The Poisson-Boltzmann equation

The aim of this chapter is to calculate the electric potential near a charged planar interface. In general, this potential depends on the distance normal to the surface x. Therefore, we consider a planar solid surface with a homogeneously distributed electric surface charge density a, which is in contact with a liquid. The surface charge generates a surface potential 

What is the potential distribution ip(x, y, z) in the solution In general, charge density and electric potential are related by the Poisson4 equation  

pe is the local electric charge density in C/m3. With the Poisson equation, the potential distribution can be calculated once the exact charge distribution is known. The complication in our case is that the ions in solution are free to move. Before we can apply the Poisson equation we need to know more about their spacial distribution. This information is provided by Boltzmann5 statistics. According to the Boltzmann equation the local ion density is given by 

4 Denis Poisson, 1781-1840. French mathematician and physicist, professor in Paris. 

5 Ludwig Boltzmann, 1844-1906. Austrian physicist, professor in Vienna. 

4) Peter Debye, 1884-1966. American physicist of Dutch origin, professor in Zurich, Utrecht, Gottingen, Leipzig, Berlin, and Ithaca. Nobel Prize for chemistry, 1936. 

5) Erich Armand Arthur Joseph Hiickel, 1896-1980. German chemist and physicist, professor in Marburg. 

Let an ion of type a be placed at the center of our coordinate system. Let p/(R/a) be the conditional density of ions of type i at R, given an ion of type a at the center (Fig. 6.6). If pi is the bulk density of ions of type /, then we have the relation 

FIGURE 6.6. The density p/(R/a) measured relative to an ion of type a at the origin. 

The essential approximation introduced in the Debye-Hiickel theory is the statement 

In calculating the activity coefficient of the species a, we shall need only the electric potential due to the ionic atmosphere, which we denoted by y/ iR/a). First we need to solve for the total potential function i//(R/of). To evaluate if/(R/a), we use the Poisson equation from electrostatics, which reads 

The first term on the rhs of (6.12.39) is the contribution to the charge density due to the point charge at the origin, where 5(R) is the Dirac delta function. The second term sums over all ionic species L Since we are interested in large distances R, we can ignore the contribution from the first term on the rhs of (6.12.39). Substituting (6.12.35) and (6.12.36) into (6.12.39) and then substituting (6.12.39) into (6.12.38), we obtain 

The Poisson-Boltzmann equation. Equilibrium is a steady state without macroscopic fluxes. As we pointed out in the Introduction, under these conditions the equations of electro-diffusion reduce to 

Recall that Aj are positive integration constants. For open systems Ai is equal to the known concentration of the charge carrier t, wherever p vanishes (e.g., at infinity). For closed systems, in which only the total number of charge carriers may be known rather than their concentration somewhere, the Ai are subject to determination in the course of the solution. (The properties of the solutions for parallel open and closed system formulations may differ quite markedly, as was exemplified in [1].) Equation (2.1.2), the Poisson-Boltzmann equation, is a particular case of the nonlinear Poisson equations 

Equation (2.1.3a) has been studied extensively in different mathematical and physical contexts ranging from differential geometry to reaction-diffusion, electrokinetics, colloid stability, theory of polyelectrolytes, etc. In 

The specific electro-diffusion phenomena, the field and force saturation and counterion condensation, as well as the corresponding features of the solutions to the Dirichlet problem for (2.1.2) to be addressed in this chapter, are closely related to those observed by Keller [7], [8] for the solutions of (2.1.3a) with f tp) positive definite, satisfying a certain growth condition. Keller considered f( p) 0, satisfying the condition 

In addition, ip(x, () satisfied (2.1.3a) in some open domain 0 with a boundary d l and assumed the value 0 on some part dfli of dfl. Keller showed that ip(x, C) approaches a finite upper limit p(x), when ( — oo, for any finite point x away from the boundary. 

