Into an extended Voorn–Overbeek (EVO) free energy model of polyelectrolyte (PE) complexation and phase behavior, we incorporate three classes of short-ranged electrostatic effects, namely counterion association–dissociation, cross-chain ion pairing (IP), and charge regulation by treating each as a reversible chemical reaction leading to a corresponding law of mass action in a self-consistent fashion. The importance of each reaction is controlled by a corresponding chemistry-dependent standard free energy input parameter. Our model also accounts for Born (or ion solvation) energy using a linear mixing rule for the effective dielectric constant. In monophasic systems, the proposed model can qualitatively explain the observed shifts in acidity and basicity observed in potentiometric titration of weak PEs in the presence of salt and oppositely charged PEs in accordance with Le Châtelier’s principle. We demonstrate how a competition between counterion condensation (CC) and IP alone can explain the complex coacervation of strongly charged PEs as well as the existence of a critical salt concentration. Binodal diagrams predicted in our model are also affected by long-ranged electrostatics and are most sensitive to IP strength both for weak and strong PEs. The extent of IP increases in the dense phase at the expense of reduced CC upon coacervation consistent with counter release view of complex coacervation. We compare binodal diagrams predicted by our model against experimental data for both weakly and strongly dissociating polyions pairs and find a plausible parameter set that leads to an acceptable and partial agreement with experiments in the two cases, respectively.

Reversible ion binding equilibria in polyelectrolyte solutions are strongly affected by interactions between dissociated ionic species. We examine how the structural correlations between ionic groups on polyelectrolytes impact the counterion binding. Treating the electrostatic correlation free energy using the classical Debye-Hückel expression leads to complete counterion dissociation in the concentrated regime. This unphysical behavior is shown to stem from improper regularization of the self-energy of dissociated ions and polyions and is mitigated by smearing point-like charges across a finite width. The influence of the self-energy on counterion binding is elaborated on by generalizing the Debye-Hückel free energy to polyelectrolytes with variable fractal dimension and stiffness. In the dilute regime, a greater propensity for binding is found for chains with more compact architectures, which in turn reduces the harsh self-repulsions of tightly packed arrangements of charge. In the concentrated regime, the effects of electrostatic correlations weaken due to screening and the extent of binding is governed by a balance of short-ranged interactions and the translational entropy of ions.

The effects of pH and salinity on both the bulk phase behavior and the layer-by-layer (LbL) growth kinetics are investigated for polyanion poly(acrylic acid) or PAA with two polycations, namely poly(N,N-dimethylaminoethyl methacrylate) or PDMAEMA and poly(diallyldimethylammonium chloride) or PDADMAC, with the goal of relating the phase behavior to the LbL growth kinetics. Depending on salinity, pH, and mixing ratio, the complex formed in the bulk is either a powdery precipitate or a gel-like coacervate, and the multilayers grow either linearly or exponentially with deposition time. In addition to primary Coulombic interactions, we observe that polymer-specific interactions have a profound effect on both bulk complexation and LbL growth of the three PE pairs studied here. The overall strength of interaction between polyelectrolytes, as indicated by their phase behavior, has a nonmonotonic effect on LbL growth rate, apparently because stronger interactions not only increase the driving force for diffusion but also reduce the effective diffusion coefficient of a polyelectrolyte molecule through the LbL multilayer. As a result, there is little correspondence between coacervation and exponential growth on one hand and precipitation and linear growth on the other. Salt concentration has a nonmonotonic effect on LbL growth kinetics at pH 7, with exponential growth found over the range 15−60% of the critical salt concentration (C S c ) needed to transition from coacervation to a clear solution in the bulk, regardless of the physical chemistry of polyelectrolytes employed, whereas salt concentrations both below and above this range result in linearly growing films. Finally, for our polyelectrolyte pairs at pH 7, we report a "universal curve" for the dependence of LbL growth rate, normalized by its maximum value, against the salt concentration, normalized by C S c . If it proves to be robust, this correlation could be used to estimate optimal salinity for LbL growth from bulk measurements of the critical salt concentration needed to suppress complexation.

The growth rates of layer-by-layer (LbL) assemblies of polyelectrolytes (PEs) with oppositely charged polystyrene (PS) nanoparticles (NPs) as a function of molecular weight (MW) of the PEs, ionic strength of the media, and NP size and charge are systematically investigated. To optimize LbL growth, the effects of suspension concentration, pH of the media, and deposition time on the growth rate of multilayers are assessed. Both linear and exponential growth behaviors are observed and, under optimal conditions, films of up to around 1 μm thick can readily be assembled after 10 or so bilayers have been deposited. For many of the cases studied, an intermediate MW of PE leads to the fastest film buildup, for both cationic poly(ethyleneimine) deposited alternately with anionic PS NPs and for anionic poly(acrylic acid) deposited alternately with cationic PS NPs. The existence of an optimal MW suggests that growth rate is determined by a balance of thermodynamic factors, including density of polymer bridges between particles, and kinetic factors, specifically the diffusivity of polymer in the film. The optimal MW, however, is very sensitive to the materials used. Moreover, depending on the MW of the PE, increasing salinity could increase or decrease the growth kinetics. Finally, the surface morphology of the films is characterized with AFM and SEM to reveal that the roughness increases less than linearly with film thickness.

We develop a general framework for transport of polyions, solvent and salt, with intended application to Layer-by-Layer (LbL) assembly of polyelectrolyte monolayers (PEMs). The formulation for the first time includes electrostatics, chemical potential gradients, and mechanical stress gradients as driving forces for mass transport. The general model allows all species to be mobile throughout the process and avoids the assumptions of stepwise instantaneous equilibrium and/or immobilized structures typical of previous approaches, while reducing to these models in appropriate limits. A simple constitutive equation is derived for a mixture of oppositely charged polyelectrolytes that accounts for network strand dilution and cross-chain ion pairing by appending reactive terms to the Smoluchowski probability diffusion equation for network strand end-to-end vectors. The resulting general framework encompasses the Poisson equation describing the electrostatic potential distribution, an osmotic pressure balance, a stress constitutive equation, and a generalized flux law of polymer transport. The computational domain is split into a PEM phase and an external solution phase with an appropriate boundary condition derived for the interface between the two. The mobile species (water and small salt ions) are taken to be in a state of dynamic equilibrium with their distributions enslaved to the perturbations in the two polyion compositions. The proposed model captures the swelling response of PEM films to external solutions. For the first time, we studied the effects of the temporal evolution of electrostatic and stress distribution on the rate of chain loss and absorption during rinsing and dipping of an idealized and arbitrarily selected and rigid brush layer into external solutions. The temporal evolution provides a kinetic basis for the ability of LbL films to grow under conditions that thermodynamics alone suggests would cause them to be washed away and to account for partial desorption during washing. The proposed transport framework constitutes a solid basis for eventual quantitative modeling of LbL assembly and transport in polyion networks more generally.