Boltzmann-Poisson approximation

The modified Poisson-Boltzmann (MPB) approximation is the product of many theoretical investigations into improvements of the PB approach of Gouy and Chapman based on classical statistical mechanics [22, 23] for the description of the electric double layer at a charged plane interface. These extensions originated in a work of Kirkwood [24] who used the Giintelberg charging process to examine the fluctuation terms neglected in the DH theory of electrolytes. 

Marmur [12] has presented a guide to the appropriate choice of approximate solution to the Poisson-Boltzmann equation (Eq. V-5) for planar surfaces in an asymmetrical electrolyte. The solution to the Poisson-Boltzmann equation around a spherical charged particle is very important to colloid science. Explicit solutions cannot be obtained but there are extensive tabulations, known as the LOW tables [13]. For small values of o, an approximate equation is [9, 14]... 

SASA), a concept introduced by Lee and Richards [9], and the electrostatic free energy contribution on the basis of the Poisson-Boltzmann (PB) equation of macroscopic electrostatics, an idea that goes back to Born [10], Debye and Htickel [11], Kirkwood [12], and Onsager [13]. The combination of these two approximations forms the SASA/PB implicit solvent model. In the next section we analyze the microscopic significance of the nonpolar and electrostatic free energy contributions and describe the SASA/PB implicit solvent model. 

To describe the simple phenomena mentioned above, we would hke to have only transparent approximations as in the Poisson-Boltzmann theory for ionic systems or in the van der Waals theory for non-coulombic systems [14]. Certainly there are many ways to reach this goal. Here we show that a field-theoretic approach is well suited for that. Its advantage is to focus on some aspects of charged interfaces traditionally paid little attention for instance, the role of symmetry in the effective interaction between ions and the analysis of the profiles in terms of a transformation group, as is done in quantum field theory. 

Instead of an exact calculation, Gouy and Chapman have assumed that (4) can be approximated by combining the Poisson equation with a Boltzmann factor which contains the mean electrical potential existing in the interface. (This approximation will be rederived below). From this approach the distribution of the potential across the interface can be calculated as the function of a and from (2) we get a differential capacitance Cqc- It has been shown by Grahame that Cqc fits very well the measurements in the case of low ionic concentrations [11]. For higher concentrations another capacitance in series, Q, had to be introduced. It is called the inner layer capacitance and it was first considered by Stern [1,2]. Then the experimental capacitance Cexp is analyzed according to ... 

Oosawa (1971) developed a simple mathematical model, using an approximate treatment, to describe the distribution of counterions. We shall use it here as it offers a clear qualitative description of the phenomenon, uncluttered by heavy mathematics associated with the Poisson-Boltzmann equation. Oosawa assumed that there were two phases, one occupied by the polyions, and the other external to them. He also assumed that each contained a uniform distribution of counterions. This is an approximation to the situation where distribution is governed by the Poisson distribution (Atkins, 1978). If the proportion of site-bound ions is negligible, the distribution of counterions between these phases is then given by the Boltzmann distribution, which relates the population ratio of two groups of atoms or ions to the energy difference between them. Thus, for monovalent counterions... 

The ideal conductor model does not account for diffuseness of the ionic distribution in the electrolyte and the corresponding spreading of the electric field with a potential drop outside the membrane. To account approximately for these effects we apply Poisson-Boltzmann theory. The results for the modes energies can be summarized as follows [89] ... 

In addition to the nearest-neighbor interaction, each ion experiences the electrostatic potential generated by the other ions. In the literature this has generally been equated with the macroscopic potential 0 calculated from the Poisson-Boltzmann equation. This corresponds to a mean-field approximation (vide infra), in which correlations between the ions are neglected. This approximation should be the better the low the concentrations of the ions. 

In the same way as described in Sec. 5.2 for a diifiise layer in aqueous solution, the differential electric capacity, Csc, of a space charge layer of semiconductors can be derived from the Poisson s equation and the Fermi distribution function (or approximated by the Boltzmann distribution) to obtain Eqn. 5-69 for intrinsic semiconductor electrodes [(Serischer, 1961 Myamlin-Pleskov, 1967 Memming, 1983] ... 