Consider, then, a fluid containing ions that are nonuniformly distributed, producing a position-dependent electric field E. Since an electric field is conservative, it is given by the gradient of a potential, E = — VtJ/. Negative ions tend to collect at locations where this potential is positive relative to some datum, and positive ions collect where it is negative. At equilibrium, the number density n, of ionic species i at each location is given by the Boltzmann distribution  

(2-41) fixes the charged species distribution, once is known. But the electric field (and hence is determined by the charge imbalance, according to Maxwell s equations, which for electrostatics reduce to the Poisson equation  

From this equation, the potential function can be determined, and therefore also the species distributions via Eq. (2-41), once suitable boundary conditions are given. One boundary condition is simply the reference position at which one chooses to set = 0. The remaining boundary conditions are usually specifications of either the surface charge density a (in coulombs/meter) or the surface potential 

Consider a uniformly charged particle immersed in a hquid containing N ionic species with valence Zt and bulk concentration (number density) rtf (/ = 1, 2,. . . , AO 

Usually we need to consider only electrolyte ions as charged species. The electric potential i//(r) at position r outside the particle, measured relative to the bulk 

Biophysical Chemistry of Biointerfaces By Hiroyuki Ohshima Copyright i 2010 by John Wiley Sons, Inc. 

FIGURE 1.1 Electrical double layer around a positively charged colloidal particle. The particle is surrounded by an ionic cloud, forming the electrical double layer of thickness 1/k, in which the concentration of counterions is greater than that of coions. 

This is the Poisson-Boltzmann equation for the potential distribution i/ (r). The surface charge density a of the particle is related to the potential derivative normal to the particle surface as 

Department of Chemistry, University of Louisville, Louisville, Kentucky 40292 

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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]... 

Fig. 1. Explanation of the principles of the finite-difference method for solution of the Poisson-Boltzmann equation...

Tanford, C., Kirkwood, J. G. Theory of protein titration curves. I. General equations for impenetrable spheres. J. Am. Chem. Soc. 79 (1957) 5333-5339. 6. Garrett, A. J. M., Poladian, L. Refined derivation, exact solutions, and singular limits of the Poisson-Boltzmann equation. Ann. Phys. 188 (1988) 386-435. Sharp, K. A., Honig, B. Electrostatic interactions in macromolecules. Theory and applications. Ann. Rev. Biophys. Chem. 19 (1990) 301-332. 

Nicholls, A., Honig, B. A rapid finite difference algorithm, utilizing successive over-relaxation to solve the Poisson-Boltzmann equation. J. Comp. Chem. 12 (1991) 435-445. 

Another way of calculating the electrostatic component of solvation uses the Poisson-Boltzmann equations [22, 23]. This formalism, which is also frequently applied to biological macromolecules, treats the solvent as a high-dielectric continuum, whereas the solute is considered as an array of point charges in a constant, low-dielectric medium. Changes of the potential within a medium with the dielectric constant e can be related to the charge density p according to the Poisson equation (Eq. (41)). 

Methods Based upon the Poisson-Boltzmann Equation... 

How can Equation (11.79) be solved Before computers were available only simple ihapes could be considered. For example, proteins were modelled as spheres or ellipses Tanford-Kirkwood theory) DNA as a uniformly charged cylinder and membranes as planes (Gouy-Chapman theory). With computers, numerical approaches can be used to solve the Poisson-Boltzmann equation. A variety of numerical methods can be employed, including finite element and boundary element methods, but we will restrict our discussion to the finite difference method first introduced for proteins by Warwicker and Watson [Warwicker and Watson 1982]. Several groups have implemented this method here we concentrate on the work of Honig s group, whose DelPhi program has been widely used. 

The Poisson-Boltzmann equation is a modification of the Poisson equation. It has an additional term describing the solvent charge separation and can also be viewed mathematically as a generalization of Debye-Huckel theory. 

