The Stability Analysis of Crystal Growth

D. I. Meiron
D. C. Sorensen
B. Wedeman
C. Yang

Imagine that a piece of crystal is placed in some under-cooled liquid. As the crystal freezes, the interface between the solid and the liquid starts to change. The solid phase grows rapidly from the crystal by sending out branching fingers. This phenomenon is also called dendritic solidification , and is responsible for the complicated interface observed in snowflakes.

There has been a great deal of interest in the simulation and modeling of crystal growth and dentridic solidification in the past few years. It is well known that the change of the temperature of both the crystal and the liquid can be described by a set of diffusion equations. At the solid-liquid interface the motion of the interface and the temperature of these two materials are related by some conservation law.

Both analytical and numerical solutions of the above problem are difficult to obtain because of the moving boundary. We are interested in analyzing the stability of a well known stationary solution that corresponds to a simple parabolic shaped moving front. The standard techniques of linear stability analysis lead to an eigenvalue problem. In particular, the rightmost eigenvalues of some convection diffusion operator are of interest. The stationary solution is said to be stable if there is no eigenvalue with positive real part.

Strategy

A stationary solution corresponding to a parabolic moving front can be derived by writing the diffusion equation in a moving frame. This is known as the Ivantsov parabola solution. The standard linear stability analysis technique yields an eigenvalue problem of the form where L is some convection diffusion operator. The eigenvalue problem is then solved numerically by discretizing L using a second order finite difference scheme and computing the rightmost eigenvalues of the corresponding algebraic eigenvalue problem.

Tools

This problem is challenging because we expect the spectrum to be unbounded. As the mesh size of the finite difference becomes small, it is not practical to use the conventional QR type of method to compute the large scale eigenvalue problem. Since the matrix derived from finite difference is sparse and structured, fast iterative methods such as the Arnoldi method is attractive in this setting.

The particular software package we used in our computation is ARPACK , a package that implements the Implicitly Restarted Arnoldi Iteration . We are able to obtain the desired eigenvalues and eigenvectors of the matrix of order 10,000 or larger in a reasonable amount of time.

Results

Click here to see computed eigenvalues and eigenfunctions.

Report

  • Numerical Computation of Linear Stability of the Crystal Growth

    TR96-04, Department of Computational and Applied Mathematics, Rice University, Houston, TX 77005. Also appears in Proceedings of the Copper Mountain Conference on Iterative Methods , April 9-13, 1996.