Dendritic solidification—II. A model for dendritic growth under an imposed thermal gradient

Abstract

A simple model for dendritic growth under an imposed thermal gradient is presented. The model satisfactorily explains the transition from a dendritic interface to a planar interface with increasing values of G L R , where G L is the thermal gradient and R the growth rate. In particular, the dendrite tip radius is shown to be a function of both R and G L . At constant growth rate, the dendrite tip radius increases with increasing G L , becoming infinite as G L R approaches the value given by the simple constitutional supercooling criterion. The tip undercooling and solute buildup also approach the corresponding values for a planar interface as the limit of stability of a planar front is reached. As the growth rate increases, the composition of the solid forming at the dendrite tip approaches the initial alloy composition at a critical growth rate. The critical growth rate is a function of G L and is shown to be intimately related to the growth velocity at which “absolute stability” is predicted by morphological stability theories. The dendrite tip undercooling, at the critical growth rate, is a function of the thermal conditions prevailing during growth and approaches ΔT 0 in the limit (where ΔT 0), is the equilibrium solidification range) as the interface becomes planar. The nature of the interface, dendritic, cellular or planar, as welt as the composition of the solid forming at large growth rates is determined by the value of a dimensionless length scale λ. The parameter λ is intimately related to another dimensionless quantity σ c , which is exactly the tip stability parameter obtained by Langer and Müller-Krumbhaar. The parameter σ c is obtained here strictly by considering steady-state behavior of a dendrite. Two estimates of λ have been provided. One of these estimates of λ yields values of σ c in remarkable agreement with the results of Langer and Müller-Krumbhaar. A comparison of the results obtained here with the Baker and Cahn analysis for partitionless solidification is also provided.