Mathematical model of electromigration allowing the deviation from electroneutrality

The structure of the double layer on the boundary between solid and liquid phases is described by various models, of which the Stern–Gouy–Chapman model is still commonly accepted. Generally, the solid phase is charged, which also causes the distribution of the electric charge in the adjacent diffuse layer in the liquid phase. We propose a new mathematical model of electromigration considering the high deviation from electroneutrality in the diffuse layer of the double layer when the liquid phase is composed of solution of weak multivalent electrolytes of any valence and of any complexity. The mathematical model joins together the Poisson equation, the continuity equation for electric charge, the mass continuity equations, and the modified G-function. The model is able to calculate the volume charge density, electric potential, and concentration profiles of all ionic forms of all electrolytes in the diffuse part of the double layer, which consequently enables to calculate conductivity, pH, and deviation from electroneutrality. The model can easily be implemented into the numerical simulation software such as Comsol. Its outcome is demonstrated by the numerical simulation of the double layer composed of a charged silica surface and an adjacent liquid solution composed of weak multivalent electrolytes. The validity of the model is not limited only to the diffuse part of the double layer but is valid for electromigration of electrolytes in general.     

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.     

Electrokinetic boundary condition compatible with the Onsager reciprocal relation in the thin double layer approximation

The boundary condition, which has been used in the conventional electrokinetic calculation in the thin double layer approximation, has a flaw that it does not give the Onsager reciprocal relation for the sedimentation of charged particle. We propose a new boundary condition, which satisfies the reciprocal relation, and derive a general form for the mobility matrix for the motion of a charged particle under the action of external force, torque, and electric field. We then calculate the mobility matrix explicitly for homogeneously charged spherical particle and discuss the effect of the surface slippage and the surface conductivity on the particle mobility and electric conductivity.     

Numerical study of the equilibrium properties of suspended particles surrounded by a permeable membrane with adsorbed charges

The network simulation method is used to calculate the electrostatic potential distribution for suspended spherical particles made of a charged core surrounded by a permeable membrane with adsorbed charges. The structure of the equilibrium diffuse double layers on both sides of the membrane-electrolyte solution interface is analyzed considering an anion adsorption process described by a Langmuir-type isotherm. It is shown that the thickness of the double layer in the membrane strongly depends on the adsorption constant, while it is almost independent of this constant in the electrolyte solution. The evolution of the electric potential on the core as a function of the electrolyte concentration is also analyzed.     

Dispersion of charged solute in charged micro‐ and nanochannel with reversible sorption

We study dispersion of a charged solute in a charged micro‐ and nanochannel with reversible sorption and derive an analytical solution for mass fraction in the fluid, transport velocity and dispersion coefficient. Electrical double layer formed on the charged surface gives rise to a charge‐dependent solute transport by modifying the transverse distribution of the solute. We discuss the effect of sorption and electrical double layer on solute transport and show that the coupling between sorption and electrical double layer gives rise to charge‐dependent transport even for a thin double layer. However, in this case, it can be reduced to a simple non‐charge‐dependent case by introducing the intrinsic sorption equilibrium constant.     

The electrokinetic behavior of charged non-spherical colloids

The response of charged colloids to electric fields is determined by combined phenomena occurring first in the electric double layer to then develop into long-range perturbations of ion concentration, local fields, and solvent flows. When particles are non-spherical, the loss of symmetry affects the short- and long-ranged processes modifying their behavior as observed through their electrophoretic mobility, dielectric permittivity, and electro-optical response. Recent measurements and theoretical developments have revealed phenomena characteristic for non-spherical particles, such as the doubling of the relaxations in the dielectric spectra, the appearance of torque-inducing hydrodynamic flows, and the anomalous perpendicular alignment. In this article we discuss in a unifying frame the recent experimental and theoretical progresses about the electrokinetic behavior of charged non-spherical colloids.     

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.     

