Phase Field Modeling on Morphology Development

In the Kobayashi phase field approach [12], a phase order parameter, ip(r, t), is introduced to express explicitly the Uquid and solid phases. It assumes a constant value in each bulk phase, for example, ip = 1 in the solid and ip = 0 in the Uquid phase. 

Basically, the time-dependent Ginzburg-Landau (TDGL) equation [12] relates the temporal change of a phase order parameter to a local chemical potential and a nonlocal interface gradient. With respect to a non-conserved phase field order parameter, the TDGL model A equation is customarily described as  

In general, the local free energy of solidification may be expressed in accordance with the Landau expression in powers of the order parameter ip as,/ocai = p + 

The total free energy is composed of local and nonlocal free energy densities. The nonlocal free energy density can be written in terms of the gradient free energy density describing the growth process as  

Xo[l + ecos(40)], where Xq is a constant and e is the strength of surface energy anisotropy. Further, Kobayashi [12] coupled the time evolution equation pertaining to the crystal phase order parameter to the energy balance (i.e., heat conduction) equation and demonstrated the evolution of side-branched dendritic structures growing into an undercooled melt. 

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