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.