Diffuse part of double layer

We have not discussed charged systems. As far as the diffuse parts of double layers are concerned, these relax very rapidly, see sec. II.2.13d, meaning that their dynamics will not be rate-controlling. Double layers do, of course, exert their influence on the diffusional transport rate and on the activation energies for the exchange of ionic surfactants between monolayers and solution (uptake in the monolayer is inhibited, release promoted). 

The Retardation of the Steady Transport of Adsorbing Ions Through the Diffuse Part of Double Layer... 

The Gouy-Chapman theory provides a better approximation of reality than does the Helmholtz theory, but it still has limited quantitative application. It assumes that ions behave as point charges, which they cannot, and it assumes that there is no physical limit for the ions in their approach to the TPB, which is not true. Stem, therefore, modified the Gouy-Chapman diffuse double layer. His theory states that ions do have finite size, so they cannot approach the TPB closer than a few nm [54, 60], The first ions of the Gouy-Chapman diffuse double layer are in the gas phase but not at the TPB. They are at some distance 8 away from the zirconia-metal-gas interface. This distance will usually be taken as the radius of the ion. As a result, the potential and concentration of the diffuse part of the layer are low enough to justify treating the ions as point charges. Stem also assumed that it is possible that some of the ions are specifically adsorbed by the TPB in the plane 8, and this layer has become known as the Stem layer. Therefore, the potential will drop by T o - Pg over the molecular condenser (i.e., the Helmholtz plane) and by T g over the diffuse layer. Pg has become known as the zeta (Q potential. 

The Stern model (1924) essentially combines the Helmholtz and Gouy-Chapman models as shown in Figure 5.4. Thus, the Stern model has two parts of double layer (a) compact layer ("rigid layer") of ions at the distance of closest approach (OHP) and (b) diffuse layer. The concentration of ions and the potential distribution from the electrode vary as shown in Figures 5.3 and 5.4. The potential has a sharp drop between the electrode and OHP beyond which the potential gradually falls to a value characteristic of bulk electrolyte. 

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. 

Simple electrolyte ions like Cl, Na+, SO , Mg2+ and Ca2+ destabilize the iron(Hl) oxide colloids by compressing the electric double layer, i.e., by balancing the surface charge of the hematite with "counter ions" in the diffuse part of the double... 

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. 

When particles or large molecules make contact with water or an aqueous solution, the polarity of the solvent promotes the formation of an electrically charged interface. The accumulation of charge can result from at least three mechanisms (a) ionization of acid and/or base groups on the particle s surface (b) the adsorption of anions, cations, ampholytes, and/or protons and (c) dissolution of ion-pairs that are discrete subunits of the crystalline particle, such as calcium-oxalate and calcium-phosphate complexes that are building blocks of kidney stone and bone crystal, respectively. The electric charging of the surface also influences how other solutes, ions, and water molecules are attracted to that surface. These interactions and the random thermal motion of ionic and polar solvent molecules establishes a diffuse part of what is termed the electric double layer, with the surface being the other part of this double layer. 

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. 

In Eq. 30, Uioo and Fi are the activity in solution and the surface excess of the zth component, respectively. The activity is related to the concentration in solution Cioo and the activity coefficient / by Uioo =fCioo. The activity coefficient is a function of the solution ionic strength I [39]. The surface excess Fi includes the adsorption Fi in the Stern layer and the contribution, f lCiix) - Cioo] dx, from the diffuse part of the electrical double layer. The Boltzmann distribution gives Ci(x) = Cioo exp - Zj0(x), where z, is the ion valence and 0(x) is the dimensionless potential (measured from the Stern layer) obtained by dividing the actual potential, fix), by the thermal potential, k Tje = 25.7 mV at 25 °C). Similarly, the ionic activity in solution and at the Stern layer is inter-related as Uioo = af exp(z0s)> where tps is the scaled surface potential. Given that the sum of /jz, is equal to zero due to the electrical... 

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 has been assumed that the decline of the current with an increase in time (not shown in the figure) is due only to the onset of a degree of diffusion control, and that the method for obtaining the desired iF depends on this assumption. However, there are two other reasons for a decline in current. First, as already stated, the effect of double-layer charging may not be finished in the early part of the f( — t plot (between B and C in Fig. 8.9) so that it may be that a straight line between 1/i and t is observed only if the earlier points are neglected. 

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. 

The Stern surface is drawn through the ions that are assumed to be adsorbed on the charged wall. (This surface is also known as the inner Helmholtz plane [IHP], and the surface running parallel to the IHP, through the surface of shear (see Chapter 12) shown in Figure 11.9, is called the outer Helmholtz plane [OHP]. Notice that the diffuse part of the ionic cloud beyond the OHP is the diffuse double layer, which is also known as the Gouy-Chapman... 

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... 

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