Piezoelectric stiffening

From a constitutive relation of the form t = t(D, ri) (here t is stress not time) it can be readily shown that, since there is no change in electric displacement in an open-circuit, thick-sample configuration, there are no secondary stresses due to electromechanical coupling. Nevertheless, the wavespeed is that of a piezoelectrically stiffened wave. 

The stiffness parameter C55 has, in effect, been increased by the factor (1 + K ) — an effect known as piezoelectric stiffening. The factor is the electromechanical coupling coefficient for the jt-propagating, z-polarized plane wave ... 

Piezoelectric stiffening increases the wave velocity from that obtained in the non-piezoelectric case. Wave velocity is given by v = (c js/p). Since the factor has a value of... 

In piezoelectric media, the strain energy is multiplied by the factor (1 + due to piezoelectric stiffening of the elastic constants. Thus, the strain energy includes electrical stored energy in this case. To calculate the magnitude of the electrical energy density Ue independent of the strain energy, the relation... 

Piezoelectrically stiffened shear modulus of quartz crystal (real part)... 

Coherent reflections at the top and bottom boundaries of the plate give way for a set of standing acoustic waves between the two main surfaces of the plate. Due to the piezoelectric nature of quartz two sets of resonance frequencies exist for each mode, depending on the electrical boundary conditions. The first set corresponds to a plate with open-circuit boundary conditions. From the physical point of view charges will be collected on the electrodes building up a potential difference and hence an electrical field from the electrical point of view the electrodes are unconnected. This resonance is termed anti-resonance in the piezoelectric literature and parallel resonance in electronics literature. The second set of resonance frequencies corresponds to a plate with short-circuit boundary conditions. The electrodes are connected and a potential difference cannot be built up. The respective names are resonance in piezoelectric and series resonance in electronics Hterature. The differences arise from piezoelectric stiffening accompanied by differences in the sound velocity. The anti-resonance (parallel) frequencies of each of the three acoustic modes are completely decoupled giving ... 

Circuit element related piezoelectric stiffening Acoustic impedance of a liquid, 2]iq = (icyjjp) ... 

The overtone order may turn into a noninteger number if piezoelectric stiffening and energy trapping are taken into account. This does not change the structure of the equations. 

Even though a/ii is an entirely acoustic quantity, the series resonance frequency is affected by the value of the electrical capacitance, Co, because of the element Z = - (j) /(icoCo), which introduces piezoelectric stiffening into the acoustic branch. Piezoelectricity adds a negative capacitor into the mechanical branch of the circuit. 

In this section, the Mason circuit does account for piezoelectric stiffening, whereas piezoelectric stiffening is neglected in Sects. 4 and 5. In order to find exact equivalence between the three models, the element Z), (deaUng with piezoelectric stiffening) must be deleted from the Mason circuit. 

Piezoelectricity and piezoelectric stiffening are rigorously accounted for in the Mason circuit. This is not the case for the mathematical and optical approaches at the level of detail presented here. 

Equation 52 shows that the areal mass density of the crystal is the only parameter connecting the load and the frequency shift, as long as the latter is small. The stiffness of the crystal (and piezoelectric stiffening, in particular) is of no influence at this level of approximation. Comparing Eqs. 51 and 39, we find ... 

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... 

Given the functional form of the term related to piezoelectric stiffening, one can define a new complex spring constant, /Cp, taking piezoelectric stiffening into account. Remembering that = (j / icoCo), one writes ... 

Fig. 1 Equivalent circuit representation of the quartz crystal including a load. Piezoelectric stiffening (described by the element 42 in Fig. 13, Chap. 2 in this volume) was neglected. The sample is represented by the load Zl...

Anyone who is not particularly interested in (high-frequency) electrochemistry is strongly advised to ground the front electrode well before starting a QCM experiment. Otherwise the electric boundary conditions will affect A/ and AT via piezoelectric stiffening. Such effects are left aside in the discussion next. 

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