Phase Field Method Equations Derived From Thermodynamic Extremal Principle

Abstract

The thermodynamic extremal principle (TEP) formulated in discrete parameters [1] represents a handy tool in modeling of thermodynamic processes in solids. As a further concept the phase field method (PFM) has gotten an established method for simulation of microstructure evolution and allows accounting parallel action of a number of thermodynamic processes in an extremely easy way. The PFM goes back to the seminal work by Ginzburg and Landau [2], dealing with the ordering of atoms within unit cells, see the discussions by Penrose and Fife [3] and Gurtin [4]. In this contribution it is shown that the elementary equations of the PFM can be directly derived from the TEP. For doing this it is necessary to express the total Gibbs energy consisting of the chemical Gibbs energy of a mixture of individual phases (two different crystalline phases and one phase characterizing the amorphous structure in the interface) in the system and of a penalty term due to square of gradient of the order parameter. The order parameter also represents an indicator determining, how the thermodynamic quantities or kinetic parameters of individual components, e.g. the chemical potentials or diffusion coefficients, must be calculated (i.e. to which phase the thermodynamic quantities are addressed). If no diffusion is assumed in the system, the dissipation is expressed exclusively as quadratic form of the rate of the order parameter.