Double electrical layer diffuse charge density

But the situation will change if the air bubble surface will be filled with densely packed adsorption monolayer of ionized surfactants. In this case, the charge density of monolayer can reach 1 C/m. And the equal quantity of charges of opposite sign forms the diffuse part of the double electric layer. Due to this phenomenon, the flotation device allows to remove some ionic impurities from treated water, which contain sufficient amount of surfactants. And the exhaustion of surfactant at air bubble surface leads to the end of ion evacuation with bubble-film flow. The dissolved ions are removed due to the action of surfactants as cofactors of flotation. 

In the second group of models, the pc surface consists only of very small crystallites with a linear parameter y, whose sizes are comparable with the electrical double-layer parameters, i.e., with the effective Debye screening length in the bulk of the diffuse layer near the face j.262,263 In the case of such electrodes, inner layers at different monocrystalline areas are considered to be independent, but the diffuse layer is common for the entire surface of a pc electrode and depends on the average charge density <7pc = R ZjOjOj [Fig. 10(b)]. The capacitance Cj al is obtained by the equation... 

So far we have talked only in terms of electrostatic potentials. Can we use this theory to find the charge density on the surface (Oo) In order for the electrical double-layer to be neutral overall, it follows that the total summed charge in the diffuse layer must equal the surface charge. Thus, it follows that... 

The Poisson-Boltzman (P-B) equation commonly serves as the basis from which electrostatic interactions between suspended clay particles in solution are described ([23], see Sec.II. A. 2). In aqueous environments, both inner and outer-sphere complexes may form, and these complexes along with the intrinsic surface charge density are included in the net particle surface charge density (crp, 4). When clay mineral particles are suspended in water, a diffuse double layer (DDL) of ion charge is structured with an associated volumetric charge density (p ) if av 0. Given that the entire system must remain electrically neutral, ap then must equal — f p dx. In its simplest form, the DDL may be described, with the help of the P-B equation, by the traditional Gouy-Chapman [23-27] model, which describes the inner potential variation as a function of distance from the particle surface [23]. 

Eq. (3.92) implies a linear relation between cosh Y0 and cosh Ym. Therefore, values for go., and Fo, can be assessed by extrapolating this relation to cosh Ym = 1. This is demonstrated in Fig. 3.46 where a plot of cosh Yg versus cosh Ym is shown. Making use of least-squares analysis we obtain a slope of 1.01 0.01 and an intercept of 1.80 0.4. Furthermore, in conformity with Eq. (3.92) the intercept yields Fo, = 119 0.03 or, respectively, (po = 30.5 0.7 mV and <70i = 1.29 0.04 mC m 2 for the diffuse electric layer potential and diffuse double layer charge density at infinite separation. These values are in... 

The method given above for calculating the zeta-potential at a diffuse double layer may be applied to the diffuse portion of the Stern double layer. If charge density on the solid and ai and are the corresponding values on the solution sides of the fixed and diffuse layers, respectively, then the condition of electrical neutrality requires that... 

The particles suspended and surfaces immersed in a liquid are usually charged by the adsorption of the ions from solution. The charge on the surface of the particle or any other surface immersed in liquid is balanced by an equal but oppositely charged layer in the adjacent liquid, resulting in a so-called electric double layer discussed earlier in Chapters 4 and 5. In a liquid with ions and molecules under constant thermal motion, one expects a diffused zone of charges in the solution and a compacted layer on the solid surface. Total charge density in the two zones must be equal and opposite in sign. When the liquid or the particle is in motion (with respect to each other) the compacted layer on the... 

Consider a microchannel filled with an aqueous solution. There is an eleetrieal doubly layer field near the interface of the channel wall and the liquid. If an electric field is applied along the length of the channel, an electrical body force is exerted on the ions in the diffuse layer. In the diffuse layer of the double layer field, the net charge density, pe is not zero. The net transport of ions is the excess counterions. If the solid surface is negatively eharged, the counterions are the positive ions. These excess counterions will move under the influenee of the... 

The Helmholtz-von Smoluchowski equation indicates that under constant composition of the electrolyte solution, the electro-osmotic flow depends on the magnitude of the zeta potential which is determined by many different factors, the most important being the dissociation of the silanol groups on the capillary wall, the charge density in the Stern layer, and the thickness of the diffuse layer. Each of these factors depends on several variables, such as pH, specific adsorption of ionic species in the compact region of the electric double layer, ionic strength, viscosity, and temperature. 

Figure 2 A very simple model of electrostatic adsorption on a negatively charged oxide surface with formation of a "double layer" (surface + diffuse layer). Small dosed drcles are cations, larger open drcles are anions, oq" surface charge density x distance from the surface into the solution k thickness of double layer < ) electric potential c ix) and c (x) local concentrations in cations and anions, respectively. The shaded area represents the excess of cations over anions in the diffuse layer, and therefore the amount of cations that are electrostatically adsorbed.

Figure 3 The solid/aqueous solution interface and electrical double layer for a hydrolyzed oxide surface o, charge density V /, electrostatic potential +, cation anion, 0, surface c, compact layer d, diffuse layer.

Certain model assumptions are necessary in order to reveal the surface concentration of specifically adsorbed ions in the total surface excess F,-. Usually, the ionic component of the electrical double layer (EDL) is assumed to consist of the dense part and the diffuse layer separated by the so-called outer Helmholtz plane. Only specifically adsorbing ions can penetrate into the dense layer close to the surface (e.g. iodide ions), with their electric centers located on the inner Helmholtz plane. The charge density of these specifically adsorbed ions ai is determined by their surface concentration F Namely, for single-charged anions ... 

If E, and 2 are assumed to be constant. Eqs. [32[ and [33] predict a linear potential drop within each part of the inner double layer, as shown in Fig. (10b). The dependence of the electric potential with distance, in the region from x = d to the bulk solution, i.e, in the diffuse part of the double layer, will be exponential if the surface potential is moderate, as predicted by Eq. 124). In the case of higher surface potentials, the dependence is that shown in Eq. [26], substituting tpn by yj at X = ti. Similarly, the surface charge density, O is related to y by an equation of the type [28]. 

Two general theoretical approaches have been applied in the analysis of heterogeneous materials. The macroscopic approach, in terms of classical electrodynamics, and the statistical mechanics approach, in terms of charge-density calculations. The first is based on the application of the Laplace equation to calculate the electric potential inside and outside a dispersed spherical particle (11, 12). The same result can be obtained by considering the relationship between the electric displacement D and the macroscopic electric field Em a disperse system (12,13). The second approach takes into account the coordinate-dependent concentration of counterions in the diffuse double layer, regarding the self-consistent electrostatic poton tial of counterions via Poisson s equation (5, 16, 17). Let us consider these approaches briefly. 

---