Effects of overlapping electric double layer on mass transport of a macro‐solute across porous wall of a micro/nanochannel for power law fluid

Effects of overlapping electric double layer and high wall potential on transport of a macrosolute for flow of a power law fluid through a microchannel with porous walls are studied in this work. The electric potential distribution is obtained by coupling the Poisson's equation without considering the Debye–Huckel approximation. The numerical solution shows that the center line potential can be 16% of wall potential at pH 8.5, at wall potential −73 mV and scaled Debye length 0.5. Transport phenomena involving mass transport of a neutral macrosolute is formulated by species advective equation. An analytical solution of Sherwood number is obtained for power law fluid. Effects of fluid rheology are studied in detail. Average Sherwood number is more for a pseudoplastic fluid compared to dilatant upto the ratio of Poiseuille to electroosmotic velocity of 5. Beyond that, the Sherwood number is independent of fluid rheology. Effects of fluid rheology and solute size on permeation flux and concentration of neutral solute are also quantified. More solute permeation occurs as the fluid changes from pseudoplastic to dilatant.     

Adhesion between a charged particle in an electrolyte solution and a charged substrate: Electrostatic and van der Waals interactions

The equilibrium separation between a charged particle in an electrolyte solution and a substrate with an initially uniform surface charge density is obtained using the classical Derjaguin-Landau-Verwey-Overbeek theory. The electrostatic free energy is obtained by coupling the electric response of the substrate with the electric potential obtained from the solution of the Debye-Hückel equation. The van der Waals free energy is calculated by integrating the 6-12 Lennard-Jones potential. Metallic, dielectric, and semiconducting substrates are considered in turn. At low ionic strength, our results demonstrate a distinct response to the charged particle in each case. For example, in the case of a metallic substrate, the attached state (corresponding to equilibrium separation at short range) is always close to the van der Waals energy minimum. In addition, the application of a surface charge of sign opposite to that of the particle facilitates the transition from the detached state (corresponding to large separation at which the interaction between the particle and the substrate is negligible) to the attached state but scarcely changes the equilibrium separation. In the case of a dielectric substrate, the attached state is located at a distance of around two orders of magnitude larger than that for a metallic substrate and this equilibrium separation decreases as the (opposing) surface charge increases. A semiconducting substrate can behave either like a metal or like a dielectric, depending on the ratio of its Debye length to that of the electrolyte solution.     

Changes in zeta potential caused by a dc electric current for thin double layers

An asymptotic solution was obtained to describe one-dimensional, steady-state transport of a symmetric binary electrolyte normal to two large parallel electrodes, in the limit in which the Debye length is infinitesimal compared to the distance separating the two electrodes. Despite the nonzero ion flux, Boltzmann's equation continues to describe the relationship between either ion concentration and the electrostatic potential inside the diffuse part of the double layer, while local electroneutrality applies outside, even for current densities approaching the limiting value. In the absence of ion adsorption or dissociation reactions at the electrodes, the magnitude of any charge or zeta potential arising on the electrodes at zero current is determined by the equilibrium constant for the redox reactions which would exchange ionic charge carriers for electric charge carriers at the electrode surface. Nonzero current causes the ionic strength of the bulk to vary with position. This perturbs the Debye length of the diffuse cloud on either electrode: it is the local ionic strength just outside the cloud which determines the Debye length for that cloud. Nonzero current also changes the zeta potential. The dimensionless rate of change dζ/dJ was as large as 30.     

