An Analytical Understanding of Reentrant Condensation of a Polyelectrolyte in the Presence of an Oppositely Charged Surfactant.

We explore the phase-transition mechanism of the reentrant condensation of a polyelectrolyte in the presence of an oppositely charged surfactant, which is of fundamental importance to the understanding of liquid-liquid phase separation (LLPS) in soft materials and biological systems. We focus on the adsorption and attraction effects of surfactants near/on polymer chains and ignore their own nonessential mixing effects if surfactant molecules are far away from polymer chains. This novel approach allows us to construct a simple mean-field equilibrium theory with closed-form analytical solutions, which can rationalize the essential features of the emergent "egg shape"-like phase diagram. The theory addresses that a strong electrostatic adsorption between the ionic monomers and surfactant ions is critical to understand the peculiar phenomenon that both the collapse and re-entry transitions of polyelectrolytes can occur when the concentration of the surfactant is lower than its bulk critical micelle concentration (CMC). Our theory also indicates that a minimum coupling energy for the nonlinear hydrophobic-aggregation effect of the adsorbed surfactant is essential for a phase transition to occur, which explains why polyelectrolytes show such a phase transition only if the surfactant chain length is beyond a minimum value. This work provides insight into the understanding of liquid-liquid phase separation in biological systems if proteins and/or peptides bound to DNAs and/or RNAs play an important role.