Modelling of Casting, Welding, and Advanced Solidification Processing XI NON-MONOTONE TEMPERATURE BOUNDARY CONDITIONS IN DENDRITIC GROWTH

When the Gibbs-Thomson-Herring anisotropic capillary boundary condition is applied at the solidliquid interface of a crystallite with shape anisotropy, such as a needle crystal or dendrite, the interface can develop periodic non-monotone temperature distributions. This surprising result was discovered recently for the case of a slender ellipsoidal crystallite with its solid-melt interfacial energy parameters chosen equivalent to those for pivalic acid—a crystal exhibiting 4-fold anisotropy of its interfacial energy. An unexpected deep minimum develops in the equilibrium temperature close to the highly curved tip. This minimum results in the tip temperature itself becoming warmer than the adjacent interface, thereby further steepening the local temperature gradient at the tip relative to nearby gradients. Numerical simulations of the solidification dynamics, using a spectrally accurate, two-dimensional boundary integral method, show that a non-monotonic temperature distribution anywhere along the crystal-melt interface leads to localized negative curvatures and, eventually, to periodic oscillations in the temperature and tip shape. Periodic changes in the tip shape and temperature distribution lead to growing protuberances that form side branches. The dynamical process acts as a “limit cycle”, stimulating a chain of wave-like disturbances behind the tip that grow sequentially and form the periodic side-branches of a dendrite. What is especially significant about these observations is that perturbations are not needed to “destabilize” the solid-melt interface, and the dendritic pattern evolves in a deterministic manner. Selective amplification of interfacial noise—the conventional explanation of dendrites—does not play an important rôle in this process, so long as sufficient shape and interfacial energy anisotropy occur together.