Electrophoresis of hydrophilic/hydrophobic rigid colloid with effects of relaxation and ion size

This article deals with a semi‐analytical study on the electrophoresis of charged spherical rigid colloid by considering the effects of relaxation and ion size. The particle surface is taken to be either hydrophilic or hydrophobic in nature. In order to consider the ion size effect we have invoked the Carnahan and Starling model (J. Chem. Phys. 1969, 51, 635‐636). The mathematical model is based on Stokes equation for fluid flow, modified Boltzmann equation for spatial distribution of ionic species and Poisson equation for electric potential. We adopt a linear perturbation technique under a weak electric field assumption. An iterative numerical technique in employed to solve the coupled set of perturbed equations. We have validated the numerically obtained electrophoretic mobility with the corresponding analytical solution derived under low potential limit. Going beyond the widely employed Debye‐Hückel linearization, we have presented the results for a wide range of surface charge density, electrolyte concentration, and slip length to Debye length ratio. We have also identified several interesting features including occurrence of local maxima and minima in the mobility for critical choice of pertinent parameters.     

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.     

Electrophoresis and electric conduction in a salt-free suspension of charged particles

The electrophoresis and electric conduction of a suspension of charged spherical particles in a salt-free solution are analyzed by using a unit cell model. The linearized Poisson-Boltzmann equation (valid for the cases of relatively low surface charge density or high volume fraction of the particles) and Laplace equation are solved for the equilibrium electric potential profile and its perturbation caused by the imposed electric field, respectively, in the fluid containing the counterions only around the particle, and the ionic continuity equation and modified Stokes equations are solved for the electrochemical potential energy and fluid flow fields, respectively. Explicit analytical formulas for the electrophoretic mobility of the particles and effective electric conductivity of the suspension are obtained, and the particle interaction effects on these transport properties are significant and interesting. The scaled zeta potential, electrophoretic mobility, and effective electric conductivity increase monotonically with an increase in the scaled surface charge density of the particles and in general decrease with an increase in the particle volume fraction, keeping each other parameter unchanged. Under the Debye-Hückel approximation, the dependence of the electrophoretic mobility normalized with the surface charge density on the ratio of the particle radius to the Debye screening length and particle volume fraction in a salt-free suspension is same as that in a salt-containing suspension, but the variation of the effective electric conductivity with the particle volume fraction in a salt-free suspension is found to be quite different from that in a suspension containing added electrolyte.     

Electric fields in an electrolyte solution near a strip of fixed potential

Electrostatic fields produced by flat electrodes are often used to manipulate particles in solution. To study the field produced by such an electrode, we consider the problem of an infinite strip of width 2a with imposed constant potential immersed in an electrolyte solution. Sufficiently close to the edge of the strip, the solution is determined by classical electrostatics and results in a field singularity. We examine two limiting cases, (a) when strip width a<1k, the Debye screening length, and (b) when strip width is much greater than the Debye screening length, a>1k. We present exact results for the two cases in the limit of small potentials where the Poisson-Boltzmann equation can be linearized. By drawing on an analogy with antiplane shear deformations of solids, and by employing the path-independent J integral of solid mechanics, we present a new method for determining the strength of the edge singularity. The strength of the singularity defines an exact near-field solution. In the far field the solution goes to that of a line of charge. The accuracy of the solution is demonstrated by comparison with the numerical solutions of the Poisson-Boltzmann equation using the finite element method.     

A new and simpler approach for the solution of the electrostatic potential differential equation. Enhanced solution for planar, cylindrical and annular geometries

The motivation of the study performed in this project is focused on deriving a more effective, accurate, and mathematically friendly solution for the prediction of the electrostatic potential, commonly used on electrokinetic research and its related applications. In this contribution, based on the Debye-Hückel approximation, a new solution strategy for the differential equations of the electrostatic potential is proposed. In fact, a simple predictor-corrector calculation is developed to achieve more accurate predictions of electrostatic potential profiles. Furthermore, in this study the authors introduce the correction function f(AO) to the inverse Debye length, lambda. The f(AO) function improves the Debye-Hückel approximation and it is a recursive function of the electrical potential. Once the inverse Debye length, lambda, has been corrected by the f(AO) function and introduced in the simplified solution of the Poisson-Boltzmann equation (i.e., the linear approximation, due to Debye and Hückel), the electrostatic potential outcome little differs from the numerical solution of the complete (nonlinear) differential equation. This new approach embraces different geometries of interest, such as planar, cylindrical, and annular, with excellent results in all the cases and for a wide range of electrostatic potential values. This new predicting semi-analytical technique can be a useful tool on electrical field applications such as the separation of a mixture of macromolecules and the removal of contaminants in soil cleaning processes. Illustrative results are presented for the geometries identified above.     

