Dilute Polyelectrolyte Solutions Research Articles

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.

Solving the constitutive equation of dilute polymeric flows: A general Fokker–Planck approach for linear elastic dumbbell models

We present a mesoscopic numerical solver for the constitutive equation of dilute polymer solutions, as described by the Fokker–Planck equation of bead-spring dumbbell models with linear spring forces, such as Oldroyd-B, FENE-P (finitely extensible nonlinear elastic with Peterlin closure), and C-FENE-P (charged FENE-P). The configuration density function of the Fokker–Planck equation is expanded into a series of Hermite polynomials, and we show that only a second order in the expansion is needed to describe the evolution of the polymer-induced stresses. The polymer-induced stress tensor is given exactly through a discrete representation of the configuration distribution function, which is obtained from a finite set of Gauss–Hermite quadrature points. In addition, we derive the conformation tensor equation, which serves as a mathematical quality check for the method. We solve the time evolution of the extra stress tensor for homogeneous flows of dilute polyelectrolyte solutions, conceptualized by the C-FENE-P model, which is a generalization of the FENE-P dumbbell model. The results are shown to be in excellent agreement with analytical and semi-analytical reference results for simple shear and extensional flows. While in this paper the focus is on linear connector forces, we believe that the principles of the derivation are extendable to other force laws.

Viscoelastic behavior of dilute polyelectrolyte solutions in complex geometries

In this work, we develop a theoretical model describing the viscoelastic behavior of dilute polyelectrolyte (PE) solutions using Onsager’s variational principle, with the effect of the polyelectrolyte charge and solvent salinity accounted for. The PE molecules are described as finitely extensible charged dumbbells suspended in a solvent containing ionic species which screen the charged sites on the macromolecules. The performance of the model was examined and compared against experimental data in simple flows, accounting for the magnitude of charge on the PE molecules and counterion screening effects. Numerical simulations of the flow past a single cylinder and an idealized porous medium with multiple cylinders were carried out and compared to flow of uncharged polymers in the same geometries to explore the effect of intramolecular electrostatic repulsion on the flow behavior. The extra rigidity offered by the repulsion between the charged monomeric units of the PE, and screening of these charged sites by the ionic species in the solvent led to significantly different behavior in complex flows when compared to that of uncharged polymers. The normal stresses generated in a complex flow show an increase with the polyelectrolyte molecule charge density, whereas enhanced screening of the charged sites by counterions suppresses this effect, showing behavior closer to that of uncharged polymers. Drag on a cylinder placed in the flow and pressure drop across the porous geometry increased with charge density of the molecules, exhibiting rheological behavior in complex geometries not shown by their uncharged counterparts. The results show the effects of the degree of hydrolysis and solvent salinity on the flow behavior of dilute PE solutions in complex geometries and has implications for biological systems in which PE species are commonly encountered, as well as operation of industrial processes such as enhanced oil recovery, which frequently involve the flow of dilute polyelectrolyte solutions in porous media. • Macroscopic flow behavior of dilute polyelectrolyte solutions is investigated. • Onsager’s Variational Principle used for developing model from first principles. • Reformulation of charged dumbbell model for simulating flows in complex geometries. • Polyelectrolyte charge density and solvent salinity influence flow behavior. • Drag coefficients and normal stresses increase with charged nature of molecules.

Complexation between Oppositely Charged Polyelectrolytes in Dilute Solution: Effects of Charge Asymmetry

We use dissipative particle dynamics to study polyelectrolyte complexation in dilute solutions under conditions of either chain-length or concentration asymmetry between the polycation and the polyanion, characterized by the macromolecular charge ratio of the minor component to the major component. Our results show that generally the systems form clusters carrying net macromolecular charges. The stoichiometry of these net-charged macromolecular clusters depends on the overall charge asymmetry, and clusters with certain stoichiometry can dominate the cluster population under a wide range of chain-length and concentration asymmetry. The morphology and stability of these net-charged clusters are closely related to their stoichiometry. When the charge ratio reaches a threshold value, the polyions condense into a single large coacervate cluster, signaling the onset of macroscopic coacervation. The threshold value is less for concentration asymmetry systems than for chain-length asymmetry systems at the same salt concentration. This threshold value decreases with added salt, as salt ions cause merging of small clusters into larger ones, eventually leading to the dominance of a single large cluster. This “salting-out” phenomenon is followed by a “salting-in” behavior, as further addition of salt dissolves the large cluster. The preferred size and composition of the net-charged clusters observed in the simulation highlight the significant role of length and concentration asymmetry on polyelectrolyte complex coacervation and hint at the possibility of a microstructured liquid.

