Electrostatic double-layer interaction between spherical particles inside a rough capillary

Predictions of electrostatic double-layer interaction forces between two similarly charged spherical colloidal particles inside an infinitely long "rough" capillary are presented. A simple model of a rough cylindrical surface is proposed, which assumes the capillary wall to be a periodic function of axial position. The periodic roughness of the wall is characterized by the wavelength and amplitude of the undulations. The electrostatic double-layer interaction force between two spherical particles located axially inside this rough capillary is determined by solving the nonlinear Poisson-Boltzmann equation employing finite element analysis. The effect of surface roughness of the cylindrical enclosure on the interaction force between two particles is extensively studied on the basis of this model. The simulations are carried out for dimensionless amplitudes (amplitude/particle radii) ranging from 0.05 to 0.15 and scaled wavelengths (wavelength/particle radii) ranging from 0.4 to 4.0. The interaction force between the particles is significantly modified by the proximity of the rough capillary wall. Generally, the interaction force for rough capillaries oscillates around the corresponding interaction force in a smooth capillary depending on the magnitudes of the scaled amplitude and wavelength of the roughness. The influence of roughness on the electrostatic interactions becomes more pronounced when the surface potential of the cylinder wall is different from the sphere surface potentials. When the cylinder and the particle surfaces have large potential differences, the axial force experienced by a particle is dominated by the capillary roughness. There are dramatic oscillations of the force, which alternately becomes repulsive and attractive as the particle moves from the crest to the trough of the rough capillary wall. These results suggest that manipulation of colloidal particles in narrow microchannels may be subject to significant force variations owing to the roughness inherent in microfabricated channels etched on metal films.     

Interaction force between two charged plates immersed in a solution of charged particles. Coupling between double layer and depletion forces

When two parallel plates are immersed in a solution of small charged particles, the center of the particles is excluded from a region of thickness D/2 near the plate, where D is their diameter. The approach which Langmuir developed for the double layer repulsion in the presence of an electrolyte with ions of negligible size is extended to the case in which one of the "ions" is a charged particle of finite, relatively small size. A general expression for the force generated between the two charged plates immersed in an electrolyte solution containing relatively small charged particles is derived. In this expression, only the electrical potential at the middle distance between the plates is required to calculate the force. A Poisson-Boltzmann equation which accounts for the volume exclusion of the charged particles in the vicinity of the surface is solved to obtain the electrical potential at the middle between the two plates. Starting from this expression, some results obtained previously for the depletion force acting between two plates or two spheres are rederived. For charged plates immersed in a solution of an electrolyte and charged small particles, the effects of the particle charge, particle charge sign, particle size, and volume fraction of the particles on the force acting between the two plates are examined.     

Dielectric force and relative motion between two spherical particles in electrophoresis

When two particles close to each other are in electrophoretic motion, each particle is under the influence of the nonuniform electric field generated by the other particle. Two particles may attract or repel each other due to the dielectric force, depending on their positions in the nonuniform electric field. In this work, the dielectric interaction and the subsequent relative motion of the two arbitrarily oriented spherical particles are analyzed. The dielectric force is obtained by integrating the Maxwell stress. The result is valid for arbitrary orientations of the particles under the thin electrical-double-layer assumption. The magnitude of the dielectric force is compared to the so-called inertia-induced force, which shows that the dielectric force is normally much greater than the inertia-induced force. The relative velocity of particles is determined by the force balance between the dielectric force and the Stokes drag. The regions of attraction and repulsion are defined. It is shown that a pair of particles eventually aligns parallel to the externally applied electric field, except in the case where the two particles are initially oriented perpendicular to the electric field. A closed-form analytical solution is obtained for the particle trajectory by using the approximate expression for the dielectric force valid for not-too-closely located particles.     

