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.

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We develop a simple inhomogeneous mean-field theory to study the interfacial structure and tension of polyelectrolyte complex coacervates in equilibrium with a supernatant solution. Our theory treats the electrostatic correlation by combining the Debye–Hückel theory with the first-order thermodynamic perturbation theory within the local density approximation, and incorporates the conformation entropy contribution for both polyions using Lifshitz’s ground-state dominance approximation. Using this theory, we systematically examine the interfacial properties of both symmetric and concentration-asymmetric coacervates. The interfacial tension γ is generally rather low, on the order of 1 mN/m or less. For asymmetric coacervates, an intricate electric double layer forms in the interfacial region, which can even contain several oscillations under certain conditions. The interfacial tension generally decreases with increasing the stoichiometric asymmetry, the added-salt concentration, and the initial polymer concentration of the mixture. We further find that the interfacial tension can be quantitatively related to the degree of phase separation S, where S is the Euclidean distance in composition between the two coexisting phases. In particular, we find that γ as a function of S for different concentration asymmetries collapses approximately to two master curves, which merge together and follow γ ∼ S 3 for small S.

The ability of polyelectrolytes to condense into a liquidlike, polyelectrolyte-rich phase out of a dilute supernatant phase through complex coacervation has led to fascinating phenomena, such as membraneless organelles and self-assembled capsules for drug delivery. Recent experiments have demonstrated that heating above a lower critical solution temperature (LCST) can drive complex coacervation. Here, we show that a coarsegrained model of electrostatic correlations is sufficient to model an LCST when accounting for the empirical decrease in the dielectric constant of the solvent upon heating. The predictions of the model agree qualitatively with experimental measurements of the compositions of the coexisting coacervate and supernatant phases. The model also achieves modest quantitative agreement with experiments, despite incorporating no other experimental parameters besides the dielectric constant and a fitted length scale. This agreement underscores the important role that can be played by electrostatic correlations in driving complex coacervation above an LCST.

Using a lattice self-consistent field (SCF) theory and the corresponding lattice Monte Carlo (MC) simulations combined with our recently proposed Z method [Zhang, P.; Wang, Q. Soft Matter 2015, 11, 862], we examined athermal homopolymer solutions confined between two parallel and nonabsorbing surfaces and in equilibrium with a bulk solution and accurately calculated the effective interaction between the two surfaces. By directly comparing our MC results with SCF predictions based on the same model system, we were able to quantitatively and unambiguously distinguish the meanfield and the fluctuation contributions to the effective interaction. We found for the first time the fluctuation-induced repulsion between the two confining surfaces at intermediate separation predicted by Semenov and Obukhov [

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