formulation, we show that VDEE computed in the Leo-Lowengrub-Jou (1998) (LLJ) model is thermodynamically consistent. We also show that the Wang-Jin-Khachaturyan (2003) (WJK) model has significant error in VDEE calculation. We proposed a first-order correction term to the WJK model which nullifies the error in VDEE and makes the model thermodynamically consistent. Also, we computed the VDEE by a numerical method with-out using any assumptions and compared the numerical VDEE with the LLJ, WJK and modified WJK models.