Further examinations of dendritic growth theories

Abstract

Two analytical theories of free dendrite growth, the microscopic solvability condition (MSC) theory and the interfacial wave (IFW) theory have been proposed during the past decade, attempting to resolve the problem of selection of dendrite growth, and explain the essence of the pattern formation. This article attempts to clarify the differences and commonalities between these two theories and compare the predictions of these theories with some latest numerical evidence and experimental data. Since the MSC theory is most well-developed for the two-dimensional case, the comparisons of the theories with the numerical simulations are made mainly by using, but not restricted to, the two-dimensional, numerical solutions for dendrite growth with anisotropy of surface tension. Such kinds of numerical simulations have been lately carried out by Wheeler et al. (Physica D 66 (1993) 243), Provatas et al. (Phys. Rev. Lett. 80 (15) (1998) 3308; 82 (22) (1999) 4496) and Karma et al. (Phys. Rev E 53 (1996) 3071; Phys. Rev. Lett. 77 (1996) 4050; J. Crystal Growth 174 (1997) 54) with the phase field model, and by Ihle and Müller-Krumbhaar with the free-boundary problem model (1994). It is seen that in a region where the anisotropy parameter is not too small, the numerical simulations yield steady needle solutions, whose side-branching structures are not self-sustaining. These results support the conclusions drawn by both the MSC and IFW theories. However, the numerical simulations also showed that there exists ‘a smallest value of the anisotropy parameter’, less than which ‘no steady needle solution was found’ (refer to Wheeler et al. Physica D 66 (1993) 243). This numerical evidence appears to be in agreement with the IFW theory and contradict the MSC theory. The prediction of the IFW theory is also compared with the latest experimental data obtained by Glicksman et al. in the microgravity of space and an excellent overall agreement is found.