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

Summary 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-Hückel approximation is also considered.

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Zusammenfassung Es wird eine Methode für numerische Berechnung der Wechselwirkung der diffusen Doppelschichten zwischen zwei parallelen gleichen Platten mit konstanter Oberflächenladungsdichte in einer symmetrischen Elektrolytlösung gezeigt. Die nichtlineare Poisson-Boltzmannsche Gleichung wird benutzt, und der Einfluß des durch die Überlappung der Doppelschichten induzierten elektrischen Feldes innerhalb der Platten wird berücksichtigt. Numerische Tabellen werden für die Kraft und die potentielle Energie der Wechselwirkung gegeben. Die Anwendbarkeit der Debye-Hückel-Näherung wird auch untersucht.

     

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.     

Diffuse double layer interaction between two parallel plates with constant surface charge density in an electrolyte solution

Summary The double layer interaction between two parallel metallic plates with adsorbed surface charge of constant density in a symmetrical electrolyte solution is considered. Numerical calculations using the nonlinearPoisson-Boltzmann equation are made. Analytic expressions for the interaction are also obtained by using theDebye-Hückel approximation. The double layer interaction is found to be influenced by true charges induced on the plate surface due to electrostatic induction. This influence becomes important at small plate separations. It is also shown that the force and potential energy of interaction between metallic plates are small compared with those between insulating plates.

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Zusammenfassung Die Wechselwirkung der Doppelschichten zwischen zwei parallelen metallischen Platten mit der konstanten Dichte der adsorbierten Oberflächenladung in einer symmetrischen Elektrolytlösung wird betrachtet. Numerische Berechnungen werden mittels der nichtlinearenPoisson-Boltzmannschen Gleichung ausgeführt. Durch Benutzung der Debye-Hückel-Näherung werden auch analytische Ausdrücke für die Wechselwirkung gewonnen. Es zeigt sich, daß die Wechselwirkung der Doppelschichten von reellen Ladungen, die an der metallischen Oberfläche durch elektrostatische Induktion induziert werden, beeinflußt wird. Dieser Einfluß wird besonders wichtig für kleine Plattenabstände. Die Kraft und die potentielle Energie der Wechselwirkung zwischen metallischen Platten sind kleiner als die zwischen Isolierplatten.

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With 4 figures and 3 tables     

The interaction energy between two parallel plates with constant surface charge density

On the basis of Langmuir's suggestion we simplify the Poisson-Boltzmann equation and derive the relation of surface potential, potential midway, and the plate distance. Thus we obtain the interaction force and energy equations between two dissimilar plates in the case of constant surface charge density. Agreement with the exact numerical values of the interaction of dissimilar plates is good. This method may not only apply to the cases of high constant potential but to the case of high constant charge density.     

The surface charge density/surface potential relationship and the Poisson-Boltzmann equation

A procedure is described which provides an approximate analytic expression for the relationship between the surface charge density and the surface potential of a spherical or cylindrical colloidal particle in a general type of electrolyte. The first approximation is found and the approximation error is given. Also derived are the fourth approximation for a symmetric z-z type electrolyte and the second approximation for a symmetric z-z, 2z-2z type electrolyte.     

Finite volume solution of the modified Poisson-Boltzmann equation for two colloidal particles

The double layer forces between spherical colloidal particles, according to the Poisson-Boltzmann (PB) equation, have been accurately calculated in the literature. The classical PB equation takes into account only the electrostatic interactions, which play a significant role in colloid science. However, there are at, and above, biological salt concentrations other non-electrostatic ion specific forces acting that are ignored in such modelling. In this paper, the electrostatic potential profile and the concentration profile of co-ions and counterions near charged surfaces are calculated. These results are obtained by solving the classical PB equation and a modified PB equation in bispherical coordinates, taking into account the van der Waals dispersion interactions between the ions and both surfaces. Once the electrostatic potential is known we calculate the double layer force between two charged spheres. This is the first paper that solves the modified PB equation in bispherical coordinates. It is also the first time that the finite volume method is used to solve the PB equation in bispherical coordinates. This method divides the calculation domain into a certain number of sub-domains, where the physical law of conservation is valid, and can be readily implemented. The finite volume method is implemented for several geometries and when it is applied to solve PB equations presents low computational cost. The proposed method was validated by comparing the numerical results for the classical PB calculations with previous results reported in the literature. New numerical results using the modified PB equation successfully predicted the ion specificity commonly observed experimentally.     

