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Modelling Elastically-Mediated Liquid-Liquid Phase Separation

We propose a continuum theory of the liquid-liquid phase separation in an elastic network where phase-separated microscopic droplets rich in one fluid component can form as an interplay of fluids mixing, droplet nucleation, network deformation, thermodynamic fluctuation, \emph{etc}. We find that the size of the phase separated droplets decreases with the shear modulus of the elastic network in the form of $\sim[\mathrm{modulus}]^{-1/3}$ and the number density of the droplet increases almost linearly with the shear modulus $\sim[\mathrm{modulus}]$, which are verified by the experimental observations. Phase diagrams in the space of (fluid constitution, mixture interaction, network modulus) are provided, which can help to understand similar phase separations in biological cells and also to guide fabrications of synthetic cells with desired phase properties.