Electrostatic Correlations in Polyelectrolyte Solutions

Dilute polyelectrolyte solutions: recent progress and open questions

Polyelectrolytes are a class of polymers possessing ionic groups on their repeating units. Since counterions can dissociate from the polymer backbone, polyelectrolyte chains are strongly influenced by electrostatic interactions. As a result, the physical properties of polyelectrolyte solutions are significantly different from those of electrically neutral polymers. The aim of this article is to highlight key results and some outstanding questions in the polyelectrolyte research from recent literature. We focus on the influence of electrostatics on conformational and hydrodynamic properties of polyelectrolyte chains. A compilation of experimental results from the literature reveals significant disparities with theoretical predictions. We also discuss a new class of polyelectrolytes called poly(ionic liquid)s that exhibit unique physical properties in comparison to ordinary polyelectrolytes. We conclude this review by listing some key research challenges in order to fully understand the conformation and dynamics of polyelectrolytes in solutions.

Fluctuations, structure, and size inside coacervates

Aqueous solutions of oppositely charged macromolecules exhibit the ubiquitous phenomenon of coacervation. This subject is of considerable current interest due to numerous biotechnological applications of coacervates and the general premise of biomolecular condensates. Towards a theoretical foundation of structural features of coacervates, we present a field-theoretic treatment of coacervates formed by uniformly charged flexible polycations and polyanions in an electrolyte solution. We delineate different regimes of polymer concentration fluctuations and structural features of coacervates based on the concentrations of polycation and polyanion, salt concentration, and experimentally observable length scales. We present closed-form formulas for correlation length of polymer concentration fluctuations, scattering structure factor, and radius of gyration of a labelled polyelectrolyte chain inside a concentrated coacervate. Using random phase approximation suitable for concentrated polymer systems, we show that the inter-monomer electrostatic interaction is screened by interpenetration of all charged polymer chains and that the screening length depends on the individual concentrations of the polycation and the polyanion, as well as the salt concentration. Our calculations show that the scattering intensity decreases monotonically with scattering wave vector at higher salt concentrations, while it exhibits a peak at intermediate scattering wave vector at lower salt concentrations. Furthermore, we predict that the dependence of the radius of gyration of a labelled chain on its degree of polymerization generally obeys the Gaussian chain statistics. However, the chain is modestly swollen, the extent of which depending on polyelectrolyte composition, salt concentration, and the electrostatic features of the polycation and polyanion such as the degree of ionization.

Tuning Net Charge in Aliphatic Polycarbonates Alters Solubility and Protein Complexation Behavior

A synthetic strategy yielded polyelectrolytes and polyampholytes with tunable net charge for complexation and protein binding. Organocatalytic ring-opening polymerizations yielded aliphatic polycarbonates that were functionalized with both carboxylate and ammonium side chains in a post-polymerization, radical-mediated thiol-ene reaction. Incorporating net charge into the polymer architecture altered the chain dimensions in phosphate buffered solution in a manner consistent with self-complexation and complexation behavior with model proteins. A net cationic polyampholyte with 5% of carboxylate side chains formed large clusters rather than small complexes with bovine serum albumin, while 50% carboxylate polyampholyte was insoluble. Overall, the aliphatic polycarbonates with varying net charge exhibited different macrophase solution behaviors when mixed with protein, where self-complexation appears to compete with protein binding and larger-scale complexation.

Theory of Charged Gels: Swelling, Elasticity, and Dynamics

The fundamental attributes of charged hydrogels containing predominantly water and controllable amounts of low molar mass electrolytes are of tremendous significance in biological context and applications in healthcare. However, a rigorous theoretical formulation of gel behavior continues to be a challenge due to the presence of multiple length and time scales in the system which operate simultaneously. Furthermore, chain connectivity, the electrostatic interaction, and the hydrodynamic interaction all lead to long-range interactions. In spite of these complications, considerable progress has been achieved over the past several decades in generating theories of variable complexity. The present review presents an analytically tractable theory by accounting for correlations emerging from topological, electrostatic, and hydrodynamic interactions. Closed-form formulas are derived for charged hydrogels to describe their swelling equilibrium, elastic moduli, and the relationship between microscopic properties such as gel diffusion and macroscopic properties such as elasticity. In addition, electrostatic coupling between charged moieties and their ion clouds, which significantly modifies the elastic diffusion coefficient of gels, and various scaling laws are presented. The theoretical formulas summarized here are useful to adequately capture the essentials of the physics of charged gels and to design new hydrogels with specified elastic and dynamical properties.

