Symmetric Electrolyte Solution Research Articles

Influence of surface charge and solution pH on the performance characteristics of a nanofiltration membrane

Acetyl cellulose nanofiltration membrane (NF) was manufactured by a dry/wet phase inversion process and studied on its performance characteristics for the electrolyte solutions NaCl, MgCl2,Na2SO4 with a concentration range of 10-3–10-2 N at various pH values. It was shown that the multivalent cation Mg2+ or sulfate anion SO2−4 present in non-symmetrical electrolytes makes the membrane more positively or negatively charged as a result of their adsorption on the amphoteric pore surface, which greatly increases the membrane selectivity up to 85–95% for these electrolyte solutions. Meanwhile,with respect to 1:1 electrolyte solution NaCl the rejection capacity is low (R=30–40%) because there is essentially no adsorption on the membrane surface. For a feed source made of the RO retentate the experimental data showed that the fabricated membrane possesses high rejection capacity for multivalent ions such as SO2−4 (R=85.7%) and Mg2+ (R=93.5%) and low selectivity for monovalent salts such as NaCl (R=33.7%). The following salts rejection sequence with respect to symmetric and nonsymmetric electrolyte solutions has been set up: RNa2SO4>RNaCl>RMgCl2 and RMgCl2>RNa2SO4>RNaCl at pH≥7 and pH≤7, respectively.The fact that at pH values above 7 rejection of sodium chloride is higher than that of magnesium chloride, although its ion size is smaller than that of the latter, can be explained only by charge exclusion effect. From these results it can be concluded that Donnan exclusion effect is the predominant mechanism for salt removal by NF membranes.

Application of the non-primitive MSA-based models in predicting the activity and the osmotic coefficients of aqueous electrolyte solutions

The non-primitive mean spherical approximation (NP-MSA) based models were used to correlate the individual and the mean ionic activity coefficients of the symmetric and asymmetric aqueous electrolyte solutions. The results of the models for the mean ionic activity coefficients were directly used to calculate the osmotic coefficient for the electrolyte solutions studied in this work. In the NP-MSA models, the Ghotbi–Vera and the Boublik–Mansoori–Carnahan–Starling–Leland hard sphere equation of state as the reference system coupled with the non-primitive mean spherical approximation model (NP-GV-MSA and NP-BMCSL-MSA models, respectively). In correlating the mean ionic activity coefficients of electrolyte solutions, while the hard sphere diameters for anion were treated as a concentration-independent adjustable parameter, the diameters for cation were assumed to be concentration dependent. In the use of the non-primitive MSA-based models in correlating the individual ionic activity coefficients of electrolyte solutions, both anion and cation diameters were considered to be concentration dependent. The results for the mean ionic activity coefficients obtained from the NP-GV-MSA model compared favorably with those of the NP-BMCSL-MSA model produced in this work and also with those of NP−BMCSL-MSA available in the literature. It was shown that the NP-GV-MSA model more accurately calculate the mean ionic activity and the osmotic coefficients than those obtained from the NP−BMCSL-MSA models.

On the Correlation of the Activity Coefficients in Aqueous Electrolyte Solutions Using the K-MSA Model

The objective of this work is to extend the Kelvin mean spherical approximation (K-MSA) model to correlate the mean and individual ionic activity coefficients for symmetric and asymmetric electrolyte solutions at different concentrations and at different temperatures. Revised values of the parameters for the mean and individual ionic activity coefficients of 1:1 electrolytes and new values of the parameters for the mean ionic activity coefficients of asymmetric electrolytes are presented. The effect of the short-range electrostatic term (Pn) at different size ratios of the ions is examined. Assuming a constant anion diameter and a composition-dependent diameter for the cation, the K-MSA model gives realistic cation-hydrated diameters, in comparison with their crystallographic Pauling diameters. Notably, for both symmetric and asymmetric electrolytes, the like and unlike ionic radial distribution function at contact value are positive over all of the concentration and temperature ranges studied. The result...

Approximate Analytic Expression for the Electrophoretic Mobility of a Spherical Colloidal Particle

An approximate analytic expression for the electrophoretic mobility of a spherical colloidal particle in a symmetrical electrolyte solution is obtained. This mobility expression, which is correct to the order of the third power of the ζ-potential of the particle, considerably improves Henry's mobility formula (Proc. R. Soc. London Ser. A133, 106 (1931)) and has a much simpler form than similar mobility formulas expressed in powers of ζ derived by Overbeek (Kolloid-Beihefte54, 287 (1943)) and by Booth (Proc. R. Soc. London Ser. A203, 514 (1950)). Comparison with the exact computer calculation by O'Brien and White (J. Chem. Soc. Faraday Trans. 274, 1607 (1978)) shows that the obtained mobility formula is an excellent approximation, especially for small κa (where κ is the Debye–Hückel parameter and a is theparticle radius) in which region no other simple analytic mobility formulas are available than Henry's formula, which is correct to order ζ and is applicable only for low ζ.

