Analyses of Two Climb Problems

Climbing Dislocations as Sinks for Excess Quenched-in Vacancies. Dislocations are generally the most important vacancy sources that act to maintain the vacancy concentration in thermal equilibrium as the temperature of a crystal changes. In the following, we analyze the rate at which the usual dislocation network in a 

The situation is illustrated in Fig. 11.12a. The loop is taken as an effective torus of large radius, RL. with much smaller core radius, R0, and the film thickness is 2d with d Rl. The vacancy concentration maintained in equilibrium with the loop 

The concentration, Cy (loop), can be found by realizing that the formation energy of a vacancy at the climbing loop is lower than at the flat surface because the loop shrinks when a vacancy is formed, and this allows the force shrinking the loop (see Section 11.2.3) to perform work. In general, Nyq = exp[—Gy/(kT) according to Eq. 3.65, and therefore 

The vacancy diffusion field around the toroidal loop will be quite complex, but at distances from it greater than about 2RL, it will appear approximately as shown in Fig. 11.12a. A reasonably accurate solution to this complex diffusion problem may be obtained by noting that the total flux to the two flat surfaces in Fig. 11.12a will not differ greatly from the total flux that would diffuse to a spherical surface of radius d centered on the loop as illustrated in Fig. 11.126. Furthermore, when d Rl, the diffusion field around such a source will quickly reach a quasi-steady state [20, 26], and therefore 

Analyses of the climbing rates of many other dislocation configurations are of interest, and Hirth and Lothe point out that these problems can often be solved by using the method of superposition (Section 4.2.3) [2]. In such cases the dislocation line source or sink is replaced by a linear array of point sources for which the diffusion solutions are known, and the final solution is then found by integrating over the array. This method can be used to find the same solution of the loopannealing problem as obtained above. 

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