Gouy Layer in the Electrolyte

As already discussed in Section 5.1, at lower ion concentrations the charges near the interface are distributed over a diffuse layer. In this case, the charge balance 

This is a differential capacity because it depends on the potential A(pQ. The total capacity for a metal-liquid interface can be considered as two capacitors, namely 

This has been verified experimentally examples are given in [4, 12]. Evaluations of corresponding data have shown that Q for ion concentrations of c  

The potential and charge distribution within the space charge region is quantitatively described by the Poisson equation as given by 

Since the Fermi level is expected to be constant within the space charge region, the position of the energy bands (, (a ) and (a ) vary with distance. Denoting the carrier density in the bulk of the semiconductor by Wg and Pq one obtains 

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It should be mentioned here, that the capacity of the space charge layer in an intrinsic semiconductor looks very similar to that of the diffuse Gouy layer in the electrolyte (compare with Eq. 5.8). This is very reasonable because the Gouy layer is also a kind of space charge layer with ions instead of electrons as mobile carriers. Q was actually derived by the same procedure as given here for Csc- Similarly as in the case of Cn and C(j, the space charge capacity Cjc and the Helmholtz capacity Ch can be treated as capacitors circuited in scries. We have then... 

Beyond the IHP is a layer of charge bound at the surface by electrostatic forces only. This layer is known as the diffuse layer, or the Gouy-Chapman layer. The innermost plane of the diffuse layer is known as the outer Helmholtz plane (OHP). The relationship between the charge in the diffuse layer, o2, the electrolyte concentration in the bulk of solution, c, and potential at the OHP, 2> can be found from solving the Poisson-Boltzmann equation with appropriate boundary conditions (for 1 1 electrolytes (13))... 

We shall use the familiar Gouy-Chapman model (3 ) to describe the behaviour of the diffuse double lpyer. According to this model the application of a potential iji at a planar solid/electrolyte interface will cause an accumulation of counter-ions and a depletion of co-ions in the electrolyte near the interface. The disposition of diffuse double layer implies that if the surface potential of the planar interface at a 1 1 electrolyte is t ) then its surface charge density will be given by ( 3)... 

The electrode roughness factor can be determined by using the capacitance measurements and one of the models of the double layer. In the absence of specific adsorption of ions, the inner layer capacitance is independent of the electrolyte concentration, in contrast to the capacitance of the diffuse layer Q, which is concentration dependent. The real surface area can be obtained by measuring the total capacitance C and plotting C against Cj, calculated at pzc from the Gouy-Chapman theory for different electrolyte concentrations. Such plots, called Parsons-Zobel plots, were found to be linear at several charges of the mercury electrode. ... 

Helmholtz [71] first described the interfacial behavior of a metal and electrolyte as a capacitor, or so-called electrical double layer, with the excess surface charge on the metallic electrode remaining separated from the ionic counter charge in the electrolyte by the thickness of the solvation shell. Gouy and Chapmen subsequently... 

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. 

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. 

Interfacial tension against electrode potential curves have a parabolic shape with a maximum value which depends on the nature and concentration of the electrolyte (see fig. 10.1). Detailed results for the mercury aqueous solution interface were initially reported by Gouy [7, G5]. Examination of these data for the alkali metal halides shows that the interfacial tension depends markedly on the nature of the electrolyte at positive potentials. On the other hand, the variation with electrolyte at negative potentials is rather small. It follows that the anions in the electrolyte strongly affect the interfacial tension when they predominate in the double layer. 

The semiconductor occupies the region to the right of the vertical solid line representing the interface (jc = 0). To the left of it, there is the Helmholtz layer formed by ions attracted to the electrode surface, and also by solvent molecules its thickness, L, is the order of the size of an ion. The space-charge region in the solution (the Gouy-Chapman layer) is adjacent to the Helmholtz layer from the electrolyte side. 

Figure 13.3.3 Potential profiles through the diffuse layer in the Gouy-Chapman model. Calculated for a 10 M aqueous solution of a 1 1 electrolyte at 25°C. 1/k = 30.4 A. See equations 13.3.14 to 13.3.16.

We have hitherto reasoned as if the potential were constant during all changes in the electrolyte concentration. This would, b.e right if we could describe the double layer completely by the theory of Gouy-Chapman. But as we have seen ( 5 of. Chapter VII), especially for larger concentrations some correction has to be made in respect of the dimensions and the specific adsorption of the-iOns, and a possible form of this cor-... 

As mentioned above, the potential difference occurs across the space charge layer of the semiconductor. This is only true provided that the ion concentration in the electrolyte is sufficiently large (Gouy layer neglected). 

The potential difference at the semiconductor-electrolyte interface is the sum of three terms Adi rising from the space-charge layer in the semiconductor A( h from the Helmholtz layer and A0qc from the Gouy-Chapman layer (Figure 3.55). 

The space charge capacity of an intrinsic s i-conductor behaves like the diffuse dovible layer (Gouy) capacity in dilute electrolytes, namely ... 

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