Diffuse Part of the Double Layer

Eleetrostatie eharaeterization of partieles is eommonly determined via their eleetrokinetie or zeta potential i.e. the potential of a slipping plane, notionally loeated slightly away from the partiele surfaee approximately at the beginning of the diffuse part of the double layer using, for example, eleetrophoresis. In some eases, zeta potential ean be used as a eriterion for aggregation. 

The model just presented describes what electrochemists call the diffuse part of the double layer and no account is made of the inner layer effects such as the plane of the closest approach. To have an idea what the impact of the effects predicted by this model on the measured capacitance could be, we assume the traditional inner and diffuse layer separation. However, we... 

Note that both before and after the experiment the sum of the charges on the metal surface and in the adsorbate layer is zero, and hence there is no excess charge in the diffuse part of the double layer. However, after the adsorption has occurred, the electrode surface is no longer at the pzc, since it has taken up charge in the process. 

The time constant, Td, for relaxation of the diffuse part of the double layer is determined by bulk properties of the medium ... 

In a qualitative way, colloids are stable when they are electrically charged (we will not consider here the stability of hydrophilic colloids - gelatine, starch, proteins, macromolecules, biocolloids - where stability may be enhanced by steric arrangements and the affinity of organic functional groups to water). In a physical model of colloid stability particle repulsion due to electrostatic interaction is counteracted by attraction due to van der Waal interaction. The repulsion energy depends on the surface potential and its decrease in the diffuse part of the double layer the decay of the potential with distance is a function of the ionic strength (Fig. 3.2c and Fig. 

Relationship between MnC>2 colloid surface area concentration and ccc of Ca2+ a stoichiometric relationship exists between ccc and the surface area concentration in case of Na+, however, this interaction is weaker, so that primarily compaction of the diffuse part of the double layer causes destabilization. 

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. 

Equation 37 gives the analytical form of the free surface energy of the diffuse part of the double layer and has been derived by a number of authors [8,9, 40] for 1 1 ionic surfactants. For systems with mixed valences, the integral in Eq. 34 is usually not available in close analytical expressions and numerical integration is often required. 

Fig. 7.17. The potential difference across the interface can be divided into the linear portion of the layer extending to the OHP, at which the ions ready to discharge are located, and a portion in the diffuse part of the double layer, which is called the elec-trokinetic or potential.

It is convenient to think of the diffuse part of the double layer as an ionic atmosphere surrounding the particle. Any movement of the particle affects the particle s ionic atmosphere, which can be thought of as being dragged along through bulk motion and diffusional motion of the ions. The resulting electrical contribution to the resistance to particle motion manifests itself as an additional viscous effect, known as the electroviscous effect. Further,... 

Beyond the Stern layer, the remaining z counterions exist in solution. These ions experience two kinds of force an electrostatic attraction drawing them toward the micelle and thermal jostling, which tends to disperse them. The equilibrium resultant of these opposing forces is a diffuse ion atmosphere, the second half of a double layer of charge at the surface of the colloid. Chapter 11 provides a more detailed look at the diffuse part of the double layer. 

The diffuse part of the double layer is of little concern to us at this point. Chapters 11 and 12 explore in detail various models and phenomena associated with the ion atmosphere. At present it is sufficient for us to note that the extension in space of the ion atmosphere may be considerable, decreasing as the electrolyte content of the solution increases. As micelles approach one another in solution, the diffuse parts of their respective double layers make the first contact. This is the origin of part of the nonideality of the micellar dispersion and is reflected in the second virial coefficient B as measured by osmometry or light scattering. It is through this connection that z can be evaluated from experimental B values. 

It is the outer portion of the double layer that interests us most as far as colloidal stability is concerned. The existence of a Stern layer does not invalidate the expressions for the diffuse part of the double layer. As a matter of fact, by lowering the potential at the inner boundary of the diffuse double layer, we enhance the validity of low-potential approximations. The only problem is that specific adsorption effects make it difficult to decide what value to use for J/6. 

FIG. 12.8 Plot of rju/e versus f/0, that is, the zeta potential according to the Helmholtz-Smoluchowski equation, Equation (39), versus the potential at the inner limit of the diffuse part of the double layer. Curves are drawn for various concentrations of 1 1 electrolyte with / = 10 15 V-2 m2. (Redrawn with permission from J. Lyklema and J. Th. G. Overbeek, J. Colloid Sci., 16, 501 (1961).)... 

