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Phase separation dynamics in deformable droplets

Phase separation can drive spatial organization of multicomponent mixtures. For instance in developing animal embryos, effective phase separation descriptions have been used to account for the spatial organization of different tissue types. Similarly, separation of different tissue types and the emergence of a polar organization is also observed in cell aggregates mimicking early embryonic axis formation. Here, we describe such aggregates as deformable two-phase fluid droplets, which are suspended in a fluid environment (third phase). Using hybrid finite-volume Lattice-Boltzmann simulations, we numerically explore the out-of-equilibrium routes that can lead to the polar equilibrium state of such a droplet (Janus droplet). We focus on the interplay between spinodal decomposition and advection with hydrodynamic flows driven by interface tensions, which we characterize by a Peclet number $Pe$. Consistent with previous work, for large $Pe$ the coarsening process is generally accelerated. However, for intermediate $Pe$ we observe long-lived, strongly elongated droplets, where both phases form an alternating stripe pattern. We show that these ``croissant'' states are close to mechanical equilibrium and coarsen only slowly through diffusive fluxes in an Ostwald-ripening-like process. Finally, we show that a surface tension asymmetry between both droplet phases leads to transient, rotationally symmetric states whose resolution leads to flows reminiscent of Marangoni flows. Our work highlights the importance of advection for the phase separation process in finite, deformable systems.