Three-dimensional discrete charge effects on the electrostatic interaction between two soft particles

An expression for the electrostatic interaction energy between two parallel plate-like soft particles (i.e., hard particles covered with an ion-penetrable surface layer of polyelectrolytes) in an electrolyte solution is derived by using the linearized Poisson-Boltzmann equation. This expression is based on a discrete charge model in which the surface layer consists of a cubic lattice of fixed point charges. We show that the deviation of the results of the discrete charge model from those of the conventional smeared charge model becomes significant as the ratio of the lattice spacing to the Debye length becomes large. As this ratio decreases, on the other hand, the discrete charge model approaches a smeared charge model, leading to the Donnan-potential regulated interaction model.     

Effects of image charges, interfacial charge discreteness, and surface roughness on the zeta potential of spherical electric double layers

We investigate the effects of image charges, interfacial charge discreteness, and surface roughness on spherical electric double layer structures in electrolyte solutions with divalent counterions in the setting of the primitive model. By using Monte Carlo simulations and the image charge method, the zeta potential profile and the integrated charge distribution function are computed for varying surface charge strengths and salt concentrations. Systematic comparisons were carried out between three distinct models for interfacial charges: (1) SURF1 with uniform surface charges, (2) SURF2 with discrete point charges on the interface, and (3) SURF3 with discrete interfacial charges and finite excluded volume. By comparing the integrated charge distribution function and the zeta potential profile, we argue that the potential at the distance of one ion diameter from the macroion surface is a suitable location to define the zeta potential. In SURF2 model, we find that image charge effects strongly enhance charge inversion for monovalent interfacial charges, and strongly suppress charge inversion for multivalent interfacial charges. For SURF3, the image charge effect becomes much smaller. Finally, with image charges in action, we find that excluded volumes (in SURF3) suppress charge inversion for monovalent interfacial charges and enhance charge inversion for multivalent interfacial charges. Overall, our results demonstrate that all these aspects, i.e., image charges, interfacial charge discreteness, their excluding volumes, have significant impacts on zeta potentials of electric double layers.     

Electric potential distributions around discrete charges in a dielectric membrane—electrolyte solution system

Within the framework of the linearized Debye-Hückel theory an exact solution of the problem of calculating the electric potential caused by discrete fixed charges located at arbitrary positions with respect to a dielectric membrane-solution interface is presented. It takes into account the existence of an electrolyte solution on both sides of the membrane. Asymmetric ionic conditions are allowed for. For some interesting typical cases of fixed charge locations and electrolyte ionic strengths electric potential distributions were calculated and discussed. It is shown that, if the fixed charges were at or in front of the membrane surface, the characteristic distance of the electric potential decay was comparable to the Debye-Hückel length. At the opposite membrane surface only very small electric potentials can be observed. If, however, the fixed charge was placed below the membrane surface the electric potential in lateral direction and towards the other membrane surface largely increased. This effect was very sensitive to the position of the fixed charge with respect to the membrane surface.     

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.     

Polarization of a diffuse soft particle subjected to an alternating current field

The polarization of a diffuse soft particle submerged in an aqueous electrolyte and subjected to a uniform alternating electric field is theoretically analyzed with the standard electrokinetic model (the Poisson-Nernst-Planck equations). The particle consists of a rigid uncharged core and a charged diffuse polyelectrolytic shell (soft layer) permeable to ions and solvent. Our focus is on the impact of the characteristics of the soft layer including the Donnan potential, the soft layer thickness, and the friction coefficient of the soft layer on the dipole coefficient, characterizing the strength of the polarization. Under the limits of thin double layers and thin polyelectrolytic shells, approximate analytical expressions to evaluate the dipole moment coefficients are derived for high-frequency and low-frequency ranges, respectively. The analytical results are compared and agree favorably with those numerically computed by the standard model. Interestingly, we discover that when the double layer is comparable to the soft layer the dipole moment behaves qualitatively differently at different Donnan potentials. When the Donnan potential is small, the dipole moment decreases as the double layer increases. In contrast, at large Donnan potentials, the dipole moment increases with the increase in the double layer. The distinct responses to Donnan potentials are attributed to the impact of the associated double layer on the charge distribution of mobile ions inside the soft layer. The theoretical model provides a fundamental basis for interpreting the polarization of heterogeneous systems, including environmental or biological colloids or microgel particles.     

The rigorous computation of the molecular electric potential

A method is presented for the rigorous computation of the electric potential of molecules of arbitrary shape, under the assumption of continuous linear dielectric media. The computational technique involves finding the distribution of induced polarization charge on the molecular surface, and proceeds by an application of the method of boundary elements. The surface, which separates the molecular interior (of low dielectric constant) from the highly polar solvent, is given a piece-wise analytic representation as a collection of curvilinear elements. Given a set of internal fixed charges, the distribution of polarization-charge is found as a continuous function over the surface elements, and the electric potential (including all polarization effects) is then easily computed at any point. The method is applied to a spherical interface, and to several small molecules of biological interest, including a hexapeptide. The resulting potentials show good convergence in all cases. The future application of the method to macromolecules is discussed.     

