The pH inside a swollen polyelectrolyte gel: poly(N-vinylimidazole)

The pH inside a dissolved polyelectrolyte coil or a swollen ionic polymer network is not accessible to direct measurement. It is here calculated through a simple model, based on Donnan equilibrium, counterion condensation (for charge density exceeding the critical value), and balance of mobile ions, without any assumption on the pKa of the ionizable groups. The data needed for the calculation with this model are polymer concentration, pH value in the initial solution, and pH value in the bath at equilibrium. All three can be determined experimentally by a batch method where the polymer is immersed in a different pot for each starting pH. The model is applied to a sample system, namely, chemically cross-linked poly(N-vinylimidazole) immersed in acidic baths of different pH values. The imidazole units are basic and become protonated by the acid, thus changing the pH of the initial bath. The model shows how the pH developed inside the swollen gel is several units higher than the pH of the bath at equilibrium, both with or without the correction for counterion condensation. Consequently, when the pKa of the polyelectrolyte is determined in the usual way (with the pH measured in the external bath), it gives an apparent value that is several units below the pKa determined from the actual pH inside the swollen gel at equilibrium. The inclusion of the counterion condensation decreases very slightly the polymer basicity. Surface effects and intramolecular association between protonated and unprotonated imidazole rings are discussed to explain the pKa behavior in the limit of low degree of ionization.     

Counterion condensation and release in micellar solutions

Counterion condensation and release in micellar solutions are investigated by direct measurement of counterion concentration with ion-selective electrode. Monte Carlo simulations based on the cell model are also performed to analyze the experimental results. The degree of counterion condensation is indicated by the concentration ratio of counterions in the bulk to the total ionic surfactant added, alpha< or =1. The ionic surfactant is completely dissociated below the critical micelle concentration (cmc). However, as cmc is exceeded, the free counterion ratio alpha declines with increasing the surfactant concentration and approaches an asymptotic value owing to counterion condensation to the surface of the highly charged micelles. Micelle formation leads to much stronger electrostatic attraction between the counterion and the highly charged sphere in comparison to the attraction of single surfactant ion with its counterion. A simple model is developed to obtain the true degree of ionization, which agrees with our Monte Carlo results. Upon addition of neutral polymer or monovalent salts, some of the surfactant counterions are released to the bulk. The former is due to the decrease of the intrinsic charge (smaller aggregation number) and the degree of ionization is increased. The latter is attributed to competitive counterion condensation, which follows the Hefmeister series. This consequence indicates that the specific ion effect plays an important role next to the electrostatic attraction.     

Theory of counter-ion condensation on flexible polyelectrolytes: adsorption mechanism

A new model is presented for counterion distribution around flexible polyelectrolytes by considering (i) free energy of the polyelectrolyte chain, (ii) translational entropy of adsorbed counterions, (iii) adsorption energy, (iv) translational entropy of unadsorbed counterions, (v) fluctuations of dissociated ions, and (vi) correlation among ion-pairs formed by adsorbed counterions on the polymer. The effective charge and size of the polymer are calculated self-consistently. The degree of ionization f of the polymer decreases continuously with 1/epsilonT (epsilon and T are the dielectric constant of the solvent and temperature, respectively), depending sensitively on local dielectric heterogeneity. Further, f decreases with an increase in salt concentration, monomer concentration, or chain flexibility. The polymer size, accompanying the changes in f, depends nonmonotonically on 1/epsilonT. The predictions of the model are consistent with all trends observed previously in simulations and are distinctly different from the Manning argument for rodlike chains.     

