Effect of energy trapping on performance of QCM

Quartz crystal microbalance (QCM) vibrating in a thickness shear mode is widely used for biosensing. Since its peripheral is sealed air or liquid tight to form a sample space, a reduction of displacement at peripheral is important in order to keep its Q value and hence its sensitivity high. A popular technique is a trapping of vibration energy in the center portion of QCM. It is realized in the thickness shear mode by making the resonant frequency of the center portion lower than the resonant frequency of the peripheral. When an operating frequency is below 10 MHz, its major surfaces are usually contoured in either plano-convex, convex-convex or bevel shape. When an operating frequency is much higher, simple mass loading by electrodes in the center portion on flat surfaces is good enough to cause energy trapping. Energy trapping, however, reduces an effective area for sensing and hence an overall sensitivity of QCM. A good design of QCM calls for a balance between two factors based on an analysis of effect of energy trapping on performance of QCM. Previous analysis based on equivalent circuit of contoured vibrators with free major surfaces is extended to include effects of mechanical loading on major surfaces due to medium to be sensed into account. A new equivalent circuit with four ports is developed for small section of a vibrator. Two ports correspond to upper and lower surfaces of the vibrator and are terminated by mechanical impedance of attached media to be sensed. Another two ports, left and right, are for connection with neighboring sections. Parameters of an equivalent circuit depend on thickness or mass loading of a section. A large number of sections with gradually varying thickness or mass loading are connected in tandem to approximate a profile of the vibrator. An application of the compensation theorem to the overall equivalent circuit shows the sensor sensitivity is proportional to the integral of square of displacement amplitude. Resonant frequencies, distribution of displacement and reduction factor of sensitivity of plano-convex vibrators are calculated as functions of width vs. thickness ratio and width vs. radius of curvature