Constant Surface Potential Research Articles

Electrical potential in a cylindrical double layer: a functional theory approach

Employing an iterative method in functional theory, the electrical potential distribution for the case of a cylindrical surface is solved. Although the analytical result derived is of an iterative nature, the second-order solution is found to be sufficiently accurate under conditions of practical significance. For the case of constant surface potential, the radius and the surface potential of a cylindrical surface can be estimated based on the extreme of the electrical potential distribution. The effects of the key parameters, including the number and the valence of the ions on a surface, the length of a particle, the relative permittivity of the liquid phase, the temperature, and the concentration of electrolyte on the surface potential, are examined. The general behavior of these effects is similar to that for a spherical surface, except that the surface potential of a cylindrical surface is independent of the electrolyte concentration. The present approach is also applicable to the case where a cylindrical surface remains at a constant charge density.

Electrical Interaction between Two Planar, Parallel Dissimilar Surfaces in a General Electrolytic Solution

The electrical interaction between two planar, parallel dissimilar surfaces, which may have different charged conditions arising from different ion-adsorption mechanisms, in an arbitrary electrolytic solution is investigated theoretically. The electrical interaction force and the interaction energy between these surfaces are evaluated, and analytical expressions for various charged conditions under the Debye-Huckel condition are derived. In general, assuming constant surface potential and assuming constant surface charge density lead respectively to the lower and the upper bounds in the electrical interaction energy between two surfaces. We show that assuming a linear relation between surface potential and surface charge density under the Debye-Huckel condition, as is often done in the literature, is appropriate for two planar parallel surfaces only but becomes inadequate for other orientations or nonplanar surfaces. The present approach does not have this limitation.

Validity of the Derjaguin Approximation on Electrostatic Effect in the Frumkin−Derjaguin Approach

The Frumkin−Derjaguin theory relates the macroscopic contact angle of a droplet with the disjoining pressure of the thin film in equilibrium with the droplet. To obtain the analytic expressions of the disjoining pressure, the Derjaguin approximation has been used, combined with the Debye−Hückel theory on the electrical double layer. As a result, the disjoining pressure can be determined without consideration of the overall geometry of the liquid surface. The validity of the Derjaguin approximation has been regarded to be limited to small contact angles, and its validity for large contact angles has rarely been assessed analytically. In this paper, the electrostatic force acting on the meniscus of a droplet (which enforces the droplet to spread) is obtained by integrating the Maxwell stress and the osmotic pressure acting on the liquid surface. The Debye−Hückel theory is employed for direct comparison with the results of the Derjaguin approximation in the two cases of the constant surface potential and the constant surface charge boundary conditions. The present electromechanical approach provides an exact result for the electrostatic contribution to the contact angle for a given model of the electrical double layer. It is shown (for the constant potential case) that if the modification of the interfacial energy at the liquid−surrounding fluid interface by the electrocapillary effect is separately considered, the Derjaguin approximation gives an exact interaction free energy, regardless of the magnitude of the contact angle. For the constant charge case, on the contrary, the interaction free energy derived based on the Derjaguin approximation for prediction of the contact angle is shown to have evident deficiency except for the case of vanishing surface charge density at the liquid surface. Such deficiency originates from the neglect of the contribution of the tangential-stress component.

The effect of many-body interactions on the electrostatic force on a finite chain of spheres confined in a long charged tube

The effect of many-body interactions on the electrostatic force between spheres in a chain of charged spheres confined in a long charged pore has been quantified by solving the non-linear Poisson–Boltzmann equation (PBE), using the adaptive finite element method (AFEM) combined with error minimisation techniques. The equation is solved for conditions of constant surface potential and constant surface charge density. The computed force indicates that the force between any two spheres in a long chain of spheres in a long charged tube does not differ significantly from the force between only two spheres in a tube. This trend is also observed for an unconfined chain of spheres, where the force between two isolated spheres is again very similar to the force between any two spheres in a long unbounded chain. The results also quantify the effect of many-body interactions on the reduction of the repulsion force between the spheres. A significant reduction in the repulsion force between the spheres is observed when the radial distance between the pore wall and the chain of spheres is reduced. The effect of the dimensionless radius of the spheres on the electrostatic force between them has been determined and significant reduction of the force observed as the dimensionless radius is reduced.

