Coarsening Dynamics of a One-Dimensional Driven Cahn-Hilliard System

TL;DR

This paper investigates the effects of an external driving field on the dynamics of domain walls in a one-dimensional Cahn-Hilliard system, revealing asymmetric solutions, bubble motion, and scaling behavior.

Contribution

It introduces a detailed analysis of driven Cahn-Hilliard dynamics, deriving reduced equations and identifying scaling laws for domain growth under external driving.

Findings

Asymmetric effects of the driving field on stationary domain walls.

Traveling-wave solutions for bubble dynamics when EL << 1.

Characteristic length scale grows as (Et)^{1/2} with dynamical scaling confirmed.

Abstract

We study the one-dimensional Cahn-Hilliard equation with an additional driving term representing, say, the effect of gravity. We find that the driving field $E$ has an asymmetric effect on the solution for a single stationary domain wall (or `kink'), the direction of the field determining whether the analytic solutions found by Leung [J.Stat.Phys.{\bf 61}, 345 (1990)] are unique. The dynamics of a kink-antikink pair (`bubble') is then studied. The behaviour of a bubble is dependent on the relative sizes of a characteristic length scale $E^{-1}$, where $E$ is the driving field, and the separation, $L$, of the interfaces. For $EL \gg 1$ the velocities of the interfaces are negligible, while in the opposite limit a travelling-wave solution is found with a velocity $v \propto E/L$. For this latter case ($EL \ll 1$) a set of reduced equations, describing the evolution of the domain lengths, is obtained for a system with a large number of interfaces, and implies a characteristic length scale growing as $(Et)^{1/2}$. Numerical results for the domain-size distribution and structure factor confirm this behavior, and show that the system exhibits dynamical scaling from very early times.