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 (ZhangP.WangZ.-G.
Zhang, P.
Wang, Z.-G.
Macromolecules20215410994), 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.