Distribution of counterions in the double layer around a cylindrical polyion

The modified Poisson-Boltzmann (MPB) equation, and the Monte Carlo (MC) method, were applied to the cell model of a polyelectrolyte solution in order to calculate the distribution of counterions around a cylindrical polyion. Both methods suggest stronger binding of counterions to the polyion than predicted by the ordinary Poisson-Boltzmann (PB) equation. The inclusion of counterion-counterion correlation being neglected in the PB equation, leads to a better agreement of the calculated osmotic coefficients with those measured.     

Statistical mechanical theories of the electric double layer

A comparison is made of present-day statistical mechanical theories of the diffuse part of the electric double layer in aqueous 1-1 electrolyte at a charged plane interface. These theories fall into three categories: (1) the modified Poisson-Boltzmann equation (MPB) based on the Kirkwood-Loeb charging of an ion; (2) the adaption to the electric double layer of the Bogoliubov-Born-Green-Yvon (BBGY) hierarchy of integral equations; (3) the use of the Ornstein-Zernike equation (OZ) for the direct correlation functions of the pair interfacial plane wall-molecular particle, as derived by Henderson, Abraham and Barker (HAB). The HAB-OZ equation is used in conjunction with the mean spherical approximation (MSA) or hypernetted chain approximation (HNC). All the theories make use of the primitive model of the bulk electrolyte, so that inhomogeneity of the dielectric permittivity next to the plane wall is neglected. Except perhaps for a variation of the BBGY theory, which uses a closure based on electrical neutrality, all the theories predict oscillatory behaviour in potential distribution as a function of distance at the higher electrolyte concentrations. The HAB-OZ model has the defect that electrostatic imaging is not consistent with the assumptions of central forces and pair-wise additivity of ionic interactions. It is found that the MPB provides the best overall agreement with Monte Carlo calculations.     

纳滤膜对电解质溶液分离特性的理论研究(II): 混合电解质溶液

假定纳滤膜具有狭缝状孔, 使用纯水透过系数、膜孔径及膜表面电势来表征纳滤膜的分离特征, 用流体力学半径和无限稀释扩散系数表征了离子特性. 采用扩展Nernst-Planck方程、Donnan平衡模型和Poisson-Boltzmann理论描述了混合电解质溶液中离子在膜孔内的传递现象, 计算了三种商用纳滤膜(ESNA1-LF, ESNA1和LES90)对同阴离子、同阳离子和含四种离子的混合电解质体系中离子的截留率, 并与实验数据进行了比较. 计算结果表明, 电解质溶液中离子在纳滤膜孔内传递的主要机理是离子的扩散和电迁移, 纳滤膜对混合电解质溶液中离子的分离效果主要由空间位阻和静电效应决定. 该模型在低浓度时对含一价离子的混合电解质溶液通过纳滤膜的截留率计算结果比较准确, 但对高浓度或含高价离子的混合电解质溶液则偏差较大.     

Electrostatic potential and electroosmotic flow in a cylindrical capillary filled with symmetric electrolyte: analytic solutions in thin double layer approximation

The electrostatic potential in a capillary filled with electrolyte is derived by solving the nonlinear Poisson-Boltzmann equation using the method of matched asymptotic expansions. This approach allows obtaining an analytical result for arbitrary high wall potential if the double layer thickness is smaller than the capillary radius. The derived expression for the electrostatic potential is compared to numerical solutions of the Poisson-Boltzmann equation and it is shown that the agreement is excellent for capillaries with radii greater or equal to four times the electrical double layer thickness. The knowledge of the electrostatic potential distribution inside the capillary enables the derivation of the electroosmotic velocity flow profile in an analytical form. The obtained results are applicable to capillaries with radii ranging from nanometers to micrometers depending on the ionic strength of the solution.     

Electrostatic effects on the migration rates of ions between charged micelles. Consequences for fluorescence quenching

The rate of exchange of countenons between micelles has been calculated from numerical solutions of the Poisson-Boltzmann equation for micelles in cells of spherical symmetry, and solutions of a steady-state diffusion problem for the ion in the resulting potential. The results indicate a rapid increase of the exchange rate with total amphiphilc concentration, as has been observed.     

