Simulations and scattering functions of polyelectrolyte–macroion complexes

Using Monte Carlo simulations of complex formation between a polyelectrolyte chain and an oppositely charged macroion, we calculated the scattering function of the polyelectrolyte chain. We investigated the case of the isolated polyelectrolyte chain and studied the effect and influence of key parameters such as the ionic concentration of the solution, polyelectrolyte length and intrinsic rigidity on the scattering function. Then, we focused on the polyelectrolyte–macroion complex by calculating the structure factor S(q) of the adsorbed polyelectrolyte chain. Typical conformations ranging from coils, extended chains to solenoids are revealed and the corresponding S(q) analysed. The effects of ionic concentration, chain length and intrinsic rigidity and relative size ratio between the polyelectrolyte and the macroion are investigated. Important effects on the structure factor of the adsorbed polyelectrolyte are observed when the macroion is partially or totally wrapped by the polyelectrolyte. Distance correlations between the polyelectrolyte monomer positions at the surface of the macroion induce the formation of peaks in the fractal regime of S(q). For semiflexible chains, when solenoid conformations are observed, the position of the peaks in the fractal regime corresponds directly to the separation distance between the turns. The formation of a protruding tail in solution is also observed through the formation in the fractal regime of a linear domain.This revised version was published online in November 2004 with corrections to the authors.     

Structuring of macroions confined between like-charged surfaces

We investigate the structuring of charged spherical nanoparticles and micelles (i.e., "macroions") between two surfaces as a function of bulk macroion concentration. Structuring is deduced from measured force profiles between a silica particle and a silica plate in the presence of an aqueous macroion (Ludox silica nanoparticle or sodium dodecyl sulfate micelle) solution, obtained with an atomic force microscope. We observe oscillatory force profiles that decay with separation. We find that the wavelength of the force profiles scales with the bulk number density as rho(-)(1/3), rather than with the effective macroion size. Only at very high silica nanoparticle concentration (above 10 vol %) in a low ionic strength solution does the wavelength become smaller than that predicted by the simple rho(-)(1/3) scaling; however, the original scaling is recovered upon the addition of a small amount of electrolyte. A comparison between the measured wavelength and the predicted spacing between the macroions in the bulk shows that the two variables differ in both magnitude and bulk density scaling. This finding suggests that confined macroions are more ordered than those in the bulk and the nature of this ordering is maintained over a relatively wide range of bulk concentration.     

Effects of structural properties of the Stern layer on the electrophoretic migration of a highly charged spherical macroion

The electrophoretic migration of a highly charged spherical macroion suspended in an aqueous solution of NaCl is studied using the molecular dynamic method. The objective is to examine the effects of the colloidal surface charge density on the electrophoretic mobility (μ) of the spherical macroion. The bare charge and the size of the macroion are varied separately to induce changes in the colloidal surface charge density. Our results indicate that μ depends on colloidal surface charge density in a nonmonotonic manner, but that this relationship is independent of the way the surface charge density is varied. It is found that an increase in colloidal surface charge density may lead to the formation of new sublayers in the Stern layer. The μ profile is also found to have a local maximum for a bare charge at which a new sublayer is formed in the Stern layer, and a local minimum for a bare charge at which the outer sublayer becomes relatively dense. Finally, the electrophoretic flow caused by the migration of the spherical macroion is studied to find that one decisive factor causing the electrophoretic flow is the ability of the macroion to carry anions in the electrolyte solution.     

Explicit ions condensation around strongly charged polyelectrolytes and spherical macroions: the influence of salt concentration and chain linear charge density. Monte Carlo simulations

The condensation of monovalent counterions and trivalent salt particles around strong rigid and flexible polyelectrolyte chains as well as spherical macroions is investigated by Monte Carlo simulations. The results are compared with the condensation theory proposed by Manning. Considering flexible polyelectrolyte chains, the presence of trivalent salt is found to play an important role by promoting chain collapse. The attraction of counterions and salt particles near the polyelectrolyte chains is found to be strongly dependent on the chain linear charge density with a more important condensation at high values. When trivalent salt is added in a solution containing monovalent salt, the trivalent cations progressively replace the monovalent counterions. Ion condensation around flexible chains is also found to be more efficient compared with rigid rods due to monomer rearrangement around counterions and salt cations. In the case of spherical macroions, it is found that a fraction of their bare charge is neutralized by counterions and salt cations. The decrease of the Debye length, and thus the increase of salt concentration, promotes the attraction of counterions and salt particles at the macroion surface. Excluded volume effects are also found to significantly influence the condensation process, which is found to be more important by decreasing the ion size.     

