The Molecular Condenser

4 MODELS FOR THE ELECTRICAL DOUBLE LAYER 9.4.1 The Molecular Condenser 

The functionality /(x) is obtained by integrating Equation 9.15 from x = 0 to x = d, which gives 

FIG U RE 9.6 Charge distribution and potential profile in a molecular condenser. 

Double layers resembling a molecular condenser are seldom met in practice, but as a limiting case they deserve attention. 

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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 molecular condenser is the simplest imaginable double layer. It is depicted in Figure 9.6. The surface solution boundary is set at x = 0, containing all the surface charge. All counterions are at x = d. In reality, such a situation could arise if the electrostatic attraction of these ions to the surface were so strong that they would rule out thermal motion and the counterions approached the surface at the shortest distance possible. That distance is determined by the radius of the ions assuming them to be either hydrated or not, that is, in Figure 9.6, d would equal the sum of the radii of surface and counterion, with or without hydration water in between. 

Figure 9.7 gives a representation of the diffuse double layer. This model is also known as the Gouy-Chapman layer (named after the persons who first developed the model). The underlying picture is that the surface is located at jc = 0 and that the counterions are not only attracted by the surface but are also subject to thermal motion. The former force tends to accumulate all counterions at the distance of closest approach to the surface (as in the molecular condenser), whereas the latter tries to spread all counterions homogeneously in the solution. The co-ions are subjected to similar counteracting tendencies. See Figure 9.1. The resulting countercharge distribution is given by the Boltzmann equation ...

The diffuse double layer model of Gouy and Chapman works reasonably well for systems of relatively low surface potential (electrolyte concentration (< 10 M). At higher surface potential and ionic strength the outer part of the double layer may still obey this model, but the inner part close to the surface tends toward the molecular condenser. Therefore, these two pictures are integrated in the Gouy-Chapman-Stern model. 

The result of Equation 10.27 is exact, and it indicates the stability of molecular BEC with respect to collapse. Compared to earlier studies, which assumed Add = 2a [37,59], Equation 10.27 gives almost twice as small a sound velocity of the molecular condensate and a rate of elastic collisions smaller by an order of magnitude. The result of Equation 10.27 has been confirmed by Monte Carlo calculations [60] and by calculations within the diagrammatic approach [61,62]. An approximate diagrammatic approach leading to Add = 0.75a has been developed in Ref. [59]. 

Next, we calculate the initial rate per unit volume at which solute molecules precipitate by condensation (i.e., the molecular condensation rate ncnd,ini), which is obtained by multiplying the initial condensation rate /end [see Eq. (65b)] by the number density of particles, N2sm, that have been generated in the first step (i.e., nucleation) of the process (2SM refers to two-step model) ... 

The total potential drop is divided into a potential over the diffuse part of the double layer and ver the molecular condenser For the diffuse part of... 

The capacity of the total double layer Cg is then simply found as the equivalent of the capacity of the diffuse layer Ca and that of the molecular condenser Cm placed in series... 

When the molecular condenser is empty (o — 0) then indeed the maximum potential of the Gouv-layer should be But when the SxBRN layer is partly or fully occupied, the distance of closest approach, averaged over the whole surface, of a Grouy-ion tp the surface, is larger than and consequently the maximum Gouv potential should be smaller than. ... 

As regards the drawbacks of the pure GouY-CKAPMAN-theory, Stern has indeed indicated the way to overcome them. The non-specificity disappears by the adsorption potentials and by the capacity of the molecular condenser. The impossibly high values of the c pzLcity in Gouy s theory cannot occur in Stern s, as the highest capacity possible here is that of the molecular condenser. 

In the original T ersion of Stern s theory the capacity of the molecular condenser is taken as a constant. In principle of course this capadty will depend upon tfie spedfit properties of the ions in the STERN-iayer. Add especially will be different 021 both sides of the ero point of charge. 

Fig. 7, Differential ca acity of the double layer according to Stern s theory. Capacity of the molecular condenser 3B [iF for positive surface and 20 pF for negative surface. (Taken from A. Frumkin, Trans. Faraday Soc., 36 (1940) 126).

Fig. 8. Differential capacity of double layer calculated by Grahame. The curve for IM NaF from Fig. 10 has served as a basis for calculating the capacity of the molecular condenser.

Fig, 8 is an example of Grahame s work. Chemisorption is assumed to be absent, the inner Helmholtz plane is empty. The capacity of the molecular condenser (outer Helmholtz plane) is taken from the experiment with the highest concentration of electrolyte (i mol/1 NaF) and is used to calculate the curves for lower concentrations. Fig, 9 shows experimental values for the mercury electrode in chloride solutions. Fig, 10 gives Grahame s experiments for NaF, to be compared with Fig, 8,... 

E. L. Mackor, Rec trav chim 70 (1951) 763, showed that instead of assuming a variable capacity Cm of the molecular condenser, an equivalent interpretation can be obtained by assuming a constant capacity and a variable > -potential which depends upon the surface charge. 

DeBruyn also determined adsorption isotherms for different salt contents, iiis results differ frem those of Van Laar and Mackor in that for high electrolyte concentrations > 0.1 TV for monovalent ions, > 0.001 N for bi- and poly-valent ions) he always found the same practically linear adsorption isotherm which would point to a constant capacity of the molecular condenser. It is not certain, however, that the complications of the sol concentration effect ( 10, p. 184) have been sufficiently avoided in his experiments. 

T he simple theory of the diffuse double layer presents no starting point for the treatment of specific influences of the ions, these being treated as point-charges. Only after the introduction of a refinement like Stern s theory (Ch. IV, 4d, p. 132 and Ch. VI, 8, p. 263) can specific effects be explained. In this theory a specific adsorption potential is introduced and the dimensions of the ions are accounted for by the thickness of the molecular condenser (or by its finite capacity). 

The potential depends — other things being equal — on the thickness of the molecular condenser and indeed is larger, the thinner the molecular condenser is. The thickness of the molecular condenser will be determined by the magnitude of the counter-ions. So we may expect that the larger counter-ions will give rise to a lower and consequently to lower stability and lower flocculation values. 

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