Charge binding and electrostatic correlations in polyelectrolyte complex coacervation
Keywords: Materials Science and Engineering.

Complex coacervation is an associative phase separation that can occur when mixing solutions of oppositely charged polyelectrolytes, the result of which is a polymer-rich coacervate phase coexisting with a dilute, aqueous supernatant. Coacervate phases are used in a wide variety of materials, such as underwater adhesives, personal care products, and encapsulation agents. More recently, the coacervation process has been recognized as an important mechanism underlying the formation of intracellular, membraneless organelles. Despite extensive efforts, coacervation remains challenging to describe theoretically. This stems from the necessity of accounting for multiple, competing interactions, including short-range, ion-specific effects and long-range Coulomb interactions. I have developed a thermodynamic model for complex coacervation that merges treatments of long-range Coulomb interactions and short-range ion binding into a single formalism. Ion binding between oppositely charged species is described as reversible ``chemical reactions'', with the degree of reaction representing the degree of binding. This allows us to study counterion adsorption and polyanion-polycation cross-linking simultaneously. The electrostatic correlation free energy is obtained by calculating the interaction due to weak fluctuations in the charge density, and incorporates charge connectivity through the polyelectrolyte structure factors. While minimizing the solution free energy to find the degree of local ion binding, we reveal that the ionic self-energy is essential for associating systems. The self-energy has been discarded in prior treatments in order to regularize the nominally divergent Coulomb interaction between a point-like charge and itself. This regularization is justified when the number of unpaired ions is conserved, but fails in associating ionic solutions. I show that, by replacing point-like charges with a local density of finite width, this unwanted sensitivity is eliminated, and ion binding and electrostatics are treated consistently. Further analysis on the effects of molecular structure reveals that more compact chain conformations produce stronger electrostatic correlations and result in a greater degree of ion binding. Working with collaborators, we validate our theory by comparison with calibrated experiments. Our first comparative study focuses on stoichiometric solutions, in which polyanions and polycations are mixed with equal concentrations. We show that with just two fitting parameters for the strength of local ion binding and polymer solubility, our theory describes the variation of coacervation phase diagrams over a wide range of molecular weight, salinity, and hydrophilicity. In particular, we find that both charge connectivity and ion binding are needed in our theory to achieve quantitative agreement with experiments. Our second comparative study focuses on non-stoichiometric solutions, in which polyanions and polycations are mixed at unequal ratios. This analysis reveals, in agreement with experimental data, the emergence of a novel ``looping-in'' behavior, wherein the polymer concentrations in the coacervate initially increase with salt addition before subsequently decreasing. By examining the phase equilibrium condition in the low-salt regime, we show that the looping-in behavior originates from the translational entropy of counterions that are needed to neutralize charge-asymmetric coacervates. One major limitation of our model is that it does not capture the rod-like structure of polyions in dilute solutions, which is needed to accurately describe the supernatant phase. To address this, I integrate a formalism for adaptive chain conformations into the free energy model. Chain stiffness, and thus the conformational properties of the chains, is obtained by optimizing the solution free energy. This leads to the first molecular model that considers adaptive chain structures, counterion adsorption, and interchain cross-linking concurrently. Future directions that build upon this theory are discussed in the perspective