The continuum treatment of electrostatics can also model salt effects by generalizing the Poisson equation (12) to the Poisson-Boltzmann equation. The finite difference approach to solving Eq. (12) extends naturally to treating the Poisson-Boltzmann equation [21], and the boundary element method can be extended as well [19]. 

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... 

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. 

This theory of the diffuse layer is satisfactory up to a symmetrical electrolyte concentration of 0.1 mol dm-3, as the Poisson-Boltzmann equation is valid only for dilute solutions. Similarly to the theory of strong electrolytes, the Gouy-Chapman theory of the diffuse layer is more readily applicable to symmetrical rather than unsymmetrical electrolytes. 

Holst, M.J. Baker, N.A. Wang, F., Adaptive multilevel finite element solution of the Poisson-Boltzmann equation I algorithms and examples, J. Comp. Chem. 2000, 21, 1319-1342... 

Here, b is the distance between the nearest unit charges along the cylinder (b = 0.34nm for the ssDNA and b = 0.17nm for the dsDNA), (+) and (—) are related to cations and anions, respectively, and a = rss for the ssDNA and a rds for the dsDNA. The expressions (5) and (6) have been obtained using the equations for the electrostatic potential derived in [64, 65], where a linearization of the Poisson-Boltzmann equation near the Donnan potential in the hexagonal DNA cell was implemented. 

Two remaining problems relating to the treatment of solvation include the slowness of Poisson-Boltzmann calculations, when these are used to treat electrostatic effects, and the difficulty of keeping buried, explicit solvent in equilibrium with the external solvent when, e.g., there are changes in nearby solute groups in an alchemical simulation. Faster methods for solving the Poisson-Boltzmann equation by means of parallel finite element techniques are becoming available, however.22 24... 

A solvated MD simulation is performed to determine an ensemble of conformations for the molecule of interest. This ensemble is then used to calculate the terms in this equation. Vm is the standard molecular mechanics energy for each member of the ensemble (calculated after removing the solvent water). G PB is the solvation free energy calculated by numerical integration of the Poisson-Boltzmann equation plus a simple surface energy term to estimate the nonpolar free energy contribution. T is the absolute temperature. S mm is the entropy, which is estimated using... 

The electrostatic methods just discussed suitable for nonelectrolytic solvent. However, both the GB and Poisson approaches may be extended to salt solutions, the former by introducing a Debye-Huckel parameter67 and the latter by generalizing the Poisson equation to the Poisson-Boltzmann equation.68 The Debye-Huckel modification of the GB model is valid to much higher salt concentrations than the original Debye-Huckel theory because the model includes the finite size of the solute molecules. 

Interactions between cationic micelles and uni- and divalent anions have been treated quantitatively by solving the Poisson-Boltzmann equation in spherical symmetry and considering both Coulombic and specific attractive forces. Predicted rate-surfactant profiles are similar to those based on the ion-exchange and mass-action models (Section 3), but fit the data better for reactions in solutions containing divalent anions (Bunton, C. A. and Moffatt, J. R. (1985) J. Phys. Chem. 1985, 89, 4166 1986,90, 538). 

The inner potentials have to be calculated by solving the Poisson-Boltzmann equations for the potentials this is done in Appendix A. 

Beyond the IHP is a layer of charge bound at the surface by electrostatic forces only. This layer is known as the diffuse layer, or the Gouy-Chapman layer. The innermost plane of the diffuse layer is known as the outer Helmholtz plane (OHP). The relationship between the charge in the diffuse layer, o2, the electrolyte concentration in the bulk of solution, c, and potential at the OHP, 2> can be found from solving the Poisson-Boltzmann equation with appropriate boundary conditions (for 1 1 electrolytes (13))... 

Sakurai M, Tamagawa H, Inoue Y, Ariga K, Kunitake T (1997) Theoretical study of intermolecular interaction at the lipid-water interface. 2. Analysis - based on the Poisson-Boltzmann equation. J Phys Chem B 101 4817—4825... 

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)... 

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