Shear-mode oscillations quartz crystal

The shear-mode oscillations in quartz crystal, which are now the solution of the three-dimensional wave equation, can be written in the form... 

Keywords AT-cut quartz crystal Sauerbrey equation Thickness shear mode Oscillation Eigenfrequency Dissipation factor Viscoelasticity Voigt model... 

If an electric held of the proper frequency is applied across the quartz crystal, the crystal wiU oscillate in a mechanically resonant mode. These condihons correspond to the creation of a standing acoustic shear wave that has a node midpoint between the two faces of the crystal and two antinodes at both faces of the disk. This is depicted schematically in Eig. 21.20b. In an EQCM experiment the crystals are operated at the fundamental resonant frequency that is a function of the thickness of the crystal. A crystal with a thickness of 330pm has a resonant frequency of 5 MHz. Crystals with these characteristics are commercially available. In an EQCM experiment, an alternating electric field of 5 MHz is applied to excite the quartz crystal into... 

AT-cut, 9 MHz quartz-crystal oscillators were purchased from Kyushu Dentsu, Co., Tokyo, in which Ag electrodes (0.238 cm2) had been deposited on each side of a quartz-plate (0.640 cm2). A homemade oscillator circuit was designed to drive the quartz at its resonant frequency both in air and water phases. The quartz crystal plates were usually treated with 1,1,1,3,3,3-hexamethyldisilazane to obtain a hydrophobic surface unless otherwise stated [28]. Frequencies of the QCM was followed continuously by a universal frequency counter (Iwatsu, Co., Tokyo, SC 7201 model) attached to a microcomputer system (NEC, PC 8801 model). The following equation has been obtained for the AT-cut shear mode QCM [10] ... 

Slip is not always a purely dissipative process, and some energy can be stored at the solid-liquid interface. In the case that storage and dissipation at the interface are independent processes, a two-parameter slip model can be used. This can occur for a surface oscillating in the shear direction. Such a situation involves bulk-mode acoustic wave devices operating in liquid, which is where our interest in hydrodynamic couphng effects stems from. This type of sensor, an example of which is the transverse-shear mode acoustic wave device, the oft-quoted quartz crystal microbalance (QCM), measures changes in acoustic properties, such as resonant frequency and dissipation, in response to perturbations at the surface-liquid interface of the device. 

More recently methods have also been developed to measure the adsorbed amount on single surfaces and not onto powders. Adsorption to isolated surfaces can, for instance, be measured with a quartz crystal microbalance (QCM) [383]. The quartz crystal microbalance consists of a thin quartz crystal that is plated with electrodes on the top and bottom (Fig. 9.11). Since quartz is a piezoelectric material, the crystal can be deformed by an external voltage. By applying an AC voltage across the electrodes, the crystal can be excited to oscillate in a transverse shear mode at its resonance frequency. This resonance frequency is highly sensitive to the total oscillating mass. For an adsorption measurement, the surface is mounted on such a quartz crystal microbalance. Upon adsorption, the mass increases, which lowers the resonance frequency. This reduction of the resonance frequency is measured and the mass increase is calculated [384-387],... 

For AT-cut quartz crystals operating in the shear mode, the oscillation frequency, /o, is inversely proportional to the thickness d of the crystal, as described by the following equation ... 

Quartz crystal microbalance — The quartz crystal microbalance (QCM) or nanobalance (QCN) is a thickness-shear-mode acoustic wave mass-sensitive detector based on the effect of an attached foreign mass on the resonant frequency of an oscillating quartz crystal. The QCM responds to any interfacial mass change. The response of a QCM is also extremely sensitive to the mass (density) and viscoelastic changes at the solid-solution interface [i-vi]. 

The crystal cut determines the mode of oscillations. Shear vibrations are generated if one large crystal face moves parallel with respect to the underlying planes as in QCMs with AT-cut a-quartz crystals. This crystal wafer is prepared by cutting the quartz at approximately 35.17° from its Z-axis. A typical crystal plate is a cylindrical disk of a diameter 10 mm and thickness about 0.7 to 0.1 mm for resonant operation in the 2 to 15 MHz frequency range. This type of crystals shows weak dependence of the resonant frequency on the temperature and stress for room temperature operation. 

In the gravimetric method, the adsorbent (usually in the form of powder) is placed into a bulb, which is mounted on a sensitive balance and the bulb is then evacuated. Next, the weight increase of the adsorbent solid as a function of the absorptive gas pressure is monitored at constant temperature. More recently, the quartz crystal microbalance (QCM) technique has been applied this is very sensitive to mass increases. Quartz is a piezoelectric material and the thin crystal can be excited to oscillate in a traverse shear mode at its resonance frequency when a.c. voltage is applied across the metal (usually gold) electrodes, which are layered on two faces of the crystal. When the mass on the crystal increases upon adsorption, its resonance frequency decreases. The increase in the mass is calculated from the reduction in resonance frequency. On the other hand, adsorption on single flat surfaces can also be measured by ellipsometry, which measures the film thickness of transparent films optically using the difference between light reflection from bare and adsorbed surfaces. 

The deposition of noble metals onto oscillating quartz crystals of the thickness shear type, for fine adjustment of their frequency, has already been carried out for many years by frequency standard manufacturers. The idea of using the frequency decrease by mass deposition to determine the weight of the coating is comparatively new. Sauerbrey [35] and Lostis [36] were the first to propose the quartz-crystal microbalance. The AT-cut crystal oscillating in a thickness shear mode was found to be best suited for this purpose. The thickness xq of an infinite quartz plate is directly related to the wavelength A. of the continuous elastic transverse wave, the phase velocity vq of that wave and the frequency vq (i.e. the period xq) of the oscillating crystal, as shown in Fig. 4 ... 

Schematic representation of a quartz crystal oscillating in the thickness shear mode (AT- or BT-cut).

The crystal cut determines the mode of oscillations. AT-cut quartz crystals, vibrating in a thickness shear mode, are almost exclusively used in EQCM devices however, it should be mentioned that attempts have been made to exploit other modes of oscillation. 

Quartz is a well-known piezoelectric material. a-Quartz belongs to the triclinic crystal system with point group 32 and has a phase transition at 537°C to its P-form, which is not piezoelectric. Quartz has a cut with a zero temperature coefficient. For instance, quartz oscillators, operated in the thickness shear mode of the AT-cut, are used extensively for clock sources in computers, frequency stabilized ones in TVs and VCRs. On the other hand, an ST-cut quartz substrate with X-propagation has a zero temperature coefficient for surface acoustic wave and so is used for SAW devices with high-stabilized frequencies. The another distinguished characteristic of quartz is an extremely high mechanical quality factor Qm >10. ... 

Generally, the cut angle of quartz crystal determines the mode of induced mechanical vibration of resonator. Resonators based on the AT-cut quartz crystal with an angle of 35.25° to the optical z-axis would operate in a thickness shear mode (TSM) (Fig. 1.1) [4]. Clearly, the shear wave is a transverse wave, that is, it oscillates in the horizontal direction (jc-axis) but propagates in the vertical direction (y-axis). When acoustic waves propagate through a one-dimensional medium, the wave function (ij/) can be described by [11] ... 

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