Motion by curvature and impurity drag: resolution of a mobility paradox Original Research Article

We rederive and explain the paradoxical prediction, confirmed by experiment, that the normal velocity, V, of a curved antiphase domain boundary (APB) in an ordered alloy tends to a finite limit at a critical point, where the APB effectively disappears, its surface energy γ tending to 0, and its thickness λ diverging. The prediction is in apparent contradiction to the expectation that the velocity of the APB should tend to 0 based on the notion that V should be proportional to γ and inversely proportional to λ. Combining an Allen–Cahn equation for the rate of change of the order parameter, a non-conserved quantity, with a Cahn–Hilliard equation for the diffusion of one of the chemical components of an alloy, a conserved quantity, we obtain a system from which we derive an expression, via formal asymptotics, for V of an APB domain boundary in an ordered alloy. The drags from the two types of quantities are found to be additive. We find that the drag from the changes in ordering accompanying the motion of the APB decreases inversely as the thickness. Thus, at the critical temperature where the thickness of an APB diverges and its γ tends to 0, the drag tends to 0. This resolves the paradoxical behavior of APB at the critical point; they continue to move up to the limiting condition where they, and their driving force for motion, disappear. The drag from the conserved component, which moves with the boundary as an adsorbed layer, exerts a drag on the boundary that is similar to what is predicted for impurity drag. In general the drag is given by a compositional integral through the interface that is not simply related to either the thickness or the total adsorption. However, for thick compositional wetting layers the drag indeed varies with thickness. We believe that the distinction we have encountered in the behavior of composition and order parameter is quite general for motion of interfaces, unless strong cross-effects influence the evolution of the system. Our results yield a unification and reconciliation of several diverse theories of interface motion.