Thermosolutal convection during directional solidification. II. Flow transitions

Abstract

The effect of thermosolutal convection on the solute segregation in crystals grown by vertical directional solidification of binary metallic alloys or semiconductors is considered. Numerical results are obtained using finite differences in a two-dimensional, time-dependent model that assumes a planar crystal–melt interface. The configuration is assumed to be periodic in the horizontal direction with a given period, and the possibility of multiple flow states sharing the same period is examined. The results are summarized in bifurcation diagrams of the nonlinear states associated with the critical points of linear theory. The use of a time-dependent numerical scheme results in gaps in the bifurcation diagram where presumed unstable states exist that cannot be computed by this procedure. As the solutal Rayleigh number is varied, multiple steady states, time-periodic states, and quasiperiodic states may occur. This case is compared to the simpler case of thermosolutal convection with linear vertical gradients and stress-free boundaries, for which a rather complete numerical treatment is possible through the use of a simple spectral representation of the nonlinear solution. Retaining a finite number of terms in the expansion results in a set of coupled nonlinear algebraic equations for the Fourier coefficients, which is solved by a quasi-Newton method. The linear stability of the nonlinear solutions is also determined from a numerical computation of the eigenvalues of the Jacobian matrix. The resulting bifurcation diagram shows qualitative similarity to the bifurcation diagram computed for the solidification system.