On secondary dendrite arm coarsening in peritectic solidification

Abstract

A model for isothermal coarsening of secondary dendrite arms in peritectic reaction and transformation (liquid + primary-phase → peritectic-phase) is proposed to evaluate the secondary dendrite arm spacing (λ₂) of the primary phase in directional solidification of peritectic alloys. The model defines three stages for thin-arm dissolution (or thick-arm coarsening), i.e. the initial, intermediate and final stages: the initial thin-arm dissolution through the primary phase is sustained solely by the Gibbs–Thomson effect; the intermediate thin-arm dissolution through the peritectic phase is driven by Gibbs–Thomson effect but retarded by the peritectic reaction and transformation; the final dissolution through the primary and peritectic phases is enhanced by the Gibbs–Thomson effect and the phase transformation. The kinetics of peritectic reaction and transformation were found to be crucial to determine the thin-arm dissolution, which were characterized by the reaction constant (f) and the diffusion coefficient of solute in solid peritectic-phase (D_S), respectively. The present model shows that λ₂V^m is constant for a given Pb–Bi peritectic alloy, where V is growth velocity, and the factor, m, ranges from 1/3 to 1/2, rather than that normally observed (e.g. 1/3) for single-phase solidification. It is also notable that the calculated λ₂ for a Zn–7.37 wt.% Cu peritectic alloy was reasonably consistent with our earlier experiments for various growth velocities.

Introduction

Peritectic solidification has attracted more attention in recent years because peritectics frequently occur in steels, bronzes, and other commercial alloys [1], [2]. It involves one solid phase reacting with a liquid phase on cooling to produce a second solid phase [1], e.g. ɛ + L → η in the Zn–Cu peritectic system [2]. Many studies on peritectic solidification have recently focused on the microstructure and phase selections [3], [4], [5], peritectic transformation kinetics [6], and formation of banded structure [7]. However, very little is known about the effects of peritectic reaction and transformation on the length scales of microstructures, such as primary or secondary dendrite arm spacing [1], [8].

During peritectic solidification, three different stages have been identified in terms of the growth of peritectic phase [1]: (i) the peritectic reaction, by direct interaction between the primary phase and liquid; (ii) the peritectic transformation, during which the peritectic phase thickens by solid-state diffusion; and (iii) direct solidification of peritectic phase from the liquid onto the previously formed layer of peritectic phase. However, the effect of each stage on the evolution of secondary dendrite arm spacing is less explored, and thus is not well understood.

The secondary dendrite arm spacing in single-phase solidification is largely determined by annealing processes occurring during growth of the dendrites, in which the thinner secondary arms melt and the thicker branches thicken [8], [9], [10]. Kattamis and Flemings [9] and Feurer and Wunderlin [10] found that the secondary dendrite arm spacing, ${\lambda }_2^{*}$, is proportional to the cube root of solidification time (t_f), and proposed

$${\lambda }_2^{*}=5.5{(M{t}_{\text{f}})}^{1/3}$$

and

$$M=\frac{\Gamma D\ln (C_{\text{l}}^{\text{m}}/C_0)}{m(k-1)(C_{\text{l}}^{\text{m}}-C_0)}$$

where Γ is the Gibbs–Thomson coefficient, D the diffusion coefficient in the liquid, m the liquidus slope, k the equilibrium distribution coefficient, C₀ the initial alloy concentration and $C_{\text{l}}^{\text{m}}$ is the maximum concentration in the final drop of liquid. In the case of directional solidification, the local solidification time is given by t_f = ΔT′/GV where G is the temperature gradient, V is the growth velocity, and ΔT′ is the non-equilibrium solidification range. Kurz and Fisher [8] suggested that ΔT′ can be evaluated by $\Delta T^{\prime }=m(C_0-C_{\text{l}}^{\text{m}})$. For a eutectic system, $C_{\text{l}}^{\text{m}}$ can be equated with the eutectic composition (C_E), i.e. $C_{\text{l}}^{\text{m}}=C_{\text{E}}$ [8]. But when there is no eutectic reaction in the system, such as for solid solution and peritectic alloys, the determination of $C_{\text{l}}^{\text{m}}$ might be difficult [8]. In this case, considering the effect of back-diffusion in the solid, Kurz and Fisher suggested that $C_{\text{l}}^{\text{m}}=C_0{(2{\alpha }^{\prime }k)}^{-p/u}$, where u = 1−2α′k, p = 1−k and α′ is a function of t_f.

However, none of the classical theories [8], [9], [10] considered the effect of peritectic reaction and transformation on the secondary dendrite arm-coarsening process, despite the fact that a new phase formed via peritectic reaction and transformation may change the solute transport between the dendrite arms, so as possibly to provoke alternative dendrite arm-coarsening mechanisms. The present paper proposes a model for isothermal coarsening of secondary dendrite arms, in which effects of the peritectic reaction and transformation have been taken into account to evaluate the secondary dendrite arm spacing of the primary phase in directional solidification of peritectic alloys.