Macroscopic description of interface migration by ledge and kink motion controlled by volume diffusion
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I.
INTRODUCTION
THE purpose
of this paper is to relate the macroscopic description of the diffusion-limited growth of a precipitate to the microstructure of the p r e c i p i t a t e / m a r x interface. The relevant interface microstmcture is taken to be given by the distribution of a subset of interface sites that can act as diffusion sinks; the remaining sites are assumed to be essentially passive. We suppose that the interface is macroscopically fiat and represent it by an adjacent mathematical plane, ~ (Figure 1). Possible extensions of the approach to curved interfaces and finite crystals are discussed in Section VI. The sink sites may be kinks [~] in monatomic surface ledges, or in some models, [e] effectively all sites along a set of ledges; the motion of these sink sites is assumed to be limited by matrix diffusion directly to the sites. The average concentration at a sink site will be assumed to be c,, corresponding to the matrix concentration in equilibrium with a flat interface; thus, the possible kinetic elevation of the concentration above c, at a moving sink site is assumed negligible compared to the elevation we shall find of the average concentration near the interface due to solute accumulation at the passive sites. The passive sites may be those on a terrace or low index plane. These models correspond to the migration of an interface below the roughening temperature. There is an extensive literature t3j of crystal growth kinetics relating crystal surface structure and processes to the rate of crystal growth. Early work devoted to crystal growth by volume diffusion directly to kinks or ledges
W.W. MULLINS, Professor, is with the Department of Metallurgical Engineering and Materials Science, Carnegie Mellon University, Pittsburgh, PA 15213. This paper is based on a presentation made in the symposium "The Role of Ledges in Phase Transformations" presented as part of the 1989 Fall Meeting of TMS-MSD, October 1-5, 1989, in Indianapolis, IN, under the auspices of the Phase Transformations Committee of the Materials Science Division, ASM INTERNATIONAL. METALLURGICAL TRANSACTIONS A
is that of Burton et al., t~l Bennema, t4] and Chernov. t2] These authors obtained approximate solutions to the matrix diffusion field by modeling kinks and ledges as hemispheres and semicylinders, respectively, with appropriate boundary conditions. More recently, analytic solutions t5'6'71 and computer simulations ts'9j of interface growth by volume diffusion to ledges have been obtained and discussed. What is offered here is a formulation in which the macroscopic diffusion process in the matrix is described by a one-dimensional diffusion equation for the average matrix concentration, ~(z, t), averaged or coarse-grained, on planes parallel to ~. The boundary condition satisfied by ff(,z, t) on E is then related to the microscopic distribution of sink sites on the interface. This boundary condition is evaluated for several atomic-scale idealized surface structures. When the boundary condition is known, growth problems may
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