Real-time monitoring of the origination of multimacroion domains in a polyelectrolyte solution

First real-time monitoring of the origination of multimacroion domains in an initially homogeneous polyelectrolyte solution was performed. Domains were generated by pH-induced increase of macroion charge in solution of poly(methacrylic acid). Monitoring was performed by static and dynamic light scatterings, in which scattering contributions from individual polyions and growing multimacroion domains were separated, such that amplitudes of both modes were expressed in absolute units. Kinetic results also yield new information regarding the nature of multimacroion domains.     

Effect of discrete macroion charge distributions in solutions of like-charged macroions

The effect of replacing the conventional uniform macroion surface charge density with discrete macroion charge distributions on structural properties of aqueous solutions of like-charged macroions has been investigated by Monte Carlo simulations. Two discrete charge distributions have been considered: point charges localized on the macroion surface and finite-sized charges protruding into the solution. Both discrete charge distributions have been examined with fixed and mobile macroion charges. Different boundary conditions have been applied to examine various properties. With point charges localized on the macroion surface, counterions become stronger accumulated to the macroion and the effect increases with counterion valence. As a consequence, with mono- and divalent counterions the potential of mean force between two macroions becomes less repulsive and with trivalent counterions more attractive. With protruding charges, the excluded volume effect dominates over the increased correlation ability; hence the counterions are less accumulated near the macroions and the potential of mean force between two macroions becomes more repulsive/less attractive.     

Effect of Discrete Macroion Charge Distributions on Electric Double Layer of a Spherical Macroion

The effect of replacing the conventional uniform macroion surface charge density with discrete macroion charge distributions on the structure of electric double layer (EDL) of a spherical macroion has been investigated by Monte Carlo (MC) simulations. Two discrete models have been investigated in addition to the central macroion charge: point charges localized on the macroion surface and finite-sized charges protruding into the solution. Both models have been studied with fixed and mobile macroion charges. The radial functions of local densities and electrostatic potential in EDL, are calculated and compared to the results obtained for the central macroion charge distribution. It is concluded that the model of charge distribution significantly affects the EDL structure close to the macroion, while the effect is much weaker at larger distances. With point charges localized on the macroion surface, counterions become stronger accumulated to the macroion, as a result the absolute values of surface potential ?₀ and zeta ξ potential are decreased. With protruding charges, the excluded volume effect dominates over the increased correlation ability; hence the counterions are less accumulated near the macroions and the absolute values of ?₀ and ξ potentials are increased.     

Dynamic light scattering by polyelectrolyte solutions

A generalized expression for the apparent diffusion coefficient (D_app) for macroions as a function of scattering vector (q) is developed. Mathematica®, a system of doing mathematics on a computer, was used to obtain the eigenvalues for a select set of polyion-electrolyte systems. It is shown that under the conditions of low electrolyte concentration D_app exhibits a marked q-dependence. The second part of this communication focuses on the so-called “ordinary-extraordinary” transition observed in some polyelectrolyte systems. The characteristic D_app versus electrolyte profile for this transition is compared with the “splitting” of relaxation times reported for many other polyelectrolyte systems. General problems associated with dynamic light scattering studies on macroion systems are discussed.     

Charge inversion of dendrimers in complexes with linear polyelectrolytes in the solutions with low pH

Complexes of fully ionized third-generation dendrimers with oppositely charged linear polyelectrolyte chains are studied by the Brownian dynamics method. A freely jointed model of a dendrimer and a linear chain is used. Electrostatic interactions are considered within the Debye-Hückel approximation with the Debye radius exceeding the dimensions of a dendrimer. In these systems, the phenomenon of charge inversion is observed, and the degree of “overcharging” is higher as compared with that taking place in analogous complexes formed by dendrimers in which only terminal groups are charged. The dependence of the amount of chain units adsorbed on a dendrimer on the polyelectrolyte chain length is nonmonotnic and agrees qualitatively with the predictions of the theory proposed by Nguyen and Shklovskii for a complex composed of a spherical macroion with an oppositely charged linear chain. This nonmonotonic character also manifests itself for certain other structural characteristics of the complexes. Upon the formation of a complex, a chain is shown to penetrate deeply into a dendrimer.     

