10 20 30 x Å Δ W x w 2 Δ W x w 2 Δ W 12 10 08 06 04 02 Initial strength

10 20 30 x å δ w x w 2 δ w x w 2 δ w 12 10 08 06

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10 20 30 x (Å) [ Δ W ( x w /2)– Δ W ( x + w /2)]/ Δ W 1.2 1.0 0.8 0.6 0.4 0.2 0 Initial strength locations in kMC Figure 5 Strengthening parameter versus dislocation position during sliding. The positions of the static solute strengthening points in the kMC simulations are as indicated. mechanism dominates, both energy and strength changes are linear in t , for example E t n , n 1, indicating single-atomic-jump transport rather than the bulk di ff usion ( n = 2 / 3 ) of equations (1a) and (1b). The strength change is also independent of the initial values, implying that the strength change is approximately additive. Similar results are found at 500 K. At early times, where cross-core di ff usion dominates, excellent quantitative agreement between the analytic model and the kMC simulations is obtained for the binding energy: Fig. 4 shows the prediction of equation (5) for E core ( t ) , with W = 0 . 13 eV and w = 7 . 5 b from Fig. 1, and with Γ c = Γ b , as used in the kMC. Good agreement is also found at 500 K using the same parameters. The magnitude of the strengthening in the kMC is also in good agreement with our model. Equation (8) predicts a ratio of strength to binding energy of τ s ( t )/ E ( t ) = α/ξ bw . Figure 5 shows the model prediction for [ W ( x w / 2 ) W ( x + w / 2 ) ] / W using the energies in the kMC model; the arrows indicate the positions of the original strengths in the kMC samples, for which the average is α = 0 . 56. The model thus predicts τ s ( t )/ E ( t ) = 13 MPa eV 1 , versus a kMC simulation value of 9 MPa eV 1 . The small discrepancy is probably due to the smearing and neglect of subtle details near x w / 2 in the model and/or to small amounts of lateral di ff usion in the kMC. A kMC study using the actual site-to-site activation enthalpies in and around the core should be carried out and compared with a more detailed analytic model before attempting to rectify the small di ff erences. At longer times, the qualitative and quantitative characteristics of the bulk continuum di ff usion models emerge from the kMC simulations 1,5,6,22,23 . The binding energy and strength continue to increase nearly in proportion, evolving towards a power law in time with n 2 / 3, and the binding energy eventually reaches 50–100 eV and the strength 300 MPa. The minimum time t cont min at which continuum di ff usion over distances > b can occur is D b t cont min / 2 b 2 1 / 6 β W (ref. 5), after which the additional core concentration and binding energy can be estimated from equations (1a), (1b) and (5) as c cont ( t ) = ( f c 0 )( 1 e (( t t cont min )/ t ) 2 / 3 ), E cont ( t ) = c cont ( t ) NW , t > t cont min . (11) The total binding-energy change is then E tot ( t ) E core ( t ) + E cont ( t ) for t > t cont min . Figure 4 shows the predicted E cont ( t ) and E tot ( t ) using W = 0 . 08 eV as a slightly adjustable parameter (theoretically, W W / 2); very good agreement is obtained to long times (Fig. 4 inset). Results at 500 K using the same parameters are also good.
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