Critical Flow and Dissipation in a Quasi-One-Dimensional Superfluid

8/10/2019 Critical Flow and Dissipation in a Quasi-One-Dimensional Superfluid

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Critical Flow and Dissipation in a Quasi-One-Dimensional Superfluid

P-F Duc1, M.Savard1 , M. Petrescu1, B. Rosenow2, A. Del Maestro3, and G. Gervais1,41Department of Physics, McGill University, Montreal, QC, H3A 2T8 Canada

2Institut fur Theoretische Physik, Universitat Leipzig, D-04103, Leipzig, Germany3Department of Physics, University of Vermont, Burlington, VT USA

4Canadian Institute for Advanced Research, Toronto, Canada andcorresponding author: [email protected]

(Dated: December 17, 2014)

In one of the most celebrated examples of the theory of universal critical phenomena, the phasetransition to the superfluid state of4He belongs to the same three dimensional O(2) universality classas the onset of ferromagnetism in a lattice of classical spins with XYsymmetry. Below the transition,the superfluid densitys and superfluid velocityvs increase as power laws of temperature describedby a universal critical exponent constrained to be equal by scale invariance. As the dimensionality isreduced towards one dimension (1D), it is expected that enhanced thermal and quantum fluctuationspreclude long-range order, thereby inhibiting superfluidity. We have measured the flow rate of liquidhelium and deduced its superfluid velocity in a capillary flow experiment occurring in single 30 nmlong nanopores with radii ranging down from 20 nm to 3 nm. As the pore size is reduced towardsthe 1D limit, we observe: i) a suppression of the pressure dependence of the superfluid velocity;ii)a temperature dependence ofvs that surprisingly can be well-fitted by a powerlaw with a singleexponent over a broad range of temperatures; and iii) decreasing critical velocities as a function

of radius for channel sizes below R

20 nm, in stark contrast with what is observed in micronsized channels. We interpret these deviations from bulk behaviour as signaling the crossover to aquasi-1D state whereby the size of a critical topological defect is cut off by the channel radius.

PACS numbers: 47.61.-k, 67.25.bf, 67.25.dg, 67.25.dr

Helium is the only known element in nature that be-comes a superfluid, with its small mass and high symme-try cooperating to prevent solidification at atmosphericpressure as the temperature approaches absolute zero.For 4He, the ability to flow without viscosity below the-transition temperature, T, is a paradigmatic manifes-tation of emergent phenomena and macroscopic quan-

tum coherence, driven by both strong interactions andbosonic quantum statistics. Its superflow with veloc-ity vs = (h/m) is caused by a quantum-mechanicalphase gradient of the wave function and a priorishouldonly be limited by the Landau criterion of superfluidity,vL due to the roton minimum in the helium excitationspectrum. However, years of experiments[1] have shownthat superfluid 4He exhibits a critical velocity that is wellbelow vL 60 m/s. The exact microscopic mechanismby which a superfluid dissipates energy remains a majorunsolved problem in condensed matter physics.

At a first glance, it would appear that this problemwould only be exacerbated as the number of spatial di-

mensions decreases, as enhanced thermal and quantumfluctuations should push T 0. However, in the onedimensional limit, the universal quantum hydrodynamicsof Luttinger liquid theory [2, 3] should apply, providinga host of theoretical predictions including the simultane-ous algebraic spatial decay of both density-density andsuperfluid correlation functions. While there is a body ofevidence of this exotic behaviour in low dimensional elec-tronic systems [47]and ultracold low density gases [8],the analogous behaviour has yet to be confirmed exper-

imentally in a highly-correlated bosonic fluid. Here, thephysics of superflow should be qualitatively altered, withthe superfluid densitysacquiring strong system size andfrequency dependence [9]. Furthermore, neutral mass-flow transport properties should be strongly modified inone dimension, with the superfluid velocity vs exhibit-ing non-universal power law dependence on temperature

and pressure. This crossover towards one dimension ismanifest in the main findings of our work: (I) a sup-pression of the pressure dependence ofvs for R 3 nmindicative of enhanced dissipation via phase slips, (II) atemperature dependence for vs that can be described bya powerlaw with a single exponent over a broad range oftemperatures, and (III) decreasing critical velocities as afunction of radius for channel sizes belowR 20 nm; be-haviour strongly deviant from what is observed in micronsized channels.

