Interface dynamics and banding in rapid solidification

Microstructural Pattern Formation during Far-from-Equilibrium Alloy Solidification

We introduce a new phase-field formulation of rapid alloy solidification that quantitatively incorporates nonequilibrium effects at the solid-liquid interface over a very wide range of interface velocities. Simulations identify a new dynamical instability of dendrite tip growth driven by solute trapping at velocities approaching the absolute stability limit. They also reproduce the formation of the widely observed banded microstructures, revealing how this instability triggers transitions between dendritic and microsegregation-free solidification. Predicted band spacings agree quantitatively with observations in rapidly solidified Al-Cu thin films.

Complex banded structures in directional solidification processes

A combination of theory and numerical simulation is used to investigate impurity superstructures that form in rapid directional solidification (RDS) processes in the presence of a temperature gradient and a pulling velocity with an oscillatory component. Based on a capillary wave model, we show that the RDS processes are associated with a rich morphology of banded structures, including frequency locking and the transition to chaos.

Domain of oscillatory growth in directional solidification of dilute binary alloys

The oscillatory growth of a dilute binary alloy has recently been described by a nonlinear oscillator equation that applies to small temperature gradients and large growth velocities in the setup of directional solidification. Based on a one-dimensional stability analysis of stationary solutions of this equation, we explore in the present paper the complete region where the solidification front propagates in an oscillatory way. The boundary of this region is calculated exactly, and the nature of the oscillations is evaluated numerically in several segments of the region.

Capillary-wave description of rapid directional solidification

A recently introduced capillary-wave description of binary-alloy solidification is generalized to include the procedure of directional solidification. For a class of model systems a universal dispersion relation of the unstable eigenmodes of a planar steady-state solidification front is derived, which readjusts previously known stability considerations. We moreover establish a differential equation for oscillatory motions of a planar interface that offers a limit-cycle scenario for the formation of solute bands and, taking into account the Mullins-Sekerka instability, of banded structures.

Diffusion-induced oscillations of extended defects

From a simple model for the driven motion of a planar interface under the influence of a diffusion field we derive a damped nonlinear oscillator equation for the interface position. Inside an unstable regime, where the damping term is negative, we find limit-cycle solutions, describing an oscillatory propagation of the interface. In the case of a growing solidification front this offers a transparent scenario for the formation of solute bands in binary alloys and, taking into account the Mullins-Sekerka instability, of banded structures.

Capillary-wave model for the solidification of dilute binary alloys

Starting from a phase-field description of the isothermal solidification of a dilute binary alloy, we establish a model where capillary waves of the solidification front interact with the diffusive concentration field of the solute. The model does not rely on the sharp-interface assumption and includes nonequilibrium effects, relevant in the rapid-growth regime. In many applications it can be evaluated analytically, culminating in the appearance of an instability that, interfering with the Mullins-Sekerka instability, is similar to that found by Cahn in grain-boundary motion.

Boundary-reaction-diffusion model for oscillatory zoning in binary crystals grown from solution

Oscillatory Zoning (OZ) is a phenomenon exhibited by many geologically formed crystals. It is characterized by quasiperiodic oscillations in the composition of a solid solution, caused by self-organization. We present a model for OZ. The growth mechanism applied includes species diffusion through the solution bulk, particle adsorption, surface diffusion, and subsequently desorption or incorporation into the crystal. This mechanism, in particular, can provide the synchronization effects necessary to reproduce the layered structure of experimentally obtained crystals, lacking in other models. We conduct a linear stability analysis combined with numerical simulations. Our results reproduce the experimental findings with respect to the patterns formed and a critical supersaturation necessary for OZ to occur.

Novel pattern forming process due to the coupling of convection and phase change

We present a novel mechanism of pattern formation behind a flat interface during directional solidification of peritectic alloys. It is shown through computational modeling that irregular oscillatory thermosolutal convection can develop in the vertical Bridgman system, even with bottom seeding and bottom cooling. The coupling of the flow oscillation near the interface with solidification leads to ordered layered structures in the solidified crystal, which agree closely with earlier experimental results.

Curvature effects in rapid alloy solidification

The growth of a cylindrical or spherical crystal into its undercooled melt is a process whose description is complicated by the lack of a stationary regime. A simple approach to the problem, justified for low growth rates and widely used in the past for both pure substances and alloy solidification, is based on a quasistatic approximation which assumes an instantaneous adaptation of the diffusional field to the interface configuration. For alloy solidification, assuming isothermal conditions and local interface equilibrium, this simplified model predicts a diffusion controlled growth, with the radius of the crystal increasing asymptotically as ~t(1/2). However, as pointed out by recent investigations, thermal diffusion and nonequilibrium effects enter as essential ingredients in rapid alloy solidification. In the present paper we use the phase-field model to simulate the cylindrical and spherical growth of a solid germ into a supersaturated alloy melt. The problem is treated in its full time-dependent characteristics, accounting for nonequilibrium effects as well as for the rejection of both heat and solute away from the advancing front. We observe a complex behavior and a rich variety of dynamic regimes: in different regions of parameter space the growth rate is limited by diffusion (either thermal or chemical) or is kinetic controlled. Traversing the boundaries which limit these regions, the process undergoes sharp transitions which leave a trace in the solidified alloy. For realistic values of the Lewis number, thermal effects drive the process into a a diffusive regime, in which the rate limiting mechanism is the rejection of solute.

Thermal and chemical diffusion in the rapid solidification of binary alloys

Solidification of binary alloys is characterized by the necessity to reject away from the advancing front two conserved quantities: the latent heat released at the solid-liquid interface and the solute atoms that cannot be accommodated in the solid phase. As thermal diffusion is much faster than chemical diffusion, the latter is generally assumed to be the rate limiting mechanism for the process, and the problem is addressed through the isothermal approximation. In the present paper we use the phase-field model to study the planar growth of a solid germ, nucleated in its undercooled melt. We focus on the effects of a noninstantaneous thermal relaxation. The steady growth predicted at large supersaturation in the isothermal limit is prevented. Depending on the value of the Lewis number the growth rate is limited by either mass or heat diffusion; in the latter case we observe a sharp transition between two different regimes, in which originates a nonmonotonic time dependence of the interface temperature. The effects of this transition reflect in the composition of the solidified alloy.

Phase and solute fields across the solid-liquid interface of a binary alloy

Solidification of binary alloys is characterized by a sharp structural transition across the solid-liquid interface. It is a typical situation which suggests a nonlocal dependence of the appropriate thermodynamic potential on its associate fields. Phase-field models with a gradient contribution for the order parameter have proved to account for nonequilibrium effects, as solute trapping and kinetic undercooling of the solid-liquid interface. The inclusion of a gradient term for the concentration field has also been theoretically investigated, but in this case the correspondence between predicted phenomena and experimental results is still rather unexplored. In the present study we analyze numerical solutions of a phase-field model in both steady and transient conditions. We focus on effects which are critically dependent on the composition field across the solid-liquid interface; the extent of the concentration gradient correction is related to measurable quantities to suggest methods for a comparison between the model predictions and experimental data.