Diffuse Double-Layer Interaction between Two Identical Spherical Colloidal Particles

A simple method is given for calculating the potential energy of the diffuse double-layer interaction between two identical spherical colloidal particles in a symmetrical electrolyte solution with the help of Derjaguin's approximation. This method uses accurate analytic expressions for the corresponding interaction energy between two parallel similar plates obtained previously (Colloids Surf. A Physicochem. Eng. Asp.146, 213 (1999); J. Colloid Interface Sci.212, 130 (1999)). Agreement with numerical data provided by Honig and Mul (J. Colloid Interface Sci.36, 258 (1971)) is excellent particularly for small particle separations.

Conductance of symmetric electrolyte solutions: Formulation of the dressed-ion transport theory (DITT)

The concentration dependence of the electrical conductance of several symmetrical electrolytes is studied using the dressed-ion transport theory (DITT) formalism. This theoretical scheme is constructed from the Fuoss–Onsager theory using the equilibrium dressed-ion theory (DIT) distribution functions for electrolyte and colloid media. The conductance equation is constructed from previous results for the relaxation of the ionic atmosphere [Varela et al., J. Chem. Phys. 110, 4483 (1999)], now completed with the calculation of the concentration-dependent electrophoretic velocity of ions on the basis of a DIT in a hydrodynamic point of view. Onsager’s limiting law for the electrophoretic velocity based on the Debye–Hückel (DH) equilibrium distribution function is reformulated in terms of the DIT renormalized quantities, effective charges and effective screening length. The electrophoretic correction to the velocity calculated in the framework of this DITT is shown to exhibit a nontrivial dependence on the renormalized dielectric constant of the medium (ε*). The concentration dependence of both the deviations of the renormalized dielectric constant from the classical limit and the electrophoretic correction to the mobility is analyzed using the modified-mean spherical approximation (MMSA) value of the DIT linear response function α̂(k). The behavior of ε* and therefore of the electrophoretic correction to the ionic (or colloidal) mobilities is studied for both the random-phase approximation (RPA) and the MMSA and they are related to ionic association through the dimensionless coupling parameter, ζ, made up from the ratio between the Bjerrum length and the mean ionic radius. The expression for the conductance of the electrolyte solutions obtained in this new theoretical approach is cast in the form of a generalized Onsager’s limiting law with a concentration-dependent slope which is predicted in terms of the formation of DIT renormalized quasiparticles in the solution. Conductance predictions of several theoretical expressions for the renormalized magnitudes of electrolyte solutions are compared to direct experimental data of KCl, and are in very good agreement with the theoretical predictions of the mean-field DITT, which confirms the convenience of understanding transport processes in terms of effective dynamical entities with renormalized ionic charges interacting by means of a renormalized screened potential.

Electrical Interactions between Two Charged Spheroids in a Symmetric Electrolyte Solution

The interactions between two charged spheroids immersed in a symmetric electrolyte solution are evaluated. We show that the more closely two particles resemble spheres, the higher the electrical interaction energy between them. The variation in the total interaction energy between two particles, which includes the electrical and the van der Waals energies, reveals that the greater the radius of curvature of a particle, the higher the primary maximum or the harder it is for two particles to undergo an irreversible coagulation. If the double layer around a particle is thin compared with its radius of curvature, the critical coagulation concentration (CCC) of counterions is independent of the latter. This implies that the relative orientation between two particles is insignificant as far as the CCC is concerned. In this case, the result obtained in the present study reduces to that predicted by the Derjaguin−Landau−Verwey−Overbeek theory.

Electrostatic Potential Distribution for Spheroidal Surfaces in Symmetric Electrolyte Solutions

The electrostatic potential distribution for a charged spheroidal surface immersed in a symmetric electrolyte solution is derived. Such surfaces simulate a wide class of dispersed entities. Two types of boundary condition at the solid surface are considered, constant surface potential and constant amount of surface charges; both conductive and nonconductive surfaces are examined for the latter. The present analysis extends the conventional one-dimensional treatment on simple geometries to a two-dimensional space. A perturbation method is adopted to solve the governing Poisson–Boltzmann equation for the case of thin to moderately thick double layers. The classic results for planar and spherical surfaces can be recovered as special cases of the present analysis. The basic thermodynamic properties of the system under consideration, such as Helmholtz free energy, entropy, and surface excess, are derived. We show that using an equivalent sphere to approximate a spheroid can lead to an appreciable deviation in the prediction of the Helmholtz free energy. For a thin double layer, assuming a planar geometry will underestimate the Helmholtz free energy.

