5 b 21 A there is a di ff erential binding energy of W 13 02 eV positive values

5 b 21 a there is a di ff erential binding energy of

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5 b = 21 ˚ A, there is a di ff erential binding energy of W 0 . 13 ± 0 . 02 eV (positive values denote binding), which provides a strong driving force for direct transport of solute atoms across the slip plane. Di ff erences in energy among other neighbouring sites are notably smaller. The contours of energy in Fig. 1 show the standard pathways for bulk di ff usion in continuum models: away from the compression side and into the tension side of the core. H values for vacancy-assisted Mg migration around the core have been computed 19 . There are reported values, 0.977 and 1.1 eV, for the two transitions indicated in Fig. 1 corresponding to Mg moving across the core from tension to compression against the thermodynamic driving force W . These values correspond to H c + W / 2, where H c 0 . 97 eV is the average activation enthalpy for transitions from tension to compression and vice versa, consistent with our independent calculations. This value is significantly lower than the activation enthalpy H b = 1 . 20 eV for bulk di ff usion computed by the same method. Migration paths perpendicular to the plane of the core on either the tension or compression side quickly attain the bulk value 19 . The above energetic and kinetic considerations show that the cross-core di ff usion has both a larger thermodynamic driving force and a reduced di ff usion activation enthalpy relative to regions outside the core. On this basis, we now model the solute motion directly across the slip plane in the core and its consequences for time-dependent dislocation binding energy and strengthening. Ignoring the small variations in cross-core energy di ff erence along the core, let the solute concentrations on the core tension and compression planes be c t ( t ) and c c ( t ) , and the average binding- energy di ff erence between these sites be W . Each site has three neighbours on the opposite side of the slip plane. Denoting a reference core transition rate as Γ c = ν 0 e β H c , the transition 876 nature materials VOL 5 NOVEMBER 2006 Nature PublishingGroup ©200 6
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ARTICLES Al Mg T C T C T C T C T C T C a b c e d f Figure 2 Schematic diagrams of the core solute positions relative to a dislocation being driven to the right by an applied stress. a , Before core diffusion; b , after core diffusion; c f , under increasing applied stress. The core region moves away from the positions of the diffused solutes on the tension (T) plane and into a region of undiffused solutes randomly distributed on the tension (T) and compression (C) planes. rate from tension to compression is Γ t c = 3 Γ c e β W / 2 and that from compression to tension is Γ c t = 3 Γ c e + β W / 2 ; the ratio is e β W , as required by detailed balance. Solution of the relevant master equations ˙ c t ( t ) = Γ c t c c ( t ) Γ t c c t ( t ) and ˙ c c ( t ) = Γ t c c t ( t ) Γ c t c c ( t ) , with initial conditions c c ( 0 ) = c t ( 0 ) = c 0 , yields the concentration change on the tension side c t ( t ) = c t ( t ) c 0 = c 0 tanh W / 2 ) 1 e 6cosh W / 2 ) Γ c t . (4) The dislocation-binding-energy change due to cross-core solute
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