Solution properties of ion-containing polymers in polar solvents

The placement of ionic groups within the molecular structure of a polymer produces marked modification in physical properties. A large number of studies have been performed on these ion-containing polymers, but few have focused on the effects of anion–cation interactions (i.e., counterion binding or ionization) on hydrodynamic volume, especially as the molecular structure of the solvent and nature of counterion are varied. In this study changes in hydrodynamic volume are followed through reduced viscosity measurements as a function of the abovementioned molecular parameters. The dilute solution properties of various polyelectrolytes that contain sulfonate and carboxylate groups were investigated as a function of the counterion structure, charge density, molecular weight, and solvent structure. The polymeric materials were selected because of their specific chemical structure and physical properties. In the first instance a (2-acrylamide-2 methylpropanesulfonic acid)-acrylamide-sodium vinyl sulfonate terpolymer was synthesized and subsequently neutralized with a series of bases. Viscometric measurements on these materials indicate that the nature of the cation affects the ability of the polyelectrolyte to expand its hydrodynamic volume at low polymer levels. The magnitude of the molecular expansion is shown to be due in part to the ability of the counterion to dissociate from the backbone chain, which, in turn, is directly related to the solvent structure. The changes in solution behaviour of these inomers lend support for the existence of ion pairs (i.e., site binding) and ionized moieties on the polymer chains. Measurements performed in a variety of solvent systems further confirm this interpretation. In addition, and acrylamide-sodium vinyl sulfonate copolymer was partially hydrolyzed with sodium hydroxide to study the effect of varying the charge density at a constant degree of polymerization and counterion structure. The results show that the charge density has a significant effect on the magnitude of the reduced viscosity and dilute solution behaviour. These observations, made in aqueous and nonaqueous solvents, are related to the interrelation of hydrodynamic volume, counterion concentration, and site binding. Again the controlling factor is the degree of site binding of the counterion onto the polymer backbone. Finally, we observe that the increased hydrodynamic volume affects viscosity behavior beyond the polyelectrolyte effect regime. If the average charge density on the macromolecule is relative high and/or the molecular weight is large (≥ 10⁶) sufficient intermolecular interactions will occur to produce rapid changes in reduced viscosity.     

Simulation Study of the Conformational Properties of Diblock Polyelectrolytes in Salt Solutions

Coarse-grained molecular dynamics simulations are performed to understand the behavior of diblock polyelectrolytes in solutions of divalent salt by studying the conformations of chains over a wide range of salt concentrations. The polymer molecules are modeled as bead spring chains with different charged fractions and the counterions and salt ions are incorporated explicitly. Upon addition of a divalent salt, the salt cations replace the monovalent counterions, and the condensation of divalent salt cations onto the polyelectrolyte increases, and the chains favor to collapse. The condensation of ions changes with the salt concentration and depends on the charged fraction. Also, the degree of collapse at a given salt concentration changes with the increasing valency of the counterion due to the bridging effect. As a quantitative measure of the distribution of counterions around the polyelectrolyte chain, we study the radial distribution function between monomers on different polyelectrolytes and the counterions inside the counterion worm surrounding a polymer chain at different concentrations of the divalent salt. Our simulation results show a strong dependence of salt concentration on the conformational properties of diblock copolymers and indicate that it can tune the self-assembly behaviors of such charged polyelectrolyte block copolymers.     

Monte Carlo simulation and molecular theory of tethered polyelectrolytes

We investigate the structure of end-tethered polyelectrolytes using Monte Carlo simulations and molecular theory. In the Monte Carlo calculations we explicitly take into account counterions and polymer configurations and calculate electrostatic interaction using Ewald summation. Rosenbluth biasing, distance biasing, and the use of a lattice are all used to speed up Monte Carlo calculation, enabling the efficient simulation of the polyelectrolyte layer. The molecular theory explicitly incorporates the chain conformations and the possibility of counterion condensation. Using both Monte Carlo simulation and theory, we examine the effect of grafting density, surface charge density, charge strength, and polymer chain length on the distribution of the polyelectrolyte monomers and counterions. For all grafting densities examined, a sharp decrease in brush height is observed in the strongly charged regime using both Monte Carlo simulation and theory. The decrease in layer thickness is due to counterion condensation within the layer. The height of the polymer layer increases slightly upon charging the grafting surface. The molecular theory describes the structure of the polyelectrolyte layer well in all the different regimes that we have studied.     

