  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 Vegard's law is obeyed, the
order-disorder transition is of a second-order kinetics. If the atomic
mismatches significantly deviate from Vegard's 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 Onsager's 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.
