The evolution and morphological stability of a spherical crystal

Abstract

The growth model of a spherical crystal in the undercooled melt including the surface energy, interfacial kinetics and convective flow is established. The effect of the convective flow induced by a small far field flow on the evolution and morphological stability of the interface of the spherical crystal is studied. The interface shape of the spherical crystal, which is affected by the far field flow, and the dispersion relation of the growth rate of amplitude of the perturbed interface are derived. It is shown that the convection induced by the far field flow makes the interface of the growing spherical crystal further grow in the upstream direction of the far field flow and inhibit growth in the downstream direction; the interface of the decaying spherical crystal further decays in the upstream direction and inhibits decay in the downstream direction. The theoretical result suggests that both the growth of the sphere in the upstream direction and the decay of the sphere in the downstream direction make the spherical crystal tend to evolve into an oval; the morphological stability of the interface depends on a certain radius R c such that the spherical crystal is unstable when its radius is greater than R c and stable when its radius is less than R c . The surface energy and interfacial kinetics have strong stabilizing effects on the growth of the spherical crystal. In the meantime interfacial kinetics is a stable factor of the interface when the interface of the sphere is growing; it is an unstable factor of the interface when the interface is decaying.solid-liquid interface, spherical crystal, convective flow, crystal growth, morphological stability Convective flow in melt has significant effect on the morphological stability of the curved solid-liquid interface in solidification. Huang and Glacksman [1] first observed the effect of buoyancy-driven convention on dendritic growth in the micro-gravity experiment. Buchholz and

The effect of far field flow on a spherical crystal growth in the undercooled melt

“…ε → 0, we derived the asymptotic solution of the evolution of the interface of the spherical crystal growth by the matched asymptotic expansion method. The analytic result shows that the convective flow induced by the far field flow makes the interface of the growing spherical crystal enhance its growth velocity in the upstream direction Compared with the case of the weak far field flow [12,13], it can be seen that the convection induced by the far field flow with the moderate order of magnitude that is relative to the interface characteristic velocity has a stronger effect on the evolution of spherical crystal growth. And the flow parameter U ∞ in the present paper is available in a lager range than that for the case of the weak far field, then we extend our work for the case of the small far field flow to the situation for the case of the moderate far field flow.…”

“…Just as the discussion in Refs. [12,13], the morphological stability of the interface of the sphere in the metastable undercooled system will depend on a critical value of its radius such that the spherical crystal is unstable when its radius is greater than the critical value and stable when its radius is less than the critical value. This will be discussed in another paper.…”

“…That is the intrinsic reason why numerical simulations conducted so far can not yield reliable information and the analytical solutions with one single scale is merely locally valid. By means of the matched asymptotic expansion method, Chen et al [12] studied the evolution and morphological stability of a spherical crystal in the undercooled melt influenced by the weak far field flow which is much less than the characteristic velocity of the interface, and showed that the interface of the growing spherical crystal further grows in the upstream direction of the far field flow and inhibits growth in the downstream direction; the interface of the decaying spherical crystal further decays in the upstream direction and inhibits decay in the downstream direction. In the present paper we extend our work [12] for the case of the weak far field flow to that in the undercooled melt influenced by the moderate far field flow which is of the same order of magnitude as the characteristic velocity of the interface, and analytically study the effect of the forced convection on the evolution of the interface of a spherical crystal.…”

Influences of Marangoni effect on solidification of curved interfaces with constant curvature

“…Based on quasi-steady heat conduction equation and interfacial conditions, including Gibbs-Thomson condition and conservation of energy, theoretical efforts have been performed to study the influences of stability, surface tension, interfacial kinetic effect on the crystal growth [1,2] . Furthermore, Chen, et al [11] examined the system with incoming flow, constructed a mathematical model of spherical crystal considering surface energy, kinetic effect, and verified the different role of convection on upstream and downstream arms. However, the variation of surface tension with temperature and the Marangoni effect induced have been neglected in the publications available.…”

“…This work aims to consider the influence of Marangoni effect on the interface development with coming flow by asymptotic method based on theoretical model [11] , and on the nonlinear evolution of cylindrical nucleus without coming flow by phase field simulation.…”