Symmetric Electrolyte Solution Research Articles

Sedimentation of a Charged Soft Sphere within a Charged Spherical Cavity.

The sedimentation of a soft particle composed of an uncharged hard sphere core and a charged porous surface layer inside a concentric charged spherical cavity full of a symmetric electrolyte solution is analyzed in a quasi-steady state. By using a regular perturbation method with small fixed charge densities of the soft sphere and cavity wall, a set of linearized electrokinetic equations relevant to the fluid velocity field, electrical potential profile, and ionic electrochemical potential energy distributions are solved. A closed-form formula for the sedimentation velocity of the soft sphere is obtained as a function of the ratios of core-to-particle radii, particle-to-cavity radii, particle radius-to-Debye screening length, and particle radius-to-porous layer permeation length. The existence of the surface charge on the cavity wall increases the settling velocity of the charged soft sphere, principally because of the electroosmotic enhancement of fluid recirculation within the cavity induced by the sedimentation potential gradient. When the porous layer space charge and cavity wall surface charge have the same sign, the particle velocity is generally enhanced by the presence of the cavity. When these fixed charges have opposite signs, the particle velocity will be enhanced/reduced by the presence of the cavity if the wall surface charge density is sufficiently large/small relative to the porous layer space charge density in magnitude. The effect of the wall surface charge on the sedimentation of the soft sphere increases with decreases in the ratios of core-to-particle radii, particle-to-cavity radii, and particle radius-to-porous layer permeation length but is not a monotonic function of the ratio of particle radius-to-Debye length.

Effect of cation species on pressure-driven electrokinetic energy conversion in charged conical nanochannels

Electrokinetic energy conversion (EEC) offers a promising avenue for transforming elusive natural energy into electric power, showing a wide application spectrum. Unlike prevalent studies that chiefly utilize symmetrical electrolyte solution (e.g., KCl) and uniformly structured negatively charged nanochannels for EEC, this paper delves into a thorough examination of EEC characteristics—streaming current, streaming potential, and output power—in both positively and negatively charged conical nanochannels with symmetrical and asymmetric (CaCl2 and LaCl3) electrolytes solutions. Operating under the assumption that the electrolyte solutions (KCl, CaCl2, and LaCl3) possess equivalent ionic strength and, thereby, a common Debye length, our findings reveal a significant dependency of EEC characteristics and their regulation parameters on the electrolyte type, nanochannel charge polarity, and ionic strength. As the ionic strength increases, both the streaming current and output power initially rise to a peak before subsequently declining, with the ionic strength at the peak being influenced by the cation valence: lower valence leads to lower ionic strength. In asymmetric electrolyte scenarios, optimal EEC characteristic is observed in positively charged conical nanochannels under a reverse pressure difference, attributed to the ion-selective and ionic concentration distribution in the charged conical nanochannels. Moreover, the complex behaviors of the regulation parameters of EEC characteristics are unveiled. Notably, a tri-valent electrolyte's regulation parameters exceed those of a bi-valent electrolyte, indicating that ionic valence asymmetry enhances the regulation effects on EEC in conical nanochannels.

Diffusiophoresis of a Charged Soft Sphere in a Charged Spherical Cavity

The quasi-steady diffusiophoresis of a soft particle composed of an uncharged hard sphere core and a uniformly charged porous surface layer in a concentric charged spherical cavity full of a symmetric electrolyte solution with a concentration gradient is analyzed. By using a regular perturbation method with small fixed charge densities of the soft particle and cavity wall, the linearized electrokinetic equations relevant to the fluid velocity field, electric potential profile, and ionic concentration distributions are solved. A closed-form formula for the diffusiophoretic (electrophoretic and chemiphoretic) velocity of the soft particle is obtained as a function of the ratios of the core-to-particle radii, particle-to-cavity radii, particle radius to the Debye screening length, and particle radius to the permeation length in the porous layer. In typical cases, the confining charged cavity wall significantly influences the diffusiophoresis of the soft particle. The fluid flow caused by the diffusioosmosis (electroosmosis and chemiosmosis) along the cavity wall can considerably change the diffusiophoretic velocity of the particle and even reverse its direction. In general, the diffusiophoretic velocity decreases with increasing core-to-particle radius ratios, particle-to-cavity radius ratios, and the ratio of the particle radius to the permeation length in the porous layer, but increases with increasing ratios of the particle radius to the Debye length.

