Crossover of the coarsening rates in demixing of binary viscous liquids

Felix Otto, Christian Seis, and Dejan Slepčev

Contact the author: Please use for correspondence this email.
Submission date: 05. Apr. 2012
Pages: 41
published in: Communications in mathematical sciences, 11 (2013) 2, p. 441-464 
DOI number (of the published article): 10.4310/CMS.2013.v11.n2.a6
Bibtex
Download full preprint: PDF (372 kB)

Abstract:
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 [24] 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 argues 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. Siggias 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 [15], 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.