Discrete charge effects on the Donnan potential and surface potential of a soft particle

The Donnan potential and surface potential of soft particles (i.e., polyelectrolyte-coated hard particles) in an electrolyte solution play an essential role in their electric behaviors. These potentials are usually derived via a continuum model in which fixed charges inside the surface layer are distributed with a continuous charge density. In this paper, for a plate-like soft particle consisting of a cubic lattice of fixed point charges, on the basis of the linearized Poisson–Boltzmann equation, we derive expressions for the electric potential distribution in the regions inside and outside the surface layer. This expression is given in terms of a sum of the screened Coulomb potentials produced by the point charges within the surface layer. We show that the deviation of the results of the discrete charge model from those of the continuous charge model becomes significant as the ratio of the lattice spacing to the Debye length becomes large.     

Heterocoagulation in Bidisperse Magnetorheological Fluids Containing Like-Charged Nanosized and Micron Particles Without a Magnetic Field

Based on extended Derjaguin–Landau–Verwey–Overbeek theory, a heterocoagulation model is proposed for magnetorheological (MR) fluids containing like-charged nanosized and micron particles without a magnetic field. This model considers three major interactions, namely van der Waals attraction, electrical double layer (EDL) interaction, and steric repulsion. The EDL interaction has been identified as the most important factor. The surface potential ratio β (ψ₂/ψ₁) between two dissimilar particles with like charge plays an important role in controlling the change of EDL interaction. At higher β ratios, the EDL interaction becomes attractive when the surface separation falls within a certain range. Two groups of MR fluid samples have been used in experimental studies based on electroacoustic measurements. In the first group, the ratio and the sum of the zeta potentials between carbonyl iron particles and ceria were 4 and ?734.57 mV, respectively. In the second group, these parameters were 1.38 and ?108.17 mV, respectively. The experimental results suggested that the second group did not undergo heterocoagulation, whereas the first group showed extensive heterocoagulation. The difference in surface potentials between particles of two different phases has been found to be critical for determining the state of dispersion or heterocoagulation in concentrated MR fluid systems.     

Molecular dynamics simulation of optically trapped colloidal particles at an oil-water interface

Using molecular dynamics simulations, we calculate the net force on a colloidal particle trapped by an optical tweezer and confined within a particle monolayer which is in motion relative to the trapped particle. The calculations are compared with recent experimental data on polystyrene particles located at an oil-water interface. Good agreement between theory and experiment is obtained over the investigated range of lattice constants for an interaction mechanism between the polystyrene particles which is dominated by an effective dipole-dipole potential. The assumed interaction mechanism is consistent with the formation of surface charge dipoles at the particle-oil interface due to the dissociaton of the hydrophilic sulfate headgroups at the surface of the polystyrene particles. A possible physical mechanism for the formation of the surface charge dipoles, involving a diffuse cloud of fully hydrated counterions, is described, and the fraction of surface groups contributing to the formation of surface charge dipoles is estimated to be of the order of 10(-1) for the present system.     

Electrostatic interaction between a cylinder and a planar surface

 An exact analytical expression for the potential energy of the electrostatic interaction between a plate-like particle 1 and a cylindrical particle 2 of radius a ₂ immersed in an electrolyte solution of Debye–Hückel parameter κ is derived on the basis of the linearized Poisson–Boltzmann equation without recourse to Derjaguin's approximation. Both particles may have either constant surface potential or constant surface charge density. In the limit of κa ₂→0, in particular, the interaction between a plate with zero surface charge density and a cylinder having constant surface charge density becomes identical to the usual image interaction between a line charge (a charged rod of infinitesimal thickness) and an uncharged plate. Received: 22 September 1998  Accepted in revised form: 27 January 1999     

Electrostatic interaction of ion-penetrable membranes. Effects of ionic solubility

An analytic, approximate expression for the electrostatic interaction between two membranes immersed in an electrolyte solution is derived on the basis of a simple membrane model. This model assumes that the membrane has a surface layer in which charged groups are uniformly distributed and that electrolyte ions can penetrate into the surface layer. The partition coefficients of cations and anions between the solution and the surface layer, which are related to their solubilities in the surface layer, may be different from unity.The electrostatic interaction depends on the ionic partition coefficients between the solution and the surface layer, and the relative permittivity of the surface layer, as well as on the membrane-fixed charges, the electrolyte concentration in the solution, and the surface layer thickness. It is shown, in particular, that even where the charge layer has no fixed charges, the electrostatic interaction force can be produced if the solubilities of cations and anions are different in the surface layer.     

