Crystal growth in a channel: Numerical study of the one-sided model

The growth of two-dimensional crystals with and without anisotropy in a channel is analyzed by a Green's-function method. A one-sided diffusional model is treated in quasistationary approximation. Our numerical results on the steady-state growth of symmetrical fingers are in agreement with approximate analytical predictions (E. Brener, M. Geilikman, and D. Temkin, Zh. Eksp. Teor. Fiz. 94, 241 (1988) [Sov. Phys. JETP 67, 1002 (1988)]). For fixed supercooling the dependence of growth velocity versus channel width is nonmonotonic, passing a maximum. We did not find stationary solutions for supercooling less than 0.5. When the width of the channel exceeds some critical value we observed that the symmetrical finger becomes unstable against tip splitting. In this case we found stable steady-state growth of nonsymmetrical fingers. We have also found an expected instability of two fingers in the common diffusion field caused by competition.