Simulation of diffusion by direct solution in the lattice-fixed frame of reference
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IN simulations of bulk diffusion, the fluxes are normally defined in a number-fixed frame of reference; i.e., the net fluxes of the components vanish over a certain point in the system. Closely related to the number-fixed frame of reference is the volume-fixed frame of reference, which is defined by the condition that the net flow of volume should vanish. Both these reference systems lead to a straightforward treatment of the fluxes, although cross-terms arise, wherein an N component system has N " 1 independent fluxes and (N " 1)2 interdiffusion coefficients. In order to take the so-called Kirkendall effect[1,2] into account, the vacancy flux must be considered. The vacancy flux opposes the net flux of atoms, but since the creation or annihilation of vacancies will cause lattice planes to grow or shrink, it is not possible to define the fluxes relative the number- (or volume-) fixed frame of reference. Instead, the lattice-fixed frame of reference, defined by rows of inert markers, must be applied. Previous theoretical treatments of the Kirkendall effect have been based on the assumption that the total vacancy content is constant and close to local equilibrium. By using this assumption and by first calculating the concentration profile in the number-fixed frame ˚ gren[3] could calculate the of reference, Ho¨glund and A Kirkendall shift and the risk for formation of Kirkendall ˚ gren was origiporosity. The approach by Ho¨glund and A nally developed for binary systems but has recently been extended to multicomponent systems.[4] In that method, the diffusion coefficients must be evaluated in both the numberfixed and the lattice-fixed frames of reference; i.e., both (N " 1)2 interdiffusion and N(N " 1) intrinsic diffusion coefficients are needed. A similar approach has also been developed by Matan et al.[5] The lattice-fixed frame of reference was recently used for simulation of diffusion by a random-walk process in order to evaluate the Kirkendall shift directly.[4,6] It was also found that the rate of a phase transformation was obtained automatically when the new method was applied to a twophase material. It was given by the same code as the Kirkendall shift. This result inspired the idea that a HENRIK STRANDLUND, Graduate Student, and HENRIK LARSSON, Researcher, are with the Division of Physical Metallurgy, Department of Materials Science and Engineering, Royal Institute of Technology (KTH), SE-100 44 Stockholm, Sweden. Contact e-mail: [email protected] This article is based on a presentation made in the ‘‘Hillert Symposium on Thermodynamics & Kinetics of Migrating Interfaces in Steels and Other Complex Alloys,’’ December 2–3, 2004, organized by The Royal Institute of Technology in Stockholm, Sweden. METALLURGICAL AND MATERIALS TRANSACTIONS A
lattice-fixed simulation based on Fick’s law should also be capable of directly yielding the Kirkendall shift as well as the progress of a diffusion-controlled phase transformation. The purpose of the present work was to develop such a simulation procedure for bulk diffusion. The application to phase t
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