Renormalization and scaling methods for quasi-static interface problems

We study the temporal evolution of an interface separating two phases for its large-time behavior by adapting renormalization group methods and scaling theory. We consider a full two-phase model in the quasi-static regime and implement a renormalization procedure in order to calculate the characteristic length of a self-similar system, R(t), that is the time-dependent length scale characterizing the pattern growth. When the dynamical undercooling is non-zero (α≠0), we find that R(t) increases as t-1/λ, where λ can take on values in the continuous spectrum, [-3,-2]. For α=0 the spectrum is [-3,0) so that the single value of λ=-1 is selected by the plane wave imposed by Jasnow and Vinals. It is also shown that in almost all of these cases, the capillarity length, d0, (arising from the surface tension, σ0) is not relevant for the large-time behavior even though it has a crucial role at the early stage evolution of an interface. The exception is λ=-3, i.e., R(t)∼t1/3, for which d0 is invariant.