Efficient solution technique for solving the Poisson-Boltzmann equation

The Poisson-Boltzmann (PB) equation has been extensively used to analyze the energetics and structure of proteins and other significant biomolecules immersed in electrolyte media. A new highly efficient approach for solving PB-type equations that allows for the modeling of many-atoms structures such as encountered in cell biology, virology, and nanotechnology is presented. We accomplish these efficiencies by reformulating the elliptic PB equation as the long-time solution of an advection-diffusion equation. An efficient modified, memory optimized, alternating direction implicit scheme is used to integrate the reformulated PB equation. Our approach is demonstrated on protein composites (a polio virus capsid protomer and a pentamer). The approach has great potential for the analysis of supramillion atoms immersed in a host electrolyte.     

General solution for Poisson-Boltzmann equation in semiinfinite planar symmetry

Self-assembled multilayer thin films have been prepared on Au substrate by alternate surface derivatization with L-cysteine hydrochloride and cupric perchlorate. The layer-by-layer structure at each step of multilayer formation was investigated by X-ray photoelectron spectroscopy. The measurements indicate that there are two structure modes in the multilayers. One is that Cu(2+) sandwiches between two amino acid groups. The other one is that Cu(+) is bonded through disulfide and thiolate. This process is also confirmed by cyclic voltammetry of Cu ion at different self-assembled multilayers. Steps further on will lead to repeated multilayer films.     

On the limiting law solution of the cylindrical Poisson-Boltzmann equation for polyelectrolytes

Upper and lower bounds are given for the behaviour of the Poisson-Boltzmann potential of a highly charged, cylindrical polyelectrolyte in excess salt solution as the salt correntration or cylinder radius tends to zero. The exact singular behaviour of the potential very close to the cylinder is given in terms of a delta function.     

Analytical solution of the Schrödinger equation for a double minimum morse potential and application to intramolecular inversion

An exact solution is obtained for the one-dimensional time-independent Schrödinger equation with a symmetric double minimum potential constructed from two Morse potentials. This model potential is used to describe the inversion motions in NH₃, ND₃, and NT₃, and its adequacy is discussed.     

Electrostatic repulsion between two parallel plates covered with polymer brush layers

An expression for the electrostatic repulsive force is obtained for two parallel similar plates immersed in an electrolyte solution at separation h covered with a uniformly charged polymer brush layer of intact thickness d _o under compression (h<2d _o) after the two brushes come into contact. It is assumed that when the two brushes come into contact, they are squeezed against each other but do not interdigitate. The electrostatic repulsive force is found to increase with decreasing h as 1/h for highly charged brushes and as 1/h ² for weakly charged brushes. This is in contrast to the interaction force between the brush layers before contact (h≥2d _o), which is essentially proportional to exp[−κ(h−2d _o)] (where κ is the Debye–Hückel parameter). It is also shown that the interaction force for highly charged brushes, which becomes independent of the electrolyte concentration, can be comparable in magnitude to the steric repulsive forces between the brushes resulting from osmotic repulsion and the elastic energy of the brushes. Received: 21 October 1998 Accepted in revised form: 25 December 1998     

Accurate solution of multi-region continuum biomolecule electrostatic problems using the linearized Poisson-Boltzmann equation with curved boundary elements