Diffusiophoresis of a cylindrical colloidal particle oriented parallel to an electrolyte concentration gradient field.

We derive the general expression for the diffusiophoretic mobility of a cylindrical particle oriented parallel to an applied electrolyte concentration gradient field in a symmetrical electrolyte solution. From the general mobility expression as combined with an approximate analytic expression with negligible error for the electric potential distribution around a cylinder, an accurate analytic mobility expression is obtained, which is applicable for arbitrary values of the particle zeta potential and the electrical double layer thickness. It is also found that the low zeta potential approximation is an excellent approximation for low-to-moderate values of the particle zeta potential.

Transport properties of anti-symmetrically charged nanochannels in symmetrical electrolyte solutions

Transport properties of anti-symmetrically charged nanochannels in symmetrical electrolyte solutions

Diffusiophoresis of a moderately charged spherical colloidal particle.

An analytic expression is obtained for the diffusiophoretic mobility of a charged spherical colloidal particle in a symmetrical electrolyte solution. The obtained expression, which is expressed in terms of exponential integrals, is correct to the third order of the particle zeta potential so that it is applicable for colloidal particles with low and moderate zeta potentials at arbitrary values of the electrical double-layer thickness. This is an improvement of the mobility formula derived by Keh and Wei, which is correct to the second order of the particle zeta potential. This correction, which is related to the electrophoresis component of diffusiophoresis, becomes more significant as the difference between the ionic drag coefficients of electrolyte cations and anions becomes larger and vanishes in the limit of thin or thick double layer. A simpler approximate mobility expression is further obtained that does not involve exponential integrals.

Revisiting the Theory of Coagulation of Colloidal Dispersions: An Improved Expression for the Stability Ratio.

The stability of a colloidal dispersion has long been expressed in terms of the stability ratio. Based on the available theories of coagulation of colloidal dispersions, a novel expression, complying with the classical definition, is developed for the stability ratio. It accounts for the contributions of both primary and secondary minimum coagulations to the overall rate of coagulations. In addition, it can also be regarded as the result of a combination of the kinetic theory of an ideal gas and the Smoluchowski theory with Fuchs' correction, considering the interaction between identical spherical particles and their surfaces immersed in a symmetrical electrolyte solution. The agreement with experimental data suggested that it is superior to the classical ones in describing the weak dependence of the stability ratio on the particle size and the valence of the counterion, by emphasizing the importance of the secondary minimum coagulation in dispersion stability and the complementation between the two modes of coagulation.

Diffusiophoresis of a highly charged dielectric fluid droplet

Diffusiophoresis of a dielectric fluid droplet with constant surface charge density in a symmetric binary electrolyte solution is investigated theoretically in this study, focusing on the chemiphoresis component, the very heart of diffusiophoresis. The resultant electrokinetic equations are solved with a pseudo-spectral method based on Chebyshev polynomial in the spirit of a computational fluid dynamic simulation. Reversions of moving directions are found for droplets less viscous than ambient solution when the electrolyte strength is increased due to the buildup of osmosis pressure in front of the moving droplets leading to an osmosis pressure gradient upon the droplet. The upward spouting effect of the spinning droplet surface is also responsible this buildup, which hinders the downward migration of ions and holds them up there. A solid particle may move faster than a gas bubble due to the energy consumption in the formation of an induced exterior vortex flow nearby surrounding the gas bubble. The less viscous the droplet is, the more severe this consumption is. A “solidification” phenomenon is observed where all the droplets move at the same speed with their surfaces and interior fluids motionless like rigid particles. Funnel-shape local extrema of mobility profiles provide solid evidence that the diffusion-induced double layer polarization is the very cause of the droplet motion in chemiphoresis. Excellent agreement with experimental data for a rigid particle is obtained. The study provides insights and guidelines in practical applications like drug delivery and other dead-end-pore types of operations such as EOR.

