Fluctuations in solidification.

Abstract

We present an analytical treatment of (i) the incorporation of thermal noise in the basic continuum models of solidification, (ii) fluctuations about nonequilibrium steady states, and (iii) the amplification of noise near the onset of morphological instability. In (i), we find that the proper Langevin formalism, consistent with both bulk and interfacial equilibrium fluctuations, consists of the usual bulk forces and an extra stochastic force on the interface associated with its local kinetics. At sufficiently large solidification rate, this force affects interfacial fluctuations on scales where they are macroscopically amplified and, thus, becomes relevant. An estimate of this rate is given. In (ii), we extend the Langevin formalism outside of equilibrium to characterize the fluctuations of a stationary and a directionally solidified planar interface in a temperature gradient. Finally, in (iii), we derive an analytic expression for the linear growth of the mean-square amplitude of fluctuations slightly above the onset of morphological instability. The amplitude of the noise is found to be determined by the small parameter ${\mathit{k}}_{\mathit{B}}$${\mathit{T}}_{\mathit{E}}$${\mathit{d}}_{0}^{\mathit{c}}$${\mathit{l}}_{\mathit{T}}$/\ensuremath{\gamma}${\ensuremath{\lambda}}_{\mathit{c}}^{4}$ where \ensuremath{\gamma} is the surface energy, ${\mathit{d}}_{0}^{\mathit{c}}$ is the chemical capillary length, ${\mathit{l}}_{\mathit{T}}$ is the thermal length, and ${\ensuremath{\lambda}}_{\mathit{c}}$ is the critical wavelength. Possible applications to experiment are discussed.