Self-consistent analytical solutions to the Voorn-Overbeek model

Electrostatically driven liquid-liquid phase separation underlies complex coacervation in solutions of oppositely charged macromolecules and plays a central role in the phase behavior of charged polymers such as nucleic acids and intrinsically disordered proteins. The Voorn-Overbeek model provides a minimal mean-field description of this phenomenon by combining polymer mixing entropy with electrostatic interactions captured at the Debye-Hückel level. Despite its long-standing importance, the Voorn-Overbeek theory does not admit closed-form analytical solutions for phase coexistence, and its phase behavior has, therefore, been studied primarily using numerical approaches or near-critical expansions. Here, we derive a self-consistent analytical solution for the binodal concentrations of the simplest Voorn-Overbeek model, describing two oppositely charged polymer species in a neutral solvent under local electroneutrality. By reformulating the coexistence conditions as a fixed-point problem, we obtain explicit analytical expressions for the phase boundaries that remain accurate across the entire phase-separated regime. These results establish an analytically tractable framework for complex coacervation and offer a foundation for future extensions incorporating additional electrostatic and compositional effects.