Mean field theory of charged dendrimer molecules

Using self-consistent field theory (SCFT), we study the conformational properties of polyelectrolyte dendrimers. We compare results for three different models of charge distributions on the polyelectrolytes: (1) a smeared, quenched charge distribution characteristic of strong polyelectrolytes; (2) a smeared, annealed charge distribution characteristic of weak polyelectrolytes; and (3) an implicit counterion model with Debye-Huckel interactions between the charged groups. Our results indicate that an explicit treatment of counterions is crucial for the accurate characterization of the conformations of polyelectrolyte dendrimers. In comparing the quenched and annealed models of charge distributions, annealed dendrimers were observed to modulate their charges in response to the density of polymer monomers, counterions, and salt ions. Such phenomena is not accommodated within the quenched model of dendrimers and is shown to lead to significant differences between the predictions of quenched and annealed model of dendrimers. In this regard, our results indicate that the average dissociated charge α inside the dendrimer serves as a useful parameter to map the effects of different parametric conditions and models onto each other. We also present comparisons to the scaling results proposed to explain the behavior of polyelectrolyte dendrimers. Inspired by the trends indicated by our results, we develop a strong segregation theory model whose predictions are shown to be in very good agreement with the numerical SCFT calculations.     

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.     

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.     

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.     

Molecular dynamics simulations of dendritic polyelectrolytes with flexible spacers in salt free solution

We present the results of molecular dynamics simulations of dendritic polyelectrolytes in dilute salt-free solutions. The dendritic polyelectrolytes are modeled as an ensemble of regular-branched bead-spring chains of neutral and charged Lennard-Jones particles with explicit counterions. A wide range of molecular variables of the dendritic polyelectrolytes such as generation number, spacer length, and charge density were considered in the simulations. The effect of dendrimer size on relaxation time, the conformation of spacers, and the size dependence of the dendrimer on molecular variables are discussed and compared with a Flory type theory. The osmotic coefficients of the dilute dendritic polyelectrolyte solutions, as well as the profiles of monomers and counterions, are calculated directly from the simulations. Our simulation results show that the inner spacers of the dendrimers are extensively stretched, and the size dependence on the molecular weight deviates from the scaling prediction that assumes a Gaussian elasticity of the spacer.     

Linear polyelectrolytes in tetravalent salt solutions

The effect of adding tetravalent salt of different sizes to a solution of linear and flexible polyelectrolytes is investigated by molecular dynamics simulations. Upon the addition of salt, a chain reexpansion takes place, following a well-known collapsed conformation. The degrees of collapse and reexpansion increase with ion size. In the solution, tetravalent counterions replace monovalent ones and condense onto the chains. The condensation for small ions displays a profile different from that for large ones. In a high-salt region, ions can form layering orders around a polyelectrolyte and locally overcompensate the charge inside. Consequently, the integrated charge distribution reveals an oscillatory behavior away from a chain. By studying the radial distribution function between monomers on different polyelectrolytes, like-charge attraction between chains is demonstrated. This attraction is a prerequisite to chain aggregation or precipitation. The results show a strong dependence of salt concentration and ion size on the properties of polyelectrolyte solutions.     

Nanoparticle adsorption on a weak polyelectrolyte. Stiffness, pH, charge mobility, and ionic concentration effects investigated by Monte Carlo simulations

Monte Carlo simulations have been used to study two different models for a weak linear polyelectrolyte in the presence of nanoparticles: (i) a rodlike and (ii) a flexible polyelectrolytes. The use of simulated annealing has made it possible to simulate a polyelectrolyte chain in the presence of several nanoparticles by improving conformation sampling and avoiding multiple minima problems when dense conformations are produced. Nanoparticle distributions along the polymer backbone were analyzed versus the ionic concentration, polyelectrolyte stiffness, and nanoparticle surface charge. Titration curves were calculated and the influences of the ionic concentration, solution pH, and number of adsorbed nanoparticles on the acid/base polyelectrolyte properties have been systematically investigated. The subtle balance of attractive and repulsive interactions has been discussed, and some characteristic conformations are presented. The comparison of the two limit models provides a good representation of the stiffness influence on the complex formation. In some conditions, overcharging was obtained and presented with respect to both the polyelectrolyte and nanoparticle as the central element. Finally, the charge mobility influence along the polyelectrolyte backbone was investigated by considering annealed and quenched polyelectrolyte chains.     

