On secondary dendrite arm coarsening in peritectic solidification

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.     

Analysis on fluid permeability of dendritic mushy zone during peritectic solidification in a temperature gradient

Compared with the growing applications of peritectic alloys,none research on the fluid permeability K of dendritic network during peritectic solidification has been reported before.The fluid permeability K of dendritic network in the mushy zone during directional solidification of Sn-Ni peritectic alloy was investigated in this study.Examination on the experimental results demonstrates that both the temperature gradient zone melting (TGZM) and Gibbs-Thomson (G-T) effects have obvious influences on the morphology of dendritic network during directional solidification.This is realized through different stages of liquid diffusion within dendritic mushy zone by these effects during directional solidification.The TGZM effect is demonstrated to play a more important role as compared with the G-T effect during directional solidification.Besides,it is shown that the evolution of dendrite network is more complex during peritectic solidification due to the involvement of the peritectic phase.Through the specific surface Sv,analytical expression based on the Carman-Kozeny model was proposed to analyze the fluid permeability of dendritic mushy zone in directionally solidified peritectic alloys.In addition,it is interesting to find a rise in permeability K after peritectic reaction in both theoretical predication and experimental results,which is different from that in other alloys.The theoretical predictions show that this rise in fluid permeability K after pedtectic reaction is caused by the remelting/resolidification process on dendritic structure by the TGZM and G-T effects during peritectic solidification.     

Solute distribution at the solid-liquid interface during steady-state peritectic solidification

The solute distributions at the solid-liquid interface during steady-state directional solidification of two contrasting peritectic systems were examined in detail by electron probe microanalysis. In the Al-Al₃Ti system the peritectic transformation plays a negligible part in the solidification reaction whereas in the Cd-Cd₃Ag system the peritectic transformation dominates. Knowing this, the solid-liquid interface profile during solidification can be predicted with reasonable accuracy on the assumption of equilibrium at all points on the solid-liquid interface. In the Cd-Cd₃Ag system the composition of the peritectic product is rapidly equilibrated by solid-state diffusion. A diffusion coefficient is estimated.     

Numerical model for nucleation of peritectic alloy during unidirectional solidification

Based on transient nucleation theory, a numerical model has been constructed to describe the nucleation process of a new phase in front of the liquid–solid interface of a prior steady-growth phase in peritectic alloy with the combination of the concentration field calculated by a self-consistent numerical model for cellular/dendritic growth. The results show that the nucleation incubation time of a new phase varies with the solidification rate during unidirectional solidification. During unidirectional solidification of the Zn–4.0 wt.% Cu alloy, the incubation time changes very slightly when the solidification rate increases from 50 to 500 μm/s, but it increases significantly when the solidification rate exceeds 500 μm/s. The calculated results show a reasonable agreement with the experimental ones. This model reveals that nucleation of a new phase is time-dependent and reasonably explains the effect of the solidification velocity on the behaviors of nucleation and growth of ɛ dendrites in the matrix of the η phase in unidirectional solidification of Zn rich Zn–Cu alloys.     

Effect of peritectic reaction on the migration of secondary dendrite arms in the presence of tertiary dendrites: analysis of a directionally solidified Sn–36 at.%Ni peritectic alloy

Directional solidification experiments have been performed on Sn–36 at.%Ni peritectic alloy in a constant temperature gradient at different growth velocities. Experimental result shows that a “sawtooth” morphology forms on secondary dendrite arms during the migration of secondary dendrites in the presence of tertiary dendrite arms. A theoretic model is therefore proposed to describe the formation of this “sawtooth” morphology in the peritectic solidification with tertiary dendrite arms taken into consideration. The migration of secondary dendrite arms is caused by remelting/solidification at the hot/cold sides of a liquid pool between secondary dendrite arms, which is a form of temperature gradient zone melting. And, the “sawtooth” morphology is ascribed to the difference in remelting velocity at the hot side of liquid pool during the migration of secondary dendrite arms due to the presence of tertiary dendrite arms. In addition, the proceeding of peritectic reaction can accelerate the formation of “sawtooth” morphology.     

