Theory of Order-Disorder Kinetics

The problem of relating fundamental atom movements to the change of the state of order of an alloy is attacked in a basic manner by introducing an infinite array of distribution functions for the occupation of all possible sets of lattice sites. These distribution functions determine all the kinds of order in the lattice, including the usual long- and short-range order. They are shown to obey general equations of motion which are linear and in which the kinetic parameters of atom movements occur as coefficients.

Solutions of the equations of motion must be found by approximations, and a variety of possible procedures are suggested. These depend on expressing higher order distribution functions by combinations of those of lower order. The simplest of these procedures is explored in detail for two mechanisms of atom movement, direct interchange and vacancy interchange, and for two common lattice types, $\mathrm{AB}$ b.c.c. and $A{B}_3$ f.c.c. The result is a calculation of the time dependence of long-range order in homogeneous systems, and is expected to be reliable whenever the long-range order is reasonably high. The simple theory leads to the Bragg-Williams result at equilibrium, and gives a fundamental derivation and limitation of the conception that ordering is a "chemical reaction" of the type $A^{\alpha }+B^{\beta }\rightleftarrows A^{\beta }+B^{\alpha }$. The vacancy and direct interchange models lead to qualitatively similar, but not identical results. Present evidence suggests predominance of the vacancy mechanism, at least in close-packed systems.

The simple vacancy model is applied to experiments of Burns and Quimby on electrical resistivity in ${\mathrm{Cu}}_3$Au. The observed relaxation times, over a range of temperatures below the critical temperature, are in reasonably good accord with theory when parameters derived from diffusion measurements in Cu and Au are employed.