Tip Oscillation of Dendritic Patterns in a Phase Field Model

Abstract

We study dendritic growth numerically with a phase field model. Tip oscillation and regular side-branching are observed in a parameter region where the anisotropies of the surface tension and the kinetic effect compete. The transition from a needle pattern to a dendritic pattern is conjectured to be a supercritical Hopf bifurcation.

Front propagation into unstable states

“…That might happen in some parameter range was already suggested by simulations of the so-called Boundary Layer Model by Pieters [346]. Recently, Sakaguchi and Tokunaga [372] observed such behavior in phase field model calculations but the data were not correlated with the parameter r introduced above. A good way to classify sidebranch regimes experimentally and to search for this possibility is to measure the dimensionless growth factor r -if this value is found to increase towards r * ≈ 3.6 then the possibility of such a regime becomes more likely.…”

“…Sometimes, it is numerically advantageous go in the opposite direction, i.e., to translate a model with sharp interfaces into what has become known as a phase field model in which the order parameter field varies continuously through the interfacial zone. Examples where this idea was exploited for a variety of physical problems can be found in [29,71,187,219,221,232,372]; for careful discussions of the derivation of a moving boundary problem for a variety of different physical systems, we refer to [30,67,132,160,219,306].…”