Hydrodynamics of Superfluid Helium in a Single Nanohole

The flow of liquid helium through a single nanohole with radius smaller than 25 nm was studied. Mass flow was induced by applying a pressure difference of up to 1.4 bar across a 50 nm thick ${\mathrm{Si}}_3{\mathrm{N}}_4$ membrane and was measured directly by means of mass spectrometry. In liquid He I, we experimentally show that the fluid is not clamped by the short pipe with diameter-to-length ratio $D/L\simeq 1$, despite the small diameter of the nanohole. This viscous flow is quantitatively understood by making use of a model of flow in short pipes. In liquid He II, a two-fluid model for mass flow is used to extract the superfluid velocity in the nanohole for different pressure heads at temperatures close to the superfluid transition. These velocities compare well to existing data for the critical superflow of liquid helium in other confined systems.

Figure 1

(a) TEM imaging of nanohole. (b) Schematics of the experimental cell. The SiN membrane (M) is epoxied to a support (S) to separate the cell between the inlet (I) and outlet (O).

Figure 2

Mass flow of liquid helium in a single nanohole of 22.8 nm radius at pressure differences of 0.069, 0.145, 0.241, 0.345, 0.483, and 1.45 bar, respectively, from bottom to top. The inset shows the pressure dependence (in bar) of the transition temperature (in kelvin) for the onset of superfluid mass flow through the nanohole; the bulk superfluid transition $T_{\lambda }$ is shown by a dashed line.

Figure 3

Mass flow measured at a pressure of 483 mbar (solid circle). The green dotted line shows the expected flow if only normal fluid flow was present (no superfluid component). The red curve [second term of Eq. (3)] is the viscous flow contribution in the two-fluid model for short pipe using the tabulated normal density. The black line is a guide to the eye.

Figure 4

Temperature dependence of the normalized critical velocity. Open triangles (from [9]) are compared to our results (full symbols) at several pressures. (Inset) Critical velocity (in $\mathrm{m}/\mathrm{s}$) of superfluid helium from experiments in many different systems [16] as a function of channel size (D), in meter. The red bar shows the range of superfluid velocities inferred from the mass flow through the single nanohole. The open circles are previous results [16] with a temperature-dependent intrinsic critical velocity, whereas the crosses are temperature-independent (extrinsic) velocities. The straight line is the Feynman model of critical velocity.