The Poisson Equation

Gauss s theorem. Equation (17.24), says that for any vector field v. 

Substituting v = DE gives Gauss s theorem applied to electric fields, 

Gauss s theorem equates the flux of the electrostatic field through a closed surface with the divergence of that same field throughout its volume. 

Equation (21.29) is one of Maxwell s four famous equations. Now substitute Equation (21.8), E = -Vqj, into Equation (21.29) to get Poisson s equation  

You now have two methods for deriving the electrostatic potential from a given distribution of fixed electrical charges. First, ou can use Coulomb s law. But this becomes unwieldy for complex constellations of charges or for systems with tw-o or more different dielectric media. Second, you can solve Poisson s equation. The second method is more general, and it can also be used for heterogeneous dielectric media (see page 401). 

Heat Conduction in Fuel Elements 6.3.1. The Poisson Equation 

Heat may be transferred from the fuel elements by conduction, convection, and radiation. For all but high-temperature gas-cooled reactors, the last of these is of little significance under normal operating conditions. Convection is the process by which the heat from the surface of the fuel element is carried into the coolant and subsequently removed from the reactor. Before this happens, however, the heat generated in the fuel must travel by conduction from the interior of the element to its surface. In the present section, we confer the application of the standard heat conduction equations to determine the temperature distribution within fuel elements of various geometries. 

We start by taking a small cubic volume, of sides Ax, Ay, Az, which is within a region of fuel where heat generation and conduction is taking place (Fig. 6.3). 

Conduction takes place as a result of a temperature gradient in the medium. The rate of flow of heat into the cube in the x direction is related to the heat flux, ql, defined as the quantity of heat transferred per unit time 

The rate of heat conduction out of the volume element in the x direction is related to the heat flux which is given by 

At the heart of all continuum solvent models is a reliance on the Poisson equation of classical electrostatics to express the electrostatic potential as a function of the charge density and the dielectric constant. The Poisson equation, valid for situations where a surrounding dielectric medium responds in a linear fashion to the embedding of charge, is written 

The Poisson equation is valid under conditions of zero ionic strength. If dissolved, mobile electrolytes are present in the solvent, the Poisson-Boltzmann (PB) equation applies instead 

Note that it is fairly common in the literature for continuum solvation calculations to be reported as having been carried out using Poisson-Boltzmann electrostatics even when no electrolyte concentration is being considered, i.e., the Poisson equation is considered a special case of the PB equation and not named separately. 

For certain ideal cavity shapes, the relevant PB equations have particularly simple analytic solutions. While such ideal cavities are not typically to be expected for arbitrary solute molecules, consideration of some examples is instructive in illustrating how more sophisticated modeling may be undertaken by generalization therefrom. 

Consider a conducting sphere bearing charge q, which may be taken as an approximation to a monatomic ion. The charge on such an object spreads out uniformly on the surface, and the charge density at any point on the surface may thus be expressed as 

For a solid or molecule subdivided into atomic cells, the Poisson equation, given in Rydberg units by 

The complementary potential Ad or generalized Madelung term is expanded in regular solid harmonics as 

Alternative equations for the coefficients Ac are given by direct matching at cellular interfaces, bypassing the need for structure constants. The two alternatives considered by [443] are the NVCM or GFCM equations, 

A factor -2 included in the last term here compensates for the use of Rydberg units and for the omission of the negative electronic charge in potential functions derived from Eq. (7.14). Hence the electrostatic multipole moments of atomic cell r/( are 

This equation is used for l = 0 to evaluate normalization integrals in the LACO program package [278], avoiding numerical volume quadrature. 

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This justifies the use of the simpler language over the one. The solution of the Poisson equation and the boundary conditions used are explained in detail elsewhere [55]. Here, we will present some selected results. 

If a gaussian function is chosen for the charge spread function, and the Poisson equation is solved by Fourier transformation (valid for periodic... 

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... 

Davis, M. E., McCammon, J. A. Dielectric boundary smoothing in finite difference solutions of the poisson equation An approach to improve accuracy and convergence. J. Comp. Chem. 12 (1991) 909-912. 

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)). 

Note that the mathematical symbol V stands for the second derivative of a function (in this case with respect to the Cartesian coordinates d fdx + d jdy + d jdz y, therefore the relationship stated in Eq. (41) is a second-order differential equation. Only for a constant dielectric Eq.(41) can be reduced to Coulomb s law. In the more interesting case where the dielectric is not constant within the volume considered, the Poisson equation is modified according to Eq. (42). 

In Ibis equation the P points R,- correspond to the grid used to solve the Poisson equation for l, i and Wi are weighting factors. 

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 ... 

In reduced electrostatic units, the factor is eliminated and the Poisson equatic becomes ... 

