Poisson equation solution electrostatic potential

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. 

The electrical state of the system is so sensitive to small changes in composition [9] that the gradient Vtp in Eq. (7) cannot be assumed, in general, to be a simple constant [12], and an additional equation is required to determine it. Given the slowness of particle motion in solution, it is justified to use the Poisson equation from electrostatics to relate the changes in electric potential to the local electric charge density pe... 

Another way of calculating the electrostatic component of solvation uses the Poisson-Boltzmann equations [22, 23]. This formalism, which is also frequently applied to biological macromolecules, treats the solvent as a high-dielectric continuum, whereas the solute is considered as an array of point charges in a constant, low-dielectric medium. Changes of the potential within a medium with the dielectric constant e can be related to the charge density p according to the Poisson equation (Eq. (41)). 

If there are ions in the solution, they will try to change their location according to the electrostatic potential in the system. Their distribution can be described according to Boltzmarm. Including these effects and applying some mathematics leads to the final linearized Poisson-Boltzmann equation (Eq. (43)). 

The final class of methods that we shall consider for calculating the electrostatic compone of the solvation free energy are based upon the Poisson or the Poisson-Boltzmann equatior Ihese methods have been particularly useful for investigating the electrostatic properties biological macromolecules such as proteins and DNA. The solute is treated as a body of co stant low dielectric (usually between 2 and 4), and the solvent is modelled as a continuum high dielectric. The Poisson equation relates the variation in the potential (f> within a mediu of uniform dielectric constant e to the charge density p ... 

The convenience of Eq. (6) is realizable only in the rather unrealistic situation where the charge distribution exhibits cylindrical or spherical symmetry. For storage silos, blenders, fluidized bed reactors, and other real vessel geometries, integral solutions are usually not possible, necessitating an alternate problem formulation. Poisson s equation serves this need, relating the volume charge distribution to the electrostatic potential. 

An alternative to the GB, COSMO, and Poisson electrostatic calculations is to model the solution to the Poisson equation in terms of pair potentials between solute atoms this procedure is based on the physical picture that the solvent screens the intra-solute Coulombic interactions of the solute, except for the critical descreening of one part of the solute from the solvent by another part of this solute. This descreening can be modeled in an average way to a certain level of accuracy by pairwise functions of atomic positions.18, M 65 One can obtain quite accurate solvation energies in this way, and it has recently been shown that this algorithm provides a satisfactory alternative to more expensive explicit-solvent simulations even for the demanding cases of 10-base-pair duplexes of DNA and RNA in water.66... 

The mutual polarization process between the solute and the polarizable medium is obtained by solving a system of two coupled equations, i.e., the QM Schrodinger equation for the solute in presence of the polarized dielectric, and the electrostatic Poisson equation for the dielectric medium in presence of the charge distribution (electrons and nuclei) of the solute. The solute occupies a molecular shaped cavity within the dielectric continuum, whose polarization is represented by an apparent surface charge (ASC) density spread on the cavity surface. The solute-solvent interaction is then represented by a QM operator, the solvent reaction potential operator, Va, corresponding to the electrostatic interaction of the solute electrons and nuclei with the ASC density of the solvent. 

In an exact calculation of the distribution of the electrostatic potential, the carrier densities and their currents, (4.81)-(4.87) are solved simultaneously, bearing in mind that only the sum of the diffusion and drift currents has physical significance. Due to the complexity of the above relations and in particular due to the coupling of electron and hole concentrations by Poisson s equation, analytical solutions exist only for a few, very specific conditions. Generally, the determination of local carrier concentrations, current densities, recombination rates, etc., requires extensive numerical procedures. This is especially true if they vary with time, but even in the steady state context. 

Poisson-Boltzmann equation — The Poisson-Boltz-mann equation is a nonlinear, elliptic, second-order, partial differential equation which plays a central role, e.g., in the Gouy-Chapman (- Gouy, - Chapman) electrical -> double layer model and in the - Debye-Huckel theory of electrolyte solutions. It is derived from the classical -> Poisson equation for the electrostatic potential... 

Now, a linear charge density-potential relation is consistent with the law of superposition of potentials, which states that the electrostatic potential at a point due to an assembly of charges is the sum of the potentials due to the individual charges. Thus, when one uses an unlinearized P-B equation, one is assuming the validity of the law of superposition of potentials in the Poisson equation and its invalidity in the Boltzmann equation. This is a basic logical inconsistency which must reveal itself in the predictions that emage from the so-called rigorous solutions. This is indeed the case, as will be shown below. 

The Ewald potential is traditionally implemented as a lattice sum (Ziman, 1972 Leeuw et al, 1980). We just outlined a conceptualization of electrostatic interactions in periodic boundary conditions that involved adding a uniform neutralizing background for each charge, and the subsequent solution of the Poisson equation in periodic boundary conditions. Here we discuss the interconnections between that conceptualization and the traditional lattice sums, as presented in many sources, e.g. (Allen and Tildesley, 1987 Frenkel and Smit, 2002 Leeuw et al, 1980). 

The Poisson Equation From classical electrostatics, the free charge density p(r)—that is, the charge density due to the solute as opposed to the polarization charges in the solvent—in a continuous medium of homogeneous dielectric constant (relative permittivity) e, where r denotes the position in space, is related to the electrostatic potential, ( )(r), by Poisson s equation, which takes the following form, in this case in Gaussian units ... 

The Poisson equation (or Gauss Law) describes the electrostatic potential of a fixed charged density of the solute Psoiuteir). The exterior charge density of the ions in the solution, Pexteriorif) is modeled by assuming a Boltzmann distribution. The Poisson-Boltzmann (PB) equation is commonly applied to molecules in aqueous solution to compute the electrostatic potential of the system. The general form of the PB equation is... 

In this discussion we limit ourselves to electrostatic interactions between point charges. The overall system is assumed to be neutral. The potential field of charge-charge interactions is in fact described by one of the classic differential equations, namely the Poisson equation with periodic boundary conditions. Because of the somewhat unusual boundary conditions it is important to realize that some care must be practiced when applying a solution strategy. 

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