Void lattice formation as a nonequilibrium phase transition

The evolution of a void ensemble in the presence of one-dimensionally migrating self-interstitials is considered, consistently taking into account the nucleation of voids via the stochastic accumulation of vacancies. Including the stochastic fluctuations of the fluxes of mobile defects caused by the random nature of diffusion jumps and cascade initiation, the evolution of the void ensemble is treated using the Fokker-Planck equation approach. A system instability signaling a nonequilibrium phase transition is found to occur when the mean free path of the one-dimensionally moving self-interstitials becomes comparable with the average distance between the voids at a sufficiently high void-number density. Due to the exponential dependence of the void nucleation probability on the net vacancy flux, the nucleation of voids is much more favored at the void lattice positions. Simultaneously, voids initially nucleated at positions where neighboring voids are nonaligned will also shrink away. These two processes leave the aligned voids to form a regular lattice. The shrinkage of nonaligned voids is not a usual thermodynamic effect, but is a kinetic effect caused entirely by the stochastic fluctuations in point-defect fluxes received by the voids. It is shown that the shrinkage of the nonaligned voids, and thus the formation of the void lattice, occurs only if the effective fraction of one-dimensional interstitials is small, less than about 1%. The formation of the void lattice in this way can be accomplished at a void swelling of below 1%, in agreement with experimental observation. The dominance of void nucleation at void-lattice positions practically nullifies the effect of void coalescence induced by the one-dimensional selfinterstitial transport.