  We investigate a family of phase field models for simulating dendritic growth
of a pure supercooled substance. The central object of interest is the reaction
term in the Allen-Cahn equation, which is responsible for spatial distribution
of latent heat release during solidification. In this context, several existing
forms of the reaction term are analyzed. Inspired by the known conclusions of
matched asymptotic analysis, we propose new variants that are simple enough to
allow mathematical and numerical analysis and robust enough to be applicable to
solidification under very large supercooling. The resulting models are tested
in a number of numerical simulations focusing on mesh-dependence and model
parameter settings. Despite the phase interface thickness being relatively
large to make numerical computations feasible, the obtained results exhibit a
good quantitative agreement with experimental data from rapid solidification of
nickel melts.