Localized adsorption at solid-liquid interfaces: the sticky site-hard wall model

The interface between a structured solid and a liquid is modelled by a flat surface with sticky sites placed on a regular lattice. It is shown that this three-dimensional adsorption model can be mapped into a two-dimensional lattice-gas model with the interaction energies of the particles on the sites related to the potential of mean force in the inhomogeneous fluid. In the simple case of a liquid of hard spheres, the first layer of liquid at contact with the solid exhibits an order-disorder phase transition and the total surface excess shows a maximum as the bulk density increases.     

Electrostatic double layer force between a sphere and a planar substrate in the presence of previously deposited spherical particles

A finite element model of the electrostatic double layer interaction between an approaching colloidal particle and a small region of a charged planar surface containing four previously deposited particles is presented. The electrostatic interaction force experienced by the approaching particle is obtained by solving the Poisson-Boltzmann equation with appropriate boundary conditions representing this complex geometry. The interaction forces obtained from the detailed three-dimensional finite element simulations suggest that for the many-body scenario addressed here, the electrostatic double layer repulsion experienced by the approaching particle is less than the corresponding sphere-plate interaction due to the presence of the previously deposited particles. The reduction in force is quite significant when the screening length of the electric double layer becomes comparable to the particle radius (kappaa approximately 1). The results also suggest that the commonly used technique of pairwise addition of binary interactions can grossly overestimate the net electrostatic double layer interaction forces in such situations. The simulation methodology presented here can form a basis for investigating the influence of several previously deposited particles on the electrostatic repulsion experienced by a particle during deposition onto a substrate.     

Particle deposition onto micropatterned charge heterogeneous substrates: trajectory analysis

A trajectory analysis of particles near a micropatterned charged substrate under radial impinging jet flow conditions is presented to investigate the effect of surface charge heterogeneity on particle trajectory and deposition efficiency. The surface charge heterogeneity is modeled as concentric bands of specified width and pitch having positive and negative surface potentials. The flow distribution is obtained using finite element analysis of the governing Navier-Stokes equations. The particle trajectory analysis takes into consideration the hydrodynamic interactions, gravity, van der Waals and electrostatic double layer interactions. The presence of surface charge heterogeneity on the substrate gives rise to an oscillating particle trajectory near the collector surface due to repulsive and attractive forces. As a result of the coupled effects of hydrodynamic and colloidal forces, the particle trajectories and deposition efficiencies are increasingly affected by surface charge heterogeneity as one moves radially away from the stagnation point. The results indicate that it is possible to render collectors with up to 50% favorable surface fraction completely unfavorable by modifying the ratio of the radial to normal fluid velocity. Utilizing the real favorable area fraction of the collector, the patch model expression for calculating the deposition efficiency is modified for impinging jet flow geometry.     

SANS study of the interactions among DNA, a cationic surfactant, and polystyrene latex particles

The compaction of DNA by a cationic surfactant both in the bulk and adsorbed on the surface of latex particles was followed for the first time by SANS. In the bulk, a decrease in the overall size of the DNA coil in the presence of the cationic surfactant was observed at a negative-to-positive charge ratio far below the phase separation region, at a negative-to-positive charge ratio of 18. Additionally, large surfactant aggregates seem to form within the DNA-surfactant complex. On the other hand, DNA adsorbs onto the surface of latex particles, forming a thick layer, as evidenced by the fitting of the SANS data to a core-shell form factor. Addition of a cationic surfactant to the DNA-coated latex particles at a negative-to-positive charge ratio of 38 induces a slight decrease in the size of the particle layer, where the cationic surfactant is evenly distributed within the adsorbed layer. A further decrease of the negative-to-positive charge ratio to 18 induces a dramatic change in the SANS data that suggests significant compaction of the adsorbed layer and the formation of large surfactant aggregates, similar to those detected in the bulk.     

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.     

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.     

Application of the method of images on electrostatic phenomena in aqueous Al2O3 and ZrO2 suspensions

A new solution for the Poisson equation for the diffuse part of the double layer around spherical particles will be presented. The numerical results are compared with the solution of the well-known DLVO theory. The range of the diffuse layer differs considerably in the two theories. Also, the inconsistent representation of the surface and diffuse layer charge in the DLVO theory do not occur in the new theory. Experimental zeta potential measurements were used to determine the charge of colloidal Al2O3 and ZrO2 particles. It is shown that the calculated charge can be interpreted as a superposition of independent H+ and OH- adsorption isotherms. The corresponding Langmuir adsorption isotherms are taken to model the zeta potential dependence on pH. In the vicinity of the isoelectric point the model fits well with the experimental data, but at higher ion concentrations considerable deviations occur. The deviations are discussed. Furthermore, the numerical results for the run of the potential in the diffuse part of the double layer were used to determine the electrostatic interaction potential between the particles in correlation with the zeta potential measurements. The corresponding total interaction potentials, including the van der Waals attraction, were taken to calculate the coagulation half-life for a suspension with a particle loading of 2 vol%. It is shown that stability against coagulation is maintained for Al2O3 particles in the pH region between 3.3 and 7 and for ZrO2 only around pH 5. Stability against flocculation can be achieved in the pH regime between 4.5 and 7 for Al2O3, while the examined ZrO2 particles are not stable against flocculation in aqueous suspensions.     

