Kinetics of B2, DO 3 , and B32 ordering: Results from pair approximation calculations and Monte Carlo simulations
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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, D0 3 , or B32 order. The theory was developed in a superposition approximation for a vacancy mechanism on a bcc lattice with first- (Inn) and second-nearest neighbor (Inn) pair interactions. Chained Inn 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 D0 3 ordering are reported for different Inn and 2nn interaction strengths. The influence of annealing temperature on the kinetic paths through the space of B2, D0 3 , 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 AB 3 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.
I. INTRODUCTION In a previous paper, we developed and implemented a kinetic theory of B2 and D0 3 ordering using a master equation method (MEM) in the point (Bragg-Williams) approximation.1 Those efforts were motivated by our experimental work on ordering kinetics in Fe3Al.2 Here, we formulate an alternate theory of ordering in bcc alloys using the pair (Bethe) approximation and the foursublattice model of the previous theory. In addition to the increased accuracy obtained by using a larger base cluster, i.e., the pair instead of the point, the new theory also provides short-range order (SRO) parameters. It is interesting to identify the qualitative similarities of the long-range order (LRO) kinetics in the point and pair approximations. The point and pair approximations were compared by examining their predictions of the "kinetic paths" taken by the alloy. To test further the reliability of these calculations, we also performed Monte Carlo simulations (MCS) of ordering.3^7 These MCS employ a single vacancy and the same mechanism of atom movement as the analytical theories, but are essentially exact for the large crystal lattices (containing up to half a million atoms) used in 348
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J. Mater. Res., Vol. 9, No. 2, Feb 1994
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the present work. Some results
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