B.T. Murray

Simulations of experimentally observed dendritic growth behavior using a phase-field model

An anisotropic phase-field model is used to simulate numerically dendritic solidification for a pure material in two dimensions. The phase-field model has been formulated to include the effect of four-fold anisotropy in both the surface energy and interfacial kinetics. The computations presented here are intended to model qualitatively experimentally observed dendritic solidification morphology. In particular, we simulate the growth into an undercooled melt of two dendrite tips which have formed as the result of a splitting event. The computation exhibits the competition between the two growing dendrite branches and the eventual predominance of one branch. Also, we simulate the effect of time-periodic forcing of an isolated dendrite tip on the mechanism of sidebranch formation. Although it is not yet computationally feasible to adequately verify convergence of the phase-field solutions, the phase-field simulations presented show many of the qualitative features observed in dendritic growth experiments.

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.

Adaptive phase-field computations of dendritic crystal growth

Phase-field models have been the subject of much interest in the last several years for the investigation of phase transitions with complicated morphology, such as dendritic growth. The phase-field method introduces a continuous transition between the two phases across a thin layer of finite thickness; the advantage of this approach is that the location of the interface does not have to be explicitly determined as part of the solution but is obtained from the solution of an additonal field equation representing the evolution of the phase-field variable over the entire domain. A brief overview is presented of the phase-field model development for a pure material using irreversible thermodynamics. The computational model includes four-fold anisotropy both in surface energy and interfacial kinetics. Numerical solutions are obtained using a general-purpose adaptive finite-difference algorithm. Adaptivity in space and time is found to extend somewhat the parameter regime where computations can be carried out. Good convergence to sharp-interface models is achieved for dimensionless undercoolings of 0.25, but a relatively small amount of solid phase grows before the thermal field is affected by the size of the computational domain. Further progress to smaller undercoolings will have to be aided by more sophisticated modeling.

Step bunching on a vicinal face of a crystal growing in a flowing solution

Abstract The effect of a parallel shear flow and anisotropic interface kinetics on the onset of (linear) instability during growth from a supersaturated solution is analyzed including perturbations in the flow velocity. The model used for anisotropy is based on the microscopic picture of step motion. A shear flow (linear Couette flow or asymptotic suction profile) parallel to the crystal-solution interface in the same direction as the step motion (negative shear) decreases interface stability. For large wavenumbers kx, the perturbed flow field can be neglected and a simple analytic approximation for the stability-instability demarcation is found. A shear flow counter to the step motion (positive shear) enhances stability and for sufficiently large shear rates (on the order of 1 s−1) the interface is morphologically stable. Alternatively, the approximate analysis predicts that the system is unstable if the solution flow velocity in the direction of the step motion at a distance(2kx)−1 from the interface exceeds the propagation rate vx of step bunches induced by the interface perturbations. The approximate results are applied to the growth of ADP and lysozyme. For sufficiently low supersaturations, the interface is stable for positive shear and unstable for negative shear. More generally, there is a critical negative shear rate for which the interface becomes unstable as the magnitude of the shear rate increases. For a range of growth conditions for ADP, the magnitude of this critical shear rate is2kxvx. Even shear rates due to natural convection may be sufficient to affect stability for typical growth conditions.

Step bunching: generalized kinetics

Our previous model of the effect of a parallel shear flow and anisotropic interface kinetics on the onset of (linear) instability during growth from a supersaturated solution is extended to allow for a kinetic coefficient which is a nonlinear function of the supersaturation and crystallographic orientation. The presence of impurities in the binary solution is also treated, although for typical impurity levels there is no explicit effect of impurities on morphological stability. As previously, a shear flow counter to the step motion enhances stability and a shear flow in the direction of the step motion promotes instability. Instability is enhanced when the kinetic coefficient increases rapidly with supersaturation. Specific calculations are carried out for parameters appropriate to the growth of ADP and KDP from solution.

Lubrication theory for reactive spreading of a thin drop

Solder drops spreading on metallic substrates are a reactive form of the wetting problem. A metallic component may diffuse in the liquid toward a metal substrate, where it is consumed by a reaction that forms a solid intermetallic phase. The resulting spatial variation in the composition of the drop may cause composition gradients along the free surface of the drop. Together with any thermal gradients along the free surface, Marangoni effects may, in turn, modify the bulk transport in the spreading drop. Motivated by this situation, we extend lubrication theory for the spreading of thin drops in the presence of gravity and thermocapillarity to include mass transport and solutocapillarity. We use an approximate solute profile in the drop to derive coupled evolution equations for the free surface shape and concentration field. Numerical solutions for the nonreactive (single component) drop agree well with previous theory. In the reactive case, we are only able to compute results for parameters outside of th...

Convective stability in the Rayleigh-Benard and directional solidification problems - High-frequency gravity modulation

The effect of vertical, sinusoidal, time‐dependent gravitational acceleration on the onset of solutal convection during directional solidification is analyzed in the limit of large modulation frequency Ω. When the unmodulated state is unstable, the modulation amplitude required to stabilize the system is determined by the method of averaging, and is O(Ω). Comparison of the results from the averaged equations with numerical solutions of the full linear stability equations (based on Floquet theory) show that the difference is O(Ω1/2). When the unmodulated state is stable, resonant modes of instability occur at large modulation amplitude. These are analyzed using matched asymptotic expansions to elucidate the boundary‐layer structure for both the Rayleigh–Benard and directional solidification configurations. The leading‐order term for the modulation amplitude is of O(Ω2); the first‐order correction of O(Ω3/2) is calculated, and the results are compared with numerical solutions of the full linear stability eq...

Prediction of solute trapping at high solidification rates using a diffuse interface phase-field theory of alloy solidification

A phase-field model for solidification of a binary alloy is developed that includes gradient energy contributions for the phase field and for the composition field. When the gradient energy coefficient for the phase field is smaller than that for the solute field, planar steady-state solutions exhibit a reduction in segregation in the liquid phase ahead of an advancing front (solute trapping), and in the limit of high solidification speeds predict alloy solidification with no redistribution of composition. Comparison is made with the Aziz model of solute trapping.

The effect of gravity modulation on thermosolutal convection in an infinite layer of fluid

In a gravitational field, the opposing effects of components of different diffusivities, for example, temperature and solute, in the density profile in a fluid may produce convective instabilities that exhibit a broad range of dynamical behavior. The effect of time periodic vertical gravity modulation on the onset of these instabilities in an infinite horizontal layer with stress free boundaries is examined. This work is viewed as a first step in expanding previous results in solidification to the full problem of characterizing the effects of gravity modulation in thermosolutal convection during the directional solidification of binary alloys. Calculations carried out both with and without steady background acceleration are presented, the latter results being relevant to microgravity conditions.

Analysis of monotectic growth: infinite diffusion in the L2 phase

The Jackson-Hunt model of eutectic solidification is applied to monotectic solidification in which a liquid (L1) transforms into rods of a different liquid (L2) in a solid matrix. Limiting cases of no diffusion and infinite diffusion (complete mixing) in the L2 phase are considered. An adaptive refinement and multigrid algorithm (MGGHAT) is used to obtain numerical solutions for the concentration field in the L1 phase; this allows consideration of a general phase diagram. Density differences between the three phases, which cause fluid flow, are treated approximately. Specific calculations are carried out for aluminum-indium alloys. Infinite diffusion in the L2 phase has only a small effect on the relationship between interface undercooling and rod spacing.