Diffuse part of the edl

FIGURE 5.67 Schematic presentation of the structure of the EDL. The surface charge is created by ionized surface groups and/or by ions tightly adsorbed in the Stem layer. The plane of closest approach of the ions from the diffuse part of the EDL is called the outer Helmholtz plane (OHP). The electric potential in the OHP plane is referred to as the surface potential, /j, in the text. The shear plane, x = x, separates the hydrodynam-ically immobile liquid that moves together with the surface, x x, which has nonzero relative velocity with respect to the surface. Note that the ions in the immobile part of the EDL can move with respect to the surface under an applied electric field, which gives rise to the anomalous surface conductivity (see Section 5.8.8). 

Ions that are involved in more or less free Brownian motion present the diffuse part of the EDL. 

Let us now examine how the potential cp changes within the diffuse part of the EDL, assuming that cp=0 in the bulk of the dispersion medium. The theory describing this part of the EDL was developed by Gouy and Chapman, who compared the energy of the electrostatic interaction of the ions with the energy of their thermal motion, assuming that the concentration of ions in the EDL was consistent with the Boltzmann distribution ... 

Fig. Ill-13. The change in the concentration of co-ions, +, counter-ions, ri, and the total ion concentration, n = ri +n, in the diffuse part of the EDL...

At the border between the diffuse part of the EDL and the dense Stem -Helmholtz layer... 

As we have seen, the structure of the diffuse part of the EDL is determined by the ratio of the potential energy of the electrostatic attraction between counter-ions and the charged surface to the thermal energy of ions. This ratio is given by a dimensionless function, zetp0 /4kT (or zecp(/ /4k7). 

Another area in which the existence of the diffuse part of the EDL plays a dominant role is represented by various electrokinetic phenomena [67, 68]. 

In the Gouy-Chapmann model based on only electrostatics (i.e., point charges are supposed and the medium is considered as a dielectric continuum), the charge potential relationship for the diffuse part of the EDL can be derived from the Poisson-Boltzmann equation. 

The charge potential relationship in the diffuse part of the EDL can be deduced. The total charge, per unit area of surface, in the diffuse layer, cr, is given by... 

It follows from the definition cited that the size of the zeta potential depends on the structure of the diffuse part of the ionic EDL. At the outer limit of the Helmholtz layer (at X = X2) the potential is j/2, in the notation adopted in Chapter 10. Beyond this point the potential asymptotically approaches zero with increasing distance from the surface. The slip plane in all likelihood is somewhat farther away from the electrode than the outer Helmholtz layer. Hence, the valne of agrees in sign with the value of /2 but is somewhat lower in absolute value. 

The structure of the EDL is neither simple nor universal it depends to a great extent on the physico-chemical properties of particles and dispersion medium. In general, it is assumed that some ions from the solvent adhere on the particle surface and partially neutralise the surface charge. This layer of immobile ions is called Stem layer. The other ions spread in the solvent by thermal motion yet are subject to the electric field generated from the charged surface. With growing surface distance the concentrations of the ionic species tend to their equilibrium values of the free solvent. The region adjacent to the Stem layer with excess of counter-ions is called the diffuse layer. In this part of the EDL, the ion distribution results from the balance of electrostatic and osmotic forces. 

Thermal motion of the ions in the EDL was included in the theories developed independently by Georges Gouy in Erance (1910) and David L. Chapman in England (1913). The combined elfects of the electrostatic forces and of the thermal motion in the solution near the electrode surface give rise to a diffuse distribution of the excess ions, and a diffuse EDL part or diffuse ionic layer with a space charge Qy x) (depending on the distance x from the electrode s surface) is formed. The total excess charge in the solution per unit surface area is determined by the expression... 

The existence of the extended diffuse layer in dilute solutions is also of crucial importance in colloidal systems formed by particles whose size is comparable to the wavelength of Kght, that is, about 1 pm. For example, electrophoresis, the movement of colloidal particles with respect to the solution under the influence of an external electric field, originates from the displacement of the particle and the mobile part of its EDL in the opposite directions under the action of this field, since these components of the system possess charges of opposite signs. A related phenomenon occurs if the solid particles move with respect... 

The second modification compared to the Stern theory was the introduction of the discreteness-of-charge factor , X. The electrostatic part of the work of ion transfer from the bulk solution into the adsorption plane across the EDL field in Stern s approximation may be represented as the sum of the contributions of the compact and diffuse layers ... 

In the MVN model, the EDL consists of two diffuse ion layers back-to-back, which produce a compact inner layer between the two phases (Fig. 3). Dielectric permittivity of the medium at any point in the diffuse layer is assumed to be constant and equal to the bulk phase value s. The compact layer or inner Helmholtz layer is located between —5 and +S. In a more detailed analysis, the dielectric permittivities in both parts of the compact layer are and... 

Figure 1 depicts a schematic diagram of the EDL next to the flat surface. The EDL consists of two parts compact and diffuse layers. The compact layer is an immobilized layer of counterions next to the surface of the material. Very strong electrostatic forces make this layer stationary. The thickness of this layer is about one ionic diameter. 

Electroosmotic flow is the bulk liquid motion that results when an externally applied electric field interacts with the net surplus of charged ions in the diffused part of an electrical double layer (EDL). In the presence of nonuniform or heterogeneous -potential, the net charge density in the EDL changes locally, resulting in an irregular... 

Physically, the hysteresis roots in that fact that the effect of the electric force on the stability of 1D conduction is different in different parts of the diffusion layer. Indeed, this force stabilizes ID conduction in the electroneutral bulk and in the quasi-equUibrium portion of EDL and destabilizes it in the ESC region. The nonlinear flow resulting from this instability reduces concentration polarization and, thus, weakens the hampering effect of the electric force in the bulk in the down way portion of the hysteresis loop. In order to verify this mechanism, a model electroosmotic formulation without electric force term in the Stokes equation was analyzed. As illustrated in Fig. 8, this modification results in shrinking of the hysteresis loop. The bifurcation still remains subcritical and the hysteresis loop still exists owing to the hampering effect of the electric force in the quasi-equilibrium portion of the EDL, implicit in the first term in the electroosmotic slip conditions (21). 

The electric double layer (EDL) of a particle is determined by the mechanisms of surface charging and ion adsorption in the Stem layer as well as by the balance of Coulombic and osmotic forces on the ions within the diffuse part of EDL. The latter is described by the nonlinear Poisson-Boltzmann equation (PBE, Eq. (3.16)), which needs to be solved numerically for particle aggregates. Additionally, one needs a model for the charge and potential regulation of overlapping EDLs which is inevitable in the case of aggregates and approaching particles. 

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