Entropy- or enthalpy-driven collapse of strongly charged polymer chains in a one-component charged fluid of counterions or coions

We applied a simulation method [T. Sumi and H. Sekino, J. Chem. Phys. 122, 194910 (2005)] to an infinitely dilute polyelectrolyte immersed in one-component charged fluids in order to investigate salt effects on its collapse. In this model system, the degree of freedom of the counterion (or the coion) is considered using a density-functional theory for polymer-solvent admixtures, while the oppositely charged ions are treated as a structureless background having the opposite charge. Results obtained by these simulations show that not only the counterion but also the coion makes the polymer chain collapsed. The effects by the coion are stronger than that by the counterion. Temperature variation of the gyration radius of the polymer chain immersed in the counterion is opposite to that in the coion: while the radius of gyration decreases as the temperature decreases in the case of the counterion, it decreases as the temperature increases in the case of the coion. From these results we conclude that the former is interpreted as an enthalpy-driven collapse caused by the screening effects of the counterion, whereas the latter is interpreted as an entropy-driven one due to the translational entropy of the coion.     

Confined polyelectrolytes: The complexity of a simple system

The interaction between polyelectrolytes and counterions in confined situations and the mutual relationship between chain conformation and ion condensation is an important issue in several areas. In the biological field, it assumes particular relevance in the understanding of the packaging of nucleic acids, which is crucial in the design of gene delivery systems. In this work, a simple coarse‐grained model is used to assess the cooperativity between conformational change and ion condensation in spherically confined backbones, with capsides permeable to the counterions. It is seen that the variation on the degree of condensation depends on counterion valence. For monovalent counterions, the degree of condensation passes through a minimum before increasing as the confining space diminishes. In contrast, for trivalent ions, the overall tendency is to decrease the degree of condensation as the confinement space also decreases. Most of the particles reside close to the spherical wall, even for systems in which the density is higher closer to the cavity center. This effect is more pronounced, when monovalent counterions are present. Additionally, there are clear variations in the charge along the concentric layers that cannot be totally ascribed to polyelectrolyte behavior, as shown by decoupling the chain into monomers. If both chain and counterions are confined, the formation of a counterion rich region immediately before the wall is observed. Spool and doughnut‐like structures are formed for stiff chains, within a nontrivial evolution with increasing confinement. © 2015 Wiley Periodicals, Inc.     

Interacting nanoparticles with functional surface groups

Functionalized nanoparticles with ionizable groups have generated a large variety of structures with important potential applications in technology. The nature of their interactions is crucial to determining their solubility and to exploring assemblies with diverse symmetries. Here, we use a molecular theory to describe the interactions between two nanoparticles coated with short polymer chains that contain ionizable (functional) end‐groups immersed in aqueous salt solution. It is shown here that the fraction of ionized functional groups in the system depends on factors such as the ionic strength and pH of solution, grafting density of polymer chains, the chain length, as well as the separation distance between the nanoparticles. The interactions between two neighboring nanoparticles influence the charge regulation of the end‐groups, which consequently induces an asymmetric distribution of these charged end‐groups on the nanoparticles, and thus confers a preferred directionality in nanoparticle–nanoparticle interactions. We show that the charge regulating system is less repulsive than an equivalent system with a fixed charge distribution. This is due to a decrease in the charge density of the weak acid end‐groups, to avoid a local increase in counterion confinement (condensation) in the region between neighboring nanoparticles, when their separation decreases. The anisotropic degree of ionization found in our results can be used to design aggregates of nanoparticles with reduced symmetries. © 2012 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys, 2012     