Electrostatic Contribution to Line Tension in a Wedge-Shaped Contact Region

In the wetting problems of very small scales, such as in nanotubes and nanoparticles, the contribution of line tension potentially becomes significant. In contrast to the molecular contribution, there rarely exists any literature that systematically considers the electrical contribution to the line tension in wetting. In this paper, the electrical double layer around a wedge-shaped confinement (which represents the region of the three-phase contact of an electrolyte droplet on a charged, or ionizable, substrate) is analyzed. An exact analytical solution for the linearized Poisson−Boltzmann equation is obtained for both the constant surface charge and the constant surface potential conditions. Comprehensive analytical formulas of the line tension are derived. An equation for predicting the contact angle is also shown, considering the dependence of the electrostatic line tension on the contact angle. It is exhibited that the line tension can have either a positive or a negative value depending on the sign o...

Approximate expression for interaction of a two-parallel-plane double layer at constant surface potential and constant surface charge

A fast convergent formula for the interactional energy between two parallel plates at constant surface potential and constant surface charge is derived by a simple method of the series expansion, and the numerical result of only the first five terms of the series is excellent for the dimensionless surface potential of colloidal particle y 0⩽4.

Electrophoresis of a Spheroid in a Spherical Cavity

The boundary effect of electrophoresis is investigated by considering the case of a spheroid in a spherical cavity; both are maintained at a low constant surface potential, and the axis of the former is aligned with the direction of the applied field. We show that the qualitative behavior of the particle depends largely on its position in the cavity and its aspect ratio. In general, the closer the particle to cavity, the smaller its electrophoretic mobility; the reverse is true, however, if it is too close to the cavity. The latter arises from the presence of an attractive force between the particle and the cavity. For a fixed particle volume, the attractive force for a prolate is greater than that for an oblate. Also, charge reversal may occur to the cavity surface when a particle is sufficiently close to it. In this case, it is possible that a particle moves in the opposite direction of that of the applied electric field. For a fixed particle volume, if the wall effect is important, the relative magnitudes of the electrophoretic mobilities for particles of various shapes follow the order prolate < sphere < oblate, and the reverse is true if the wall effect is unimportant.

Evaluation of the “DSPM” model on a titania membrane: measurements of charged and uncharged solute retention, electrokinetic charge, pore size, and water permeability

The DSPM (Donnan steric partitioning pore model) was evaluated in the case of a titania membrane with “nanofiltration properties” by measuring the electrokinetic charge, pore size, and water permeability of the membrane, along with charged and uncharged solute retention. The zeta potential values ( ζ) were determined from measurements of the electrophoretic mobility (EM) of titania powder forming the filtering layer of the membrane. Zeta potential values were converted into membrane volume charge ( X) by assuming two limiting cases: a constant surface charge ( σ s cst) and a constant surface potential ( ψ s cst). The mean pore radius and thickness/porosity ratio of the membrane were determined by permporometry and from water permeability measurements, respectively. Retention measurements were carried out as a function of the permeate volume flux for both neutral solutes (polyethylene glycol PEG of different size) and salts (KCl, MgSO 4, K 2SO 4, and MgCl 2) at various pH values. Ionic retentions showed minimum values near the IEP of the membrane. Retention data were analyzed using the DSPM. Very good agreement was found between the pore radius calculated by the model and that determined by permporometry. X values calculated from fitting retention data using the DSPM were also in satisfactorily agreement with X values calculated from EM measurements assuming a constant surface potential for a large pH range. Furthermore, the DSPM leads to X values ( X DSPM) between those calculated from EM ( X EM) using the two limiting bounds. In other words, X DSPM was higher than X EM assuming ψ s cst at pH values far from the isoelectric point (IEP) and lower than X EM assuming σ s cst. These results show that the DSPM is in qualitative agreement with the charge regulation theory (increase of the pore surface potential and decrease of the pore surface charge density with decreasing the pore size). On the other hand, the thickness/porosity ratio of the membrane calculated from solute retention data differed significantly from that determined from water permeability measurements. Moreover, a single value of Δ x/ A k could not be determined from PEG and salt retention data. This means that the Δ x/ A k parameter loses its physical meaning and includes physical phenomena which are not taken into account by the DSPM. Nevertheless, the model satisfactorily predicted the limiting retention, as this is not influenced by the Δ x/ A k parameter.

