Poisson-type equation

If the first box is initially contaminated by an amount Mq, the analytical solution of the differential equation [3] leads to a Poisson-type equation [4] which describes the nuclide... 

By insertion of Eq. [52b] for j(r, ) into this equation, it is easy to see that a Poisson-type equation is obtained (actually a Laplace equation, with the rhs equal to zero). If the diffusion coefficient is a constant over the domain, then that variable drops out. Notice that, in Eq. [52b], an effective dielectric constant [exp(—pct)(r))] appears, and this expression can vary over orders of magnitude due to large variations in the potential. Finally, if we seek a solution to the time-dependent problem, we must solve the full Smoluchowski equation iteratively. The particle density then depends explicitly on time. 

Weinert M 1981 Soiution of Poisson s equation beyond Ewaid-type methods J. Math. Phys. 22 2433... 

The device model describes transport in the organic device by the time-dependent continuity equation, with a drift-diffusion form for the current density, coupled to Poisson s equation. To be specific, consider single-carrier structures with holes as the dominant carrier type. In this case,... 

One of the simplest equations is obtained using the Debye-Hiickel approximation (for low potentials) and the superposition principle. The latter assumes that the unperturbed potential near a charged surface can be simply added to that potential due to the other (unperturbed) surface. Thus, for the example shown in the Figure 6.12, it follows that /m = 2 /d/2- This is precisely valid for Coulomb-type interactions, where the potential at any point can be calculated from the potentials produced by each fixed charge, individually. However, the Poisson-Boltzmann equation is non-linear (this has to do with the fact that in the diffuse double-layer the ions are not fixed but move because of their kinetic energy) and so this is formally not correct although it still offers a useful approximation. 

Another related phenomenon to be discussed in 2.3 is known in the polymer literature as counterion condensation. This term refers to a phase transition-like switch of the type of singularity, induced by a line charge to solutions of (2.1.2), occurring at some critical value of the linear charge density. Counterion condensation as a limiting property of the solutions of the Poisson-Boltzmann equation was studied in detail in [11]—[19]. Presentation of 2.3 follows that of [17]. 

The electric field within each cell is determined in the mean field approximation from the Poisson Boltzmann equation (2.3.1), written for the prototypical case of a symmetric low molecular electrolyte of valency z added to a polyelectrolyte with a single type of proper counterion of va-... 

Metals, such as platinum, are usually introduced to improve the electron-hole separation efficiency. In order to analyze the energy structure of the metal-loaded particulate semiconductor, we solved the two-dimensional Poisson-Boltzmann equation.3) When the metal is deposited to the semiconductor by, for example, evaporation, a Schottky barrier is usually formed.45 For the Schottky type contact, the barrier height increases with an increase of the work function of the metal,4 which should decrease the photocatalytic activity. However, higher activity was actually observed for the metal with a higher work function.55 This results from the fact that ohmic contact with deposited metal particles is established in photocatalysts when the deposited semiconductor is treated by heat65 or metal is deposited by the photocatalytic reaction.75 Therefore, in the numerical computation we assumed ohmic contact at the energy level junction of the metal and semiconductor. 

Fig. 5.2 Contour map of energy bending for metal deposited n-type semiconductor obtained by numerical solving of 2D Poisson-Boltzmann equation.31 Parameters d= 20 nm, ND = 10 24 nr3, and e = 80. Plausible pathways for electron and hole generated by photoabsorption are shown by arrows.

In order to calculate the potential distribution in the gap we need not only the Poisson-Boltzmann equation, but in addition, boundary conditions must be specified. Two common types of boundary conditions are ... 

Once a finite element formulation has been implemented in conjunction with a specific element type — either 1D, 2D or 3D — the task left is to numerically implement the technique and develop the computer program to solve for the unknown primary variables — in this case temperature. Equation (9.19) is a form that becomes very familiar to the person developing finite element models. In fact, for most problems that are governed by Poisson s equation, problems solving displacement fields in stress-strain problems and flow problems such as those encountered in polymer processing, the finite element equation system takes the form presented in eqn. (9.19). This equation is always re-written in the form... 

