Size, shape, and fluctuations of condensates in non-equilibrium liquid-liquid phase separation*

Equilibrium phase separation, in the absence of chemical reactions, leads at long times, to a condensate of system size due to the interfacial tension of smaller-sized domains. In contrast, additional long-range (Coulomb) interactions competing with interfacial tension are known to stabilize the condensate size at intermediate-length scales. Examples of such tension long-range interacting (TLR) systems are – Rayleigh instability of charged liquids, block copolymer melts, binary solvent with antagonistic salt, and biomolecular condensates. In the latter case, the chemical reactions (production and degradation of proteins, RNA molecules, etc.) involve the input of energy (activity) and are coupled to the equilibrium aspects of phase separation. In such non-equilibrium phase separation, the slow chemical kinetics of the constituents play an antagonistic role to fast molecular diffusion (Ostwald ripening) and lead to a non-equilibrium steady state. For first-order chemical kinetics, the non-equilibrium term maps to a Coulomb interaction in the effective free energy. In the mean-field limit, for infinite periodic systems, the transition between various morphologies (sphere, cylinder, lamellar, etc.) depends on the relative concentrations of the solute-rich and the solute-poor domains (chemical composition in the other equilibrium cases). An important finding of our theory is that for finite (but very large) systems with lamellar microstructure, the sample aspect ratio enters the system with free energy and the steady-state domain size. While the lamellar phase is locally stable, its restoring force to undulations is related to the curvature of the undulations and does not depend on the extra area of the layer (effectively zero tension) for needle-like and periodic systems.