Thermodynamically consistent incorporation of the Schneider rate equations into two-phase models

We formulate a solid-liquid two-phase model including viscous stresses, heat conduction in the two phases, as well as heat exchange through the interface, and a phase change in the structure of nonequilibrium thermodynamics described by a general equation for the nonequilibrium reversible-irreversible coupling (GENERIC). The evolution of the microstructure is studied in terms of the Schneider rate equations introducing the nucleation rate and the radial growth rate of the solid phase. The application of the GENERIC structure shows that this radial growth factor is not an additional, independent material function but is to be expressed in terms of the difference in the chemical potentials, in the temperatures, and in the pressures between the two phases. The contribution due to the pressure difference appears in conjunction with the surface tension in such a way, that a driving force results only if deviations from a generalized version of the Laplace equation occur. Furthermore, it is found that for conditions under which the radial growth rate is zero, the nucleation rate must vanish.