RNA-Mediated Feedback Control of Transcriptional Condensates

Regulation of biological processes typically incorporates mechanisms that initiate and terminate the process and, where understood, these mechanisms often involve feedback control. Regulation of transcription is a fundamental cellular process where the mechanisms involved in initiation have been studied extensively, but those involved in arresting the process are poorly understood. Modeling of the potential roles of RNA in transcriptional control suggested a non-equilibrium feedback control mechanism where low levels of RNA promote condensates formed by electrostatic interactions whereas relatively high levels promote dissolution of these condensates. Evidence from in vitro and in vivo experiments support a model where RNAs produced during early steps in transcription initiation stimulate condensate formation, whereas the burst of RNAs produced during elongation stimulate condensate dissolution. We propose that transcriptional regulation incorporates a feedback mechanism whereby transcribed RNAs initially stimulate but then ultimately arrest the process.

SAXS methods for investigating macromolecular and self-assembled polyelectrolyte complexes

Polyelectrolyte complexation is driven by associative interactions between oppositely charged polyelectrolytes, resulting in formation of a macroscopic polymer dense phase and a polymer dilute phase with applications in coatings, adhesives, and purification membranes. Beyond macroscale phase separation, precision polymer synthesis has enabled further development of polyelectrolyte complex (PEC)-based self-assembled micelles and hydrogels with applications in biotechnology. Interestingly, it has been suggested that mechanisms similar to polyelectrolyte complexation drive formation of biological condensates that play an indispensable role in cellular biogenesis. The formation pathways and functionality of these complex materials is dependent on the physical properties that are built into polymer structure and the resulting physical conformation in the dilute and dense phase. Scattering techniques have enabled in situ investigation of structure-function relationships in PEC materials that may address unresolved biophysical questions in cellular processes as well as catalyze the development of novel materials for diverse applications. We describe preparation of PEC materials with controlled polymer characteristics (length, blockiness, charge density), small-angle X-ray scattering (SAXS) techniques employed to probe appropriate length scales, and the data analysis routines from a practical standpoint for new users. This article deals with bulk complexes and not with the related, important and interesting area of non-equilibrium layer-by-layer assembly of polyelectrolytes.

Counterion Condensation, the Gibbs Equation, and Surfactant Binding: An Integrated Description of the Behavior of Polyelectrolytes and Their Mixtures with Surfactants at the Air-Water Interface

By applying the Gibbs equation to the bulk binding isotherms and surface composition of the air-water (A-W) interface in polyelectrolyte-surfactant (PE-S) systems, we show that their surface behavior can be explained semiquantitatively in terms of four concentration regions, which we label as A, B, C, and D. In the lowest-concentration range A, there are no bound PE-S complexes in the bulk but there may be adsorption of PE-S complexes at the surface. When significant adsorption occurs in this region, the surface tension (ST) drops with increasing concentration like a simple surfactant solution. Region B extends from the onset of bulk PE-S binding to the end of cooperative binding, in which the slow variation of surfactant activity with cooperative binding means that the ST changes relatively little, although adsorption may be significant. This leads to an approximate plateau, which may be at high or low ST. Region C starts where the binding in the bulk complex loses its cooperativity leading to a rapid change of surfactant activity with the total concentration. This, combined with significant adsorption, often leads to a sharp drop in ST. Region D is where precipitation and redissolution of the bulk PE-S complex occur. ST peaks may arise in region D because of loss of the solution complex that matches the value of the preferred surface stoichiometry, which seems to have a well-defined value for each system. The analysis is applied to the experimental systems, sodium polystyrene sulfonate-alkyltrimethylammonium bromides and poly(diallyldimethyl chloride)-sodium alkyl sulfates, with and without the added electrolyte, and includes data from bulk binding isotherms, phase diagrams, aggregation behavior, and direct measurements of the surface excess and stoichiometry of the surface. The successful fits of the Gibbs equation to the data confirm that the surfaces in these systems are largely equilibrated.

A unified analytical theory of heteropolymers for sequence-specific phase behaviors of polyelectrolytes and polyampholytes

The physical chemistry of liquid-liquid phase separation (LLPS) of polymer solutions bears directly on the assembly of biologically functional dropletlike bodies from proteins and nucleic acids. These biomolecular condensates include certain extracellular materials and intracellular compartments that are characterized as "membraneless organelles." Analytical theories are a valuable, computationally efficient tool for addressing general principles. LLPS of neutral homopolymers is quite well described by theory, but it has been a challenge to develop general theories for the LLPS of heteropolymers involving charge-charge interactions. Here, we present a theory that combines a random-phase-approximation treatment of polymer density fluctuations and an account of intrachain conformational heterogeneity based on renormalized Kuhn lengths to provide predictions of LLPS properties as a function of pH, salt, and charge patterning along the chain sequence. Advancing beyond more limited analytical approaches, our LLPS theory is applicable to a wide variety of charged sequences ranging from highly charged polyelectrolytes to neutral or nearly neutral polyampholytes. This theory should be useful in high-throughput screening of protein and other sequences for their LLPS propensities and can serve as a basis for more comprehensive theories that incorporate nonelectrostatic interactions. Experimental ramifications of our theory are discussed.

