Velocity decay length

The experimental dependence of the quantity AF/f p on the velocity decay length exhibits a sharp increase at low values of S, followed by a gentle growth at large values of S. This effect becomes more pronounced with increasing roughness (open circles). 

Fig. 4 Dependence a of the parameter AF/pf and b Afjpp on the velocity decay length in different liquids, for an ideally smooth surface (lines 1), and experimental data for two real surfaces vacuum-sputtered gold (closed circles) and electrochemically deposited gold (open circles). Lines 2 and 3 represent results of parameter fitting, see text. (From [27])...

Fig. 5 Dependence of the parameter AF/pf on the velocity decay length points experimental data, line 1 in both plots indicates an ideally smooth surface, a Influence of strong roughness according to Eq. 44 for different values of -2 69, 3 172, 4 276 nm) and L = 506 nm. b The same for different values of L 2 460, 3 506, 4 690 nm and fn = 172 nm. Line 5 in both plots was calculated for slight roughness (roughness factor R = 1.3, Eqs. 34 and 39). (From [27])...

Here we should emphasize only one point, of major importance for electrochemical use of the QCM. The velocity decay length of most solvents of interest for electrochemical and analytical purposes happen to be at the lower end of the values of S shown in Figs. 4 and 5. This is the region where the interplay between the two types of roughness is the strongest, and it is the most difficult to fit the data to either model. This inherent difficulty should be borne in mind whenever an attempt is made to interpret the impedance response of the QCM operating in typical solvents such as water, alcohols, or many of the other non-aqueous solvents employed in electrochemistry. 

The properties (the effective viscosity and density) of the liquid layer in close vicinity to the interface can differ from their bulk values. There are various reasons for these phenomena. For example, the properties of a thin liquid layer confined between solid walls are determined by interactions with the solid walls [58,59]. In electrochemical system the structuring of a solvent induced by the substrate and a nonuniform ion distribution in the diffuse double layer can significantly influence the properties of the solution at the interface. The nonuniform distribution of species, which influences the properties of the liquid near the electrode, also occurs in the case of diffusion kinetics. The latter was considered in Ref. 60, where the ferro/ferri redox system was studied by the EQCM. This was the case where the velocity decay length ( >25 pm) was much less than the thickness of the diffusion layer ( >100 pm), in which the composition of the solution is different from the bulk composition. 

Equations (26) and (27) show that the influence of the slippage on the response of the QCM in liquid is determined by the ratio of the slip length A to the velocity decay length, 6. Even for a small value of A 1 nm, the slippage-induced correction to the frequency shift, A/ /, will be of the order of 6.5 Hz for the fundamental frequency of o = 5 MHz. This value far exceeds the resolution of the QCM, but it is difficult to separate it from the overall QCM signal. 

In order to describe both the effect of the electrostatic adsorption of ions and the effect of the viscosity inside the diffuse double layer on the response of the EQCM, one can use the thin-layer model described in Sec. II.C. Since the thickness of the diffuse double layer is much less than the velocity decay length, the corresponding equation of the model is Eqs. (23) and (24), which can be rewritten in the following form ... 

FIG. 19. Dependence of the parameter Tjpf on the velocity decay length Points—experimental data. Lines la and lb, ideally smooth surface 2a-4a, influence of strong roughness according to Eq. (49) for different values of (2a—... 

The response of the EQCM on rough surfaces cannot be treated in terms of the electrochemically defined roughness factor R, which is obtained from adsorption phenomena, e.g., from data such as presented in Fig. 22. This quantity can be considered as representing all adsorption sites on the surface, which is equivalent to the surface roughness on the atomic scale. However, the response of the EQCM depends on roughness on a mesoscopic scale, which is comparable to the hydrodynamic velocity decay length rather than to the double layer thickness. The width of the resonance is an important characteristic of the surface, as seen in Fig. 23b, and can serve as a semi-quantitative measure of its roughness on the scale relevant to the response of the EQCM. Unfortunately, only very few publications so far contain this information. 

---