Double layer theories Stern

One other aspect of nonprimitive electric double layer theories which is particularly relevant to the inner Stern region are the models for the water molecule and the ions. The simplest models for a water molecule and an ion are a hard-sphere point dipole and point charge, respectively. A more realistic model of the hard-sphere water molecule would include quadrupoles and octupoles and also polarizability. However the hard-sphere property is best avoided and replaced, for example, by a Lennard-Jones potential. An alternative to a multipolar water model are three point charge sites associated with the atoms within the water molecule. 

Though a bit artificial, the subdivision into a Stern and a diffuse part has proven its value. One reason Is that the diffuse part can be described with relatively simple analytical equations that become exact at sufficiently large distance from the surface. These two parts play central roles in electrokinetics, colloid stability and many other phenomena where diffuse double layer theory is found to apply well. From the more theoretical side, the diffuse part is characterized by relatively low potentials, so that deviations from ideality are... 

Stern 1 has therefore altered the model underlying the double layer theory for a solid wall by dividing the liquid charge into two parts. One part is thought of as a layer of ions adsorbed to the wall, and is represented in the theory by a surface charge concentrated in a plane at a small distance 5... 

Several models that depart partially or totally from DLVO have been proposed. One such model is the Spitzer dissociative electrical double layer theory (Spitzer 1984, 2003 and references therein). It essentially uses the linearized PB equation (which is consistent with Maxwellian electromagnetism) along with a coion exclusion boundary, which prevents ions of the same charge as the surface becomes too close in practice, this avoids negative concentrations, which will be predicted by the linear PB theory. It also includes a double layer association parameter a, which gives the fractions of counterions that are associated to the surface forming the Stern layer (Spitzer 1992). This theory, however, has not been further developed. 

It is evident now why the Helmholtz and Gouy-Chapman models were retained. While each one alone fails completely when compared with experiment, a series combination of the two yields reasonably good agreement. There is room for improvement and refinement of the theory, but we shall not deal with that here. The model of Stern brings theory and experiment close enough for us to believe that it does describe the real situation at the interface. Moreover, later work of Grahame shows that the diffuse-double-layer theory, used in the proper context, yields consistent results and can be considered to be correct, within the limits of the approximations used to derive it. 

Grahame therefore suggests that a test of double layer theories should in the first place be directed to a case, where specific anion-adsorption may be expected to be absent, that is practically to the case of fluorides. Owing to the strong electronegativity of the F ion this ion will not be dehydrated and thus not enter into the Stern plane . 

On the basis of the theory of the diffuse double layer it is possible to show that already a compression of the double layer by electrolytes is sufficient to explain loss of stability. It may be remarked that in order to explain certain specific effects in flocculation it is necessary to extend the double layer theory by some modification such as Stern s theory (c/. chapter VI, 8, p. 263) which in essence is a partial return lo elements of the chemical theories and the adsorption theory of Freundlich. 

It has been mentioned in chapter VI, 8, p, 263 that Stern s correction results in a lower potential drop in the diffuse double layer (9 instead of The potential in the diffuse layer, instead of being oply dependent on the amount of potential-determining ions, now also depends upon the total electrolyte concentration and is lower, the higher the electrolyte content. This explains why in several cases a sort of critical u-potential has been found and it shows how in the refinements of the double layer theory of stability, conceptions of the older theories (like discharge by adsorption of counter ions) again play a role. 

The Stern theory is difficult to apply quantitatively because several of the parameters it introduces into the picture of the double layer cannot be evaluated experimentally. For example, the dielectric constant of the water is probably considerably less in the Stern layer than it would be in bulk because the electric field is exceptionally high in this region. This effect is called dielectric saturation and has been measured for macroscopic systems, but it is difficult to know what value of e6 applies in the Stern layer. The constant K is also difficult to estimate quantitatively, principally because of the specific chemical interaction energy . Some calculations have been carried out, however, in which the various parameters in Equation (97) were systematically varied to examine the effect of these variations on the double layer. The following generalizations are based on these calculations ... 

As the electrode surface will, in general, be electrically charged, there will be a surplus of ionic charge with opposite sign in the electrolyte phase in a layer of a certain thickness. The distribution of jons in the electrical double layer so formed is usually described by the Gouy— Chapman—Stern theory [20], which essentially considers the electrostatic interaction between the smeared-out charge on the surface and the positive and negative ions (non-specific adsorption). An extension to this theory is necessary when ions have a more specific interaction with the electrode, i.e. when there is specific adsorption of ions. 