We present a boundary-element method (BEM) implementation for accurately solving problems in biomolecular electrostatics using the linearized Poisson-Boltzmann equation. Motivating this implementation is the desire to create a solver capable of precisely describing the geometries and topologies prevalent in continuum models of biological molecules. This implementation is enabled by the synthesis of four technologies developed or implemented specifically for this work. First, molecular and accessible surfaces used to describe dielectric and ion-exclusion boundaries were discretized with curved boundary elements that faithfully reproduce molecular geometries. Second, we avoided explicitly forming the dense BEM matrices and instead solved the linear systems with a preconditioned iterative method (GMRES), using a matrix compression algorithm (FFTSVD) to accelerate matrix-vector multiplication. Third, robust numerical integration methods were employed to accurately evaluate singular and near-singular integrals over the curved boundary elements. Fourth, we present a general boundary-integral approach capable of modeling an arbitrary number of embedded homogeneous dielectric regions with differing dielectric constants, possible salt treatment, and point charges. A comparison of the presented BEM implementation and standard finite-difference techniques demonstrates that for certain classes of electrostatic calculations, such as determining absolute electrostatic solvation and rigid-binding free energies, the improved convergence properties of the BEM approach can have a significant impact on computed energetics. We also demonstrate that the improved accuracy offered by the curved-element BEM is important when more sophisticated techniques, such as nonrigid-binding models, are used to compute the relative electrostatic effects of molecular modifications. In addition, we show that electrostatic calculations requiring multiple solves using the same molecular geometry, such as charge optimization or component analysis, can be computed to high accuracy using the presented BEM approach, in compute times comparable to traditional finite-difference methods.     

Approximate expression for interaction of a two-parallel-plane double layer at constant surface potential and constant surface charge

A fast convergent formula for the interactional energy between two parallel plates at constant surface potential and constant surface charge is derived by a simple method of the series expansion, and the numerical result of only the first five terms of the series is excellent for the dimensionless surface potential of colloidal particle y(0)< or =4.     

Boundary element solution of the linear Poisson-Boltzmann equation and a multipole method for the rapid calculation of forces on macromolecules in solution

The Poisson-Boltzmann equation is widely used to describe the electrostatic potential of molecules in an ionic solution that is treated as a continuous dielectric medium. The linearized form of this equation, applicable to many biologic macromolecules, may be solved using the boundary element method. A single-layer formulation of the boundary element method, which yields simpler integral equations than the direct formulations previously discussed in the literature, is given. It is shown that the electrostatic force and torque on a molecule may be calculated using its boundary element representation and also the polarization charge for two rigid molecules may be rapidly calculated using a noniterative scheme. An algorithm based on a fast adaptive multipole method is introduced to further increase the speed of the calculation. This method is particularly suited for Brownian dynamics or molecular dynamics simulations of large molecules, in which the electrostatic forces must be calculated for many different relative positions and orientations of the molecules. It has been implemented as a set of programs in C++, which are used to study the accuracy and speed of this method for two actin monomers.     

Analytical solution for quenching of hot rolled aluminium plates without passing through C-curve

Journal of Thermal Analysis and Calorimetry - At the end of the hot rolling process, quenching is deployed for obtaining the desired microstructure and mechanical properties. During the quenching...     

Electrostatic binding energy calculation using the finite difference solution to the linearized Poisson-Boltzmann equation: Assessment of its accuracy

A full account of how to calculate the electrostatic binding energy using the finite difference solution to the linearized Poisson-Boltzmann equation (FDPB) for protein-ligand systems is described. The following tests show that the statistical and systematic errors due to discrete grid representation of molecular shape and charges amount to about 1% and 5% of calculated binding energy difference, respectively. The greater accuracy results from a three-stage error cancellation: first in ΔG_s, then ΔΔGd_s, and finally ΔΔG_ele. We conclude in this study that the intrinsic error of FDPB is mostly canceled in computing binding energy differences. Among the parameters examined, the partial charge, dielectric constant, and radius of solvent can influence the calculated results most. © 1996 by John Wiley & Sons, Inc.