The charging of a liquid dielectric upon its flow past a flat-plate

The charging of a liquid dielectric upon its flow past a flat-plate is considered. Analytical expressions for the density distribution of electric charge and charging current are derived. The dependences of the current and charge density distribution on the system parameters are investigated. The effect of the electric field that emerges owing to charge separation on the charging process is taken into account. Consideration of the electric field is shown to lead to lower values of the electric charge density and charging current. As the Debye number decreases, the charge density also decreases. The charging current increases with an increase in the Debye number.     

The Goldstone mode in mixtures containing ferroelectric liquid crystals

Measurements of complex electric permittivity of room temperature ferroelectric liquid crystal mixtures have been made on aligned samples with the electric measuring field being parallel to the layer planes. The spontaneous polarization, the tilt angles and pitch have been measured in these mixtures. By theoretical fitting of the experimental points of electric permittivity for the Cole-Cole modification of the Debye equation dielectric parameters, the dielectric strength, relaxation frequency, and distribution parameter for the Goldstone mode have been computed. The dielectrically observed Goldstone mode in our mixtures is shown to have both DC bias field and AC field dependences.     

Transient electrophoresis in a suspension of charged particles with arbitrary electric double layers

The startup of electrophoretic motion in a suspension of spherical colloidal particles, which may be charged with constant zeta potential or constant surface charge density, due to the sudden application of an electric field is analytically examined. The unsteady modified Stokes equation governing the fluid velocity field is solved with unit cell models. Explicit formulas for the transient electrophoretic velocity of the particle in a cell in the Laplace transforms are obtained as functions of relevant parameters. The transient electrophoretic mobility is a monotonic decreasing function of the particle-to-fluid density ratio and in general a decreasing function of the particle volume fraction, but it increases and decreases with a raise in the ratio of the particle radius to the Debye length for the particles with constant zeta potential and constant surface charge density, respectively. On the other hand, the relaxation time in the growth of the electrophoretic mobility increases substantially with an increase in the particle-to-fluid density ratio and with a decrease in the particle volume fraction but is not a sensitive function of the ratio of the particle radius to the Debye length. For specified values of the particle volume fraction and particle-to-fluid density ratio in a suspension, the relaxation times in the growth of the particle mobility in transient electrophoresis and transient sedimentation are equivalent.     

Estimation of zeta potential by electrokinetic analysis of ionic fluid flows through a divergent microchannel

The streaming potential is generated by the electrokinetic flow effect within the electrical double layer of a charged solid surface. Surface charge properties are commonly quantified in terms of the zeta potential obtained by computation with the Helmholtz-Smoluchowski (H-S) equation following experimental measurement of streaming potential. In order to estimate a rigorous zeta potential for cone-shaped microchannel, the correct H-S equation is derived by applying the Debye-Hückel approximation and the fluid velocity of diverging flow on the specified position. The present computation provides a correction ratio relative to the H-S equation for straight cylindrical channel and enables us to interpret the effects of the channel geometry and the electrostatic interaction. The correction ratio decreases with increasing of diverging angle, which implies that smaller zeta potential is generated for larger diverging angle. The increase of Debye length also reduces the correction ratio due to the overlapping of the Debye length inside of the channel. It is evident that as the diverging angle of the channel goes to nearly zero, the correction ratio converges to the previous results for straight cylindrical channel.     

