Analytical solution of Poisson–Boltzmann equation for interacting plates of arbitrary potentials and same sign

Efficient calculation of electrostatic interactions in colloidal systems is becoming more important with the advent of such probing techniques as atomic force microscopy. Such practice requires solving the nonlinear Poisson–Boltzmann equation (PBE). Unfortunately, explicit analytical solutions are available only for the weakly charged surfaces. Analysis of arbitrarily charged surfaces is possible only through cumbersome numerical computations. A compact analytical solution of the one-dimensional PBE is presented for two plates interacting in symmetrical electrolytes. The plates can have arbitrary surface potentials at infinite separation as long they have the same sign. Such a condition covers a majority of the colloidal systems encountered. The solution leads to a simple relationship which permits determination of surface potentials, surface charge densities, and electrostatic pressures as a function of plate separation H for different charging scenarios. An analytical expression is also presented for the potential profile between the plates for a given separation. Comparison of these potential profiles with those obtained by numerical analysis shows the validity of the proposed solution.     

Hybrid boundary element and finite difference method for solving the nonlinear Poisson-Boltzmann equation

A hybrid approach for solving the nonlinear Poisson-Boltzmann equation (PBE) is presented. Under this approach, the electrostatic potential is separated into (1) a linear component satisfying the linear PBE and solved using a fast boundary element method and (2) a correction term accounting for nonlinear effects and optionally, the presence of an ion-exclusion layer. Because the correction potential contains no singularities (in particular, it is smooth at charge sites) it can be accurately and efficiently solved using a finite difference method. The motivation for and formulation of such a decomposition are presented together with the numerical method for calculating the linear and correction potentials. For comparison, we also develop an integral equation representation of the solution to the nonlinear PBE. When implemented upon regular lattice grids, the hybrid scheme is found to outperform the integral equation method when treating nonlinear PBE problems. Results are presented for a spherical cavity containing a central charge, where the objective is to compare computed 1D nonlinear PBE solutions against ones obtained with alternate numerical solution methods. This is followed by examination of the electrostatic properties of nucleic acid structures.     

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.     

359 - The electrochemical oxidation of three proteins: RNAase A,bovine serum albumin and concanavalin A at solid electrodes

The electrochemical oxidation of three proteins: RNAase A, bovine serum albumin (BSA) and Concanavalin A, has been investigated on solid electrodes (gold, platinum, vitreous carbon, carbon paste electrode) using a.c. voltammetry and cyclic voltammetry. All the three proteins are adsorbed on these surfaces, but oxidation peaks on voltammograms are only well developed on carbon paste electrode (CPE), therefore the main experimental results presented here will refer to CPE. Tyrosinyl and tryptophanyl residues can first be oxidized over the range of pH 1 to 7; when oxidation potentials are more positive, other amino acids residues (histidine, cystine…) can also be oxidized. In both cases proteins adsorption is prerequisite. The adsorption mainly takes place at negative potentials (less than potential of zero charge) by the intermediary of electrostatic complex with the surface, this electrostatic adsorption is immediately followed by a rearrangement of the proteins at the electrode surface which leads to the formation of non electrostatic irreversible adsorbed complex.     

The Calculation of Interfacial Tension and Electrostatic Free Energy of Spherical Ionic Micelles with High Surface Potentials

Based on extended Langmuir's method on the dressed micelles, approximate expressions for the calculation of interfacial tension and electrostatic free energy of spherical ionic micelles with high surface potentials have been presented. These expressions are derived from nonlinear Poisson‐Boltzmann equation. The present formulae for the calculation of interfacial tension and electrostatic free energy of spherical ionic micelles are in quite good agreement with Hayter's results.     

