End Mass Effects on the Frequency Response of Cantilevers: Analytical Results

International audienc

Deflection of a Viscoelastic Cantilever under a Uniform Surface Stress: Applications to Static-mode Microcantilever Sensors Undergoing Adsorption

The equation governing the curvature of a viscoelastic microcantilever beam loaded with a uniform surface stress is derived. The present model is applicable to static-mode microcantilever sensors made with a rigid polymer, such as SU-8. An analytical solution to the differential equation governing the curvature is given for a specific surface stress representing adsorption of analyte onto the viscoelastic beam’s surface. The solution for the bending of the microcantilever shows that, in many cases, the use of Stoney’s equation to analyze stress-induced deflection of viscoelastic microcantilevers (in the present case due to surface analyte adsorption) can lead to poor predictions of the beam’s response. It is shown that using a viscoelastic substrate can greatly increase sensitivity (due to a lower modulus), but at the cost of a longer response time due to viscoelasticcreep in the microcantilever. In addition, the effects of a coating on the cantilever are considered. By defining effective moduli for the coated-beam case, the analytical solution for the uncoated case can still be used. It is found that, unlike the case of a silicon microcantilever, the stress in the coating due to bending of a polymer cantilever can be significant, especially for metalcoatings. The theoretical results presented here can also be used to extract time-domain viscoelasticproperties of the polymermaterial from beam response data

Resonant Microcantilevers for the Determination of the Loss Modulus of Thin Polymer Films

The increasing interest in polymer materials creates the need for accurate tools to characterize their mechanical properties. Due to energy dissipation in polymers during deformation, these materials exhibit viscoelastic behavior. Accurate determination of these viscoelastic properties and, more specifically, viscous losses, remains challenging and mainly unknown for thin polymer films. In this paper, a straightforward method to determine the loss modulus of organic materials using resonating microcantilevers has been developed. The extracted results for polyisobutylene show the variation of viscous losses over a large range of frequencies (7-350 kHz)

Cantilever-based Resonant Microsensors with Integrated Temperature Modulation for Transient Chemical Analysis

This work introduces a resonant cantilever platform with integrated temperature modulation for real-time chemical sensing. Embedded heaters allow for rapid thermal cycling of individual sensors, thereby enabling real-time transient signal analysis without the need for a microfluidic setup to switch between analyte and reference gases. Compared to traditional mass-sensitive microsensors operating in steady state, the on-chip generation of signal transients provides additional information for analyte discrimination

Transient Analysis of Analyte Desorption Due to Thermal Cycling with Varying Pulse Duration

This paper introduces heating pulse duration modulation on a chemically sensitive, polymer-coated resonant cantilever platform for analyte discrimination during the desorption phase. As in our previous work, the embedded heaters enable real-time measurements of analyte sorption into the polymer film, without the need for traditional valve systems and reference gases [1-2]. This work particularly looks at the effects of varying pulse lengths on the sensor responses, while holding the heating power constant. A model differential equation is developed for the sensor response based on both the device sensitivity and transient response. This model can then be used together with estimation theory for analyte identification and quantification, even in mixtures

Generalized Model of Resonant Polymer-Coated Microcantilevers in Viscous Liquid Media

Expressions describing the resonant frequency and quality factor of a dynamically driven, polymer-coated microcantilever in a viscous liquid medium have been obtained. These generalized formulas are used to describe the effects the operational medium and the viscoelastic coating have on the device sensitivity when used in liquid-phase chemical sensing applications. Shifts in the resonant frequency are normally assumed proportional to the mass of sorbed analyte in the sensing layer. However, the expression for the frequency shift derived in this work indicates that the frequency shift is also dependent on changes in the sensing layer’s loss and storage moduli, changes in the moment of inertia, and changes in the medium of operation’s viscosity and density. Not accounting for these factors will lead to incorrect analyte concentration predictions. The derived expressions are shown to reduce to well-known formulas found in the literature for the case of an uncoated cantilever in a viscous liquid medium and the case of a coated cantilever in air or in a vacuum. The theoretical results presented are then compared to available chemical sensor data in aqueous and viscous solutions

A Multi-Modal Continuous-Systems Model of a Novel High-Q Disk Resonator in a Viscous Liquid

International audienc

Development of Analytical Models of T- and U-shaped Cantilever-based MEMS Devices for Sensing and Energy Harvesting Applications

Dynamic-mode cantilever-based structures supporting end masses are frequently used as MEMS/NEMS devices in application areas as diverse as chemical/biosensing, atomic force microscopy, and energy harvesting. This paper presents a new analytical solution for the free vibration of a cantilever with a rigid end mass of finite size. The effects of both translational and rotational inertia as well as horizontal eccentricity of the end mass are incorporated into the model. This model is general regarding the end-mass distribution/geometry and is validated here for the commonly encountered geometries of T- and U-shaped cantilevers. Comparisons with 3D FEA simulations and experiments on silicon and organic MEMS are quite encouraging. The new solution gives insight into device behavior, provides an efficient tool for preliminary design, and may be extended in a straightforward manner to account for inherent energy dissipation in the case of organic-based cantilevers

Geometrical Considerations for the Design of Liquid-phase Biochemical Sensors Using a Cantilever\u27s Fundamental In-plane Mode

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 in-plane 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 liquid-phase biochemical sensors using in-plane cantilevers