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.     

Molecular Dynamics Simulation of the Electrical Double Layer of Spherical Macroion

The structure of the electrical double layer (EDL) of a spherical macroion with a total charge of 60 elementary charges is studied by molecular dynamics methods. In calculations we used two models: continuous and discrete. In the continuous model, the total charge was concentrated in the center of the macroion; in the discrete model, elementary charges were randomly distributed over the surface of the macroion. The radial profiles of local densities and electric potential in EDL, as well as the degree of counterion binding by the macroion, are calculated with allowance for the Lennard-Jones and electrostatic interactions. It is established that the character of charge distribution significantly affects the EDL structure near the macroion, whereas its effect is much weaker at larger distances. The results obtained are compared with the experimental data on the surface potential and the diffuse part of EDL of sodium dodecyl sulfate micelles in aqueous solution, as well as on the micelle-bound charge. It is shown that even weak specific interaction between counterions and a macroion can substantially influence the structure of its EDL.     

Electrophoretic properties of highly charged colloids: a hybrid molecular dynamics/lattice Boltzmann simulation study

Using computer simulations, the electrophoretic motion of a positively charged colloid (macroion) in an electrolyte solution is studied in the framework of the primitive model. In this model, the electrolyte is considered as a system of negatively and positively charged microions (counterions and coions, respectively) that are immersed into a structureless medium. Hydrodynamic interactions are fully taken into account by applying a hybrid simulation scheme, where the charged ions (i.e., macroion and electrolyte), propagated via molecular dynamics, are coupled to a lattice Boltzmann (LB) fluid. In a recent electrophoretic experiment by Martin-Molina et al. [J. Phys. Chem. B 106, 6881 (2002)], it was shown that, for multivalent salt ions, the mobility mu initially increases with charge density sigma, reaches a maximum, and then decreases with further increase of sigma. The aim of the present work is to elucidate the behavior of mu at high values of sigma. Even for the case of monovalent microions, a decrease of mu with sigma is found. A dynamic Stern layer is defined that includes all the counterions that move with the macroion while subjected to an external electrical field. The number of counterions in the Stern layer, q(0), is a crucial parameter for the behavior of mu at high values of sigma. In this case, the mobility mu depends primarily on the ratio q(0)/Q (with Q the valency of the macroion). The previous contention that the increase in the distortion of the electric double layer (EDL) with increasing sigma leads to the lowering of mu does not hold for high sigma. In fact, it is shown that the deformation of the EDL decreases with the increase of sigma. The role of hydrodynamic interactions is inferred from direct comparisons to Langevin simulations where the coupling to the LB fluid is switched off. Moreover, systems with divalent counterions are considered. In this case, at high values of sigma the phenomenon of charge inversion is found.     

Effect of a Dynamic Stern Layer on the Sedimentation Velocity and Potential in a Dilute Suspension of Colloidal Particles

In this paper the theory of the sedimentation velocity and potential (gradient) in a dilute suspension of charged spherical colloidal particles developed by Ohshima et al. (H. Ohshima, T. W. Healy, L. R. White, and R. W. O'Brien, J. Chem. Soc., Faraday Trans. 2, 80, 1299 (1984)) has been modified to include the presence of a dynamic Stern layer on the particle surfaces. The starting point has been the theory that Mangelsdorf and White (C. S. Mangelsdorf, and L. R. White, J. Chem. Soc., Faraday Trans. 86, 2859 (1990)) developed to calculate the electrophoretic mobility of a colloidal particle allowing for the lateral motion of ions in the inner region of the double layer (dynamic Stern layer). The effects of varying the different Stern layer parameters on the sedimentation velocity and potential are discussed and compared to the case when a Stern layer is absent. The influence of electrolyte concentration and zeta potential of the particles is also analyzed. The results show that regardless of the chosen set of Stern layer and solution parameters, the presence of a dynamic Stern layer causes the sedimentation velocity to increase and the sedimentation potential to decrease, in comparison with the standard case (no Stern layer present). These changes are almost negligible when sedimentation velocity is concerned, but they are very important when it comes to the sedimentation potential. A justification for this fact can be given in terms of an Onsager reciprocal relation, connecting the magnitudes of the sedimentation potential and the electrophoretic mobility. As previously reported, the presence of a dynamic Stern layer exerts a great influence on the electrophoretic mobility of a colloidal particle, and by means of the Onsager relation, the same is confirmed to occur when the sedimentation potential is concerned. Copyright 2000 Academic Press.     