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     

Electric double layer for spherical particles in salt-free concentrated suspensions including ion size effects

The equilibrium electric double layer (EDL) that surrounds colloidal particles is essential for the response of a suspension under a variety of static or alternating external fields. An ideal salt-free suspension is composed of charged colloidal particles and ionic countercharges released by the charging mechanism. Existing macroscopic theoretical models can be improved by incorporating different ionic effects usually neglected in previous mean-field approaches, which are based on the Poisson-Boltzmann equation (PB). The influence of the finite size of the ions seems to be quite promising because it has been shown to predict phenomena like charge reversal, which has been out of the scope of classical PB approximations. In this work we numerically obtain the surface electric potential and the counterion concentration profiles around a charged particle in a concentrated salt-free suspension corrected by the finite size of the counterions. The results show the high importance of such corrections for moderate to high particle charges at every particle volume fraction, especially when a region of closest approach of the counterions to the particle surface is considered. We conclude that finite ion size considerations are obeyed for the development of new theoretical models to study non-equilibrium properties in concentrated colloidal suspensions, particularly salt-free ones with small and highly charged particles.     

Direct measurement of the viscous force between two spherical particles trapped in a thin wetting film

Here we present the first direct measurement of the viscous drag force between two spherical particles of millimeter size trapped in a thin wetting film. Each particle is constrained by the liquid/air interface and the solid substrate. The viscous force is counterbalanced by another known force, the attractive capillary immersion force between identical particles protruding from the film surface. The results of the measurements provide evidence for an increased hydrodynamic force due to a non-Stokesian resistance to the particle motion. Our findings can be applied to the self-assembly of colloidal particles in a two-dimensional array for coating and to the friction between small species and a solid. Received: 19 March 1999 Accepted in revised form: 11 May 1999     

Electrostatic double layer forces in the case of extreme charge regulation

We analyze the interaction forces between charged surfaces across aqueous solutions under the conditions of extreme charge regulation. Under such conditions, interactions may be weaker than those given by the constant potential (CP) boundary conditions. Thermodynamically, even vanishing electrostatic interactions are conceivable. Within the constant regulation approximation, the known results can be extended to this sub-CP regime by adopting regulation parameters outside of the common range. A mean-field lattice model of an adsorbed layer shows that such conditions are most likely found near critical points within the adsorbed layer.     

The wall and multivalent counterion effects on the electrostatic force between like-charged spherical particles confined in a charged pore

The effect of wall confinement (wall charge and wall-sphere separation distance) on the electrostatic force between two charged spheres confined in a long charged pore in symmetric and asymmetric electrolytes have been quantified by solving the nonlinear Poisson-Boltzmann equation (PBE), using adaptive finite elements combined with error minimization techniques. The computed force indicated the strong effect of the wall potential on the reduction of the repulsive force for all type of electrolytes. The influence of the wall effect was reduced when the valence of the electrolyte was increased. A significant reduction in the repulsive force between the two spheres was also observed when the distance between the pore wall and the sphere surface was reduced. A smaller long-range repulsive interaction was observed between spheres when the solutions contained multivalent counterions as compared with a monovalent solution. However, at short ranges of separation distances multivalent counterions increase the electrostatic repulsive force between the spheres. The effect of the dimensionless radius of the spheres on the electrostatic force between them has been determined and a significant reduction observed as the dimensionless radius was reduced.     

Direct measurement of the binding force between microfabricated particles and a planar surface in aqueous solution by force-sensing piezoresistive cantilevers

