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.
Bibliographical noteFunding Information:
The authors thank Professors Vittorio Cristini and Robert Sekerka for stimulating discussions. S. Li is supported by a Doctoral Dissertation Fellowship from the Graduate School, University of Minnesota. The authors also acknowledge the generous computing resources from the Minnesota Supercomputer Institute (MSI), the Network and Academic Computing Services at University of California at Irvine (NACS), and computing resources of the BME department (U.C. Irvine). S. Li also thanks the support from the Department of Biomedical Engineering and the Department of Mathematics at U.C. Irvine where he is a visiting researcher. J. Lowengrub thanks the National Science Foundation (Division of Mathematical Sciences) for partial support.
- Crystal growth
- Morphological stability
- Mullins-Sekerka instability
- Pattern formation