电毛细管中非线性PB积分方程及其数值解

依据电毛细管非线性Ｐｏｉｓｓｏｎ Ｂｏｌｔｚｍａｎｎ微分方程的物理原理,导出其积分形式的ＰＢ方程．并采用数值迭代法给出相应方程的数值解．数值计算只用到电势Ψ的离散值,不需要Ψ的导数值,从根本上解决了因电势在管壁陡然变化引起数值解法的困难．文中给出的计算实例表明该算法是正确的、有效的和高精度的（相对误差小于０．０１％）,且在ＰＣ机上容易实现．     

用积分方程方法决定胶粒之间的空耗势

Binary components Ornstein-Zernike integral equation with the concentration of large particle component being set to zero was employed to study the depletion potential behavior between two large neutral colloid particles （modeled as hard spheres）immersed in a sea of small neutral solvent particles. The prediction for the depletion potential behavior compared well with simulation data and experimental data available in the literature. It is found that the Hansen-Verlet one phase criterion,based on the effective one component system with the present depletion potential,for the freezing transition is completely not suitable for the real binary components system. It is disclosed that the unsuitability is due to the volume term of the solid phase and liquid phase which can not be treated selfconsistently in the Hansen-Verlet one phase criterion.     

Solution of the Poisson-Boltzmann Equation for a Cylindrical Particle with a Limited Length： Functional Theoretical Approach

Introduction The electrostatic potentialΨis the most importantproperty for the electrical double layer( EDL) of acharged particle in an electrolyte solution[1—4]. Thispotential is characterized by the so-called Poisson-Bolt-zmann(PB) equation. The PB equation is a second-or-der nonlinear differential equation with a constant coef-ficient, except a flat-plate model, which cannot besolved analytically by the traditional method. To ourknowledge, apart from the numerical solution to thisequa…     

Electrical Properties of Highly Charged Spherical Reverse Micelle

A simpler expression of the surface charge density/surface potential relationship for spherical reverse micelles is obtained. In order to illustrate its application the approximate solution was used to calculate the thermodynamic characteristics and adsorption excesses. The approximate solution is based on the extended Langmuir's approach to the Poisson‐Boltzmann equation. The solutions for spherical reverse micelle are quite accurate provided that their radius and surface charge density are relatively larger. It is anticipated that this solution will be much easier to use in applications.     

泛函数迭代法求解反胶束内双电层电势

用泛函数分析理论中的迭代法求解了反胶束内关于双电层势的Poisson- Boltzmann（PB）方程，导出了泛电位下的第一、二次迭代的解析表达式。与 Debye-Huckel（DH）线性近似及计算机的数值解进行对比表明，迭代解不仅在低电 位条件下能与两者相符合，而且在高电位下与数值解在相吻合。     

Electrical Properties of a Highly Charged Cylindrical Particle in a Symmetric Electrolyte Solution

Based on the extended Langmuir's method, accurate analytic expressions for the electrical properties of cylindrical colloidal particle in a solution of symmetric electrolyte are derived from the nonlinear cylindrical Poisson-Boltzmann equation. Comparison is made with previous literature results. The present formulae have good agreement with literature values and are quite simple for practical use.     

球型胶体颗粒的表面电位和表面电荷密度的关系

胶体颗粒的表面电荷密度和表面电位之间的关系是颗粒表面的基本性质之一．要确定这个关系,需要解Poisson-Boltzmann(PB)方程,求出颗粒外的电位分布．然而对于球形颗粒,PB方程却没有解析解．Loeb等,求出了数值解,近似解析表达式虽然很多,也比较复杂,     

Transient behaviors of interacting electrical double layers

 The unsteady-state potential and space charge distributions between two identical, planar parallel charged surfaces immersed in an a:b electrolyte solution are examined theoretically. The effects of the ratio of the diffusivities of counterions and coions, D _con/D _co, the mean diffusivity (D _con D _co)^1/2, and the separation distance between two surfaces, H, on the transient distributions of electrical potential and space charges are investigated. The result of numerical simulation reveals that the extent of a system to reach its new equilibrium state depends largely on the magni-tude of a scaled time ν(=Dtκ²). For a fixed H, the greater the value of ν, the closer a system to its new equilibrium state. For constant H and ν, the smaller the ratio (D _con/D _co), the greater the deviation of a system from its new equilibrium state. In addition, the effect of D _con on this deviation is greater than that of D _co. Received: 3 September 1997 Accepted: 16 October 1997     

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)).     

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.     