A more detailed view of the dynamies of a ehromatin chain was achieved in a recent Brownian dynamics simulation by Beard and Schlick [65]. Like in previous work, the DNA is treated as a segmented elastic chain however, the nueleosomes are modeled as flat cylinders with the DNA attached to the cylinder surface at the positions known from the crystallographic structure of the nucleosome. Moreover, the electrostatic interactions are treated in a very detailed manner the charge distribution on the nucleosome core particle is obtained from a solution to the non-linear Poisson-Boltzmann equation in the surrounding solvent, and the total electrostatic energy is computed through the Debye-Hiickel approximation over all charges on the nucleosome and the linker DNA. 

Recent advances in computational power have made it possible to use efficient approximations of the Poisson-Boltzmann equation also to estimate the electrostatic component of protein-ligand interactions [56]. 

For our work, expressions of Ohshlma et. al. (37) obtained from an approximate form of the Poisson-Boltzmann equation were used. These analytical expressions agree with the exact solution for xRp 2. (All of our calculations meet this criterion.) The relation between the surface potential and the surface charge density Is (37)... 

The geometry. It is clear that the geometry of the system is much simplified in the slab model. Another possibility is to model the protein as a sphere and the stationary phase as a planar surface. For such systems, numerical solutions of the Poisson-Boltzmann equations are required [33]. However, by using the Equation 15.67 in combination with a Derjaguin approximation, it is possible to find an approximate expression for the interaction energy at the point where it has a minimum. The following expression is obtained [31] ... 

Other molecular thermodynamic models for protein-reverse micelle complexes have also emerged. Bratko et al. [171] presented a model for phase transfer of proteins in RMs. The shell and core model was combined with the Poisson-Boltzmann approximation for the protein-RM complex and for the protein-free RM. The increase in entropy of counterions released from RMs on solubilization of a protein was the main contribution to the decrease in free energy of com-plexation. Good agreement was found with SANS results of Sheu et al. [151] for cytochrome C solubilization and the effect of electrolytes on it. However, this model assumes that filled and empty RMs are of the same size, independent of salt strength and pH, which is not true according to experimental evidence available since then. 

One of the simplest equations is obtained using the Debye-Hiickel approximation (for low potentials) and the superposition principle. The latter assumes that the unperturbed potential near a charged surface can be simply added to that potential due to the other (unperturbed) surface. Thus, for the example shown in the Figure 6.12, it follows that /m = 2 /d/2- This is precisely valid for Coulomb-type interactions, where the potential at any point can be calculated from the potentials produced by each fixed charge, individually. However, the Poisson-Boltzmann equation is non-linear (this has to do with the fact that in the diffuse double-layer the ions are not fixed but move because of their kinetic energy) and so this is formally not correct although it still offers a useful approximation. 

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 derivation of the Poisson equation implies that the potentials associated with various charges combine in an additive manner. The Boltzmann equation, on the other hand, involves an exponential relationship between the charges and the potential. In this way a fundamental inconsistency is introduced when Equations (26) and (28) are combined. Equation (29) does not have an explicit general solution anyhow and must be solved for certain limiting cases. These involve approximations that —at the same time —overcome the objection just stated. 

The above equation is known as the linearized Poisson-Boltzmann equation since the assumption of low potentials made in reaching this result from Equation (29) has allowed us make the right-hand side of the equation linear in p. This assumption is also made in the Debye-Hiickel theory and prompts us to call this model the Debye-Hiickel approximation. Equation (33) has an explicit solution. Since potential is the quantity of special interest in Equation (33), let us evaluate the potential at 25°C for a monovalent ion that satisfies the condition e p = kBT ... 

For studying the stability of colloidal particles in suspension (Chapter 13) or for determining the potential at the surface of particles (Chapter 12), one often needs expressions for potential distributions around small particles that have curved surfaces. Solving the Poisson-Boltzmann equation for curved geometries is not a simple matter, and one often needs elaborate numerical methods. The linearized Poisson-Boltzmann equation (i.e., the Poisson-Boltzmann equation in the Debye-Hiickel approximation) can, however, be solved for spherical electrical double layers relatively easily (see Section 12.3a), and one obtains, in place of Equation (37),... 

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