Electrostatic potential distribution Poisson-Boltzmann equation

If there are ions in the solution, they will try to change their location according to the electrostatic potential in the system. Their distribution can be described according to Boltzmarm. Including these effects and applying some mathematics leads to the final linearized Poisson-Boltzmann equation (Eq. (43)). 

The next step is to determine the electrical charge and potential distribution in this diffuse region. This is done by using relevant electrostatic and statistical mechanical theories. For a charged planar surface, this problem was solved by Gouy (in 1910) and Chapman (in 1913) by solving the Poisson-Boltzmann equation, the so called Gouy-Chapman (G-C) model. 

Rahaman and Hatton [152] developed a thermodynamic model for the prediction of the sizes of the protein filled and unfilled RMs as a function of system parameters such as ionic strength, protein charge, and size, Wq and protein concentration for both phase transfer and injection techniques. The important assumptions considered include (i) reverse micellar population is bidisperse, (ii) charge distribution is uniform, (iii) electrostatic interactions within a micelle and between a protein and micellar interface are represented by nonlinear Poisson-Boltzmann equation, (iv) the equilibrium micellar radii are assumed to be those that minimize the system free energy, and (v) water transferred between the two phases is too small to change chemical potential. 

The purpose of the present chapter is to introduce some of the basic concepts essential for understanding electrostatic and electrical double-layer pheneomena that are important in problems such as the protein/ion-exchange surface pictured above. The scope of the chapter is of course considerably limited, and we restrict it to concepts such as the nature of surface charges in simple systems, the structure of the resulting electrical double layer, the derivation of the Poisson-Boltzmann equation for electrostatic potential distribution in the double layer and some of its approximate solutions, and the electrostatic interaction forces for simple geometric situations. Nonetheless, these concepts lay the foundation on which the edifice needed for more complicated problems is built. 

The ionic groups on the micellar surface and the counterions will give rise to a nonuniform electrostatic potential according to the Poisson equation. If furthermore the electrostatic effects dominate the counterion distribution the ion concentration is determined by following a Boltzmann distribution. These approximations lead to the Poisson-Boltzmann equation. 

A thorough discussion of the basic theory describing electrostatic interactions can be found in [7] the pertinent points are discussed below. Electrostatic forces arise from the osmotic pressure difference between two charged surfaces as a result of the local increase in the ionic distribution around each charged surface. For a single electrified interface, the local ion distribution is coupled to the potential distribution near that surface and can be described using the Poisson-Boltzmann equation. The solution of this equation shows that for low surface potentials the potential follows an exponential function with distance from the interface, D, given by... 

The ion and electrical potential distributions in the electrical double layer can be determined by solving the Poisson-Boltzmann equation [2,3]. According to the theory of electrostatics, the relationship between the eleetrieal potential ij/ and the local net charge density per unit volume at any point in the solution is deseribed by the Poisson equation ... 

Dependencies of AG on the average separation l between SC>3 groups and on the distance a between the plane of proton transfer and the negatively charged interface were studied theoretically in Ref. 43, 44. Using an approach based on the Poisson-Boltzmann equation, modulations of the electrostatic potential and of the distribution of mobile protons in the proximity of the charged anionic sites were calculated. 

Equation 5.178 demonstrates that for two identically charged surfaces n, is always positive, i.e., corresponds to repulsion between the surfaces. In general, we have 0 < m < 1, because the coions are repelled from the film due to the interaction with the film surfaces. To find the exact dependence of riel oil tho film thickness, h, we solve the Poisson-Boltzmann equation for the distribution of the electrostatic potential inside the film. The solution provides the following connection between riel 2nd h for symmetric electrolytes " i ... 

Here s(r) is the dielectric constant. We use s(r) to emphasize that the constant is a function of position for example, it is different inside a solvent and inside a solute. The electrostatic potential to be determined and p(r) is the charge distribution of the solute. Equation (25) is an exact equation of the electrostatic potential in a dielectric. If the solvent contains some dissolved salt (i.e., the concentration of positive and negative charges is the same) and we assume that the distribution of salt ions follows a Boltzmann distribution, then the electrostatic potential of the system is described by the Poisson-Boltzmann equation... 

The external electric field perturbs the ionic distribution and the electrostatic potential around the central ion. The distribution functions fij and the electrostatic potential y)ij which are related by the Poisson-Boltzmann equation... 

The functional depends on three types of fields, namely, surface fluctuations (R), electrostatic potential < (z, R), and the ionic concentrations rtf (z, R). Minimizing it with respect to < i and nf at a given f(R), one obtains Poisson-Boltzmann equations that describe the distribution of the electrostatic potential and ionic concentrations. Substituting the results into the density functional, one can minimize that with respect to f (R). This... 

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. 

The use of molecular dynamics to study the electric double-layer structure started a little over a decade ago, with the hope of determining more accurate structures because the classical description of an electric double layer based on the Poisson-Boltzmann equation is accurate only for low surface potential and dilute electrolytes. The Poisson-Boltzmann equation only considers the electrostatic interactions between the charged surface and ions in the solution, but not the ion-ion interactions in the solution and the finite molecule size, which can be taken into account in molecular dynamics simulations. It was shown [6, 7] that the ion distribution in the near-wall region could be significantly different from the prediction of classical theory. Typical molecular dynamics simulation results of counterion and co-ion concentrations in a nanochannel are shown in Fig. 2a. The ion distribution obtained... 

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.)... 

Gels should, in general, be considered to be composed of heterogeneous structures on different orders from a few angstroms to several micrometers (structure hierarchy). This makes it extremely difficult to calculate accurately the electrostatic potential distribution in the gel by the Poisson-Boltzmann equation. 

The last equation is known as the Boltzmann distribution of the ions in the external electrostatic potential. Combining eqns [33] and [40], one arrives at the nonlinear Poisson-Boltzmann equation describing the distribution of the reduced electrostatic potential around a polyelectrolyte chain... 

Under conditions relevant for fuel cell operation, the reaction current density of the ORR is small compared to separate flux contributions caused by proton diffusion and migration in Equation 3.62. Therefore, the electrochemical flux termNjj+ on the left-hand side of the Nernst-Planck equation in Equation 3.62 can be set to zero. In this limit, the PNP equations reduce to the Poisson-Boltzmann equation (PB equation). This approach allows solving for the potential distribution independently and isolating the electrostatic effects from the effects of oxygen transport. 

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