Frequency Response of Microcantilevers in Viscous Fluids
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FREQUENCY RESPONSE OF MICROCANTILEVERS IN VISCOUS FLUIDS A.M. Schilowitz1, D.G. Yablon1 and F. Zypman2 ExxonMobil Research and Eng. Co., Annandale, NJ; 2Yeshiva University, NY, NY
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ABSTRACT This paper presents a new, general reconstruction algorithm that enables modeling of the experimental resonance spectrum of a prismatic microcantilever in a viscous fluid. A closedform solution is obtained for the microcantilever frequency response from the equation of motion with fluid damping and internal friction terms, which allows direct calculation of the fluid damping and internal friction damping constants. In principle, the fluid damping constant has a simple relationship to viscosity thus potentially simplifying the process of obtaining viscosity from experimental data. Finally, the model is compared to experimental data. INTRODUCTION Vibration of microcantilevers (MC) has been modeled over a wide range of damping conditions in fluids.1, 2 Resonant frequency shifts and changes in the quality factor, Q, are effectively predicted and have been used to measure fluid properties such as density and viscosity.3, 4 Models used to reconstruct MC motion range from a linear spring model which incorporates damping, effective mass and geometric terms 2 to more rigorous models that simulate viscous damping of a non-flexing resonating beam.4 This latter approach is most relevant for large values of Q. In a few cases, available models account for internal energy dissipation in the microcantilever, due to anelastic effects.5 In this paper, we use the equation of motion for an elastic rectangular microcantilever that includes a novel expression to account for internal friction dissipation. In addition, we solve this equation and present an explicit closed form solution for the frequency response of the rectangular MC. This solution offers several benefits over previous efforts for deriving fluid viscosity from MC vibration spectra. First, the frequency spectrum is explicitly dependent on the damping constant β defined below. This allows β to be derived directly from the resonance spectrum. In addition, under typical microcantilever conditions, β has a simpler relationship to viscosity than Q, which is more commonly used as a measure of viscosity.3,4 Most importantly, without complicating the data analysis process, the model explicitly accounts for internal friction. Comparison of data generated in vacuum and air suggests that, in air, damping due to internal friction can be neglected. Within the framework of this model, we propose an algorithm that can bound the internal friction of the oscillating cantilever and extract the damping coefficient of the embedding liquid. Finally, we present experimental results and apply the proposed method. THEORY In the present work we develop a continuum-damping model for a cantilever driven by an external sinusoidal force such as a piezoelectric oscillator. Internal friction is accounted for explicitly as a damping term caused by beam flexing. Fluid damping is included as a velocity-
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