Poisson-Boltzmann equation definition

The question to be discussed is whether saturation of the electric field (asserted by Proposition 2.1) implies saturation of the interparticle force of interaction. Consider for definiteness repulsion between two symmetrically charged particles in a symmetric electrolyte solution. In the onedimensional case (for parallel plates) the answer is known—the force of repulsion per unit area of the plates saturates. (This follows from a direct integration of the Poisson-Boltzmann equation carried out in numerous works, primarily in the colloid stability context, e.g., [9]. Recall that again in vacuum, dielectrics, or an ionic system with a linear screening, the appropriate force grows without bound with the charging of the particles.)... 

According to Katchalsky et al. [21] the resolution of the Poisson-Boltzmann equation in the presence of added salt leads to a definition of a minimum distance of approach, R, for a free counter-ion such that ... 

Attempts to improve the theory by solving the Poisson-Boltzmann equation present other difficulties first pointed out by Onsager (1933) one consequence of this is that the pair distribution functions g (r) and g (r) calculated for unsymmetrically charged electrolytes (e.g., LaCl or CaCl2) are not equal as they should be from their definitions. Recently Outhwaite (1975) and others have devised modifications to the Poisson-Boltzmann equation which make the equations self-consistent and more accurate, but the labor involved in solving them and their restriction to the primitive model electrolyte are drawbacks to the formulation of a comprehensive theory along these lines. The Poisson-Boltzmann equation, however, has found wide applicability in the theory of polyelectrolytes, colloids, and the electrical double-layer. Mou (1981) has derived a Debye-Huckel-like theory for a system of ions and point dipoles the results are similar but for the presence of a... 

Kotin and Nagasawa [6] defined the counter-ion binding in analogy to the definition of Bjerrum on ion-pair formation [36]. That is, it is assumed that a polyion is placed in an infinite volume of a neutral salt solution of uni-uni valent type and the polyion is a rod of infinite length having a charge density N/L, Moreover, it is assumed that the ionic distribution around the rod is determined from the Poisson-Boltzmann equation. Then, if one plots the distribution of counter-ions PcC ) against the distance from the axis of the polyion r,... 

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... 

The applicability of continuum theories, such as the Poisson-Boltzmann model, in nanoscale is the most concerned issue in this field yet. Numerous conflicting results were reported in literature. We have done careful MD simulations of EOF and compared the ion distributions with the PB predictions rigorously to clarify the applicability of the continuum theory. To compare the descriptions from two different scales, above all, the observers have to be stand on the same base to avoid definition gaps. First, when presenting the MD results, the bin size should not be smaller than the solvent molecular diameter in comparison with the continuum theory otherwise, the MD results are not the macroscopic properties at the same level of the continuum. A second gap which departs the MD results from the PB predictions is the effect of the Stem layer. As well known, the PB equation describes only the ion distribution in diffusion (outer) layer of the electric double layer (EDL) [1]. In the continuum theory, the compact (iimer) layer of EDL is too thin (molecular scale) to be considered, and therefore, the PB equation almost governs the ion distribution in the whole domain. However, in nanofluidics the iimer layer which is also termed as Stem layer is comparable to the channel in size. The PB equation is not able to govern the ion behavior in the Stem layer in theory. Therefore, if one compares the MD... 

Poisson equation (7) and in the exponent of the Boltzmann factor in Eq. 8, which requires the definitions of the potential y and the screening constant k to incorporate an additional factor of v. This implies a replacement t — in Eqs. 13 , 15 , and 16. The remarkable consequence... 

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