Electrostatic potential Poisson equation

This is a beautiful theory, in that it allows us to calculate the excess surface-charge density and the double-layer capacitance from well-known principles of electrostatics (the Poisson equation and the Gauss theorem) and thermodynamics (the Boltzmann equation). It has, however, one major drawback it does not predict the correct experimental results Perhaps it would be more accurate to state that agreement between theory and experiment is found only in dilute solutions and over a limited range of potentials, near the potential of zero charge, as seen in Figure 8.4. 

The model used is the RPM. The average electrostatic potential ifr) at a distance r away from an ion / is related to tire charge density p.(r) by Poisson s equation... 

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 Poisson equation relates the electrostatic potential ([) to the charge density p. The Poisson equation is... 

The Poisson equation describes the electrostatic interaction between an arbitrary charge density p(r) and a continuum dielectric. It states that the electrostatic potential ([) is related to the charge density and the dielectric permitivity z by... 

Solving the one-dimensional Poisson equation with the charge density profile pc z), the electrostatic potential drop near the interface can be calculated according to... 

The profile of the electrostatic potential V in an MS junction can be calculated by solving Poisson s equation... 

Calculation of Electrostatic Potential by the Poisson-Boltzmann Equation... 

The polyelectrolyte chain is often assumed to be a rigid cylinder (at least locally) with a uniform surface charge distribution [33-36], On the basis of this assumption the non-linearized Poisson-Boltzmann (PB) equation can be used to calculate how the electrostatic potential

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In order to describe the effects of the double layer on the particle motion, the Poisson equation is used. The Poisson equation relates the electrostatic potential field to the charge density in the double layer, and this gives rise to the concepts of zeta-potential and surface of shear. Using extensions of the double-layer theory, Debye and Huckel, Smoluchowski,... 

The relation between the spatial distribution of the electrostatic potential /(jc) and the spatial distribution of charge density Qy(x) can be stated, generally, in terms of Poisson s differential equation. 

In addition to the nearest-neighbor interaction, each ion experiences the electrostatic potential generated by the other ions. In the literature this has generally been equated with the macroscopic potential 0 calculated from the Poisson-Boltzmann equation. This corresponds to a mean-field approximation (vide infra), in which correlations between the ions are neglected. This approximation should be the better the low the concentrations of the ions. 

Similar equations can be derived for the other ions. The charge density p x) is then obtained by adding the charge densities pertaining to all four kinds of ions. Both p x) and the electrostatic potential (f> x) are calculated by solving the Poisson equation self-consistently, and the particle distributions then follow by substituting (j)(x) into Eq. (25) and the respective equations for the other ions. 

For small deviations from electroneutrality, the charge density at x is proportional to -(x)/kT9 where < is the difference of the electrostatic potential from its (constant) value when there is no charge density (the density of a species of charge z is proportional to 1 - zkT on linearizing the Boltzmann exponential). Then the Poisson equation [Eq. (44)] becomes the linearized Poisson-Boltzmann equation ... 

It should be noted that the electrostatic potential V(r) is related rigorously to the total charge density D(r) through Poisson s equation, Eq. (3.4). 

The Finite Difference Method (FD)168 169. This is a general method applicable for systems with arbitrary chosen local dielectric properties. In this method, the electrostatic potential (RF) is obtained by solving the discretized Poisson equation ... 

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. 

Here, b is the distance between the nearest unit charges along the cylinder (b = 0.34nm for the ssDNA and b = 0.17nm for the dsDNA), (+) and (—) are related to cations and anions, respectively, and a = rss for the ssDNA and a rds for the dsDNA. The expressions (5) and (6) have been obtained using the equations for the electrostatic potential derived in [64, 65], where a linearization of the Poisson-Boltzmann equation near the Donnan potential in the hexagonal DNA cell was implemented. 

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 second term in Equation 1, , involves carrying out a Poisson-Boltzmann calculation and evaluating the exposed surface area of all atoms for all the snapshots for C, M, and L. Currently, we use Hartree-Fock (HF)/6-31G restrained electrostatic potential (RESP)13 charges and PARSE14 radii for the PB calculation within DELPHI15 and the program... 

Physically meaningful ionic radii may be obtained from Poisson equation for anions, and from electrostatic potentials defined in the the context of DFT for cations [17,18], However, there remains the problem of being forced to use different mathematical criteria in both cases, because the electrostatic potential of anions and cations display a different functional behaviour with respect to the radial variable. 

For a spherically symmetric charge distribution, an exact relationship between the electrostatic potential and the electron density is the Poisson equation ... 

The second term in equation (9) is the usual electrostatic term. Here vA is the valency of the unit and e is the elementary charge, and ip(z) is the electrostatic potential. This second term is the well-known contribution accounted for in the classical Poisson-Boltzmann (Gouy -Chapman) equation that describes the electric double layer. The electrostatic potential can be computed from the charge distribution, as explained below. 

The next step is to generate all possible and allowed conformations, which leads to the full probability distribution F). The normalisation of this distribution gives the number of molecules of type i in conformation c, and from this it is trivial to extract the volume fraction profiles for all the molecules in the system. With these density distributions, one can subsequently compute the distribution of charges in the system. The charges should be consistent with the electrostatic potentials, according to the Poisson equation ... 

The concept of Green s functions can be illustrated by the example of the electrostatic problem. The potential U(r) for a given charge distribution p(r) is determined by the Poisson equation... 

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