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

doi:10.1016/S0167-2789(97)00297-2

Physica D: Nonlinear Phenomena

Volume 116, Issues 3–4, 1 June 1998, Pages 363-391

https://doi.org/10.1016/S0167-2789(97)00297-2 Get rights and content

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.

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The directional solidification in the undercooled pure melt influenced by a transverse far field flow was studied by using the multiple scale method. The result shows that in the boundary layer near the liquid-solid interface, when affected by a transverse far field flow, the temperature distribution in the direction of crystal growth presents an oscillatory and decay front in the side of liquid phase. The crucial distinguishing feature of a temperature pattern due to the transverse convection is the additional periodic modulation of the pattern in the growth direction. The wave number and eigenvalue that satisfy the Mullins-Sekerka dispersion relation are suppressed by the transverse far field flow.
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This is particularly evident for crystal growth problems where it has been recognized that growth morphologies are determined by the interaction between macroscopic driving forces (applied supercooling or far-field flux) and microscopic interfacial forces (surface tension, kinetics of atomic attachment, anisotropies, etc.) and their associated time scales. The interaction among these forces produces complex morphologies including dendrites and side-branches, which have been extensively studied theoretically (e.g. [2–18]) and experimentally (e.g. [26–31]). The complex morphologies are a result of the Mullins-Sekerka instability.
In this paper, we study the nonlinear stability of growing crystals and the development of protocols for controlling their shapes. We focus on the effects of surface tension anisotropy on the morphology of a 2D, non-circular crystal growing in a supercooled melt. We use a spectrally accurate boundary integral method to simulate the long-time, nonlinear dynamics of evolving crystals and to characterize the nonlinear morphological stability during growth. Our analysis and simulations reveal that under critical conditions of an applied far-field heat flux, there is nonlinear stabilization leading to the growth of compact crystals even though unstable growth may be significant at early times. The crystal morphologies are determined by a complex competition between the heat transport (far-field flux) and the strength of surface tension anisotropy. In particular, we observe three types of behavior: universal, limiting and oscillatory. Universal behavior occurs when the shape of the evolving crystal tends to a shape that is independent of both time and the initial condition. Limiting behavior occurs when the shape of the crystal tends to a shape that is independent of time, but depends on the initial condition. Oscillatory behavior occurs when the shape depends on both time and the initial condition, though its growth is bounded. Unlike the isotropic case where universal shapes have an arbitrary symmetry that depends only on the far-field flux [S. Li, J. Lowengrub, P. Leo, V. Cristini, Nonlinear Stability Analysis of Self-similar crystal growth: control of Mullins-Sekerka instability, J. Crystal Growth 277 (2005) 578–592.], here the surface tension anisotropy selects the symmetry of the universal shape. The results are presented as a phase diagram that characterizes the long time evolution behavior in a plane parameterized by the far-field flux and the strength of surface tension anisotropy. The phase diagram delineates the parameter ranges for the observed behaviors and can be used to design a nonlinear protocol to control the shapes of growing crystals that might be able to be carried out in an experiment. Finally, the analysis developed here may provide insights on the control of pattern formation in other physical and biological systems.
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Effect of anisotropy on morphological instability in the freezing of a hypercooled melt

Abstract

J. Crystal Growth

J. Crystal Growth

J. Crystal Growth

J. Crystal Growth

Acta Metall. Mater.

Acta Metall.

J. Cryst. Growth

J. Cryst. Growth

Physica D

Physica D

J. Cryst. Growth

Morphological stability of a particle growing by diffusion or heat flow

J. Appl. Phys.

Stability of a planar interface during solidification of a dulite binary alloy

J. Appl. Phys.

Morphological instability