In this work, the mass flow rate of superfluid heliumis measured in a capillary experiment through channelswith radii as small as R 3 nm and lengths L = 30 nm.

To determine the effective dimensionality of this geome-try, it is imperative to perform a comparative analysis ofall possible relevant length scales. Unlike superconduc-tors and superfluid 3He which undergo a BCS pairing,4He has a very small coherence length, on the angstromscale: 4(T) 0(1T /T)

, with0 3.45A and2/3, making it technically difficult to fabricate a trans-verse confinement dimension with R 4 approachingthe truly one dimensional limit, as, for example, 4 0.5 1.5 nm in the temperature range considered here.

a

rXiv:1412.5124v1

[cond-mat.mes-hall]16Dec2014


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ForT= 0.52 K, Rcan also be compared to the thermal

de Broglie wavelength, (T) =

2h2/mkBT 1 nm

and a thermal length LT = hc1/kBT 1 nm, wherec1 235 m/s is the first sound velocity of

4He. An al-ternative measure of one-dimensionality can be obtainedby computing the thermal energy needed to populatetransverse angular momentum states for a single helium

atom confined inside a long hard cylinder of radius R:T /kB 3.5/R2 nm2 K 0.4 K for R = 3 nm.

These estimates, which mostly neglect interaction effects,would place our flow experiments in a mesoscopic regime,with confinement length and energy scales on the orderof the intrinsic ones in the problem. However, recentab initio simulations of 4He confined inside nanopores[10,11] have demonstrated that classical adsorption be-haviour leads to an effective phase separation, between aquasi-1D superfluid core of reduced radius and concen-tric shells of quasi-solid helium near the pore walls. Thiseffect, which is likely also present in our channels, wouldtend to provide additional confinement, allowing us to

approach an effectively quasi-one-dimensional state.Previous investigations of helium confined at the

nanometer scale have focused on porous media such asin Vycor [12]and more recently in the zeolites and othermesoporous media. These studies have shown a possiblenew thermodynamic phase of 4He stabilized at low tem-perature[13]as well a nuclear magnetic resonance signa-ture (NMR) of a one-dimensional crossover for 3He[14].While these advances are certainly considerable in thesearch for a strongly-interacting 1D neutral quantum liq-uid, our approach differs much in spirit from those citedabove. In our experiment, the helium atoms are confinedinside a single, nearly cylindricalpore, rather than in anextremely large number of them necessary to gain signalfor a macroscopic probe. This lone pore, or channel, istailor-made from a pristine Si3N4 membrane that can befabricated with radii ranging from R 1 100 nm. Themain advantage of our approach is that there is no ensem-ble averaging over pore distributions and/or potential de-fects of the sample. Its main drawback, however, is thattraditional bulk measurement techniques, such as specificheat or NMR most likely cannot be performed in a sin-gle nanopore containing only 104 to 105 helium atoms.Taken as a whole, these two approaches are complemen-tary to one another and similar in spirit to bottom-up

vs. top-down or single-molecule vs. ensemble aver-aged studies in other fields, such as nanoelectronics ormolecular biology.

The experiment is configured in a similar fashion, andfollows the same procedure as previously reported in Ref.[15]. However, the present work is performed in a newly-designed experimental cell made out of coin silver andshown in Fig. 1(A). The single nanopores were fabri-cated in the Si3N4 membrane using an electron beamfrom a Field-Effect Transmission Electron Microscope

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FIG. 1: Design of the capillary flow experiment. (A)CAD drawing of the coin silver experimental cell. The in-let (I) and outlet (O) reservoir are connected to the top andbottom parts of the cell, and sealed with indium o-rings (R).The SiN membrane (M) is itself sealed to the bottom part ofthe cell with an indium o-ring and a push-on plate (P). (B)Illustration of the flow experiment where the source reservoiris kept at a pressure PS > PD 0 and the flow measured bymass spectrometry (Am) in a series experiment. (C)and(D)TEM picture of two nanopores used in this experiment. Thebars represent 5 nm in both pictures. The diameter shownhere is only an upper bound since the pore undergoes relax-ation and its diameter decreases in size post-fabrication. Inthe experiment, the experimental pore radius was determined

in situusing both Knudsen effusion in the gas phase and vis-cous flow measurements in the normal phase of liquid helium.

