Analytic model for planar growth of a solid germ from an undercooled melt

Planar growth of a solid germ of infinitesimal initial thickness from an undercooled melt is addressed within a continuum model with linear interfacial kinetics. For this problem, we present an analytic solution that describes the global features of the process beyond the initial and long-time limits. In particular, it yields analytic expressions for the main characteristics of the transient regime. This solution is obtained within the heat balance integral method with a relatively simple boundary layer approximation for the temperature profile in the melt. The analytic solution is validated by the known asymptotic results, as well as by comparison with the numerical solution of the problem considered. The emerging physical picture is formulated in terms of the evolution characteristics of the phase-change front and those of the thermal boundary layer.