On coherency-induced ordering in substitutional alloys—I. Analytical Original Research Article

As pointed out by Linus Pauling in his classic work on the relationship between crystal packing and ionic radius ratio, a difference in atomic size can be accommodated more readily by an ordered structure than by a disordered one. Because of mathematical complexity, however, very few works have been reported for substitutional alloys. In this work, coherency-induced ordering in substitutional alloys is examined through a simple model based on a two-dimensional square lattice. Within the assumption of nearest-neighbor interactions on a square lattice, both modified Bragg–Williams and Onsager approaches show that coherency strain arising due to atomic mismatch can exert profound effects on order–disorder transitions in substitutional alloys. If the alloy system is elastically homogeneous and Vegardʹs law is obeyed, the order–disorder transition is of a second-order kinetics. If the atomic mismatches significantly deviate from Vegardʹs law, however, the transition may become a first-order kinetics, as the configurational free energy surface is composed of double wells. At the transition of a first-order kinetics, the lattice parameter can either increase or decrease upon heating, i.e. the lattice parameter of an ordered state can be less or greater than that of a disordered state. The results of Onsagerʹs approach are independently confirmed with those of the Discrete Atom Method, a Monte Carlo technique predicated upon the combination of statistical mechanics and linear elasticity.