The potential at the inner limit of the diffuse part of the double layer enters Equation (1) through T0, defined by Equation (11.65) with p0 in place of ip. For large values of ip0, T0 1, so sensitivity to the value of p0 decreases as ip0 increases. Figure 13.7 shows the effect of variations in the value of ip0 on the total interaction potential energy with k (109 m -l or 0.093 M for a 1 1 electrolyte) and A (2 10 19 J) constant. The height of the potential energy barrier is seen to increase with increasing values of ip0, as would be expected in view of the... 

The next layer beyond the specifically adsorbed layer is rich in cations attracted by the negative electrode. The excess of cations decreases with increasing distance from the electrode. This region, whose composition is different from that of bulk solution, is called the diffuse part of the double layer and is typically 0.3-10 nm thick. The thickness is controlled by the balance between attraction toward the electrode and randomization by thermal motion. 

When a species is created or destroyed by an electrochemical reaction, its concentration near the electrode is different from its concentration in bulk solution (Figure 17-12 and Color Plate 12). The region containing excess product or decreased reactant is called the diffusion layer (not to be confused with the diffuse part of the double layer). 

Electrode-solution interface. The tightly adsorbed inner layer (also called the compact, Helmholtz, or Stem layer) may include solvent and any solute molecules. Cations in the inner layer do not completely balance the charge of the electrode. Therefore, excess cations are required in the diffuse part of the double layer for charge balance. 

Figure 26-20 (a) Electric double layer created by negatively charged silica surface and nearby cations. (fc>) Predominance of cations in diffuse part of the double layer produces net electroosmotic flow toward the cathode when an external field is applied. 

In an electric field, cations are attracted to the cathode and anions are attracted to the anode (Figure 26-20b). Excess cations in the diffuse part of the double layer impart net momentum toward the cathode. This pumping action, called electroosmosis (or electroen-dosmosis), is driven by cations within — lOnm of the walls and creates uniform pluglike electroosmotic flow of the entire solution toward the cathode (Figure 26-21a). This process is in sharp contrast with hydrodynamic flow, which is driven by a pressure difference. In hydro-... 

Figure 26-24 Charge reversal created by a cationic surfactant bilayer coated on the capillary wall. The diffuse part of the double layer contains excess anions, and electroosmotic flow is in the direction opposite that shown in Figure 26-20. The surfactant is the didodecyldimethylammonium ion, (n-C,2H25)N(CH3)2, represented as in the illustration.

Cations striking a cathode liberate electrons. A series of dynodes multiplies the number of electrons by 105 before they reach the anode, electroosmosis Bulk flow of fluid in a capillary tube induced by an electric field. Mobile ions in the diffuse part of the double layer at the wall of the capillary serve as the pump. Also called electroendosmosis. electroosmotic flow Uniform, pluglike flow of fluid in a capillary tube under the influence of an electric field. The greater the charge on the wall of the capillary, the greater the number of counterions in the double layer and the stronger the electroosmotic flow. 

The electric double layer can be regarded as consisting of two regions an inner region which may include adsorbed ions, and a diffuse region in which ions are distributed according to the influence of electrical forces and random thermal motion. The diffuse part of the double layer will be considered first. 

The ions in the diffuse part of the double layer are assumed to be point charges distributed according to the Boltzmann distribution. 

The treatment of the diffuse double layer outlined in the last section is based on an assumption of point charges in the electrolyte medium. The finite size of the ions will, however, limit the inner boundary of the diffuse part of the double layer, since the centre of an ion can only... 

Specifically adsorbed ions are those which are attached (albeit temporarily) to the surface by electrostatic and/or van der Waals forces strongly enough to overcome thermal agitation. They may be dehydrated, at least in the direction of the surface. The centres of any specifically adsorbed ions are located in the Stern layer - i.e. between the surface and the Stern plane. Ions with centres located beyond the Stern plane form the diffuse part of the double layer, for which the Gouy-Chapman treatment outlined in the previous section, with 0o replaced by (f/d, is considered to be applicable. 

Stern assumed that a Langmuir-type adsorption isotherm could be used to describe the equilibrium between ions adsorbed in the Stern layer and those in the diffuse part of the double layer. Considering only the adsorption of counter-ions, the surface charge density cr, of the Stern layer is given by the expression... 

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