Kinetics of short- and long-range B2 ordering in the pair approximation
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The kinetic master equation is developed in the pair approximation to study disorder -» B2 order transformations in bcc binary alloys. Coupled sets of rate equations for the pair variables are obtained for atom movements by either the direct interchange mechanism or the vacancy mechanism. Numerical integrations provide the nonequilibrium relaxations of short- and long-range order (SRO and LRO) and the vacancy balances between the two sublattices. For binary alloys, disorder -*• order transformations were studied for all combinations of interatomic interaction strengths, activation barrier heights, and alloy stoichiometry variations. After a transient vacancy relaxation, all cases began with a quick relaxation of SRO, followed later and independently by the growth of LRO and additional SRO. There were some variations in kinetic path through SRO and LRO, moderate variations in overall kinetics, and large variations in vacancy balance. Some nonphysical aspects of kinetics in the pair approximation are discussed. I. INTRODUCTION
All materials in engineering service are somewhat out of thermodynamic equilibrium, but in recent years new experimental techniques such as rapid quenching, physical vapor deposition, laser processing, and ion beam bombardment have been used to prepare materials that are far-removed from thermodynamic equilibrium. Various forms of kinetic constraints, often related to diffusive phenomena, prevent the state of equilibrium from being attained in these highly nonequilibrium materials. As the kinetic constraint is relieved so that the material is allowed to relax toward equilibrium, for example by low temperature annealing, it passes through nonequilibrium states that are selected by the kinetic processes. It is a goal of nonequilibrium materials science to predict and control the nonequilibrium states of a material, and toward this end kinetic theories play a greater predictive role than thermodynamic theories. It is from this viewpoint of kinetic control that I approach the kinetics of disorder -> order transformations. In recent work, we showed how the chain of nonequilibrium states taken by a material en route to equilibrium (the "kinetic path" followed by the alloy) depends on the energetics of diffusive exchanges of atoms and vacancies. These previous formulations were in the mean field approximation, so they required either ternary alloys1'2 or alloys with secondary ordering3 to allow for two or more independent order parameters. In the pair approximation, however, even binary alloys have at least two order parameters that are partially independent: short-range order (SRO) and long-range or-
der LRO). Furthermore, when vacancies are introduced into a binary A-B alloy, vacancy-solute pairs as well as vacancy populations on the sublattices are additional, independent SRO and LRO parameters. , Two formulations of the kinetic master equation for the pair variables of a bcc binary alloy with B2 order are presented in Sec. II. Both formulations use activated state rate theory for their transitio
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