Sedimentation velocity and potential in concentrated suspensions of charged porous spheres

The body-force-driven migration in a homogeneous suspension of polyelectrolyte molecules or charged flocs in an electrolyte solution is analyzed. The model used for the particle is a porous sphere in which the density of the hydrodynamic frictional segments, and therefore also that of the fixed charges, is constant. The effects of particle interactions are taken into account by employing a unit cell model. The overlap of the electric double layers of adjacent particles is allowed and the relaxation effect in the double layer surrounding each particle is considered. The electrokinetic equations which govern the electrostatic potential profile, the ionic concentration (or electrochemical potential energy) distributions, and the fluid velocity field inside and outside the porous particle in a unit cell are linearized by assuming that the system is only slightly distorted from equilibrium. Using a regular perturbation method, these linearized equations are solved for a symmetrically charged electrolyte with the density of the fixed charges as the small perturbation parameter. An analytical expression for the settling velocity of the charged porous sphere is obtained from a balance among its gravitational, electrostatic, and hydrodynamic forces. A closed-form formula for the sedimentation potential in a suspension of identical charged porous spheres is also derived by using the requirement of zero net electric current. The dependence of the sedimentation velocity and potential of the suspension on the particle volume fraction and other properties of the particle-solution system is found to be quite complicated.     

Analysis of the response of suspended colloidal soft particles to a constant electric field

A network model, originally designed for an electrokinetic study of soft particle suspensions, has been used for an in-depth analysis of the physical behavior of these systems under the action of an externally applied DC electric field. The versatility of the network simulation method used makes it possible to obtain information readily not only about the electrophoretic mobility, but also about any physical variable of interest at all points around the suspended particle: electric potential, ion concentrations, fluid velocity. The field-induced polarization of the double layer is described in terms of the dependence of these and other derived variables (volume charge density, electric field components, ion flux components) on the distance to the membrane-solution interface. In contrast to colloidal suspensions of hard particles, which basically depend on just two parameters (the reciprocal Debye length multiplied by the particle radius, kappaa, and the zeta potential, zeta), soft particle suspensions require a wider parameter set. First, there are two characteristic diffusion lengths in the system (one inside the membrane and the other in the solution) and two geometrical lengths (the core radius a and the membrane thickness (b-a)). Furthermore, there is the fixed charge density inside the membrane (and possibly a surface charge density over the core) that cannot be represented by a zeta potential. Finally, the parameter that characterizes the interaction between the fluid and the permeable membrane, gamma, strongly influences the behavior of the system. Dependences on all these parameters (except the geometrical ones) are included in this study.     

Effect of ion solvation on the diffuse layer structure in mixed electrolytes

A “solvionic” model of a multicomponent electrochemical system (mixed electrolyte) is considered. An ion in the solution is considered as a point charge rigidly fixed inside its solvation shell. The corresponding equations for the diffuse layer on an ideally polarizable electrode are derived, and an effective method of their numerical solution is formulated. The calculations are performed in order to follow the changes in the diffuse layer structure with variations in the electrode charge and electrolyte composition. Far from the zerocharge potential of solution, the dependences of distributions of solution components over the diffuse layer on the electrode charge radically differ from those within the classic Gouy-Chapman theory. Analytical equations (asymptotics at large electrode charges) for concentrations of solvated ions in the plane of their maximum approach and for their “surface excesses” (diffuse adsorption) are determined. Results of numerical calculations for a 0.2 M LiCl + 0.05 M BaCl₂ solution are plotted.     

Surface properties of fluids of charged platelike colloids

Surface properties of mixtures of charged platelike colloids and salt in contact with a charged planar wall are studied within density functional theory. The particles are modeled by hard cuboids with their edges constrained to be parallel to the Cartesian axes corresponding to the Zwanzig model [J. Chem. Phys. 39, 1714 (1963)] and the charges of the particles are concentrated at their centers. The density functional applied is an extension of a recently introduced functional for charged platelike colloids. It provides a qualitative approach because it does not determine the relation between the actual and the effective charges entering into the model. Technically motivated approximations, such as using the Zwanzig model, are expected not to influence the results qualitatively. Analytically and numerically calculated bulk and surface phase diagrams exhibit first-order wetting for sufficiently small macroion charges and isotropic bulk order as well as first-order drying for sufficiently large macroion charges and nematic bulk order. The asymptotic wetting and drying behaviors are investigated by means of effective interface potentials which turn out to be asymptotically the same as for a suitable neutral system governed by isotropic nonretarded dispersion forces. Wetting and drying points as well as predrying lines and the corresponding critical points have been located numerically. A crossover from monotonic to nonmonotonic electrostatic potential profiles upon varying the surface charge density has been observed. Nonmonotonic electrostatic potential profiles are equivalent to the occurrence of charge inversion. Due to the presence of both the Coulomb interactions and the hard-core repulsions, the surface potential and the surface charge do not vanish simultaneously, i.e., the point of zero charge and the isoelectric point of the surface do not coincide.     