Modeling constrained dendrite growth in rapidly directional solidification

Incorporating the effect of nonlinear liquidus and solidus, a constrained dendrite growth model was established for rapidly directional solidification, where both the interface and the bulk liquid are under local non-equilibrium conditions. The effect of nonlinear liquidus and solidus was first introduced into the interface kinetics, based on which the marginal stability criterion was then derived. By this way, not only the temperature dependent equilibrium partition coefficient k _e, equilibrium liquidus slope m _l and equilibrium solidus slope m _s but also the derivation of k _e with temperature are considered. The model is more physically realistic than the works in which the effect of nonlinear liquidus and solidus is not incorporated into the interface kinetics and hence the temperature dependent m _s is not embodied. Applying the model to rapidly directional solidification of Ag–15 wt% Cu alloy, the effects of nonlinear liquidus and solidus and local non-equilibrium on the constrained dendrite growth were studied.     

Microstructure and orientation variation during cell/dendrite transition in directional solidification of a single crystal nickel-base superalloy

The crystallographic orientation and microstructure variation during cell/dendrite transition of a nickel-base single crystal superalloy have been analyzed. The crystal with the off-<0 1 1> orientation grew with a cellular interface at 6 μm s^?1 while with a dendrite interface at 100 μm s^?1. The crystal kept the orientation of bottom seed in the dendritic growth stage, and there was a short transitional zone. The deviation angle between the cellular orientation and the heat flow direction became small in the cellular growth stage. The cellular growth direction was in accordance with the heat flow direction and was independent of the crystallographic orientation. At the same crystallographic orientation, the dendrites had a large primary dendrite arm spacing during the cell/dendrite transition. The cellular crystal with a smaller orientation deviation from the cylindrical axis showed a larger primary arm spacing.     

Secondary and tertiary dendrite arm spacing relationships in directionally solidified Al-Si alloys

Secondary, ₂, and tertiary, ₃, dendrite arm spacings have been measured from Al-Si alloys which were directionally solidified as functions of growth velocity,V, temperature gradient,G, and composition,C _o. Both ₂ and ₃ decreased as the imposed growth velocity and silicon concentrations were increased, and for each function a systematic variance in the rate was seen. Complications with measuring secondary arm spacings are shown and it was found that the tertiary arm data agree much better with coarsening theory, the implication being that ₃, when measurable, is a more representative and reliable measure of the solidification history than ₂.     

Simulation of Si concentration effect on the permeability for columnar dendrite structures during solidification of Al–Si alloy

A numerical model has been developed for the determination of liquid flow permeability through columnar dendrite during growth and segregation in Al–Si mushy alloys. The model is inclusive of two stages, first numerical evolution of the dendrite shape during growth, and second numerical determination of the permeability. Simulation results shown which Si concentration by transform of dendrite shape dendrite could reduce of the interdendritic liquid permeability.     

A boundary element method analysis of temperature fields and stresses during solidification

Summary The subject of this paper is the analysis of transient temperature fields and associated stress fields in solidifying bodies. A mathematical model for such problems, together with a solution procedure using the boundary integral equation method, are presented here. Numerical and some experimental results, for the one-dimensional solidification of an aluminum bar, are presented and discussed.With 9 Figures     

Kinetic growth characteristics of a single dendrite during crystallization from a solution

The behavior kinetics of the transition rate during free quasitwo-dimensional growth of a single dendrite from an aqueous NH₄Cl solution is studied experimentally. It is established that the integral curves are described by a Weibull distribution and the possibility of applying the Kolmogorov-Avrami theory to describe the kinetics of dendritic crystallization is discussed. Small-amplitude oscillations of the growing dendrite mass are observed against the background trend, having a quasiperiodic nature with a frequency around 0.1 Hz. The nature of these oscillations is related to the interaction between the diffusion fields of existing and newly forming secondary branches. Pis’ma Zh. Tekh. Fiz. 25, 64–70 (October 26, 1999)     

Electric domains created in polarized molecular condensates during condensation onto a substrate with a one dimensional temperature gradient

The distribution of the condensate surface potential was studied during condensation of a polarized molecular beam onto a substrate with a one-dimensional temperature gradient. It was found that on the free surface of the condensate layer, electric regions of different sign appeared: electric domains. Estimation of the surface charge density within a single domain in the case of a poly-domain structure of the condensate gives considerably smaller values than that obtained for the mono-domain condensate with a uniform polarized structure.     

Effect of a temperature gradient on the heat transfer during laminar natural convection along a vertical wall

Results are shown of calculations and of an experimental study concerning the heat transfer during laminar natural convection along a vertical wall, in the general case when the temperature excess varies along the wall height according to an arbitrary law.