Upper Bounds on Coarsening Rates in Demixing Binary Viscous Liquids

We consider the demixing process of a binary mixture of two liquids after a temperature quench. Thermodynamics favors two domains of the two different equilibrium volume fractions with a characteristic interfacial layer. After an initial stage, the dynamics are thus driven by the reduction of the interfacial area. Therefore in large systems, one observes an increase in the typical length scale ` of the domains as a function of time t, a phenomenon called coarsening. In an intermediate stage, coarsening is mediated by cross-diffusion of the two species (“evaporation-recondensation process”). Siggia [21] argued that at a later stage, coarsening should be mediated by viscous flow of the mixture. Simple scaling arguments based on the assumption of statistical selfsimilarity of the domain morphology in time show that this hypothesis leads to a crossover in the coarsening rate: from ` ∼ t1/3 for diffusion-mediated to ` ∼ t for flow-mediated. We consider a simple sharp-interface model which just allows for flowmediated coarsening. For this model, we prove rigorously that coarsening cannot proceed faster than ` ∼ t — without assumption of statistical selfsimilarity. The analysis follows closely a method proposed in [15], which is based on the gradient flow structure of the evolution. This method translates bounds on the energy landscape into bounds on the dynamics. The bounds on the energy landscape relate the energy functional (given by the driving thermodynamics — here interfacial energy) to the intrinsic distance in configuration space (determined by the limiting dissipation mechanism — here viscous dissipation). As opposed to the diffusion-mediated case, the intrinsic distance is not known explicitly; instead, we use a Monge-Kantorowicz-Rubinstein transportation distance with logarithmic cost functional as a proxy. CNRS, Université de Nice (FR 2800 W. Döblin), Institut Universitaire de France Institute of Applied Mathematics, University of Bonn, 53115 Bonn, Germany 1