(FE-TEM), with images taken shortly after fabricationshown in Fig. 1. While the single pores have a well-defined diameter, we have observed in previous work thattheir structure has a tendency to relax at room temper-ature, with the pore radius decreasing as a function oftime (see supplementary information). To circumventthe uncertainty in the pore dimension, Knudsen effusionmeasurements in the gas phase of helium were conducted

at low temperature (77 K) using the protocol discussedin Ref. [16]. The respective values obtained for each ofthe pores were determined to beRKn = 8.20.5 nm andRKn = 3.10 0.35 nm (see supplementary information).

In a second step, the experimental cell was cooled downto liquid helium temperature (below 4.5 K). Above T,in the normal phase of helium, the flow through thenanopore is viscously dissipative, and expected to followthe model developed for a short pipe by Langhaar [17].In this phase, we have conducted pressure sweeps at con-


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FIG. 2: Flow measurements raw data. (A)Mass flowmeasurements as a function of pressure for a 7.81 nm and(B)a 3.15 nm pore radius in the normal state. The blue lineis a fit of the data using Eq.(1) and the dashed and dash-dotted lines are one standard variation from the mean valuefor the radius, with all other parameters kept constant. Thefinite intercept value at zero pressure is a spurious signal (seetext). (C) and (D) Temperature dependence of the massflow at several pressures. The dashed line shows the knownsuperfluid transition temperature (T) at saturated vapourpressure.

stant temperature while monitoring the mass flow rateQm, as shown in Fig. 2(A) and (B). In the absence of achemical potential difference, the mass flow rate shouldgo to zero. However, we observe a spurious signal asP 0 arising from evaporation at the walls of thechannel. To determine this offset, the data were fittedwith the flow equation for short pipe,

Qn=8L

1 + R4

322L2P 1

(1)

where is the viscosity and is a coefficient to takeinto account the acceleration of the fluid at the pipe end(see supplementary information). In Fig.2 (A) and (B),the solid line is a fit to to the data with a radius ofRHeI = 7.810.15 nm andRHeI = 3.140.11 nm. Thesevalues are in excellent agreement with those determinedindependently via Knudsen effusion measurements. Im-portantly, it demonstrates de facto that our experimentcan quantitatively determine the mass flow near the -transition in very small channels.

The mass flow was measured as a function of temper-ature across the superfluid phase transition T at severalpressures for both pores. These data are displayed inFig. 2 (C) and (D) with the offset previously discussedsubtracted. Previous work in Vycor[12] have found thesuperfluid transition to be suppressed to 1.95K, however,the superfluid transition in our channels is observed atthe temperature corresponding to the bulk value, 2.17K.

This is not surprising since we measure the total conduc-tance of the nanopore channel andof the source reservoirin series, so the onset of superfluidity in the bulk is firstobserved atT. Considering only data below T, we canextract the superfluid velocities using the two-fluid modelwhere we assume Qtot = Qn+ Qs = (nvn+ svs)R

2.Subtracting Qn from the total mass flow using Eq. (1)yields the superfluid portion of the flow with a velocityvs = Qs/R

2s. The superfluid density is taken fromthe bulk, as justified by previous work in Vycor (with asimilar network pore size), albeit with a lower transitiontemperature [12]. The extracted superfluid velocities areshown in Fig. 3 for the lower pressure datasets, wherelinear response is expected to be a better approximation,and where the datasets were taken over a large range oftemperatures. An inspection by eye readily shows thatthe superfluid velocities are smaller in theR 3 nm poreat similar pressures and temperatures. Such suppressionof the flow velocity as the radius is decreased is in starkcontrast with the bulk behaviour and shows that dissi-pation is increasing as the radius of the pore approachesa few nanometers.

