Poisson potential

An equation in the distributions sense, equivalent with the Thomas-Fermi equation, can be introduced by taking into account the expression of the Coulomb-Poisson potential, resulting the Thomas-Fermi differential equation with 0(x) instead of ( ), thus ... 

Poisson Potential Profile and Double Layer Structure for the ILE... 

Fig. 7.27 Poisson potential and Li free energy as a function of position with respect to the interface with LiFeP04 in an ionic liquid electrolyte...

The potential of mean force (POME), i.e., the free energy as a function of the position of a Li relative to the LiFeP04 surface, is shown in Fig. 7.27 for the ILE/ LiFeP04 system and in Fig. 7.28 for the 0LE/LiFeP04 system. For both the OLE and ILE systems the POMF is qualitatively similar to the Poisson potential. First,... 

By considering that polymers have a porous nature, Barker, Albery et al., Buck, and Paasch have shown that to calculate the impedance of electrode/film/solution systems there is a full equivalence of the transport of species by diffcsion-migration and Poisson potential distribution and a transmission line equivalent circuit, like those shown in Fig. 2. Their distributed components are... 

The mathematics is completed by one additional theorem relating the divergence of the gradient of the electrical potential at a given point to the charge density at that point through Poisson s equation... 

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... 

The Poisson equation relates spatial variation of the potential 4> at position r to the density of the charge distribution, p, in a medium with a dielectric constant e... 

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... 

These four equations, using the appropriate boundary conditions, can be solved to give current and potential distributions, and concentration profiles. Electrode kinetics would enter as part of the boundary conditions. The solution of these equations is not easy and often involves detailed numerical work. Electroneutrahty (eq. 28) is not strictly correct. More properly, equation 28 should be replaced with Poisson s equation... 

J-M Mouesca, JL Chen, F Noodleman, D Bashford, DA Case. Density functional/Poisson-Boltzmann calculations of redox potentials for iron-sulfur clusters. J Am Chem Soc 116 11898-11914, 1994. 

In the simplest case of one-dimensional steady flow in the x direction, there is a parallel between Eourier s law for heat flowrate and Ohm s law for charge flowrate (i.e., electrical current). Eor three-dimensional steady-state, potential and temperature distributions are both governed by Laplace s equation. The right-hand terms in Poisson s equation are (.Qy/e) = (volumetric charge density/permittivity) and (Qp // ) = (volumetric heat generation rate/thermal conductivity). The respective units of these terms are (V m ) and (K m ). Representations of isopotential and isothermal surfaces are known respectively as potential or temperature fields. Lines of constant potential gradient ( electric field lines ) normal to isopotential surfaces are similar to lines of constant temperature gradient ( lines of flow ) normal to... 

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