Disappearance of the polyelectrolyte peak in salt-free solutions

We investigate the nature of the polyelectrolyte peak in salt-free solutions by molecular dynamics simulations using a minimal model of polyelectrolyte solutions that includes an explicit solvent and counterions and small angle scattering experiments. It is found that the polyelectrolyte peak progressively disappears as the strength of solvation for the charged species is increased and the scattering profiles start to resemble those of neutral polymer solutions. The disappearance of the polyelectrolyte peak coincides with the emergence of attractive interchain interactions over a wide range of length scales. These findings provide insights into the microscopic origin of the polyelectrolyte peak.

DEAE-dextran hydrochloride behaviour in aqueous solution-The effect of ionic strength and concentration

DEAE-dextran hydrochloride is a positively charged biocompatible polyelectrolyte. Its behaviour in aqueous solutions - the changes in its colloidal characteristics and resulting conformation changes - were investigated using a combination of light scattering, densitometry and ultrasound spectrometry. The results indicated the formation of a voluminous, hydrated and coiled conformation of chains having average particle sizes in the range of units of microns at low ionic strength. This phenomenon was supported by an increase in DEAE-dextran hydrochloride concentration in the solution. The increase in ionic strength caused the shielding of DEAE-dextran hydrochloride charges on its chain, resulting in the destabilization of the conformation arrangement of the polyelectrolyte and the formation of denser, more compact and stiffer structures having smaller particle sizes. The proposed model of DEAE-dextran hydrochloride density in the studied environments (distilled water, NaCl solutions) showed the possibility of using a single equation model to calculate the overall density of the system.

Enhanced scattering induced by electrostatic correlations in concentrated solutions of salt-free dipolar and ionic polymers

We present a generalized theory for studying the static monomer density-density correlation function (structure factor) in concentrated solutions and melts of dipolar as well as ionic polymers. The theory captures effects of electrostatic fluctuations on the structure factor and provides insights into the origin of experimentally observed enhanced scattering at ultralow wavevectors in salt-free ionic polymers. It is shown that the enhanced scattering can originate from a coupling between the fluctuations of electric polarization and monomer density. Local and non-local effects of the polarization resulting from finite sized permanent dipoles and ion-pairs in dipolar and charge regulating ionic polymers, respectively, are considered. Theoretical calculations reveal that, similar to the salt-free ionic polymers, the structure factor for dipolar polymers can also exhibit a peak at a finite wavevector and enhanced scattering at ultralow wavevectors. Although consideration of dipolar interactions leads to attractive interactions between monomers, the enhanced scattering at ultralow wavevectors is predicted solely on the basis of the electrostatics of weakly inhomogeneous dipolar and ionic polymers without considering the effects of any aggregates or phase separation. Thus, we conclude that neither aggregation nor phase separation is necessary for observing the enhanced scattering at ultralow wavevectors in salt-free dipolar and ionic polymers. For charge regulating ionic polymers, it is shown that electrostatic interactions between charged monomers get screened with a screening length, which depends not only on the concentration of "free" counterions and coions, but also on the concentration of "adsorbed" ions on the polymer chains. Qualitative comparisons with the experimental scattering curves for ionic and dipolar polymer melts are presented using the theory developed in this work.

Polyelectrolyte association and solvation

There has been significant interest in the tendency of highly charged particles having the same charge to form dynamic clusters in solution, but an accepted theoretical framework that can account for this ubiquitous phenomenon has been slow to develop. The theoretical difficulties are especially great for flexible polyelectrolytes due to the additional complex coupling between the polyelectrolyte chain configurations and the spatial distribution of the ionic species in solution. For highly charged polyelectrolytes, this leads to the formation of a diffuse "polarizable" cloud of counter-ions around these polymers, an effect having significant implications for the function of proteins and other natural occurring polyelectrolytes, as emphasized long ago by Kirkwood and co-workers. To investigate this phenomenon, we perform molecular dynamics simulations of a minimal model of polyelectrolyte solutions that includes an explicit solvent and counter-ions, where the relative affinity of the counter-ions and the polymer for the solvent is tunable through the variation of the relative strength of the dispersion interactions of the polymer and ions. In particular, we find that these dispersion interactions can greatly influence the nature of the association between the polyelectrolyte chains under salt-free conditions. We calculate static and dynamic correlation functions to quantify the equilibrium structure and dynamics of these complex liquids. Based on our coarse-grained model of polyelectrolyte solutions, we identify conditions in which three distinct types of polyelectrolyte association arise. We rationalize these types of polyelectrolyte association based on the impact of the selective solvent affinity on the charge distribution and polymer solvation in these solutions. Our findings demonstrate the essential role of the solvent in the description of the polyelectrolyte solutions, as well as providing a guideline for the development of a more predictive theory of the properties of the thermodynamic and transport properties of these complex fluids.

