Poisson-Boltzmann theory of the diffuse double layer

Poisson-Boltzmann theory of the diffuse double layer... 

Poisson-Boltzmann Theory of the Diffuse Double Layer... 

Poisson-Boltzmann Theory of the Diffuse Double Layer 99 The left-hand side is equal to E (S). We insert this and integrate ... 

Regardless of whether the transfer or transport approach is followed, the flux equation can, in principle, be solved for any given potential profile in the diffuse layers. Usually, the potential profile is quite well described using the Poisson-Boltzmann equation, which in the case of a z z supporting electrolyte can be solved analytically. More advanced theories may also be applied [81, 82]. In a recent theoretical study based on the idea of determining the permeabilities of the diffuse double layers and the interface independently, different potential profiles were analyzed. The Poisson-Boltzmann potential profile, a stepwise linear potential profile, and a potential step profile were compared. Interestingly, even a very simple stepwise potential profile is hardly distinguishable from the linear potential profile and the relatively complicated Poisson-Boltzmann... 

The Debye-Hiickel approximation to the diffuse double-layer problem produces a number of relatively simple equations that introduce a variety of double-layer topics as well as a number of qualitative generalizations. In order to extend the range of the quantitative relationships, however, it is necessary to return to the Poisson-Boltzmann equation and the unrestricted Gouy-Chapman theory, which we do in Section 11.6. 

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. 

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... 

The situation is still more complex in the presence of surfactants. Recently, a self-consistent electrostatic theory has been presented to predict disjoining pressure isotherms of aqueous thin-liquid films, surface tension, and potentials of air bubbles immersed in electrolyte solutions with nonionic surfactants [53], The proposed model combines specific adsorption of hydroxide ions at the interface with image charge and dispersion forces on ions in the diffuse double layer. These two additional ion interaction free energies are incorporated into the Boltzmann equation, and a simple model for the specific adsorption of the hydroxide ions is used for achieving the description of the ion distribution. Then, by combining this distribution with the Poisson equation for the electrostatic potential, an MPB nonlinear differential equation appears. 

The presence of the diffuse layer determines the shape of the capacitance-potential curves. For a majority of systems, models describing the double-layer structure are oversimplified because of taking into account only the charge of ions and neglecting their specific nature. Recently, these problems have been analyzed using new theories such as the modified Poisson-Boltzmann equation, later developed by Lamper-ski. The double-layer capacitanties calculated from these equations are... 

This model is based on the Gouy-Chapman theory (diffuse double-layer theory). The theory states that in the area of the boundary layer between solid and aqueous phase, independently of the surface charge, increased concentrations of cations and anions within a diffuse layer exists because of electrostatic forces. In contrast to the constant-capacitance model, the electrical potential does not change up to a certain distance from the phase boundaries and is not immediately declining in a linear manner (Fig. 14 a). Diffusion counteracts these forces, leading to dilution with increasing distance from the boundary. This relation can be described physically by the Poisson-Boltzmann equation. 

We use the Gouy-Chapman theory for the diffuse layer which is based on the Poisson-Boltzmann (P.B.) equation for the potential distribution. Although the different corrections to the P.B. equation in double-layer theory have been investigated (20, 21, 22, 23), it is difficult to state precisely the range of validity of this equation. In the present problem the P.B. equation seems a reasonable approximation at 0.1M of a 1-1 electrolyte to 50mV for the mean electrostatic potential pd at the ohp (24) this upper limit for pd increases with a decrease in electrolyte concentration. All the values for pd calculated in Tables I-IV are less than 50 mV— most of them are well below. If n is the volume density of each ion type of the 1-1 electrolyte in the substrate, c the dielectric constant of the electrolyte medium, and... 

Diffuse double layer (DDL) theory as applied to the surface chemistry of soils refers to the description of ion charge and inner potential contained in the Poisson-Boltzmann equation ... 

The most widely used theory of the stability of electrostatically stabilized spherical colloids was developed by Deryaguin, Landau, Verwey, and Overbeek (DLVO), based on the Poisson-Boltzmann equation, the model of the diffuse electrical double layer (Gouy-Chapman theory), and the van der Waals attraction [60,61]. One of the key features of this theory is the effective range of the electrical potential around the particles, as shown in Figure 25.7. Charges at the latex particles surface can be either covalently bound or adsorbed, while ionic initiator end groups and ionic comonomers serve as the main sources of covalently attached permanent charges. 

These theories are based on the interaction of the solute ion with the charged surface layer established by the adsorbed counterion and by adsorbed competing ions. The nonstochiometric models apply the Poisson-Boltzmann equation to estimate retention from an electrostatic point of view. The electrical double-layer model applied uses different approaches such as liquid partition , surface adsorption, diffuse layer ion-exchange , and sru face adsorption doublelayer models. It is not possible to draw conclusions about the ion pair process from chromatographic retention data, but each model and theory may find use in describing experimental results under the particular conditions studied. 

Stern combined the ideas of Helmholtz and that of a diffuse layer [64], In Stern theory we take a pragmatic, though somewhat artificial, approach and divide the double layer into two parts an inner part, the Stern layer, and an outer part, the Gouy or diffuse layer. Essentially the Stern layer is a layer of ions which is directly adsorbed to the surface and which is immobile. In contrast, the Gouy-Chapman layer consists of mobile ions, which obey Poisson-Boltzmann statistics. The potential at the point where the bound Stern layer ends and the mobile diffuse layer begins is the zeta potential (C potential). The zeta potential will be discussed in detail in Section 5.4. 

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