Late-time theory for the effects of a conserved field on the kinetics of an order-disorder transition

The dynamics of an order-disorder transition is investigated through a nonlinear Langevin model known as model C. This model describes the dynamics of an ordering nonconserved field (e.g., sublattice concentration), {φ}, coupled to a nonordering conserved field (e.g., absolute concentration), c. An approximate asymptotic time-dependent solution is presented for both fields through a singular perturbative solution of the coupled nonlinear-dynamical system. In particular, analytic expressions for the dynamic structure factors [i.e., S_φ(k,t)==〈{φ}(k,t)φ*(k,t)〉, and Sc(k,t)==〈c(k,t)c*(k,t)〉, where k is the wave vector and t is time] of both fields are presented. In the late-time regime these expressions reduce to the scaling forms S_φ(k,t){}td/2{f}_{{{φ}}}$(Q) and ${S}c(k,t){t}^{d/2{{-}}1}fc(Q), where Q=kt1/2. Furthermore it is shown that f_φ(Q){∝}Q^-d-1, fc(Q){∝}Q^-d+1 for Q{}1 and fc(Q){∝}Q⁴ for Q{}1. Intermediate-time corrections, due to a finite interfacial width, to the asymptotic solutions of both fields are also obtained. Many of these predictions are experimentally accessible.