Structure and rheology of polyelectrolyte complex coacervates

Scattering investigations of the structure and chain conformations, and the rheological properties of polyelectrolyte complexes (PECs) comprising model polyelectrolytes are presented. The use of charged polypeptides - (poly)-lysine and (poly)-glutamic acid with identical backbones allowed for facile tuning of the system parameters, including chain length, side-chain functionality, and chirality. Systematic studies using small-angle X-ray scattering (SAXS) of liquid PEC coacervates revealed a physical description of these materials as strongly screened semidilute polyelectrolyte solutions comprising oppositely charged chains. At the same time, solid PECs were found to be composed of hydrogen-bonding driven stiff ladder-like structures. While the coacervates behaved akin to semidilute polyelectrolyte solutions upon addition of salt, the solids were largely unaffected by it. Rheology measurements of PEC coacervates revealed a terminal relaxation regime, with an unusual plateauing of the storage modulus at low oscillation frequencies. The plateau may be ascribed to a combination of instrumental limitations and the long-range electrostatic interactions contributing to weak energy storage modes. Excellent superposition of the dynamic moduli was achieved by a time-salt superposition. The shift factors, however, varied more strongly than previously reported with added salt concentration.

Communication: Counter-ion solvation and anomalous low-angle scattering in salt-free polyelectrolyte solutions

We investigate the influence of counter-ion solvation on the homogeneity of salt-free polyelectrolyte solutions based on a coarse-grained model that includes an explicit solvent. We show that the solvation of the counter-ions can cause a transformation between a nearly homogeneous to a non-uniform polymer solution, in which there is both a chain clustering and the formation of large charge-free domains, i.e., "voids." The emergence of these heterogeneous structures induced by counter-ion solvation is accompanied by the localization and formation of counter-ion rich domains that are symptomatic of emergent effective long-range attractive interchain interactions.

50th Anniversary Perspective: A Perspective on Polyelectrolyte Solutions

From the beginning of life with the information-containing polymers until the present era of a plethora of water-based materials in health care industry and biotechnology, polyelectrolytes are ubiquitous with a broad range of structural and functional properties. The main attribute of polyelectrolyte solutions is that all molecules are strongly correlated both topologically and electrostatically in their neutralizing background of charged ions in highly polarizable solvent. These strong correlations and the necessary use of numerous variables in experiments on polyelectrolytes have presented immense challenges toward fundamental understanding of the various behaviors of charged polymeric systems. This Perspective presents the author's subjective summary of several conceptual advances and the remaining persistent challenges in the contexts of charge and size of polymers, structures in homogeneous solutions, thermodynamic instability and phase transitions, structural evolution with oppositely charged polymers, dynamics in polyelectrolyte solutions, kinetics of phase separation, mobility of charged macromolecules between compartments, and implications to biological systems.

Two regions of microphase separation in ion-containing polymer solutions

The phenomenon of spinodal decomposition in weakly charged polyelectrolyte solutions is studied theoretically within the random phase approximation. A novel feature of the theoretical approach is that it accounts for the effects of ionic association, i.e. ion pair and multiplet formation between counterions and ions in polymer chains, as well as the dependence of local dielectric permittivity on the polymer volume fraction Φ. The main focus is on the spinodal instability of polyelectrolyte solutions towards microscopic phase separation. It has been shown that increasing the binding energy of ions decreases the classical microphase separation region (possible at low polymer concentrations) due to the effective neutralization of the chains. A qualitatively new type of microphase separation is found in the presence of a dielectric mismatch between polymer and solvent. This new branch of microphase separation is realized at high polymer concentrations where ion association processes are the most pronounced. Typical microstructures are shown to have a period of a few nanometers like in ionomers. The driving force for the microphase formation of a new type is more favourable ion association in polymer-rich domains where ionomer-type behavior takes place. Effective attraction due to ion association promotes microscopic as well as macroscopic phase separation, even under good solvent conditions for uncharged monomer units of polymer chains. Polyelectrolyte-type behavior at low Φ and ionomer-type behavior at high Φ result in the presence of two critical points on the phase diagrams of polyelectrolyte solutions as well as two separate regions of possible microscopic structuring. Our predictions on the new type of microphase separation are supported by experimental data on polymer solutions, membranes and gels.