Synthesis and properties in solution of rodlike polyelectrolytes

Efficient synthetic strategies are described for the preparation of rodlike polyelectrolytes based on the intrinsically rigid poly(p-phenylene). Uncharged precursors were first prepared via the Suzuki coupling and then characterized by different methods of polymer analysis. Finally, they were transformed into polyelectrolytes using macromolecular substitution reactions. Depending on the substitution pattern, the obtained polyelectrolytes are either soluble or insoluble in water. Using water-soluble derivatives, the Poisson-Boltzmann cell model was tested by osmotic measurements and small-angle X-ray scattering. It is shown that the cell model provides a good first approximation of the distribution of the counterions around the macroion but still underestimates their correlation. Moreover, the PPP polyelectrolytes show a very pronounced polyelectrolyte effect. Since the rodlike PPPs are very rigid in shape, this observation proves that the polyelectrolyte effect is caused by long-range intermolecular electrostatic repulsion of the dissolved macroions rather than due to conformational changes.     

Investigation of the interaction of a polyelectrolyte with low molecular weight electrolytes containing nonidentical counterions by differential refractometry

If the counterion of a polyelectrolyte is not identical with any of the ions of a low molecular weight electrolyte added to the solution, the system may be regarded as a four-component system. Relations for the refractive index increments have been derived which allow the determination of the coefficient of selective sorption of the added electrolyte from the refractive index increments of the components independent of the original counterion of the polyelectrolyte. Equilibrium dialysis and differential refractometry were used to study the interaction of KCl, KBr and NaI with poly[-1(2-hydroxyethyl)pyridiniumbenzenesulfonate methacrylate] or with an analogous polymer containing a toluenesulfonate counterion in aqueous solutions. The coefficient of selective sorption increases in the series Cl^? < Br^? < I^? from strongly negative to strongly positive values; the specific interaction of these counterions with the macroion increases in the same order.     

Interaction between oppositely charged polyelectrolytes in aqueous solution

Oppositely charged polyelectrolytes interact in solution, forming polyelectrolyte complexes, which often appear as gel-like precipitates. This kind of complex formation was studied by means of calorimetric and rheological measurements. The enthalpy effects, though being fairly small, give some information about the binding strength of counterions to the macroion. We studied the system poly(p-styrene sulfonate)/poly(trimethylammonium-2-ethyl methacrylate) (PSS-PTMA), varying systematically the low molar mass counterions of PSS. In every case, the maximum of enthalpy was found around a 1:1 (mol:mol monomer units) composition of the complexes, with the shape of enthalpy versus composition-curve indicating a stoichiometric interaction. The maximum enthalpy decreased with increasing atomic mass of the counterion when the alkaline metal salts of PSS were used and no change was made on the side of the cationic polyelectrolyte. The salts of the alkaline earth metals gave a distinctly higher enthalpy. On the contrary, viscosity measurements showed a very broad minimum as a function of composition, indicating that the formation of non-stoichiometric complexes is also occurring. The conclusion of these observations is that the complex formation is stoichiometric with respect to the monomeric units, but not necessarily stoichiometric with respect to the entire macromolecules.     

Structure of spherical electric double layers: a density functional approach

A density functional theory is presented for the structure of spherical electric double layers within the restricted primitive model, where the macroion is considered as a hard sphere having uniform surface charge density, the small ions as charged hard spheres, and the solvent is taken as a dielectric continuum. The theory is partially perturbative as the hard-sphere contribution to the one-particle correlation function is evaluated using suitably averaged weighted density and the ionic part is obtained through a second-order functional Taylor expansion around the uniform fluid. The theory is in quantitative agreement with Monte Carlo simulation for the density profiles and the zeta potentials over a wide range of macroion sizes and electrolyte concentrations. The theory is able to provide interesting insights about the layering and the charge inversion phenomena occurring at the interface.     

Correlations and Fluctuations of Charged Colloids as Determined by Anomalous Small‐Angle X‐Ray Scattering

Summary: We performed molecular dynamics simulation of a charged colloidal particle with explicit counterions. Our work provides a direct comparison between simulations and ASAXS‐experiments, offering insight into the counterion distribution of charged colloidal suspensions. We give a detailed constitution of the appearing scattering terms with their physical meaning. It is shown that the cross‐correlation between a macroion and its counterions gives the meanfield approximation of the counterion density even if the counterion system is highly fluctuating. Furthermore, it is shown that cross‐correlations can be negative due to oscillations of the density amplitudes of the macroion and counterions and, therefore, must be distinguished from other scattering contributions. These oscillations become more pronounced if the counterions exhibit a fixed shape and if the size of the macroion and that of the counterion system are different.

