Stress-Assisted Precipitation on Dislocations

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.