Degenerate Nucleation in the Cahn--Hilliard--Cook Model

Abstract

Phase separation in metal alloys is an important pattern forming physical process with applications in materials science, both for understanding materials structure and for the design of new materials. The Cahn-Hilliard equation is a deterministic model for the dynamics of alloys which has proven to be fundamental for the understanding of several types of phase separation behavior. However, stochastic effects occur in any physical experiment, and thus need to be incorporated into models. While white noise is one standardly chosen option, it is not immediately clear in which way the noise characteristics affect the resulting patterns. In this paper we study the effects of not necessarily small colored noise on pattern formation in a stochastic Cahn-Hilliard model in the nucleation regime. More precisely, we focus on degenerate noise which acts either on isolated eigenmodes, or with a well-defined spatial wavelength. Our studies show that the types of resulting patterns depend critically on the spatial noise frequency, and we can explain via numerical continuation methods that if this frequency is too high, then pattern formation is significantly impaired by the underlying structure of the system. In addition, we provide rigorous bounds on the nucleation time frame in the degenerate stochastic setting.

Analysis and Optimal Velocity Control of a Stochastic Convective Cahn–Hilliard Equation

“…The stochastic version of the model was then confirmed multiple times (Binder 1981;Pego 1989) to be the only one that can genuinely describe phase-separation in alloys. Since then, the random version of the equation has been increasingly studied, both in the physics literature (Rogers et al 1988;Elder et al 1988;Grant et al 1985;Langer et al 1975;Milchev et al 1988) and in the direction of model validation and numerical simulations (Blömker et al 2001(Blömker et al , 2008(Blömker et al , 2016Hawick 2010;Hawick and Playne 2010;Hawick 2008;Lee et al 2014).…”

Stochastic Cahn–Hilliard equation in higher space dimensions: the motion of bubbles

“…In this regime, an initially constant solution is locally stable, and one has to wait for a large deviation event that leads to the nucleation of droplets. See, for example, [10,11,[13][14][15].…”

A multilevel Monte Carlo finite element method for the stochastic Cahn–Hilliard–Cook equation

“…The system allows for considering thermal fluctuations directly in terms of the Cahn-Hilliard-Cook (CHC) equation by a conserved noise source term. The thermal fluctuations play an essential role in the early stage of phase dynamics such as initial dynamics of nucleation [22,23]. Some authors, such as Binder [24] and Pego [25], have expressed the belief that only the stochastic version can correctly describe the whole decomposition process in a binary alloy [26].…”