Assuming that mobile solute-vacancy complexes migrating to grain boundaries (vacancy sinks) were responsible for the non-equilibrium grain-boundary segregation of solute atoms in an alloy, analytical expressions were derived which described such segregation. The driving force for the segregation was a decrease in the free energy of the system, which was caused by the annihilation of vacancies at grain boundaries during cooling. From the theoretical expressions, a sufficient condition for the non-equilibrium segregation to occur was shown to be that the ratio of the diffusion coefficients for free solute atoms to that for solute vacancy complexes should be less than unity. The expressions predicted that the grain-boundary segregation enrichment increased with a decrease in the above ratio, in the vacancy formation energy or in the cooling rate (above a critical value), or with an increase in the initial temperature or the binding energy of complexes. The relative enrichment of solute atoms at grain boundaries decreased with an increase in the bulk concentration of solutes. An analytical relationship was also derived for the cooling rate which gave the maximum grain boundary enrichment. The relationship was such that the critical cooling rate increased with an increase in the diffusion coefficient for solutes or complexes, in the initial temperature or in the binding energy for complexes, or with a decrease in the vacancy formation energy. It was independent of the bulk solute concentration.

A Theoretical Investigation of Non-Equilibrium Grain Boundary Segregation. D.Lin, Y.Zhang: Materials Science and Engineering A, 1998, 256[1-2], 39-50