Cellular solidification as a bifurcation problem

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Abstract
The spontaneous formation and evolution of stationary cellular interfaces arising in directional solidification of a binary alloy are discussed in terms of bifurcation theory. Algebraic bifurcation equations for the amplitudes of the cells are derived from the nonlinear equations of motion with the interface velocity as a control parameter. It is shown that the form of these bifurcation equations is determined by the system’s symmetry and by nonflux boundary conditions imposed at the sidewalls. The generic transitions from planar to cellular interfaces and between cellular interfaces of different wavelengths are determined and the effects of variations of system parameters on the bifurcation diagrams are analyzed. A variety of new phenomena, such as various types of hysteresis and cellular island formation, secondary bifurcations describing beats and mode jumping among interfacial cells, and tertiary Hopf bifurcations to standing waves, is discovered, which one can expect to find in solidification experiments on purely topological grounds.