Near the bulk superfluid transition, it is well-established that the superfluid density follows a uni-versal powerlaw form s = 0(1 T /T)

, where is

a correlation length critical exponent found experimen-tally to be close to 2/3. Considering a slowly-varyingquantum-mechanical wavefunction with a phase , thekinetic energy of the superfluid is given by sv2s/2 =s(h

2/2m2)||2. From scale invariance, we expectthat near T, the mean square of the superfluid veloc-ity should scale with the correlation length 4(T) asv2s 1/4(T)

2 (1 T /T)2. This result is strictly

speaking valid only at temperatures very closed to T,(1 T /T)


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displays a significantly distinct exponent 0.66 0.05 andzero-temperature critical velocity vc0 = 30.1 2.4 m/s.While not a proof, given the limited range in tempera-ture explored, this non-universal powerlaw behaviour asthe dimensionality is reduced is consistent with expec-tations from quantum hydrodynamics in 1D where in-creased fluctuations should prohibit long range order.

Other important features of the flow data not previ-

ously observed are (I) the extremely weak pressure de-pendence below T for the smaller pore, and (II) anoverall decrease in critical velocity as the channel sizeis reduced, in contrast to the behaviour vc

hmRln(

Ra0

),with a0 the size of the vortex core, predicted by Feyn-man and found in larger channels (see supplementaryinformation). The former is a hallmark of the macro-scopic phase coherence that exists in a superfluid phase,in sharp contrast with the Euler prediction of a classicalinviscid fluid, vs =

2P/s. Using the Gibbs-Duhem

relation to convert a pressure to chemical potential differ-ence, energy conservation dictates that there must exist adissipation mechanism in the channel with a rate suchthath = mPs

12

mv2s . From our data, it is clear that

the dissipation rate must be flow (pressure) dependent.The question of how energy is dissipated in superfluidshas a long history, beginning with the proposal of An-derson[18]that, in analogy with the Josephson effect insuperconductors, a steady state non-entropic flow maybe achieved at a critical velocity vc via a mechanism thatunwinds the phase of the order parameter in quanta of2. Such phase slips, occurring at rate , correspondsto a process whereby the amplitude of the order param-eter is instantaneously suppressed to zero at some pointalong the channel, and can be driven by either thermal or

quantum fluctuations. Momentum conservation dictatesthat such events can only occur in the presence of brokentranslational invariance along the pore [19].

Microscopically, dissipation occurs through the cre-ation of quantized vortex rings, the topological defectsof superfluid hydrodynamics. In our experiments, thesize of critical vortex ring, Rc plays a crucial role, and itis determined by the equilibrium condition between therelative frictional force between the normal and super-fluid component and the hydrodynamic forces acting onthe ring in the presence of flow. Energetically, this man-ifests as a competition between a positive vortex energythat scales linearly with radius and a negative kinetic

core energy scaling like its area. Langer and Fisher [20]found Rc 3 nm below T, exactly the length scale ofthe smallest pore considered here. When R < Rc, themaximum size of a vortex ring is constrained by the ra-dius of the channel, and thus the energy barrier for theircreation is lowered, leading to increased dissipation andan upper bound onvs set by the Feynman critical veloc-ity. The suppression in the observed critical velocity atT = 1.5 K as a function of decreasing radius shown inFig.3 can then be interpreted as a crossover to a regime

FIG. 3: Superfluid velocities. (A) The superfluid veloci-ties are shown at several pressures below 1 bar for the 7.81 nmpore (open symbols) and 3.14 nm (closed symbols), and 20 nmpore (half-filled symbols)[15]. The dashed lines are fit usingthe powerlaw vs(T) = vc0(1

TT

) (see text). (B) Log-log

plot of the superfluid velocity versus the reduced temperaturefor the 3.14 nm pore data. The closed symbols indicate thedata used for the power-law fit of the data taken at 482 (solidline) and 827 (dashed line) mbar pressure. (C) Critical ve-locity in three nanopores extracted at 1.5K temperature (atseveral pressures less than 1 bar) in order to compare withprevious work in much larger channels (see supplementaryinformation). The superfluid velocities are assumed to be

reaching the critical velocity. The dotted line is a blind linearfit shown here only as a guide-to-the-eye.

where flow is dominated by the physics of the channel.As the channel radius continues to decrease further, itis expected that backscattering of helium atoms at lowtemperature in the guise of quantum phase slips will in-crease, resulting in a continued suppression of the criticalvelocity.


