Electric Debye Layer

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... 

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 basic difference between metal-electrolyte and semiconductor-electrolyte interfaces lies primarily in the fact that the concentration of charge carriers is very low in semiconductors (see Section 2.4.1). For this reason and also because the permittivity of a semiconductor is limited, the semiconductor part of the electrical double layer at the semiconductor-electrolyte interface has a marked diffuse character with Debye lengths of the order of 10 4-10 6cm. This layer is termed the space charge region in solid-state physics. 

Electrochemical impedance measurements of the physical adsorption of ssDNA and dsDNA yields useful information about the kinetics and mobihty of the adsorption process. Physical adsorption of DNA is a simple and inexpensive method of immobilization. The ability to detect differences between ssDNA and dsDNA by impedance could be applicable to DNA biosensor technology. EIS measurements were made of the electrical double layer of a hanging drop mercury electrode for both ssDNA and dsDNA [34]. The impedance profiles were modeled by the Debye equivalent circuit for the adsorption and desorption of both ssDNA and dsDNA. Desorption of denatured ssDNA demonstrated greater dielectric loss than desorption of dsDNA. The greater flexibility of the ssDNA compared to dsDNA was proposed to account for this difference. 

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),... 

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 ... 

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). 

AF4 is the change in the free energy of the electrical double layer accompanying the adsorption of charged trains on the charged surface, and if the Debye-Hiichel approximation is applied, it is given by... 

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],... 

It is instructive to compare this to the capacitance of a plate capacitor o A/d. Here, A is the cross-sectional area and d is the separation between the two plates. We see that the electric double layer behaves like a plate capacitor, in which the distance between the plates is given by the Debye length The capacity of a double layer — that is the ability to store charge — rises with increasing salt concentration because the Debye length decreases. 

We can observe electro-osmosis directly with an optical microscope using liquids, which contain small, yet visible, particles as markers. Most measurements are made in capillaries. An electric field is tangentially applied and the quantity of liquid transported per unit time is measured (Fig. 5.13). Capillaries have typical diameters from 10 fim up to 1 mm. The diameter is thus much larger than the Debye length. Then the flow rate will change only close to a solid-liquid interface. Some Debye lengths away from the boundary, the flow rate is constant. Neglecting the thickness of the electric double layer, the liquid volume V transported per time is... 

This simple equation is, however, only valid for R Xp- If the radius is not much larger than the Debye length we can no longer treat the particle surface as an almost planar surface. In fact, we can no longer use the Gouy-Chapman theory but have to apply the theory of Debye and Hiickel. Debye and Hiickel explicitly considered the electric double layer of a sphere. A result of their theory is that the total surface charge and surface potential are related by... 

Electrostatic forces, acting when the electric double layers of two drops overlap, play an important role. As mentioned above, oil drops are often negatively charged because anions dissolve in oil somewhat better than cations. Thus, the addition of salt increases the negative charge of the oil drops (thus their electrostatic repulsion). At the same time it reduces the Debye length and weakens the electrostatic force. For this reason, emulsion stability can exhibit a maximum depending on the salt concentration. 

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... 

When one first thinks of the electrical double layer (edl) one imagines the description conceived by the originators, Debye and Huckel [2], Gouy and Chapman [3], Verwey and Overbeek [1], of a sharp and well-defined boundary between two phases. One of the phases usually being an aqueous medium in which a strong electrolyte is dissolved to a molar concentration of cs. The other phase is usually a solid, impermeable to either the electrolyte... 

To determine the spatial variation of a static electric field, one has to solve the Poisson equation for the appropriate charge distribution, subject to such boundary conditions as may pertain. The Poisson equation plays a central role in the Gouy-Chapman (- Gouy, - Chapman) electrical - double layer model and in the - Debye-Huckel theory of electrolyte solutions. In the first case the one-dimensional form of Eq. (2)... 

For relatively wide channels with negligible electrical double-layer overlap (r/8 > 10), a nearly flat flow profile is expected. It has often been stated that when the channel size and the Debye length are of similar dimensions (r 8), complete electrical double-layer overlap occurs and the EOF is negligible. However, when r 8, a significant EOF can still be created the EOF velocity in the central part of the channel is approximately 20% of that in an infinitely wide channel. Only at conditions where r/8 1 is the EOF fully inhibited by double-layer overlap [25], It should be noted here that the approximations made by using the Rice and Whitehead theory at r/8 < 10 may lead to significant errors in the calculation of the velocity distribution and magnitude of the EOF [17] compared to more sophisticated models. 

Figure 1.4 shows y(x) for several values of yo calculated from Eq. (1.37) in comparison with the Debye-Hlickel linearized solution (Eq. (1.25)). It is seen that the Debye-Hiickel approximation is good for low potentials (lyol< 1). As seen from Eqs. (1.25) and (1.37), the potential i//(x) across the electrical double layer varies nearly... 

Debye-Hiickel parameter k (the Debye length), which has the dimension of length, serves as a measure for the thickness of the electrical double layer. Figure 1.5 plots the... 

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