Acoustic load impedance

Electrical admittance Electrical impedance Acoustic load (impedance)... 

The acoustic load impedance, Zl, of a film measured at frequency ao- 2nf, is ... 

For example, Ft(s) could be pressure in an acoustic tube and 1 (s) the corresponding volume velocity. In the parallel case, the junction reduces to the unloaded case when the load impedance Rj(s) goes to infinity. 

To determine Wt one could also measure W2 and adopt a known power transformation ratio. This ratio of course depends on the acoustic load and has to be determined in each and every particular case. Again, such a method does not take into account any possible aging of the system, and calibration should be made from time to time. Furthermore, it will be of little use in chemistry as the acoustic impedance of the load will almost certainly change as the reaction proceeds. 

Equation 2 can be rewritten in a way that Z x can be presented as a parallel arrangement of Co, the only genuine electrical parameter in Eq. 2 (formed by the two electrodes with quartz as dielectric), and a so-called motional impedance, Z Z = Co Z (Pig. 4a). Zm contains two elements in series. The first summand, Z q, includes only crystal parameters and describes the motional impedance of the quartz crystal as a fimction of frequency co = litf. The second summand expresses the transformation of the acoustic load, Zl, into the (electrical) motional load impedance, Z il- We therefore call the fraction in front of Zl transformation factor. Applying some assumptions reasonable in most sensor applications Z becomes ... 

Consequently, the second term in Eq. 3 is the important one for sensor applications. Obviously a small coupling factor and a small wave velocity increase the electrical representation of the acoustic load. On the other hand, the quartz motional impedance also alters upon changes in Vq and K. It is therefore helpful to rewrite Eq. 3 for our purposes ... 

Impedance analysis is also suggested when properties of an attached film, a liquid, or interfaces are of interest. Due to the weak frequency dependence of the acoustic load within a typical measurement range of some 10 kHz at fundamental mode, one measurement point would be sufficient to calculate Zl (Eq. 2). An effective method to decrease statistical errors is to first fit a theoretical curve to the experimental curve or a specific segment, secondly to calculate Zl from the fit, and finally to extract (material) parameters of interest using separate models describing how the acoustic load is generated [37]. 

Figure 6a shows the transmission hne representing a viscoelastic layer [64]. Every layer is represented by a T . The apphcation of the Kirchhoff laws to the Ts reproduces the wave equation and the continuity of stress and strain. The detailed proof is provided in [4]. To the left and to the right of the circuit are open interfaces (ports). These can be exposed to external shear waves. They can also be connected to the ports of neighboring layers (Fig. 6b). Alternatively, they may just be short-circuited, in case there is no stress acting on this surface (left-hand side in Fig. 6c). Finally, if the stress-speed ratio Zl (the load impedance, see below) of the sample is known, the port can be short-circuited across an element of the form AZl, where A is the active area (right-hand side in Fig. 6c). Figure 6c shows a viscoelastic layer which is also piezoelectric. This equivalent circuit was first derived by Mason [4,55]. We term it the Mason circuit. The capacitance, Co, is the electric capacitance between the electrodes. The port to the right-hand side of the transformer is the electrical port. The series resonance frequency is given by the condition that the impedance of the acoustic part (the stress-speed ratio, aju) be zero, where the acoustic part comprises all elements connected to the left-hand side of the transformer.

In the following, we derive the Butterworth-van Dyke (BvD) equivalent circuit (Fig. 7) from the Mason circuit (Fig. 6c). The Mason circuit itself is derived in detail in [4]. The BvD circuit approximates the Mason circuit close to the resonances. The BvD circuit accounts for piezoelectric stiffening and can also be extended in a simple way to include an acoustic load on one side of the crystal. In the derivation of the BvD circuits, one assiunes small frequency shifts as well as small loads and apphes Taylor expansions in the frequency shift (or the load) whenever these variables occur. The condition of A/// load impedance of the sample, Zi, is much smaller than the impedance of crystalhne quartz, Zq (where the latter, as opposed to Zl, is a material constant). Zq sets the scale of the impedances contained in the Mason circuit. Generally speaking, the QCM only works properly if ZL Zq.ii... 

We have performed FLIC microscopy for all cell types that have been Hsted in Table 2 in order to quantify their individual distance to the growth substrate. It is an inherent problem of this approach that the cells had to be grown on micropatterned silicon for FLIC measurements and not on a quartz resonator as used in QCM. But the imcertainty whether the cells behave differently on either substrate cannot be bypassed in principle. Figure 11 shows the change in load impedance A Zl for the different cell types studied here as a function of their individual cell-substrate separation distance extracted from FLIC measurements. If the hypothesis appUes that cells provide a more sustained QCM response the closer they are to the surface, one would have to expect a decrease of A Zl with increasing distance d. The graph in Fig. 11, however, shows no obvious correlation between the acoustic load of the resonator and the distance between lower cell membrane and its surface. Please note that, for instance, BAEC and NRK cells show very similar distances from the surface of approximately 75 nm but the change in load impedance A Zl differs by more than a factor of four. 

