Approximate expressions of determining the double layer energy and force of interaction between two charged colloidal spheres

The approximate expressions have been obtained to calculate the electrical double layer energy and force between two spherical colloidal particles based on the improved Derjaguin approximation. Results for identical spheres interacting under constant surface potential, constant surface charge are given. Comparison of present results with numerical results calculated by Carnie and Chan is made. The expressions are found to work quite well for the constant surface potential case, and for the constant charge case, we make correction for the expressions. The results given are satisfactory providedkh0.4.     

Simple Approximate Expressions for Electrical Double-Layer Interaction at Constant Moderate Potentials

Simple yet accurate expressions for the electrical double-layer interaction force and energy between the particles that hold for a wide range of surface potential is required in the modeling, simulation, and optimization of many processes employed in industry. In this paper, simple approximate expressions for the interaction are obtained based on the asymptotic results and the numerical solution to the Poisson-Boltzmann equation for identical parallel plates with constant surface potential up to 180/z(i) mV at 25 degrees C, and the Derjaguin approximation. Within the moderate surface potential range, the semianalytical expressions agree well with the exact numerical results and are convenient to use for the purpose of process modeling and simulation. Copyright 2000 Academic Press.     

Approximate analytical expressions for the electrical potential between two planar, cylindrical, and spherical surfaces

Approximate analytical expressions for the electrical potential of planar, cylindrical, and spherical surfaces are derived for the case in which the dispersion medium contains counterions only. On the basis of the results for single surfaces, those for two identical surfaces can be derived. The curvature effect of a surface on the electrical potential distribution can be neglected when the order of its radius exceeds approximately 100 times the thickness of the corresponding double layer. If this effect needs to be considered, it can be taken into account by multiplying a correction function by the electrical potential of a planar surface. The electrical potential at the center between two derived surfaces is readily applicable to the evaluation of the electrostatic force per unit area between two surfaces, or the osmotic pressure. For the same set of parameters, the magnitudes of the osmotic pressure for various types of surfaces rank as follows: planar surface > cylindrical surfaces > spherical surfaces.     

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

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

Approximate analytic expressions for the electrophoretic mobility of spheroidal particles

Approximate analytic expressions are derived for the electrophoretic mobility of spheroidal particles (prolate and oblate) carrying low zeta potential in an electrolyte solution under an applied tangential or transverse electric field. The present approximation method, which is based on the observation that the electrophoretic mobility of a particle is determined mainly by the distortion of the applied electric field by the presence of the particle. The exact expression for the equilibrium electric potential distribution around the particle, which can be expressed as an infinite sum of spheroidal wave functions, is not needed in the present approximation. The electrophoretic mobility values calculated with these approximate expressions for spheroidal particles with constant surface potential or constant surface charge density are in excellent agreement with the exact numerical results of previous reports with the relative errors less than about 4%.     

Critical coagulation concentration of a salt-free colloidal dispersion

Both exact and approximate analytical solutions of the Poisson-Boltzmann equation for two planar, parallel surfaces are derived for the case when a dispersion medium contains counterions only, and the results obtained are used to evaluate the critical coagulation concentration of a spherical dispersion. A correction factor, which is a function of the valence of counterions, the surface potential of a particle, and the potential on the midplane between two particles at the onset of coagulation, is derived to modify the classic Schulze-Hardy rule for the dependence of the critical coagulation concentration on the valence of counterions. The correction factor is found to increase with the increase in the valence of counterions and/or with the increase in the surface potential. However, it approaches a constant value of 0.8390 if the surface potential is sufficiently high.     

Principles of Surface Potential Estimation in Mixed Electrolyte Solutions:Taking into Account Dielectric Saturation

The dielectric properties between in-particle/water interface and bulk solution are significantly different, which are ignored in the theories of surface potential estimation. The analytical expressions of surface potential considering the dielectric saturation were derived in mixed electrolytes based on the nonlinear Poisson-Boltzmann equation. The surface potentials calculated from the approximate analytical and exact numerical solutions agreed with each other for a wide range of surface charge densities and ion concentrations. The effects of dielectric saturation became important for surface charge densities larger than 0.30 C/m\begin{document}$ ^2 $\end{document}. The analytical models of surface potential in different mixed electrolytes were valid based on original Poisson-Boltzmann equation for surface charge densities smaller than 0.30 C/m\begin{document}$ ^2 $\end{document}. The analytical model of surface potential considering the dielectric saturation for low surface charge density can return to the result of classical Poisson-Boltzmann theory. The obtained surface potential in this study can correctly predict the adsorption selectivity between monovalent and bivalent counterions.     