Analytical solution to dielectric droplet diffusiophoresis under Debye-Hückel approximation.

A simple analytical formula is obtained for the diffusiophoresis of a dielectric fluid droplet in symmetric binary electrolyte solutions under Debye-Hückel approximation valid for weakly charged droplets. The chemiphoresis is found to yield negative mobilities most of the time for droplets of constant surface charge density, which implies that the droplets tend to move away from the source releasing ionic chemicals. This is undesirable in some practical applications like drug delivery with liposomes in terms of conveying the drug-carrying liposomes to the desired area in the human body releasing specific ionic chemicals utilizing the self-guiding nature of diffusiophoresis. The further involvement of the electrophoresis component, however, may change the scenario via the oriented electric field generated by the induced diffusion potential. The lesson here is that while the impact of the chemiphoresis component is determined by nature and uncontrollable, the electrophoresis component serves as an artificially adjustable factor via choosing droplets with the surface charge of appropriate sign in practical applications. The results here have potential use in practical applications such as drug delivery. The portable simple analytical formula is a powerful asset to experimental researchers and design engineers in colloid science and technology to facilitate their works.

Simple formulas for prediction of the sizes of worm-like and globular micelles in symmetrical electrolyte solutions

The micro/macroscale properties of micellar solutions and the nanoscale characteristics of aggregates are controlled by a complex combination of the properties of the surfactant, electrolyte solution, and thermodynamic conditions. The science and practice of micellar solutions have relied predominantly on various experimental characterization methods to determine their physicochemical properties among which the size of micelles is the most crucial factor as other properties of the solutions are mainly influenced by it. In contrast to the extensive experimental work, so far only very few theoretical studies have been devoted to the prediction of phase behavior and properties of surfactant solutions. Our thorough review of the literature reveals that the vast majority of the previous theoretical models have been based on molecular thermodynamic approaches with different conditions for achievement of an equilibrium state during the micellization process. The lack of general models capable of predicting the contribution of each aggregation step to the free energy of micellization under different conditions is a critical factor that severely limits the practical application of such models. The main purpose of the present study is to circumvent this challenge by approaching the problem from a different theoretical point of view. In addition, it is aimed at increasing our understanding and capability for accurate prediction of the behavior and properties of micellar solutions. To do so, novel models are proposed to quantify adsorption of ionic surfactant molecules onto the surface of normal micelles in aqueous media based on the concepts of the electrical double layer (EDL) and surface charging. These models are developed in cylindrical and spherical configurations that are the most commonly observed structures of surfactant aggregates and validated by comparison with available experimental data. Also, the cylindrical model is used to obtain a criterion of the necessary condition for the formation and evolution of less frequently seen worm-like micelles. The results of this study allow academic researchers and industrial practitioners to calculate the size of micelles formed in aqueous solutions by ionic surfactants using a few simple parameters whose exact or estimated values are readily available.

Diffusiophoresis of a Highly Charged Soft Particle in Electrolyte Solutions.

