Phase Field Modelling - Department of Materials Science and ...

e chosen to represent strain and hence can be used to model phasechanges and the associated microstructure evolution.In the phase field modelling of solidification, there is no distinctionmade between the solid, liquid and the interface. All regions are describedin terms of the order parameter. This allows the whole domainto be treated simultaneously. In particular, the interface is not trackedbut is given implicitly by the chosen value of the order parameter as afunction of time and space. The classical formulation of the free boundaryproblem is replaced by equations for the temperature and phasefield.Appendix 1Recall (MP4–6) that a Taylor expansion for a single variable aboutX = 0 is given byJ{X} = J{0} + J ′ {0} X 1! + J ′′ {0} X22!...A Taylor expansion like this can be generalised to more than one variable.Cahn assumed that the free energy due to heterogeneities in asolution can be expressed by a multivariable Taylor expansion:g{y, z, . . .} =g{c 0 } + y ∂g∂y + z ∂g∂z + ...+ 1 []y 2 ∂2 g2 ∂y 2 + z2 ∂2 g∂z 2 +2yz ∂2 g∂y∂z + ... + ...in which the variables, y, z, . . . in our context are the spatial compositionderivatives (dc/dx, d 2 c/dx 2 , etc). For the free energy of a small volumeelement containing a one–dimensional composition variation (and