X-ray study of the electric double layer at the n-hexane/nanocolloidal silica interface

The spatial structure of the transition region between an insulator and an electrolyte solution was studied with x-ray scattering. The electron-density profile across the n-hexane/silica sol interface (solutions with 5, 7, and 12 nm colloidal particles) agrees with the theory of the electrical double layer and shows separation of positive and negative charges. The interface consists of three layers, i.e., a compact layer of Na(+), a loose monolayer of nanocolloidal particles as part of a thick diffuse layer, and a low-density layer sandwiched between them. Its structure is described by a model in which the potential gradient at the interface reflects the difference in the potentials of "image forces" between the cationic Na(+) and anionic nanoparticles and the specific adsorption of surface charge. The density of water in the large electric field (approximately 10(9)-10(10) Vm) of the transition region and the layering of silica in the diffuse layer is discussed.     

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 a concentrated dispersion of spherical particles covered by an ion-penetrable membrane layer

The electrophoretic behavior of a concentrated dispersion of soft spherical particles is investigated theoretically, taking the effects of double-layer overlapping and double-layer polarization into account. Here, a particle comprises a rigid core and an ion-penetrable layer containing fixed charge, which mimics biocolloids and particles covered by artificial membrane layers. A cell model is adopted to simulate the system under consideration, and a pseudo-spectral method based on Chebyshev polynomials is chosen for the resolution of the governing electrokinetic equations. The influence of the key parameters, including the thickness of the double layer, the concentration of particles, the surface potential of the rigid core of a particle, and the thickness, the amount of fixed charge, and the friction coefficient of the membrane layer of a particle on the electrophoretic behavior of the system under consideration is discussed. We show that while the result for the case of a dispersion containing rigid particles can be recovered as the limiting case of a dispersion containing soft particles, qualitative behaviors that are not present in the former are observed in the latter.     

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.     

Direct calculation of lattice sums. A method to account for the crystal field effects

A method of direct calculation of lattice sums in three-dimensional crystals is reported. The method is based on annihilation of some lowest multipole moments of the unit cell by a redefinition of the unit cell content. As a result, properties of the infinite crystal can be calculated as usual by taking a finite cluster of unit cells, but surrounded by an additional surface layer of a charge density (e.g., a layer of point charges). This charge density distribution produces the electric field approximating that one of the rest of the infinite crystal. The method proposed is easily applicable in the SCFLCAO procedure as well as in any method using a cluster representation for an infinite crystal. The validity of the infinite crystal model for a finite crystal is also discussed.     

Charge heterogeneity of surfaces: mapping and effects on surface forces

The DLVO theory treats the total interaction force between two surfaces in a liquid medium as an arithmetic sum of two components: Lifshitz–van der Waals and electric double layer forces. Despite the success of the DLVO model developed for homogeneous surfaces, a vast majority of surfaces of particles and materials in technological systems are of a heterogeneous nature with a mosaic structure composed of microscopic and sub-microscopic domains of different surface characteristics. In such systems, the heterogeneity of the surface can be more important than the average surface character. Attractions can be stronger, by orders of magnitude, than would be expected from the classical mean-field DLVO model when area-averaged surface charge or potential is employed. Heterogeneity also introduces anisotropy of interactions into colloidal systems, vastly ignored in the past. To detect surface heterogeneities, analytical tools which provide accurate and spatially resolved information about material surface chemistry and potential — particularly at microscopic and sub-microscopic resolutions — are needed.Atomic force microscopy (AFM) offers the opportunity to locally probe not only changes in material surface characteristic but also charges of heterogeneous surfaces through measurements of force–distance curves in electrolyte solutions. Both diffuse-layer charge densities and potentials can be calculated by fitting the experimental data with a DLVO theoretical model. The surface charge characteristics of the heterogeneous substrate as recorded by AFM allow the charge variation to be mapped. Based on the obtained information, computer modeling and simulation can be performed to study the interactions among an ensemble of heterogeneous particles and their collective motions. In this paper, the diffuse-layer charge mapping by the AFM technique is briefly reviewed, and a new Diffuse Interface Field Approach to colloid modeling and simulation is briefly discussed.     