Electrophoretic evidence for a new type of counterion condensation

A systematic capillary electrophoresis study uncovered how polyelectrolyte effective charge density varies with backbone charge spacing and solvent dielectric constant. The study primarily focused on aliphatic ionenes, a special class of polyelectrolytes, which possess regularly spaced quaternary ammonium groups in the backbone. Complete ionization of functional units and good solvency in water or mixtures of water with lower dielectric constant solvents (methanol, acetonitrile) enabled continuous measurements of ionene effective charge density through the onset of counterion condensation. Ionenes with both uniform and alternating charge spacing were examined. As expected, effective charge density rose linearly with fixed charge density to a critical value, above which effective charge density remained constant. Deviating from expectation, the onset of condensation did not occur at a critical fixed charge density. Instead, condensation initiated at a constant critical Bjerrum length. The same onset condition was found for quaternized poly(vinyl pyridine)s. These experimental results suggest a new form of condensation, one driven by ion-pairing of polyelectrolyte with counterions. In support of this hypothesis, the onset of condensation appeared to correlate with counterion size. © 2004 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 42: 3616–3627, 2004     

Aqueous solutions of ionenes: interactions and counterion specific effects as seen by neutron scattering

Aqueous solutions of ionenes with bromide and fluoride counterions have been investigated using small angle neutron scattering for the first time. Ionenes are a class of cationic polyelectrolytes based on quaternary ammonium atoms and, considering the very low solubility of their uncharged part (hydrocarbon chain), would be formally classified as hydrophobic. Ionenes present important structural differences over previously studied polyelectrolytes: (a) charge is located on the polyelectrolyte backbone, (b) the distance between charges is regular and tunable by synthesis, (c) hydrophobicity comes from methylene groups of the backbone and not from bulky side groups. Results for Br ionenes feature a disappearance of the well-known polyelectrolyte peak beyond a given monomer concentration. Below this concentration, the position of the peak depends on the chain charge density, f(chem), and scales as f(chem)(0.30±0.04). This is an indication of a hydrophilic character of the ionene backbone. In addition, osmotic coefficients of ionene solutions resemble again other hydrophilic polyelectrolytes, featuring no unusual increase in the water activity (or a significant counterion condensation). We conclude that despite the hydrophobicity of the hydrocarbon chain separating charged centers on ionenes, these chains behave as hydrophilic. In contrast to Br ionenes, the polyelectrolyte peak remains at all concentrations studied for the single F ionene investigated. This strong counterion effect is rationalized in terms of the different hydrating properties and ion pairing in the case of bromide and fluoride ions.     

Theory of competitive counterion adsorption on flexible polyelectrolytes: divalent salts

The counterion distribution around an isolated flexible polyelectrolyte in the presence of a divalent salt is evaluated using the adsorption model [M. Muthukumar, J. Chem. Phys. 120, 9343 (2004)] that considers the Bjerrum length, salt concentration, and local dielectric heterogeneity as physical variables in the system. Self-consistent calculations of effective charge and size of the polymer show that divalent counterions replace condensed monovalent counterions in competitive adsorption. The theory further predicts that at modest physical conditions for a flexible polyelectrolytes such as sodium polystyrene sulfonate in aqueous solutions polymer charge is compensated and reversed with increasing divalent salt. Consequently, the polyelectrolyte shrinks and reswells. Lower temperatures and higher degrees of dielectric heterogeneity between chain backbone and solvent enhance condensation of all species of ions. Complete diagrams of states for the effective charge calculated as functions of the Coulomb strength and salt concentration suggest that (a) overcharging requires a minimum Coulomb strength and (b) progressively higher presence of salt recharges the polymer due to either electrostatic screening (for low Coulomb strengths) or coion condensation (for high Coulomb strengths). Consideration of ion-bridging by divalent counterions leads to a first-order collapse of polyelectrolytes in modest presence of divalent salts and at higher Coulomb strengths. The authors' theoretical predictions are in agreement with the generic results from experiments and simulations.     

