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.

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 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. The interaction of propagating extended defects with a diffusion field frequently leads to oscillations or jerky motions of the defects. A prime example of such an effect is the oscillation of a solidification front, induced by the diffusion of the solute component in a dilute binary alloy, which is growing in the setup of directional solidification. In a large number of metallic materials this leads to the formation of banded structures [1], reflecting a periodic array of layers with high and low solute concentrations where the former ones show a dendritic microstructure. The appearance of similar banding effects has recently been discussed [2] in rapid solidification of colloids.Layered structures are also generated by the oscillatory nucleation of a solid phase under the action of a diffusion field [3]. A related phenomenon is the oscillatory zoning, observed in solid solutions [4] and in natural minerals [5]. Another notable scenario is that of diffusion-controlled jerky motions of a driven grain boundary [6]. A similar behavior of dislocations in metallic alloys leads to the Portevin-Le Chatelier effect [7], denoting the appearance of jerky plastic deformations. We, finally, mention the oscillatory motion of a crack tip, which is coated by the nucleus of a new phase [8], replacing the attached cloud of a diffusion field.Theoretical discussions of such effects either are of a phenomenological type, like those in Ref. [7], and partly in Refs.[1] and [2], or they rely on a Fokker-Planck [6], or a diffusion equation with non-equilibrium boundary conditions [9]. In all approaches the source of oscillatory defect motions is identified as an unstable regime where a reduction of the driving force leads to an increase of the defect velocity. Additional information is provided by simulations, based on phase-field models for directional solidification [10] and for nucleation [3] processes.In the present Letter we introduce an extremely simple but powerful model for the diffusion-induced oscillatory motion of a planar interface, using the language adapted to the directional solidification of a dilute binary alloy. A major advantage of our approach is that it allows a transparent and, to a large extend, analytical evaluation. This includes a readjustment of the stability analysis by Merchant and Davis [11] who discovered an oscillatory instability, similar to that, discussed earlier by Coriell and Sekerka [12]. Also included is a clarifying analysis of the so far barely understood low-velo...

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.

Close to a bulk phase transition, a moving planar defect can be covered by a layer of the ordered phase. This, in fact, happens above the transition point in some finite region of the temperature-velocity diagram. In the case of a first-order transition this region is furnished with a net of nonequilibrium phase-transition lines. The topology of this net resembles that of the phase diagram of a first-order wetting transition in thermal equilibrium. In particular, there appears a kinetic complete-wetting line where a significant change of the drag coefficient of the defect is predicted.

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.