Self-diffusion along twist grain boundaries in Cu
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In a previous paper we studied vacancy diffusion in two high-angle twist grain boundaries in Cu, using the EAM. In this paper, we discuss vacancy diffusion along four additional twist grain boundaries, from 8.8-43.6°. Vacancy formation energies in all the possible sites were calculated (0.14-1.42 eV) and found to be directly related to the degree of coincidence with the neighboring crystal planes. The optimal migration paths were found to coincide with the screw dislocations which comprise the boundary. Vacancy migration energies were found to be low (0.02-0.52 eV). The activation energies for self-diffusion at the boundaries were found to be less than half of the bulk value, in general agreement with experiment. Calculated diffusion rates, 3D, for medium-high angle twist grain boundaries were in reasonable agreement with experimental data for polycrystalline material. Diffusion rates were found to decrease with increasing twist angle, in contrast with two sets of conflicting experimental data.
I. INTRODUCTION Grain boundary diffusion is important in many processes involving material rearrangement, such as Coble creep, sintering, diffusion-induced grain boundary migration, discontinuous precipitation, discontinuous dissolution, recrystallization, and grain growth.1'2 At low temperatures (
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FIG. 4. Structure of a vacancy in various twist boundaries, (a) 217 (28.07°), (b) 229 (43.06°), (c) 265 (14.25°), and (d) 285 (8.79°). The oval indicates the delocalized vacancy.
Monte-Carlo procedure to calculate correlation factors in periodic grain boundaries; they found / to be roughly 0.5 for grain boundary migration behavior similar to ours (diffusion in 2D). Nv can be written as Nv = exp
- 9L kT
(6)
Hi is the formation enthalpy of the vacancy (equivalent to E{ at zero pressure), and S{ is the formation entropy of the vacancy. S( can be roughly estimated using Zener's31 approximation. \ o pi
S'v =
v
(i = formation or migration)
where G{, is the free-energy of formation of a vacancy given by, G{ = Hi- TS{ (7)
For Cu, A = 0.5, /3 = 0.4, and Tmp is the melting temperature of 1356 K. 3207
J. Mater. Res., Vol. 7, No. 12, Dec 1992 http://journals.cambridge.org
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M. Nomura and J. B. Adams: Self-diffusion along twist grain boundaries in Cu
FIG. 5. Optimal paths for delocalized vacancy (indicated by the ovals) migration, (a) 2 8 5 (8.79°), path 1: intracell diffusion and E™ = 0.26 eV; path 2: intercell diffusion and E™ = 0.02 eV. (b) 2 2 9 (43.06°), path 1: intercell diffusion and E™ = 0.052 eV; path 2: interplanar diffusion and EJ1 = 0 . 1 1 eV. The squares represent planes two layers away from the boundary and the circles represent planes directly adjacent to the boundary. The filled (or the open) symbols are on the same side of the boundary, (c) 2 1 3 (22.62°), path 1: intracell diffusion and £™ = 0.04 eV; path 2: intercell and diffusion and E™ = 0.02 eV.
To calculate T, one must first determine the vacancy jump rate, k, which is normally expressed as
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