Dynamic Similarity of Oscillatory Flows Induced by Nanomechanical Resonators

Abstract

Rarefied gas flows generated by resonating nanomechanical structures pose a significant challenge to theoretical analysis and physical interpretation. The inherent noncontinuum nature of such flows obviates the use of classical theories, such as the Navier-Stokes equations, requiring more sophisticated physical treatments for their characterization. In this Letter, we present a universal dynamic similarity theorem: The quality factor of a nanoscale mechanical resonator at gas pressure P 0 is α times that of a scaled-up microscale resonator at a reduced pressure αP 0 , where α is the ratio of nanoscale and microscale resonator sizes. This holds rigorously for any nanomechanical structure at all degrees of rarefaction, from continuum through to transition and free molecular flows. The theorem is demonstrated for a series of nanomechanical cantilever devices of different size, for which precise universal behavior is observed. This result is of significance for research aimed at probing the fundamental nature of rarefied gas flows and gas-structure interactions at nanometer length scales. Miniaturization of resonant mechanical structures has driven progress across a wide range of advanced technologies, including sensors for mass detection and imaging with atomic resolution [1-3], monitoring of biological processes such as DNA hybridization [4], and mass spectrometry at the molecular scale [5]. While the constituent elastic properties of mechanical resonators remain identical to bulk values upon miniaturization to nanometer length scales [6][7][8][9], operation in fluid environments gives rise to physical phenomena not normally seen at macroscopic levels [10][11][12][13][14]. In liquid, these include transport due to the electrical nature of surfaces [15] and the possibility of violation of the usual no-slip condition [16]. Breach of no-slip is exacerbated for operation in gas, because the relevant molecular length scale is orders of magnitude larger than in liquid [17][18][19]. This leads to failure of the continuum hypothesis at relatively large length scales (microns), obviating the use of classical theories such as the Navier-Stokes equation.Continuum theories are used ubiquitously in the design and characterization of macroscale and microscale mechanical structures. In the context of mechanical resonators in gas, this has provided insight into the fundamental physical processes underpinning their operation. For example, it is known that the quality factor (scaled inverse rate-ofenergy dissipation) of a mechanical resonator decreases as the size of the structure is reduced [10,20]. This is due to growth of the viscous penetration depth, where vorticity is prevalent, relative to the structure size. Yet this knowledge is predicated on the assumption that the gas mean free path λ is much smaller than the size of the structure. The mean free path of air at 1 atm and room temperature is approximately 70 nm [21]. Thus, miniaturization of nanomechanical resonators to several hundred nanometers induces strong noncontinuu...

Uncertainty in least-squares fits to the thermal noise spectra of nanomechanical resonators with applications to the atomic force microscope

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Generalized Knudsen Number for Unsteady Fluid Flow

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Dynamical similarity and instabilities in high-Stokes-number oscillatory flows of superfluid helium

“…Historically, experiments on oscillatory flows of classical viscous fluids have been studied since the days of Stokes [1], with many notable developments made in the last century [2][3][4][5]. Recently, oscillating flows have reemerged thanks to developments in micromechanical and nanomechanical engineering, where access to nano electromechanical systems (NEMS) [6][7][8][9][10] has offered unprecedented sensitivity and resolution in fluid dynamical experiments, allowing the transition from continuum to ballistic (molecular) regime to be probed at easily attainable pressures, directly probe fluid boundary layers [9], or formulate universality relations [6][7][8] for classical oscillatory flows. This work extends such universality relations to superfluids, concentrating on the hydrodynamic regime; the transitional and ballistic regimes will represent the subject of a later publication.…”