Electrical double layer potential variation

FIGURE 1-12 Variation of the potential across the electrical double layer. 

The electroviscous effect present with solid particles suspended in ionic liquids, to increase the viscosity over that of the bulk liquid. The primary effect caused by the shear field distorting the electrical double layer surrounding the solid particles in suspension. The secondary effect results from the overlap of the electrical double layers of neighboring particles. The tertiary effect arises from changes in size and shape of the particles caused by the shear field. The primary electroviscous effect has been the subject of much study and has been shown to depend on (a) the size of the Debye length of the electrical double layer compared to the size of the suspended particle (b) the potential at the slipping plane between the particle and the bulk fluid (c) the Peclet number, i.e., diffusive to hydrodynamic forces (d) the Hartmarm number, i.e. electrical to hydrodynamic forces and (e) variations in the Stern layer around the particle (Garcia-Salinas et al. 2000). 

The electrified interface is generally referred to as the electric double layer (EDL). This name originates from the simple parallel plate capacitor model of the interface attributed to Helmholtz.1,9 In this model, the charge on the surface of the electrode is balanced by a plane of charge (in the form of nonspecifically adsorbed ions) equal in magnitude, but opposite in sign, in the solution. These ions have only a coulombic interaction with the electrode surface, and the plane they form is called the outer Helmholtz plane (OHP). Helmholtz s model assumes a linear variation of potential from the electrode to the OHP. The bulk solution begins immediately beyond the OHP and is constant in potential (see Fig. 1). 

The variation of the electric potential in the electric double layer with the distance from the charged surface is depicted in Figure 6.2. The potential at the surface ( /o) linearly decreases in the Stem layer to the value of the zeta potential (0- This is the electric potential at the plane of shear between the Stern layer (and that part of the double layer occupied by the molecules of solvent associated with the adsorbed ions) and the diffuse part of the double layer. The zeta potential decays exponentially from to zero with the distance from the plane of shear between the Stern layer and the diffuse part of the double layer. The location of the plane of shear a small distance further out from the surface than the Stem plane renders the zeta potential marginally smaller in magnitude than the potential at the Stem plane ( /5). However, in order to simplify the mathematical models describing the electric double layer, it is customary to assume the identity of (ti/j) and The bulk experimental evidence indicates that errors introduced through this approximation are usually small. 

Fig. 1. Variation of the eiectric potential near a surface in the presence of an electrolyte solution, (a) Electrical double layer at the surface of a solid positively charged, in contact with an electrolyte solution, (b) The variation of the electrical potential when the measurement is made at an increasing distance from the surface, and when the liquid phase is mobile at a given flow rate. The zeta potential [) can be calculated from the streaming potential, which can be measured according to the method described by Thubikar et al. [4].

It is important to note that the concept of osmotic pressure is more general than suggested by the above experiment. In particular, one does not have to invoke the presence of a membrane (or even a concentration difference) to define osmotic pressure. The osmotic pressure, being a property of a solution, always exists and serves to counteract the tendency of the chemical potentials to equalize. It is not important how the differences in the chemical potential come about. The differences may arise due to other factors such as an electric field or gravity. For example, we see in Chapter 11 (Section 11.7a) how osmotic pressure plays a major role in giving rise to repulsion between electrical double layers here, the variation of the concentration in the electrical double layers arises from the electrostatic interaction between a charged surface and the ions in the solution. In Chapter 13 (Section 13.6b.3), we provide another example of the role of differences in osmotic pressures of a polymer solution in giving rise to an effective attractive force between colloidal particles suspended in the solution. 

Equation (1.35) is known as the Debye-Hiickel or Gui-Chapman equation for the equilibrium double layer potential. In terms of the original variable x (1.34), (1.35) suggest e1/2(r(j) is the correct scale of ip variation, that is, the correct scale for the thickness of the electric double layer. At the same time, it is observed from (1.32) that for N 1 the appropriate scale depends on N, shrinking to zero when N — oo (ipm — — oo). This illustrates the previously made statement concerning the meaningfulness of the presented interpretation of relectric potential

(—oo) — 0, < (oo) — —oo). 