A very similar result was found when we coated the resonator with a layer of collagen. In a typical experiment we measured an increase for A Zl of 236 18 relative to a medium-loaded resonator. Decomposing the complex load impedance in real and imaginary components revealed a load resistance of 43 15 compared to a load reactance of232 18 f2. Thus, the layer of purified collagen shows similar acoustic properties as the remaining ECM on the surface [16]. 

Acoustic shear waves and shear-wave resonators, in particular, have a long tradition in interfacial sensing. Typically one infers the thickness and softness of an adsorbate layer from the shifts of resonant frequency and bandwidth. Laterally heterogeneous samples (vesicles, cells, adsorbed particles) can be modeled numerically. The shifts of frequency and bandwidth are proportional to the stress-velocity ratio (the "load impedance") at the interface, and this load impedance can be calculated by (for instance) the finite element method (FEM). 

An interpretation of the fact that, for some explosives at least, the detonation velocity does not continue to rise with rise in density, but goes thru a maximum and detonation finally fails when the density exceeds a critical value is reptd by Dunkle (Ref 5) and Price Refs 9 10). Roth (Ref 4), on the basis of results reported in Refs 1, 2 St 3, suggests the existence of a property he calls Widerstand ("resistance or "impedance ) of value equal to the product of loading density and detonation velocity, analogous to acoustic impedance and shock impedance (See abstract of Roth s paper at the end of this item)... 

When selective layers are deposited, the whole structure must be treated as a multiple resonator in which the reflection and/or refraction of the acoustic energy occurs at each interface. For example, when a polymer film is deposited on top of the gold electrode of the QCM, it is the polymer-Au interface with which we are concerned. When the mass loading of multiple structures becomes too high, the effect of the impedance mismatches becomes significant and the crystal ceases to oscillate. Even approximate treatment of the multiple resonator is difficult because densities, as well as thicknesses and shear moduli, of the individual layers must be known. 

An ideal mass layer is assumed to have an infinitesimal thickness, yet contribute a finite areal mass density to the device surface. In Section 3.1.1, we noted that this criterion holds as long as the acoustic phase shift across the film is small compared with ir. The equivalent-circuit model for the mass-loaded resonator can be determined from the surface mechanical impedance contributed by a surface perturbation. The surface stress required to sinusoidally accelerate a mass layer is [14]... 

The data derived from calorimetric measurements reflect acoustic power delivery for fairly well matched loads. This is not always the case under normal working conditions. If the calorimeter is used as reaction vessel, and if a matching system is used, the difference in acoustical impedance between the medium inside the calorimeter and the coupling liquid must be known in order to introduce a correction factor. 

The equivalent circuit of a hquid-immersed quartz crystal. Fig. 10, consists of the same basic components. The motional acoustic impedance caused by a (sensing) rigid film and the liquid load can be separated into an inductance, Lc, (coating), an inductance, Tuq, and a resistance, ituq, (liquid). All additional elements are in series to the quartz motional elements. Cuq and Gnq account... 

However, with such high mass loads, the mechanical stability of the system quartz-crystal deposited thick films decreases. Thus the fact that materials with different elastic properties will obey different mass-frequency relations requiring correction by the acoustic impedance ratio Z = Zq / Zf, eqn. (5), is of less importance practically. 

There are two electrical equivalent circuits in common usage, the transmission line model (TLM) and a lumped element model (LEM) commonly referred to as the Butterworth-van Dyke (BvD) model these are illustrated in Figs. 2(a and b), respectively. In the TLM, there are two acoustic ports that represent the two crystal faces one is exposed to air (i.e. is stress-free, indicated by the electrical short) and the other carries the mechanical loading (here, a film and the electrolyte solution, represented below by the mechanical loading Zs). These acoustic ports are coimected by a transmission line, which is in turn connected to the electrical circuitry by a transformer representing the piezoelectric coupling. For the TLM, one can show [18, 19] that the motional impedance (Zj ) associated with the surface loading can be related to the mechanical impedances of... 

Fig. 6 Summary of data interpretation strategy (described in main text) for extracting viscoelastic parameters from film-loaded TSM resonator frequency response. Input parameters (at left of diagram) are resonator impedance and any selected parameter representative of film thickness (here, charge, interpreted using Faraday s law). Upper part of the scheme relates to acoustically thin films (yielding hf and pf). Lower part of diagram relates to acoustically thick films (yielding, with the help of hf and pf, G and C"). (Reproduced from Ref. [24] with permission from the American Chemical Society.)...

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