Liquid-induced frequency shift

Here d and pf are the thickness and the density of the film. These equations are valid in a particular case, when d < S. The general case for arbitrary df was given in [44]. The first terms in Eqs. 16 and 17 yield the liquid-induced frequency shift and half-width of the resonance in the absence of a film. The terms in brackets describe the influence of fhe viscosity and density of a film of thickness df. According to Eqs. 16 and 17, the ratio of the film-induced halfwidth to the film-induced frequency shift is proportional to d(/S. Thus, for d /S < 1, the contribution of the thin interfacial film to the width is much smaller than its contribution to the frequency shift. For the film acts... 

The liquid-induced frequency shift and the half-width of the resonance have the following form [83]... 

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. 

It should be noted that for 1/5 1 the roughness-induced frequency shift includes a term that does not depend on the viscosity of the liquid. 

Nematic liquid crystals (LCs) are a classical example of complex fluids. If we trust the small-load approximation as well as the matured theory of nema-todynamics [76], we must be able to predict the frequency shift induced by nematic LCs. The theory of nematic LCs in contact with the QCM has been worked out in detail by people who did not know about the QCM as a tool to probe these phenomena. These authors performed ultrasonic reflectometry. As we know from Sect. 5, the results of these studies can be transported to the QCM in a straightforward way by just using Eq. 39. 

Even though there is a big shift in the resonance frequency, the mass sensitivity to thin film is observed to be the same in a fluid as in a vacuum. In addition to the viscous motion, many other factors still affect the resonance frequency of QCM in a high-density fluid. Roughness of the surface is one important factor. Based on the measurement, resonance frequencies turned out to be dependent on the surface roughness [14,15]. The roughness seems to induce a shift of the resonant frequency by both the inertial contribution from the liquid mass rigidly coupled to the surface, and the viscous contribution from the viscous energy dissipation caused by the nonlaminar motion in the liquid [16]. 

The variables A/, /o, Aw, Apie o. and Fq represent, respectively, the crystal s measured frequency shift, initial resonant frequency, mass change, surface area, shear modulus, and density [6]. TSM resonators immersed in liquid present a special case because acoustic energy dissipation through the fluid lowers Q as well. Additional formulas need to be applied to account for the liquid s viscosity and loading [6]. Figure 2 shows a comparison between shifts induced by solid and hquid loading on resonators. 

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. 

Figure 8.4 Frequency shift (A/, top) and shift of bandwidth (AF, bottom, both normalized to overtone order) observed upon producing small bubbles by electrochemical means. After the potential is set to —1 V vs. SCE (t = 10 min, vertical arrow), a thin layer of hydrogen evolves at the gold surface. The layer is visually observable via a slight coloration. Individual bubbles cannot be discerned by the naked eye. The voltage is turned off after 0.5 s, and the bubbles subsequently dissolve. Importantly, A/ is negative, as expected for the deposition of solid objects. A laterally continuous slipping layer would have induced A/ >0. The same would have happened with macroscopic bubbles. Bubble formation here is accompanied by in increase in bandwidth, AF, which occurs because the bubbles deviate the liquid flow. Abbreviation SCE, standard calomel electrode.

A typical sequence of dipping events in water is displayed in Fig. 9.14. The liquid is approached with a step motor toward the nanoneedle. When the tip is far from the surface, that is, more than a few nanometers, there is no interaction between the needle and the air-liquid interface and thus no frequency shift (phase 1). As soon as the tip touches the liquid, the elastic response of the meniscus induces a positive shift of the resonance frequency (phase 2). Because of the water evaporation, the average contact angle decreases with time leading to a decrease of the frequency shift (phase 3). The phase 4 is identical to phase 1 with no significant interaction between the tip and the liquid. 

Just as in our abbreviated descriptions of the lattice and cell models, we shall not be concerned with details of the approximations required to evaluate the partition function for the cluster model, nor with ways in which the model might be improved. It is sufficient to remark that with the use of two adjustable parameters (related to the frequency of librational motion of a cluster and to the shifts of the free cluster vibrational frequencies induced by the environment) Scheraga and co-workers can fit the thermodynamic functions of the liquid rather well (see Figs. 21-24). Note that the free energy is fit best, and the heat capacity worst (recall the similar difficulty in the WR results). Of more interest to us, the cluster model predicts there are very few monomeric molecules at any temperature in the normal liquid range, that the mole fraction of hydrogen bonds decreases only slowly with temperature, from 0.47 at 273 K to 0.43 at 373 K, and that the low... 

A substantial dependence of the induced absorption on density has been seen in liquid nitrogen when high pressure was applied to the liquid [252], Furthermore, with increasing density, a shift of the peak absorption to higher frequencies is observed while the low-frequency profile is not much affected by increasing density, significantly increased absorption is... 

Ultrasonic Pressure Transducers. Advantage is taken of the fact that pressure influences sound propagation in solids, liquids, and gases, but in different ways. In solids, applied pressure leads to so-called stress-induced anisotropy, In liquids, the effects of pressure are usually small (relative to effects in gases), but the frequency of relaxation peaks can be shifted significantly,... 

In nematic liquid crystals, the viscosity depends on the relative orientation between the shear gradient and the orientation of the nematic phase. Close to a surface, the orientation is usually governed by surface orientational anchoring [77]. Anchoring transitions, for instance induced by the adsorption of an analyte molecule to the surface [78], can therefore be easily detected with the QCM [79,80]. This reorientation induced by adsorption amounts to an amplification scheme the expected shift in the resonance frequency and bandwidth... 

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