Diffusion in the Presence of Grain Boundaries: a Variable Length Scale Simulation Method
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Diffusion in the Presence of Grain Boundaries: a Variable Length Scale Simulation Method Irina V Belova and Graeme E Murch Diffusion in Solids Group, Dept. Mechanical Engineering The University of Newcastle, Callaghan, NSW 2308 AUSTRALIA ABSTRACT In this paper we introduce a Monte Carlo method for numerical analysis of the problem of tracer diffusion in the presence of isolated grain boundaries with a variable mesh in the direction perpendicular to the grain boundary plane. A number of isoconcentration contour profiles were studied. Two different expressions for the angle between the isoconcentration contours and the grain boundary were derived for the instantaneous tracer source following procedures analogous to Fisher’s and Whipple’s approximations for the constant tracer source. Comparison with the numerical data was made. Both these approximations overestimate the grain boundary diffusivity. INTRODUCTION Many phenomenological diffusion problems cannot be easily solved analytically and one needs to invoke numerical techniques. One of the most significant of the numerical techniques is the Monte Carlo method. A number of Monte Carlo strategies are possible in principle. One strategy that has enjoyed some success is the grid method [1-8]. The system is overlaid with a grid which is explored by particles that sample the various diffusivities. For example, the problem of determining tracer concentration depth profiles from an instantaneous tracer source in the presence of parallel grain boundary slabs of high diffusivity can be solved in the following way [3, 8]. Particles are permitted to diffuse using Monte Carlo methods, one at a time, on a grid for some time t after having been released from a random position on the tracer source plane. Their jump rates depend on position i.e. grain or grain boundary. The tracer concentration depth profile can be assembled from a large number of particle histories (typically 106) by accumulating the final position of each particle after the same `anneal' time. The grid is purely an artifice for conveniently representing the various diffusivities and geometry of the system and has no atomistic significance. The method gives excellent agreement with exact solutions when available. A shortcoming of this method is that for large grain boundary spacings a particle can spend a great deal of time within a grain sampling a diffusivity that does not change for large distances. Methods which attempt to speed up the process by altering the time step length carry a large computational overhead and can lead to difficulties at short times with overshooting the time permitted for each particle to diffuse. In this paper, a different strategy is suggested which is based on a variable grid spacing. It permits large spatial steps through regions where the diffusivity is unchanging but on the other hand permits small spatial steps in regions where there is functional detail to be captured, for example, where the diffusivity changes rapidly. The method is exemplified with the grain boundary problem a
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