Macroion solutions in the cell model studied by field theory and Monte Carlo simulations

Aqueous solutions of charged spherical macroions with variable dielectric permittivity and their associated counterions are examined within the cell model using a field theory and Monte Carlo simulations. The field theory is based on separation of fields into short- and long-wavelength terms, which are subjected to different statistical-mechanical treatments. The simulations were performed by using a new, accurate, and fast algorithm for numerical evaluation of the electrostatic polarization interaction. The field theory provides counterion distributions outside a macroion in good agreement with the simulation results over the full range from weak to strong electrostatic coupling. A low-dielectric macroion leads to a displacement of the counterions away from the macroion.     

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.     

Transport numbers of sodium polyacrylate in aqueous solutions

The moving boundary technique has been used to measure independently the transference numbers of macroion and counterions in water solutions of sodium polyacrylate at 25° and in the concentration range 0·050-0·8 eq/l. The experimental results allow us to compute, without any arbitrary assumption, the fraction of counterions which contribute to the transport of current towards the cathode and which may be called the “fraction of free ions”.     

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 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.     

Simulation of the electrical double layer of a spherical micelle of an anionic substance with regard to the solvent structure

Computer simulation methods are employed to consider the structure of the electrical double layer of a spherical micelle in aqueous surfactant solutions with allowance for the contribution of the solvent. Three micelle models were used in the calculations, namely, a macroion with discretely distributed charges and a continual solvent, a spherical model micelle with a coarse-grained representation of the solvent, and a spherical model micelle in an aqueous phase with an explicit account for water molecules. Based on these three models, the radial profiles of the local densities and electric potentials in the electrical double layer, as well as the degrees of binding single-, double-, and triple-charged counterions by the macroion in aqueous surfactant solutions, are calculated with regard to the Lennard-Jones and electrostatic interactions. The allowance for the molecular structure of the solvent leads to qualitatively different local dependences of the electric potential as compared to both the continual and coarse-grained representation of the solvent.     

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.     

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.     

Equilibrium and Transport Characteristics of Perfluorinated Cation-Exchange Membranes Containing Carboxyl Groups

Equilibrium (the exchange capacity and moisture content) and transport (the ion transport numbers and the electrical conductance) properties of perfluorinated cation-exchange membranes containing carboxyl groups were studied as a function of pH and concentration of KCl solutions (10^–4–1 M). Dissociation constants of carboxyl groups, adsorption potentials of K⁺ ions, transport numbers and mobilities of counterions in the membranes, concentrations of fixed ions, co- and counterions, as well as the Donnan potentials were calculated.     

Generalizations for the potential of mean force between two isolated colloidal particles from Monte Carlo simulations

A substantial amount of experimental and numerical evidence has shown that the Derjaguin-Landau-Verwey-Overbeek theory is not suitable for describing those colloidal solutions that contain multivalent counterions. Toward improved understanding of such solutions, the authors report Monte Carlo calculations wherein, following Rouzina and Bloomfield, they postulate that, in the absence of van der Waals forces, the overall force between two isolated charged colloidal particles in electrolyte solutions is determined by a dimensionless parameter Gamma=z(2)l(B)/a, which measures the electrostatic repulsion between counterions adsorbed on the macroion surface, where z = counterion valence, l(B)=Bjerrum length, and a = average separation between counterions on the macroion surface calculated as if the macroion were fully neutralized. The authors find, first, that the maximum repulsion between like-charged macroions occurs at Gamma approximately 0.5 and, second, that onset of attraction occurs at Gamma approximately 1.8, essentially independent of the valence and concentration of the surrounding electrolyte. These observations might provide new understanding of interactions between electrostatic double layers and perhaps offer explanations for some electrostatic phenomena related to interactions between DNA molecules or proteins.     

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.     

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.     

Theory of the effect of small ions on the electrical admittance of macroion solutions

A general theory is given on the electrical admittance of a parallel plate capacitor filled with a conductive medium containing charged particles by considering both their electric atmospheres and the electric field due to charges induced on the electrodes. The theory is applied to a dilute macroion solution with added salt and it is found that the low frequency dielectric response of the solution reflects the motions of the macroion and bound small ions and the effect of motion of the ion atmosphere around the macroion is canceled out by the current due to the induced charges on the electrodes.     

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.     

Electric transport properties of ultra-and nanoporous glasses in electrolyte solutions

Electric transport characteristics (conductivity, specific surface conductance, and transport numbers of counterions) for nano-and ultraporous glass membranes with pore radii of 1.3–160 nm are studied and compared for chloride solutions containing single-, double-, and triple-charged cations.     

Transport properties of a cation exchange membrane with sodium and potassium as counterions

The measured transport properties for the cation exchange membrane Ionics 61AZL389 in the Na+ + K+ mixed counterion states are presented. The transport numbers of counterions and of water as well as the specific conductivity of the membrane showed approximately linear behaviour as a function of ionic fraction in the membrane. It is therefore concluded that it is possible to estimate the transport numbers of counterions from known selectivity coefficients and concentrations in the external solution. The transport number of water and the specific conductivity of the membrane can also be evaluated reasonably accurately using data for membranes in pure homoionic states. The separation ratio was found to be constant over a wide range of concentration ratios under the hydrodynamic conditions used experimentally.     

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.     

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).