Influence of charged intramolecular hydrogen bonds in weak polyelectrolytes: A Monte Carlo study of flexible and extendible polymeric chains in solution and near charged spheres

For weak polyelectrolytes, the interplay between pH, solvent properties, and polymer structure affects the amount of charges, their distribution, and hence their conformations via Coloumb repulsion. Attractive interactions can also develop between charged and neutral sites counteracting the expected Coulomb‐induced expansion. To gauge how such competition affects polyelectrolyte structure and ionization, the titration of a single polyelectrolyte chain, isolated or close to a charged sphere, mimicked with a novel many‐body potential model is simulated with Monte Carlo. Apart from showing a 10‐fold higher ionization than isolated monomers at low pH, interacting species contracted forming short‐range clusters of charged and neutral ionizable groups. The presence of a charged sphere synergically boosted both effects due to monomer interactions, forcing the chains to condense onto its surface at much lower pH. Structural properties, however, seem to be controlled only by the ionization degree despite the presence of the topological restraint represented by the spherical surface. Using Monte Carlo titration results, the equilibrium ionization of isolated chains is also estimated; the results evidence that even weak interactions can easily lead to a doubling of the total charge. © 2015 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2015 , 53, 650–663     

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.     

Molecular dynamics simulations of end‐grafted polymers with charged side chains

We report molecular dynamics simulations on bottle‐brush polyelectrolytes end‐grafted to a planar surface. For each bottle‐brush polyelectrolyte, flexible charged side chains are anchored to one neutral main chain. The effects of the counterion valence and the grafting density on the density profiles and the structural characteristics of the brush were studied in this work. It is found that the electrostatic repulsion between charged monomers in the side chains leads an extended conformation of the brush in a solution containing monovalent counterions, while strong electrostatic binding of multivalent counterions to the side chains has a significant contribution to the collapse of the brush. For the trivalent case, the distribution of end monomers in the main chains becomes broader upon decreasing the grafting density, as compared with the monovalent case. However, the position of the distribution for the monovalent case is relatively insensitive to the change of the grafting density. Additionally, with increased counterion valence, enhanced electrostatic correlation between counterions and charged side chains also weakens the diffusive ability of counterions. © 2011 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys, 2011     

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.     

Diffusion of cationic polyelectrolytes into cellulosic fibers

The penetration of cationic polyelectrolytes into anionic cellulosic fibers was evaluated with fluorescent imaging techniques in order to clarify the mechanism and time scales for the diffusion process. The bulk charge of the cellulosic fibers indirectly creates a driving force for diffusion into the porous fiber wall, which is entropic in nature due to a release of counterions as the polyelectrolyte adsorbs. The individual bulk charges in the fiber cell wall also interact with the diffusing polyelectrolyte, such that the polyelectrolyte diffuses to the first available charge and consequently adsorbs and remains fixed. Thus, subsequent polyelectrolyte chains must first diffuse through the adsorbed polyelectrolyte layer before adsorbing to the next available bulk charges. This behavior differs from earlier suggested diffusion mechanisms, by which polyelectrolytes were assumed to first adsorb to the outermost surface and then reptate into the pore structure. The time scales for polyelectrolyte diffusion were highly dependent on the flexibility of the chain, which was estimated from calculations of the persistence length. The persistence length ultimately depended on the charge density and electrolyte concentration. The charge density of the polyelectrolyte had a greater influence on the time scales for diffusion. High charge density polyelectrolytes were observed to diffuse on a time scale of months, whereas the diffusion of low charge density polyelectrolytes was measured on the order of hours. An influence of the chain length, that is, steric interactions due the persistence length of the polyelectrolyte and to the tortuosity of the porous structure of the fiber wall, could only be noted for low charge density polyelectrolytes. Increasing the electrolyte concentration increased the chain flexibility by screening the electrostatic contribution to the persistence length, in turn inducing a faster diffusion process. However, a significant change in the diffusion behavior was observed at high electrolyte concentrations, at which the interaction between the polyelectrolyte charges and the fiber charges was almost completely screened.     

Counterion localization in solutions of starlike polyelectrolytes and colloidal polyelectrolyte brushes: a self-consistent field theory

A quantitative analysis of the distribution of counterions in salt-free solutions of colloidal polyelectrolyte brushes and starlike polyelectrolytes is performed on the level of the Poisson-Boltzmann approximation. Exact numerical solutions are obtained for starlike polyelectrolyte molecules composed of f = 20, . . ., 50 arms with a fixed fractional charge alpha per segment by applying the self-consistent field method of Scheutjens and Fleer (SF-SCF). The Wigner-Seitz cell dimension defines the concentration of polyelectrolyte stars in the system. The numerical results are compared to predictions of an analytical mean field theory and related to experimental observations on the osmotic pressure in solutions of starlike polyelectrolytes and colloidal polyelectrolyte brushes.     

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.     

Dynamic Electrophoretic Mobility of a Soft Particle

A theory of the dynamic electrophoretic mobility of a spherical soft particle (that is, a polyelectrolyte-coated spherical particle) in an oscillating electric field is presented. In the absence of the polyelectrolyte layer a spherical soft particle becomes a spherical hard particle, while in the absence of the particle core it tends to a spherical polyelectrolyte. The present theory thus covers two extreme cases, that is, dynamic electrophoresis of hard particles and that of spherical polyelectrolytes. Simple analytic mobility expressions are derived. It is shown how the dynamic electrophoretic mobility of a soft particle depends on the volume charge density distributed in the polyelectrolyte layer, on the frictional coefficient characterizing the frictional forces exerted by the polymer segments on the liquid flow in the polyelectrolyte layer, on the particle size, and on the frequency of the applied oscillating electric field. Copyright 2001 Academic Press.     