Calculations of Double-Layer Electrostatic Interactions for the Sphere/Plane Geometry

A numerical scheme for solving the nonlinear Poisson–Boltzmann equation for the sphere/sphere and plane/sphere geometries has been developed. The method is based on an alternating direction overrelaxation procedure using the Newton–Raphson iteration to solve the nonlinear equation stemming from finite-difference discretization. The novelty of the algorithm consists in using the grid transforming functions that allow a more uniform distribution of mesh points in the vicinity of the particle. The method was used to perform extensive calculations for opposite surface potentials of the interface and the particle immersed in a symmetric electrolyte solution. The electric potential distribution (within and outside the sphere) was calculated, as well as the force and energy of interaction (from the integral of the force over separation distance), for the constant potential, the constant charge, and the mixed cases. The energy profiles calculated for various κawere compared with the analytical approximations derived using the Hogg–Healy–Fuerstenau method and the linear superposition approach (LSA). These calculations enabled us to conclude that the LSA can be used as a good estimate of interaction energy for a broad range of κavalues at distances greater than κ−1, i.e., for problems pertinent to colloid particle adsorption.

The Critical Coagulation Concentration of Counterions: Spherical Particles in Asymmetric Electrolyte Solutions

The ratio of the critical coagulation concentration (CCC) of counterions is evaluated for spherical particles and asymmetric electrolytes. A perturbation method is adopted to solve the Poisson–Boltzmann equation governing the electrical potential distribution of the system under consideration. On the basis of the result obtained, an approximate expression for the CCC is derived. Another approach based on the Derjaguin approximation is also used to estimate the CCC. We show that the CCC ratio of counterions is a complicated function of the valences of the ion species in the liquid phase and the sizes of particles. Depending upon the thickness of the Debye length, the CCC ratio of counterions for various combinations of electrolytes can be estimated. The classic Schulze–Hardy rule for planar particles in a symmetric electrolyte solution can be recovered as a limiting case of the present model. If the surface potential is low, the effect of curvature on the CCC ratio of counterions is negligible.

Effective Surface Potential and Double-Layer Interaction of Colloidal Particles

The double-layer interaction energy between two colloidal particles at large separations is proportional to the product of the scaled effective surface potentials of the respective particles (Bell et al., J. Colloid Interface Sci. 33, 335 (1970); Brenner, S. L., and Parsegian, V. A., Biophys. J. 14, 327 (1974)). The exact expression is obtained for the scaled effective surface potential of a plate in a solution of general electrolytes, which agrees with Wilemski's results (J. Colloid Interface Sci. 88, 111 (1982)). An accurate approximate analytic expression is derived for the scaled effective surface potential of a sphere in a solution of general electrolytes. For the special case of a sphere in a symmetrical electrolyte solution, the obtained expression reduces to that derived previously (Ohshima et al., J. Colloid Interface Sci. 90, 17 (1982)). The obtained expressions are found to be excellent approximations to the exact numerical values given by Loeb et al. ("The Electrical Double Layer Around a Spherical Colloidal Particle." MIT Press, Cambridge, MA, 1961), the relative error being less than 1% for κa > 1 (a = sphere radius and κ = Debye-Hückel parameter) at all values of the surface potential.

Partial purification of a potassium channel with low permeability for sodium from tonoplast membranes of Hordeum vulgare cv. Gerbel

A potassium-specific tonoplast channel was identified by reconstitution of tonoplast polypeptides into planar lipid bilayer membranes. Highly purified tonoplast membranes were solubilized in Triton X-100-containing buffer and fractionated by size-exclusion chromatography. The protein fractions were assayed for ion channel activity in a planar bilayer system, and the potassium channel was routinely recovered in specific fractions corresponding to an apparent molecular mass of 80 kDa. In symmetrical electrolyte solutions of 100 mM potassium chloride, the potassium channel had a single-channel conductance of 72 pS. Substates of the channel with conductances of 17, 33 and 52 pS were frequently observed. After identification of the channel in low or high KCl, addition of sodium acetate or sodium chloride caused only insignificant conductance changes. This result suggested that the channel was not or little permeable for sodium or chloride, whereas it had similar single-channel conductance for rubidium and caesium ions as compared with potassium ions. The channel is presumably responsible for the equilibration of potassium between the vacuole and the cytosol. The role of the channel in the physiology of the barley cell under salt stress is discussed.

