ABSTRACT
For practical rates of solidification, it is often desirable for phase field simulations to reproduce the results of the standard sharp interface models of solidification [58]. This occurs in the limit when there is a clear separation of scales between the interface width (W) and the solute or thermal diffusion fields around a solidifying front. One way to do this in principle is to make the interface width small (denote this W ≪ d o https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003204312/34e3ad47-9f0c-49dc-acd7-5caec98ace28/content/math6_1.tif"/> , where d o is the capillary length of the solid-liquid interface as shown by Caginalp [59, 60]). This is not practical however, as the grid resolution and numerical time scales that result would be computationally intractable for numerical simulations at low to even moderate undercooling. A common strategy for making phase field simulations involving this solidification regime is to smear the interfaces of the ϕ α fields. Doing so in an uncontrolled way however creates spurious kinetics at the interface, excessive levels of solute trapping in bulk phases and alters the flux conservation across interfaces from its classic form due to lateral diffusion and interface stretching. These effects are physically relevant at rapid rates of solidification where interface kinetics across even a microscopic interface control solidification. For slow to moderate rates of solidification, however, these effects are negligible, and should be eliminated, or reduced, when using diffuse interface models to simulate solidification. The work of Refs. [9, 10, 61] has shown that these effects can be countered numerically—at least in two-phase binary alloys—by 30adopting special choices of the interpolation function for solute diffusion and the chemical potential, as well as an addition of a so-called anti-trapping flux in the mass transport equations. While the latter adoption precludes the phase field equations from being derived from variational derivatives of the grand-potential, it is suitable if all we care about is emulating the appropriate sharp-interface kinetics across the solid-liquid interface. We will implement analogous modifications to the multi-order parameter phase field models derived in the previous chapters. 1 These are discussed as follows.