Ion-Exchange Chromatographic Supports Obtained by Formation of Polyelectrolyte Multi-Layers for the Separation of Proteins

Summary High performance liquid chromatography (HPLC) was used to study the mechanism of formation of polyelectrolyte multilayers on porous silicas. The coatings were produced by alternating the adsorption of positively and negatively charged polymers. The stationary phases formed by adsorbing a single layer, double layers and triple layers were tested by studying the elution behavior of model proteins. The double polymer coating was achieved by adsorbing first a polycation such as hexadimethrine bromide (HB) on the HPLC silica support and then a polyanion such as dextran sulfate (DS) on the cationic layer formed. The retention properties of this support are mainly those of a cation exchanger as the negatively charged proteins were strongly retained while positively charged ones were weakly adsorbed. This work demonstrated the importance of the first underlying layer as the retention behavior of proteins was greatly affected by the properties of this coating. The triple polymer coating was achieved by adsorbing the polycation (HB) on the double layer coating (HB-DS). Its retention behavior was that of an anion exchange support. The HB-DS stationary phase displayed good chromatographic performances, with an adsorbed layer relatively stable. The polyelectrolyte multilayer coating procedure was useful to easily synthesize cation-exchange supports for the separation of basic proteins.     

Double layer in ionic liquids: The nature of the camel shape of capacitance

Monte Carlo simulations of a model ionic liquid show that if ions have charged heads and neutral counterparts, the latter give rise to the camel shape of the voltage dependence of the double layer capacitance. Neutral ‘tails’ of ions play the role of latent voids that can be replaced by charged groups via rotations and translations of ions. This provides extra degrees of freedom for the field-induced charge rearrangements in the double layer which results in the peculiar double-hump capacitance profile.     

The interplay of diffusional and electrophoretic transport mechanisms of charged solutes in the liquid film surrounding charged nonporous adsorbent particles employed in finite bath adsorption systems

A model that describes the diffusive and electrophoretic mass transport of the cation and anion species of a buffer electrolyte and of a charged adsorbate in the liquid film surrounding nonporous adsorbent particles in a finite bath adsorption system, in which adsorption of the charged adsorbate onto the charged surface of the nonporous particles occurs, is constructed and solved. The dynamic behavior of the mechanisms of this model explicitly demonstrates (a) the interplay between the diffusive and electrophoretic molar fluxes of the charged adsorbate and of the species of the buffer electrolyte in the liquid film surrounding the nonporous adsorbent particles, (b) the significant effect that the functioning of the electrical double layer has on the transport of the charged species and on the adsorption of the charged adsorbate, and (c) the substantial effect that the dynamic behavior of the surface charge density has on the functioning of the electrical double layer. It is found that at equilibrium, the value of the concentration of the charged adsorbate in the fluid layer adjacent to the surface of the adsorbent particles is significantly greater than the value of the concentration of the adsorbate in the finite bath, while, of course, the net molar flux of the charged adsorbate in the liquid film is equal to zero at equilibrium. This result is very different than that obtained from the conventional model that is currently used to describe the transport of a charged adsorbate in the liquid film for systems involving the adsorption of a charged adsorbate onto the charged surface of nonporous adsorbent particles; the conventional model (i) does not consider the existence of an electrical double layer, (ii) assumes that the transport of the charged adsorbate occurs only by diffusion in the liquid film, and (iii) causes at equilibrium the value of the charged adsorbate in the liquid layer adjacent to the surface of the particles to become equal to the value of the concentration of the charged adsorbate in the liquid of the finite bath. Furthermore, it was found that a maximum can occur in the dynamic behavior of the concentration of the adsorbate in the adsorbed phase when the value of the free molecular diffusion coefficient of the adsorbate is relatively large, because the increased magnitude of the synergistic interplay between the diffusive and electrophoretic molar fluxes of the adsorbate in the liquid film allows the adsorbate to accumulate (to be entrapped) in the liquid layer adjacent to the surface of the adsorbent particles faster than the concentrations of the electrolyte species, whose net molar fluxes are significantly hindered due to their opposing diffusive and electrophoretic molar fluxes, can adjust to account for the change in the surface charge density of the particles that arises from the adsorption of the charged adsorbate. The results presented in this work also have significant implications in finite bath adsorption systems involving the adsorption of a charged adsorbate onto the surface of the pores of charged porous adsorbent particles, because the diffusion and the electrophoretic migration of the charged solutes (cations, anions, and charged adsorbate) in the pores of the adsorbent particles will depend on the dynamic concentration profiles of the charged solutes in the liquid film surrounding the charged porous adsorbent particles. The results of the present work are also used to illustrate how the functioning of the electrical double layer could contribute to the development of inner radial humps (concentration rings) in the concentration of the adsorbate in the adsorbed phase of charged porous adsorbent particles.     

