Counterion adsorption theory of dilute polyelectrolyte solutions: apparent molecular weight, second virial coefficient, and intermolecular structure factor

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.

Effect of counterion size on polyelectrolyte conformations and thermodynamics

We present a theoretical model to study the effect of counterion size on the effective charge, size, and thermodynamic behavior of a single, isolated, and flexible polyelectrolyte (PE) chain. We analyze how altering counterion size modifies the energy and entropy contributions to the system, including the ion-pair free energy, excluded volume interactions, entropy of free and condensed ions, and dipolar attraction among monomer-counterion pairs, which result in competing effects challenging intuitive predictions. The PE self-energy is calculated using the Edwards-Muthukumar Hamiltonian, considering a Gaussian monomer distribution for the PE. The condensed ions are assumed to be confined within a cylindrical volume around the PE backbone. The dipolar and excluded volume interactions are described by the second and third virial coefficients. The assumption of freely rotating dipoles results in a first-order coil-globule transition of the PE chain. A more realistic, weaker dipolar attraction, parameterized in our theory, shifts it to a second-order continuous transition. We calculate the size scaling-exponent of the PE and find exponents according to the relative dominance of the electrostatic, excluded volume, or dipolar effects. We further identify the entropy- and energy-driven regimes of the effective charge and conformation of the PE, highlighting the interplay of free ion entropy and ion-pair energy with varying electrostatic strengths. The crossover strength, dependent on the counterion size, indicates that diminishing sizes favor counterion condensation at the expense of free ion entropy. The predictions of the model are consistent with trends in simulations and generalize the findings of the point-like counterion theories.

Intramolecular structural heterogeneity altered by long-range contacts in an intrinsically disordered protein

Short-range interactions and long-range contacts drive the 3D folding of structured proteins. The proteins' structure has a direct impact on their biological function. However, nearly 40% of the eukaryotes proteome is composed of intrinsically disordered proteins (IDPs) and protein regions that fluctuate between ensembles of numerous conformations. Therefore, to understand their biological function, it is critical to depict how the structural ensemble statistics correlate to the IDPs' amino acid sequence. Here, using small-angle X-ray scattering and time-resolved Förster resonance energy transfer (trFRET), we study the intramolecular structural heterogeneity of the neurofilament low intrinsically disordered tail domain (NFLt). Using theoretical results of polymer physics, we find that the Flory scaling exponent of NFLt subsegments correlates linearly with their net charge, ranging from statistics of ideal to self-avoiding chains. Surprisingly, measuring the same segments in the context of the whole NFLt protein, we find that regardless of the peptide sequence, the segments' structural statistics are more expanded than when measured independently. Our findings show that while polymer physics can, to some level, relate the IDP's sequence to its ensemble conformations, long-range contacts between distant amino acids play a crucial role in determining intramolecular structures. This emphasizes the necessity of advanced polymer theories to fully describe IDPs ensembles with the hope that it will allow us to model their biological function.

Studying the physics of charged macromolecules by single molecule fluorescence spectroscopy

It is well documented that conventional methods such as dynamic light scattering have encountered difficulties in characterizing charged macromolecules and, therefore, it is desirable that new methods and techniques are introduced. With the ultra-high sensitivity, single molecule fluorescence spectroscopy has successfully lowered the detection limit considerably and enabled measurement under extreme dilution conditions-around the concentration of 10^-9M-at which the effect of inter-chain electrostatic repulsion is suppressed. Furthermore, the excellent spatial and temporal resolution as well as the capacity of molecular recognition of these methods help in obtaining rich information of charged macromolecules. This paper summarizes the applications of single molecule fluorescence spectroscopy, especially fluorescence correlation spectroscopy and photon counting histogram, in the studies on charged macromolecules in aqueous solutions and plenty of new information has been revealed on the molecular conformation, counterion distribution, and a few important governing factors. The powerfulness and effectiveness of single molecule fluorescence spectroscopy make it promising in the investigations of charged macromolecules.

Apparent repulsion between equally and oppositely charged spherical polyelectrolytes in symmetrical salt solutions

Ion-mediated interactions are very important for the properties of colloids and biomacromolecules such as nucleic acids and proteins. In this work, the ion-mediated interactions between equally and oppositely charged spherical polyelectrolytes (SPEs) in symmetrical divalent electrolytes have been investigated by Monte Carlo simulations, and an unexpected apparent repulsion was observed at high divalent salt concentration. Our investigations also show that the effective repulsion becomes more pronounced for SPEs with higher charge densities and for counterions with larger sizes and was found to be tightly accompanied with the over-neutralization to SPEs by condensed counterions and their release upon the approach of SPEs. Such attractive interaction can be reproduced by our proposed modified Poisson-Boltzmann model and is mainly attributed to the increase in the electrostatic repulsion between on charged SPE and the over-neutralized counterions around the other oppositely SPE with the approach of the two SPEs.

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.

Spectrum of hydrodynamic volumes and sizes of macromolecules of linear polyelectrolytes versus their charge density in salt-free aqueous solutions

Viscous flow was studied in salt-free solutions of random N-methyl-N-vinylacetamide copolymers that varied in the average number of charged units. The ranges are determined where the effect of the average charge density manifests itself in different ways.

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.

Electrostatic Correlations in Polyelectrolyte Solutions

The major attribute of polyelectrolyte solutions is that all chains are strongly correlated both electrostatically and topologically. Even in very dilute solutions such that the chains are not interpenetrating, the chains are still strongly correlated. These correlations are manifest in the measured scattering intensity when such solutions are subjected to light, X-ray, and neutron radiation. The behavior of scattering intensity from polyelectrolyte solutions is qualitatively different from that of solutions of uncharged polymers. Using the technique introduced by Sir Sam Edwards, and extending the earlier work by the author on the thermodynamics of polyelectrolyte solutions, extrapolation formulas are derived for the scattering intensity from polyelectrolyte solutions. The emergence of the polyelectrolyte peak and its concentration dependence are derived. The derived theory shows that there are five regimes. Published experimental data from many laboratories are also collected into a master figure and a comparison between the present theory and experiments is presented.