Crossover of the coarsening rates in demixing of binary viscous liquids
Félix Otto, Christian Seis, Dejan Slepčev · Communications in Mathematical Sciences · 2013
We consider a model for phase separation in binary viscous liquids that allows for material transport due to cross-diffusion of unlike particles and convection by the hydrodynamic bulk flow.Typically, during the evolution, the average size of domains of the pure phases increases with time -a phenomenon called coarsening.Siggia [Eric D. Siggia, Phys.Rev. A, 20(2), 595-605, Aug. 1979] predicts that at an initial stage, coarsening proceeds mainly by diffusion, which leads to the well-known evaporation-recondensation growth law ℓ ∼ t 1/3 , when ℓ denotes the average domains size and t denotes time.Furthermore, he argued that at a later stage, convection by the bulk flow becomes the dominant transport mechanism, leading to a crossover in the coarsening rates to ℓ ∼ t.Siggia's predictions have been confirmed by experiments and numerical simulations.In this work, we prove the crossover in the coarsening rates in terms of time-averaged lower bounds on the energy, which scales like an inverse length.We use a method proposed by Kohn and the first author [Robert V. Kohn and Felix Otto, Commun.Math.Phys., 229(3), 375-395, 2002], which exploits the gradient flow structure of the dynamics.Our adaption uses techniques from optimal transportation.Our main ingredient is a dissipation inequality.It measures how the optimal transportation distance changes under the effects of convective and diffusive transport.