Charge density: diffuse layer metal

Unfortunately, the value of the diffuse-layer potential ( 2) which is needed in the corrected equation, is not readily available. However, the diffuse-layer potential is related to the potential of zero charge (pzc), a quantity more readily available in the literature.The 2 potential is related to the charge density on the metal surface through a relation given by double-layer theory ""... 

The density of the diffuse layer charge,, depends on the concentration of the ionized groups on the surface of the metal oxide ... 

Gouy length — The width of the diffuse double layer at an electrode depends on a number of factors among which the charge density q on the surface of the metal and the concentration c of electrolyte in the solution are paramount. Roughly speaking, the charge in, and the potential of, the double layer falls off exponentially as one proceeds into the solution from the interface. 

Here, as = -Vrs is the charge density on the -S groups of a hypothetical close-packed monolayer of thiolate anions, P is the distance of the center of charge of the sulfur atoms from the metal surface, d = p + y is the length of the thiol molecule, and p and y are the dielectric constants accounting for the distortional polarization in the (0 < x< P) and (psurface dipole potential due to electron spillover and Xs is that due to any polar groups of the thiol molecule. For simplicity, the small potential difference (4/ across the diffuse layer is disregarded. 

Figure 11. Plots of log Z and / v.v. log / for a thiol-hexapeptide-coated mercury drop immersed in 5xlO 3M (a), I.3xlO 2M (b), 3.6xlO 2M (c), and 0.1M (d) KC1, as obtained at -1.000 V over the frequency range from 0.1 to 105 Hz. At frequencies <102 Hz all Bode plots coincide hence, only the experimental points for the lower KC1 concentration were reported. The solid curves are least-squares fits to the simple equivalent circuit of inset (1), which consists of the electrolyte resistance Ra, with in series a RSCS mesh representing the self-assembled monolayer and a further RjiCji mesh representing the diffuse layer. Rs = 0.14 Mfi cm2 C, = 11 pF cm-2 Ra = 4.53 (a), 4.17 (b), 1.27 (c) and 0.87 KO cm2 (d). CW 68 (a), 61 (b), 80 (c) and 84 pF cm 2 (d). Inset (2) shows the reciprocal, 1/Cji, of the experimental diffuse-layer capacitance vs. the l/C fajj = 0) value corresponding to the same KC1 concentration, as calculated on the basis of the Gouy-Chapman (GC) theory. The solid curves are 1 /Ca(OM) vs 1 /C,ii(ctm = 0) plots calculated from the GC theory for different charge densities afo on the metal, whose values are reported on each curve. (Reprinted from Ref.114 with permission from the Am. Chem. Soc.)...

The profile of an ideally smooth interface is sketched in Fig. 13.Thehalf-spacez < 0 is occupied by the ionic skeleton of the metal. This can be described, roughly, in a jellium model, as a continuum of positive charge n+ and the effective dielectric constant due to the polarizability of the bound electrons (this quantity is, with rare exceptions (Hg Sh = 2, Ag 5 = 3.5), typically close to 1 [125]). The gap 0 < z < a accounts for a nonzero distance of the closest approach of solvent molecules to the skeleton. The region of a < z < a + d stands for the first layer of solvent molecules, while z > a + d is the diffuse-layer region. n(z) denotes the profile of the density of free electrons. This is, of course, an extremely crude picture, but it eventually helps to rationalize the results of the various theoretical models and simulations. 

Fig. 13 A cartoon of a profile of a smooth electrochemical interface. The half-space z < 0 is occupied by the metal ionic skeleton that, within the jellium model, is described as a continuum of positive charge density (n+) and the dielectric constant due to bound electrons (ei,), the value of which lies typically between 1 and 2. The gap accounts for a finite distance of closest approach of solvent molecules to the skeleton the gap is determined by the balance offerees that attract the molecules to the metal and the Pauli repulsion of the closed shells of the molecules from the free electron cloud of the metal of density n(z). The regions a < z < a + d and z> a + d correspond, respectively, to the first layer of solvent molecules (which can be roughly characterized by charge-dependent effective dielectric constant) and the diffuse-layer part.

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