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This argument does not address the actual rate, orprobability per unit space time that topological defectsare created, and experimental estimates of were firstmade by Trela and Fairbank [21], who found 1 Hz forsuperfluid flow through constrictions with R 104 m.For the nanoscale pores considered here, we estimatethat 3 5 GHz, well below the flow rate of 7.5 1012 atoms/s measured in our smaller pore, yet approach-

ing the quantum of mass flow q= m2/h 1010 atom/sat one bar differential pressure and fluid density taken atsaturated vapour pressure.

The behaviour of superfluid helium flow was studied incapillary channels down to 3 nm radius. For the smallerpore, the superfluid velocity can be well described by apowerlaw and it was found to be significantly smallerthan in larger channels. This likely signals the crossingover to a quasi-one-dimensional state whereby increasedfluctuations and interaction renormalization are modify-ing superfluidity. As the channel size is reduced evenfurther, near, or into the sub-nanometer range, we ex-pect to observe physics characteristic of the truly one-dimensional limit. In this regime, the algebraic decay ofthe superfluid order parameter will manifest itself as a re-duction in the superfluid density as a function of channellength and the appearance of non-universal powerlawsin the massflow dependence on pressure (Q1D P)and temperature (Q1D T

) . Such observations wouldbe strikingly different than that seen due to the macro-scopic quantum coherence of bulk helium, and wouldsignal the experimental discovery of a one-dimensionalbosonic quantum fluid.

We thank I. Affleck for illuminating discussions andcomments. We also thank the (CM)2 facility at Poly-

technique Montreal for providing access and help withthe TEM. This work was funded by Natural Sciences andEngineering Research Council of Canada (NSERC), theFonds Quebecois de la Recherche sur la Nature et lesTechnologies FRQNT (Quebec) and the Canadian Insti-tute for Advanced Research (CIFAR). All data, analysisdetails and material recipes presented in this work areavailable upon request to G.G.

[1] E. Varoquaux, C. R. Phys. 7, 1101 (2006).[2] F. D. M. Haldane, J Phys C: Solid State Phys 14, 2585

(1981);Phys. Rev. Lett. 47 , 1840 (1981).[3] A. Del Maestro, M. Boninsegni, and I. Affleck, Phys. Rev.

Lett.106 , 105303 (2011), and references therein.[4] H. Ishii, H. Kataura, H. Shiozawa, H. Yoshioka, H. Ot-

subo, Y. Takayama, T. Miyahara, S. Suzuki, Y. Achiba,M. Nakatake, T. Narimura, M. Higashiguchi, K. Shimada,H. Namatame M. Taniguchi, Nature426, 540 (2003).

[5] O. M. Auslaender, H. Steinberg, A. Yacoby, Y.Tserkovnyak, B. I. Halperin, K. W. Baldwin, L. N. Pfeif-fer, K. W. West, Science308, 88-92 (2005).

[6] Y. Jompol, C. J. B. Ford, J. P. Griffiths, I. Farrer, G. A.

C. Jones, D. Anderson, D. A. Ritchie, T. W. Silk, A. J.Schofield,Science325, 597-601 (2009).

[7] D. Laroche, G. Gervais, M. P. Lilly and J. L. Reno,Science343, 631-634 (2014).

[8] M. A. Cazalilla, R. Citro, T. Giamarchi, E. Orignac, andM. Rigol, Rev. Mod. Phys. 83, 1405 (2011).

[9] T. Eggel, M. A. Cazalilla, and M. Oshikawa, Phys. Rev.Lett.107, 275302 (2011).

[10] B. Kulchytskyy, G. Gervais, and A. Del Maestro, Phys.

Rev. B 88, 064512 (2013).[11] L. Pollet and A. B. Kuklov, Phys. Rev. Lett.113, 045301

(2014).[12] G. M. Zassenhaus and J. D. Reppy, Phys. Rev. Lett. 83,

4800 (1999), and references therein.[13] R. Toda, M. Hieda, T. Matsushita, N. Wada, J.

Taniguchi, H. Ikegami, S. Inagaki, and Y. Fukushima,Phys. Rev. Lett. 99, 255301 (2007).

