Modified Poisson-Boltzmann models

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... 

Eqs. (1), (3) and (4) represent a Modified Poisson-Boltzmann approach, which, when the ions interacts only via the mean field potential ip(z) (e.g. AWa=AWc=0), reduces to the well-known Poisson-Boltzmann equation. The only change in the polarization model is the replacement of the constitutive equation Eq. (1) by Eq. (2), which accounts for the correlation in the orientation of neighboring dipoles. However, since independent functions, four boundary conditions are needed to solve the system composed of Eqs. (2) (3) and (4). For only one surface immersed in an electrolyte, t]/(z-co)-0, m(z-co)=0 whereas for two identical surfaces, the symmetry of ip and m requires that = 0 and m(z=0)=0 where z- 0 represents... 

Das, T., Bratko, D., Bhuiyan, L.B., and Outhwaite, C.W. Polyelectrolyte solutions containing mixed valency ions in the cell model A simulation and modified Poisson-Boltzmann study. Journal of Chemical Physics, 1997,107, No. 21, p. 9197-9207. 

There have been considerable efforts to move beyond the simplified Gouy-Chapman description of double layers at the electrode-electrolyte interface, which are based on the solution of the Poisson-Boltzmann equation for point charges. So-called modified Poisson-Boltzmann (MPB) models have been developed to incorporate finite ion size effects into double layer theory [61]. An early attempt to apply such restricted primitive models of the double layer to the ITIES was made by Cui et al. [62], who treated the problem via the MPB4 approach and compared their results with experimental data for the more problematic water-DCE interface. This work allowed for the presence of the compact layer, although the potential drop across this layer was imposed, rather than emerging as a self-consistent result of the theory. The expression used to describe the potential distribution across this layer was... 

These deviations were first explained by the presence of a compact, ion-free layer at the interface this is known as the modified Verwey-Niessen model. Obviously, the presence of an ion-free layer can only reduce the capacity, so the theory had to be modified further. For a few systems a consistent interpretation of the experimental capacity was achieved [78-80] by combining this model with the soolled modified Poisson-Boltzmann (MPB) theory [81], which attempts to correct the GC theory by accounting for the finite size of the ions and for image effects, while the solvent is still treated as a dielectric continuum. The combined model has an adjustable parameter, so it is difficult to judge whether the agreement with experimental data is significant. The existence... 

AppKcation of MPB model to ITIES was made first by Torrie and Val-leau [28]. Cui and coworkers [29] applied the modified Poisson-Boltzmann theory (MPB4) to the ITIES and found... 

Li B, Kwok DY (2004) Electrokinetic microfluidic phenomena by a lattice Boltzmann model using a modified Poisson-Boltzmann equation with an excluded volume effect. J Chem Phys 120 947-953... 

In addition to the solvent contributions, the electrochemical potential can be modeled. Application of an external electric field within a metal/vacuum interface model has been used to investigate the impact of potential alteration on the adsorption process [111, 112]. Although this approach can model the effects of the electrical double layer, it does not consider the adsorbate-solvent, solvent-solvent, and solvent-metal interactions at the electrode-electrolyte interface. In another approach, N0rskov and co-workers model the electrochemical environment by changing the number of electrons and protons in a water bilayer on a Pt(lll) surface [113-115]. Jinnouchi and Anderson used the modified Poisson-Boltzmann theory and DFT to simulate the solute-solvent interaction to integrate a continuum approach to solvation and double layer affects within a DFT system [116-120]. These methods differ in the approximations made to represent the electrochemical interface, as the time and length scales needed for a fiilly quantum mechanical approach are unreachable. 

In the two cases, it is interesting to know how the electric potential varies from one phase to the next and therefore what the charge distribution is on either side of the interface. At present, molecular dynamics is not powerful enough to treat a system containing enough ions to evaluate the electric potential variation. Theoretical approaches have to more rely on classical models such as the modified Poisson-Boltzmann equation. 

C. W. Outhwalte and L. B. Bhuiyan, J. Chem. Phys., 84, 3461 (1986). A Modified Poisson-Boltzmann Equation In Electric Double Layer Theory for a Primitive Model Electrolyte with Size-Asymmetric Ions. 

Primitive Model Electrolytes in the Modified Poisson-Boltzmann Theory. 

It seems that we are not far from a new and this time quantitative understanding of specific ion effects in colloid and surface chemistry. The key is the use of ion-surface potentials and water profiles near surfaces inferred from molecular dynamics simulation and their appropriate use in solvent-averaged models in order to derive an efficient, but physically well-based alternative to DLVO. The modified Poisson-Boltzmann equation is shown to be useful approach to calculate thermodynamic properties that depend on energies and structures at different length scales. 

What remains to be done is two-fold first, the PMFs coming from MD simulations must be checked and probably further improved. Therefore, the presented results must still be taken with a grain of salt. Second, the same model for ion hydration should be able to describe both bulk properties (activity coefficients) and surfaces properties (surface tensions). If this is possible, we can have confidence that the modified Poisson-Boltzmann approach is robust enough to capture the most relevant features of specific ion effects. 

A new theory of electrolyte solutions is described. This theory is based on a Debye-Hiickel model and modified to allow for the mutual polarization of ions. From a general solution of the linearized Poisson-Boltzmann equation, an expression is derived for the activity coefficient of a central polarized ion in an ionic atmosphere of non-spherical symmetry that reduces to the Debye-Hiickel limiting laws at infinite dilution. A method for the simultaneous charging of an ion and its ionic cloud is developed to allow for ionic polarization. Comparison of the calculated activity coefficients with experimental values shows that the characteristic shapes of the log y vs. concentration curves are well represented by the theory up to moderately high concentrations. Some consequences in relation to the structure of electrolyte solutions are discussed. 

The polarization model is extended to account for the ion-ion and ion-surface interactions, not included in the mean field electrical potential. The role of the disorder on the dipole correlation length A, is modeled through an empirical relation, and it is shown that the polarization model reduces to the traditional Poisson Boltzmann formalism (modified to account for additional interactions) when X, becomes sufficiently small. 

Despite the difficulties in quantitative treatment, there exist theoretical models based on the classical treatment initiated by Gouy, Chapman, Debye, and Hiickel and later modified by Stem and Cjrahame. As shown in Figure 7.3, a reasonable representation of the potential distribution by the Poisson-Boltzmann equation can be given as... 

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