Rough surfaces frequency shift

It should be noted that for l/S 2> 1 the roughness-induced frequency shift includes a term that does not depend on the viscosity of the liquid, the first term in Eqs. 37 and 33. It reflects the effect of the non-imiform pressure distribution, which is developed in the liquid under the influence of a rough oscillating surface [80]. The corresponding contribution has the form of the Sauerbrey equation. This effect does not exist for smooth interfaces. The second term in Eq. 37 and Eq. 39 describe a viscous contribution to the QCM response. Their contribution to A/ has the form of the QCM response at a smooth liquid-solid interface, but includes an additional factor R that is a roughness factor of the surface. The latter is a consequence of the fact that for l/S 1 the liquid sees the interface as being locally fiat, but with R times its apparent surface area. 

Results obtained in [80,82] show that the influence of slight surface roughness on the frequency shift cannot be explained in terms of the mass of liquid trapped by surface cavities, as proposed in [76,77]. This statement can be illustrated by consideration of the sinusoidal roughness profile. The mass of the liquid trapped by sinusoidal grooves does not depend on the slope of the roughness, h/l, and is equal to S h, where S is the area of the crystal. However, Eq. 33 demonstrates that the roughness-induced frequency shift does increases with increasing slope. 

Fig. 10.18. Effects of surface roughness on EHD impedance (amplitude ratio, H(p)IH(p- 0), and phase lag, 9, against scaled frequency, p and comparison with the behaviour of a uniform disc—asymptotic line marked (a) —and an array of UMEs— asymptotic line marked (b). The frequency shift is deduced from the displacement between the two sections of the phase angle diagram where the data superimpose for different n . The modulation frequency, to2, at which the data deviate from that of a uniform electrode, is related to the amplitude of the surface roughness or the spacing between the elements of the UME array. Data from Reference [121], for Fe(CN)i reduction on smooth Pt at 120 rpm 4 240 rpm, and on a rough, Pt-coated silver electrode (roughness scale 5 (im, disc diameter 6 mm) at O 120 rpm + 240 rpm A 500 rpm and x 1000 rpm.

Figure 5 shows that there is no way to fit the experimental data assuming that only one type of roughness is presented on the surface. We are thus forced to conclude that, in these experiments the surface has a multiscale roughness, shown schematically in Fig. 6. The structure of this rough surface is a combination of a slight and a strong roughness shown in Fig. 3a,b. When this is taken into account, it is possible to use Eqs. 33, 34,43, and 44 to calculate the shift in resonance frequency and shift in the width of the resonance, and fit the experiments to the calculated curves with properly chosen values of the parameters of strong roughness. The result of such a fit is shown in Fig. 4, curves 2 and 3. For details of the fitting procedure, the limitations associated with the use of a simplified model, and the comparison with STM data see [27].

Figure 4.17 Shift and change of the resonance frequency of a quartz crystal microbalance, real part of the admittance versus frequency, /q, Wq, resonance frequency and full width at half maximum (FWHM) of the initial gold electrode,/j, w, resonance frequency and FWHM of a gold electrode after formation of a rigid and smooth surface film (no damping), resonance frequency and FWHM of a gold electrode after formation of a viscoelestic and/or rough surface film (strong damping).

The Sauerbrey constant Cjb for a 10 MHz AT-quartz is 226.01 Hz cm pg (AT describes the type of cut, which is like a slice from a quartz crystal (Section 4.2.3)). It was mentioned that the application of the simple Sauerbrey equation is limited to rigid films. Polymer films can have viscoelastic properties or can develop a high surface roughness. In these cases the mechanical impedance contains terms that take into account viscoelasticity and surface roughness, and the resonance frequency shift becomes a complex quantity. 

Another aspect of the admittance spectrum is shown in Fig. Ic. Here the same metal deposition was conducted as in Fig. la, but the conditions were purposely chosen to produce a very rough surface (by plating at a current density close to the mass-transport limited value). The width of the resonance is increased and the frequency is shifted to lower values with increasing roughness. 

Comparison of Fig. 21c,d for a highly rough surface (line 3) shows that a decrease in width is associated with a positive shift in resonance frequency in the region of surface oxide formation. This is consistent with the notion that both effects result from a weakening of the interaction between the vibrating surface and the liquid under surface oxidation. 

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