NMR on polyelectrolytes

Nuclear magnetic resonance (NMR) is a versatile tool for the investigation of structure and molecular mobility in soft mater. It is a standard technique for structure determination of polymers and polyelectrolytes. In addition NMR provides information on both the polyelectrolyte, the counterions and often valuable information originating from the surrounding medium. High-resolution NMR spectroscopy enables the observation of counterion interaction in particular π interactions as well as the information about spatial proximity of functional groups in polyelectrolyte complexes. Combinations of PFG NMR and electrophoresis NMR permit the direct observation of counterion condensation. Cryoporometry showed different states of water interacting with polyelectrolytes in multilayers. Solid-state NMR has been applied to investigate both packing effects and local molecular dynamics in polyelectrolyte multilayers. The current research in the field is reviewed.     

Determination of charge and molecular weight of rigid-rod polyelectrolytes

Rod-like polyelectrolytes are an interesting model system because their persistence length is independent of the ionic strength and pH of the surrounding medium and they permit the investigation of polyelectrolytes in the absence of conformational degrees of freedom. In this work, rigid-rod poly(aramide) polyelectrolytes were synthesized by the Higashi method. Electrophoresis NMR spectroscopy in conjunction with diffusion NMR spectroscopy has been applied to determine the effective charge of the polymer. The charge was determined from the balance between the force in the electric field and the hydrodynamic friction in the steady-state electrophoretic motion. Because only organic counterions were present, and were identified in the proton NMR spectra, the counterions were investigated as well, and the fraction of condensed counterions was determined directly. From the effective charge per molecule and the knowledge of the fraction of condensed counterions, the total charge per molecule was determined. Finally, from the total charge, the number of repeat units and thus the molecular weight were inferred.     

Chain stiffness, salt valency, and concentration influences on titration curves of polyelectrolytes: Monte Carlo simulations

Monte Carlo simulations have been used to study two different models of a weak linear polyelectrolyte surrounded by explicit counterions and salt particles: (i) a rigid rod and (ii) a flexible chain. We focused on the influence of the pH, chain stiffness, salt concentration, and valency on the polyelectrolyte titration process and conformational properties. It is shown that chain acid-base properties and conformational properties are strongly modified when multivalent salt concentration variation ranges below the charge equivalence. Increasing chain stiffness allows to minimize intramolecular electrostatic monomer interactions hence improving the deprotonation process. The presence of di and trivalent salt cations clearly promotes the chain degree of ionization but has only a limited effect at very low salt concentration ranges. Moreover, folded structures of fully charged chains are only observed when multivalent salt at a concentration equal or above charge equivalence is considered. Long-range electrostatic potential is found to influence the distribution of charges along and around the polyelectrolyte backbones hence resulting in a higher degree of ionization and a lower attraction of counterions and salt particles at the chain extremities.     

On electrophoresis of a pH‐regulated nanogel with ion partitioning effects

The electrophoresis of a polyelectrolyte nanoparticle, whose charge condition depends on the salt concentration and pH of the suspended medium as well as the dielectric permittivity difference, is analyzed. The present nonlinear model for the electrophoresis of this pH‐regulated polyelectrolyte (PE) particle is based on the consideration of full set of governing equations of fluid and ion transport coupled with the equation for electric field. The Born energy of the ions are incorporated to account for the difference in the dielectric permittivity of the PE and the electrolyte. The governing equations are computed numerically through a control volume approach. The nonlinear effects are highlighted by comparing with the existing linear model as well as results based on the first‐order perturbation analysis valid for a weak applied field. The ion partitioning effect arising due to the difference in self energy of ions between the two media, have a strong impact on the mobility of the PE. The ion partitioning effect attenuates the penetration of counterions in the PE, which enhances the electric force and hence, results in a larger mobility of the PE. The nonlinear effects due to the double layer polarization and relaxation are intensified due to the ion partitioning effect. The ion partitioning effect influences the association/dissociation of PE functional group by tuning the hydrogen/hydroxide ions. Present study shows that the ion partitioning effect is profound for higher salt concentration and/or higher volume density of PE functional groups.     

Polyelectrolyte stars in planar confinement

We employ monomer-resolved molecular dynamics simulations and theoretical considerations to analyze the conformations of multiarm polyelectrolyte stars close to planar, uncharged walls. We identify three mechanisms that contribute to the emergence of a repulsive star-wall force, namely, the confinement of the counterions that are trapped in the star interior, the increase in electrostatic energy due to confinement as well as a novel mechanism arising from the compression of the stiff polyelectrolyte rods approaching the wall. The latter is not present in the case of interaction between two polyelectrolyte stars and is a direct consequence of the impenetrable character of the planar wall.