The previous section used the constant strain three-noded element to solve Poisson s equation with steady-state as well as transient terms. The same problems, as well as any field problems such as stress-strain and the flow momentum balance, can be formulated using isoparametric elements. With this type of element, the same (as the name suggests) shape functions used to represent the field variables are used to interpolate between the nodal coordinates and to transform from the xy coordinate system to a local element coordinate system. The first step is to discretize the domain presented in Fig. 9.12 using the isoparametric quadrilateral elements as shown in Fig. 9.15. 

Equation (4) is thus a time-dependent boundary condition to Eqs. (6, 7), which, supplemented by the remaining boundary conditions (which also involve external constraints resulting from the operation mode of the experiment, s.b.) and possibly by the incorporation of convection, form the most basic Ansatz for modeling patterns of the reaction-transport type in electrochemical systems. However, so far, there are no studies on electrochemical pattern formation that are based on this generally applicable set of equations. Rather, one assumption was made throughout that proved to capture the essential features of pattern formation in electrochemistry and greatly simplifies the problem it is assumed that the potential distribution in the electrolyte can be calculated by Laplace s equation, i.e. Poisson s equation (6) becomes ... 

In all of the discussion above, comparisons have been made between various types of approximations, with the nonlinear Poisson-Boltzmann equation providing the standard with which to judge their validity. However, as already noted, the nonlinear Poisson-Boltzmann equation itself entails numerous approximations. In the language of liquid state theory, the Poisson-Boltzmann equation is a mean-field approximation in which all correlation between point ions in solution is neglected, and indeed the Poisson-Boltzmann results for sphere-sphere [48] and plate-plate [8,49] interactions have been derived as limiting cases of more rigorous approaches. For many years, researchers have examined the accuracy of the Poisson-Boltzmann theory using statistical mechanical methods, and it is... 

Even with these useful results from statistical mechanics, it is difficult to specify straightforward criteria delineating when the Poisson-Boltzmann or linear Poisson-Boltzmann equations can be expected to yield quantitatively accurate results for particle-wall interactions. As we have seen, such criteria vary greatly with different types of boundary conditions, what type of electrolyte is present, the electrolyte concentration and the surface-to-surface gap and double layer dimensions. However, most of the evidence supports the notion that the nonlinear Poisson-Boltzmann equation is accurate for surface potentials less than 100 mV and salt concentrations less than 0.1 M, as stated in the Introduction. Of course, such a statement might not hold when, for example, the surface-to-surface separation is only a few ion diameters. We have also seen that the linear Poisson-Boltzmann equation can yield results virtually identical with the nonlinear equation, particularly for constant potential boundary conditions and with surface potentials less than about 50 mV. Even for constant surface charge density conditions the linear equation can be useful, particularly when Ka < 1 or Kh > 1, or when the particle and wall surfaces have comparable charge densities with opposite signs. 

There is another family of continuum models51-53 where the influence of the reaction field is modeled by a set of polarized charges distributed on the surface of a molecular cavity. Models of this type are usually referred to as polarized continuum models (PCMs), and they have as their origin the solution of Poisson s equation. 

The potential distribution, and hence the extent of the band bending, within the space charge layer of a planar macroscopic electrode may be obtained by solution of the one-dimensional Poisson-Boltzmann equation [95]. However, since the particles may be assumed to have spherical geometry, the Poisson-Boltzmann for a sphere must be solved. This has been done by Albery and Bartlett [131] in a treatment that was recently extended by Liver and Nitzan [125]. For an n-type semiconductor particle of radius r0, the Poisson-Boltzmann equation for the case of spherical symmetry takes the form ... 

This transformed the nonlinear Poisson-Boltzmann equation into the linear Helmholtz-type equation... 

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