The formation of an electrical double layer at a metal-solution interface brings about a particular arrangement of atoms, ions and molecules in the region near the electrode surface, and an associated variation in electrical potential with distance from the interface. The double layer structure may significantly affect the rates of electrochemical reactions. 

We start this chapter with electrocapillarity because it provides detailed information of the electric double layer. In a classical electrocapillary experiment the change of interfacial tension at a metal-electrolyte interface is determined upon variation of an applied potential (Fig. 5.1). It was known for a long time that the shape of a mercury drop which is in contact with an electrolyte depends on the electric potential. Lippmann1 examined this electrocapillary effect in 1875 for the first time [68], He succeeded in calculating the interfacial tension as a function of applied potential and he measured it with mercury. 

Figure 1. Potential variation through the electrical double layer for a higher concentration of potential-determining ion (a), a lower concentration of potential-determining ion (b), and in the presence of a specifically adsorbed counter ion with a potential-determining ion below the point-of-zero charge (c). Note that the potential in all three instances could be identical. (Reproduced, with permission, from Ref. 1. Copyright 1970, International Union of Pure and Applied Chemists.)...

This electrokinetically driven micro mixer uses localized capacitance effects to induce zeta potential variations along the surface of silica-based micro channels [92], The zeta potential variations are given near the electrical double layer region of the electroosmotic flow utilized for species transport. Shielded ( buried ) electrodes are placed underneath the channel structures for the fluid flow in separate channels, i.e. they are not exposed to the liquid. The potential variations induce flow velocity changes in the fluid and thus promote mixing [92],... 

Using Equation 4.16 to estimate g in region I at first appears to be a strange procedure because the case Od = 0 corresponds to no overlap of the double layers. The variation of electric potential with distance is then as sketched by the thin line in Figure 4.2. The thicker line in Figure 4.2 represents an exponentially decaying potential function (see method (b) below). Let us first pursue the mathematics of the Od = 0 case. The crude approximation... 

Once we know the value of the inner potential difference, — 0, we can correlate A using Equations (10) and (11) and then calculate the variation of y with due to the presence of the electrical double layers. In the present model, we simply neglect — < °, which is known to be small and shows no strong dependence on [24]. 

Fig.l Schematic representation of the electric double layer at a solid-liquid interface and variation of potential with the distance from the solid surface if/Q, surface potential potential at the Stern plane potential at the plane of share (zeta potential) 8, distance of the Stern plane from the surface (thickness of the Stern layer) k, thickness of the diffuse region of the double layer. 

FIGURE 2-1 Helmholtz model of the electrical double layer, (a) Distribution of counterions in the vicinity of the charged surface. (b) Variation of electrical potential with distance from the charged surface. 

The measurement of the variation of A(AV(t)) often does not give us a direct information on changes in the surfactant or polymer adsorption. So far the models are more or less qualitative. A quantitative model should be able to describe the complex structure of the electric double layer in the interfacial region and to explain peculiarities, such as the surface potential of a bare interface and changes in the potential due to adsorption of nonionic surfactants. 

Poisson-Boltzmann Equation A fundamental equation describing the distribution of electric potential around a charged species or surface. The local variation in electric-field strength at any distance from the surface is given by the Poisson equation, and the local concentration of ions corresponding to the electric-field strength at each position in an electric double layer is given by the Boltzmann equation. The Poisson-Boltzmann equation can be combined with Debye-Hiickel theory to yield a simplified, and much used, relation between potential and distance into the diffuse double layer. 

Three models for the structure of the electric double layer, showing the variations of electric potential negative charges on the surface ... 

EDLCs store energy within the variation of potential at the electrode/electrolyte interface. This variation of potential at a surface (or interface) is known as the electric double layer or, more traditionally, the Helmholtz layer. The thickness of the double layer depends on the size of the ions and the concentration of the electrolyte. For concentrated electrolytes, the thickness is on the order of 10 A, while the double layer is 1000 A for dilute electrolytes (5). In essence, this double layer is a nanoscale model of a traditional capacitor where ions of opposite charges are stored by electrostatic attraction between charged ions and the electrode surface. EDLCs use high surface area materials as the electrode and therefore can store much more charge (higher capacitance) compared to traditional capacitors. 

Figure 3 Electrical double layer and potential variation vs. distance from interface.

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