Surface conductivity reveals counterion condensation within grafted polyelectrolyte layers

Surface conductivity (SC) has been demonstrated to be a valuable parameter for the characterization of surface-bound polyelectrolyte layers (PLs). The measurement of the SC in dependence of the pH and solution concentration yields information about the Donnan potential, PsiD, the intrinsic charge, the potential of the PL electrolyte interface, Psi0, the pK of the ionizable groups within the PLs, and the concentration of segments, n. We discuss herein that SC measurements may additionally provide information about counterion condensation. The mobility of the counterions within grafted poly(acrylic acid) (PAA) layers was estimated from the density of COOH groups and SC data to be only 14% of that of free ions (Zimmermann, et al. Langmuir 2005, 21, 5108). In view of this large deviation and the limited sterical constraints within the brushes, we conclude that the number of freely moving counterions is decreased due to counterion condensation. This interpretation agrees well with the measurement of the osmotic pressure for PAA solution (Boisvert, et al. Polymer 2002, 43, 141), which can be exclusively attributed to the remaining mobile counterions of the polyelectrolyte.     

关于聚电解质单链抗衡离子浓度径向分布的研究

采用单分子荧光显微统计光谱技术,通过将pH响应型荧光探针分子精确标记于聚苯乙烯磺酸钠分子链末端,并通过不同长度的多肽链调节分子链与探针分子间的距离,有效测量了聚苯乙烯磺酸钠单分子链的抗衡离子浓度的空间分布.实验结果清晰展示了聚电解质分子链的抗衡离子云结构,并确定了抗衡离子浓度随着距离分子链末端长度的不同而发生变化的规律,为描述聚电解质抗衡离子浓度的径向分布特征提供了实验信息.     

Interaction of cationic surfactants with carboxymethylcellulose in aqueous media

We have examined the polymer-surfactant interaction in mixed solutions of the cationic surfactants, i.e., dodecyltrimethylammonium chloride, dodecyltrimethylammonium bromide, tetradecyltrimethylammonium bromide, hexadecyltrimethylammonium bromide, tetradecyltriphenylphosphonium bromide, and tetradecylpyridinium bromide and a semiflexible anionic polyelectrolyte carboxymethylcellulose in water and aqueous salt solutions by various techniques: tensiometry, viscosimetry or ion-selective electrode method, and dynamic light scattering. We have investigated the effect of varying surfactant chain length, head group size, counterion, and ionic strength on the critical aggregation concentration (CAC) of mixed polymer surfactant systems and the collapse of the polymer molecule under different solution conditions. The CAC decreases with increasing alkyl chain length. Above a certain surfactant concentration, mixed aggregates start growing until their macroscopic phase separation. The growth is more rapid with greater surfactant tail length and with increasing head group size. This is attributed in both cases to the increasing hydrophobic interaction between polymer and surfactant. Among surfactants with monovalent halide counterions, iodide induces the strongest binding, reflected by the onset of growth of the mixed aggregates at low surfactant concentration. This is perhaps related to the decreasing hydration of the counterion from chloride to iodide. The surfactant concentration at which the viscosity of the solution starts to decrease sharply is smaller than the CAC, and probably reflects polymer chain shrinkage due to noncooperative binding.     

Dynamic exchange of counterions of polystyrene sulfonate

Adopting a cationic fluorescent molecule, rhodamine 6G, as the probe of the counterions of the model anionic polyelectrolyte (sodium polystyrene sulfonate, PSSNa), the diffusion of the counterion probes inside the solution of PSSNa was studied by fluorescence correlation spectroscopy. Two species of the counterion probes with different diffusion coefficient were discovered--the freely diffusing probes and the probes bound to the PSS(-) chains. The concentration fraction of these two species was found to change with the concentration and molecular weight of PSSNa. The results show that the counterion binding to the PSS(-) chain is enhanced with the increase of polymer concentration, attributed to the result of the lowered translational entropic penalty at higher polymer concentrations. The counterion binding is also enhanced with the increase of molecular weight, and the origin was attributed to the chain end effect to the counterion distribution. The results indicate the dynamic exchange process between the free counterions and the bound ones, which is further evidenced by the replacement of the bound probes by the elevated salt levels in the solution.     