[14] B. Yager, J. Nyeki, A. Casey, B. P. Cowan, C. P. Lusher,and J. SaundersPhys. Rev. Lett.111, 215303 (2013).

[15] M. Savard, G. Dauphinais and G. Gervais, Phys. Rev.Lett.107, 254501 (2011).

[16] M. Savard, C. Tremblay-Darveau and G. Gervais, Phys.Rev. Lett.103, 104502 (2009).

[17] H. L. Langhaar, J. Appl. Mech.9, A55-A58 (1942).[18] P. W. Anderson,Rev. Mod. Phys.38, 298 (1966).[19] S. Khlebnikov,Phys. Rev. Lett. 93, 090403 (2004).[20] J. S. Langer and M. E. Fisher, Phys. Rev. Lett. 19, 560

(1967).[21] W. J. Trela and W. M. Fairbank, Phys. Rev. Lett. 19,

822 (1967).


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SUPPLEMENTARY INFORMATION

Flow measurements

Design

Figure 1(A) shows a CAD drawing of the experimentalcell used for the gas flow measurements and Fig. 1(C)and (D) shows a field-emission transmission electron mi-croscope (FE-TEM) images of the nanoholes used. TheSi3N4 wafer is installed in a coin silver cell and sealed byan indium o-ring separating two reservoirs (inlet and out-let) in an experimental cell designed such that any masstransfer between the two reservoirs is restricted to occurthrough the nanohole. Capillaries connect the extremi-ties of the experimental cell to a gas handling system suchthat pressurized helium can be introduced in the cell, flow

through the nanohole, and be pumped from the outlet ofthe cell by a mass spectrometer, see Fig. 1(B). We usepacked silver powder heat exchangers to condense heliumbefore it enters the inlet of the cell and to ensure a goodthermal anchor to the cryostat. The temperature wasdetermined using two calibrated ruthenium oxide ther-mometers apposed on the experimental cell and the 3Hepot. The temperature control was achieved with the PIDloop of a LakeShore 340 AC resistance bridge. The co-ordination of the measurement and the PID control werehandled in a homemade Python interface.

The drain pressure below the membrane (PD) is kept

at vacuum through continuous pumping and helium gasis introduced in the top part of the cell creating a pres-sure gradient P = PSPD PS which induces amass flow Qm. This flow was detected with a Pfeiffervacuum Smart Test HLT560 calibrated with an externalstandard leak of 2.79 108 atm cc/s 10 15%. Acartoon representation of the whole experiment is shownin Fig. 1(B). The two reservoirs are depicted by capillaryconductancesGS and GD in series before and after thenanohole with a conductance Gnh. The mass spectrom-eter is denoted byAMand measures the volumetric flowwhen the drain side of the set-up is kept under vacuum,typically below 2 103 mbar. With our technique, the

total conductance G1

T = G1

S +G1

D +G1

nh of the cir-cuit is measured. The source and drain conductance canbe estimated using the infinite pipe approximation forPoiseuille flow (GS 10

11 ms at 1 bar) and Knud-sen free-molecular diffusion (GD 10

13 ms at 103

mbar). These conductances are several orders of magni-tude larger than the nanohole conductance which has atypical value Gnh 10

18 ms (see Fig. 2(B)). We cantherefore neglect the source and drain conductance to avery good approximation.

300 600

Time (s)

Time (s)

3000 9000 15000

Flo

wQv(a.u.)

FlowQv(a.u.)

(A)

(B)

FIG. 4: Flow measurements. (A)Volumetric flow through

a single 101 nm nanopore at 77K in the gas phase of he-lium when the pressure differential is decreased in a stepwisefashion. The red line is a fit of the function Qf(t, Pf) =

Qi(Pi) + Qet/ used to extract the equilibrium value. (B)

Similar measurements in the 6 nm diameter nanopore for thesuperfluid phase of helium when the temperature is increasedin a stepwise fashion. The red line is obtained in the samefashion as in (A).

