Statistical Mechanics Approach to Phase Transitions in Binary Alloys

Abstract

Order-disorder transformations in binary substitutional alloys are conventionally described by mean-field theory,
which predicts a sharp critical temperature but systematically misestimates it because it neglects short-range
correlations between neighboring atoms. We performed Metropolis Monte Carlo simulations of a nearest-neighbor
Ising-type lattice model for a body-centered-cubic A-B binary alloy at equiatomic composition, using simple-cubic
sublattice geometry with periodic boundary conditions, across 34 temperatures spanning 0.55 to 1.35 times the
mean-field critical temperature T_c^MF, with lattice sizes from L = 16 to L = 64 used for finite-size scaling. The
long-range order parameter fell from 0.94 at the lowest temperature tested to below 0.03 by T/T_c^MF = 1.05,
locating the simulated critical temperature at T_c/T_c^MF = 0.7524 ± 0.0038, 24.8% below the mean-field
prediction. Finite-size scaling of the order-parameter susceptibility across four lattice sizes yielded a critical
exponent ratio gamma/nu = 1.963 ± 0.041, consistent with the three-dimensional Ising universality class value of
1.963 to within 0.2%, and heat-capacity data scaled with alpha/nu = 0.183 ± 0.027. The short-range order
parameter, tracked separately from long-range order, remained substantial (0.31) even 15% above T_c, showing
that local chemical ordering persists well into the nominally disordered phase, a feature entirely absent from
mean-field predictions, which force short-range order to vanish discontinuously at T_c. A finite-size scaling collapse
of the specific-heat curves for all four lattice sizes onto a single master curve, using the extracted critical exponents,
achieved a mean squared residual of 0.0019 in reduced units, supporting the assignment of the transition to the 3D
Ising universality class rather than a mean-field-like transition. Sensitivity tests varying the ratio of next-nearest to
nearest-neighbor interaction energy from 0 to 0.25 shifted T_c/T_c^MF by up to 6.1% but left the extracted critical
exponents unchanged within uncertainty, indicating that the universality class of the transition is robust to moderate
changes in interaction range even though the transition temperature itself is not. These results quantify how much
standard mean-field treatments overestimate ordering temperatures in real alloys and provide benchmark
critical-exponent values for validating more computationally expensive atomistic alloy models.