The Diffuse Double Layer

The colloidal particle is assumed to be so large compared to the ions that its surface may be regarded as a plane. The electric potential and the charge density then become a function of the distance / from the charged surface (a flat charged plane). Hence, the Poisson equation (7.2) is written 

The key difference between this presentation and the Debye-Hiickel theory is that the electrical surface potential is not always smaller than the 

Multiplying both sides of (7.32) by 2dy/dx and integrating once with respect to X under the boundary conditions, y = 0 and dy/dx = 0 for x = oo, we obtain 

The minus sign results from the slope of y, which is assumed to be positive here (Fig. 7.2). After a second integration of (7.34) under the conditions = 0 or y = yo for X = 0, we obtain 

The electrical potential iff decays exponentially against larger I, irrespective of the magnitude of if/o. When the electrical potential is much less than 25.7 mV, on the other hand, the Poisson-Boltzmann equation takes the form 

This portion is controlled solely by electrostatic forces, the distribution of ions in relation to the charged electrode surface is a function of their charge 

The diffuse double layer in the region bounded by the conditions = po and // = 0. 

Let the charge density at the electrode surface (i.e. at x = 0) be a/unit area. This then is equal in magnitude, but of opposite sign, to the total volume charge in solution, i.e.. 

When this latter expression for A is substituted into that for Equation (7.2), we obtain 

In the condition that x approaches a, i.e. the outer limit of the Helmholtz layer, (a — x) 0. Under these conditions will be designated 4io i.e. 

FIGURE 9.7 A diffuse electrical double layer according to Gouy and Chapman. 

The space charge density p(x) at the solution side in the electrical double layer follows from the excess of counterions and the deficit of co-ions, 

Both the counterions and the co-ions are considered to be point charges that have no volume, so that a diffuse distribution is obtained up to the boundary with the solid snrface, x = 0. 

In Equations 9.27 and 9.28 K is the so-called reciprocal Debye length it is related to the ionic strength as (for symmetrical electrolytes) 

According to Equation 9.28, the Debye length (k ) equals the distance over which / reduces from /q to By convention, that distance is referred to as the thickness of the electrical double layer. It follows from Equation 9.29 that /(x) decays more steeply, or, in other words, that the thickness of the double layer decreases as the ionic strength of the solution increases. 

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The quantity 1 /k is thus the distance at which the potential has reached the 1 je fraction of its value at the surface and coincides with the center of action of the space charge. The plane at a = l//c is therefore taken as the effective thickness of the diffuse double layer. As an example, 1/x = 30 A in the case of 0.01 M uni-univalent electrolyte at 25°C. 

By analogy with the Helmholtz condenser formula, for small potentials the diffuse double layer can be likened to an electrical condenser of plate distance /k. For larger yo values, however, a increases more than linearly with o, and the capacity of the double layer also begins to increase. 

In the case of a charged particle, the total charge is not known, but if the diffuse double layer up to the plane of shear may be regarded as the equivalent of a parallel-plate condenser, one may write... 

Derive the general equation for the differential capacity of the diffuse double layer from the Gouy-Chapman equations. Make a plot of surface charge density tr versus this capacity. Show under what conditions your expressions reduce to the simple Helmholtz formula of Eq. V-17. 

Rheology. Flow properties of latices are important during processing and in many latex appHcations such as dipped goods, paint, inks (qv), and fabric coatings. For dilute, nonionic latices, the relative latex viscosity is a power—law expansion of the particle volume fraction. The terms in the expansion account for flow around the particles and particle—particle interactions. For ionic latices, electrostatic contributions to the flow around the diffuse double layer and enhanced particle—particle interactions must be considered (92). A relative viscosity relationship for concentrated latices was first presented in 1972 (93). A review of empirical relative viscosity models is available (92). In practice, latex viscosity measurements are carried out with rotational viscometers (see Rpleologicalmeasurement). 

F r d ic Current. The double layer is a leaky capacitor because Faradaic current flows around it. This leaky nature can be represented by a voltage-dependent resistance placed in parallel and called the charge-transfer resistance. Basically, the electrochemical reaction at the electrode surface consists of four thermodynamically defined states, two each on either side of a transition state. These are (11) (/) oxidized species beyond the diffuse double layer and n electrons in the electrode and (2) oxidized species within the outer Helmholtz plane and n electrons in the electrode, on one side of the transition state and (J) reduced species within the outer Helmholtz plane and (4) reduced species beyond the diffuse double layer, on the other. 

Measurements based on the Gouy-Chapman-Stem theory to determine the diffuse double-layer capacitance 10, 24,72, 74... 

