A kinetic theory of ordering based on the path probability method was implemented in the pair (Bethe) approximation and used to study the kinetics of short- and long-range ordering in alloys with equilibrium states of B2, DO3, or B32 order. The theory was developed in a superposition approximation for a vacancy mechanism on a bcc lattice with first- (1nn) and second-nearest neighbor (2nn) pair interactions. Chained 1nn conditional probabilities were used to account for the entropy of 2nn pair configurations. Monte Carlo simulations of ordering were also performed and their results compared to predictions of the pair approximation. Comparisons are also made with predictions from an earlier kinetic theory implemented in the point (Bragg-Williams) approximation. For all three calculations (point, pair, and Monte Carlo), critical temperatures for B2 and DO3 ordering are reported for different 1nn and 2nn interaction strengths. The influence of annealing temperature on the kinetic paths through the space of B2, DO3, and B32 order parameters was found to be strong when the thermodynamic preferences for the ordered states were of similar strengths. Transient states of intermediate order were also studied. A transient formation of B32 order in an AB3 alloy was found when 2nn interactions were strong, even when B32 order was neither a Richards-Allen-Cahn ground state nor a stable equilibrium state at that temperature. The formation of this transient B32 order can be argued consistently from a thermodynamic perspective. However, a second example of transient B2 order in an AB alloy with equilibrium B32 order cannot be explained by the same thermodynamic argument, and we believe that its origin is primarily kinetic.
Kinetics of Catalytic Reactions Occurring in a Small Reaction Volume