Langevin dynamics of semiflexible polyelectrolytes: rod-toroid-globule-coil structures and counterion distribution

We have investigated the nature of counterion condensation on uniformly charged semiflexible polyelectrolyte chains and the concomitant configurations by monitoring the role of chain stiffness, chain length, counterion valency, and the strength of electrostatic interaction. The counterion condensation is seen to follow the adsorption process and the effective polymer charge increases with chain stiffness. Size and shape, as calculated through the radius of gyration, effective persistence length, and hydrodynamic radius, are studied. Stable coil-like, globular, folded-chain, toroidal, and rodlike configurations are possible at suitable combinations of values of chain stiffness, chain length, electrostatic interaction strength, and the valency of counterion. For high strengths of electrostatic interactions, sufficiently stiff polyelectrolytes form toroids in the presence of multivalent counterions, whereas flexible polyelectrolytes form disordered globules. The kinetic features of the nucleation and growth of toroids are monitored. Several metastable structures are found to frustrate the formation of toroids. The generic pathway involves the nucleation of one primary loop somewhere along the chain contour, followed by a growth process where the rest of the chain is folded continuously on top of the primary loop. The dependence of the average radii of toroids on the chain length is found to be roughly linear, in disagreement with existing scaling arguments.     

Electrophoresis of a spherical dispersion of polyelectrolytes in a salt-free solution

The electrophoretic behavior of a spherical dispersion of polyelectrolytes of arbitrary concentration is analyzed theoretically under a salt-free condition, that is, the liquid phase contains only counterions which come from the dissociation of the functional groups of polyelectrolytes. We show that, in general, the surface potential of a polyelectrolyte increases nonlinearly with its surface charge. A linear relation exists between them, however, when the latter is sufficiently small; and the more dilute the concentration of polyelectrolytes, the broader the range in which they are linearly correlated. If the amount of surface charge is sufficiently large, counterion condensation occurs, and the rate of increase of surface potential as the amount of surface charge increases declined. Also, it leads to an inverse in the perturbed potential near the surface of a polyelectrolyte, and its mobility decreases accordingly. For a fixed amount of surface charge, the lower the concentration of polyelectrolytes and/or the lower the valence of counterions, the higher the surface potential. The qualitative behavior of the mobility of a polyelectrolyte as the amount of its surface charge varies is similar to that of its surface charge.     

Counterion condensation on charged spheres, cylinders, and planes

We use the framework of counterion condensation theory, in which deviations from linear electrostatics are ascribed to charge renormalization caused by collapse of counterions from the ion atmosphere, to explore the possibility of condensation on charged spheres, cylinders, and planes immersed in dilute solutions of simple salt. In the limit of zero concentration of salt, we obtain Zimm-Le Bret behavior: a sphere condenses none of its counterions regardless of surface charge density, a cylinder with charge density above a threshold value condenses a fraction of its counterions, and a plane of any charge density condenses all of its counterions. The response in dilute but nonzero salt concentrations is different. Spheres, cylinders, and planes all exhibit critical surface charge densities separating a regime of counterion condensation from states with no condensed counterions. The critical charge densities depend on salt concentration, except for the case of a thin cylinder, which exhibits the invariant criticality familiar from polyelectrolyte theory.     

Functionalized microgel swelling: comparing theory and experiment

A comprehensive gel swelling model accounting for the effects of added salt, counterion/polyelectrolyte charge condensation, inter-cross-link chain length distribution, polyelectrolyte chain stiffness, and direct charge-charge repulsion between fixed polymer network charges has been applied to predict water fraction profiles in -COOH-functionalized microgels based on poly(N-isopropylacrylamide). The model can successfully order the microgels according to their rheologically measured water fractions and explains key differences in observed microgel swelling according to the different functional group and cross-linker distributions in the microgels. The cross-linking efficiency is used as an adjustable variable to match the magnitude of the different model predictions with the experimental water contents from rheological measurements. The resulting cross-linking efficiency predictions are correlated with the ability of the different comonomers to facilitate chain transfer and/or radical termination in the polymerization environment. The model can capture the differing responses of the microgels in the presence of different salt concentrations and can account for the impact of many key physical parameters and heterogeneities in microgel swelling which the Flory-Huggins model cannot directly address.