Design of An Efficient Quartz Crystal Nanobalance: a Finite Element Study
Reetu Singh, Chemical Engineering Department, University of South Florida, 4202 E. Fowler Avenue, ENB 118, Tampa, FL 33620 and Venkat Bhethanabotla, Chemical and Biomedical Engineering, University of South Florida, 4202 E. Fowler Avenue, ENB 118, Tampa, FL 33620.
Abstract: I. Motivation/Background The mass sensitivity (> 1 micrograms) of current commercial analytical instruments, such as thermal gravimetric analyzers, severely limits their utility for measurement of valuable but poorly soluble materials like synthetic proteins or DNA fragments. A quartz crystal microbalance (QCM), based on a transverse shear mode piezoelectric crystal operating at high frequencies, is gaining immense popularity in chemical and bio-sensing applications due to higher mass sensitivities as compared to the traditional analyzers and lesser sensitivity to vibrations. However, these devices suffer from non-uniformity of sensitivity distribution along the sensor surface thereby limiting their use for the determination of mass. Overcoming this limitation would lead to the development of a robust sensor with improved mass sensitivities and reduced sensitivity to vibrations, as compared to the currently available microbalances. The sensitivity profile can be influenced by a number of factors like the electrode design and surface properties of the crystal. In the current work, we developed a finite element (FE) model of the QCM to investigate the mass sensitivity and its radial distribution on the sensor surface for various electrode designs. Such a model will aid in the development of versatile nano-balances with a uniform sensitivity distribution. II. Computational details The QCM device (operating frequency 9 MHz) was based on an AT-cut quartz crystal, cut at 35.35 degrees to the z-axis, with gold electrodes on the top and bottom surfaces. The device was 8.0 mm diameter and 0.185 mm thick with 5.0 mm diameter gold electrodes having a thickness of 0.15*10-3 mm on either side of the QCM. The density of gold and QCM were 18500 kg/m3 and 2649 kg/m3, respectively. The model was meshed using 24, 8131 coupled field piezoelectric elements (solid226). The meshed model is depicted in Fig. 1. The applied voltage was 1 V and a damping coefficient of 2% was used. A harmonic analysis was carried out to study the frequency response in the range of 8.9 to 10.1 MHz, using a total of 200 steps/intervals. The frequency response was then utilized to obtain the displacement profiles on the device surface, which in turn were used to compute the sensitivity distribution on the surface. III. Results/Discussion The radial distribution of the displacement is shown in Figs. 2 and 3, which also indicate the first thickness shearing mode of the device. Our results indicate that the displacement peaks at the center of the electrode and decays rapidly on moving away from the center, which is in qualitative agreement with the experimental results reported in the literature . In addition, the distribution of the displacement is elliptical rather than circular and is largely bounded by the electrode surface. The peak displacements are of the order of pico-meter and are mainly in the x direction. The displacements in the y and z directions are almost 2 orders of magnitude smaller than in the x-direction. Considering the influence of mechanical resonance, the dynamic displacements can be obtained from the static displacements by using the Quality factor (Q)  Udyn=Ustat*Q For a 9 MHz crystal, Q in air is reported to be 84195 . This gives the value of peak displacement to be Udyn=0.923*10-6 (ým)*84195 =77.7 nm Thus, peak displacement for the current 9 MHz device, at a peak driving voltage of 1 V is 77.5 nm, which is in agreement with the experimentally observed displacements of a QCM obtained using scanning tunneling microscopy under vacuum . The dynamic displacements for the device surface, after incorporating the Quality factor, Q, are shown in Fig. 3. The corresponding sensitivity distribution is depicted in Fig. 4 and shows a qualitative agreement with the sensitivity distribution obtained using analytical models  as evident from the distributions normalized using the corresponding maximum values (Fig. 5). Transient simulations will be performed to incorporate the influence of mechanical resonance in the QCM device. Further simulations are underway to evaluate the influence of various electrode designs, like the ring design. The sensitivity distributions for various ring designs will be evaluated. The results will be presented in detail. Comparisons with experimental as well as analytic results based on Helmholtz wave equation, will be presented. The overall goal is to obtain a device with uniform sensitivity distribution and high sensitivity leading to an efficient and robust nano-balance for chemical and bio sensing applications.