A macroion electrokinetics algorithm

A numerical algorithm is presented for the standard model of macroion electrokinetics and certain generalizations of it. The macroion consists of a cylindrical section with identical, hemispheroidal endcaps, each piece having arbitrary length. The system of one macroion and adjoining salt solution is subjected to an arbitrary sequence of pulsed electrical fields and pulsed translational and rotational velocities. Numerical solutions are obtained for the time dependent electrostatic and mobile ion concentration fields and the solvent velocity. From these fields the dielectric response, force, and torque are calculated. Generalizations of the standard model include the diffusive motion of macroion surface charges, partial slip of solvent motion at the macroion surface, and a simple model for the reactive exchange of surface charge with solution ions. The primary illustrative application is to recent measurements of electric birefringence versus applied field frequency for poly-(tetrafluorothylene) colloidal particles, but a few results are presented for the dielectric response of DNA fragments and of spherical colloidal particles. The source code and additional details are provided as supplementary documentation.     

Simulation of the electrical double layer of a macroion with different counterion charges

An electrical double layer of a spherical macroion with single-, double-, and triple-charged counterions in aqueous solution of 1: 1 background electrolyte at different concentrations are studied by the molecular dynamics method for models with discrete and continuous surface charge distribution. Radial profiles of ion partial densities and the electric potential distribution in the double layer are calculated. The degree of counterion binding with a macroion is determined. The effect of water permittivity on the structure of electrical double layer is studied.     

Viscosity and electrophoretic mobility of cesium fullerenehexamalonate in aqueous solutions--comparing experiments and theories on nanometer-sized spherical polyelectrolyte

The viscosity of aqueous solutions of cesium fullerenehexamalonate T h -C 66(COOCs) 12, a rigid spherical nanometer-sized polyvalent salt, was measured by the Ubbelohde-type viscometer. The measurements were performed without added salt at 25 degrees C in the concentration range between 7 and 320 g/dm (3). THe concentration dependence of the obtained reduced viscosity was compared with the theoretical prediction, taking into account contributions stemming from the intrinsic viscosity, hydrodynamic perturbations of the hypothetically bare fullerenehexamalonate macroion, the primary electroviscous effect, and the secondary electroviscous effect. Using the geometric radius of the bare macroion from the previous measurements of the estimated effective charge of the macroion and from the small-angle X-ray scattering data of the estimated thickness of the compact shell of counterions electrostatically bound to the macroion, a good agreement between theory and the experiment was obtained in the range of the lowest and of the highest concentrations. Electrostatic interactions are identified as the main cause of the increased reduced viscosity at the lowest measured concentrations. At the highest concentrations, electrostatic interactions are effectively screened, and the influence of binary hydrodynamic interactions and perturbations of the hypothetical bare macroion prevails over electrostatic contributions to the increased viscosity. The electrophoretic mobility of the fullerenehexamalonate ion in aqueous salt-free medium was computed with the same value for the radius of the fullerenehexamalonate macroion as that used in the calculation of viscosity. The numerical solution of Ohshima's equation agreed well with the experimental values.     

Electrophoretic mobility of a colloidal particle with constant surface charge density

When the electrophoretic mobility of a particle in an electrolyte solution is measured, the obtained electrophoretic mobility values are usually converted to the particle zeta potential with the help of a proper relationship between the electrophoretic mobility and the zeta potential. For a particle with constant surface charge density, however, the surface charge density should be a more characteristic quantity than the zeta potential because for such particles the zeta potential is not a constant quantity but depends on the electrolyte concentration. In this article, a systematic method that does not require numerical computer calculation is proposed to determine the surface charge density of a spherical colloidal particle on the basis of the particle electrophoretic mobility data. This method is based on two analytical equations, that is, the relationship between the electrophoretic mobility and zeta potential of the particle and the relationship between the zeta potential and surface charge density of the particle. The measured mobility values are analyzed with these two equations. As an example, the present method is applied to electrophoretic mobility data on gold nanoparticles (Agnihotri, S. M.; Ohshima, H.; Terada, H.; Tomoda, K.; Makino, K. Langmuir 2009, 25, 4804).     

Renormalized charge in a two-dimensional model of colloidal suspension from hypernetted chain approach

The renormalized charge of a simple two-dimensional model of colloidal suspension was determined by solving the hypernetted chain approximation and Ornstein-Zernike equations. At the infinite dilution limit, the asymptotic behavior of the correlation functions is used to define the effective interactions between the components of the system and these effective interactions were compared to those derived from the Poisson-Boltzmann theory. The results we obtained show that, in contrast to the mean-field theory, the renormalized charge does not saturate, but exhibits a maximum value and then decays monotonically as the bare charge increases. The results also suggest that beyond the counterion layer near to the macroion surface, the ionic cloud is not a diffuse layer which can be handled by means of the linearized theory, as the two-state model claims, but a more complex structure is settled by the correlations between microions.     

