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Analysis on Irreversible Processes using the Phase-Field Variational Approach with the Entropy or Energy Functional

The variational approach usually used in phase field models (PFVA) is applied here to analyse complex irreversible processes such as thermoelectric (TE) effects and thermally driven mass transport (TDMT). Complex irreversible processes arise from the coupling effects between simple irreversible processes. Each simple irreversible process is assiciated with an entropy or energy density function. During complex irreversible processes with multiple fields present, this entropy or energy density function is assumed to be dependent on all independent field variables. Using the total entropy functionals, the TE effects and TDMT are analysed and important kinetic coefficients such as the Seebeck coefficient and the heat of transport are determined with straightforward physical contents. Using the total energy functionals, the linear irreversible processes are analysed with the Onsager approach and the nonlinear irreversible processes with PFVA. It is found both the Onsager's relations and the fluxes defined using PFVA guarantee the satisfaction of the first law of thermodynamics during the process of conversion of energies. To analyze the diffusion process under the influence of elasticity, PFVA is also modified to incorporate the reversible evolution of elastic fields. It is shown energies are conserved via both the irreversible diffusion process and the reversible evolution of the elastic fields. Finally, PFVA is generalized to study nonequilibrium thermodynamics using an extra kinetic contribution to the entropy density function. The analyses can be extended to a nonequilibrium thermodynamic system with multiple physical fields present. Thus, it is believed PFVA has the potential of not only significantly advancing our understanding of the thermodynamics of irreversible processes, but also making thermodynamics as a discipline and the study of it truly dynamic.