Influence of nanoporous carbon electrode thickness on the electrochemical characteristics of a nanoporous carbon|tetraethylammonium tetrafluoroborate in acetonitrile solution interface

Electrochemical characteristics for the nanoporous carbon|Et₄NBF₄+acetonitrile interface have been studied by cyclic voltammetry and impedance spectroscopy methods. The influence of the electrolyte concentration and thickness of the nanoporous electrode material on the shape of the cyclic voltammetry and impedance curves has been established and the reasons for these phenomena are discussed. A value of zero charge potential, depending slightly on the structure and concentration of the electrolyte, the region of ideal polarizability and other characteristics have been established. The nanoporous nature of the carbon electrodes introduces a distribution of resistive and capacitive elements, giving rise to complicated electrochemical behaviour. Analysis of the complex plane plots shows that the nanoporous carbon|Et₄NBF₄+acetonitrile electrolyte interface can be simulated by an equivalent circuit, in which two parallel conduction paths in the solid and liquid phases are interconnected by the double-layer capacitance in parallel with the complex admittance of the hindered reaction of the charge transfer or of the partial charge transfer (i.e. adsorption stage limited) process. The values of the characteristic frequency depend on the electrolyte concentration and electrode potential, i.e. on the nature of the ions adsorbed at the surface of the nanoporous carbon electrode. The value of the solid state phase resistance established is independent of the thickness of the electrode material.     

Study on Electric Double Layer of a Cylindrical Particle with Functional Theoretical Approach

A new method, i.e. the iterative method in functional theory, was introduced to solve analytically the nonlinear Poisson-Boltzmann （PB） equation under general potential ψ condition for the electric double layer of a charged cylindrical colloid particle in a symmetrical electrolyte solution. The iterative solutions of ψ are expressed as functions of the distance from the axis of the particle with solution parameters： the concentration of ions c, the aggregation number of ions in a unit length m, the dielectric constant e, the system temperature T and so on. The relative errors show that generally only the first and the second iterative solutions can give accuracy higher than 97%. From the second iterative solution the radius and the surface potential of a cylinder have been defined and the corresponding values have been estimated with the solution parameters, Furthermore, the charge density, the activity coefficient of ions and the osmotic coefficient of solvent were also discussed,     

An algorithm for the calculation of the electrostatic repulsion between surface coated with a charge membrane

We propose a numerical procedure for the calculation of the electrostatic repulsion force between two identical, parallel surfaces immersed in anab electrolyte solution. These surfaces are coated with an ion-penetrable membrane carrying fixed charges. The amount of fixed charges is governed by the dissociation of the functional groups in the membrane phase. The effect of pH on the degree of dissociation of these functional groups is taken into account. The difficulty of extensive use of Jacobi elliptic function in the numerical treatment of Poisson-Boltzmann equation can be circumvented by resorting to the present algorithm.     

Double Layer at the Pt(111)–Aqueous Electrolyte Interface: Potential of Zero Charge and Anomalous Gouy–Chapman Screening

We report, for the first time, the observation of a Gouy–Chapman capacitance minimum at the potential of zero charge of the Pt(111)‐aqueous perchlorate electrolyte interface. The potential of zero charge of 0.3 V vs. NHE agrees very well with earlier values obtained by different methods. The observation of the potential of zero charge of this interface requires a specific pH (pH 4) and anomalously low electrolyte concentrations (<10^?3 m ). By comparison to gold and mercury double‐layer data, we conclude that the diffuse double layer structure at the Pt(111)‐electrolyte interface deviates significantly from the Gouy–Chapman theory in the sense that the electrostatic screening is much better than predicted by purely electrostatic mean‐field Poisson–Boltzmann theory.     

Colloidal chemistry as a guide to design intended dispersions of carbon nanomaterials

In this review, some established concepts from Colloidal Science and their application to graphene and carbon nanotubes dispersions in organic or aqueous media are highlighted to rationalize alternatives for some issues in terms of colloidal properties. Recent applications for carbon-based dispersions are presented, as well as van der Waals interactions in carbon materials and strategies to overcome these interactions, such as increasing electrostatic repulsion between dispersed particles, surface functionalization, or adsorption of passivation agents such as macromolecules, which are the basis of many dispersion and exfoliation procedures. The demonstration of how knowledge and fine control of colloidal interactions have been used to overcome several limitations, such as the preparation of stable and concentrated dispersions of carbon materials and keeping appreciable electrical conductivity, is presented. It is also showed that the same knowledge can help the development of more environmentally friendly carbon-based colloids as well as the improvement of similar systems as dispersions of two-dimensional materials.