Electrical double-layer effects on the deposition of zeolite A on surfaces

Zeolite particles formed from an aluminosilicate solution possess a negative surface charge due to the substitution of aluminum atoms into a SiO4 tetrahedral structure making it difficult to form a continuous layer in solution. The particle interactions with surfaces and each other can be studied using the Derjaguin-Landau-Verwey-Overbeek (DLVO) theory. The interaction energy between zeolite-zeolite and zeolite-substrate on various materials can be estimated in this fashion. The zeolite LTA particles show a stronger repulsion interaction on all substrates and on the each other as compared to the ZSM-5 particles. This repulsive energy also increases as the particles size increases. This results in the formation of conglomerate in the solution rather than forming an adhered layer on the substrate.     

Non-equilibrium electric surface phenomena

Non-equilibrium aspects of traditional electrokinetic phenomena (electrophoresis, electroosmosis, streaming potential, sedimentation potential), electrostatic interaction of particles and new electrokinetic phenomena are considered. The significance of non-equilibrium electric surface phenomena for many major areas of modern colloid science (characterization of colloids, membrane science, transport phenomena and separation, particle interaction and coagulation) is established.The study of non-equilibrium electric surface phenomena is connected with the validation of the standard electrokinetic model (SEM), the development of a non-standard model and the development of an extensive programme of disperse system characterization based on integrated electrokinetic investigations. Experimental and theoretical studies of systems with a smooth, non-porous impermeable surface (mica in Anderson's experiments, and quartz microcapillaries with a molecule-smooth surface in Churaev's experiments) have shown that usually there are no significant difficulties in interpreting electrokinetic investigations despite the possible anomaly in the water structure near the surface and the possibility of maximum shear stress (yield stress), i.e. the anomalous viscosity and decreased dissolving power with respect to ions. However, systems which do not satisfy the conditions of the SEM are widely distributed, owing to the porosity, roughness or permeability of the boundary layer of the surface of the solid body which simultaneously belongs to the solid and liquid phases. In this layer, enclosed between the outer Helmholtz plane and the slipping plane, the motion of the liquid strongly slows down and the tangential flow of ions is characterized purely by the mobility which is close to the normal. Thus, a general property of a non-standard electrokinetic model is the presence of an anomalous (additional) surface conductivity in excess of the surface conductivity determined according to Bikerman's equation based on the ζ -potential alone.Confidence in modelling the electrokinetic phenomena has grown with the development of methods for modifying the surface such that its properties approach those of the SEM (Bijsterbosch and co-workers; Saville and co-workers).Extension of the particle characterization concept requires the measurement of both the mobile charge and the electrokinetic charge and from this an estimate of the thickness of the additional conductivity zone can be made. With the additional measurement of a titratable charge, it is possible to estimate the ion distribution between the dense and diffuse parts of the double layer (DL) and to estimate the decreased mobility of ions in the Stern layer or in the immobilized part of the DL.Quantitative laws governing the interaction of particles and corresponding to the non-standard model substantially differ from the traditional laws described by the DVLO theory as applied to the SEM. This is also true for adsorption properties which are characterized without sufficient reason by means of the ζ-potential. Therefore both the development of models of interaction and adsorption of ions, allowing for the non-standard electrokinetic model, and the extension of the particle characterization programme to integrated investigations of electric surface phenomena are required.Further generalization of the theory of electrokinetic phenomena is achieved. In addition to the surface charge another variety of surface force can be the origin of the electrokinetic phenomena.     

Substantivity and After‐Rinsing Hair Growth Promotion of Cationic Microparticles of Minoxidil

Cationic minoxidil (MXD) particles were prepared by passing a suspension containing MXD and distearyldimethylammonium chloride (DSDMAC) through a high pressure microfluidizer, operating at 500 bar to 1000 bar. The size of the particles is a few micrometers and the surface charge was+42 mV to+44 mV. The cationic MXD particles were included in a hair cleansing shampoo, of which a major detergent is sodium lauryl ether sulfate (SLES). On an UV spectrophotometer, the turbidity of the cationic MXD particles suspension increased with increasing amount of the anionic surfactant. At the same time, the surface charge of the cationic MXD particles was neutralized around equi‐molar ratio of SLES/DSDMAC, and the value became negative in the excess amount of SLES. These mean that DSDMAC adsorbed on MXD particles is complexed with SLES in the hair shampoo by an ionic interaction. Interestingly, even though the MXD particles contained in a shampoo exhibited negative surface charge, the skin‐retentive amount of MXD was appreciable and the after‐rinsing hair growth promotion effect was remarkable. One of possible mechanisms is that SLES would be desorbed from the complexed MXD particles during the rinsing step, and the charge of the particles might change from a positive value to a negative one, leading to an ionic interaction between the cationic particles and negatively charged skin.     

