Double layer Poisson-Boltzmann equation

More sophisticated approaches to describe double layer interactions have been developed more recently. Using cell models, the full Poisson-Boltzmann equation can be solved for ordered stmctures. The approach by Alexander et al shows how the effective colloidal particle charge saturates when the bare particle charge is increased [4o]. Using integral equation methods, the behaviour of the primitive model has been studied, in which all the interactions between the colloidal macro-ions and the small ions are addressed (see, for instance, [44, 45]). 

The presence of the diffuse layer determines the shape of the capacitance-potential curves. For a majority of systems, models describing the double-layer structure are oversimplified because of taking into account only the charge of ions and neglecting their specific nature. Recently, these problems have been analyzed using new theories such as the modified Poisson-Boltzmann equation, later developed by Lamper-ski. The double-layer capacitanties calculated from these equations are... 

Gouy and Chapman (1910-13) independently used the Poisson-Boltzmann equations to describe the diffnse electrical double-layer formed at the interface between a charged snrface and an aqueous solution. 

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. 

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),... 

The Debye-Hiickel approximation to the diffuse double-layer problem produces a number of relatively simple equations that introduce a variety of double-layer topics as well as a number of qualitative generalizations. In order to extend the range of the quantitative relationships, however, it is necessary to return to the Poisson-Boltzmann equation and the unrestricted Gouy-Chapman theory, which we do in Section 11.6. 

Solve the Poisson-Boltzmann equation for a spherically symmetric double layer surrounding a particle of radius Rs to obtain Equation (38) for the potential distribution in the double layer. Note that the required boundary conditions in this case are at r = Rs, and p - 0 as r -> oo. (Hint Transform p(r) to a new function y(r) = r J/(r) before solving the LPB equation.)... 

Figure 6.10 Electrostatic double-layer force between a sphere of R = 3 /um radius and a flat surface in water containing 1 mM monovalent salt. The force was calculated using the nonlinear Poisson-Boltzmann equation and the Derjaguin approximation for constant potentials (tpi = 80 mV, ip2 = 50 mV) and for constant surface charge (i/2/Ad so that at large distances both lead to the same potential.

Two approaches for the calculation of the double-layer contribution are explored. Hogg el of. (16) linearized the Poisson- -Boltzmann equation to compute the double-layer force between two dissimilar plane surfaces, then used Derjaguin s approximation to extend this result to the interaction of two spheres of different radii. When the radius of one sphere is infinite, their result becomes... 

A second and more rigorous approach to the calculation of the double-layer contribution is to solve the nonlinear Poisson-Boltzmann equation, taking into account the following dissociation equilibria to determine the surface charge and potential at each value of x ... 

To calculate the double-layer force, the nonlinear Poisson-Boltzmann equation was solved for the case of two plane parallel plates, subject to boundary conditions which arise from consideration of the simultaneous dissociation equilibria of multiple ionizable groups on each surface. Deijaguin s approximation is then used to extend these results to calculate the force between a sphere and a plane. Details of the method can be found in Ref. (6). 

In this Appendix, equations will be derived for the double layer interaction between two charged, planar surfaces in an electrolyte-free system. W e assume that the potential ip(x) obeys the Poisson—Boltzmann equation... 

Ruckenstein and Schiby derived4 an expression for the electrochemical potential, which accounted for the hydration of ions and their finite volume. The modified Poisson-Boltzmann equation thus obtained was used to calculate the force between charged surfaces immersed in an electrolyte. It was shown that at low separation distances and high surface charges, the modified equation predicts an additional repulsion in excess to the traditional double layer theory of Deijaguin—Landau—Verwey—Overbeek. 

IV.3. Modification of the Double Layer Repulsion Due to the Finite Volume of Ions. In the linear approximation of the Poisson—Boltzmann equation, the potential between two surfaces, separated by the distance l is given by... 

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]

Figure 5. Ratio between the double layer force with site exclusion (modified Poisson-Boltzmann) and the double layer force provided by the Poisson-Boltzmann equation (a)constant (small) surface potential (Vs = 0.02 V) (b) constant (small) surface charge density (<7S = 0.032 C/m2), n/V = 1.0 M and T = 300 K.

After the solution of the Poisson—Boltzmann equation is obtained, the total double layer free energy per unit area is obtained by adding the electrostatic... 

It has been long known that the over-simplified Poisson-Boltzmann equation is accurate in predicting the double layer interaction only in a relatively narrow range of electrolyte concentrations. One obvious weakness of the treatment is the prediction that the ions of the same valence produce the same results, regardless of their nature. In contrast, experiment shows marked differences when different kinds of ions are used. The ion-specific effects can be typically ordered in series (the Hofmeister series [36]), and the placement of ions in this series correlates well with the hydration properties of the ions in bulk water. 

For separation distances d > d, the Poisson-Boltzmann equation is obeyed within d1< <2d—d1. The system is completely equivalent with a traditional double layer, but formed between the surfaces S ( = d1) and not between the real surfaces S( = 0). The corresponding surface potential is the potential at S, % = ( =A), and the corresponding surface charge density, a, is given by ... 

In this section, the effect of the ion-dispersion forces on the double layer interactions will be investigated, by assuming that they provide the only correction to the traditional Poisson-Boltzmann equation. 

The traditional theory of the double layer is based on a combination of the Poisson equation and Boltzmann distribution. While this involves the approximation that the potential of mean force used in the Boltzmann expression equals the mean value of the electrical potential [9], the results thus obtained are satisfactory at least for 1 1 electrolytes. The equations proposed in the present paper use the approximations inherent in the Poisson—Boltzmann equation, but also include the effect of the polarization field of the solvent which is caused by a polarization source assumed uniformly distributed on the surface and by the double layer itself. 

Another mechanism for the hydration repulsion between lipid bilayers was more recently proposed by Marcelja.22 It is based on the fact that in water the ions are hydrated and hence occupy a larger volume. The volume exclusion effects ofthe ions are important corrections to the Poisson— Boltzmann equation and modify substantially the doublelayer interaction at low separation distances. The same conclusion was reached earlier by Ruckenstein and Schiby,28 and there is little doubt that the hydration of individual ions modifies the double-layer interaction, providing an excess repulsion force.28 However, while the hydration of ions affects the double-layer interactions, the hydration repulsion is strong even in the absence of an electrolyte, or double-layer repulsion. 

For the sake of simplicity, in what follows it will be considered that the double layer potential is sufficiently small to allow the linearization of the Poisson—Boltzmann equation (the Debye—Hiickel approximation). The extension to the nonlinear cases is (relatively) straightforward however, it will turn out that the differences from the DLVO theory are particularly important at high electrolyte concentrations, when the potentials are small. In this approximation, the distribution of charge inside the double layer is given by... 

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