Morphological stability of a planar interface under rapid solidification conditions

The linear perturbation theory of Mullins and Sekerka for the stability of a planar interface is extended to the case of large thermal peclet numbers. It is shown that an absolute stability criterion for a planar interface exists for undercooled melts also. In light of these results, the conventional constitutional supercooling criterion is reexamined and a more general planar interface stability criterion is proposed which is valid for low as well as high growth rate conditions. The results of this stability analysis are applied to dendritic growth from pure undercooled melt.     

An analytical model for solute redistribution during solidification of planar,columnar, or equiaxed morphology

Existing models for solute redistribution (microsegregation) during solidification were reviewed. There are no analytical models that take into account limited diffusion in both the liquid and the solid phases. A new analytical mathematical model for solute redistribution was developed. Diffusion in liquid and in solid was considered. This model does not require a prescribed movement of the interface. It can be used for one-dimensional (1-D) (plate), two-dimensional (cylinder), or three-dimensional (3-D) (sphere) calculations. Thus, it is possible to calculate microsegregation at the level of primary or secondary arm spacing for columnar dendrites or for equiaxed dendrites. The solution was compared with calculations based on existing models, as well as with some available experimental data for the segregation of base elements in as cast Al-4. 9 wt pct Cu, INCONEL 718, 625, and plain carbon (0. 13 wt pct C) steel.     

Solute diffusion model for equiaxed dendritic growth

A new approach to the modeling of the equiaxed solidification of dendritic alloys is proposed. It is assumed that, in metallic alloys, microstructure formation is primarily controlled by solute diffusion (i.e. there is complete “thermal mixing” at the scale of one grain), and that the dendrite interface is an iso-concentrate at all times. The evolution of one dendritic grain is therefore modelled as follows: (i) complete mixing of solute within the interdendritic liquid; (ii) no back-diffusion in the solid; (iii) spherical solute diffusion in the liquid around the grain envelope; (iv) overall solute balance; (v) overall thermal balance; (vi) growth velocity, υ_g, of dendrite tips governed by the kinetic equation derived for the isolated dendrite case. By using an explicit finite difference scheme to solve these coupled equations, the concentration profiles, cooling curve, fraction of solid and evolution of dendritic grain envelope can be calculated. The intitial conditions used to start the calculation are provided by two parameters related to nucleation: the initial undercooling and the density of gains. The effect of nucleation and thermal conditions on equiaxed growth are studied. The theoretical predictions of recalescence and of the distribution of an interdendritic eutectic phase are in good agreement with experimental observations.     

The energy and solute conservation equations for dendritic solidification

The energy equation for solidifying dendritic alloys that includes the effects of heat of mixing in both the dendritic solid and the interdendritic liquid is derived. Calculations for Pb-Sn alloys show that this form of the energy equation should be used when the solidification rate is relatively high and/or the thermal gradients in the solidifying alloy are relatively low. Accurate predictions of transport phenomena in solidifying dendritic alloys also depend on the form of the solute conservation equation. Therefore, this conservation equation is derived with particular consideration to an accounting of the diffusion of solute in the dendritic solid. Calculations for Pb-Sn alloy show that the distribution of the volume fraction of interdendritic liquid (g _L) in the mushy zone is sensitive to the extent of the diffusion in the solid. Good predictions ofg _L are necessary, especially when convection in the mushy zone is calculated. Formerly Research Associate, The University of Arizona. Formerly Graduate Student, The University of Arizona.     

On the transition from short-range diffusion-limited to collision-limited growth in alloy solidification

Short-range diffusion-limited growth, collision-limited growth, and the transition between the two regimes are explained as natural consequences of a single model for the kinetics of alloy solidification. Analytical expressions are developed for the velocity-undercooling function of a planar interface during dilute alloy solidification, using Turnbull's collision-limited growth model and the Continuous Growth Solute Trapping Model of Aziz and Kaplan both with and without a solute drag effect. The interface mobility, −dv/dT, is shown to be very high (proportional to the speed of sound) if the alloy is sufficiently dilute or if the growth rate is sufficiently rapid for nearly complete solute trapping. The interface mobility is reduced by about three orders of magnitude (becoming proportional to the diffusive speed) at intermediate growth rates where partial solute trapping occurs. Differences in low velocity predictions of the models with and without solute drag are also discussed. Comparison of the results of the analytical expressions to numerical solutions of the non-dilute kinetic model for AlBe alloys shows that the dilute approximation breaks down at melt compositions on the order of 10 at.%. Similar variations in the interface mobility are shown for the disorder-trapping model of Boettinger and Aziz.     

An analytical model for nodular eutectic grain predictions during solidification

In this work, analytical expressions are derived for the globular eutectic transformation, which occurs during the solidification of nodular iron. The model incorporates heat and mass balance equations in considering nucleation and growth of nodular eutectic grains. In particular, an expression is found that relates the volumetric density of nodular eutectic grains to the maximum degree of undercooling. The proposed expression includes various experimental constants, which are well defined in nodular iron. Accordingly, good agreement is found to exist between the experimental and predicted data on the density of nodular eutectic grains and the maximum degree of undercooling.     

An experimental and analytical study of the solidification of a binary dendritic system

Experimental measurements are reported on the controlled solidification of a 30 pct aqueous solution of ammonium chloride in a two-dimensional slot. The actual measurements taken include the transient temperature profiles, the transient velocity fields (using tracers) and photographic observations. Through the statement of the differential energy balance and the laminar Navier-Stokes equations, written for both the liquid and the two phase regions, a mathematical model has been developed for the system. The theoretically predicted temperature and velocity profiles were found to be in good agreement with the experimentally measured values for both the two phase regions and the liquid region. For the experimental conditions the velocities in the liquid region were of the order of 0.4 to 0.8 cm/s, while the corresponding values for the two phase region were at least an order of magnitude smaller.     

An experimental and analytical study of the solidification of a binary dendritic system

Experimental measurements are reported on the controlled solidification of a 30 pct aqueous solution of ammonium chloride in a two-dimensional slot. The actual measurements taken include the transient temperature profiles, the transient velocity fields (using tracers) and photographic observations. Through the statement of the differential energy balance and the laminar Navier-Stokes equations, written for both the liquid and the two phase regions, a mathematical model has been developed for the system. The theoretically predicted temperature and velocity profiles were found to be in good agreement with the experimentally measured values for both the two phase regions and the liquid region. For the experimental conditions the velocities in the liquid region were of the order of 0.4 to 0.8 cm/s, while the corresponding values for the two phase region were at least an order of magnitude smaller.