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., ${\mathit{S}}_{\mathrm{\phi }}$(k,t)==〈φ(k,t)${\mathrm{\phi }}^{\mathrm{*}}$(k,t)〉, and ${\mathit{S}}_{\mathit{c}}$(k,t)==〈c(k,t)${\mathit{c}}^{\mathrm{*}}$(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 ${\mathit{S}}_{\mathrm{\phi }}$(k,t)≊${\mathit{t}}^{\mathit{d}/2\mathrm{}}$ ${\mathit{f}}_{\mathrm{\phi }}$(Q) and ${\mathit{S}}_{\mathit{c}}$(k,t)≊${\mathit{t}}^{\mathit{d}/2\mathrm{}\mathrm{-}1}$ ${\mathit{f}}_{\mathit{c}}$(Q), where Q=${\mathit{kt}}^{1/2}$. Furthermore it is shown that ${\mathit{f}}_{\mathrm{\phi }}$(Q)∝${\mathit{Q}}^{\mathrm{-}\mathit{d}\mathrm{-}1}$, ${\mathit{f}}_{\mathit{c}}$(Q)∝${\mathit{Q}}^{\mathrm{-}\mathit{d}+1\mathrm{}}$ for Q≫1 and ${\mathit{f}}_{\mathit{c}}$(Q)∝${\mathit{Q}}^4$ 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.