Flow Conductance of a Single Nanohole

The mass flow conductance of single nanoholes with a diameter ranging from 75 to 100 nm was measured using mass spectrometry. For all nanoholes, a smooth crossover is observed between single-particle statistical flow (effusion) and the collective viscous flow emanating from the formation of a continuum. This crossover is shown to occur when the gas mean free path matches the size of the nanohole diameter. As a consequence of the pinhole geometry, the breakdown of the Poiseuille approximation is observed in the power-law temperature exponent of the measured conductance.

Figure 1

(a) Experimental cell for the mass flow conductance experiment. Gas inlet and outlet are connected through stainless steel capillaries with conductance $G_S$ and $G_D$. (b) TEM image of the nanopore with diameter $101\pm 2\mathrm{nm}$. (c) Cartoon of the experimental setup. A gas reservoir held at pressure $P_S$ induces a flow through the nanohole with a conductance $G_{\mathrm{nh}}$ (not drawn to scale), whereas a second reservoir is kept under vacuum, $P_D=0$. The mass current flow is measured with a mass spectrometer analyzer $A_M$.

Figure 2

(a) Volumetric flow (triangles) of helium gas versus the pressure $P_S$ measured by mass spectrometry at three different range of pressure (Knudsen number Kn). The dotted line is a linear fit to the low-pressure data. (b) Conductance $G_{\mathrm{nh}}$ (red diamonds) of the nanohole shown in Fig. 1 versus the Knudsen number at 77 K. The dotted red line is a fit of the data at high Knudsen number $\mathrm{Kn}>10$ using a free-molecular Knudsen flow model (with cutoff) and a nanopore radius of 50.0 nm. The blue dashed line is a fit using a unified flow model [see text Eq. (1)] for a finite-size pipe. The dot-dashed green line shows contribution to the conductance from viscous collective flow. The inset shows the crossover transition for a 77 nm diameter nanohole at 297 K (blue data) and 77 K (red data).

Figure 3

Power-law exponent $n$ extracted from the conductance data taken upon warming (red triangles) and cooling (blue circles). The gray dashed line shows the inverse square root behavior expected for Knudsen free-molecular flow. The vertical gray bar indicates the critical Knudsen number ${\mathrm{Kn}}_c$ (and its width the uncertainty) flagging the pressure at which the finite-size effects become predominant. The inset shows a log-log plot of the conductance versus the temperature at two Knudsen numbers and the dashed lines are fit giving $n$.