Influence of squeeze-film damping on higher-mode microcantilever vibrations in liquid
© Bircher et al.; licensee Springer on behalf of EPJ. 2014
Received: 25 July 2014
Accepted: 13 November 2014
Published: 12 December 2014
The functionality of atomic force microscopy (AFM) and nanomechanical sensing can be enhanced using higher-mode microcantilever vibrations. Both methods require a resonating microcantilever to be placed close to a surface, either a sample or the boundary of a microfluidic channel. Below a certain cantilever-surface separation, the confined fluid induces squeeze-film damping. Since damping changes the dynamic properties of the cantilever and decreases its sensitivity, it should be considered and minimized. Although squeeze-film damping in gases is comprehensively described, little experimental data is available in liquids, especially for higher-mode vibrations.
We have measured the flexural higher-mode response of photothermally driven microcantilevers vibrating in water, close to a parallel surface with gaps ranging from ~200 μm to ~1 μm. A modified model based on harmonic oscillator theory was used to determine the modal eigenfrequencies and quality factors, which can be converted into co-moving fluid mass and dissipation coefficients.
The range of squeeze-film damping between the cantilever and surface decreased for eigenfrequencies (inertial forces) and increased for quality factors (dissipative forces) with higher mode number.
The results can be employed to improve the quantitative analysis of AFM measurements, design miniaturized sensor fluid cells, or benchmark theoretical models.
PACS: 07.10.Cm (Micromechanical devices and systems), 46.40.Ff (Resonance and damping of mechanical waves), 07.79.-v (Scanning probe microscopes and components), 07.07.Df (Sensors (chemical, optical, electrical, movement, gas, etc.); remote sensing).
KeywordsMicrocantilever Dissipation Squeeze-film damping Higher eigenmode Photothermal excitation Eigenfrequency Quality factor Fluid–structure interaction
Damping is an important design criterion for micro- and nanometer sized resonators, because surface forces dominate body forces at small dimensions . Immersing a resonator, e.g., a microcantilever, in fluid drastically changes its dynamic properties. The eigenfrequencies and quality factors decrease due to hydrodynamic forces, which can be decomposed into an inertial (added mass) and dissipative (viscous damping) term . Additionally, placing the resonator close to a solid surface leads to squeeze-film damping, where displacement of the fluid between the resonator and the surface during each vibration period introduces additional added mass and viscous damping . The damping occurring by both mechanisms has direct impact on atomic force microscopy (AFM) and dynamically operated nanomechanical sensors. With progressing miniaturization, squeeze-film damping starts to dominate other dissipative effects and, thus, needs to be considered and characterized .
Furthermore, higher modes of vibration are increasingly used. In multifrequency AFM imaging, higher modes allow the material characteristics, e.g., mechanical, magnetic or electrical properties, of the substrate to be measured . To reduce squeeze-film damping, AFM samples have been placed on pillars , or cantilever geometries have been optimized by focused-ion beam milling . In cantilever-based sensor applications, the use of higher vibrational modes provides increased mass sensitivity  and allows the elastic properties  and the position of adsorbates  to be disentangled. Squeeze-film damping needs to be considered below a certain critical dimension of the AFM cantilever tip or container in which the cantilever sensor is mounted.
Experimental investigations of cantilevers with dimensions ranging from centimeters to micrometers, immersed in water, buffer, organic solvents and oils are reported in the literature ,-. However, all experimental studies on microcantilevers in liquid and close to a surface, were limited to the fundamental mode (n = 1) -. Here we present the full spectral response of microcantilevers vibrating in water at different distances from a polydimethylsiloxane (PDMS) surface. PDMS was selected because of its abundant use for the fabrication of microfluidic devices. Spurious-free resonance spectra were obtained by driving the microcantilevers photothermally , and several higher flexural modes of vibration were characterized.
Results and discussion
Vacuum frequencies, added mass coefficients and damping coefficients measured far from the surface (mean ± SD)
f n,vac /kHz
a m,H »1
cH »1/mPa · s
44.8 ± 5.4
9.17 ± 0.20
67.8 ± 1.6
277 ± 15
6.78 ± 0.03
116.2 ± 11.1
768 ± 34
5.75 ± 0.01
186.7 ± 4.43
1512 ± 52
5.05 ± 0.01
372.0 ± 26.9
We have measured the squeeze-film damping on higher flexural mode vibrations of microcantilevers placed in proximity to a parallel surface in liquid. Due to the strong damping only a direct excitation method, such as the employed photothermal excitation , obtains spurious-free resonance spectra. A model consisting of a sum of harmonic oscillators was employed to extract the modal eigenfrequencies and quality factors from the phase spectra, and described the measured data well. Correct alignment of the data, i.e., calibration of the gap g, was crucial and limited the precision of the measurements. As predicted ,, strong squeeze-film damping of the fundamental mode was observed for normalized gaps H < 1. With increasing mode number the range of squeeze-film damping decreased for the eigenfrequencies (inertial forces) and increased for the quality factors (dissipative forces). Furthermore, the effect seems to depend on the length of the cantilever that determines the spatial wavelength of each mode. These findings should be considered for the design of sensor containers and cantilever tip geometries, because the quality factor is directly related to the sensitivity of the sensor . The observed behavior is likely due to the three-dimensional nature of the flow field generated by higher modes, where gradients along the length of the cantilever must not be neglected . For theoretical models, this entails the introduction of another parameter, besides the normalized gap H and the Reynolds number Re, related to the spatial wavelength of the cantilever, i.e., depending on the mode number as well as the cantilever length (similar to the normalized mode number in ). Finally, added mass and damping coefficients were calculated to support the comparability of the data. The shift in added mass decreased with mode number as predicted by numerical models . The opposite was observed for the damping coefficients, which increased. More work is required to identify the underlying mechanisms governing squeeze-film damping acting on higher modes. Nevertheless, our data from microcantilevers with common dimensions, allows the magnitude of the squeeze-film damping effect to be assessed.