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。     

Electrokinetic microfluidic phenomena by a lattice Boltzmann model using a modified Poisson-Boltzmann equation with an excluded volume effect

A lattice Boltzmann model (LBM) for electrokinetic microfluidics recently proposed by us [J. Colloid Interface Sci. 263, 144 (2003); Langmuir 19, 3041 (2003)] is employed by consideration of a modified Poisson-Boltzmann equation including an excluded volume effect. In our study, pressure is considered as the only external force for liquid flow. As commonly used, the Poisson-Boltzmann equation assumes only point charge, the predicted microfluidic phenomena of KCl and LiCl electrolyte solutions are theoretically the same; these phenomena, however, have been found to be experimentally different. Our LBM in conjunction with the modified Poisson-Boltzmann equation are capable of explaining this discrepancy and the results are in good agreement with recent experimental data for KCl and LiCl electrolyte solutions in pressure-driven microchannel flow, suggesting that our proposed LBM can be employed to predict the more complex microfluidic systems that might be problematic using conventional methods and electrokinetic models.     

Comparison of the Nernst-Planck model and the Poisson-Boltzmann model for electroosmotic flows in microchannels

In the analysis of electroosmotic flows, the internal electric potential is usually modeled by the Poisson-Boltzmann equation. The Poisson-Boltzmann equation is derived from the assumption of thermodynamic equilibrium where the ionic distributions are not affected by fluid flows. Although this is a reasonable assumption for steady electroosmotic flows through straight microchannels, there are some important cases where convective transport of ions has nontrivial effects. In these cases, it is necessary to adopt the Nernst-Planck equation instead of the Poisson-Boltzmann equation to model the internal electric field. In the present work, the predictions of the Nernst-Planck equation are compared with those of the Poisson-Boltzmann equation for electroosmotic flows in various microchannels where the convective transport of ions is not negligible.     

A boundary element method for molecular electrostatics with electrolyte effects

A boundary element method is developed to compute the electrostatic potential inside and around molecules in an electrolyte solution. A set of boundary integral equations are derived based on the integral formulations of the Poisson equation and the linearized Poisson-Boltzmann equation. The boundary integral equations are then solved numerically after discretizing the molecular surface into a number of flat triangular elements. The method is applied to a spherical molecule for which analytical solutions are available. Use is made of both constant and linearly varying unknowns over the boundary elements, and the method is tested for various values of parameters such as the dielectric constant of the molecule, ionic strength, and the location of the interior point charge. The use of the boundary integral method incorporating the nonlinear Poisson-Boltzmann equation is also briefly discussed.     

Transient finite element analysis of electric double layer using Nernst-Planck-Poisson equations with a modified Stern layer

A finite element implementation of the transient nonlinear Nernst-Planck-Poisson (NPP) and Nernst-Planck-Poisson-modified Stern (NPPMS) models is presented. The NPPMS model uses multipoint constraints to account for finite ion size, resulting in realistic ion concentrations even at high surface potential. The Poisson-Boltzmann equation is used to provide a limited check of the transient models for low surface potential and dilute bulk solutions. The effects of the surface potential and bulk molarity on the electric potential and ion concentrations as functions of space and time are studied. The ability of the models to predict realistic energy storage capacity is investigated. The predicted energy is much more sensitive to surface potential than to bulk solution molarity.     

Specific ion effects in solutions of globular proteins: comparison between analytical models and simulation

Monte Carlo simulations have been performed for ion distributions outside a single globular macroion and for a pair of macroions, in different salt solutions. The model that we use includes both electrostatic and van der Waals interactions between ions and between ions and macroions. Simulation results are compared with the predictions of the Ornstein-Zernike equation with the hypernetted chain closure approximation and the nonlinear Poisson-Boltzmann equation, both augmented by pertinent van der Waals terms. Ion distributions from analytical approximations are generally very close to the simulation results. This demonstrates that properties that are related to ion distributions in the double layer outside a single interface can to a good approximation be obtained from the Poisson-Boltzmann equation. We also present simulation and integral equation results for the mean force between two globular macroions (with properties corresponding to those of hen-egg-white lysozyme protein at pH 4.3) in different salt solutions. The mean force and potential of mean force between the macroions become more attractive upon increasing the polarizability of the counterions (anions), in qualitative agreement with experiments. We finally show that the deduced second virial coefficients agree quite well with experimental results.     

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.     

