S.R. Coriell

Solidification Microstructures: Recent Developments, Future Directions

The status of solidification science is critically evaluated and future directions of research in this technologically important area are proposed. The most important advances in solidification science and technology of the last decade are discussed: interface dynamics, phase selection, microstructure selection, peritectic growth, convection effects, multicomponent alloys, and numerical techniques. It is shown how the advent of new mathematical techniques (especially phase-field and cellular automata models) coupled with powerful computers now allows the following: modeling of complicated interface morphologies, taking into account not only steady state but also non-steady state phenomena; considering real alloys consisting of many elements through on-line use of large thermodynamic data banks; and taking into account natural and forced convection effects. A series of open questions and future prospects are also given. It is hoped that the reader is encouraged to explore this important and highly interesting field and to add her/his contributions to an ever better understanding and modeling of microstructure development.

Convective and interfacial instabilities during unidirectional solidification of a binary alloy

Abstract The onset of coupled convective and constitutional interfacial instabilities during the directional solidification of a single phase binary alloy at constant velocity vertically upwards (positive z -direction) is treated by a linear stability analysis. We consider a system for which the temperature gradient alone would cause a negative density gradient and the solute gradient alone would cause a positive density gradient. The temperature and concentration fields are coupled through the hydrodynamic equations. The solidification boundary conditions at the solid-liquid interface couple the hydrodynamic and interfacial stability phenomena. Specific calculations were made for physical properties appropriate to the solidification of lead containing tin. Results indicate that the stability-instability criterion differs substantially from the criterion of a net neutral density gradient. For a temperature gradient in the liquid of 200 K/cm and for velocities in the range 1–40 μm/s, a convective-like long wavelength instability occurs at a critical concentration that increases with velocity; whereas for V > 40 μm/s, the concentration at which instability occurs decreases as velocity is increased and the values of concentration and wavelength at the onset of instability correspond to the predictions of previous morphological stability theory in which density changes and convection are neglected. Application of a vertical static magnetic field increases the critical concentration for convective instabilities but a field of a tesla (10 4 gauss) is needed to cause an order of magnitude change.

Stability of liquid zones

The criterion for the stability of a liquid zone of volume V and length L between two parallel circular coaxial plates of radius R and Ru is formulated via the calculus of variations. The liquid in the zone is static or uniformly rotating with angular velocity Ω and is subject to a gravitational acceleration g along the axis of the plates. For Ω = 0, V = πR2L, R = Ru, numerical computations give the maximum stable value of LR as a function of the Bond number e ≡ ρgR 2γ, where ρ is the density difference between the liquid and the surrounding fluid and γ is the liquid-fluid surface tension. Experimental measurements of the stability of water zones are in good agreement with the numerical results. Experimental and theoretical results for horizontal liquid zones (gravitational field perpendicular to the axis of the plates) show that for e < 0.5 horizontal zones are more stable than vertical zones.

Oscillatory morphological instabilities due to non-equilibrium segregation

Abstract Linear perturbation theory is used to study morphological instability for rapid directional solidification at constant velocity under conditions where there is significant departure from local equilibrium at an initially planar solid-liquid interface. Under conditions where the segregation coefficient k depends significantly on velocity v , the stability criterion depends explicitly on both k and ∂ k /∂ v and instabilities that are oscillatory in time can occur for solute concentrations that are much smaller than those that would be necessary to cause non-oscillatory instability for the same k if ∂ k /∂ v were simply neglected. Such oscillatory instabilities seem to b e related to a “solute pump” mechanism according to which local changes in k , due to periodic changes in local interface velocity v , can occur out of phase with local interface position, thus resulting in lateral inhomogeneity of concentration on a length scale large enough that the resulting instabilities will not be suppressed by capillarity. Such instabilities can, however, be suppressed by a sufficiently large dependence of interface undercooling on v . When present, oscillatory instabilities lead to a three-dimensional segregation pattern in which periodic solute variations in the two transverse directions are modulated by a periodic variation in the direction of growth.

Mechanisms of microsegregation-free solidification

Abstract At high rates of solidification, two mechanisms can produce microsegregation-free crystalline alloys: planar growth and partitionless solidification. For growth at high velocities, but still with equilibrium partitioning of solute, capillarity can stabilize a planar liquid-solid interface. This type of stability, known as absolute stability, has been confirmed experimentally for AgCu alloys and should apply only when the net heat flow is towards the solid. Another possibility for producing microsegregation-free alloys is partitionless solidification which can occur at high velocities and arises from the kinetics of interface motion. These kinetics involve the trapping of solute by the moving interface, causing the partition coefficient to be unity. A unified model for the variation in the interface temperature and partition coefficient with interface velocity is presented. This model spans the range from slow velocities, where local equilibrium is usually valid, to high velocities where partitionless solidification occurs. Considerations necessary to predict the conditions of microsegregation-free solidification for concentrated alloys are also discussed.