The Goldstone mode in mixtures containing ferroelectric liquid crystals

Abstract

Measurements of complex electric permittivity of room temperature ferroelectric liquid crystal mixtures have been made on aligned samples with the electric measuring field being parallel to the layer planes. The spontaneous polarization, the tilt angles and pitch have been measured in these mixtures. By theoretical fitting of the experimental points of electric permittivity for the Cole–Cole modification of the Debye equation dielectric parameters, the dielectric strength, relaxation frequency, and distribution parameter for the Goldstone mode have been computed. The dielectrically observed Goldstone mode in our mixtures is shown to have both DC bias field and AC field dependences.     

Diffusioosmosis of electrolyte solutions along a charged plane wall

The steady diffusioosmotic flow of an electrolyte solution along a dielectric plane wall caused by an imposed tangential concentration gradient is analytically examined. The plane wall may have either a constant surface potential or a constant surface charge density of an arbitrary quantity. The electric double layer adjacent to the charged wall may have an arbitrary thickness, and its electrostatic potential distribution is determined by the Poisson-Boltzmann equation. The macroscopic electric field along the tangential direction induced by the imposed electrolyte concentration gradient is obtained as a function of the lateral position. A closed-form formula for the fluid velocity profile is derived as the solution of a modified Navier-Stokes equation. The direction of the diffusioosmotic flow relative to the concentration gradient is determined by the combination of the zeta potential of the wall and the properties of the electrolyte solution. For a given concentration gradient of an electrolyte along a plane wall, the magnitude of fluid velocity at a position in general increases with an increase in its electrokinetic distance from the wall, but there are exceptions. The effect of the lateral distribution of the induced tangential electric field in the double layer on the diffusioosmotic flow is found to be very significant and cannot be ignored.     

Electrostatic stretching of a charged vesicle

We present a closed-form solution of electrostatic potential self-induced by a uniformly charged micro/nanovesicle and the corresponding elastic deformation of the vesicle membrane due to Maxwell stress. At equilibrium, the electrostatic force induced on both sides of the membrane is balanced by the elastic force of the stretched membrane. We develop differential and integral solutions of the coupled Poisson-Boltzmann system for a spherical vesicle and demonstrate that the integral solution is relatively flexible in formulating asymmetric configurations. Analytical results are formulated in terms of vesicle size, Debye length, and the surface charge density. The membrane stretching is characterized by the dimensionless group that defines the relative strength of the net electric force with respect to the membrane stiffness. We found that the self-induced electrostatic interaction will lead to a pre-stressed membrane although the small displacement is often negligible compared with the vesicle size. Quantitative analysis also reveals that the electric force can assist the vesicle in recovering its opening pore.     

Transport properties of long straight nano-channels in electrolyte solutions: a systematic approach

The principle of local thermodynamic equilibrium is systematically employed for obtaining various transport properties of long straight nano-channels. The concept of virtual solution is used to describe situations of non-negligible overlap of diffuse parts of electric double layers (EDLs) in nano-channels. Generic expressions for a variety of transport properties of long straight nano-channels are obtained in terms of quasi-equilibrium distribution coefficients of ions and functionals of quasi-equilibrium distribution of electrostatic potential. Further, the Poisson-Boltzmann approach is used to specify these expressions for long straight slit-like nano-channels. In the approximation of non-overlapped diffuse parts of double electric layers in nano-channels, simple analytical expressions are obtained for the apparent electrophoretic mobilities of (trace) analytes of arbitrary charge as well as for the salt reflection coefficient (osmotic pressure), salt diffusion permeability and electro-viscosity (electrokinetic energy conversion). The approximate solutions are compared with the results of rigorous solution of non-linearized Poisson-Boltzmann equation, and the accuracy of approximation is shown to be typically excellent when the nano-channel half-height exceeds ca.3 Debye screening lengths. Due to non-negligible electrostatic adsorption of ions by nano-channels, the apparent electrophoretic mobilities of counter-ionic analytes in nano-channels are smaller than in micro-channels whereas those of co-ionic analytes are larger. This dependence on the charge is useful for the separation of analytes of close electrophoretic mobilities. The osmotic pressure is shown to be positive, negative or pass through maxima as a function of applied salt-concentration difference within a fairly narrow range of ratios of nano-channel height to the Debye screening length. The diffusion permeability of charged nano-channels to single salts is demonstrated (for the first time) to be typically larger than that of neutral nano-channels of the same dimensions due to electrical facilitation of salt diffusion.     