The conditions governing the specific adsorption of counter-ions in the interaction of two parallel plate-like colloidal particles

Frens and Overbeek have proposed that during the Brownian collision of two colloidal particles in a hydrophobic sol, the surface charge density due to potential-determining (p.d.) ions remains virtually unchanged. It is argued here that the cause of this behaviour is the low concentration of p.d. ions in the diffuse layer. However, equilibrium can be maintained with respect to counter-ions adsorbed into the Stern region from the supporting electrolyte, because the concentration of such electrolyte in the dispersion medium is considerably greater than that of p.d. ions.A general expression is quoted from earlier work for the electric double layer interaction between two parallel plate-like particles in the case where surface charge due to p.d. ions is fixed, but where counter-ions adsorbed into the Stern region can equilibrate with ions of the same species in the diffuse layer. Incorporating discreteness-of-charge and ion-size effects into the adsorption isotherm of the counter-ions, the double layer interaction energy of the two plates is calculated at contact of the two outer Helmholtz planes (o.h.p.'s). It is shown that although this energy exceeds the classical expression obtained by assuming the potential at the o.h.p. to be independent of plate separation, it remains finite.     

Exact solution for polarons on the anharmonic lattice and charge transfer in biopolymers

The possible mechanism of charge transfer via polarons to long distances in biopolymers was considered. A set of accurately integrable equations was obtained for a lattice with a potential of interaction of neighboring particles with cubic nonlinearity and at a certain ratio of the parameters of the problem. This system has many-soliton solutions, while the polaron is a one-soliton solution. Numerical modeling proved high stability of the obtained solutions. A new class of stable polarons with several peaks were detected for arbitrary values of the parameters. The behavior of polarons on a lattice with defects was studied by numerical methods. The applicability of the results to rationalization of recent experiments on the effective charge transfer in biopolymers was analyzed.     

Dynamics of electrical double layer formation at a charged solid surface

A theoretical study of the dynamics of electrical double layer formation near a charged solid surface is presented. A microscopic expression for the time dependent inhomogeneous charge density of an ionic solution next to a newly charged surface is derived by using linear response theory and molecular hydrodynamics. The presence of interionic correlations is included through ionic structure factors. The rate of electrical double layer formation is found to depend rather strongly on ion concentration and on the dielectric constant of the medium. It is also found that the formation of double layer becomes slower with increase in distance from the charged surface.     

Recent study on counterion-mediated attraction between colloidal particles

Recent experimental results were reviewed. The 1D- and 2D-USAXS studies gave higher orders of Bragg diffraction for single crystals of colloidal silica particles, allowing more accurate determinations of the lattice constant, lattice symmetry, and direction. The closest interparticle spacing thus determined was confirmed to be smaller than the average spacing. The most closely packed planes ((110) planes for bcc) of negatively charged particles were found to be parallel to the likewise negatively charged capillary surface, inconsistently with the accepted double layer interaction theory but consistently with a recent experimental finding of positive adsorption. Shaking caused disruption of the single crystals but newly formed microcrystals retained the lattice constant and the preference of the (110) planes. The liquid-solid-liquid transition, a re-entrant phase transition, was found for silica particles and latex particles at given particle volume fraction and salt concentration, when the charge density of particles was varied. It was demonstrated that the purely repulsive Yukawa potential and the concept of renormalized charge cannot account for the re-entrant behavior. The Monte-Carlo simulation using the Sogami potential, which contains short-range repulsion and long-range attraction, was found to account for the fcc–bcc transition, which was earlier claimed to be explainable only by the Yukawa potential. Furthermore, the homogeneous-inhomogeneous phase transition and void formation could be accounted for by the simulation using the Sogami potential; the Yukawa potential could not reproduce the experiments. Attention was drawn to the experimental conditions in direct measurements of interparticle forces; only short interparticle distance and low charge density particles were covered, which make it practically impossible to detect the long-range counterion-mediated attraction. It is hoped that, by technical improvements, these shortcomings may be made up and quantitative argument become possible on the attraction.