Dynamic Similarity of Oscillatory Flows Induced by Nanomechanical Resonators

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 $\alpha$ times that of a scaled-up microscale resonator at a reduced pressure $\alpha P_0$, where $\alpha$ 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.

Figure 1

SEM micrographs of cantilever devices showing plan view geometries. All devices uniformly scaled in all dimensions on 100, 300, and 500 nm SiN films. Devices are highlighted in yellow. Triangular devices (A) at all three size scales shown in top row. Smallest versions of all other devices fabricated from 100 nm thick films shown in middle and bottom rows. Scale bars for top row (device $A$) are $20\mu \mathrm{m}$. Scale bars for all other devices are $2\mu \mathrm{m}$.

Figure 2

Measured quality factors (due to gas) for three different sizes of device $E$ which possess identical rectangular geometries (length to $\text{width}=5$), as a function of gas pressure. These devices have a nominal thickness 100 (blue), 300 (red), and 500 nm (green), with the plan view dimensions and coatings scaled accordingly. Continuum theory, Eq. (8) (solid lines at high pressure). Free molecular theory with diffuse reflection, Eq. (9) (dashed lines at low pressure). These complementary theories are plotted up to their point of intersection, which occurs in the transition regime, $\mathrm{Kn}\sim 1$.

Figure 3

Scaled quality factor $H(\mathrm{Kn})$ for cantilever devices with three different sizes: nominal thicknesses 100 (blue), 300 (red), and 500 nm (green). Reference gas density and pressure specified at 1 atm. Data for devices $A$, $B$, $C$, and $D$. Other device data display similar agreement; see Supplemental Material [32]. Analytical solutions for device $D$ shown for continuum (solid lines) and free molecular flows (dashed lines), as per Fig. 2. Upper horizontal axis gives length of device for uniform reduction in size at a gas pressure of 1 atm; setting ${\rho }_0=\rho$ in vertical axis gives the required quality factor at 1 atm for this size rescaling. Observed enhanced scatter in $H(\mathrm{Kn})$ at high Knudsen number is due to a finite intrinsic quality factor limiting precision at low gas pressure. Identical scales are used for the $H(\mathrm{Kn})$ and Kn axes.