Surface-Induced Near-Field Scaling in the Knudsen Layer of a Rarefied Gas

We report on experiments performed within the Knudsen boundary layer of a low-pressure gas. The noninvasive probe we use is a suspended nanoelectromechanical string, which interacts with ${}^4\mathrm{He}$ gas at cryogenic temperatures. When the pressure $P$ is decreased, a reduction of the damping force below molecular friction $\propto P$ had been first reported in Phys. Rev. Lett. 113, 136101 (2014) and never reproduced since. We demonstrate that this effect is independent of geometry, but dependent on temperature. Within the framework of kinetic theory, this reduction is interpreted as a rarefaction phenomenon, carried through the boundary layer by a deviation from the usual Maxwell-Boltzmann equilibrium distribution induced by surface scattering. Adsorbed atoms are shown to play a key role in the process, which explains why room temperature data fail to reproduce it.

Figure 1

(a) Schematic of the experimental situation with in brown the container’s surface (on left), in light yellow the adsorbed atoms and then the gas (bulk on the far right). The pressure $P_0$ remains constant while the density $n(x)$ drops from $n_0$ as we approach the wall, within a thickness of order a few mean free paths $\lambda$ (Knudsen layer). Our NEMS local probe of width $w$ lies a distance $g$ from the surface (red rectangle). Close to the wall, macroscopic theories based on simple expansions fail to describe the physics, and delicate microscopic modeling or numerical simulations are required (shaded area). (b) Actual device used in this work (SEM false color image), in which oscillatory motion is orthogonal to the wall (arrow). Note the spongy nature of the interface.

Figure 2

Main: Deviation form molecular damping (proportional to $P$) measured at low pressures, for three devices at 4.2 K as a function of the Knudsen number $\lambda /g$. The lines are fits to Eq. (1), see text. Inset: Raw data for the $150\mu \mathrm{m}$ long device (intrinsic damping subtracted); the full line is the standard molecular damping expression, and the vertical dashed line shows the crossover to Navier-Stokes damping (see text).

Figure 3

Main: Effective density calculated at three different temperatures for the $150\mu \mathrm{m}$ long device. Thin lines correspond to fits obtained from Eq. (1) while the thick lines at high pressure ($\propto P$) and the dashed lines at low pressures ($\propto P^2$) are the asymptotes. Inset: Same data at 4.2 K for the three different devices, showing the independence towards $g$.

Figure 4

Main: Leading-order coefficients extracted from the fits of Fig. 2. Different symbols stand for different fitting routines (see Supplemental Material [19]). Dashed lines are guides for the eye. Inset: Measured frequency shifts due to adsorbed layers onto the NEMS surface for three temperatures. For 1.7 K and 4.2 K, data obtained on the different devices have been normalized to the $L=150\mu \mathrm{m}$ one [19]. At 1.7 K, the rapid growth when $P$ is increased corresponds to the condensation of a thin superfluid film [19, 27, 28], while at higher temperatures the weaker growth is a signature of the crossover towards the Navier-Stokes regime. Lines at low pressures are fits based on Dubinin-Astakhov isotherms [19, 29, 30, 31, 32], while the horizontal dashes correspond to a dense adsorbed monolayer [33].