Diffusiophoresis of a soft particle suspended in an infinite medium of symmetric binary electrolyte solution is investigated theoretically in this study, focusing on the chemiphoresis component when there is no global diffusion potential in the bulk solution. The general governing electrokinetic equations are solved with a pseudo-spectral method based on Chebyshev polynomials, and particle mobility, defined as the particle velocity per unit concentration gradient, is calculated. Parameters of electrokinetic interest are examined, in general, to explore their respective impact upon particle motion, such as the fixed charge density and permeability in the outer porous layer, the surface charge density and size of the inner rigid core, and the electrolyte strength in the solution. Nonlinear phenomena such as the motion-deterring double-layer polarization and the counterion condensation effects are scrutinized, in particular, for highly charged soft particles. Mobility reversal is observed in some range of electrolyte strength for highly charged particles. The generation of an axisymmetric counterclockwise vortex flow across the porous layer is found to be responsible for it. The onset of the mobility reversal is synchronized with the appearance or disappearance of this vortex flow. Mobility reversal may happen more than once, with particle moving toward or away from the region of higher solute concentration. The latter is undesirable in the application of drug delivery and thus should be avoided by delicate control of the electrokinetic environment. A local micro diffusion potential is discovered, which always speeds up the migration of coions and slows down that of counterions to guarantee that there is no net electric current across the double layer. Moreover, multilayer structure of the double-layer polarization is discovered when the electrolyte strength is high. The study presented here provides insight and crucial information for practical applications of soft particles, such as drug delivery.

Impedance Response of Electrochemical Interfaces: Part I. Exact Analytical Expressions for Ideally Polarizable Electrodes

In this work, we revisit the impedance response of the electrical double layer (EDL) at an ideally polarizable electrode which is immersed in a dilute, symmetric, and monovalent electrolyte solution. Analytical impedance expressions are derived from the standard Poisson-Nernst-Planck (PNP) theory. These expressions are formally exact under the linear polarization approximation. Frequency dispersion of the double- layer capacitance (C dl), ascribed to finite-rate ion transport in the electrolyte phase, is revealed. At the potential of zero charge (pzc), we compare the new impedance expression with that derived from the orthodox Gouy-Chapman-Stern (GCS) model. We reveal that the series connection of a compact layer and a diffuse layer in the GCS model tacitly prescribes a zero potential gradient at the solution-side boundary, which is, rigorously speaking, problematic. The bearing of this problematic assumption becomes more significant when the double layer is confined in narrower space. The analytical results derived at the pzc are good approximations (with a relative error in terms of C dl less than 14% for reasonably-valued parameters) when the electrode potential is away from the pzc less than 0.2 V.

Effect of hydrophobic patch on the modulation of electroosmotic flow and ion selectivity through nanochannel

In this article we made a systematic study of the electrokinetically driven flow through a long nanochannel patterned with charged hydrophobic patch embedded along the walls. The hydrophobic patch possess either similar or opposite charge to that of the hydrophilic portion of the channel wall. The hydrodynamic slip length is considered to be a function of the surface charge distributed along the patch. We consider the aqueous medium as binary symmetric electrolyte solution with constant property Newtonian fluid. We adopt a nonlinear model based on the Poisson-Nernst-Planck equations coupled with Navier–Stokes equations. The coupled set of governing equations are solved using a finite volume method. We use a higher order upwind based total variation diminishing (TVD) scheme to discretize the convective and electromigration terms (i.e., hyperbolic terms) and the central difference scheme is used to discretize the diffusion term. We have identified several interesting key features of the modulation of electroosmotic flow (EOF) through such a nanochannel. An enhancement or a reduction in average flow rate may be achieved depending on the polarity of the charges distributed along the hydrophobic or hydrophilic portion of the channel wall. We have also shown that the reversal in EOF may occur depending on the critical choice of the pertinent parameters. We have further studied the selectivity of mobile ions by regulating the charge properties of the channel walls.

Surface charging parameters of charged particles in symmetrical electrolyte solutions.

