The Roughness Factor

Practicing electrochemists who, for years, have worked with dropping or hanging mercury drops never had to face this problem, because the roughness factor for the very smooth surface of the mercury is unity. On the contrary, for solid electrodes, the surface is never perfectly smooth and a roughness factor has to be introduced in calculations. 

It was proposed, for determination of the roughness factor, that the value of the capacity at very negative density of charge (where there is no longer adsorption of anions) be compared to that of mercury. This does not take into account that at such negative density of charge there are metal-solvent interactions and these are not only metal specific but also face specific. 

Another way of determining the roughness factor was proposed by Valette and Hamelin when there is no adsorption. For this case. 

The plot of C (o-) (a charge density) has to be made for several supposed values (for instance, 1.00, 1.05, 1.10, etc.) of R. For unity (obviously a too-small value) the diffuse-layer part is compensated too much and gives rise to a sharp maximum on the C ((r) curve 

Most of the time the experimenter has to use these two last methods to have an idea of the situation. 

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Consider the roughness factor, r (Eq. 19), for such a fractal surface ... 

Obviously the roughness factor is similarly arbitrary, but it is of interest to use Eq. 25 to compute its value for some trial values of D and a. This is done in Table 2. In order to map the surface features even crudely, the probe needs to be small. It can be seen that high apparent roughness factors are readily obtained once the fractal dimension exceeds 2, its value for an ideal plane. 

As long as the liquid actually wets the rough surface, a less contentious approach linking the roughness factor to the extent of contact would seem to be via the spreading coefficient as shown in Eq. 20 and summarised in Table 1. If air is trapped within pits by the liquid, a composite surface is produced. 

This conclusion was additionally confirmed by Palczewska and Janko (67) in separate experiments, where under the same conditions nickel-copper alloy films rich in nickel (and nickel films as well) were transformed into their respective hydride phases, which were proved by X-ray diffraction. The additional argument in favor of the transformation of the metal film into hydride in the side-arm of the Smith-Linnett apparatus consists of the observed increase of the roughness factor ( 70%) of the film and the decrease of its crystallite size ( 30%) after coming back from low to high temperatures for desorbing hydrogen. The effect is quite similar to that observed by Scholten and Konvalinka (9) for their palladium catalyst samples undergoing the (a — j8) -phase transformation. 

Leikis et al,223 used the Parsons-Zobel method to obtain the roughness factor fpz for pc/Ag electrodes. It was found that /pz 1.2, which was explained by the geometric inhomogeneity of the pc-Ag electrode surface. A more detailed analysis is given in Section II.2. Thus it should be noted that in the case of pc electrodes with appreciable differences of EamQvalues for the various planes (AEff o > 100 mV), it is impossible to obtain the true roughness coefficient, the actual Ea=0, and the inner-layer capacity. 

Data for the pc-Au/DMF + LiC104 interface have been collected by Borkowska and Jarzabek.109 The value of ffa0was found to be 0.27 V (SCE in H20) and the roughness factor / = 1.3 (Table 8). Unlike Hg, Bi, In(Ga), and Tl(Ga) electrodes and similarly to the Ga/DMF interface, the inner-layer capacity for pc-Au in DMF depends weakly on a, and thus the effect of solvent dipole reorientation at pc-Au is less pronounced than at In(Ga), Bi, and other interfaces. 

The R of electropolished Zn single-crystal face electrodes has been obtained from the shape of the adsorption-desorption peak of cyclohex-anol at various Zn and Hg surfaces.154 The roughness factor of Zn electrodes has been found to increase in the order Zn(0001) < Zn(lOlO) < Zn(llZO) with values in the range 1.1 to 1.25. 

Such an approach revealed objective limitations as it became evident that the equality in the capacitance values for different metals was only a first approximation. The case of Ga is representative. Ga is a liquid metal and the value of capacitance cannot depend on the exact determination of the surface area as for solid metals (i.e., the roughness factor is unambiguously =1). 

Such information can be obtained from cyclic voltammetric measme-ments. It is possible to determine the quantity of electricity involved in the adsorption of hydrogen, or for the electrooxidation of previously adsorbed CO, and then to estimate the real surface area and the roughness factor (y) of a R-C electrode. From the real surface area and the R loading, it is possible to estimate the specific surface area, S (in m g ), as follows ... 

Electrodes of 2 x 2 cm2 geometric area were used in the experiments. In the case of platinum, a pretreatment by fast triangular potential scans (200-300 V/s between 0.05 V and 1.5 V RHE) followed by heating up to 900 K in a 3 x 10 6 mbar 02 atmosphere was carried out. Electrodes, pre-treated in this way, can be emersed with a thin liquid film, which can easily be evaporated in the vacuum chamber. The heat treatment drastically reduces contamination of the platinum electrode by carbon. The roughness factor is usually in the order of three. 

Figure 2-41 is a cyclic voltanunogram of an electrochemically platiniaed platinum electrode in 3 M sulfuric add. It is almost the same as that on a smooth platinum electrode. The roughness factor was calculated as ... 

Chemically platinized platinum with the roughness factor ISO was used for the same experiment. The resTilt is shown in Fig. 2-43. The oxidation cmrent for eoCOad forms a small peak rather than just a shoulder. Since the roughness factors are almost same for both chemically and electrochemically platinized platinum electrodes, the difference probably stems from the morphology of the platinum. 

The lack of knowledge of precise values of the roughness factor makes it difficult to compare data reported from different studies. This applies in particular to the double-layer capacity data, the values of surface concentration of the adsorbates, and the rates of electrochemical reactions. Therefore, the question of how to determine the real surface of the electrode is of cmcial importance. A survey of various methods for determining roughness was given by Trasatti and Petrii. For noble metal electrodes, the charges of hydrogen deposition and surface oxide formation can be utilized in real-surface determination." ... 

In terms of the pellet size, dp, the ratio Vp/Sx is dp/6 for spheres and cubes Thus, for most practical catalyst pellets, the average pore length is 2 dp/6 according to Wheeler s model. The roughness factor rappearing in eqn. (27) is assumed to be 2 for practical purposes. 

The Wheeler model thus provides an average pore radius, r and pore length, L, in terms of the experimentally determinable parameters v, and fj. The only adjustable parameter is the roughness factor, r. The usefulness of this model was demonstrated by Wheeler, who successfully predicted the reaction rates of several catalytic reactions of industrial importance. 

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