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The characteristics of resonant cantilevers in viscous liquids are analyzed. Various rectangular cantilevers geometries are studied in pure water, glycerol and ethanol solutions of different concentrations, and the results are described in terms of the added displaced liquid mass and the liquid damping force for both, the resonance frequency and the quality factor (Q-factor). Experimental results using a set of magnetically actuated resonant cantilevers vibrating in the out-of-plane ("weak-axis bending") mode are presented and compared to theoretical calculations. The importance of the study is in the use of resonant cantilevers as biochemical sensors in liquid environments.

Abstract:At the microscale, cantilever vibrations depend not only on the microstructure's properties and geometry but also on the properties of the surrounding medium. In fact, when a microcantilever vibrates in a fluid, the fluid offers resistance to the motion of the beam. The study of the influence of the hydrodynamic force on the microcantilever's vibrational spectrum can be used to either (1) optimize the use of microcantilevers for chemical detection in liquid media or (2) extract the mechanical properties of the fluid. The classical method for application (1) in gas is to operate the microcantilever in the dynamic transverse bending mode for chemical detection. However, the performance of microcantilevers excited in this standard out-of-plane dynamic mode drastically decreases in viscous liquid media. When immersed in liquids, in order to limit the decrease of both the resonant frequency and the quality factor, and improve sensitivity in sensing applications, alternative vibration modes that primarily shear the fluid (rather than involving motion normal to the fluid/beam interface) have been studied and tested: these include inplane vibration modes (lateral bending mode and elongation mode). For application (2), the classical method to measure the rheological properties of fluids is to use a rheometer. However, such systems require sampling (no insitu measurements) and a relatively large sample volume (a few milliliters). Moreover, the frequency range is limited to low frequencies (less than 200Hz). To overcome the limitations of this classical method, an alternative method based on the use of silicon microcantilevers is presented. The method, which is based on the use of analytical equations for the hydrodynamic force, permits the measurement of the complex shear modulus of viscoelastic fluids over a wide frequency range.

The characteristics of microcantilevers vibrating laterally in viscous liquid media are investigated and compared to those of similar microcantilevers vibrating in the out-of-plane direction. The hydrodynamic loading on the vibrating beam is first determined using a numerical model. A semi-analytical expression for the hydrodynamic forces in terms of the Reynolds number and the aspect ratio (beam thickness over beam width) is obtained by introducing a correction factor to Stokes' solution for a vibrating plate of infinite area to account for the effects of the thickness. The results enable the effects of fluid damping and effective fluid mass on the resonant frequency and the quality factor (Q) to be investigated as a function of both the beam's geometry and liquid medium's properties and compared to experimentally determined values given in the literature. The resonant frequency and Q are found to be higher for laterally vibrating microcantilevers compared to those of similar geometry experiencing transverse (out-of-plane) vibration. Compared to transversely vibrating beams, the resonant frequency of laterally vibrating beams is shown to decrease at a slower rate (with respect to changes in viscosity) in media having higher viscosities than water. The theoretical results are compared to experimental data obtained for cantilevers completely immersed in solutions of varying aqueous percent glycerol. The increases in resonant frequency and Q are expected to yield much lower limits of detection in liquid-phase chemical sensing applications.

Absorption of a chemical analyte into a polymer coating results in an expansion governed by the concentration and type of analyte that has diffused into the bulk of the coating. When the coating is attached to a microcantilever, this expansion results in bending of the device. Assuming that absorption ͑i.e., diffusion across the surface barrier into the bulk of the coating͒ is Fickian, with a rate of absorption that is proportional to the difference between the absorbed concentration and the equilibrium concentration, and the coating is elastic, the bending response of the coated device should exhibit a first-order behavior. However, for polymer coatings, complex behaviors exhibiting an overshoot that slowly decays to the steady-state value have been observed. A theoretical model of absorption-induced static bending of a microcantilever coated with a viscoelastic material is presented, starting from the general stress/strain relationship for a viscoelastic material. The model accounts for viscoelastic stress relaxation and possible coating plasticization. Calculated responses show that the model is capable of reproducing the same transient behavior exhibited in the experimental data. The theory presented can also be used for extracting viscoelastic properties of the coating from the measured bending data.

To achieve higher quality factors (Q) for microcantilevers used in liquid-phase sensing applications, recent studies have explored the use of the lateral (inplane) flexural mode. In particular, we have recently shown that this mode may be excited electrothermally using integrated heating resistors near the micro-cantilever support, and that the resulting increase in Q helps to make low-ppb limits of detection a possibility in liquids. However, because the use of electrothermally excited, liquid-phase, microcantilever-based sensors in lateral flexure is relatively new, theoretical models are lacking. Therefore, we present here a new analytical model for predicting the vibratory response of these devices. The model is also used to successfully confirm the validity of our previously derived Q formula, which was based on a single-degree-offreedom (SDOF) model and a harmonic tip force. Comparisons with experimental data show that the present model and, thus, the analytical formula provide excellent Q estimates for sufficiently thin beams vibrating laterally in water and reasonable upper-bound estimates for thicker beams.

Abstract:The influence of the beam geometry on the quality factor and resonance frequency of resonant silicon cantilever beams vibrating in their fundamental in-plane flexural mode in water has been investigated. Compared to cantilevers vibrating in their first out-of-plane flexural mode, utilizing the inplane mode results in reduced damping and reduced mass loading by the surrounding fluid. Quality factors as high as 86 have been measured in water for cantilevers with a 20 μm thick silicon layer. Based on the experimental data, design guidelines are established for beam dimensions that ensure maximal Q-factors and minimal mass loading by the surrounding fluid, thus improving the limit-of-detection of mass-sensitive biochemical sensors. Elementary theory is also presented to help explain the observed trends. Additional discussion focuses on the tradeoffs that exist in designing liquidphase biochemical sensors using in-plane cantilevers.