Surface electric charge of dispersed particles is an essential determinant of physicochemical properties, coagulation and flocculation processes, and stability of colloidal solutions. Size-dependence of surface potential, charge density, and total surface charge of suspended charged particles has recently received attention in the literature. Despite the clear significance of understanding such dependence, very few studies have been devoted to this problem, with contradictory results of the relationship type. Currently, there is no analytical formula to represent explicit relationships between surface charging parameters and particle size. This research work is directed at development of accurate physics-based formulas for quantification of curvature-dependence of surface potential, surface charge density, and total surface charge for cylindrical and spherical charged particles immersed in a symmetrical electrolyte solution. First, a non-dimensional approach is adopted to simplify the problems, overcoming the difficulty of dealing with multiple influential variables. Then, to reduce the degrees of freedom of the problems under consideration, Gauss's law is combined with the condition of electro-neutrality in an electrical double layer (EDL). Next, the resulting complex integral equations are solved to construct characteristic curves and to express the dimensionless surface charging parameters explicitly as a function of the dimensionless particle radius. The new theoretical expressions are founded on approximate analytical and numerical solutions of the nonlinear Poisson-Boltzmann (PB) equation in cylindrical and spherical geometries. Afterwards, the solutions of the non-dimensionalized problems are dimensionalized to derive accurate explicit closed-form expressions, describing how surface charging parameters are related to the radius of a charged particle, properties of the solution, and thermodynamic conditions. These analytical formulas enable researchers to properly determine surface potential, surface charge density, total surface charge, and radius of dispersed particles by characterizing only one of them. Finally, the validity of the commonly-held hypothesis that surface charge density is independent of particle size is examined at the end of this study.

Calculation of re-defined electrical double layer thickness in symmetrical electrolyte solutions

An electrical double layer (EDL) has crucial roles to play in diverse chemical/physical/biological phenomena and technological processes. The thickness of an EDL is one of the most important characteristics significantly affecting the value of zeta and streaming potentials, physicochemical properties of solutions, concentration polarization, the extent of stability of colloidal systems, coagulation and flocculation of colloids, etc. Although such thickness seems to be a straightforward characteristic of charged particles in contact with electrolyte solutions, there is no universal consensus among specialists on its definition and quantification. In spite of being incorporated in interface and colloid science for a century, the EDL thickness still has remained a dubious concept as there exist a variety of perceptions between scientists, researchers, and engineers.Unfortunately, the quantification of EDL thickness in current practices is founded primarily on rules of thumb with poor scientific justifications. Our comprehensive review of the literature shows that the EDL thickness is taken to be the Debye–Hückel length (i.e., κ-1) in a lot of applications, but sometimes the thickness is assumed to be equal to a few times κ-1. Such an assumption ignores the fact that the distribution of electric potential, and consequently EDL thickness, around a charged particle is affected by surface properties such as surface charge density and particle size. In other words, the common practice of κ-1-EDL-thickness suffers from not taking into account several other key factors contributing to the spatial extension of an EDL.This study is directed at development of theoretical physics-based formulas for accurate quantification of EDL thickness in symmetrical electrolyte solutions in different coordinate systems. The new analytical expressions are founded on the basis of exact or approximate solutions of the Poisson-Boltzmann (PB) equation for plate-like, cylindrical, and spherical charged particles. In fact, one of the targets of the present research work is to analytically address factors other than κ-1 affecting EDL thickness. Eventually, the degree of deviation of the commonly-used rule-of-thumb κ-1-thickness from the corresponding exact value is investigated by conducting sensitivity analyses over wide ranges of influential parameters.

Hydrodynamic dispersion in a combined magnetohydrodynamic- electroosmotic-driven flow through a microchannel with slowly varying wall zeta potentials

The effective dispersion coefficient of a neutral solute in the combined electroosmotic (EO) and magnetohydrodynamic (MHD)-driven flow of a Newtonian fluid through a parallel flat plate microchannel is studied. The walls of the microchannel are assumed to have modulated and low zeta potentials that vary slowly in the axial direction in a sinusoidal manner. The flow field required to obtain the dispersion coefficient is solved using the lubrication approximation theory. The solution of the electrical potential is based on the Debye-Hückel approximation for a symmetric (Z:Z) electrolyte solution. The EO and MHD effects, together with the variations in the zeta potentials of the walls, are observed to notably modify the axial distribution of the effective dispersion coefficient. The problem is formulated for two cases of the zeta potential function. Note that the dispersion coefficient primarily depends on the Hartmann number, on the ratio of the half height of the microchannel to the Debye length, and on the assumed variation in the zeta potentials of the walls.