Experimental procedure

The procedure for making the measurements is as fol-

lows: we first empty both sides of the cell at a tempera-ture well above the helium boiling point so as to ensurethat no residual helium is present in either reservoir. Themass spectrometer is then connected to the outlet of thecell to determine a background signal that is treated as anoffset to the pressure-driven flow of interest in this study.This background signal was found to be always less than 5101 pg/s in the liquid phase and less than 3102

pg/s in the gas phase. In the liquid phase it is less thanthe measured mass flow by a few orders of magnitude


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FIG. 5: Nanopore structural stability. The TEM im-age shows the nanopore at several days apart while kept atroom temperature in a clean environment. This structuralrelaxation has been observed in several samples with differentR/L ratios.

whereas in the gas phase at extremely low pressures iteventually becomes comparable to the flow signal. In the

next step, the whole apparatus is cooled below the -transition so that gaseous helium introduced from the gashandling system condenses and fills the heat exchangerand inlet of the experimental cell. Once condensation isachieved, the higher pressure above the membrane forcesthe liquid helium to flow through the nanohole. Whenatoms reach the very low pressures in the drain reservoir,they evaporate and are pumped out to the mass spec-trometer. The volumetric flow signal is then monitored asthe temperature of the cell is slowly increased. The mea-surement is then repeated at different pressure gradientsacross the nanopore. The volumetric flow is convertedinto a mass flow usingQm= (QvQv,bkg)/(10RsTroom).

The factor of ten here comes from the transformationof liters to cubic meters and mbars to Pascals. Finally,Troom is in Kelvin and the specific gas constant for he-lium Rs = R/Mm,He is in Joules per kilogram KelvinJ/(kgK).

Time constants and flow

In analogy with an electrical circuit with a time con-stant =RC, the time required for the mass flow signal

to stabilize upon a pressure of temperature variation is in-versely proportional to the conductance of the nanohole.Figure4shows measurements of the volumetric flow ver-sus time and a fit of the signal with an exponential decayfunction of the form Qf(t, Pf) = Qi(Pi) + Qe

t/. Thetime constantfor the superfluid flow through the small-est nanopore is typically of order of 2000 seconds. Wehave verified that waiting over a period of time longerthan 2 did not improved the accuracy of the fit in asignificant fashion.

Radii determination

Nanopore structural stability

The nanopores used in the present study were fabri-cated by focusing a TEM-FE beam on 30 nm thickSi3N4membranes as in Ref. [15, 16]. For the smaller nanopores,we have have found that it had a tendency to relax dur-ing post-fabrication. An example of such relaxation isshown in Fig. 5where the nanopore was imaged at differ-ent times following the fabrication. While this structuralrelaxation is more acute for the smaller pore, we havefound that the relaxation process stopped at cryogenictemperature, below 5 K. This was verified by perform-ing Knudsen effusion measurements before and after longperiod of time during which the membrane was kept athelium temperature. However, because of the deadtimebetween the fabrication and the cooling procedure in thecryostat, this relaxation process causes an uncertainty inthe radii determination of the pore. For this reason we

have developed two independent ways to determine theradii in situ using both Knudsen effusion and classicalfluid dynamics.

Radius determination from Knudsen effusion

The methodology is similar to that reported in Ref.[16] where the conductance of the nanopore is measuredas a function of the Knudsen number (defined here asthe ratio of the atoms mean free path to the nanoporediameter). For the smaller nanopore, the Knudsen num-ber is sufficiently high that we can therefore neglect the

contribution from the viscous regime. The Knudsen con-

ductance is given by Gth= R2(R,L,)

2RsTcellwhere

R is the radius of the nanopore, L its length and theopening angle of the nanopore (when= 0 the nanoporeis a cylinder). The opening angle of other nanoporeswith similar dimensions were measured using a TEM to-mography technique in[2] and was found to be close to30. In our case, an angle near 15 was found to best fitthe Knudsen effusion data. While this angle is consistentwith the profile reconstructed from the TEM picture, theuncertainty in its precise determination will lead to anuncertainty in the radius. The Clausing factor (R,L,)

is a number between 0 and 1 that express the probabil-ity for an atom to go from one side of the nanopore tothe other by bouncing on the walls. Figure6shows thevalue of the conductance as a fonction of Knudsen num-ber Kn. The data at higherKn have larger uncertain-ties because they correspond to very low pressures/flowregimes. The radius of the nanopore and its uncertaintywere extracted from the minimization ofGth(R) Gexpand Gth(R) (Gexp Gexp) for which L = 30nm and = (15 5) and where Gexp is the weighted average