Conformational properties of complexes of poly(propylene imine) dendrimers with linear polyelectrolytes in dilute solutions.

This study investigates the conformational properties of complexes of poly(propylene imine) dendrimers with a linear polyelectrolyte (LPE) at neutral pH in an aqueous solution via molecular dynamics simulations. Various conformational properties, such as the atomic density profile, counterion density distribution, charge distribution, cavity volume, and the static structure factor are studied as a function of the charge and chain length of the LPE. The lower generation dendrimer complexes encapsulate the shorter linear PE chains, while the longer PE chains are adsorbed on the dendrimer surface that screen the surface charge and prevent the penetration of the counterions and water molecules. However, the overall charge of the higher generation dendrimers is not neutralized by the charge of the PE chains, which results in chloride counterion penetration within the dendrimers. The adsorption of the PE chains on the dendrimers is also verified from the charge distribution of the dendrimer-PE complexes. The charge on the lower generation dendrimer complexes is overcompensated by the longer PE chains resulting in an overall negative charge on the complexes, while the PE chains do not completely neutralize the charge of the higher generation dendrimers and produce positively charged complexes. The results of the structure factor indicate a conformational transition of the dendrimer-PE complexes from a dense compact structure to an open one with an increase in the PE chain length. This transition is characterized by an increase in the cavity volume in dendrimers with an increase in the PE chain length.

A charged finitely extensible dumbbell model: Explaining rheology of dilute polyelectrolyte solutions

A robust non-Newtonian fluid model of dilute polyelectrolyte solutions is derived from kinetic theory arguments. Polyelectrolyte molecules are modeled as finitely elongated nonlinear elastic dumbbells, where effective charges (interacting through a simple Coulomb force) are added to the beads in order to model the repulsion between the charged sections of polyelectrolyte chains. It is shown that the relative strength of this repulsion is regulated by the electric-to-elastic energy ratio, E, which is one of the key parameters of the model. In particular, E accounts for the intrinsic rigidity of polyelectrolyte molecules and can be used to explain the impact of solvent salinity on polyelectrolyte rheology. With two preaveraging approximations, the constitutive equations of the resulting fluid model are formulated in closed form. Material functions predicted by the model for steady shear flow, steady extensional flow, small-amplitude oscillatory shear flow, and start-up and cessation of steady shear flow are obtained and investigated using a combination of analytical and numerical methods. In particular, it is shown how these material functions depend on E. The two limiting cases of the model—uncharged dumbbells (E = 0) and rigid dumbbells (E → ∞)—are included in the analysis. It is found that despite its simplicity, the model predicts most of the experimentally observed rheological features of polyelectrolyte solutions.

Reaction enthalpy from the binding of multivalent cations to anionic polyelectrolytes in dilute solutions.

Dilute solutions of sodium poly(styrene sulfonate) (NaPSS) in the presence of Al3+, Ca2+, and Ba2+ were analysed by means of isothermal titration calorimetry (ITC) in order to investigate the heat effect of bond formation between those cations and the anionic SO3- residues of NaPSS. The selection of the cations was guided by the solution behavior of the corresponding PSS salts from a preceding study [M. Hansch et al., J. Chem. Phys. 148(1), 014901 (2018)], where bonds between Ba2+ and anionic PSS showed an increasing solubility with decreasing temperature and Al3+ exhibited the inverse trend. Unlike to Al3+ and Ba2+, Ca2+ is expected to behave as a purely electrostatically interacting bivalent cation and was thus included in the present study. Results from ITC satisfactorily succeeded to explain the temperature-dependent solution behavior of the salts with Al3+ and Ba2+ and confirmed the non-specific behavior of Ca2+. Additional ITC experiments with salts of Ca2+ and Ba2+ and sodium poly(acrylate) complemented the results on PSS by data from a chemically different polyanion. Availability of these joint sets of polyanion-cation combinations not only offers the chance to identify common features and subtle differences in the solution behavior of polyelectrolytes in the presence of multi-valent cations but also points to a new class of responsive materials.