Electrokinetic actuation of an uncharged polarizable dielectric droplet in charged hydrogel medium

Electrokinetic transport of an uncharged nonconducting microsized liquid droplet in a charged hydrogel medium is studied. Dielectric polarization of the liquid drop under the action of an externally imposed electric field induces a non-homogeneous charge density at the droplet surface. The interactions of the induced surface charge of the droplet with the immobile charges of the hydrogel medium generates an electric force to the droplet, which actuates the drop through the charged hydrogel medium. A numerical study based on the first principle of electrokinetics is adopted. Dependence of the droplet velocity on its dielectric permittivity, bulk ionic concentration, and immobile charge density of the gel is analyzed. The surface conduction is significant in presence of charged gel, which creates a concentration polarization. The impact of the counterion saturation in the Debye layer due to the dielectric decrement of the medium is addressed. The modified Nernst–Planck equation for ion transport and the Poisson equation for the electric field is considered to take into account the dielectric polarization. A quadrupolar vortex around the uncharged droplet is observed when the gel medium is considered to be uncharged, which is similar to the induced charge electroosmosis around an uncharged dielectric colloid in free-solution. We find that the induced charge electrokinetic mechanism creates a strong recirculation of liquid within the droplet and the translational velocity of the droplet strongly depends on its size for the dielectric droplet embedded in a charged gel medium.     

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.     

Calculation of the dynamic impedance of the double layer on a planar electrode by the theory of electrokinetics

Applications of microelectromechanical systems in the biotechnological arena (bioMEMS) are a subject of great current interest. Accurate calculation of electric field distribution in these devices is essential to the understanding and design of processes such as dielectrophoresis and AC electroosmosis that drive MEMS-based devices. In this paper, we present the calculation of the electrical double-layer impedance (Z(el)) of an ideally polarizable plane electrode using the standard model of colloidal electrokinetics. The frequency variation of the electrical potential drop across the double layer above a planar electrode in a general electrolyte solution is discussed as a function of the electrode zeta potential zeta, the Debye length kappa(-1), the electrolyte composition and the bulk region thickness L.     

Effective Debye length in closed nanoscopic systems: a competition between two length scales

The Poisson-Boltzmann equation (PBE) is widely employed in fields where the thermal motion of free ions is relevant, in particular in situations involving electrolytes in the vicinity of charged surfaces. The applications of this non-linear differential equation usually concern open systems (in osmotic equilibrium with an electrolyte reservoir, a semi-grand canonical ensemble), while solutions for closed systems (where the number of ions is fixed, a canonical ensemble) are either not appropriately distinguished from the former or are dismissed as a numerical calculation exercise. We consider herein the PBE for a confined, symmetric, univalent electrolyte and quantify how, in addition to the Debye length, its solution also depends on a second length scale, which embodies the contribution of ions by the surface (which may be significant in high surface-to-volume ratio micro- or nanofluidic capillaries). We thus establish that there are four distinct regimes for such systems, corresponding to the limits of the two parameters. We also show how the PBE in this case can be formulated in a familiar way by simply replacing the traditional Debye length by an effective Debye length, the value of which is obtained numerically from conservation conditions. But we also show that a simple expression for the value of the effective Debye length, obtained within a crude approximation, remains accurate even as the system size is reduced to nanoscopic dimensions, and well beyond the validity range typically associated with the solution of the PBE.