Dendrite tip selection during isothermal free growth in multi-component alloys: Marginal stability theories and insights from phase-field simulations

Abstract

The principal length scales associated with dendritic solidification, which being the dendritic tip-radius (Rtip) and the primary dendritic arm spacing, are functions of the relative solutal interdiffusivities of the different components in a multi-component alloy. In this paper, we firstly derive marginal stability based theories to predict Rtip during isothermal free growth in any generic multi-component alloy for any generic diffusivity matrix. Here, we extend the Ivanstov solution of the diffusion problem in the liquid with a parabolic solid-liquid interface and propose the closure conditions between the Rtip and the selected dendrite tip velocity (V) based on modification of the classical marginal stability criteria for multi-component situations. Additionally, we derive the constant Rtip2V from phase-field simulations and using the Ivantsov parabola as the approximate interface shape compute the Rtip and the V, which we term as “Approximate microsolvability prediction (AMP)”. Thereafter, we compare our predictions of Rtip,V as well as phase compositions with independent phase-field simulation results for different choices of the diffusivity matrix, where we show that the analytical theories based upon the marginal stability theory are unable to accurately predict variations of the dendrite tip velocity V and the Rtip. We thereby utilize the phase-field simulation results to investigate the tip selection in multi-component alloys and derive an appropriate solvability constant σ∗. We find that the σ∗ in a multi-component alloy is a function of the diffusivity ratios of the components for the case of a diagonal diffusivity matrix.