Transient electrophoresis in a suspension of charged particles with arbitrary electric double layers

The startup of electrophoretic motion in a suspension of spherical colloidal particles, which may be charged with constant zeta potential or constant surface charge density, due to the sudden application of an electric field is analytically examined. The unsteady modified Stokes equation governing the fluid velocity field is solved with unit cell models. Explicit formulas for the transient electrophoretic velocity of the particle in a cell in the Laplace transforms are obtained as functions of relevant parameters. The transient electrophoretic mobility is a monotonic decreasing function of the particle-to-fluid density ratio and in general a decreasing function of the particle volume fraction, but it increases and decreases with a raise in the ratio of the particle radius to the Debye length for the particles with constant zeta potential and constant surface charge density, respectively. On the other hand, the relaxation time in the growth of the electrophoretic mobility increases substantially with an increase in the particle-to-fluid density ratio and with a decrease in the particle volume fraction but is not a sensitive function of the ratio of the particle radius to the Debye length. For specified values of the particle volume fraction and particle-to-fluid density ratio in a suspension, the relaxation times in the growth of the particle mobility in transient electrophoresis and transient sedimentation are equivalent.     

Effect of Electrolyte Concentration on the Stern Layer Thickness at a Charged Interface

The chemistry and physics of charged interfaces is regulated by the structure of the electrical double layer (EDL). Herein we quantify the average thickness of the Stern layer at the silica (SiO₂) nanoparticle/aqueous electrolyte interface as a function of NaCl concentration following direct measurement of the nanoparticles’ surface potential by X‐ray photoelectron spectroscopy (XPS). We find the Stern layer compresses (becomes thinner) as the electrolyte concentration is increased. This finding provides a simple and intuitive picture of the EDL that explains the concurrent increase in surface charge density, but decrease in surface and zeta potentials, as the electrolyte concentration is increased.     

Approximate analytic expressions for the electrophoretic mobility of spherical soft particles

Approximate analytic expressions are derived for the electrophoretic mobility of a weakly charged spherical soft particle consisting of the particle core covered with a surface layer of polymers in an electrolyte solution. The particle core and the surface polymer layer may be charged or uncharged. The obtained electrophoretic mobility expressions, which involve neither numerical integration nor exponential integrals, are found to be in excellent agreement with the exact numerical results. It is also found that the obtained mobility expressions reproduce all the previously derived limiting expressions and approximate analytic expressions for the electrophoretic mobility of a weakly charged spherical soft particle.     

Electrophoresis of a membrane-coated sphere in a spherical cavity

The boundary effect on the electrophoresis of particles covered by a membrane layer is discussed by considering a spherical particle in a spherical cavity under the conditions where the effect of double-layer polarization can be significant. The influence of the key parameters of the system under consideration on the electrophoretic mobility of a particle is investigated. These include the surface potential; the thickness of the double layer; the relative size of the cavity; and the thickness, the fixed charge density, and the friction coefficient of the membrane layer. The fixed charge in the membrane layer of a particle is found to have a significant influence on its electrophoretic behavior. For instance, depending upon the amount of fixed charge in the membrane layer, the mobility of a particle may exhibit a local minimum as the thickness of the double layer varies.     

Electrokinetics of concentrated suspensions of spherical colloidal particles with surface conductance, arbitrary zeta potential, and double-layer thickness in static electric fields

In this paper the electrophoretic mobility and the electrical conductivity of concentrated suspensions of spherical colloidal particles have been numerically studied under arbitrary conditions including zeta potential, particle volume fraction, double-layer thickness (overlapping of double layers is allowed), surface conductance by a dynamic Stern layer model (DSL), and ionic properties of the solution. We present an extensive set of numerical data of both the electrophoretic mobility and the electrical conductivity versus zeta potential and particle volume fraction, for different electrolyte concentrations. The treatment is based on the use of a cell model to account for hydrodynamic and electrical interactions between particles. Other theoretical approaches have also been considered for comparison. Furthermore, the study includes the possibility of adsorption and lateral motion of ions in the inner region of the double layers (DSL model), according to the theory developed by C. S. Mangelsdorf and L. R. White (J. Chem. Soc. Faraday Trans.86, 2859 (1990)). The results show that the correct limiting cases of low zeta potentials and thin double layers for dilute suspensions are fulfilled by our conductivity formula. Moreover, the presence of a DSL causes very important changes, even dramatic, on the values of both the electrophoretic mobility and the electrical conductivity for a great range of volume fractions and zeta potentials, specially when double layers of adjacent cells overlap, in comparison with the standard case (no Stern layer present). It can be concluded that in general the presence of a dynamic Stern layer causes the electrophoretic mobility to decrease and the electrical conductivity to increase in comparison with the standard case for every volume fraction, zeta potential, and double-layer thickness.     

