MPB equation

The popular Poisson-Boltzmann equation considers the mean electrostatic potential in a continuous dielectric with point charges and is therefore, an approximation of the actual potential. An improved model and mathematical solution resulted in the MPB equation (26). This equation is based on a restricted primitive electrolyte model that considers ions as charged hard spheres with diameter d in a continuous uniform structureless dielectric medium of constant dielectric permittivity s. The sphere representing an ion has the same permittivity e. The model initially was developed for an electrolyte at a hard wall with dielectric permittivity and surface charge density a. The charge is distributed over the surface evenly and continuously. 

In comparison with the Gouy-Chapman theory, the MPB equation for point charges contains two improvements in that the Debye-Huckel parameter k depends on distance, and the ion image is screened. 

In solving the MPB equations, different estimates of fj(x) were considered. Depending on the type of fi(x), the equations were called MPBl, MPB2, MPB3 and... 

The MPB approximation has been used recently, in 1991, by Bhuiyan et al. [25] for the surface tension problem. The excess surface tension was determined by numerical integration of the Gibbs adsorption isotherm (the Gibbs equation), with the electrolyte activity obtained from the bulk MPB approximation. A simple model of the interface, used previously in another study [16], was adopted. In an earlier study [26], this version of the MPB equation had been reported to be... 

Kirkwood gave the first detailed derivation of the nonlinear PB equation, including corrections due to correlation (fluctuation) and finite ion size. These were elaborated on by Levine, culminating in a paper with Bell" ° dealing with the electric double layer near a polyelectrolyte surface. The leading terms in the Bell-Levine treatment were kept and some restrictions and approximations were applied to obtain a modified Poisson-Boltzmann (MPB) equation in the form of an integrodifference/differential equation that took into account the fluctuation potential of Kirkwood. 

S. G. Johnson, and J. D. Joanuopoulos, Block-iterative frequency-domain methods for Maxwell s equations in a planewave basis. Optics Express 8, 173-190 (2001) http //ab-initio.mit.edu/mpb/download.html... 

Linear. Since mass and energy are linearly related between modules, purely linear flowsheet calculations can be formulated as a solution to a set of linear equations once linear models for the modules can be constructed. Linear systems, especially for material balance calculations can be very useful (16). Two general systems, based on linear models, SYMBOL (77) and MPB II (7 ) are indicated in Table 1. MPB II is based on a thesis by Kniele (79). If Y is the vector of stream outputs and the module stream inputs are X, then as discussed by Mahalec, Kluzik and Evans (80)... 

The kinetic procedure used for calculation of C from N-data (Sect. 4.1.2.1) can be modified to yield relevant equations applied when MPB-data are available. Thus, if Rp and Rtl are time-independent, then equations ... 

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... 

This theory takes into account the finite size of ions, the fluctuation potential, and image forces in the electrolyte solution next to a rigid electrode, but it still an approximation. The MPB theory begins with the Poisson equation for the mean electrostatic potential ir in solution ... 

In the case of electrode-electrolyte solution interfaces, the Poisson-Boltzmann equation has been modified for integrating many effects as, for example, finite ion size, concentration dependence of the solvent, ion polarizability, and so on. More often, this modification consists in the introduction of one or several supplementary terms to the energetic contribution in the distribution, which leads to modified Poisson-Boltzmann (MPB) nonlinear differential equations [52],... 

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. 

Outhwaite and coworkers (Outhwaite 1978 Bhuiyan, Outhwaite, and Levine 1979 Outhwaite, Bhuiyan, and Levine 1980 Outhwaite and Bhuiyan 1983) introduced a MPB approach, in which a modified PB equation was proposed in the RPM, always in the infinite plane surface geometry ... 

FIGURE 3.15 Dimensionless mean electrostatic potential (a) and surface-ion distribution function (b) as predicted by the Gouy-Chapman-Stern (GCS) and modified Poisson-Boltzmann (MPB) theories for a 1 1 electrolyte with a = 0.425 nm and c = 0.197 M. (Outhwaite, Bhuiyan, and Levine, 1980, Theory of the electric double layer using a modified Poisson-Boltzmann equation. Journal of the Chemical Society, Faraday Transactions 2 Molecular and Chemical Physics, 76, 1388-1408. Reproduced by permission of The Royal Society of Chemistry.)... 

There have been other attempts to apply integral equations methods derived from Equation 3.51 or other similar expressions. One of them is the hypemetted chain (Henderson 1983), which is a generalization of the MSA theory, applicable to higher charge/potential values. It gives results comparable to the MPB ones but requires more extensive numerical evaluations. Another proposal is so-called dressed-ion theory of Kjellander and Mitchell (1994, 1997). 

Thus the total number of metal-polymer bonds, MPB, in the polymerization system will increase with time and may be described by the equation... 

In addition to neglecting ion correlation, using the mean electrostatic potential has the undesirable consequence that the (nonlinear) PB equation no longer satisfies a reciprocity condition that use of the potential of mean force would obey. Linearization of the equation by Debye and Hiickel regained this condition. These considerations led Outhwaite and others to propose modifications of the PB equation to treat these problems. Within this modified Poisson-Boltzmaim (MPB) theory, the effect of ion correlation is expressed in terms of a fluctuation potential for which a first-order (local) expression, written as an activity coefficient, can be derived. Their result for bulk hard-sphere electrolyte ions of valence z, and common radius a gives the formula ... 

Figure 52 A comparison of cation concentration profiles for a bulk 0.05 M 1 1 electrolyte with 1 and 2-A radius hard-sphere ions in the presence of a charged cylinder of radius 10 A with surface charge density = —0.01 eo/A calculated according to the PB equation (Eq. [389]) without activity corrections (dashed line), the PB equation with MPB activity corrections of Eqs. [429] and [430] (solid line), and Metropolis Monte Carlo simulations (filled circles —1 A open circles 2 A).

Integral Equation and Eield-Theoretic Approaches In addition to theories based on the direct analytical extension of the PB or DH equation, PB results are often compared with statistical-mechanical approaches based on integral equation or density functional methods. We mention only a few of the most recent theoretical developments. Among the more popular are the mean spherical approximation (MSA) and the hyper-netted chain (HNC) equation. Kjellander and Marcelja have developed an anisotropic HNC approximation that treats the double layer near a flat charged surface as a series of discrete layers.Attard, Mitchell and Ninham have used a Debye-Hiickel closure for the direct correlation function to obtain an analytical extension (in terms of elliptic integrals) to the PB equation for the planar double layer. Both of these approaches, which do not include finite volume corrections, treat the fluctuation potential in a manner similar to the MPB theory of Outhwaite. 

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