Phase-field study of solidification in three-dimensional channels

Three-dimensional solidification of a pure material with isotropic properties of the solid phase is studied in cylindrical capillaries of various cross sections (circular, hexagonal, and square). As the undercooling is increased, we find, depending on the width of the capillary, a number of different growth modes and dynamical behaviors, including stationary symmetric single fingers, stationary asymmetric fingers, and oscillating double and quadruple fingers. Chaotic states are also observed, some of them in unexpected parameter regions. Our simulations suggest that the bifurcation from symmetric to asymmetric fingers is supercritical. We discuss the nature of the oscillatory states, one of which is chirality breaking, and the origin of the unexpected chaotic finger. Bifurcation diagrams are given comparing three different ratios of capillary length to channel width in the hexagonal channel as well as the three different geometries.