We now highlight key features of the above analysis We find c t E t τ s t

We now highlight key features of the above analysis

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We now highlight key features of the above analysis. We find c ( t ) E ( t ) τ s ( t ) without assumptions. Equations (7)–(9) predict that the strength change is additive to τ s0 . τ 0 is governed by the material parameters c 0 , W and b , but not w . t = [ 3 ν 0 e β( H c W / 2 ) ] 1 and ˙ ε = Ω / t are governed by an e ff ective activation enthalpy H e ff c = H c W / 2 H b . The exponent in equation (2) is n = 1. The phenomenology of equations (2) and (3) is thus put on a firm theoretical foundation. To explicitly show the cross-core mechanism and to support the analytic model, we use a kinetic Monte Carlo (kMC) model (see Methods). We do not take explicit account of the cross-core migration enthalpy H c ; all transitions use the bulk nature materials VOL 5 NOVEMBER 2006 877 Nature PublishingGroup ©200 6
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ARTICLES Δ τ s (MPa) τ τ b t = D b t /2 b 2 Cross-core diffusion dominates Bulk diffusion contributes Δ E cont Δ E core Δ E tot 0 10 20 30 40 50 0 0.05 0.10 0.15 0 1 2 3 4 5 6 Δ E (eV) s 0.10 0.30 0.50 0.70 3 4 5 6 7 8 Δ Δ E (eV) Γ b t = D b t /2 b 2 Γ Figure 4 Change in binding energy (squares) and change in strength (circles) versus dimensionless time, as computed by kMC simulations at 300K. Core and bulk activation enthalpies and transition rates are assumed equal ( H b = H c , Γ b = Γ c ) in these simulations. Also shown are predictions for the binding energy due to the cross-core diffusion mechanism equation (5), which dominates for D b t / 2 b 2 (3e β W / 2 ) 1 0 . 025 (thick black line), the continuum bulk diffusion mechanism of equation (11) at longer times D b t / 2 b 2 > D b t cont min / 2 b 2 = (6 β W ) 1 0 . 055 (thin black line) and the sum of the two (grey line). Inset: Simulated and predicted total energies versus time, to longer times. migration enthalpy H b modified only by the thermodynamic driving force. Nonetheless, the large cross-core driving force W accelerates the cross-core di ff usion su ciently to separate the timescales for the cross-core and bulk mechanisms. Figure 3 shows the Mg concentration distribution at a time where the core di ff usion is complete but the bulk di ff usion has barely started; the concentration in the core on the first compressive slip plane is nearly 0%, whereas that on the first tensile slip plane is nearly doubled to 10%, with little change elsewhere. Proceeding quantitatively, as di ff usion occurs in the kMC model we compute the binding energy of the dislocation as a function of the virtual displacement x from its initial position x = 0, by summing the individual Mg–dislocation energies as E ( x , t ,τ) = i E s ( x i ( t ) x , y i ( t )) τ b ξ x . (10) Equation (10) is accurate because Mg–Mg interactions are negligible, even in the core 26 . As in equation (6), the time- dependent strength is computed as τ s ( t ) = d E ( x , t , 0 )/ d x | max / b ξ . Figure 4 shows the change in binding energy E ( t ) = E ( 0 , t 0 ) E ( 0 , 0 0 ) and the change in strength τ s ( t ) = τ s ( t ) τ s0 versus dimensionless time Γ b t = D b t / 2 b 2 , obtained by averaging 100 kMC runs for each of six initial random Mg configurations at c 0 = 5% and 300 K. Consistent with the predictions of equations (4), (5) and (7), at early times Γ b t ( 1 / 3e β W / 2 ) 0 . 025, where the cross-core 0
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