Electrostatic Double Layer Force between Two Spherical Particles in a Straight Cylindrical Capillary: Finite Element Analysis

A numerical solution of the nonlinear Poisson−Boltzmann equation based on finite element approximation is developed to assess the interaction force between two identical spherical particles located inside a straight cylindrical capillary. The interaction forces for constant potential and constant charge particle surfaces are calculated for different surface potentials on the cylinder wall, various particle sizes, and different cylinder radii. The interaction force between the spheres was affected significantly by the proximity of the charged cylinder wall, particularly for constant surface potential particles. The influence of the cylinder wall on the particle−particle interaction force was most pronounced when the particle radius was comparable to the cylinder radius. The results suggest that when the cylinder and the particle surfaces are like-charged (bearing the same sign of the charge or surface potential), the electrostatic repulsion between the two spherical colloids is considerably suppressed. Con...

On the Chemical Free Energy of the Electrical Double Layer

The free energy of interaction between colloidal particles due to the overlap of their double layers was traditionally calculated either for systems of arbitrary geometry interacting at constant surface potential or at constant surface charge or for parallel plates interacting under arbitrary surface conditions. An expression is obtained for the change in the chemical contribution to the free energy of the double layers during a general interaction, which allows the calculation (within the Poisson−Boltzmann formalism) of the interaction free energy for systems of arbitrary geometry and surface conditions. The change in chemical free energy depends not only on the values of the surface charge and potential at the final state but also on their values at each distance between infinity and the final state. A simple approximate expression for the change in the chemical free energy contribution is also proposed, which involves only the states at infinite and final distances. Its accuracy is tested for planar and parallel surfaces, with charges generated via the dissociation of surface groups.

Electrophoresis of a Sphere at an Arbitrary Position in a Spherical Cavity

The boundary effect on electrophoresis is of both fundamental and practical significance. Here, we consider the case when a sphere is placed at an arbitrary position in a spherical cavity. We show that the qualitative behavior of the particle depends largely on the charged conditions on both particle surface and cavity surface, and the presence of the latter may have the effect of increasing the mobility of the particle. For example, for the case when both particle and cavity are maintained at constant surface potential, if the particle is close to the cavity surface, charge reversal may occur on the latter. In this case, the closer the particle to cavity surface, the greater its mobility. It is found that the thicker the double layer surrounding the particle and/or the greater the difference between the surface potential of the particle and that of the cavity, the easier it is for the charge reversal to occur.

Effect of chemical stress on diffusion in a hollow cylinder

The effect of chemical stress on diffusion in a hollow cylinder for plane strain and zero axial force has been investigated. Two diffusion processes of constant surface stress and constant surface potential are studied. No matter what the plane strain or zero axial force is the influence of chemical stress on the diffusion process of constant surface potential is similar to that of constant surface stress. Chemical stress enhances both the diffusion coefficient and the concentration. For a given time, the level of concentration becomes lower with a greater ratio of outer radius to inner radius for constant surface potential with given FXf and for constant surface stress with given FXi. F=2EV̄/[9(1−v)RT] where E, V̄, R, and T are the Young’s modulus, partial molal volume, gas constant, and absolute temperature, respectively; Xi and Xf are the initial and final mole fractions at boundary surfaces. The results are also compared with those in thin plates and solid cylinders reported in the literature.

