0.95 1 1.05 0 1 2 3 4 5 6 7 8 Normalized Frequency Amplitude Increasing Excitation Amplitude Figure 3: Simulated nonlinear frequency response of a representative piezoelectrically-actuated cantilever. The frequency response exhibits further softening as the excitation amplitude is increased. term(χ) vary strongly with system’s geometry and elastic material properties and exhibit a weak dependence on material nonlinearities. In contrast, the other terms in the expression are strong functions of the linear and nonlinear piezoelectric material coefficients. Note that though nonlinear constitutive relations were used to develop this model, if the constitutive relations are assumed to be linear, the system resembles a classic Duffing resonator. Detailed expressions for each of the terms presented here are omitted for the sake of brevity. Equation (8) is highly nonlinear and fails to feature a tractable closed-form solution. Accordingly, the equation is analyzed using a numerical continuation program, AUTO [19]. This preliminary analysis is carried out by assuming that all the nonlinear material coefficients, with the exception of α1, are zero. This follows the approach previously detailed in [17]. Figure 3 depicts the frequency response structure associated with the system, under various excitation amplitudes, for a representative value of α1. As evident at comparatively-large excitation amplitudes, the system exhibits a softeninglike behavior, with coexistent stable solutions and hysteresis. Accordingly, if the system is excited at a comparativelylow excitation frequency and frequency is swept to a higher value, the response amplitude will slowly increase until the excitation frequency crosses the (lower) point at which the stable and unstable solution branches meet (a saddlenode bifurcation). At this point, the response amplitude jumps to the large-amplitude response from the low amplitude state, and then decreases with increasing excitation frequency. In contrast, if the system is excited at a comparatively high excitation frequency and frequency is swept to a lower value, the response amplitude will rapidly increase until the excitation frequency crosses the (upper) point at which the stable and unstable solution branches meet (a second saddlenode bifurcation). At this point, the response amplitude jumps to the small-amplitude response from the high amplitude state, and then decreases with decreasing excitation frequency. 2.1 Bifurcation-Based Mass Sensing As previously noted, in a traditional linear resonance based mass sensor, chemomechanically-induced shifts in the system’s resonance frequency are tracked and used to signal an analyte detection event. In contrast, the proposed bifurcation-based resonant mass sensors exploit the rich nonlinear frequency response structure detailed above. Specifically, these sensors are designed to operate near the saddle-node bifurcation point designated A in Fig. 1(b). As with linear sensors, chemomechanical interactions with a target analyte are used to alter the effective mass of the resonator. However, in this instance, the added mass serves to not only shift the system’s natural frequency, but also to drive the system across the saddle-node bifurcation. This yields a rapid and dramatic jump in the system’s response amplitude which can be easily detected, even in the absence of significant readout electronics. 3 Experimental Investigation To validate the feasibility and merits of the proposed nonlinear, bifurcation-based mass sensors a succinct experimental investigation was initiated. To begin, the nonlinear frequency response behavior of a representative device was characterized via laser vibrometry. Select microcantilevers were then selectively functionalized and tested within a carefully controlled environment using a custom test apparatus. Veeco DMASP probes, initially designed for use in scanning probe microscopy applications, were used as a test platform. These devices consist of a silicon cantilever and an integrated piezoelectric actuator incorporating a ZnO layer sandwiched between two Au/Ti electrodes. 60
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