Sequential coating of nanopores with charged polymers: A general approach for controlling pore properties of self-assembled block copolymer membranes

A general approach is developed for controlling the pore size and pore chemistry of integral isoporous membranes derived from the assembly of polystyrene-b-poly-4-vinyl pyridine (PS-b-P4VP) diblock copolymer. In this approach, initially, the sub-50 nm pore surface, decorated with poly-4-vinyl pyridine (P4VP) polymer brush, is coated with polyacrylic acid (PAA). PAA offers a majority of anionic and a minority of neutral acid functionalities. The neutral acid groups adhere to the P4VP segments of the pore wall through hydrogen bonding interactions and the anionic sites remain free. The availability of the anionic sites sets the stage for a layer-by-layer deposition of cationic and anionic polymers, in a sequential manner, on the pore-wall surface through a continuous flow of a dilute polyelectrolyte solution through the nanoporous membrane. In this way, multiple layers, stabilized through electrostatic interactions, can be deposited leading to a continuous decrease in the pore size and a known surface charge. Due to the known facile nature of the large area isoporous asymmetric membrane formation and modular nature of the polyelectrolyte assembly, the present approach is anticipated to yield new block copolymer membranes with tailored separation, sensing, and catalytic properties.

Ionization of short weak polyelectrolytes: when size matters

The ionization of short weak polyelectrolytes in dilute solutions is investigated by 1H nuclear magnetic resonance (NMR). The increase of the pH of the medium causes both the ionization of the polyacids and, due to the intra-molecular repulsions, the swelling of the chains. As a result, a noticeable effect of the pH change on the NMR chemical shift δ, and on the self-diffusion coefficient D is observed. The ionization of the polyelectrolyte occurs at higher pH as the length of the chain increases. The variation of the self-diffusion coefficient with pH exhibits the opposite dependence on the length of the chain. However, no detectable effect of the length of the polyelectrolyte chain on the evolution of the chemical shift with pH is observed. These apparent contradictions are examined to clarify the impact of the chain length on the polyelectrolyte properties, and the counterion role in the ionization process.

CDPD: A new dissipative particle dynamics method for modeling electrokinetic phenomena at the mesoscale.

We develop a "charged" dissipative particle dynamics (cDPD) model for simulating mesoscopic electrokinetic phenomena governed by the stochastic Poisson-Nernst-Planck and the Navier-Stokes equations. Specifically, the transport equations of ionic species are incorporated into the DPD framework by introducing extra degrees of freedom and corresponding evolution equations associated with each DPD particle. Diffusion of ionic species driven by the ionic concentration gradient, electrostatic potential gradient, and thermal fluctuations is captured accurately via pairwise fluxes between DPD particles. The electrostatic potential is obtained by solving the Poisson equation on the moving DPD particles iteratively at each time step. For charged surfaces in bounded systems, an effective boundary treatment methodology is developed for imposing both the correct hydrodynamic and electrokinetics boundary conditions in cDPD simulations. To validate the proposed cDPD model and the corresponding boundary conditions, we first study the electrostatic structure in the vicinity of a charged solid surface, i.e., we perform cDPD simulations of the electrostatic double layer and show that our results are in good agreement with the well-known mean-field theoretical solutions. We also simulate the electrostatic structure and capacity densities between charged parallel plates in salt solutions with different salt concentrations. Moreover, we employ the proposed methodology to study the electro-osmotic and electro-osmotic/pressure-driven flows in a micro-channel. In the latter case, we simulate the dilute poly-electrolyte solution drifting by electro-osmotic flow in a micro-channel, hence demonstrating the flexibility and capability of this method in studying complex fluids with electrostatic interactions at the micro- and nano-scales.

Flexible polyelectrolyte chain in a strong electrolyte solution: Insight into equilibrium properties and force-extension behavior from mesoscale simulation.

Macromolecules with ionizable groups are ubiquitous in biological and synthetic systems. Due to the complex interaction between chain and electrostatic decorrelation lengths, both equilibrium properties and micro-mechanical response of dilute solutions of polyelectrolytes (PEs) are more complex than their neutral counterparts. In this work, the bead-rod micromechanical description of a chain is used to perform hi-fidelity Brownian dynamics simulation of dilute PE solutions to ascertain the self-similar equilibrium behavior of PE chains with various linear charge densities, scaling of the Kuhn step length (lE) with salt concentration cs and the force-extension behavior of the PE chain. In accord with earlier theoretical predictions, our results indicate that for a chain with n Kuhn segments, lE ∼ cs (-0.5) as linear charge density approaches 1/n. Moreover, the constant force ensemble simulation results accurately predict the initial non-linear force-extension region of PE chain recently measured via single chain experiments. Finally, inspired by Cohen's extraction of Warner's force law from the inverse Langevin force law, a novel numerical scheme is developed to extract a new elastic force law for real chains from our discrete set of force-extension data similar to Padè expansion, which accurately depicts the initial non-linear region where the total Kuhn length is less than the thermal screening length.