Electrokinetic Flow through an Elliptical Microchannel: Effects of Aspect Ratio and Electrical Boundary Conditions

The electrokinetic flow of an electrolyte solution through an elliptical microchannel is studied theoretically. The system under consideration simulates the flow of a fluid, for example, in vein. We show that, for a constant cross-sectional area, both the electroosmotic volumetric flow rate and the streaming potential increase monotonically with an increase in the aspect ratio, and both the total electric current and the electroviscous effect may exhibit a local minimum as the aspect ratio varies. Also, for a constant average potential on the channel wall, the difference between the results based on three kinds of boundary conditions, which include constant surface charge, constant surface potential, and charge-regulated surface, is inappreciable if the hydraulic diameter is larger than 1 μm.

Strong and Weak Interactions of Colloidal Particles with High Surface Potentials

A simple method for calculating the interaction force and energy per unit area between two dissimilar plates with high potentials at constant surface potential presented. Using Derjaguin's method and the improved Derjaguin' method, the expressions of the interaction free energy between two dissimilar spheres with high surface potentials are derived. These formulae may be divided into two groups: those for the strong interaction and those for the weak interaction. The juncture of strong and weak interactions is at κh=4 for dissimilar plates and at κh=2.8 for spheres The relative error is largest at this point, about 10%.

Influence of Ionic Surfactants on the Drainage Velocity of Thin Liquid Films

The process of thinning of liquid films in the presence of ionic surfactants and background electrolytes is studied. The lubrication approximation is used to obtain the influence of spatial dynamic distribution of the different ions and the electric potential on the mobility of film interfaces and on the additional nonequilibrium part of the electrostatic disjoining pressure. The material properties of the interfaces (Gibbs elasticity and surface viscosity) are taken into account. The bulk and surface diffusivities of ionic species tend to restore the equilibrium and suppress the role of the Gibbs elasticity. The problem is linearized under the assumptions of small deviations of the adsorption and the concentration from their equilibrium values and of small Peclet number. In the case of plane parallel films an analytical solution is obtained. The correctness of the assumptions, widely used in the literature, of constant surface charge, of constant surface potential, and of the quasi-equilibrium approach is discussed. Numerical analysis of the governing equations shows that the ionic surfactants influence the film drainage in two ways. For small surfactants and background electrolyte concentrations the main effect is a reduction in surface mobility, which decelerates the drainage of the film. At high surfactant or salt concentrations the interfaces become tangentially immobile and then dynamic changes in the concentration, adsorption, electric charge, and middle plane potential affect the film thinning due to the change in the nonequilibrium part of the electrostatic disjoining pressure.

Polarizability of conducting and dielectric agglomerates:: theory and experiment

A method has been described for calculation of the dipole moment, when polarized in an electric field, of agglomerates of spherical particles of any specified size and arrangement, and of both conducting and dielectric materials. The polarization of each spherical component of an agglomerate of conducting spheres is represented by a real charge and a dipole moment in a general direction. Within the constraints of zero net charge the problem can be solved self-consistently. As a check on the validity of the method it is possible to plot contours of the constant potential surface for such an agglomerate, and these are shown to coincide closely with the geometric surface. The theoretical predictions are, for the case of linear agglomerates of limited length, verified by direct measurement of force on large scale models in spatially varying electric fields designed so that the polarization force is only weakly dependent on position. The theory is also applied to dielectric agglomerates but in this situation no net charge develops on any of the component spheres. The induced dipole moment, especially in the situation where the agglomerate is aligned parallel to the field, is much smaller than that of a conductor with the same shape. The functional relationship of the induced dipole moment per unit volume of an infinitely long linear agglomerate aligned perpendicular to the field lies between the theoretical form for an isolated sphere and that for an infinitely long cylinder. That for a sheet of spheres lies between the form for an infinite cylinder and an infinite sheet of dielectric.

