A. A. Golovin

A convective Cahn-Hilliard model for the formation of facets and corners in crystal growth

Abstract We consider solidification into a hypercooled melt in which kinetic undercooling and anisotropic surface energy are present. We allow the anisotropy to be strong enough that missing orientations would be present in equilibrium configurations, and track the unstable evolution of an initially planar front to a facetted front. Regularization by curvature-dependent surface energy is posed, and in the nonlinear regime a convective Cahn-Hilliard equation is derived. The emergence of facets is thus related to spinodal decomposition and subsequent coarsening. The presence of convective terms generated by the effect of kinetics destroys the binodal construction and leads to a fast coarsening, that for large times t goes as t 1 2 .

Step-flow growth of a nanowire in the vapor-liquid-solid and vapor-solid-solid processes

Consider the growth of a nanowire by a step-flow mechanism in the course of vapor-liquid-solid and vapor-solid-solid processes. The growth is initiated by the nucleation of a circular step at the nanowire-catalyst interface near the edge of the nanowire (the triple junction) and proceeds by the propagation toward the center by the Burton–Cabrera–Frank mechanism. Two cases are considered: (i) bulk transport, where the interfacial diffusion of adatoms and the step motion are coupled to the diffusion flux of atoms from the bulk of the catalyst particle, and (ii) surface transport, where atoms from the vapor phase are adsorbed at the surface of the catalyst particle and diffuse along the surface toward the triple line, whence they diffuse to the nanowire-catalyst interface. The attachment kinetics of adatoms at the step, the adsorption kinetics of atoms from the bulk phase, the exchange kinetics at the triple contact line, and the capillarity of the step are taken into account. In case (i) the problem is redu...

Oscillatory instability in super-diffusive reaction – diffusion systems: Fractional amplitude and phase diffusion equations

Non-linear evolution of a reaction – super-diffusion system near a Hopf bifurcation is studied. Fractional analogues of the complex Ginzburg-Landau equation and Kuramoto-Sivashinsky equation are derived, and some of their analytical and numerical solutions are studied.

Effect of anisotropy on morphological instability in the freezing of a hypercooled melt

Abstract Considered is the morphological instability of a rapid-solidification front propagating in a hypercooled melt when the solidification process is controlled by kinetics and there are cubic anisotropies of surface tension and attachment kinetics. It is shown that, due to anisotropy, the threshold of morphological instability depends on the direction of the crystal growth and generates, in the general case, traveling cells (waves) propagating on the solidification front in a preferred direction determined by the anisotropy coefficients. Weakly nonlinear analysis of the waves is carried out in the vicinity of the instability threshold and it is shown that the evolution of the waves is usually governed by an anisotropic dissipation-modified Korteweg-de Vries equation. In special cases it is governed by an anisotropic Kuramoto-Sivashinsky equation that describes stationary cells. Regions in the parameter space are found where the stationary and traveling cells are stable and could be observed in experiment. The characteristics of the cells are studied as functions of the direction of the crystal growth.

Stability of a two-layer binary-fluid system with a diffuse interface

The phase separation of a binary fluid can lead to the creation of two horizontal fluid layers with different concentrations resting on a solid substrate and divided by a diffuse interface. In the framework of the Cahn–Hilliard equation, it is shown analytically and numerically that such a layered system is subject to a transverse instability that generates a slowly coarsening multidomain structure. The influence of gravity, solutocapillary effect at the free boundary, and Korteweg stresses inside the diffuse interface on the stability of the layers is studied using the coupled system of the hydrodynamic equations and the nonlinear equation for the concentration (H model). The parameter regions of long-wave instabilities are found.

Coupled KS—CGL and coupled burgers—CGL equations for flames governed by a sequential reaction

Abstract We consider the nonlinear evolution of the coupled long-scale oscillatory and monotonic instabilities of a uniformly propagating combustion wave governed by a sequential chemical reaction, having two flame fronts corresponding to two reaction zones with a finite separation distance between them. We derive a system of coupled complex Ginzburg–Landau and Kuramoto–Sivashinsky equations that describes the interaction between the excited monotonic mode and the excited or damped oscillatory mode, as well as a system of complex Ginzburg–Landau and Burgers equations describing the interaction of the excited oscillatory mode and the damped monotonic mode. The coupled systems are then studied, both analytically and numerically. The solutions of the coupled equations exhibit a rich variety of spatio-temporal behavior in the form of modulated standing and traveling waves, blinking states, traveling blinking states, intermittent states, heteroclinic cycles, strange attractors, etc.

Traveling and spiral waves for sequential flames with translation symmetry: coupled CGL-Burgers equations

Abstract The dynamics of traveling and spiral waves resulting from the oscillatory instability of a uniformly propagating planar flame front governed by a sequential reaction is investigated. The nonlinear dynamics of this system, which enjoys translation symmetry, is described by two coupled equations: a complex Ginzburg–Landau equation for the oscillation amplitude, and a Burgers equation for the slow Goldstone mode due to the symmetry, describing frontal deformation. The pure CGL equation in 2D exhibits spiral wave solutions as well as uniformly propagating wave solutions. We present a stability analysis and numerical computations of the coupled system to show how the coupling modifies traveling and spiral wave solutions of the pure CGL equation. In particular, we show that the coupled system exhibits new types of instabilities as well as new dynamical behavior, including bound states of two or four spirals, “liquid spiral” states, superspiral structures, oscillating cellular structures separated by chaotically merging and splitting domain walls, and others.

A complex Swift-Hohenberg equation coupled to the Goldstone mode in the nonlinear dynamics of flames

Abstract The nonlinear dynamics of a propagating flame front governed by a two-stage sequential chemical reaction is considered in the parameter range where the uniformly propagating front is unstable. We show that near the transition from the short wave to the long wave oscillatory instability the nonlinear dynamics is described by a Swift–Hohenberg equation with dominant dispersive term, coupled to an evolution equation for the zero mode associated with the translation symmetry of the propagating wave. The nonlinear dynamics described by this system of equations is studied both analytically and numerically. In the case of weak coupling between the two equations, we observe the spontaneous formation of spiral waves with rapidly moving cores, while strong coupling leads either to chaotic dynamics or to the formation of oscillons—spatially localized oscillating structures.

Modeling the formation of facets and corners using a convective Cahn–Hilliard model

Abstract We consider solidification into a hypercooled melt in the presence of kinetic undercooling and anisotropic surface energy. We allow the anisotropy to be strong enough that equilibrium configurations would contain facets divided by corners, and track the unstable evolution of an initially planar front to a facetted front. Regularization by curvature-dependent surface energy is posed, and in the nonlinear regime a convective Cahn–Hilliard equation is derived. The emergence of facets is thus related to spinodal decomposition and subsequent coarsening. The presence of convective terms generated by the effect of kinetics destroys the binodal construction and leads to a fast coarsening, that for large times t goes as t1/2.

Step-flow growth of a crystal surface by Lévy flights

Step-flow growth of a crystal surface is considered in the case when the adatom diffusion on the terraces is governed by Levy flights. A superdiffusive analog of the Burton-Cabrera-Frank (BCF) theory is developed, and the step-flow velocity is found as a function of the terrace width and the anomalous-diffusion exponent. It is shown that the Levy-flights–controlled step-flow velocity is lower than that in the case of normal diffusion.