Barrier-induced dielectric counterion relaxation at super-low frequencies in salt-free polyelectrolyte solutions.

Based on the thermally activated diffusion of counterions over the barrier of the electrostatic binding potential, we construct a scaling theory for the slow dielectric response in dilute and semi-dilute polyelectrolyte solutions. The theory is based on an analytic evaluation of the mean-escape time of a single counterion from the surface of a polyelectrolyte chain and uses a variational expression for the electrostatic potential of a charged cylinder including counterion condensation. This mean-escape time shows a range of characteristic power-law dependencies on the polyelectrolyte length and the polyelectrolyte monomer concentration. The existence of this novel dielectric mode at super-low frequencies reflects the wide spectrum of experimental findings for the super-low-frequency dielectric relaxation mode and thereby helps to reconcile conflicting interpretations of experimental data in terms of conventional scaling laws. We also devise a scaling theory for the counterion condensation of finite-length polyelectrolyte chains at finite concentration, which allows us to include polyelectrolyte charge renormalization in dilute as well as semi-dilute solutions in a unified theoretical framework.

Polyelectrolytes in dilute solution: viscometric access to coil dimensions and salt effects

Viscometric and light scattering radii in water containing 0.25 mol NaCl per liter as a function of the reduced polymer concentration.

Lipid Monolayers with Adsorbed Oppositely Charged Polyelectrolytes: Influence of Reduced Charge Densities

Polyelectrolytes in dilute solutions (0.01 mmol/L) adsorb in a two-dimensional lamellar phase to oppositely charged lipid monolayers at the air/water interface. The interchain separation is monitored by Grazing Incidence X-ray Diffraction. On monolayer compression, the interchain separation decreases to a factor of two. To investigate the influence of the electrostatic interaction, either the line charge density of the polymer is reduced (a statistic copolymer with 90% and 50% charged monomers) or mixtures between charged and uncharged lipids are used (dipalmitoylphosphatidylcholine (DPPC)/ dioctadecyldimethylammonium bromide (DODAB)) On decrease of the surface charge density, the interchain separation increases, while on decrease of the linear charge density, the interchain separation decreases. The ratio between charged monomers and charged lipid molecules is fairly constant; it decreases up to 30% when the lipids are in the fluid phase. With decreasing surface charge or linear charge density, the correlation length of the lamellar order decreases.

Diffusion behavior of polyelectrolytes in dilute solution: coupling effects of hydrodynamic and Coulomb interactions

The diffusion behavior of polyelectrolytes in dilute salt-free solution is studied through a hybrid mesoscale simulation technique that combines the molecular dynamics method and the multiparticle collision dynamics approach. To elucidate the effects of hydrodynamic interactions (HI), we compare results for hydrodynamic and random solvents. When HI are taken into account, we find that the chain diffusivity decreases initially and then increases gradually with the increasing strength of the Coulomb interaction. By contrast, when HI are switched off, the electrostatic-dependent diffusivity shows three distinct regions, and a plateau of approximately constant diffusivity manifests between two decreasing regions. The findings reveal that the dynamics of polyelectrolytes in dilute solution depend on the coupling effects of hydrodynamic and Coulomb interactions, and that these dynamics can be understood by considering the conformational changes of chains, the counterion condensation, and the dynamics of counterions.

Limiting law excess sum rule for polyelectrolytes

We revisit the mean-field limiting law screening excess sum rule that holds for rodlike polyelectrolytes. We present an efficient derivation of this law that clarifies its region of applicability: The law holds in the limit of small polymer radius, measured relative to the Debye screening length. From the limiting law, we determine the individual ion excess values for single-salt electrolytes. We also consider the mean-field excess sum away from the limiting region, and we relate this quantity to the osmotic pressure of a dilute polyelectrolyte solution. Finally, we consider numerical simulations of many-body polymer-electrolyte solutions. We conclude that the limiting law often accurately describes the screening of physical charged polymers of interest, such as extended DNA.