Stability of the Shape of a Solid Cylinder Growing in a Diffusion Field

The stability of the shape of an infinite cylinder undergoing radial growth controlled by diffusion is studied by a method originated by Mullins and Sekerka (MS). It is found that the circular cross section of a cylinder is stable when its radius is less than and unstable when its radius is greater than a certain radius Rc. This result is analogous to the MS result that a sphere is stable below and unstable above a certain radius Rc, which is seven times the critical radius R* of nucleation theory. However, in the present case, the ratio Rc/R* is not equal to seven, but is a function of S=(c∞‐cs)/(C‐cs), where c∞, cs, and C are the concentrations of the solute at infinity, at the surface of the cylinder, and in the precipitate, respectively.The case of perturbations in the radius along the length of the cylinder is also treated. Potential application of the result to such problems as the growth of branches on dendrites is discussed.

Lateral solute segregation during undirectional solidification of a binary alloy with a curved solid—Liquid interface

Abstract The lateral solute segregation associated with a slightly curved solid-liquid interface during steady-state unidirectional solidification of a binary alloy is calculated under the assumption of no convection in the liquid. This is done to first order in the deviation of the interface from planarity by an adaptation of the perturbation methods of the theory of morphological stability. The calculation is based on an assumed knowledge of the solid-liquid interface shape; no attempt is made to relate this shape to the thermal field. The results are valid for the case where the solute boundary layer is thick compared to the deviation of the solid-liquid interface from planarity. In the limiting case of a very thick boundary layer, the lateral segregation in the solid is simply the product of the distribution coefficient, the deviation from planarity and the concentration gradient applicable to a planar interface. Numerical estimates of the relative lateral segregation are in reasonable agreement with experimental data of Witt et al. on crystals of gallium-doped germanium grown in space.

Thermosolutal convection during directional solidification

During solidification of a binary alloy at constant velocity vertically upward, thermosolutal convection can occur if the solute rejected at the crystal-melt interface decreases the density of the melt. We assume that the crystal-melt interface remains planar and that the flow field is periodic in the horizontal direction. The time-dependent nonlinear differential equations for fluid flow, concentration, and temperature are solved numerically in two spatial dimensions for small Prandtl numbers and moderately large Schmidt numbers. For slow solidification velocities, the thermal field has an important stabilizing influence: near the onset of instability the flow is confined to the vicinity of the crystal-melt interface. Further, for slow velocities, as the concentration increases, the horizontal wavelength of the flow decreases rapidly — a phenomenon also indicated by linear stability analysis. The lateral in-homogeneity in solute concentration due to convection is obtained from the calculations. For a narrow range of solutal Rayleigh numbers and wavelengths, the flow is periodic in time.

Morphological stability of a vicinal face induced by step flow

Abstract For growth from a supersaturated solution, the linear stability with respect to step bunching of a step train forming a vicinal face is considered accounting for both capillarity and anisotropy of interface kinetics. It is found that the step motion with respect to a stagnant solution provides stabilization at sufficiently large wavelengths for which the typical diffusion rate is comparable to the rate of incorporation of the crystallizing species at the steps, i.e., to the kinetic coefficient. Since capillarity can stabilize the interface against short wavelength perturbations, the combined action of both kinetic anisotropy and capillarity provides complete linear stability at sufficiently high growth rates.

The effect of anisotropic crystal-melt surface tension on grain boundary groove morphology

Abstract The shape of a stationary solid-liquid interface in a temperature gradient near a grain boundary in a pure material is calculated for anisotropic crystal-melt surface tension and equal thermal conductivities of crystal and melt. Results are compared with those for the well-known problem of the two-dimensional equilibrium shape of a crystal. For small anisotropy, the resulting interface shapes have continuously turning tangents but differ in detail from the grain boundary groove shapes that have been calculated for isotropic surface tension. For larger anisotropy, the interface shapes have discontinuities in slope as a result of missing orientations; these missing orientations are the same as those that would be missing on the corresponding equilibrium interface shape. In cases where a normal to the grain boundary or to the macroscopic interface is in the range of missing orientations on the corresponding equilibrium shape, the groove shape may contain some of these orientations as well as having varifold surfaces. Detailed numerical results are presented for a surface tension with fourfold symmetry.