平板型高电位胶粒双电层的相互作用

利用线性迭加法,提出了平行平板型高电位颗粒之间的弱相互作用的近似表达式.结合文献[3]给出的强相互作用表达式,对高电位平行平板型颗粒的相互作用给出了完整的描述,和精确数值解吻合相当好.强弱相互作用的接合点在κh=4,误差在接合点处最大,～10％.根据Derjaguin法和改进的Derjaguin法,求出了高电位球颗粒在恒电位条件下的相互作用能.     

Exact analytical expressions for the potential of electrical double layer interactions for a sphere-plate system

Exact, closed-form analytical expressions are presented for evaluating the potential energy of electrical double layer (EDL) interactions between a sphere and an infinite flat plate for three different types of interactions: constant potential, constant charge, and an intermediate case as given by the linear superposition approximation (LSA). By taking advantage of the simpler sphere-plate geometry, simplifying assumptions used in the original Derjaguin approximation (DA) for sphere-sphere interaction are avoided, yielding expressions that are more accurate and applicable over the full range of κa. These analytical expressions are significant improvements over the existing equations in the literature that are valid only for large κa because the new equations facilitate the modeling of EDL interactions between nanoscale particles and surfaces over a wide range of ionic strength.     

Analytical evaluation of Fukui functions and real-space linear response function

Many useful concepts developed within density functional theory provide much insight for the understanding and prediction of chemical reactivity, one of the main aims in the field of conceptual density functional theory. While approximate evaluations of such concepts exist, the analytical and efficient evaluation is, however, challenging, because such concepts are usually expressed in terms of functional derivatives with respect to the electron density, or partial derivatives with respect to the number of electrons, complicating the connection to the computational variables of the Kohn-Sham one-electron orbitals. Only recently, the analytical expressions for the chemical potential, one of the key concepts, have been derived by Cohen, Mori-Sánchez, and Yang, based on the potential functional theory formalism. In the present work, we obtain the analytical expressions for the real-space linear response function using the coupled perturbed Kohn-Sham and generalized Kohn-Sham equations, and the Fukui functions using the previous analytical expressions for chemical potentials of Cohen, Mori-Sánchez, and Yang. The analytical expressions are exact within the given exchange-correlation functional. They are applicable to all commonly used approximate functionals, such as local density approximation (LDA), generalized gradient approximation (GGA), and hybrid functionals. The analytical expressions obtained here for Fukui function and linear response functions, along with that for the chemical potential by Cohen, Mori-Sánchez, and Yang, provide the rigorous and efficient evaluation of the key quantities in conceptual density functional theory within the computational framework of the Kohn-Sham and generalized Kohn-Sham approaches. Furthermore, the obtained analytical expressions for Fukui functions, in conjunction with the linearity condition of the ground state energy as a function of the fractional charges, also lead to new local conditions on the exact functionals, expressed in terms of the second-order functional derivatives. We implemented the expressions and demonstrate the efficacy with some atomic and molecular calculations, highlighting the importance of relaxation effects.     

Surface Charge Density/Surface Potential Relationship for a Cylindrical Particle in an Electrolyte Solution

An accurate analytic expression of the surface charge density/surface potential relationship for an infinitely long cylindrical colloidal particle in a solution of general electrolytes is derived from an approximate solution to the nonlinear cylindrical Poisson– Boltzmann equation. The mathematical procedure is based on a method developed previously by Ohshima, Healy, and White for the case of a sphere (J. Colloid Interface Sci.90, 17 (1982)). Comparison is made with exact numerical results. Accurate expressions for the potential distribution around a cylinder and the effective surface potential of a cylinder are also derived. Finally, expressions for the double-layer interaction energy and force between two cylinders at large separations are derived on the basis of the method of Brenner and Parsegian (Biophys. J.14, 327 (1974)).     

On approximate analytical solutions of differential equations in enzyme kinetics using homotopy perturbation method

Homotopy perturbation method is used to extend the approximate analytical solutions of non-linear reaction equations describing enzyme kinetics for combinations of parameters for which solutions obtained in previous works are not valid. Also, by constructing a new homotopy, alternative approximate analytical expressions for substrate, substrate-enzyme complex and product concentrations are found. These first-order approximate solutions give more accurate results than the second-order approximations derived in previous works.     

Electrostatic Interaction between Two Planar Double Layers: A Further Extension to Langmuir's Method

A simple method for calculating the interaction force and interaction free energy per unit area between parallel flat plates with high surface potential in the case of constant surface potential and constant surface charge is presented. It is a supplement to previous works (Langmuir, 17: 2167 (2001) and J. Colloid Interface Sci., 241: 81 (2001)). Although this approximation is essentially from simplifying the nonlinear PB equation in the case of high surface potentials, these approximate expressions work quite well for small plate separations for all values of the surface potentials.     

