Density of Kink Site Positions

The residence times presented in Table 7.1 are relative values, but they provide a feeling for the large differences in the surface dynamics of these metals under deposition conditions. While the differences in the bond energies are inconspicuous, the differences in the residence times cover several orders of magnitude. 

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In the following treatment, the density of kink site positions is considered to be in equihbrium with the ad-atom concentration and with the ion concentration in the electrolyte. This is shown in Figure 7.11. 

DENSITY OF KINK SITE POSITIONS 7.4.1 Equilibrium conditions... 

In a first approximation it can be assumed that the density of kink site positions is in equilibrium with the ad-atom concentration. More generally, all partial reactions are in equilibrium at the Nemst potential. The following equation describes equilibrium between ions in the electrolyte and atoms in kink site positions ... 

The density of kink site positions A and AA at equilibrium potential is [ksp]Q. For equilibrium conditions one gets, Thus it follows with Eq. (7.25) ... 

With Avogadro s number and F the Faraday constant. The density of kink site positions at the equilibrium potential [ksp] is given by the product of exchange current density and residence time. This equation will be discussed in the next section. 

Dividing Eq. (7.35) through the density of kink site positions [A p], the number of atoms arriving during the residence time per kink site position is obtained... 

If is much smaller than [ksp], the number of kink site positions decrease, because the renewal of the kink site positions by accumulation of metal ions is so slow that more and more kink site positions disappear because of the natural surface reconstruction. If is much larger than [ksp], however, the number of kink positions will increase because new surface structures can develop. Thus one can say for stationary conditions that N,should approach the value = 1. The density of kink site positions [fop] finally therefore approach the value... 

A special situation that can cause stationary conditions is at the electrochemical equilibrium potential (Nemst potential). The current density at the Nemst potential is the exchange current density The density of kink site positions [fopJo is therefore proportional to the product of exchange current density Iq and residence time, as was already shown in Eq. (7.34). 

Exchange current densities have been determined for many metals. Using ig values from the literature and the values for the residence times presented in Table 7.1 a rough estimation of the order of magnitude of the density of kink site positions at the Nemst potential is possible. Examples are presented in Table 7.2. 

The calculation of [fop], takes into account the nature of the metal (by r) and the electrochemical conditions (by if). The value of ig reflects the different experimental parameters like electrolyte composition, and additives, etc. Therefore, the values of the density of kink site positions provide an image of the surface dynamic at equilibrium conditions. While the absolute values might be questionable, the comparative nature of the described procedures allows at least a comparison of the dynamics of experimental systems. The values for Ag reflect the density of kink site positions for three experimental conditions. The largest value of 5 X lO cm corresponds to one of the highest experimental exchange current densities observed. In this case, [fopJo approaches the surface density of atoms on... 

Density of kink site positions at the Nemst potential calculated with Eq. (7.34) for some metals, residence times taken from Table 7.1, and exchange current densities from the literature ... 

With increasing step density the softness increases. Furthermore, the softness increases with the surface density of kink site positions, or, quite generally, with increasing surface roughness. On polycrystalline surfaces, the same rules apply and the softness increases with an increasing amount of surface roughness. 

Substituting the surface densities of kink site positions by equilibrium constants and ion concentrations in the electrolyte bath one obtains... 

The part of the potential drop across the double layer acting on the transition state of separation is described by a p. If the density of kink site positions in the first approximation is independent of the potential, the Tafel equation would be... 

Surface density of kink site positions Frequency shift Complex frequency shift Gibbs energy of alloy A j ... 

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