Debye Poisson equation

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. 

The first assumption of the Debye-Hiickel theory is that is spherically symmetric. With the elimination of any angular dependence, the Poisson equation (expressed in spherical-polar coordinates) reduces to... 

In order to describe the effects of the double layer on the particle motion, the Poisson equation is used. The Poisson equation relates the electrostatic potential field to the charge density in the double layer, and this gives rise to the concepts of zeta-potential and surface of shear. Using extensions of the double-layer theory, Debye and Huckel, Smoluchowski,... 

At the final concentration c, the potential j)k at distance r is given not only by the potential of the ion, )°ky but also by the potentials of the surrounding ions. Debye and Hiickel assumed that a spherical ionic atmosphere of statistically prevailing ions with opposite charge forms around each ion, giving rise to the potential ipa. Thus xpk = ipk + The potential of the space charge density p is given by the Poisson equation... 

The electrostatic methods just discussed suitable for nonelectrolytic solvent. However, both the GB and Poisson approaches may be extended to salt solutions, the former by introducing a Debye-Huckel parameter67 and the latter by generalizing the Poisson equation to the Poisson-Boltzmann equation.68 The Debye-Huckel modification of the GB model is valid to much higher salt concentrations than the original Debye-Huckel theory because the model includes the finite size of the solute molecules. 

Debye and Huckel applied the Boltzmann statistical distribution law and the Poisson equation for electrostatics in the model above (1,6, 10). In the calculations using the model above they considered one particular ion (the reference ion, or central ion) with... 

A modification of GB that includes the effects of dissolved electrolytes in the formalism, i.e., an extension analogous to the Poisson-Boltzmann extension of the Poisson equation, has been proposed by Srinivasan et al. (1999). In this model, the dielectric constant is a function of the interatomic distance and the Debye-Huckel parameter (Eq. (11.7)). 

The Poisson equation (see Equation (11.18)) gives the fundamental differential equation for potential as a function of charge density. The Debye-Hiickel approximation may be used to express the charge density as a function of potential as in Equation (11.28) if the potential is low. Combining Equations (11.24) and (11.32) gives... 

Stochastic aggregation does not emerge for oppositely charged particles, when electroneutrality holds due to conditions nk(t) = nB(f) = n(f), particle charge ea = — eB = e. Let us introduce, following the Debye-Hiickel method, the self-consistent potential (J> through Poisson equation... 

The Debye-Huckel theory was developed to extend the capacitor model and is based on a simplified solution of the Poisson equation. It assumes that the double layer is really a diffuse cloud in which the potential is not a discontinuous function. Again, the interest is in deriving an expression for the electrical potential function. This model states that there is an exponential relationship between the charge and the potential. The distribution of the potential is ... 

The primary particle involved in the screening process is the mobile electron. One has then the problem of a self-consistent calculation of the charge distribution in the neighborhood of a test charge. The Thomas-Fermi approach to this problem is the analog of the Debye-Huckel calculation wherein allowance has been made for the Pauli exclusion principle. From any standard text one can obtain the Poisson equation (19)... 

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... 

Taking the surface potential to be xp°, the potential at a distance x as rp, and combining the Boltzmann distribution of concentrations of ions in terms of potential, the charge density at each potential in terms of the concentration of ions, and the Poisson equation describing the variation in potential with distance, yields the Pois-son-Boltzmann equation. Given the physical boundary conditions, assuming low surface potentials, and using the Debye-Huckel approximation, yields... 

Begin with the Poisson equation but keep the e matrix inside the divergence operation V (eV0) = -4jrpext (see Fig. L3.24). The net electric-charge density pext at a given point depends on the magnitude of potential as in Debye-Huckel theory. As before in relation (L3.175),... 

To determine the spatial variation of a static electric field, one has to solve the Poisson equation for the appropriate charge distribution, subject to such boundary conditions as may pertain. The Poisson equation plays a central role in the Gouy-Chapman (- Gouy, - Chapman) electrical - double layer model and in the - Debye-Huckel theory of electrolyte solutions. In the first case the one-dimensional form of Eq. (2)... 

In an electrolyte solution the ions are interspersed by water molecules, moreover they are subject to thermal motion. Debye and Huckel simplified the description of this problem to a mathematically manageable one by considering one isolated ion in a hypothetical, uniformly smeared-out sea of charge, the ionic cloud, with the total charge just opposing that of the ion considered. For this case the Poisson equation in terms of spherical coordinates is given by... 

Equation (1.9) is the linearized Poisson-Boltzmann equation and k in Eq. (1.10) is the Debye-Htickel parameter. This linearization is called the Debye-Hiickel approximation and Eq. (1.9) is called the Debye-Hiickel equation. The reciprocal of k (i.e., 1/k), which is called the Debye length, corresponds to the thickness of the double layer. Note that nf in Eqs. (1.5) and (1.10) is given in units of m . If one uses the units of M (mol/L), then must be replaced by IQQQNAn, Na being Avogadro s number. 

Results for 2-2 electrolytes were of considerable importance since he showed that the numerical integration which uses the complete Poisson-Boltzmann equation fitted well with the Debye-Hiickel equation coupled with Bjerrum Ks for association, provided that the Bjerrum q was used instead of a in the denominator, i.e. 1 + Bq Jl must be used. 

This is an excellent assumption for the plasmas of interest since the Debye length is exceedingly small (10s of pm) compared to the reactor dimensions. Of course, the electroneutrality constraint can t be applied in the sheath, where the Poisson equation... 

Poisson-Boltzmann Equation A fundamental equation describing the distribution of electric potential around a charged species or surface. The local variation in electric-field strength at any distance from the surface is given by the Poisson equation, and the local concentration of ions corresponding to the electric-field strength at each position in an electric double layer is given by the Boltzmann equation. The Poisson-Boltzmann equation can be combined with Debye-Hiickel theory to yield a simplified, and much used, relation between potential and distance into the diffuse double layer. 

In a spherically symmetric situation, such as that in the Debye-Hiickel model described in Section 16.7, the potential is a function only of r and the Poisson equation, Eq. (AIL 19), becomes... 

The typical space-size characterizing a plasma is the Debye radius, which is a linear measure of electroneutrality and shielding of external electric fields. The typical plasma time scale and typical time of plasma response to the external fields is determined by the plasma frequency illustrated in Fig. 3-19. Assume in a one-dimensional approach that all electrons at X > 0 are initially shifted to the right on the distance xq, whereas heavy ions are not perturbed and remain at rest. This results in an electric field, which pushes the electrons back. If = 0 at X < 0, this electric field acting to restore the plasma quasi-neutrality can be found at x > xq from the one-dimensional Poisson equation as... 

The opposite limiting case of a PE star with a small number p -c of arms in a salt-free solution was considered in [121], In the latter case, the counterions can be disregarded and the Poisson equation allowed for an exact numerical solution for the polymer density profile, which confirmed the uniform stretching of the arms in the interior region of the star. The LEA may be applied for analysis of conformations of stars with a small number of arms in salt-added solution, provided the bulk Debye length ro is smaller than the overall size of the star [28]. 

---