Approximate expression for the potential energy of double-layer interaction between two parallel similar plates with constant surface potential

An approximate equation for the electric potential distribution is obtained by linearizing the Poisson–Boltzmann with respect to the deviation of the electric potential from the surface potential. On the basis of the solution to this linearized Poisson–Boltzmann equation, an approximate expression is derived for the potential energy of the double layer interaction per unit area between two parallel similar plates at constant surface potential. It is found that this linearization approximation works quite well for small plate separations for all values of the surface potentials.     

The stability of a salt-free colloidal dispersion

The electrical potential for the case of two identical, planar parallel particles immersed in a salt-free medium, where the ionic species in the counterions come solely from those that dissociated from the surfaces, is evaluated. Analytical expressions for the electrical potential, the concentration of counterions, and the electrical energy are derived. We show that in a salt-free dispersion, if the separation distance between two particles is sufficiently far, the electrical repulsive force dominates, that is, the total energy is positive and does not have a secondary minimum, which is not the case for a dispersion where both coions and counterions are present. Also, the conditions used to calculate the critical coagulation concentration in the classic Derjaguin-Landau-Verwey-Overbeek theory become inappropriate and the Derjaguin approximation is inapplicable. We show that if the surface charge density exceeds approximately 0.04 Cm(2), the stability of a salt-free dispersion remains essentially the same. If the surface charge density is sufficiently high, the maximum separation distance between two particles below which coagulation occurs is in the ranges of [0,1 nm] and [1,7 nm] for the cases where the Hamaker constant is 10(-20) and 10(-19) J, respectively.     

Temporal modulation of a spatially periodic potential for kinetically governed oriented motion

This theoretical paper introduces an experimental protocol derived from the concept of Brownian motors in order to selectively confer an oriented motion to given charged reactants. Instead of maintaining permanently the system in nonequilibrium conditions, we propose a simple experimental trick to restore periodically a transient out-of-equilibrium regime: the reactive medium is alternately submitted to a sawtooth potential and to a potential ramp. The model provides approximate analytical expressions for the operating conditions allowing us to design the extraction from a mixture of any desired reactant characterized by its rate constants. The orders of magnitude suggest a possible implementation in microsystems where the present approach could be used for separation and analysis.     

圆盘微电极的稳态电化学行为

本文利用“均匀表面浓度”处理法研究了微盘电极稳态电化学行为, 给出了溶液中O和R均存在时可逆极化曲线的精确式, 及仅有O存在时不可逆和准可逆极亿曲线的近似式。还讨论了微电极的半径与速率常数K°的测量上限之间的关系。     

A complementarity for extended Langmuir method

The interaction free energy per unit area between parallel flat plates with high surface potential in the case of constant surface potential is again presented. It is complementarity for extended Langmuir method [G. Luo, H. Wang, J. Jin, Langmuir 17 (2001) 2167 and G. Luo, R. Feng, J. Jin, H. Wang, J. Colloid Interface Sci. 241 (2001) 81]. These approximate expressions work quite well for all values of the surface potentials so as the plates separations are small.     

Adsorption relaxation for nonionic surfactants under mixed barrier-diffusion and micellization-diffusion control

Relaxation processes of surfactant adsorption and surface tension, which are characterized by two specific relaxation times, are theoretically investigated. We are dealing with fluid interfaces and small initial deviations from equilibrium. For surfactant concentrations below the critical micellization concentration (CMC), we consider adsorption under mixed barrier-diffusion control. General analytical expressions are derived, which are convenient for both numerical computations and asymptotic analysis. Series expansions for the short- and long-time limit are derived. The results imply that the short-time asymptotics is controlled by the adsorption barrier, whereas the long-time asymptotics is always dominated by diffusion. Furthermore, for surfactant concentrations above the CMC, adsorption under mixed micellization-diffusion control is considered. Again, a general analytical expression is derived for the relaxation of surfactant adsorption and surface tension, whose long- and short-time asymptotics are deduced. The derived equations show that at the short times the relaxation is completely controlled by the diffusion, whereas the long-time asymptotics is affected by both demicellization and diffusion. The micellar effect is manifested as an exponential (rather than square-root) decay of the perturbation. The derived expressions are applied to process available experimental data for the nonionic surfactant Triton X-100 and to determine the respective demicellization rate constant.     

Approximate analytic expressions for the electrophoretic mobility of spherical soft particles

Approximate analytic expressions are derived for the electrophoretic mobility of a weakly charged spherical soft particle consisting of the particle core covered with a surface layer of polymers in an electrolyte solution. The particle core and the surface polymer layer may be charged or uncharged. The obtained electrophoretic mobility expressions, which involve neither numerical integration nor exponential integrals, are found to be in excellent agreement with the exact numerical results. It is also found that the obtained mobility expressions reproduce all the previously derived limiting expressions and approximate analytic expressions for the electrophoretic mobility of a weakly charged spherical soft particle.