We propose a force measurement method for evaluating the binding force between microscale flat surfaces in an aqueous solution. Using force-sensing piezoresistive cantilevers with sub-nanonewton force resolution, we have directly measured binding forces between SiO2-SiO2 microcontacts, which were created by gravity-driven random collision between microfabricated SiO2 cylindrical particles and a planar SiO2 substrate in a HCl solution. First, to examine our method we measured the pH dependence of the binding force. The binding forces were 12 and 5.8 nN at pH 1.0 and 2.0, respectively. As the pH increased, the binding force decreased and became zero at pH greater than 3.0. We confirmed that the bindings were based on the van der Waals' (VDW) force at pH 2.0 or less whereas a repulsive double-layer force acted between the surfaces at pH 3.0 or more. Second, the binding forces were categorized into a friction force or an adhesion force between the particles and the substrate. In the measurement, the friction force between the particle and the substrate was measured in the case when the particle slid on the substrate. On the contrary, the adhesion force was measured when the particle came off the substrate. Whether the particle slid or came off depended on the aspect ratio of the particle. We fabricated cylindrical particles with an aspect ratio of 0.03-2.0 and distinguished the friction force from the adhesion force by changing the aspect ratio of the particles. As a result, the friction force per unit contact area between SiO2-SiO2 flat surfaces was found to be 330 pN/microm2 +/- 20% when we used particles with a low aspect ratio (<0.1), and the adhesion force per unit contact area was 90 pN/microm2 +/- 20% for particles with a high aspect ratio (>0.4). For fluidic self-assembly that utilizes microscale surface contact in a liquid, our measurement method is an effective tool for studying and developing systems.     

Electrical double layer around a spherical colloid particle: the excluded volume effect

The influence of the excluded volume effect on both the spatial distribution of ionic species and the electrostatic potential distribution in the neighborhood of a suspended spherical particle is examined on the basis of a modified Poisson-Boltzmann equation, which takes into account the finite ion size by means of a Langmuir-type correction. We find that kappaa (kappa and a being the reciprocal Debye length and the particle radius, respectively) ceases to be a valid parameter for the characterization of the electrical double layer, and that it is necessary to use both parameters kappa and a to characterize adequately the system. We also find that the excluded volume effect considerably increases the surface potential (for a given value of the surface charge density) as compared to the case when ideal ion behavior is assumed. This suggests the use of the particle charge rather than the surface potential in order to characterize the system. Because of this, an approximate equation for the surface charge density of spherical colloid particles, valid for a wide range of system parameter values, is also reported.     

Electric double layer and electrostatic interaction of hydrophobic particles

An hypothesis regarding the impact of water density near hydrophobic surfaces on the electrostatic component of their interaction was offered. A theoretical model of the electric double layer and the interparticle interaction under conditions of the variable density and, consequently, variable dielectric permittivity of water has been developed. It was shown that reduction of the dielectric permittivity near interfaces determined by their hydrophobicity resulted in compression of double electrical layers and weakening of their overlapping. This, in its turn, results in reduction of the electrostatic repulsion of hydrophobic disperse particles as compared with nonhydrophobic ones.     

Electrophoresis of a sphere normal to a plane at arbitrary electrical potential and double layer thickness

The electrophoretic movement of a sphere normal to an uncharged, planar surface is analyzed theoretically, taking the effect of double layer polarization into account. Here, both the surface potential of the particle and the thickness of the double layer surrounding it can be arbitrary. We show that if double layer polarization is neglected, the effect of the surface potential of a particle on its electrophoretic velocity is inappreciable. On the contrary, it becomes significant if double layer polarization is present. However, if the distance between the particle and the surface is sufficiently close, since the hydrodynamic effect dominates, the influence of the surface potential and double layer polarization becomes insignificant.     

The electric double layer capacitance for a hard sphere ion-dipole electrolyte in the mean field approximation

The electric double layer capacitance for a hard sphere ion-dipole system in the neighbourhood of a plane charged wall is calculated in the mean field approximation. To order ka the capacitance predicts the same structural features as the MSA capacitance.     

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.     

Approximate expressions of determining the double layer energy and force of interaction between two charged colloidal spheres

The approximate expressions have been obtained to calculate the electrical double layer energy and force between two spherical colloidal particles based on the improved Derjaguin approximation. Results for identical spheres interacting under constant surface potential, constant surface charge are given. Comparison of present results with numerical results calculated by Carnie and Chan is made. The expressions are found to work quite well for the constant surface potential case, and for the constant charge case, we make correction for the expressions. The results given are satisfactory providedkh0.4.