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.     

Hydrolysis versus ion correlation models in electrokinetic charge inversion: establishing application ranges

In this article, we investigate experimentally a wide range of situations where charge inversion (i.e., overcompensation of the surface charge of a colloidal particle by the countercharge) can occur. To that end, the electrophoretic mobility of sodium montmorillonite, silica, and polystyrene latex as functions of pH and concentration of different salts is presented, and conditions are established where charge inversion occurs. The reason for this study is to provide experimental evidence for distinguishing between two existing models for the explanation of charge inversion. One of these is the specific adsorption of ions located in the Stern layer in combination with a Gouy-Chapman diffuse part of the double layer. The other ion-correlation theories explain the phenomenon in terms of purely physical arguments based on Coulombic pair interactions between ions and surface charges and on excluded volume effects. In distinguishing between these two interpretations, the influence of the pH plays a central role because of its effect on the hydrolysis of multivalent cations. In our experiments, it is found that although 1-2 and 2-2 electrolytes provoke a decrease in the absolute values of the electrophoretic mobilities when their concentration in solution is increased, they never lead to charge inversion, whatever the surface charge or the pH. However, in the case of salts of trivalent cations, electrokinetic charge reversal is often observed above a certain critical electrolyte concentration. In addition, the extent of overcharging increases when the concentration is raised above the critical value. This trend occurs for any system in which the surface charge is pH-independent, as in polystyrene latex and montmorillonite. Most of the results presented here are compatible with the specific adsorption of hydrolyzed metal ions as the main driving force for charge inversion. At low pH, when the hydrolysis of trivalent cations is likely to be absent, overcharging can be attributed to ion correlation effects.     

Ion size effects on the electrokinetics of salt-free concentrated suspensions in ac fields

We analyze the influence of finite ion size effects in the response of a salt-free concentrated suspension of spherical particles to an oscillating electric field. Salt-free suspensions are just composed of charged colloidal particles and the added counterions released by the particles to the solution that counterbalance their surface charge. In the frequency domain, we study the dynamic electrophoretic mobility of the particles and the dielectric response of the suspension. We find that the Maxwell-Wagner-O’Konski process associated with the counterions condensation layer is enhanced for moderate to high particle charges, yielding an increment of the mobility for such frequencies. We also find that the increment of the mobility grows with ion size and particle charge. All these facts show the importance of including ion size effects in any extension attempting to improve standard electrokinetic models.     

Approximate expression for the electrophoretic mobility of a spherical colloidal particle covered with an ion-penetrable uncharged polymer Layer

An approximate analytic expression is derived for the electrophoretic mobility of a charged spherical colloidal particle covered with an ion-penetrable uncharged polymer layer in an electrolyte solution by taking into account the relaxation effects. This expression is applicable for all values of zeta potentials at large a(aca. 30), where is the Debye–Huckel parameter and a is the radius of the particle core. A simple expression for the ratio of the electrophoretic mobility of a polymer-coated particle to that of a bare particle without a polymer layer is also given.     

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.     

Electrophoresis of solid particles at large Peclet numbers

A theory of concentration polarization of a thin electrical double layer (DL) on a spherical particle is developed for the regime of large Peclet numbers which is realized in strong electric fields. In this regime, the concentration field arising outside DL is estimated under influence of diffusion and convection. According to the theory developed, polarization of DL at large Peclet numbers causes a change in the Stern potential, the formation of a dipole moment and the long-range potential. The diffuse layer deviates strongly from spherical symmetry and electroneutrality, and the screen of the surface charge is provided not only by the diffuse atmosphere but also by the charge induced in the convective-diffusion layer. The effect of electric field on the induced charge gives rise to the additional electroosmotic slip, that was called "secondary electroosmosis". Thus, a nonlinear additional term for the Smoluchowski formula of electrophoretic velocity is based on the changes of zeta-potential and on the secondary electroosmotic slip. The comparison of theory with experimental results revealed considerable fitting.