Lyotropic effects at the silver iodide/electrolyte solution interface

The lyotropic effect in the flocculation of a negatively charged AgI sol is explained on the basis of a double layer model without specific adsorption of cations. The constant potential model for the double layer interaction gives better results than the constant charge model. The theoretical results indicate furthermore that repeptization of the flocculated sol is possible.     

Electrophoresis of a rigid sphere in a Carreau fluid normal to a large charged disk

The electrophoresis of a rigid sphere in a Carreau fluid normal to a large disk is analyzed theoretically under the conditions of low surface potential and weak applied electric field. Previous analyses are extended to the case where a disk can be charged, and a more realistic electrostatic force formula is applied. We show that the qualitative behavior of a sphere depends largely on its distance from a disk, the thickness of double layer, and the nature of a fluid. In general, the presence of a disk has the effect of increasing the conventional hydrodynamic drag on a sphere, and a decrease in the thickness of the double layer surrounding a sphere has the effect of enhancing the shear-thinning effect. However, this might not be the case if a sphere is uncharged and a disk is charged, where the osmotic pressure field and the induced charge on the sphere surface can be significant. The shear-thinning effect is important only if the thickness of double layer is sufficiently thick. This result can play a significant role in practice such as in electrophoretic deposition, where the deposition electrode is charged and the fluid medium is usually of shearing-thinning nature.     

Finite volume solution of the modified Poisson-Boltzmann equation for two colloidal particles

The double layer forces between spherical colloidal particles, according to the Poisson-Boltzmann (PB) equation, have been accurately calculated in the literature. The classical PB equation takes into account only the electrostatic interactions, which play a significant role in colloid science. However, there are at, and above, biological salt concentrations other non-electrostatic ion specific forces acting that are ignored in such modelling. In this paper, the electrostatic potential profile and the concentration profile of co-ions and counterions near charged surfaces are calculated. These results are obtained by solving the classical PB equation and a modified PB equation in bispherical coordinates, taking into account the van der Waals dispersion interactions between the ions and both surfaces. Once the electrostatic potential is known we calculate the double layer force between two charged spheres. This is the first paper that solves the modified PB equation in bispherical coordinates. It is also the first time that the finite volume method is used to solve the PB equation in bispherical coordinates. This method divides the calculation domain into a certain number of sub-domains, where the physical law of conservation is valid, and can be readily implemented. The finite volume method is implemented for several geometries and when it is applied to solve PB equations presents low computational cost. The proposed method was validated by comparing the numerical results for the classical PB calculations with previous results reported in the literature. New numerical results using the modified PB equation successfully predicted the ion specificity commonly observed experimentally.     

Sedimentation of a composite particle in a spherical cavity

The boundary effect on the sedimentation of a colloidal particle is investigated theoretically by considering a composite sphere, which comprises a rigid core and an ion-penetrable membrane layer, in a spherical cavity. A pseudo-spectral method is adopted to solve the governing electrokinetic equations, and the influences of the key parameters on the sedimentation behavior of a particle are discussed. We show that both the qualitative and quantitative behaviors of a particle are influenced significantly by the presence of the membrane layer. For example, if the membrane layer is either free of fixed charge or positively charged and the surface potential of the rigid core is sufficiently high, the sedimentation velocity has a local minimum and the sedimentation potential has a local maximum as the thickness of the double layer varies. These local extrema are not observed when the membrane layer is negatively charged. If the double layer is thin, the influence of the fixed charge in the membrane layer on the sedimentation potential is inappreciable.     