Electrodiffusion: a continuum modeling framework for biomolecular systems with realistic spatiotemporal resolution

A computational framework is presented for the continuum modeling of cellular biomolecular diffusion influenced by electrostatic driving forces. This framework is developed from a combination of state-of-the-art numerical methods, geometric meshing, and computer visualization tools. In particular, a hybrid of (adaptive) finite element and boundary element methods is adopted to solve the Smoluchowski equation (SE), the Poisson equation (PE), and the Poisson-Nernst-Planck equation (PNPE) in order to describe electrodiffusion processes. The finite element method is used because of its flexibility in modeling irregular geometries and complex boundary conditions. The boundary element method is used due to the convenience of treating the singularities in the source charge distribution and its accurate solution to electrostatic problems on molecular boundaries. Nonsteady-state diffusion can be studied using this framework, with the electric field computed using the densities of charged small molecules and mobile ions in the solvent. A solution for mesh generation for biomolecular systems is supplied, which is an essential component for the finite element and boundary element computations. The uncoupled Smoluchowski equation and Poisson-Boltzmann equation are considered as special cases of the PNPE in the numerical algorithm, and therefore can be solved in this framework as well. Two types of computations are reported in the results: stationary PNPE and time-dependent SE or Nernst-Planck equations solutions. A biological application of the first type is the ionic density distribution around a fragment of DNA determined by the equilibrium PNPE. The stationary PNPE with nonzero flux is also studied for a simple model system, and leads to an observation that the interference on electrostatic field of the substrate charges strongly affects the reaction rate coefficient. The second is a time-dependent diffusion process: the consumption of the neurotransmitter acetylcholine by acetylcholinesterase, determined by the SE and a single uncoupled solution of the Poisson-Boltzmann equation. The electrostatic effects, counterion compensation, spatiotemporal distribution, and diffusion-controlled reaction kinetics are analyzed and different methods are compared.     

Some comments and corrections regarding the calculation of electrostatic potential derivatives using the Ewald summation technique

A review of the literature on the calculation of electrostatic potentials, fields, and field gradients in systems consisting of charges and dipoles using the Ewald summation technique is presented. Discrepancies between the previous formulas are highlighted, and an error in the derivation of the reciprocal contributions to the electrostatic field and field gradient is corrected. The new formulas for the field and field gradient are shown to exhibit a termwise identity with the ones for the electrostatic energy.     

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.     

Fitting properties from density functional theory based molecular dynamics simulations to parameterize a rigid water force field

In the quest towards coarse-grained potentials and new water models, we present an extension of the force matching technique to parameterize an all-atom force field for rigid water. The methodology presented here allows to improve the matching procedure by first optimizing the weighting exponents present in the objective function. A new gauge for unambiguously evaluating the quality of the fit has been introduced; it is based on the root mean square difference of the distributions of target properties between reference data and fitted potentials. Four rigid water models have been parameterized; the matching procedure has been used to assess the role of the ghost atom in TIP4P-like models and of electrostatic damping. In the former case, burying the negative charge inside the molecule allows to fit better the torques. In the latter, since short-range interactions are damped, a better fit of the forces is obtained. Overall, the best performing model is the one with a ghost atom and with electrostatic damping. The approach shown in this paper is of general validity and could be applied to any matching algorithm and to any level of coarse graining, also for non-rigid molecules.     

Long-Range Electrostatic Interaction between a Charged Wall and Two Similarly Charged Colloidal Spheres at Low Surface Potentials

The long-range electrostatic interaction between a pair of similarly charged colloidal spheres and a charged planar wall at low surface potentials is theoretically investigated. The linear Poisson-Boltzmann equation (PBE) and the point charge approximation of the charged sphere are used. The electrical potential distribution in the electrolyte solution is found from the PBE at the constant surface potentials using the image charge method. The electrostatic forces acting on the spheres are then calculated. The results show that the repulsive interaction between a pair of similarly charged colloidal spheres clearly decreases when a charged wall appears nearby, but it is impossible for an attractive force to emerge at the scaled surface potentials less than 1. There is, however, an attractive force between the charged wall and the similarly charged colloidal spheres, when the surface potential zetap on the wall is sufficiently higher than the surface potential zetas on the spheres to make zetap > zetasexp(kappah) (h is the distance from the wall to the sphere center). In this case, there are negative surface charges on the spheres at positive surface potential zetas. It is these negative charges that produce the above attraction. Copyright 1999 Academic Press.     

