Deterministic side-branching during thermal dendritic growth

The accepted view on dendritic side-branching is that side-branches grow as the result of selective amplification of thermal noise and that in the absence of such noise dendrites would grow without the development of side-arms. However, recently there has been renewed speculation about dendrites displaying deterministic side-branching [see e.g. ME Glicksman, Metall. Mater. Trans A 43 (2012) 391]. Generally, numerical models of dendritic growth, such as phase-field simulation, have tended to display behaviour which is commensurate with the former view, in that simulated dendrites do not develop side-branches unless noise is introduced into the simulation. However, here we present simulations at high undercooling that show that under certain conditions deterministic side-branching may occur. We use a model formulated in the thin interface limit and a range of advanced numerical techniques to minimise the numerical noise introduced into the solution, including a multigrid solver. Not only are multigrid solvers one of the most efficient means of inverting the large, but sparse, system of equations that results from implicit time-stepping, they are also very effective at smoothing noise at all wavelengths. This is in contrast to most Jacobi or Gauss-Seidel iterative schemes which are effective at removing noise with wavelengths comparable to the mesh size but tend to leave noise at longer wavelengths largely undamped. From an analysis of the tangential thermal gradients on the solid-liquid interface the mechanism for side-branching appears to be consistent with the deterministic model proposed by Glicksman.