Poisson-Boltzmann equation, counterion

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... 

Another related phenomenon to be discussed in 2.3 is known in the polymer literature as counterion condensation. This term refers to a phase transition-like switch of the type of singularity, induced by a line charge to solutions of (2.1.2), occurring at some critical value of the linear charge density. Counterion condensation as a limiting property of the solutions of the Poisson-Boltzmann equation was studied in detail in [11]—[19]. Presentation of 2.3 follows that of [17]. 

The electric field within each cell is determined in the mean field approximation from the Poisson Boltzmann equation (2.3.1), written for the prototypical case of a symmetric low molecular electrolyte of valency z added to a polyelectrolyte with a single type of proper counterion of va-... 

A plot of equilibrium r/e as a function of the external parameter a is schematically presented in Fig. 2.3.5a. The plot in Fig. 2.3.5a is markedly different from that in Fig. 2.3.4a by its lack of bifurcation. (Uniqueness of the appropriate solutions of the Poisson-Boltzmann equation for any values of a is proved in [18].) In the (F, ri) or (F, aeS) plane this corresponds to the existence of solutions of the Poisson-Boltzmann equations with finite F (bounded norm of the appropriate solution with a subtracted singular part due to the effective line charge) only for aeS < with adetermined by Conjecture 2.1. This is schematically illustrated in Fig. 2.3.5b. Note that F as a function of creS is constructed in a single counterion case by solving (2.3.3a) with a = of J e rdr and with the boundary conditions tp(a) = -aeS lna, = 0, and by going to the limit a- 0. 

Study of counterion condensation as a limiting property of the solutions of the Poisson-Boltzmann equation for arbitrary, charged cylindrical manifolds in H3 (see 2.3). 

I. Rubinstein, Counterion condensation as an exact limiting property of the Poisson-Boltzmann equation, SIAM J. Appl. Math., 46 (1986), pp. 1024-1038. 

For counterions, Eq. (5.7) predicts a larger rate constant for the association process than Eq. (5.6) since the ions are attracted by the micelle. An estimate using Eq. (5.7) and a solution of the Poisson-Boltzmann equation (cf. Sect. 6) gives k2 - lO M s-1 for the counterion association. 

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. 

The micellar surface has a high charge density and the stability of the aggregate is heavily dependent on the binding of counterions to the surface. From the solution of the Poisson-Boltzmann equation one finds that a large fraction (0.4—0.7) of the counterions is in the nearest vicinity of the micellar surface300. These ions could be associated with the Stern layer, but it seems simpler not to make a distinction between the ions of the Stern layer and those more diffusely bound. They are all part of the counterions and their distribution is primarily determined by electrostatic effects. 

A number of methodologies have been developed and generalized in recent years to quantitatively describe the ion atmosphere around nucleic acids [11, 12, 17, 28, 29]. These include models based on Poisson-Boltzmann equation [11, 12], counterion condensation [17], and simulation methods, such as Monte Carlo, molecular dynamics, and Brownian dynamics [28, 29]. 

Let us examine now the effect of the excluded volume at low surface potentials. In the linear approximation of the Poisson—Boltzmann expression, the increase in the number of counterions in the vicinity of the interface equals the decrease in the number of co-ions. If the co-ions have a larger size, one expects the available volume near the surface to be larger than that in the bulk. As a result, a concentration of ions in excess to that predicted by the Poisson—Boltzmann equation is expected to occur in the vicinity of the surface, when the volume exclusion is taken into account. 

Of course, when the potential of the surface becomes sufficiently large, the linear approximation fails, and the (exponential) increase in the counterions density in the vicinity of the surface predicted by the Poisson—Boltzmann equation largely exceeds the depletion of co-ions and the available volume is expected to become smaller than that in the bulk. 

When Di > i>2, the effective Debye—Hiickel length X (which now depends on ip(x)) is larger than that obtained for the Poisson—Boltzmann equation. Consequently, the diffuse double layer is larger in the vicinity of a charged surface, as predicted earlier.4 7 9 However, when V2 > Vi (small counterions), X < X and the diffuse double layer is compressed. The effect is proportional to the ionic strength and is, in general, small for typical electrolyte concentrations, since n(v — v[Pg.337]

In all of the AFM studies mentioned above, the system contained monovalent counterions, and hence the Poisson-Boltzmann theory could be expected to be accurate. Kekicheff et al. [62] studied interactions between mica surfaces and between silicon nitride and mica in Ca(N03) solutions by using both the SFA and AFM methods. As discussed above, the presence of a divalent counterion complicates particle-surface interactions significantly. Both experimental methods showed that there is a strong, attractive force at very small surface separations, a result that could not be explained by the Poisson-Boltzmann equation. The authors interpreted their results by using the AHNC described above, with the primitive model... 

In this chapter, mathematical procedures for the estimation of the electrical interactions between particles covered by an ion-penetrable membrane immersed in a general electrolyte solution is introduced. The treatment is similar to that for rigid particles, except that fixed charges are distributed over a finite volume in space, rather than over a rigid surface. This introduces some complexities. Several approximate methods for the resolution of the Poisson-Boltzmann equation are discussed. The basic thermodynamic properties of an electrical double layer, including Helmholtz free energy, amount of ion adsorption, and entropy are then estimated on the basis of the results obtained, followed by the evaluation of the critical coagulation concentration of counterions and the stability ratio of the system under consideration. 

Here zm denotes the valency of the counterions with the highest absolute charge. The function G is dependent upon the zeta potential of the particle as well as the bulk concentrations, valencies and diffusivities of the ions. The equilibrium electrical potential can be obtained using the Runge-Kutta method to solve the Poisson-Boltzmann equation numerically, and then the integral in Eq. (52) can be evaluated. 

The Poisson-Boltzmann equation for the potential distribution around a cylindrical particle without recourse to the above two assumptions for the limiting case of completely salt-free suspensions containing only particles and their counterions was solved analytically by Fuoss et al. [1] and Afrey et al. [2]. As for a spherical particle, although the exact analytic solution was not derived, Imai and Oosawa [3,4] smdied the analytic properties of the Poisson-Boltzmann equation for dilute particle suspensions. The Poisson-Boltzmann equation for a salt-free suspension has recently been numerically solved [5-8]. 

In this chapter, we first discuss the case of completely salt-free suspensions of spheres and cylinders. Then, we consider the Poisson-Boltzmann equation for the potential distribution around a spherical colloidal particle in a medium containing its counterions and a small amount of added salts [8]. We also deals with a soft particle in a salt-free medium [9]. 

Note that the right-hand side of Eq. (6.6) contains only one term resulting from counterions, unlike the usual Poisson-Boltzmann equation. Since cr (or Q) and z are of the same sign, the product zff (or zQ) is always positive. Note that k depends on cr (or Q), a, and R unlike the case of salt solutions, where k is essentially independent of these parameters. 

Here we have assumed that the relative permittivity is assumed to take the same value inside and outside the membrane. We also assume that the distribution of counterions n(x) obeys Eq. (18.4) and thus the charge density PeiW is given by Eq. (18.5).Thus, we obtain the following Poisson-Boltzmann equations for the scaled potential y x) = ze l/(x)/kT ... 

Electrostatic interactions in solutions containing charged particles and ions can be described using the Poisson-Boltzmann equation. A charged surface attracts counterions into a double layer of thickness defined by the Debye length, which depends on counterion concentration and solvent dielectric constant. From simplified theories, expressions can be derived for the attractive interaction potential between charged spheres. 

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