Bulk solute di ff usion around a static dislocation is well established in

Bulk solute di ff usion around a static dislocation

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Bulk solute di ff usion around a static dislocation is well established in continuum models 1,3–6 . Using the continuum pressure field for an edge dislocation, p ( x , y ) = ( 1 + ν)μ b 3 ( 1 ν) π y /( x 2 + y 2 ) with Burgers vector b , shear modulus μ and Poisson’s ratio ν , the change in core solute concentration versus time, t , is c cont ( t ) = c ( t ) c 0 = ( f c 0 )( 1 e ( t / t ) 2 / 3 ) (1a) t = ( 2 b 2 / D b β W )( f / 3 c 0 ) 3 / 2 , (1b) where c 0 is the bulk solute concentration, D b the bulk solute di ff usion coe cient, W ≈ | p ( x = 0 , y = ± b ) V | the maximum solute binding energy, β = 1 / kT , and f = c 0 e β W /( 1 + c 0 e β W ) the saturation concentration. The di ff usion coe cient is D b = 2 b 2 ν 0 e β H b , where H b is the activation enthalpy and ν 0 the attempt frequency. Note that the pressure is singular and changes nature materials VOL 5 NOVEMBER 2006 875 Nature PublishingGroup ©200 6
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ARTICLES Tension Compression 10 Å –0.017 –0.033 –0.070 0.116 0.098 0.093 0.090 –0.084 –0.056 –0.028 0.028 0.056 0.084 Figure 1 Binding energy of a Mg substitutional solute and an edge dislocation in Al versus Mg solute position. Contours in eV; positive energies indicate binding, that is lower total energy. Cores of the split partials are evident on the compression side of the dislocation 24 . Arrows denote flux of solutes around the dislocation core in traditional continuum models. Right: magnification of the core region, with specific binding energies as indicated in eV, and arrows indicating the two cross-core transitions for which the activation enthalpy has been computed in ref. 19. sign at y = 0, precluding continuum treatment of di ff usion within the core. Connecting solute di ff usion to DSA and nSRS requires further approximations and assumptions. The form of equation (1a) is assumed to apply to a time-dependent extra strength τ s ( t ) (resolved shear stress to move the dislocation) of τ s ( t ) = τ 0 ( 1 e ( t / t ) n ), n = 2 / 3 . (2) Solute di ff usion occurs over the waiting time t w when dislocations are pinned at obstacles 7–10 . t w is related to the strain rate via t w = Ω / ˙ ε , where Ω = ρ m b / ρ f 10 4 –10 3 , and ρ m and ρ f are the mobile and forest dislocation densities 9–16 . Setting t = t w = Ω / ˙ ε in equation (2) and defining ˙ ε = Ω / t then leads to a strain-rate- dependent strengthening of τ s ( ˙ ε) = τ 0 ( 1 e ( ˙ ε / ˙ ε) n ), n = 2 / 3 . (3) τ 0 and ˙ ε , and sometimes n , are then treated as fitting parameters 15,16 . Equation (3) predicts nSRS by construction, as anticipated by Penning 17 . Use of equation (3) with τ 0 = 12 MPa and ˙ ε = 5 × 10 5 s 1 at 300 K in a homogenized crystal plasticity model leads to the prediction of serrated flow and the various types of macroscopic Portevin–LeChatelier band versus strain rate observed in an Al–2.5% Mg alloy 16 . This is strong evidence that the phenomenological form of equation (3) is basically correct.
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