A phase field model for isothermal solidification of multicomponent alloys

With the ability to model the kinetics and the pattern formation for solidification, a phase field model has been studied by many scientists. Currently available models, however, are restricted not only to binary alloys but also to those with substitutional solute elements. In this work, a new phase field model is developed to study solidification of a multicomponent alloy containing substitutional as well as interstitial solute elements. By employing the number of moles per unit volume as the concentration variable, the evolution equations of both the phase field and the concentration fields are derived from the free energy functional in the thermodynamically consistent way. In the model, the interfacial region is assumed to be a mixture of solid and liquid with the same composition, but with different chemical potentials. Based on this assumption, the phase field parameters are matched to the alloy properties and an interface thickness limitation is also deduced. Using the chemical rate theory, the phase field mobility is determined at a thin-interface limit condition under the assumption of negligible diffusivity in the solid phase. Another advantage of the model is that any thermodynamic database available in the literature can be directly ported to the model such that quantitative results for solidification of the real alloy systems could be made. As an example, a dendritic growth in an Fe-Mn-C ternary alloy is examined with the thermodynamic data from the commercial software Thermo-Calc code.