The charge density is simply the distribution of charge throughout the system and has 1 units of Cm . The Poisson equation is thus a second-order differential equation (V the usual abbreviation for (d /dr ) + (f /dx/) + (d /dz )). For a set of point charges in constant dielectric the Poisson equation reduces to Coulomb s law. However, if the dielectr... 

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

This may be solved numerically or within some analytic approximation. The Poisson equation is used for obtaining the electrostatic properties of molecules. 

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... 

The Poisson equation has been used for both molecular mechanics and quantum mechanical descriptions of solvation. It can be solved directly using numerical differential equation methods, such as the finite element or finite difference methods, but these calculations can be CPU-intensive. A more efficient quantum mechanical formulation is referred to as a self-consistent reaction field calculation (SCRF) as described below. 

The Poisson equation assumes that the solvent is completely homogeneous. However, a solvent can have a significant amount of charge separation. An example of a heterogeneous solution would be a polar solute molecule surrounded by water with NaCl in solution. The positive sodium and negative... 

The Poisson-Boltzmann equation is a modification of the Poisson equation. It has an additional term describing the solvent charge separation and can also be viewed mathematically as a generalization of Debye-Huckel theory. 

Olefin distribution in the Albemarle stoichiometric process tends to foUow the Poisson equation, where is the mole fraction of alkyl groups in whichp ethylene units have been added, and n is the average number of ethylene units added for an equal amount of aluminum. 

The continuum treatment of electrostatics can also model salt effects by generalizing the Poisson equation (12) to the Poisson-Boltzmann equation. The finite difference approach to solving Eq. (12) extends naturally to treating the Poisson-Boltzmann equation [21], and the boundary element method can be extended as well [19]. 

The original particle mesh (P3M) approach of Hockney and Eastwood [42] treats the reciprocal space problem from the standpoint of numerically solving the Poisson equation under periodic boundary conditions with the Gaussian co-ion densities as the source density p on the right-hand side of Eq. (10). Although a straightforward approach is to... 

IV. CLASSICAL CONTINUUM ELECTROSTATICS A. The Poisson Equation for Macroscopic Media... 

The continuum electrostatic approximation is based on the assumption that the solvent polarization density of the solvent at a position r in space is linearly related to the total local electric field at that position. The Poisson equation for macroscopic continuum media... 

Continuum electrostatic approaches based on the Poisson equation have been used to address a wide variety of problems in biology. One particularly useful application is in the determination of the protonation state of titratable groups in proteins [46]. For... 

In this chapter, the elution curve equation and the plate theory will be used to explain some specific features of a chromatogram, certain chromatographic operating procedures, and some specific column properties. Some of the subjects treated will be second-order effects and, therefore, the mathematics will be more complex and some of the physical systems more involved. Firstly, it will be necessary to express certain mathematical concepts, such as the elution curve equation, in an alternative form. For example, the Poisson equation for the elution curve will be put into the simpler Gaussian or Error function form. 

The effect of space charge can be taken into account by means of the Poisson equation, which in the case of a cylindrical geometry is expressed in the form... 

Instead of an exact calculation, Gouy and Chapman have assumed that (4) can be approximated by combining the Poisson equation with a Boltzmann factor which contains the mean electrical potential existing in the interface. (This approximation will be rederived below). From this approach the distribution of the potential across the interface can be calculated as the function of a and from (2) we get a differential capacitance Cqc- It has been shown by Grahame that Cqc fits very well the measurements in the case of low ionic concentrations [11]. For higher concentrations another capacitance in series, Q, had to be introduced. It is called the inner layer capacitance and it was first considered by Stern [1,2]. Then the experimental capacitance Cexp is analyzed according to ... 

Both entropic and coulombic contributions are bounded from below and it can be verified that the second variation of is positive definite so that the above equations correspond to a minimum [27]. Using conditions in the bulk we can eliminate //, from the equations. Then we get the Boltzmann equation in which the electric potential verifies the Poisson equation by construction. Hence is equivalent within MFA to the... 

Now, combining the equations for both kinds of ion, and using the Poisson equation for the average potential 8V f), we can write our equations in the following matrix form [28] ... 

Since the potential verifies the Poisson equation the nonlinear Gouy-Chapman theory is recovered. In what follows we summarize some results of the nonlinear Gouy-Chapman (NLGC) theory that are useful for the subsequent part of this work. 

We will use the hatted symbols without a hat as, in practice, it does not lead to confusion. With these notations we have g z) = Qxpi v z)/2) and the Poisson equation is... 

The electrical potential is obtained by solving the Poisson equation... 

To solve the Poisson equation, we must express as a p function of the coordinates of the system and solve the resulting second-order differential equation to obtain ip(x,y, z), from which trci and hence, AG, p - p°, and 7 can be calculated. 

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