Apparent Molar Mass of a Polyelectrolyte in an Organic Solvent in the Low Ionic Strength Limit As Revealed by Light Scattering

The apparent molar mass of a partially quaternized poly-2-vinylpyridines (degree of quaternization 4.3% < Q < 35%) in 1-propanol is measured by light scattering at low ionic strength (10–6 M < cs < 10–3 M) as a function of polyion repeat unit concentration (7 × 10–5 monomol/L < cm,p < 4 × 10–2 monomol/L). No dialysis was applied prior to measurements. The apparent molar mass under “salt-free” conditions is smaller than the true molar mass by an order of magnitude, while approaching the true value at higher ionic strength. Concomitant data on the dependence of scattering intensity on scattering wave vector show that the dilute polyelectrolyte solutions are strongly correlated. A recent theory by Muthukumar for light scattering of dilute polyelectrolyte solutions, developed for correlated multicomponent systems, accounts for interchain electrostatic correlations and regularization of polymer charge by counterion binding isotherm. The experimental results on the relation between the apparent and true molar masses as a function of salt concentration are compared with the predictions of Muthukumar’s counterion adsorption theory. Taking binding equilibrium constant as a single fitting parameter, this theory is demonstrated to describe the experimental data as long as the chains do not interact significantly.

Low-frequency collective exchange mode in the dielectric spectrum of salt-free dilute polyelectrolyte solutions

We study the frequency-dependent dielectric response of flexible polyelectrolyte chains in aqueous salt-free dilute solution as a function of polymer concentration and length by means of Brownian dynamics simulations including hydrodynamic interactions. We show that condensed and uncondensed counter ions are characterized by different dielectric response spectra and thus confirm long-held scaling assumptions used for the interpretation of experimental data. In addition to the well-known relaxation modes of condensed and uncondensed counter ions we observe for a single polyelectrolyte chain, we find in many-chain simulations a novel spectral feature at low frequencies that essentially corresponds to an exchange of counter ions between neighboring polyelectrolytes. Our results suggest that the experimental low-frequency dielectric mode might not be due to condensed counter ions, as is commonly assumed, but rather due to a collective many-chain process. This could resolve a long-standing puzzle in the comparison of experimental results and scaling predictions.

Conformations and hydration structure of hydrophobic polyelectrolyte atactic poly(ethacrylic acid) in dilute aqueous solution as a function of neutralisation

Chain conformations, counter-ion structure, intermolecular hydrogen bonding structure and dynamics of atactic polyethacrylic acid (PEA) in salt-free aqueous dilute solution at 25°C are studied via molecular dynamics (MD) simulations with explicit-solvent and explicit-ion description for the first time. The intermolecular structure was analysed by the radial distribution functions (RDF) for specific atom types between PEA chain, water molecules and Na+ counter-ions, as well as by the hydration near the PEA chain in the solvated system. An increase in f provides an increase in 〈Rg〉 of the chain, consistent with the existence of the compact form of PEA. The simulations show expansion for radius-of-gyration with increase in f, as expected for flexible polyelectrolytes under salt-free condition. The extent of intermolecular hydrogen bonds (H-bonds) between PEA and water is enhanced by increase in f. Chains having a higher counter-ion density show higher values of 〈Rg〉, influenced by intermolecular interactions between PEA and water. The coordination of Na+ counter ions and water molecules to carboxyl oxygens of polyacrylic acid (PAA) increases with charge density of the chain. A comparison of the structure aspects is made with PAA and PMA polyelectrolytes in dilute solution, which brings out the hydrophobic effect of the ethyl side-groups in PEA on conformational properties and counter-ion condensation structure.

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

Polyelectrolyte chains are well known to be strongly correlated even in extremely dilute solutions in the absence of additional strong electrolytes. Such correlations result in severe difficulties in interpreting light scattering measurements in the determination of the molecular weight, radius of gyration, and the second virial coefficient of charged macromolecules at lower ionic strengths from added strong electrolytes. By accounting for charge-regularization of the polyelectrolyte by the counterions, we present a theory of the apparent molecular weight, second virial coefficient, and the intermolecular structure factor in dilute polyelectrolyte solutions in terms of concentrations of the polymer and the added strong electrolyte. The counterion adsorption of the polyelectrolyte chains to differing levels at different concentrations of the strong electrolyte can lead to even an order of magnitude discrepancy in the molecular weight inferred from light scattering measurements. Based on counterion-mediated charge regularization, the second virial coefficient of the polyelectrolyte and the interchain structure factor are derived self-consistently. The effect of the interchain correlations, dominating at lower salt concentrations, on the inference of the radius of gyration and on molecular weight is derived. Conditions for the onset of nonmonotonic scattering wave vector dependence of scattered intensity upon lowering the electrolyte concentration and interpretation of the apparent radius of gyration are derived in terms of the counterion adsorption mechanism.