Poisson equation of electrostatics

If the Poisson equation is modified in this way by using such time-averaged rjr and p, and then is applied to the electrolyte solution, there has been a smearing out of the discrete charges of the ionic atmosphere into a continuous distribution of charge. Only then is it legitimate to use the Poisson equation of electrostatics to discuss the theory of electrolyte solutions, i.e. 

Guoy and Chapman combined Eq. (13.4-1) with the Poisson equation of electrostatics and found that if the potential is taken equal to zero at large distances, the electric potential in the diffuse double layer is given as a function of x, the distance from the electrode, by... 

We have just described the linearized theory of capillarity. In the electrostatic analogy the field M(r ) is identified with a 2D electrostatic potential ( capillary potential ) and Il(r ) with a charge density ( capillary charge ) Equation 2.8 reduces to the Poisson equation of electrostatics and Equation 2.9 relates the tensor Tn, which has the form of Maxwell s stress tensor, with the electric force exerted on the capillary charge n(rn) (also the usual boundary conditions imposed on the interface have a close electrostatic analogy [34,35]). 

A second relation between p(r) and 4>(r) may be obtained by noting that the sources of 0(r) are the point charge Ze of the nucleus, located at the origin and the charge distribution due to the N electrons. Treating the charge density —ep(r) of the electrons as continuous, Poisson s equation of electrostatics may be used to write... 

Now, we also assume that, even on a microscopic scale, Poisson s equation of electrostatics is valid ... 

An important equation of electrostatics, which follows directly from Maxwell s equations (Jackson 1975) is Poisson s equation. It relates the divergence of the gradient of the potential charge density at that point ... 

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... 

In continuum notation, this relation would constitute one form of Poisson s equation of electrostatics. The continuum forms of E(x) and V (x) are valid if the charge density planes are so close together that over small regions of space the charge density can be viewed as a continuous function p(x) of position x. [The local space charge density p(x) has units of Coulombs m-3]. In such cases, the sums in eqns. (37) and (40) for E (x) and V (x) can be approximated by integrals to give... 

This latter relation constitutes the integral form of Poisson s equation of electrostatics. 

The electrostatic field in the stationary state is described by the Poisson-Boltzmann equation. The PB model constitutes the fundamental equation of electrostatics and is based on the differential Poisson equation which describes the electrostatic potential 4>(r) in a medium with a charge density p(r) and a dielectric scalar field e(r) ... 

The Debye-Hiickel theory focuses on and by using Poisson s equation of electrostatics finds an explicit equation for yfrj, firom which the potential, at the surface of the central j ion... 

Here c° is the bulk concentration of a z Z valent electrolyte. However, a fundamental equation of electrostatics (Poisson s equation) relates this density to the way in which ip varies with the distance from the surface ... 

If one accepts the continuum, linear response dielectric approximation for the solvent, then the Poisson equation of classical electrostatics provides an exact formalism for computing the electrostatic potential (r) produced by a molecular charge distribution p(r). The screening effects of salt can be added at this level via an approximate mean-field treatment, resulting in the so-called Poisson-Boltzmann (PB) equation [13]. In general, this is a second order non-linear partial differential equation, but its simpler linearized form is often used in biomolecular applications ... 

Poisson s equation of electrostatics relates pe to the variation of i/>with distance to the charged surface, x, in the form... 

In the general case the electrostatic potential is not constant but varies within the resin phase. To be able to calculate the electrostatic potential at a point as a function of the ionic strength, the geometry and the charge density of the system must be specified. The electrostatic potential at a point is usually obtained by solving the Poisson-Boltzmann equation, which is a combination of the Boltzmann equation, eqn [4], with the Poisson equation from electrostatics. The... 

The Poisson equation in electrostatics relates the charge concentration contained in a system to the Laplacian of the electric potential. 

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... 

The simplest model of charge shielding and colloidal stability against aggregation was developed around 1910 independently by L-G Gouy (18.34-1926), a French physicist, and DL Chapman (1869-1958), a British chemist. They combined Poisson s equation of electrostatics with the Boltzmann distribution law. 

In calculating the activity coefficient of the species a, we shall need only the electric potential due to the ionic atmosphere, which we denoted by y/ iR/a). First we need to solve for the total potential function i//(R/of). To evaluate if/(R/a), we use the Poisson equation from electrostatics, which reads... 

This equation is known as the Poisson equation for electrostatics. The knowledge of charge density distribution pg) is required for obtaining potential distribution. If the charge density is zero, we get Laplace equation. These equations will be used in the electrohydrodynamic analysis of microfluidics. 

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)). 

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 ... 

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

These early results of Coulomb and his contemporaries led to the full development of classical electrostatics and electrodynamics in the nineteenth cenmry, culminating with Maxwell s equations. We do not consider electrodynamics at all in this chapter, and our discussion of electrostatics is necessarily brief. However, we need to introduce Gauss law and Poisson s equation, which are consequences of Coulomb s law. 

Although the continuum model of the ion could be analyzed by Gauss law together with spherical symmetry, in order to treat more general continuum models of electrostatics such as solvated proteins we need to consider media that have a position-specific permittivity e(r). For these a more general variant of Poisson s equation holds ... 

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]. 

SASA), a concept introduced by Lee and Richards [9], and the electrostatic free energy contribution on the basis of the Poisson-Boltzmann (PB) equation of macroscopic electrostatics, an idea that goes back to Born [10], Debye and Htickel [11], Kirkwood [12], and Onsager [13]. The combination of these two approximations forms the SASA/PB implicit solvent model. In the next section we analyze the microscopic significance of the nonpolar and electrostatic free energy contributions and describe the SASA/PB implicit solvent model. 

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