Poisson equation, meaning

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 ... 

Interaction ofthe electrons in the framework of the self-consistent field approximation is accounted for by considering the induced density fluctuations as a response of independent particles to Oext + Poissons equation [2], This means, physically, that collective excitations of the electrons can occur, taken into account via a chain of electron-holeexcitations. These collective excitations show up in S(q, ) as a distinct energy loss feature. Figure 2 shows the shape of the real and imaginary parts of the dielectric function in RPA (er(q, ), Si(q, )) and the resulting dielectric response... 

A governing equation for the mean pressure field appearing in (2.93) can be found by Reynolds averaging (1.29). This leads to a Poisson equation of the form... 

The mean pressure field (p) appears as a closed term in the conditional acceleration (A, V, ip). Nevertheless, it must be computed from a Poisson equation found by taking the divergence of the mean velocity transport equation ... 

Eq. (B.4), the Green s function of the Poisson equation, has an obvious physical meaning. By labeling the point charge at Fo as the "cause," the potential it creates at a point f is its "effect," and the influence of the point... 

The Poisson equation is a fundamental relationship of classical electrostatics and really need not be proved here. However, since we are using it as a starting point, it seems desirable to explore the meaning of this important equation to some extent. 

The history of PB theory can be traced back to the Gouy-Chapmann theory and Debye-Huchel theory in the early of 1900s (e.g., see Camie and Torrie, 1984). These two theories represent special simplified forms of the PB theory Gouy-Chapmann theory is a one-dimensional simplification for electric double-layer, while the Debye-Huchel theory is a special solution for spherical symmetric system. The PB equation can be derived based on the Poisson equation with a self-consistent mean electric potential tj/ and a Boltzmann distribution for the ions... 

The first theoretical description of the double layers assumed that the ions interact via a mean potential, which obeys the Poisson equation.2 Such a simple theory is clearly only approximate and sometimes predicts ionic concentrations in the vicinity of the surface that exceed the available volume.3 There were a number of attempts to improve the model, by accounting for the variation of the dielectric constant in the vicinity of the surface,4 for the volume-exclusion effects of the ions,5 or for additional interactions between ions and surfaces, due to the screened image force potential,6 to the van der Waals interactions of the ions7 with the system, or to the change in hydration energy when an ion approaches the interface.8... 

The traditional theory of the double layer is based on a combination of the Poisson equation and Boltzmann distribution. While this involves the approximation that the potential of mean force used in the Boltzmann expression equals the mean value of the electrical potential [9], the results thus obtained are satisfactory at least for 1 1 electrolytes. The equations proposed in the present paper use the approximations inherent in the Poisson—Boltzmann equation, but also include the effect of the polarization field of the solvent which is caused by a polarization source assumed uniformly distributed on the surface and by the double layer itself. 

The traditional double-layer theory combines the Poisson equation with the assumption that the polarization is proportional to the macroscopic electric field, and uses Boltzmann distributions for the concentrations of the ions. The potential of mean force, which should be used in the Boltzmann distribution, is approximated by the mean value of the electrical potential. The macroscopic field E and the polarization P are related via the Poisson equation... 

The potential occurring in the Poisson equation is basically the mean potential whereas that in the Boltzmann equation is the potential of mean force. In the Poisson-Boltzmann equation this distinction Is not made. The error made in this approximation is quantified by the r.h.s. of 13.6.18], excluding the first term. 

Rotta (Rl) studied the Poisson equation in some detail, and proposed Eq. (49) for the portion of P, independent of the mean-fluctuation inter-... 

In a typical experiment the solute charge distribution is assumed to change abruptly, at t = 0, say, from pi(r) to P2(r), then stays constant. This means that the dielectric displacement, related to p(r) by the Poisson equation V T> = 47rp, is also switched from to T>2 at t = 0. In the process that follows the solvent structure adjusts itself to the new charge distribution. In our continuum model this appears as a local relaxation of the solvent polarization, which over time changes from Pl to P2. 

The one-particle equations for the wave functions p z, p) are solved by means of the fully numerical mesh method described in refs. [3,18,20]. In our first works on the helium atom in magnetic fields [18,20] we calculated the Coulomb and exchange integrals by means of a direct summation over the mesh nodes. But this direct method is very expensive with respect to the computer time and due to this reason we obtained in the following works [28-31] these potentials as solutions of the corresponding Poisson equation. The problem of the boundary conditions for the Poisson equation as well as the problem of simultaneously solving Poisson equation on the same meshes with Schrodinger-like equations for the wave functions p z, p) have been discussed in ref. [20],... 

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