Stern theory

A comparison between the Stern theory and experiment was reported by Grahame (5), who found very close agreement between experimental and calculated doublelayer capacities when one and the same solution is considered for example, Figure 4.10. However, when solutions of different electrolytes are compared, the theory fails. Thus, once more, a new model is needed. 

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 ... 

By allowing for surface saturation, the Stern theory overcomes the objection to the Gouy-Chapman theory of excessive surface concentrations. In so doing, however, it trades off one set of difficulties for another. In the Gouy-Chapman theory the functional dependence of 0 on x involves only the parameters k and 0O- The former is known and the latter may be... 

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. 

In the present investigation the Stern theory (42) is assumed to hold and consequently,... 

In a number of instances a decision had to be taken as to what to call "fundamentals and what "advanced". In the case of electric double layers, this decision related to the classical Gouy-Stern theory versus modem statistical theories. For pragmatic reasons we decided to emphasize the former the equations are simple and analytical, and can account for the great majority of situations met in practice. However, a section is Included to give an impression of more a priori statistical approaches. In the domain of electrokinetics the decision was between simple theories on the level of Helmholtz-Smoluchowski (HS), that may apply to perhaps 30-50% of all systems studied in practice, or on... 

Figure 3.22, Zeroth-order Stern theory. Differential capacitances (a) and surface charges (b) for various values of the (charge-free) inner layer capacitance, C,. Salt 10 = M (1-1) electrolyte. Temperature, 25°C.

The Gouy-Chapman-Stern theory relates the surface potential, to the charge density of a membrane, , as follows (for a symmetrical electrolyte) [44, 48, 49] ... 

The second modification compared to the Stern theory was the introduction of the discreteness-of-charge factor , X. The electrostatic part of the work of ion transfer from the bulk solution into the adsorption plane across the EDL field in Stern s approximation may be represented as the sum of the contributions of the compact and diffuse layers ... 

Proceeding now to the problem of the interaction of two parallel flat double layers, we shall base our considerations, as a first approximation, on the same picture as that underlying the Gouy-Chapman theory. Later on we shall consider possible corrections of the theory by taking into account the finite dimensions of the ions in the sense of the Stern-theory. 

In the absence of specifically adsorbable ions, the double-layer structure is adequately described in terms of the Gouy-Chapman-Stern theory. The outer Helmholtz plane potential (distinguished by Grahame) is associated with a surface charge density a through the relation... 

FIGURE 3.11 Schematic representation of the Gouy-Chapman-Stern theory. Compare with Figure 3.3 an additional plane, the Stem plane, is defined as the distance of closest ionic approach to the surface. The region between the Stern plane and the surface behaves as a dielectric. 

Grahame Modification of the Gouy-Chapman-Stern Theory... 

Gouy-Chapman-Stern Theory for Nonoverlapping EDLs... 

In 1946, D. Grahame (27) introduced a refinement of the Stern theory, in which he distinguish between hydrated ions and ions which are adsorbed by covalent bonds or van der Waals forces (or both). Grahame introduced the terms inner-Helmholtz plane for the locus of the electrical centres of the adsorbed ions and outer-Helmholtz plane for the locus of the electrical centres for the hydrated ions in contact with the charged surface. This is illustrated in Figure 1.8. A number of more sophisticated models on the same theme as Grahame s model have been suggested (see ref. (28),... 

The inner Helmholtz plane may be treated in the same way as in the Stern theory. The number of anions present per cm in that layer is... 

In order to arrive at a further and more quantitative interpretation of charge and adsorption of ions it is necessary to consider a rather detailed model of the double layer. It has already been mentioned that the simple Gouy picture leads to inconsistencies and that it is necessary to take account of the finite dimensions of the ions by the Stern theory or a modification of it. 

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