Micrometer-Scale Ion Current Rectification at Polyelectrolyte Brush-Modified Micropipets.

Here we report for the first time that ion current rectification (ICR) can be observed at the micrometer scale in symmetric electrolyte solution with polyimidazolium brush (PimB)-modified micropipets, which we call micrometer-scale ion current rectification (MICR). To qualitatively understand MICR, a three-layer model including a charged layer, an electrical double layer, and a bulk layer is proposed, which could also be extended to understanding ICR at the nanoscale. Based on this model, we propose that when charges in the charged layer are comparable with those in the bulk layer, ICR would occur regardless of whether the electrical double layers are overlapped. Finite element simulations based on the solution of Poisson and Nernst-Planck equations and in situ confocal laser scanning microscopy results qualitatively validate the experimental observations and the proposed three-layer model. Moreover, possible factors influencing MICR, including the length of PimB, electrolyte concentration, and the radius of the pipet, are investigated and discussed. This study successfully extends ICR to the micrometer scale and thus opens a new door to the development of ICR-based devices by taking advantage of ease-in-manipulation and designable surface chemistry of micropipets.

Approximate Analytic Expression for the Electrophoretic Mobility of Moderately Charged Cylindrical Colloidal Particles.

An approximate analytic expression for the electrophoretic mobility of an infinitely long cylindrical colloidal particle in a symmetrical electrolyte solution in a transverse electric field is obtained. This mobility expression, which is correct to the order of the third power of the zeta potential ζ of the particle, considerably improves Henry's mobility formula correct to the order of the first power of ζ (Proc. R. Soc. London, Ser. A 1931, 133, 106). Comparison with the numerical calculations by Stigter (J. Phys. Chem. 1978, 82, 1417) shows that the obtained mobility formula is an excellent approximation for low-to-moderate zeta potential values at all values of κa (κ = Debye-Hückel parameter and a = cylinder radius).

On the maximum of the magnitude of the electrophoretic mobility of a spherical colloidal particle in an electrolyte solution

The magnitude of the electrophoretic mobility μ of a spherical colloidal particle in an electrolyte solution with κa > 3 (κ = the Debye-Huckel parameter of the electrolyte solution and a = particle radius), when plotted as a function of the particle zeta potential ζ, exhibits a maximum μmax at ζ = ζmax. Analytic expressions applicable for large κa (κa ≥ 30) are derived for μmax and ζmax for a spherical particle in a symmetrical electrolyte solution. Analytic expressions for μmax and ζmax are also derived for a spherical particle in a 2:1 or 1:2 electrolyte solution. Finally, it is to be noted that μmax and ζmax for a cylindrical particle of radius a when the particle is oriented perpendicular to the applied electric field are the same as those for a spherical particle of radius a for large κa (κa ≥ 30).

Synthetic ion transporters: Pore formation in bilayers via coupled activity of non-spanning cobalt-cage amphiphiles

Three amphiphilic cobalt-cage congeners bearing a diaza-crown bridge and varying alkyl chains (1:2:3; n=12, 16, 18) have been assessed for their ion transport across planar lipid bilayer membranes. In symmetrical electrolyte solutions, a range of ion transport activity is provoked: 1 disrupts painted (fluid) bilayers in a detergent-like mode of action; 2 forms conducting “pores” in folded (rigid) membranes with long open lifetimes (>2min) while 3 requires the larger auxiliary solvent volume and lower lateral stress of painted membranes to effect ion transport via long-lived pores. Hill analysis of the conductance variation with monomer concentration yields coefficients (2:3; n=2.3, 1.9) in support of dimeric (n=2) membrane-active structures, for which the derived “pore” radii are correlated with charge-density of the transported cations and their affinity for the crown moiety. A toroidal-pore model is invoked to account for the flux of guest ions through planar bilayer membranes without a fast-diffusing intermediary or direct membrane-spanning structure.