Simulation sanpshot of a charged colloid (big central sphere) with counterions (small spheres).     

Polyelectrolyte behavior and conformation of poly-L-methionine S-methylsulfonium salts in aqueous solution

The influence of the species of counterion on the polyelectrolyte behavior and the conformation of poly-L -methionine S-methylsulfonium salts in aqueous solution was studied by viscometric, electrochemical, and optical measurements. The degree of binding of small counterions to charged polyions increases in the sequence: chloride ? bromide < iodide < thiocyanate. The conformations of chloride and bromide salts are independent of polymer concentration. On the contrary, iodide and thiocyanate salts indicate a conformational transition, probably from a random-coil conformation to an intermolecularly stabilized β-form, with the increase of polymer concentration. The results suggest the existence of a strong specific interaction between counterion and macroion in iodide and thiocyanate salt solutions at high polymer concentration.     

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.     

Combining molecular dynamics with Lattice Boltzmann: a hybrid method for the simulation of (charged) colloidal systems

We present a hybrid method for the simulation of colloidal systems that combines molecular dynamics (MD) with the Lattice Boltzmann (LB) scheme. The LB method is used as a model for the solvent in order to take into account the hydrodynamic mass and momentum transport through the solvent. The colloidal particles are propagated via MD and they are coupled to the LB fluid by viscous forces. With respect to the LB fluid, the colloids are represented by uniformly distributed points on a sphere. Each such point [with a velocity V(r) at any off-lattice position r] is interacting with the neighboring eight LB nodes by a frictional force F = xi0(V(r)-u(r)), with xi0 being a friction coefficient and u(r) being the velocity of the fluid at the position r. Thermal fluctuations are introduced in the framework of fluctuating hydrodynamics. This coupling scheme has been proposed recently for polymer systems by Ahlrichs and Dunweg [J. Chem. Phys. 111, 8225 (1999)]. We investigate several properties of a single colloidal particle in a LB fluid, namely, the effective Stokes friction and long-time tails in the autocorrelation functions for the translational and rotational velocity. Moreover, a charged colloidal system is considered consisting of a macroion, counterions, and coions that are coupled to a LB fluid. We study the behavior of the ions in a constant electric field. In particular, an estimate of the effective charge of the macroion is yielded from the number of counterions that move with the macroion in the direction of the electric field.     

On the interaction between macroions in solution at infinite dilution

The pair interaction potential for charged macromolecules in solution is calculated from the Ornstein-zernike equation with MSA-closure for the salt-ion/salt-ion and salt-ion/macroion interaction, and HNC-closure for the macroion/macroion interaction. Two different limiting cases are considered: low salt-ion concentration or zero hardcore volume of the macroion. It is found that if the Debye screening length is much larger than the smallest linear dimension of the macroion, the potential is simply a superposition of point-point interaction potentials (screened Coulomb potentials).     

Poisson-Boltzmann theory of the charge-induced adsorption of semi-flexible polyelectrolytes

A model is suggested for the structure of an adsorbed layer of a highly charged semi-flexible polyelectrolyte on a weakly charged surface of opposite charge sign. The adsorbed phase is thin, owing to the effective reversal of the charge sign of the surface upon adsorption, and ordered, owing to the high surface density of polyelectrolyte strands caused by the generally strong binding between polyelectrolyte and surface. The Poisson-Boltzmann equation for the electrostatic interaction between the array of adsorbed polyelectrolytes and the charged surface is solved for a cylindrical geometry, both numerically, using a finite element method, and analytically within the weak curvature limit under the assumption of excess monovalent salt. For small separations, repulsive surface polarization and counterion osmotic pressure effects dominate over the electrostatic attraction and the resulting electrostatic interaction curve shows a minimum at nonzero separations on the Angstrom scale. The equilibrium density of the adsorbed phase is obtained by minimizing the total free energy under the condition of equality of chemical potential and osmotic pressure of the polyelectrolyte in solution and in the adsorbed phase. For a wide range of ionic conditions and charge densities of the charged surface, the interstrand separation as predicted by the Poisson-Boltzmann model and the analytical theory closely agree. For low to moderate charge densities of the adsorbing surface, the interstrand spacing decreases as a function of the charge density of the charged surface. Above about 0.1 M excess monovalent salt, it is only weakly dependent on the ionic strength. At high charge densities of the adsorbing surface, the interstrand spacing increases with increasing ionic strength, in line with the experiments by Fang and Yang [J. Phys. Chem. B 101, 441 (1997)].