Using a simple liquid-state theory, we study the phase behaviors of concentration-asymmetric mixtures of polycation and polyanion solutions. We construct a three-dimensional (3D) phase diagram in terms of the concentrations of the three independent charged components: polycation, polyanion, and small cation (ρ p+ −ρ p− −ρ +), for a given Bjerrum length. This phase diagram yields rich and complex phase-separation scenarios. To illustrate, we sequentially examine the following three systems that are directly relevant to experiments: a symmetric mixture, an asymmetric mixture with one type of small ions, and an asymmetric mixture with both types of small ions. We re-express the information in the 3D phase diagram using three experimentally more easily controllable parameters-the asymmetry factor r, the initial extra-salt concentration ρ s,0 , and the initial polyelectrolyte (PE) concentration ρ p,0 of both solutions prior to mixing. We construct three reduced phase diagrams in the ρ p,0 −r, r−ρ s,0 , and ρ s,0 −ρ p,0 planes, respectively, and examine the evolution of the volume fraction of the coexisting phases, concentration of the PE and small-ion species in each phase, and the Galvani potential Ψ G , as functions of these experimental controlling parameters. We rationalize our findings in terms of the key thermodynamic factors, namely, the translational entropy of the small ions, the electrostatic correlation energy, and the requirement for charge neutrality.

We perform a general thermodynamic analysis for the salt partitioning behavior in the coexisting phases for symmetric mixtures of polycation and polyanion solutions. We find that salt partitioning is determined by the competition between two factors involving the ratio of the polyelectrolyte concentration in the coacervate phase to that in the supernatant phase and the difference in the exchange excess chemical potential Δμexthe excess chemical potential difference between PE segments and small ionsbetween the coexisting phases. The enrichment of salt ions in the coacervate phase predicted by the Voorn–Overbeek theory is shown to arise from its neglect of chain connectivity in the excess free energy which results in Δμex = 0 under all conditions. We argue that chain connectivity in general leads to a finite value of Δμex, which decreases with increasing PE concentration. Explicit calculations using theories that include the chain connectivity correlationsa simple liquid-state theory and a renormalized Gaussian fluctuation theoryshow nonmonotonic behavior of the salt-partitioning coefficient (the ratio of salt ion concentration in the coacervate phase to that in the supernatant phase): it is larger than 1 at very low salt concentrations, reaches a minimum at some intermediate salt concentration, and approaches 1 at the critical point. This behavior is consistent with recent computer simulation and experimental results.

We study the phase behavior of polyelectrolyte (PE) solutions with salt using a simple liquid-state (LS) theory. This LS theory accounts for hard-core excluded volume repulsion by the Boublik–Mansoori–Carnahan–Starling–Leland equation of state, electrostatic correlation by the mean-spherical approximation, and chain connectivity by the first-order thermodynamic perturbation theory. We predict a closed-loop binodal curve in the PE concentration-salt concentration phase diagram when the Bjerrum length is smaller than the critical Bjerrum length in salt-free PE solution. The phase-separated region shrinks with decreasing Bjerrum length, and disappears below a critical Bjerrum length. These results are qualitatively consistent with experiments, but cannot be captured by the Voorn–Overbeek theory. On the basis of the closed-loop binodal curve and the lever rule, we predict three scenarios of salting-out and salting-in phenomena with addition of monovalent salt into an initially salt-free PE solution. The Galvani potentialthe electric potential difference between the coexisting phasesis found to depend nonmonotonically on the overall salt concentration in some PE concentration range, which is related to the partition of the co-ions in the coexisting phases. Free energy analysis suggests that phase separation is driven by a gain in the electrostatic energy, at the expense of a large translational entropy penalty, due to significant counterion accumulation in the PE-rich phase.