transport is then E core t c t t N W 5 where N 2 w ξ 3 b 2 is the number of

Transport is then e core t c t t n w 5 where n 2 w ξ

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transport is then E core ( t ) = c t ( t ) N W , (5) where N = 2 w ξ/ 3 b 2 is the number of atoms in the interaction core width w of a dislocation segment of length ξ on a (111) slip plane in a face-centred-cubic crystal. Prediction of the strength change due to cross-core solute di ff usion requires a model for the dislocation/solute-cloud energy E ( x , t ,τ) versus dislocation position x and applied stress τ . Starting at x = 0, the dislocation attempts to glide away from the core region w / 2 x w / 2 under an applied stress. As Fig. 2 schematically shows, the energy changes because di ff used solutes on the tension side are left behind as the core moves, whereas new undi ff used solutes on both tension and compression sides enter the core region. Denoting the solute binding energy change at position x in the core by W ( x ) , the total energy is E ( x , t ,τ) = E 0 ( x ) τ b ξ x i W ( x i x ), where the sum is over the di ff used solutes { i } at positions {− w / 2 x i w / 2 } , E 0 ( x ) is the energy well for the dislocation before any solute di ff usion and τ b ξ x is the work done by the applied stress. The time-dependent strength τ s ( t ) is then the applied stress at which the energy barrier becomes zero or, equivalently, at which the energy versus position has no local minimum, d E ( x , t s )/ d x 0 for all x 0. The strength is then τ s ( t ) = max x ( d E 0 ( x )/ d x )/ b ξ + ( 1 / b ξ) j d W ( x j x )/ d x . (6) 0.10 0.09 0.08 0.06 0.04 0.02 0.01 Figure 3 Contour map of Mg solute concentration after diffusion for a time D b t / 2 b 2 = 0 . 18 at 500K as computed in the kMC model. Solute transport is predominantly across the slip plane from compression (bottom) to tension (top). Initial Mg concentration was 0.05 throughout the system. Changing the sum to an integral, and noting that by symmetry W ( x ) = W ( x ) , leads to τ s ( t ) = max x [ ( d E 0 ( x )/ d x )/ b ξ + ( 2 c t ( t )/ 3 b 3 ) × [ W ( x w / 2 ) W ( x + w / 2 ) ]] . Approximating the solute binding energy outside the core as zero and a constant W within w / 2 x w / 2 yields τ s ( t ) = d E 0 ( x )/ d x | max / b ξ + α( 2 c t ( t )/ 3 b 3 ) W = τ s0 + τ s ( t ), (7) where τ s0 = d E 0 ( x )/ d x | max / b ξ is the initial strength and τ s ( t ) = α( 2 c t ( t )/ 3 b 3 ) W is an additive strengthening due to solute di ff usion, with α = 1. A more accurate result, recognizing that the initial strength is largely independent of the di ff used solutes in the core, has α = W ( x w / 2 ) W ( x + w / 2 ) x / W , where x denotes an average over the range of positions where d E 0 ( x )/ d x is maximized. Using equation (4) in (7) leads to our main result for the time-dependent additive strengthening, τ s ( t ) = α( 2 c 0 W / 3 b 3 ) tanh W / 2 ) [ 1 e 6cosh W / 2 ) Γ c t ] , (8) which is identical in form to equation (2) with τ 0 = 2 α c 0 W tanh W / 2 )/ 3 b 3 2 α c 0 W / 3 b 3 t = [ 6cosh W / 2 ) Γ c ] 1 ≈ [ 3 ν 0 e β( H c W / 2 ) ] 1 n = 1 . (9) Equation (3) follows by substituting t = t w = Ω / ˙ ε and ˙ ε = Ω / t into equation (8). Equations (7)–(9) are the main results of this paper.
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