A theoretical investigation on non-equlibrium grain boundary segregation1

Based on the concept that mobile solute-vacancy complexes migrating to grain boundaries (vacancy sink) is responsible for the non-equilibrium grain boundary segregation of solute atoms in an alloy, analytical expressions describing such a segregation process are presented. The driving force for the segregation is a decrease in the free energy of the system caused by the annihilation of vacancies at grain boundaries during cooling from an initial temperature. From the theoretical expressions, a sufficient condition for the non-equilibrium segregation to occur is determined to be that Dsf/Dp<1 (Dsf and Dp are diffusion coefficients for free solute atoms and solute-vacancy complexes, respectively). Besides, the expressions predict that the grain boundary segregation enrichment increases with a decrease in the ratio Dsf/Dp, the vacancy formation energy or the cooling rate (above a critical cooling rate) and an increase in the initial temperature or the binding energy of complexes, while the relative enrichment of solute atoms at grain boundaries decreases with an increase in the bulk concentration of solutes. Furthermore, an analytical relationship for the critical cooling rate giving the maximum grain boundary enrichment is also presented. The relationship exhibits that the critical cooling rate increases with an increase in the diffusion coefficient for solutes or complexes, the initial temperature or the binding energy of complexes and a decrease in vacancy formation energy, while it is independent of the bulk concentration of solute. On the other hand, the calculated results are compared with the computer simulations done by Karlsson in boron-doped Austenite. Although good agreement between the two studies on the characteristics of the non-equilibrium grain boundary segregation behavior is achieved, real segregation profiles of solutes can only be calculated quantitatively by the functional expression without spending much computation time.