Stress-Assisted Precipitation on Dislocations

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Abstract
The kinetic theory for the rate of stress‐assisted precipitation on dislocations is re‐examined in order to extend the work of Cottrell and Bilby. Two approximate expressions are used for the elastic interaction between a solute atom and edge and screw dislocations: V=−A sinθ/r and V=−B/r, respectively. The resulting partial differential diffusion equations are integrated numerically to get the time‐dependent rate of precipitation on an isolated dislocation. These results are used to calculate the short‐time part of the precipitation curve for an array of dislocations. Exact steady‐state solutions to the diffusion equations are derived for both interactions and are used with a variational procedure to establish the long‐time part of the curve for a regular array. For very short times the precipitated fraction W is proportional to t⅔ as derived by Cottrell and Bilby, but this result is not accurate over the range of t during which most of the precipitation occurs. The long‐time part of the curve is given accurately by W=1−exp(−t/τ), where τ can be calculated by replacing each dislocation and its stress field by a cylinder with an ``effective capture radius,'' the value of which is calculated for each form of interaction. Complete precipitation curves are obtained for a regular array by combining the short‐ and long‐time results, and it is shown that these are changed but slightly if the array is random. These results differ significantly from a formula suggested by Harper to extend Cottrell and Bilby's work, and they thereby indicate that the interpretation of the precipitation process for carbon in alpha‐iron is not yet entirely established.

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