A computational analysis of the electrostatic potentials and relative bond strengths of hydrazine and some of its 1,1-dimethyl derivatives

We have carried out a computational study of hydrazine and five of its 1,1-dimethyl derivatives, focusing on their electrostatic potentials and relative bond strengths. Our approach has involved the calculation of ab initio self-consistent-field molecular orbital wave functions and molecular properties using the GAUSSIAN 82 system of programs. The electrostatic potentials of the hydrazines possess negative regions of varying sizes and strengths associated with the nitrogens of the α-diamino linkages. Through an analysis of the positions of the most negative potentials of these regions, we have obtained directly the dihedral angles between the nitrogen lone pairs in these systems. Our use of the electrostatic potential to obtain these angles is a direct and general approach, in contrast to indirect procedures used in the past. We find this dihedral angle to be close to 90° in hydrazine, with variations in the substituted hydrazines that depend on the nature of the substituents. A highly polar structure is found for 1-chloromethyl-1-methylhydrazine, which involves a delocalization of electronic charge from the substituted nitrogen towards the CH₂Cl group. We find that substituents able to withdraw significant amounts of electronic density from the central nitrogen lone pair regions, either through resonance or by induction, have a slight bond strengthening effect on the central N-N bond. This is attributed to a decrease in the repulsion between the weakened nitrogen lone pair regions. The difficulties encountered in seeking the controlled oxidation of hydrazine to nitro derivatives may be due, in part, to the fact that two factors which would favor this, highly negative nitrogen potentials and strong N-N bonds, are opposing in nature.     

Cumulative and discrete similarity analysis of electrostatic potentials and fields

A new index suitable for computing molecular similarity based upon the similarity of molecular properties such as electrostatic potentials or electrostatic fields is presented in two forms. For one form of the present index, general conditions are established for which a linear measure of similarity is obtained. An illustrative example is provided in which the electrostatic field and electrostatic potential of guanine obtained from different wave functions are compared. © 1993 John Wiley & Sons, Inc.     

SANS and SAXS analysis of charged nanoparticle adsorption at oil-water interfaces

A systematic study of the adsorption of charged nanoparticles at dispersed oil-in-water emulsion interfaces is presented. The interaction potentials for negatively charged hexadecane droplets with anionic polystyrene latex particles or cationic gold particles are calculated using DLVO theory. Calculations demonstrate that increased ionic strength decreases the decay length of the electrostatic repulsion leading to enhanced particle adsorption. For the case of anionic PS latex particles, the energy barrier for particle adsorption is also reduced when the surface charge is neutralized through changes in pH. Complementary small-angle scattering experiments show that the highest particle adsorption for PS latex occurs at moderate ionic strength and low pH. For cationic gold particles, simple DLVO calculations also explain scattering results showing that the highest particle adsorption occurs at neutral pH due to the electrostatic attraction between oppositely charged surfaces. This work demonstrates that surface charges of particles and oil droplets are critical parameters to consider when engineering particle-stabilized emulsions.     

Suitability of the PM3-derived molecular electrostatic potentials

A systematic study of the suitability of PM3-derived molecular electrostatic potentials (MEPs) is presented. Forty-six MEP minima, 81 electrostatic charges, and 17 electrostatic dipoles were determined at the PM3 level and compared with those obtained from the ab initio 6-31G* wave function, as well as from the semiempirical MNDO and AM1 wave functions. The statistical results of the comparison analysis between semiempirical and ab initio 6-31G* MEPs show that PM3 is in general reliable for the study of the MEP minima but a mediocre method as a source of electrostatic charges. © 1993 John Wiley & Sons, Inc.     

Concentration Polarization of Interacting Solute Particles in Cross-Flow Membrane Filtration

A theoretical approach for predicting the influence of interparticle interactions on concentration polarization and the ensuing permeate flux decline during cross-flow membrane filtration of charged solute particles is presented. The Ornstein-Zernike integral equation is solved using appropriate closures corresponding to hard-spherical and long-range solute-solute interactions to predict the radial distribution function of the solute particles in a concentrated solution (dispersion). Two properties of the solution, namely the osmotic pressure and the diffusion coefficient, are determined on the basis of the radial distribution function at different solute concentrations. Incorporation of the concentration dependence of these two properties in the concentration polarization model comprising the convective-diffusion equation and the osmotic-pressure governed permeate flux equation leads to the coupled prediction of the solute concentration profile and the local permeate flux. The approach leads to a direct quantitative incorporation of solute-solute interactions in the framework of a standard theory of concentration polarization. The developed model is used to study the effects of ionic strength and electrostatic potential on the variations of solute diffusivity and osmotic pressure. Finally, the combined influence of these two properties on the permeate flux decline behavior during cross-flow membrane filtration of charged solute particles is predicted. Copyright 1999 Academic Press.     