Electrostatic Interaction and Partitioning of Ion-Penetrable Spheres

Electrostatic interaction between two ion-penetrable spheres near a horizontal plate or in a slit pore is investigated theoretically. The orientation of the line connecting the two particle centers can be arbitrary relative to the plate(s). The electrostatic interaction energy and force on each particle are obtained analytically by the method of images. Emphasis is placed on the effect of the presence of the second particle, compared to the case of a single particle or the case without any plate(s). It is found that the horizontal electrical force on each particle is always repulsive. This repulsive force is enhanced by the plate(s) of constant surface charge density, while it is reduced by the plate(s) of constant surface potential. The electrostatic interaction together with the steric effect is used to determine the partition coefficient for the case of a slit pore, correct to O(C∞), where C∞ is the volume fraction of particles in the bulk solution. The positive correction coefficient is larger for conducting plates than for insulating plates.

Strong Interaction of Colloidal Particles: An Extension of Langmuir's Method

According to Langmuir's suggestion, that is, sinh y ≈ ey/2 in the nonlinear Poisson−Boltzmann equation for high surface potentials of the particles, we derive simple approximate expressions for the repulsive energy and force between two dissimilar plates with constant high surface potentials. By use of Derjaguin's method and the improved Derjaguin method, the expressions of the repulsive energy between two dissimilar spheres with constant high surface potentials have been derived. These formulas are applicable in the regime of a repulsive system and can only be used at κh < 2π, and the accurate location is at ∼κh < 4. These formulas are considerably in agreement with the exact numerical values of the interaction of dissimilar plates given by Devereux and de Bruyn for high surface potentials.

Absolute determination of the Fermi level pinning at the dielectric/semiconductor interface in GaAs based heterostructures

This paper deals with semiconductor/dielectric interface characterization, and in particular with GaAs/dielectric interface. A new electrical method of investigation, constant Fermi level transient spectroscopy (CFTS), is explained. It is based on MIS-H devices, an acronym that we propose to design gated Hall effect devices incorporating a 2DEG channel in an otherwise depleted heterostructure. The main characteristic of CFTS is that transients are obtained under constant potential conditions in the semiconductor and constant surface potential. This allows a rigorous theoretical analysis of capture/emission process by the interface states. This method is comparable in its spirit to feedback methods such as constant-capacitance DLTS (CC-DLTS). The results obtained prove, firstly, the validity of the CFTS and, secondly, that MIS-H samples can be used both for the determination of D it and of its energetic position within the bandgap, and thus with a good accuracy and a high level of confidence despite difficulties usually met with GaAs/dielectric interfaces.

A Model for Calculating Electrostatic Interactions between Colloidal Particles of Arbitrary Surface Topology

A numerical model for calculating the electrostatic interaction between two particles of arbitrary shape and topology is described. A key feature of the model is a generalized discretization program, capable of simulating any desired analytical shape as a set of flat, triangular elements. The relative sizes of the elements are adjusted using a density function to better match the desired shape and the spatial variation of the electrical surface properties on each particle. The distribution of either surface potential or surface charge density is then calculated using a boundary element approach to solve the linearized Poisson–Boltzmann equation. Example interaction energy profiles are calculated for three different types of roughness—bumps, pits, and surface waves. It is found that the interaction energy between rough particles remains different from that between two equivalent smooth spheres at all separations, even for gap widths much larger than either the solution Debye length or the characteristic roughness size. This behavior at large gap widths arises from the nature of the decay of the electric potential away from each particle. In addition, the magnitude of the roughness effect is found to depend greatly on the size and shape of the nonuniformity as well as the electrostatic boundary conditions. For example, for a sphere containing asperities of height equal to 0.2 times the particle radius, the interaction energy can be as much as 50% greater than that between two equivalent spheres under the condition of constant surface potential. At constant surface charge density, the ratio of the interaction energies between rough and smooth spheres was found to either diverge or become zero as contact between the two particles is approached, depending on the nature of the roughness. Changes of this magnitude could clearly have a substantial impact on the stability behavior of a dispersion of such particles.