Supernatant Phase in Polyelectrolyte Complex Coacervation: Cluster Formation, Binodal, and Nucleation

This work studies the structure and thermodynamics of the supernatant phase in polyelectrolyte complex coacervation, a relatively unexplored area. By combining the cluster theory in dilute solution with our recently developed mean-field theory for inhomogeneous polyelectrolyte solutions ( Zhang, P. ; Wang, Z.-G. Macromolecules 2021, 54, 10994 ), we systematically investigate the structure of finite-sized clusters formed by oppositely charged polyions in symmetric dilute solutions and how these clusters affect the binodal, spinodal, and nucleation for polyelectrolyte complex coacervation. We find that both the polyion concentration deep inside the cluster and the interfacial tension decrease with increasing the cluster size, reaching their respective bulk coexistence values with corrections inversely proportional to the cluster radius. The polyion concentration in the supernatant phase at coexistence is several orders of magnitude higher than that obtained under the uniform mixing approximation. For most relevant conditions away from the critical point, the supernatant phase consists predominantly of polyion pairs. By examining the nucleation barrier in supersaturated solutions, we can determine a pseudo-spinodal when the barrier is a few multiples of the thermal energy. The location of this pseudo-spinodal is similarly shifted to much higher concentrations than predicted under the uniform mixing approximation. Making the volume approximation for the clusters, we obtain simple analytical expressions for the cluster formation free energy, the modified binodal, and the pseudo-spinodal. In particular, we propose a simple approximate formula for estimating the concentration of the coexisting supernatant phase in terms of the chain length, interfacial tension, and the polyion concentration in the coacervate phase.