The same system has been studied previously by Boguslavsky et al. [29], who also used the drop weight method. While qualitatively the same behavior was observed over the broad concentration range up to the solubility limit, the data were fitted to a Frumkin isotherm, i.e., the ions were supposed to be specifically adsorbed as the interfacial ion pair [29]. The equation of the Frumkin-type isotherm was derived by Krylov et al. [31], on assuming that the electrolyte concentration in each phase is high, so that the potential difference across the diffuse double layer can be neglected. 

FIG. 8 Inverse differential capacity at the zero surface charge vs. inverse capacity Cj of the diffuse double layer for the water-nitrobenzene (O) and water-1,2-dichloroethane (, ), interface. The diffuse layer capacity was evaluated by the GC ( ) or the MPB (0,)> theory. (From Ref. 22.)... 

To evaluate the contribution of the SHG active oriented cation complexes to the ISE potential, the SHG responses were analyzed on the basis of a space-charge model [30,31]. This model, which was proposed to explain the permselectivity behavior of electrically neutral ionophore-based liquid membranes, assumes that a space charge region exists at the membrane boundary the primary function of lipophilic ionophores is to solubilize cations in the boundary region of the membrane, whereas hydrophilic counteranions are excluded from the membrane phase. Theoretical treatments of this model reported so far were essentially based on the assumption of a double-diffuse layer at the organic-aqueous solution interface and used a description of the diffuse double layer based on the classical Gouy-Chapman theory [31,34]. 

It is also possible for ions in the water, especially positively charged ions, or cations, to be attracted to the negatively charged surfaces. This leads to a zone of water and ions surrounding the clay particles, known as the diffuse double layer. 

When an aqueous solution containing an irreducible cation M+ is electrolyzed, H2 evolves at the cathode with the overall reaction Haq+ + e(cathode) — (l/2)H2(gas). The detailed mechanism of this reaction is somewhat ambiguous, as it could be attributed either to absorbed H atoms or absorbed H2+ ions. According to Walker (1966, 1967), the basic cathodic reaction is (6.II) followed by (6.1) to give H2. There are several possibilities for reaction (6.II) (Walker, 1968) (1) direct electron donation by the cathodic metal to water, (2) electron liberation from the diffuse double layer, and (3) neutralization of the irreducible cation M+ (e.g., Na+) at the cathode, followed by the reaction of the neutral atom with water ... 

The definition of the electrosorption valence involves the total surface excess, not only the amount that is specifically adsorbed. It is common to correct the surface excess F, for any amount that may be in the diffuse double layer in order to obtain the amount that is specifically adsorbed. This can be done by calculating the excess in the... 

To obtain the dipole moment we set o = —o% in Eq. (18.14) so that the diffuse double layer is free of excess charge (see Section 4.3). 

While the lipid bilayer has a very low water content, and therefore behaves quite hydrophobically, especially in its core (see Chapter 2 of this volume), the cell wall is rather hydrophilic, with some 90% of water. Physicochemically, the cell wall is particularly relevant because of its high ion binding capacity and the ensuing impact on the biointerphasial electric double layer. Due to the presence of such an electric double layer, the cell wall possesses Donnan-like features, leaving only a limited part of the interphasial potential decay in the diffuse double layer in the adjacent medium. For a detailed outline, the reader is referred to recent overviews of the subject matter [1,2]. 

Cations form a diffuse layer of ions called the diffuse double layer or the electrical double layer around soil particles as depicted in Figure 5.10. The existence of the diffuse double layer means that the ions are not evenly distributed throughout the solution rather, cations are more concentrated close to soil particle surfaces and are less concentrated further away. This phenomenon must be kept in mind, particularly when electrochemical analytical methods of analysis are developed [5,7],... 

Figure 5.10. A soil particle with a diffuse layer of hydrated ions around it. The dashed line represents the boundary of the layer of tightly held cations. This diagram is not an exact representation of the diffuse double layer around a soil particle.

The electrical characteristics, along with the salts, their movement through soil, and the diffuse double layer must be kept in mind when making any soil measurement using electricity or electrodes. 

Soil has electrical characteristics associated with its components, salts, ions in solution, and the diffuse double layer. All, singly or in combination, can affect electrical measurements in soil. Electrodes inserted into soil are used to... 

In this situation, the tumor cells (or a group of cells viz. microtumor) are the immobile phase and the electroosmotic flow causes the water to move as a plug, the entire velocity gradient being concentrated at the cell surface in a layer of the same order of thickness as the diffuse double layer (Figure 5). In concentrated solutions, the thickness of the diffuse double is quite small (< 10 A) whereas in very dilute solutions (as are indeed... 

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