We address complex coacervation, the liquid-liquid phase separation of a solution of oppositely charged polyelectrolyte chains into a polyelectrolyte rich complex coacervate phase and a dilute aqueous phase, based on the general premise of spontaneous formation of polycation-polyanion complexes even in the homogeneous phase. The complexes are treated as flexible chains made of dipolar segments and uniformly charged segments. Using a mean field theory that accounts for the entropy of all dissociated ions in the system, electrostatic interactions among dipolar and charged segments of complexes and uncomplexed polyelectrolytes, and polymer-solvent hydrophobicity, we have computed coacervate phase diagrams in terms of polyelectrolyte composition, added salt concentration, and temperature. For moderately hydrophobic polyelectrolytes in water at room temperature, neither hydrophobicity nor electrostatics alone is strong enough to cause phase separation, but their combined effect results in phase separation, arising from the enhancement of effective hydrophobicity by dipolar attractions. The computed phase diagrams capture key experimental observations including the suppression of complex coacervation due to increases in salt concentration, temperature, and polycation-polyanion composition asymmetry, and its promotion by increasing the chain length, and the preferential partitioning of salt into the polyelectrolyte dilute phase. We also provide new predictions such as the emergence of loops of instability with two critical points.

In light of recent experimental observations of lower critical solution temperature (LCST) in polyelectrolyte complex coacervates (Ali, S. et al. ACS Macro Lett. 2019, 8, 289−293), we explore its possible mechanisms on the basis of a slight modification of our theory (Adhikari, S. et al. J. Chem. Phys. 2018, 149, 163308). We explore the consequences of the temperature dependence of the solvent dielectric constant (ε) and the solvent−polymer interaction parameter (χ) on the complex coacervates' phase behavior. The results show that the temperature dependence of the solvent dielectric constant and solvent−polymer interaction parameter can result in a complex phase behavior involving two disjoint unstable regions on the temperature (T)− polyelectrolyte concentration (ϕ p ) plane. Comparison of phase diagrams constructed for different possible temperature dependencies of ε and χ shows that the experimentally observed LCST behavior is obtained only if the solvent dielectric constant decreases and the solvent−polymer interaction parameter increases with increasing temperature. Preferential partitioning of salt into the polyelectrolyte poor phase is predicted for all possible combinations of temperature dependencies of χ and ε considered in this work.

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Polymer crystals grown from melts consist of alternating lamellar crystalline regions and amorphous regions. We study the statistics of ties: chains which bridge the adjacent lamellae, loops: chains which come out of one lamella and enter back into the same lamella before reaching the other lamellae, and tails: chains which end in an amorphous region. We develop a theory to calculate the probabilities of formation of ties, loops, and tails with consideration of finite chain length and cooperative incorporation of a chain into multiple lamellae. The results of our numerical calculations based on a field-theoretic formalism show that the fraction of ties increases with increasing chain length, and it decreases with increasing interlamellar separation. In the limiting case of an infinite chain confined between only two walls, we recover the classical results of the gambler’s ruin model. We show that the density anomaly encountered in previous theories is avoided naturally in the present theory without forcing the majority of stems to form tight loops. The derived results on the probability of tie chains in the amorphous regions are pertinent to the mechanical properties of semicrystalline polymers.

We rationalize the unusual gas transport behavior of polymer-grafted nanoparticle (GNP) membranes. While gas permeabilities depend specifically on the chemistry of the polymers considered, we focus here on permeabilities relative to the corresponding pure polymer, which show interesting, “universal” behavior. For a given NP radius, R c, and for large enough areal grafting densities, σ, to be in the dense brush regime, we find that gas permeability enhancements display a maximum as a function of the graft chain molecular weight, M n. Based on a recently proposed theory for the structure of a spherical brush in a melt of GNPs, we conjecture that this peak permeability occurs when the densely grafted polymer brush has the highest, packing-induced extension free energy per chain. The corresponding brush thickness is predicted to be h max = √3R c, independent of chain chemistry and σ, i.e., at an apparently universal value of the NP volume fraction (or loading), ϕNP, ϕNP,max = [R c/(R c + h max)]3 ≈ 0.049. Motivated by this conclusion, we measured CO2 and CH4 permeability enhancements across a variety of R c, M n, and σ and find that they behave in a similar manner when considered as a function of ϕNP, with a peak in the near vicinity of the predicted ϕNP,max. Thus, the chain length-dependent extension free energy appears to be the critical variable in determining the gas permeability for these hybrid materials. The emerging picture is that these curved polymer brushes, at high enough σ, behave akin to a two-layer transport mediumthe region in the near vicinity of the NP surface is comprised of extended polymer chains that speed up gas transport relative to the unperturbed melt. The chain extension free energy increases with increasing chain length, up to a maximum, and apparently leads to an increasing gas permeability. For long enough grafts, there is an outer region of chain segments that is akin to an unperturbed melt with slow gas transport. The permeability maximum and decreasing permeability with increasing chain length then follow naturally.