Stern combined the ideas of Helmholtz and that of a diffuse layer [64], In Stern theory we take a pragmatic, though somewhat artificial, approach and divide the double layer into two parts an inner part, the Stern layer, and an outer part, the Gouy or diffuse layer. Essentially the Stern layer is a layer of ions which is directly adsorbed to the surface and which is immobile. In contrast, the Gouy-Chapman layer consists of mobile ions, which obey Poisson-Boltzmann statistics. The potential at the point where the bound Stern layer ends and the mobile diffuse layer begins is the zeta potential (C potential). The zeta potential will be discussed in detail in Section 5.4. 

Figure 7.2 Schematic representation of the structure of the electric double layer according to Stern s theory... 

HLC have suggested that the solvent dipoles near the colloidal particles are preferentially aligned. This effect is well known in theories of the electrical double layer. One simple way of accounting for this effect is through the use of a Stern layer of low dielectric constant near the colloidal particles. It is difficult to calculate this correction for spherical particles. As a result, HLC considered a hard sphere fluid between two hard walls and with a region of low dielectric constant near the walls. They found that Eq. (62) should be generalized to... 

Finally we shall argue that present-day theories of the nonprimitive models of the electric double layer have considerable difficulty in treating properly ion adsorption in the Stern inner region at metal-aqueous electrolyte interfaces and we suggest that this region is a useful concept which should not be dismissed as unphysical. Indeed Stern-like inner region models continue to be used in colloid and electrochemical science, for example in theories of electrokinetics and aqueous-non-metallic (e.g., oxide) interfaces. 

Earlier theories by Gouy, Chapman, and Hcrzfeld discussed the double layer as wholly of this diffuse type but Stem points out that these give far too high values for the capacity of the double layer, partly because in them the ions are supposed mathematically to be able to approach indefinitely close to the solid surface, which is impossible physically owing to the size of the ions. Stern s theory gives a complicated expression for the capacity of the double layer, but accounts reasonably well for the experimental values. Though the layer is largely diffuse in many cases, the capacity is usually of the same order as if the layer were of the plane parallel type, because most of the ions are fairly close to the fixed part of the layer. 

Up till about 1921, it was often supposed that the potential could be identified with the single potential difference at the phase boundary. Freundlich and his collaborators1 showed that this is quite impossible, since the variation with concentration, and the influence of adsorbed substances, are entirely different in the two cases sometimes indeed the two potentials may have different signs. The phase boundary potential, if defined as the Volta potential, is the difference between the energy levels of the charged component, to which the phase boundary is permeable, inside the two phases when these are both at the same electrostatic potential. We have seen that it is difficult, or impossible, to define the phase boundary potential in any other way (see 2 and 3). It includes the work of extraction of the charged component from each phase, and this includes the part of the double layer which according to Stern s theory is fixed. The potential is merely the potential fall in the mobile, diffuse part of the double layer, and is wholly within one phase. 

Parsons and Zobel plot — In several theories for the electric - double layer in the absence of specific adsorption, the interfacial -> capacity C per unit area can formally be decomposed into two capacities in series, one of which is the Gouy-Chapman (- Gouy, - Chapman) capacity CGC 1/C = 1 /CH + 1 /CGC. The capacity Ch is assumed to be independent of the electrolyte concentrations, and has been called the inner-layer, the - Helmholtz, or Stern layer capacity by various authors. In the early work by Stern, Ch was attributed to an inner solvent layer on the electrode surface, into which the ions cannot penetrate more recent theories account for an extended boundary region. In a Parsons and Zobel plot, Ch is determined by plotting experimental values for 1/C vs. 1/Cgc- Specific adsorption results in significant deviations from a straight line, which invalidates this procedure. 

Conversely, according to the description of the electrical double layer based on the Stern-Gouy-Chapman (S-G-C) version of the theory [24], counter ions cannot get closer to the surface than a certain distance (plane of closest approach of counter ions). Chemically adsorbed ions are located at the inner Helmholtz plane (IHP), while non-chemically adsorbed ions are located in the outer Helmholtz plane (OHP) at a distance x from the surface. The potential difference between this plane and the bulk solution is 1 ohp- In this version of the theory, Pqhp replaces P in all equations. Two regions are discernible in the double layer the compact area between the charged surface and the OHP in which the potential decays linearly and the diffuse layer in which the potential decay is almost exponential due to screening effects. 

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