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FIG. 6: Determination of the radius by Knudsen ef-fusion. Conductance values obtained from the ratio of themeasured mass flow and the applied pressure gradient for thesmaller nanopore. The green circles and the blue squares are

Knudsen effusion measurements at 77K made prior and af-ter superfluid flow measurements, respectively, two monthsapart from each other. The dashed and dashed-dotted linescorrespond to the maximum value ofGexp+ Gexp, and theminimum ofGexp Gexp, respectively. The data points aredisplayed here with three standard deviations.

.

of the measured values. The deviation Gexp here corre-sponds here to three standard deviation from Gexp. Theextracted values for the radii are RKn = 8.2 0.5 nmand RKn = 3.1 0.35 nm.

Radius determination from the viscous normal flow

Pressure sweeps were performed in the normal phaseof the liquid helium and the data were fitted against aslightly modified model of short pipe viscous flow fromLanghaar [17],

Qm=8L

1 + R4

322L2P 1

+Qm,offset (2)

.

The last term, Qm,offset, is required here because weobserve a spurious signal as P0. This signal is be-lieved to arise from evaporation at the walls on the drainside. The free parameters in Eq.(2) are the radius R, themass flow offset Qm,offset, and which is a geometry-dependent factor accounting for the acceleration of thefluid at the nanopore boundary. The best fit valueswere determined using a least squares method, evaluating

(Qm,modelQm,meas)2 over a cube in parameter space

in order to find a global minimum. In Fig. 2 (A) and

FIG. 7: Critical velocities versus channel size. Criticalvelocities at temperature T = 1.5K from this work (colour)are displayed alongside with critical velocities from previouswork (black and white) and summarized by Varoquaux [1].The open squares correspond mainly to pressure driven ACflow experiment with circular nanopores in 100nm thicknickel foil and in nanoporous 5m thick mica. The open trian-gles represent an heterogenous set of data collected from var-ious type of experiments (heat flow, oscillations). The outliertriangle data point at small radius is a thin film experiment.Because these measurements were performed at temperatureT 0.95K and above, the superfluid critical values is quotedat 1.5K.

(B), the solid line is a fit to to the data with a radius ofRHeI = 7.810.15nm andRHe1 = 3.140.11nm. These

values are in excellent agreement with those determinedindependently via the Knudsen effusion measurementsdiscussed above.

It is interesting to note that, as and arenearly constant in the normal phase, for sufficientlysmall values of R the influence of the parame-

ter becomes negligible: 8L

1 + R

4

322L2 P 1

8L

1 + R

4

642L2 P 1

= R4

8L P. This is the case

for the smaller nanopore of radius R 3 nm, but not forthe larger nanopore ofR 8 nm. The parameter wasdetermined from a fit using Knudsen effusion data and

found to be equal to 4.7. We have verified that modifica-tions in Qm,model arising from a variation of between0 and 10 was negligable, i.e. the value of had little orno influence on the data.

Critical velocities

In the two-fluid model proposed by Landau and Tiszathe total mass current is given by Jtotal = svs+ nvn


9/9

9

and total density is given by the sum of the superfluidand normal component = s +n. Per symmetry of thepore, we consider the flow to be in the axial direction ofthe nanohole so Qm,total = JtotalR

2. The normal partis well-modeled by the function given in equation (1),with here replaced byn. The superfluid velocities aregiven byvs= Qs/R

2s= (Qm,totalQn)/R2.s. The

values from our work are reported in Fig. 7at 1.5 K, per

previous convention in the literature. Finally, we havemade the assumption that the superfluid velocities were

reaching the critical velocities,i.e. the superfluid velocitywas only limited by dissipation.

[1] E. Varoquaux, C. R. Phys. 7, 1101 (2006).[2] M. J. Kim, M. Wanunu, D. C. Bell, and A. Meller, Ad-

vanced Materials 18, 3149-3153 (2006).

Critical Flow and Dissipation in a Quasi-One-Dimensional Superfluid

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