Debye-Hiickel approximation

Equation (7.25) can be substituted into equation (7.20) to give a second order differential equation in ijj. In theory, the resulting equation can be solved to give ip as a function of r. However, it has an exponential term in -ip, that makes it impossible to solve analytically. In the Debye-Hiickel approximation, the exponential is expanded in a power series to give... 

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. 

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. 

We introduce the first of the Debye-Hiickel approximations by considering only those situations for which < kBT). In this case the exponentials in Equation (28) may be expanded (see Appendix A) as a power series. If only first-order terms in z,eyp/kBT) are... 

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),... 

The solution of the linearized Poisson-Boltzmann equation around cylinders also requires numerical methods, although when cylindrical symmetry and the Debye-Hiickel approximation are assumed the equation can be solved. The solution, however, requires advanced mathematical techniques and we will not discuss it here. It is nevertheless useful to note the form of the solution. The potential for symmetrical electrolytes has been given by Dube (1943) and is written in terms of the charge density a as... 

The Debye-Hiickel approximation is strictly applicable only in the case of low potentials. Nevertheless, there are several reasons why the significance of Equation (37) should be fully appreciated ... 

FIG. 11.5 Fraction of double-layer potential versus distance from a surface according to the Debye-Hiickel approximation, Equation (37) (a) curves drawn for 1 1 electrolyte at three concentrations and (b) curves drawn for 0.001 M symmetrical electrolytes of three different valence types. 

Equation (45) provides a relationship between the surface charge density and the slope of the potential at the surface. Next, we turn to Equation (37) —the Debye-Hiickel approximation for p — to evaluate (dip/dx)0. Differentiation leads to the value... 

Even allowing for the fact that the Debye-Hiickel approximation applies only for low potentials, the above analysis reveals some features of the electrical double layer that are general and of great importance as far as stability with respect to coagulation of dispersions and electrokinetic phenomena are concerned. In summary, three specific items might be noted ... 

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. 

The theoretical inconsistencies inherent in the Poisson-Boltzmann equation were shown in Section 11.4 to vanish in the limit of very small potentials. It may also be shown that errors arising from this inconsistency will not be too serious under the conditions that prevail in many colloidal dispersions, even though the potential itself may no longer be small. Accordingly, we return to the Poisson-Boltzmann equation as it applies to a planar interface, Equation (29), to develop the Gouy-Chapman result without the limitations of the Debye-Hiickel approximation. 

Equation (62) describes the variation in potential with distance from the surface for a diffuse double layer without the simplifying assumption of low potentials. It is obviously far less easy to gain a feeling for this relationship than for the low-potential case. Anticipation of this fact is why so much attention was devoted to the Debye-Hiickel approximation in the first place. Note that Equation (62) may be written... 

The Derjaguin approximation illustrated in the above example is suitable when kR > 10, that is, when the radius of curvature of the surface, denoted by the radius R, is much larger than the thickness of the double layer, denoted by k 1. (Note that for a spherical particle R = Rs, the radius of the particle.) Other approaches are required for thick double layers, and Verwey and Overbeek (1948) have tabulated results for this case. The results can be approximated by the following expression when the Debye-Hiickel approximation holds ... 

Obtain the corresponding Poisson-Boltzmann equation and the linearized version based on the Debye-Hiickel approximation. 

The Poisson equation (see Equation (11.18)) gives the fundamental differential equation for potential as a function of charge density. The Debye-Hiickel approximation may be used to express the charge density as a function of potential as in Equation (11.28) if the potential is low. Combining Equations (11.24) and (11.32) gives... 

Equations (6.4.43a-c) yield the central result of this section—the following expression for the electro-osmotic slip velocity ua under an applied potential and concentration gradient, in the Debye-Hiickel approximation for a thin double layer... 

The integration of the spherical analog of Equation 4 can be done analytically only in the Debye-Hiickel approximation, u0 << 1, and leads to Equations 10a and 10b. 

Unfortunately, the Debye-Hiickel approximation (ztp < c. 25 mV) is often not a good one in the treatment of colloid and surface phenomena. Unapproximated, numerical solutions of equation (7.11) have been computed.88... 

If the Debye-Hiickel approximation, zetf/JkT < 1, is made, equation (8.7) reduces to equation (8.5). 

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... 

The entropic contribution to the free energy (per unit area) becomes in the Debye—Hiickel approximation... 

In what follows, a detailed comparison will be carried out in the Debye-Hiickel approximation, for which simple analytical expressions can be derived. 

The purpose of this chapter is to get a better insight of the first problem listed above, i.e., the polarization of interfaces (colloidal particles) during their interaction. Because of tutorial reasons, the electrolyte solution will be described using a rather simple, mean field approximation, that, however, allows to obtain an analytical solution of the problem. It is clear that this elaboration can easily be followed, and one can extend our model on more advanced situations. This model is identical with so-called weak-coupling theory for point ions treated in a course of the Debye-Hiickel approximation. Before going to make an elaboration for two interacting macrobodies immersed into an electrolyte solution, we would like to introduce a method, which is usually used to model this polarization, and to compute the electrical field next to a polarized medium. Then we will also discuss consequences of the polarization for the ion distribution at the particle-solution interface. 

The linearization of the PB equation is often called the Debye-Hiickel approximation and it is valid when qfo/kT < 1. At room temperature this corresponds to surface potentials, 0o, below 25 mV. In the case of flat surfaces and if symmetry is considered as in the Gouy-Chapman equation... 

The Runge-Kutta method then can be used in order to calculate at the closest distance r = a, a and (d/dx)XmBXa(xa = xa). However, at short distances, the Debye-Hiickel approximation is no longer valid (a > 1) it then is assumed that at the micelle surface, the field must be coulombian, so that ... 

For the spherical Debye-Hiickel approximation, the Poission-Boltz-mann Equation reduces to... 

Equation (1.9) is the linearized Poisson-Boltzmann equation and k in Eq. (1.10) is the Debye-Htickel parameter. This linearization is called the Debye-Hiickel approximation and Eq. (1.9) is called the Debye-Hiickel equation. The reciprocal of k (i.e., 1/k), which is called the Debye length, corresponds to the thickness of the double layer. Note that nf in Eqs. (1.5) and (1.10) is given in units of m . If one uses the units of M (mol/L), then must be replaced by IQQQNAn, Na being Avogadro s number. 

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