A free-boundary model for diffusion-induced grain boundary motion

Abstract

On the basis of a phase field model previously proposed in Cahn et al. (Acta Mater. 45, 4397-4413 (1997)) to describe the phenomenon of diffusion-induced grain boundary motion (DIGM), we present a formal asymptotic reduction to a moving free-boundary problem. This problem is one of enhanced motion by curvature. The enhancement depends on the local concentration jump, across the grain boundary, of a solute species that is diffusing along the boundary. The reduction depends on the material diffusivity vanishing outside the grain boundary; this also introduces important mathematical and conceptual difficulties which are addressed in detail. A rigorous theory is given for two-dimensional solutions of the free-boundary problem when there is steady motion of the grain boundary spanning a plate from face to face, solute being supplied at the faces of the plate. The motion is in the direction parallel to the face. This situation represents well known experiments designed to illustrate DIGM. The corresponding theory is also given for grain boundaries which do not span the plate, but rather trail behind the moving intersection with one of the faces, never reaching the other one.

Stress- and diffusion-induced interface motion: Modelling and numerical simulations

“…In this paper we replace (2.9) by v = c. It was demonstrated in [9,13] that this does not change the limiting sharp interface problem.…”

“…We refer to the papers [14,13,17,21,2,27] for more details about this method. Obtaining the equations in the bulk is straightforward and we only discuss the derivation of the equations on the interface.…”

On the elastic driving force in diffusion-induced grain boundary motion

“…It was concluded that the elastic driving force was the best candidate, though perhaps not the only one. In that paper, however, the elastic energy changes were modelled in an A mathematical model detailing the way in which the elastic driving force and the capillary force due to curvature of the grain boundary interact with diffusion in the grain boundary to determine the shape and velocity of the moving grain boundary is studied in refs [16], [17]. The elastic driving force seems also to be important for the theory of discontinuous preciptiation; see ref.…”

Chemically induced grain boundary dynamics, forced motion by curvature, and the appearance of double seams

“…This phase-field model was further developed in Fife et al [6], especially in the context of curved boundaries in a 2D geometry, and then through a systematic approximation procedure reduced to a free boundary model. In this latter version, the grain boundary has zero thickness.…”

“…[3], Fife et al [6] and Cahn & Penrose [4] is extended, in the case of thin metallic films, to account for bidirectional motion, together with the appearance of S-shapes and double seam configurations. These are often observed in the laboratory.…”