The Poisson-Boltzmann model for tRNA: Assessment of the calculation set-up and ionic concentration cutoff

Using tRNA molecule as an example, we evaluate the applicability of the Poisson-Boltzmann model to highly charged systems such as nucleic acids. Particularly, we describe the effect of explicit crystallographic divalent ions and water molecules, ionic strength of the solvent, and the linear approximation to the Poisson-Boltzmann equation on the electrostatic potential and electrostatic free energy. We calculate and compare typical similarity indices and measures, such as Hodgkin index and root mean square deviation. Finally, we introduce a modification to the nonlinear Poisson-Boltzmann equation, which accounts in a simple way for the finite size of mobile ions, by applying a cutoff in the concentration formula for ionic distribution at regions of high electrostatic potentials. We test the influence of this ionic concentration cutoff on the electrostatic properties of tRNA.     

A novel method for measuring zeta potentials of solid–liquid interfaces

A novel method for measuring the zeta potential of PDMS–electrolyte solution interface is presented in this paper. When an electrolyte solution passes the electrode coated with PDMS film, an electrical current will be induced due to the electrostatic potential difference between the PDMS–air interface and the PDMS–electrolyte solution interface. Based on this principle, the relationship between the measured electrical signal and the zeta potential of PDMS–electrolyte solution interface is experimentally investigated. The results show that the magnitude of the measured signal is linearly proportional to this potential difference. An empirical correlation between the zeta potential and the measured voltage signal was obtained. Good agreement was found when comparing the zeta potential calculated by this empirical equation with that reported in the published journal papers. The zeta potential measurement method presented in this paper is simple and accurate and can be used for measuring zeta potentials of other dielectric–electrolyte interfaces.     

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.     

Microflotation Suppression and Enhancement Caused by Particle/Bubble Electrostatic Interaction

The processes of attachment and detachment of small or medium-sized particles to relatively large bubbles during microflotation are considered in terms of the heterocoagulation theory. Calculations are made for the conditions that the surface potentials are of similar sign and constant, that one of the surface potentials is small, that hydrophobic attraction is absent, and that there are no surface deformations. Under these conditions bubble-particle aggregates may form as a result of an electrostatic attraction which exceeds the repulsive van der Waals force at intermediate distances. Next to electrostatic and van der Waals forces, hydrodynamic and gravitational forces are considered. These forces may overcome the electrostatic repulsion at large distances and promote particle bubble attachment. Strong electrostatic attraction at small distances, arising at a large difference of the surface potentials of the bubble and the particle and of low electrolyte concentrations, can prevent subsequent detachment by hydrodynamic and gravitational forces. With increasing electrolyte concentration the electrostatic barrier increases and the attractive electrostatic force diminishes. As a result, a critical electrolyte concentration for microflotation exists. Above this concentration attachment may still occur but it is followed by detachment. At lower electrolyte concentrations the electrostatic attractive force prevents the detachment. The dependence of the critical electrolyte concentration on the values of the bubble and particle potentials and the Hamaker constant is calculated. The critical concentration does not depend on particle or bubble size if the absolute values of the total detachment force and the total pressing force coincide, which is the case for Stokes and potential flow. For every electrolyte concentration lower than the critical value there are two critical particle sizes that limit the flotation possibility. For small particle sizes attachment is impossible because the pressing force is smaller than the electrostatic barrier. For large particle sizes detachment cannot be prevented because the detachment force exceeds the maximum electrostatic attraction. A microflotation domain of intermediate particle sizes exists in which irreversible heterocoagulation occurs. Copyright 2001 Academic Press.