Debye Layer Potential Distribution

the concentration far from the surface is c Cg where 0 - 0. The charge density can be written as 

Equation (6.64) can be integrated explicitly. The other approach is to use small potential approximation, zFct) RT. This expression is valid when the electrical energy is small compared to the thermal energy. Using the power series expansion of exponential function in equation (6.64), we get 

The small potential approximation is termed as the Debye-Hiickel approximation. The boundary conditions can be written as 

Solution of the governing equation (6.65) and use of above boundary conditions give 

---

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),... 

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... 

Figure 3.13. Potential distribution in a spherical diffuse double layer. Symmetrical electrolyte. Dashed curves Debye-Huckel approximation. (Back-scaling in terms of concentrations. radii and valencies can be done by using 13.5.8] for K and a (in nm) = 0.3042 Ka / z-Jc with c in M at 25 C.)...

Using the Debye-Hiickel solution for the potential distribution in the double layer enables Eq. (7.2.13) to be integrated to give a result that may be written in the form... 

The Debye-Huckel approximation gives the potential distribution in the electrical double layer exp(—x/2d). Then we have at the symmetry axis... 

The zeta-potential is frequently used to predict the stability of a suspension or the adhesion of suspended particles on macroscopic surfaces (e.g. cellulose fibres, tubing). This is because double layer interaction between particles or between particles and surfaces is governed by the ion distribution in the diffuse layer, which primarily depends on the Debye-Hiickel parameter k and the diffuse layer potential i/ d- The latter, however, is commonly approximated by the zeta potential f (Lyklema 2010, cf. Fig. 3.3). It is quite obvious that repulsion requires high zeta-potential values of equal sign, whereas adhesion occurs in the absence of surface charge or for oppositely charged surfaces. 

Fig. 1 compares the potential distribution at a semicon-ductor/electrolyte and a metal/electrolyte interface. The extension of the space charge layer in the semiconductor is a function of the mobile charge concentration in the bulk and can be approximated by a "Debye-length" L ... 

In the earlier section, we have assumed constant potential distribution of electric potential inside the electric double layer for the estimation of EO velocity. Let us use the Debye-Hiickel approximation for potential distribution as... 

Equation (6.163) gives the velocity distribution assuming constant potential inside the Debye layer. Let us derive the velocity profile assuming the Debye-Htickel linear approximation as... 

A polyelectrolyte solution contains the salt of a polyion, a polymer comprised of repeating ionized units. In dilute solutions, a substantial fraction of sodium ions are bound to polyacrylate at concentrations where sodium acetate exhibits only dissoci-atedions. Thus counterion binding plays a central role in polyelectrolyte solutions [1], Close approach of counterions to polyions results in mutual perturbation of the hydration layers and the description of the electrical potential around polyions is different to both the Debye-Huckel treatment for soluble ions and the Gouy-Chapman model for a surface charge distribution, with Manning condensation of ions around the polyelectrolyte. 

The Debye-Huckel theory was developed to extend the capacitor model and is based on a simplified solution of the Poisson equation. It assumes that the double layer is really a diffuse cloud in which the potential is not a discontinuous function. Again, the interest is in deriving an expression for the electrical potential function. This model states that there is an exponential relationship between the charge and the potential. The distribution of the potential is ... 

In the years 1910-1917 Gouy2 and Chapman3 went a step further. They took into account a thermal motion of the ions. Thermal fluctuations tend to drive the counterions away form the surface. They lead to the formation of a diffuse layer, which is more extended than a molecular layer. For the simple case of a planar, negatively charged plane this is illustrated in Fig. 4.1. Gouy and Chapman applied their theory on the electric double layer to planar surfaces [54-56], Later, Debye and Hiickel calculated the potential and ion distribution around spherical surfaces [57],... 

The history of PB theory can be traced back to the Gouy-Chapmann theory and Debye-Huchel theory in the early of 1900s (e.g., see Camie and Torrie, 1984). These two theories represent special simplified forms of the PB theory Gouy-Chapmann theory is a one-dimensional simplification for electric double-layer, while the Debye-Huchel theory is a special solution for spherical symmetric system. The PB equation can be derived based on the Poisson equation with a self-consistent mean electric potential tj/ and a Boltzmann distribution for the ions... 

Measurement of the potential by means other than electro-kinetic measurement. G. S. Hartley and J. W. Roe1 point out that the potential determines the distribution of ions near a surface in the same manner as the potential just outside an ion controls the ionic atmosphere in the Debye-Huckel theory of strong electrolytes. There is a simple relation between the concentration of an ion in the layer next to a surface and in the bulk solution at a distance from the surface and the potential, so that if a means can be found of measuring the concentration of an ion in the surface and in the solution, it should be possible to estimate the potential of that surface. 

---