Poisson-Boltzmann equation structure

The presence of the diffuse layer determines the shape of the capacitance-potential curves. For a majority of systems, models describing the double-layer structure are oversimplified because of taking into account only the charge of ions and neglecting their specific nature. Recently, these problems have been analyzed using new theories such as the modified Poisson-Boltzmann equation, later developed by Lamper-ski. The double-layer capacitanties calculated from these equations are... 

A more detailed view of the dynamies of a ehromatin chain was achieved in a recent Brownian dynamics simulation by Beard and Schlick [65]. Like in previous work, the DNA is treated as a segmented elastic chain however, the nueleosomes are modeled as flat cylinders with the DNA attached to the cylinder surface at the positions known from the crystallographic structure of the nucleosome. Moreover, the electrostatic interactions are treated in a very detailed manner the charge distribution on the nucleosome core particle is obtained from a solution to the non-linear Poisson-Boltzmann equation in the surrounding solvent, and the total electrostatic energy is computed through the Debye-Hiickel approximation over all charges on the nucleosome and the linker DNA. 

The purpose of the present chapter is to introduce some of the basic concepts essential for understanding electrostatic and electrical double-layer pheneomena that are important in problems such as the protein/ion-exchange surface pictured above. The scope of the chapter is of course considerably limited, and we restrict it to concepts such as the nature of surface charges in simple systems, the structure of the resulting electrical double layer, the derivation of the Poisson-Boltzmann equation for electrostatic potential distribution in the double layer and some of its approximate solutions, and the electrostatic interaction forces for simple geometric situations. Nonetheless, these concepts lay the foundation on which the edifice needed for more complicated problems is built. 

A new theory of electrolyte solutions is described. This theory is based on a Debye-Hiickel model and modified to allow for the mutual polarization of ions. From a general solution of the linearized Poisson-Boltzmann equation, an expression is derived for the activity coefficient of a central polarized ion in an ionic atmosphere of non-spherical symmetry that reduces to the Debye-Hiickel limiting laws at infinite dilution. A method for the simultaneous charging of an ion and its ionic cloud is developed to allow for ionic polarization. Comparison of the calculated activity coefficients with experimental values shows that the characteristic shapes of the log y vs. concentration curves are well represented by the theory up to moderately high concentrations. Some consequences in relation to the structure of electrolyte solutions are discussed. 

The Structured Ionic Cloud. We base our treatment on the general solution of the linearized Poisson-Boltzmann equation (LPBE, Equation 1). 

Metals, such as platinum, are usually introduced to improve the electron-hole separation efficiency. In order to analyze the energy structure of the metal-loaded particulate semiconductor, we solved the two-dimensional Poisson-Boltzmann equation.3) When the metal is deposited to the semiconductor by, for example, evaporation, a Schottky barrier is usually formed.45 For the Schottky type contact, the barrier height increases with an increase of the work function of the metal,4 which should decrease the photocatalytic activity. However, higher activity was actually observed for the metal with a higher work function.55 This results from the fact that ohmic contact with deposited metal particles is established in photocatalysts when the deposited semiconductor is treated by heat65 or metal is deposited by the photocatalytic reaction.75 Therefore, in the numerical computation we assumed ohmic contact at the energy level junction of the metal and semiconductor. 

The mean field potential for this system, a solution of the linear Poisson-Boltzmann equation, Eq. (32), will appropriately have the same periodic structure as the surface boundary condition. Thus, we expect that if/ will have the Fourier series,... 

A hierarchy of approximations now exists for calculating interactions between a charged particle and a charged, planar interface in electrolyte solutions. At moderate surface potentials less than approximately 2(kT/e the linear Poisson-Boltzmann equation provides a good approximation in many circumstances, provided the solution is a 1 1 electrolyte at low to moderate ionic strength. The relative simplicity of the linear equation makes it particularly useful for examining problems that are complicated in other ways, such as interactions involving many particles, interactions with deformable interfaces, and interactions where the detailed structure and properties of the particle (or macromolecule) play an important role. 

Fogolari F, Brigo A, Molinari H (2002) The Poisson-Boltzmann equation for biomolecular electrostatics a tool for structural biology, J Mol Recognit, 15 377—392... 

Thus, the electrostatic potential at the interface can be calculated with high accuracy, using either the Gouy-Chapman-Stern method or the precise nonlinear Poisson-Boltzmann equations, depending on the level of structural accuracy of the domain under study. 

Those using analytical methods (often via perturbation approaches, which necessitate some level of approximation such as linearization of the Poisson-Boltzmann equation) to incorporate increasing complexity into the model of interfacial structure. 

Since pairwise electrostatic interaction energies can be calculated from protein structures using Poisson-Boltzmann Equation (PBE) solvers [71,72], we can attempt to forge a unique link between protein structure and protein titration curves... 

The limitations imposed on DDL theory as a molecular model by these four basic assumptions have been discussed frequently and remain the subject of current research.In Secs. 1.4 and 3.4 it is shown that DDL theory provides a useful framework in which to interpret negative adsorption and electrokinetic experiments on soil clay particles. This fact suggests that the several differences between DDL theory and an exact statistical mechanical description of the behavior of ion swarms near soil particle surfaces must compensate one another in some way, at least in certain applications. Evidence supporting this conclusion is considered at the end of the present section, whose principal objective is to trace out the broad implications of Eq. 5.1 as a theory of the interfacial region. The approach taken serves to develop an appreciation of the limitations of DDL theory that emerge from the mathematical structure of the Poisson-Boltzmann equation and from the requirement that its solutions be self-consistent in their physical interpretation. TTie limitations of DDL theory presented in this way lead naturally to the concept of surface complexation. 

The use of molecular dynamics to study the electric double-layer structure started a little over a decade ago, with the hope of determining more accurate structures because the classical description of an electric double layer based on the Poisson-Boltzmann equation is accurate only for low surface potential and dilute electrolytes. The Poisson-Boltzmann equation only considers the electrostatic interactions between the charged surface and ions in the solution, but not the ion-ion interactions in the solution and the finite molecule size, which can be taken into account in molecular dynamics simulations. It was shown [6, 7] that the ion distribution in the near-wall region could be significantly different from the prediction of classical theory. Typical molecular dynamics simulation results of counterion and co-ion concentrations in a nanochannel are shown in Fig. 2a. The ion distribution obtained... 

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