Electrostatic interactions in colloidal suspensions: Tests of pairwise additivity

AbstractThe electrical double‐layer free energy is computed numerically for a model colloidal suspension of charged, hexagonally‐packed monodisperse cylinders in a symmetric electrolyte solution using the Poisson‐Boltzmann equations. Comparison is made to the energies estimated from sums of pair interactions and from two versions of a radially‐symmetric cell model: one in which the outer boundary of the cylindrical cell is chosen to make the particle volume fraction Φ in the approximate model agree with that of the full lattice and the other in which the position of the outer boundary is chosen as half the nearest neighbor separation in the lattice. Calculations are performed for ranges of particle volume fractions from infinite dilution to 98% of closest packing, ionic strengths (in terms of the ratio of Debye length to particle radius λ/a) from 0.1 to 10.0, and dimensionless surface potentials and charge densities from 0.1 to 10.0.The pair approximation is valid only when typical interparticle spacings are greater than the Debye length. The first implementation of the cell model proves quite accurate, while the second is only qualitatively correct. The magnitude of particle surface charge density or potential has little effect on these conclusions.

Electroviscous effect when charged surfaces approach one another in electrolyte solutions

The electroviscous effect arising when charged surfaces of two similar plane discs or spherical particles approach each other in symmetrical electrolyte solutions under the effect of a compressive force, has been theoretically investigated. In the case of discs, the expression for the effective viscosity coincides with that characteristic of the electrolyte solution flow in a charged flat slit in the direction of a preset pressure gradient. In the case of spheres the viscosity increases to a much weaker extent and its maximum shifts toward the range of gaps whose thickness does not exceed the Debye length. The dependence of the apparent viscosity on the surface potential and the gap width has been derived, and the maximum influence of the surfaces charging mechanism on the effect under consideration has been assessed. In the case of a slow coagulation, this effect should decrease the coefficient of mutual diffusion of particles and thereby cause an additional retardation of coagulation.

Theory of symmetric electrolyte solutions: Field-theoretic approach.

Presentation de traitements perturbatifs, non perturbatifs, et de groupe de renormalisation pour le probleme de l'electrolyte symetrique, qui permettent d'obtenir les corrections au resultat connu de Debye-Huckel. De plus, des methodes non perturbatives permettent d'obtenir des resultats dans le cas d'electrolytes denses

Secondary minimum analysis in the DLVO-theory

A simple method for determining the secondary minimum distance and energy for two parallel plane surfaces interacting with constant potential and constant surface charge density in a symmetrical electrolyte solution is presented. Both cases are treated, namely, non-retarded and retarded interaction. Two graphics which provide a quick survey for determining the secondary minimum are reported.

The solution of the nonlinear Poisson—Boltzmann equation for thin, spherical double layers

A solution to the nonlinear Poisson—Boltzmann equation has been obtained for the region exterior to a spherical, colloidal particle in a binary, symmetric electrolyte solution. Using the Debye length as the expansion parameter, uniformly valid asymptotic expansions for the potential and charge density distributions have been obtained. The results are valid for arbitrary values of the surface potential.

Approximate analytic expressions for the electrophoretic mobility of spherical colloidal particles and the conductivity of their dilute suspensions

An approximate analytic expression is obtained for the electrophoretic mobility of a spherical colloidal particle in a symmetric electrolyte solution. This expression includes, consistently to order 1/κa(where κ is the Debye–Hückel parameter and a is the particle radius), all correction terms to Smoluchowski's formula and is found to be in good agreement with the exact computer calculation by the method of O'Brien and White (J. Chem. Soc., Faraday Trans. 2, 1978, 74, 1607). On the basis of this expression a semi-empirical formula with relative errors < 1% for κa 10 is derived. The same approximation method is employed to obtain a large κa approximate expression for the conductivity of a dilute suspension of spherical colloidal particles.

Statistical mechanics of electrolytes and polyelectrolytes. I. Symmetric electrolyte solutions

Simple ionic solutions are studied using the BBGKY hierarchy. An iterative estimation technique is developed which enables one to derive various well-known approximations. It is shown that in dilute symmetric electrolytes the Poisson–Boltzmann and the hypernetted chain equations are valid to a higher degree of accuracy than in asymmetric electrolytes and plasmas. The DHX, EXP, and the PY approximations are derived from the HNC equation. A perturbative correction to the EXP approximation is obtained which gives contact values of the correlation functions and excess internal energies close to those obtained by numerical solutions to the HNC equation for 1-1 electrolytes over a wide range of molar concentrations and for 2-2 electrolytes of relatively high concentration.

Diffuse double layer interaction between two parallel plates with constant surface charge density in an electrolyte solution IV. Numerical calculation of the interaction between similar plates using the non-linear Poisson-Boltzmann equation

A method is presented for numerical calculation of the diffuse double layer interaction between two parallel similar plates with constant surface charge density in a symmetrical electrolyte solution. The non-linear Poisson-Boltzmann equation is used and the influence of the electric field induced within the plates by the overlapping of the double layers is taken into account. Numerical tables are given for the force and potential energy of interaction. The applicability of the Debye-Huckel approximation is also considered.