Diffusioosmosis and electroosmosis in a capillary slit with surface charge layers

This study analytically examines the steady diffusioosmotic and electroosmotic flows of an electrolyte solution in a fine capillary slit with each of its inside walls covered by a layer of adsorbed polyelectrolytes. In this solvent-permeable and ion-penetrable surface charge layer, idealized polyelectrolyte segments are assumed to distribute at a uniform density. The electric double layer and the surface charge layer may have arbitrary thicknesses relative to the gap width between the slit walls. The electrostatic potential distribution on a cross section of the slit is obtained by solving the linearized Poisson–Boltzmann equation, which applies to the case of low potentials or low fixed-charge densities. Explicit formulas for the fluid velocity profile due to the imposed electrolyte concentration gradient or electric field through the slit are derived as the solution of a modified Navier–Stokes/Brinkman equation. The results demonstrate that the structure of the surface charge layer can lead to an augmented or a diminished electrokinetic flow (even a reversal in direction of the flow) relative to that in a capillary with bare walls, depending on the characteristics of the capillary, of the surface charge layer, and of the electrolyte solution. For the diffusioosmotic flow with an induced electric field, competition between electroosmosis and chemiosmosis can result in more than one reversal in direction of the flow over a range of the Donnan potential of the adsorbed polyelectrolyte in the capillary.     

Electrokinetic flow in a capillary with a charge-regulating surface polymer layer

An analytical study of the steady electrokinetic flow in a long uniform capillary tube or slit is presented. The inside wall of the capillary is covered by a layer of adsorbed or covalently bound charge-regulating polymer in equilibrium with the ambient electrolyte solution. In this solvent-permeable and ion-penetrable surface polyelectrolyte layer, ionogenic functional groups and frictional segments are assumed to distribute at uniform densities. The electrical potential and space charge density distributions in the cross section of the capillary are obtained by solving the linearized Poisson-Boltzmann equation. The fluid velocity profile due to the application of an electric field and a pressure gradient through the capillary is obtained from the analytical solution of a modified Navier-Stokes/Brinkman equation. Explicit formulas for the electroosmotic velocity, the average fluid velocity and electric current density on the cross section, and the streaming potential in the capillary are also derived. The results demonstrate that the direction of the electroosmotic flow and the magnitudes of the fluid velocity and electric current density are dominated by the fixed charge density inside the surface polymer layer, which is determined by the regulation characteristics such as the dissociation equilibrium constants of the ionogenic functional groups in the surface layer and the concentration of the potential-determining ions in the bulk solution.     

A multipole approximation of the electrostatic potential of molecules

The problem of approximating three-dimensional spatial distributions of quantum-mechanical electrostatic potentials of molecules by analytic potentials on the basis of atomic charges, real dipoles, and atomic multipoles up to quadrupoles inclusive was considered. Real dipole potentials are created by pairs of point charges of opposite signs, and the search for their arrangement in the volume of a molecule is part of the approximation problem. A FitMEP program was developed for the optimization of the parameters of models of the types specified taking into account molecular symmetry. It was shown for the example of several molecules (HF, CO, H₂O, NH₃, CH₄, formaldehyde, methanol, formamide, ethane, cyclopropane, cyclobutane, cyclohexane, tetrahedrane, cubane, adamantane, ethylene, and benzene) that the real dipole and atomic multipole models gave errors in approximated quantum-mechanical electrostatic potential values smaller by one or two orders of magnitude compared with the atomic charge model. The atomic charge model was shown to be virtually inoperative as applied to saturated hydrocarbons. Real dipole models were slightly inferior to atomic multipole models in quality but had all the advantages of the potential of point charges as concerned simplicity and compactness, and their use in potential energy calculations did not require changes in the existing program codes.     

Crystal habit of fcc metal particles controlled by electrode potential in solution

The crystal habit of fcc metal particles formed on an amorphous carbon film electrode in solution at different electrode potentials is discussed. The fcc metal particles have different crystallographic habits depending on applied electrode potential; that is, icosahedral and/or decahedral particles are formed at lower potentials, and fcc single-crystalline or polycrystalline particles at higher potentials. It was found that decahedra and icosahedra of Cu-Au alloy particles are formed in the potential region of underpotential deposition (UPD) of Cu at which only fcc Au single-crystalline particles and Au polycrystalline particles appear. This is attributed to the charge transfer from the UPD Cu ions to the Au overlayer of Cu-Au alloy particles. The formation of decahedral and icosahedral Cu-Au alloy particles depends on the composition of the Cu-Au alloy. On the basis of these results it was deduced that the contraction of the surface lattice of the growing particles is responsible for the formation of icosahedral and decahedral particles. Received: 25 February 1997 / Accepted: 21 April 1997