Determination of the dynamic electrophoretic mobility of a spherical colloidal particle through a novel numerical solution of the electrokinetic equations

The standard equations developed to describe the electrophoretic motion of a charged particle immersed in an electrolyte subjected to an oscillating electric field are solved numerically with a new technique suitable for stiff systems. The focus of this work is to use this solution to determine the dynamic particle mobility, one of several quantities that can be extracted from these equations. This solution is valid from low frequencies to indefinitely high frequencies and has no restriction on zeta potential, double-layer thickness, or electrolyte composition. The solution has been used to calculate the dynamic electrophoretic mobility of a particle for a wide range of double-layer thicknesses and zeta potentials. The solution agrees with analytic approximations obtained previously by other authors under the conditions of a thin double layer and low zeta potential. The results are also consistent with calculations valid at frequencies where the ion diffusion length extends a significant distance beyond the double layer as obtained by another numerical technique.     

The effects of unequal ionic sizes for an electrolyte in a charged slit

The modified Gouy-Chapman (MGC) theory has been used to study the electrical double layer near two charged plates immersed in a model electrolyte. The effects of assigning to the cations and anions different distances of closest approach to the charged surfaces are examined. The dependence of overcharging and charge reversal on the system parameters such as concentration, ion size and valence, is investigated both inside and outside the charged slit.     

Screening and separation of charges in microscale devices: complete planar solution of the Poisson-Boltzmann equation

The Poisson-Boltzmann (PB) equation is widely used to calculate the interaction between electric potential and the distribution of charged species. In the case of a symmetrical electrolyte in planar geometry, the Gouy-Chapman (GC) solution is generally presented as the analytical solution of the PB equation. However, we demonstrate here that this GC solution assumes the presence of a bulk region with zero electric field, which is not justified in microdevices. In order to extend the range of validity, we obtain here the complete numerical solution of the planar PB equation, supported with analytical approximations. For low applied voltages, it agrees with the GC solution. Here, the electric double layers fully absorb the applied voltage such that a region appears where the electric field is screened. For higher voltages (of order 1 V in microdevices), the solution of the PB equation shows a dramatically different behavior, in that the double layers can no longer absorb the complete applied voltage. Instead, a finite field remains throughout the device that leads to complete separation of the charged species. In this higher voltage regime, the double layer characteristics are no longer described by the usual Debye parameter kappa, and the ion concentration at the electrodes is intrinsically bound (even without assuming steric interactions). In addition, we have performed measurements of the electrode polarization current on a nonaqueous model electrolyte inside a microdevice. The experimental results are fully consistent with our calculations, for the complete concentration and voltage range of interest.     

Electrophoresis of two identical rigid spheres in a charged cylindrical pore

The electrophoresis of two identical spheres moving along the axis of a long cylindrical pore under the conditions of low surface potential and weak applied electric field is investigated. The geometry considered allows us to examine simultaneously the effects of boundary and the presence of a nearby entity on the behavior of a particle. The influences of the separation distance between two spheres, the thickness of a double layer, the ratio (radius of sphere/radius of pore), and the charged conditions on the surfaces of the spheres and the pore on the mobility of a particle are investigated. Several interesting results that are not reported in the literature are observed. For instance, although for the case of two positively charged spheres in an uncharged pore the qualitative behavior of a sphere depends largely on its size relative to that of a pore and the thickness of the double layer, this might not be the case when two uncharged spheres are in a positively charged pore. In addition, in the latter, the mobility of a sphere increases with the increases in the separation distance between two spheres, and this effect is pronounced when the ratio (radius of sphere/radius of pore) takes a medium value or the thickness of the double layer is either sufficiently thin or sufficiently thick.