Tipless silicon microcantilevers (NSC12/tipless/noAl, MikroMasch) with nominal dimensions of 250 μm × 35 μm × 2 μm and calculated spring constants of 0.76 N/m were employed. The data reported in the Additional file 1 was obtained using longer microcantilevers (300 μm × 35 μm × 2 μm and 350 μm × 35 μm × 2 μm) following the same protocol. A comparison of the different cantilevers is provided in the Additional file 1: Table S1. To improve reflectivity and avoid unspecific adsorption, 20 nm gold was coated at the bottom side of the cantilevers and they were passivated with short polyethylene glycol chains, as described previously .
The cavity containing the water was formed using PDMS (SYLGARD 184, Dow Corning) and a glass microscope slide (AA00000112E, Menzel-Gläser), exploiting surface tension forces (see Figure 8). The base was fabricated by reversibly bonding a 150 μm-thick PDMS sheet with a 10 mm wide circular hole at its center to the glass slide. The 300 μm-thick cantilever chip was attached to the glass slide at the center of the hole using UV curable glue (F-UVE-61, Newport). The thickness of the chip was sufficient (H = 8.6) to exclude any influence of the glass surface on cantilever dynamics. Furthermore, as the thickness of the PDMS sheet (150 μm) was less than the thickness of the chip, access from above was retained. A flat upper cavity surface was fabricated by pouring degassed PDMS onto a silicon wafer to a thickness of about 5 mm and baking for 4 hours at 60°C. The PDMS was subsequently removed from the wafer and cut to give a circular disk with a diameter of 15 mm. The diameter exceeded all dimensions of the microcantilevers by at least an order of magnitude to avoid edge effects. The rougher surface of the disc was fixed to a kinematic mirror mount (KM05/M, Thorlabs), which was in turn mounted on an encoded piezo motor linear stage (CONEX-AG-LS25-27P, Newport) with a nominal precision of 0.2 μm. The cavity allowed the cantilever to be immersed in ~200 μL of water.
The flat upper PDMS surface was manually aligned parallel to the cantilever. To do this, a piece of silicon wafer was attached to the surface by adhesion forces to render it reflective. The read-out laser was then focused on the silicon surface and detected by the PSD otherwise used to measure the cantilever deflection. The residual angular misalignment was estimated to be less than 1 mrad (0.06°). The same procedure was repeated after rotating the PSD by 90° to align the angle perpendicular to the longitudinal axis of the cantilever.
where m is a data point in the spectrum ranging from 0 to M-1, M the total number of points, f m=0 the lowest and f m=M the highest frequency in the measured data and p the power of the transformation required for each mode of vibration to be assigned an equal number of data points. The value of p was estimated to be 0.514 from the calculated widths of the resonance peaks of all employed cantilevers in an unbounded fluid . The phase values corresponding to the transformed frequencies f* were linearly interpolated from the measured data.
where the cantilever response ϕ c is the sum of damped harmonic oscillators with f n and Q n over all recorded modes N, ϕ th is the linear thermal lag due to photothermal excitation with time constant τ th  and ϕ el (center frequency f el and offset c off ) is an empirical first-order filter that considers the phase responses of the measurement electronics. The filter center frequency f el and the time constant τ th were determined on the first spectrum recorded far from the surface (H » 1) and then held constant.
Parameters for the employed silicon cantilevers immersed in water
2330 kg · m−3
Mass per unit length
0.163 mg · m−1
Area moment of inertia
Quality factor of mode n
Eigenfrequency of mode n
Vacuum frequency of mode n
Added mass coefficient
Damping per unit length
Pa · s
998.25 kg · m−3
1.005 mPa · s
H = g/b
Atomic force microscopy
The authors gratefully acknowledge Henning Stahlberg (C-CINA, Biozentrum, University of Basel) for financial support and providing facilities, Shirley Müller (C-CINA, Biozentrum, University of Basel) for critically reading and discussing the manuscript, Francois Huber and Hans Peter Lang (SNI, Institute of Physics, University of Basel) for their support on cantilever preparation, Stefan Arnold and Andrej Bieri (C-CINA, Biozentrum, University of Basel) for fruitful discussions.
This work was supported by ARGOVIA grant NoViDeMo and Swiss National Science Foundation grant SNF 200020_146619.
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