Transient analysis of electroosmotic flow in a slit microchannel

The transient aspects of electroosmotic flow in a slit microchannel are studied. Exact solutions for the electrical potential profile and the transient electroosmotic flow field are obtained by solving the complete Poisson-Boltzmann equation and the Navier-Stokes equation under an analytical approximation for the hyperbolic sine function. The characteristics of the transient electroosmotic flow are discussed under influences of the electric double layer and the geometric size of the microchannel.     

A new time dependent approach for solving electrochemical interfaces Part I: theoretical considerations using lie group analysis

In this paper we introduce a nonlinear partial differential equation (nPDE) of the third order to the first time. This new model equation allows the extension of the Debye-Hückel Theory (DHT) considering time dependence explicitly. This also leads to a new formulation of the meaning of the nonlinear Poisson-Boltzmann Equation (PBE) and therefore we call it the modified Poisson-Boltzmann Equation (mPBE). In the present first part of this extensive study we derive the equation from the electromagnetics from a quasistatic perspective, or more precisely the electroquasistatic approximation (EQS). Our main focus will be the analysis via the Lie group formalism and since that up to now no symmetry calculation is available we believe that it seems indispensable to apply this method yielding a deeper insight into the behaviour of the solution manifold of this new equation following electrochemical considerations. We determine the classical Lie point symmetries including algebraic properties. Similarity solutions in a most general form and suitable nonlinear transformations are obtained. In addition, a note relating to potential and generalized symmetries is drawn. Moreover we show how the equation leads to approximate symmetries and we apply the method to the first time. The second part appearing shortly after will deal with algebraic solution methods and we shall show that closed-form solutions can be calculated without any numerical methods. Finally the third part will consider appropriate electrochemical experiments proving the model under consideration.     

Highly accurate biomolecular electrostatics in continuum dielectric environments

Implicit solvent models based on the Poisson-Boltzmann (PB) equation are frequently used to describe the interactions of a biomolecule with its dielectric continuum environment. A novel, highly accurate Poisson-Boltzmann solver is developed based on the matched interface and boundary (MIB) method, which rigorously enforces the continuity conditions of both the electrostatic potential and its flux at the molecular surface. The MIB based PB solver attains much better convergence rates as a function of mesh size compared to conventional finite difference and finite element based PB solvers. Consequently, highly accurate electrostatic potentials and solvation energies are obtained at coarse mesh sizes. In the context of biomolecular electrostatic calculations it is demonstrated that the MIB method generates substantially more accurate solutions of the PB equation than other established methods, thus providing a new level of reference values for such models. Initial results also indicate that the MIB method can significantly improve the quality of electrostatic surface potentials of biomolecules that are frequently used in the study of biomolecular interactions based on experimental structures.     

Simplifications of the Poisson-Boltzmann Equation for the Electrostatic Interaction of Close Hydrophilic Surfaces in Water

Simple solutions of the Poisson-Boltzmann (PB) equation for the electrostatic double-layer interaction of close, planar hydrophilic surfaces in water are evaluated. Four routes, being the weak overlap approximation, the Debye-Hückel linearization based on low electrostatic potentials, the Ettelaie-Buscall linearization based on small variations in the potential, and a new approach based on the fact that concentrations are virtually constant in the gap between close surfaces, are discussed. The Ettelaie-Buscall and constant-concentration approach become increasingly accurate for closer surfaces and are exact for touching surfaces, while the weak overlap approximation is exact for an isolated surface. The Debye-Hückel linearization is valid as long as potentials remain low, independent of separation. In contrast to the Ettelaie-Buscall approach and the weak overlap approximation, the Debye-Hückel linearization and constant-concentration approach can also be used for systems containing multivalent ions. Simulations in which the four approaches are compared with the PB equation for the constant-charge model, the constant-potential model, as being used in the DLVO theory, and the charge-regulation model are presented. Copyright 2001 Academic Press.     

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.     

The surface potential of a spherical colloid particle: functional theoretical approach

By using the iterative method in functional analysis, the potential of the electrical double layer of a spherical colloid particle, which is represented by the so-called Poisson-Boltzmann (PB) equation, has been solved analytically under general potential conditions. With the help of the diagram method in mathematics, the surface potential of the particle has been defined from the second iterative solution. The influence of the parameters included in the solutions on the surface potential has been studied. The results show that the surface potential of the particle increases as the temperature of the system, the aggregation number, and the concentration of ions increase, but decreases with an increase in the dielectric constant and the valence of the ions. The corresponding space charge density also has been illustrated in this work.