Proceedings of the 29th International Conference on Low Temperature Physics (LT29)
Downloaded from journals.jps.jp by 157.1.136.5 on 10/10/24
Proc. 29th Int. Conf. Low Temperature Physics (LT29)
JPS Conf. Proc. 38, 011006 (2023)
https://doi.org/10.7566/JPSCP.38.011006

Superﬂow of 4He through an oriented nanometer-sizedporous membrane
Masato Kuribara and Junko Taniguchi and Masaru Suzuki
1

Department of Engineering Science, University of Electro-Communications, Tokyo 182-8585,
JapanE-mail: tany@phys.uec.ac.jp(Received July 31, 2022)
In a quasi-one dimension (1D) 4 He system, not only the suppression of superﬂuid growth but also
the behavior of critical velocity has attracted great interest. We have observed the superﬂow of 4 He
through straight uniform nanochannels, by means of the bundle of highly oriented channels synthesized in a porous alumina membrane. Under 0.13 MPa, the mass velocity of superﬂow shows a small
rise at around 1.5 K, which can be attributed to the superﬂow appearance in the nanochannel. The
superﬂuid velocity evaluated from the mass velocity is at most in the order of 10−1 m·s−1 , which isnearly two digits smaller than the value which was previously reported for a single 6.3-nm hole.
KEYWORDS: superﬂow, low dimension, critical velocity

1.

Introduction

Superﬂow of 4 He through channels has been understood to be limited by quantized vortices. The
massive experiments demonstrate that its critical velocity increases with decreasing channel diameter
down to the micrometer size. In this size region, the excitation of vortex ring restricts the superﬂow,
as described by the Feynman criterion. Below the micrometer size, the critical velocity reaches aplateau, which is due to the phase slippage of nucleated vortex line. One of the most interestingquestion about the critical velocity is what happens when the channel diameter is further decreased
close to the coherence length. Maestro et al. theoretically predicted that the critical velocity will drop
drastically due to the decrease in the energy barrier for the nucleation [1].
On the experimental side, Duc et al. measured the superﬂow through a single nanohole, whose radius ranged from 3 to 20 nm, and ﬁrst observed the reduction of critical velocity [2]. Their nanoholes
are fabricated in the Si3 N4 membrane (30 nm in thickness) using an electron beam, and thus its
sectional view along the hole is like an oriﬁce.
Motivated by the progress of the study on the critical velocity behavior, we have started preliminary measurements of superﬂow through the bundle of straight uniform nanochannels with a smaller
diameter than that of the previous work. The critical velocity at a much lower temperature than that ofsuperﬂuid appearance is evaluated at most in the order of 0.1 m·s−1 , which is two orders of magnitude
smaller than the previously observed values [2].

2.

Experiments

The adopted mesoporous medium was the porous-alumina membrane with perpendicularly oriented nanochannels [3]. The host porous-alumina membrane is 60 µm in thickness and has 100-nm
pores, whose density is ∼10 per 1 µm2 . Nanochannels are synthesized in the 100-nm pores. Inside thepore, hundreds of nanochannels get together to form a bundle, as shown in Fig. 1 (a). The nanochan1011006-1©2023 The Author(s)This article is published by the Physical Society of Japan under the terms of the Creative Commons Attribution 4.0 License. Any further distribution of this work
must maintain attribution to the author(s) and the title of the article, journal citation, and DOI.

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JPS Conf. Proc. 38, 011006 (2023)

nel is 5-20 µm in length and is located near the edge of the pores. From the TEM photograph, thenumber of nanochannels per one pore is estimated to be n =350±20. The diameter of the nanochannel
is uniform and is evaluated as 3.4 ± 0.2 nm from a nitrogen adsorption pressure isotherm at 77 K bymeans of the Barret-Joyner-Halenda (BJH) method [3].
(a)(b)
PGrd
bundle of 3.4-nmnanochannels
5-20mmPGru

vb

Prd, Vrd

Pru, Vru

vdvu
gas 4HeLTliquid 4He
100-nm poreof alumina

Pld, Vld
PGld

Fig. 1.
nels.

Plu, Vlu
PGlu

Schematic view of (a) an experimental cell, and (b) a 100-nm pore containing a bundle of nanochan-

We attached this membrane to a copper ﬁlm with a 0.7-mm hole in the center using the Stycast
1266. Due to the meniscus of Stycast, the diameter of the membrane where gas or liquid can permeate
was 0.4 mm. Thus, the numbers of 100-nm pores and nanochannels inside the 0.4-mm hole were
estimated to be N =1.26 ×106 and N × n = 4.4 × 108 , respectively. The experimental cell was madeof beryllium copper and was separated by the copper ﬁlm, as shown in Fig. 1 (b). Capacitive pressuregauges were located at both sides of the ﬁlm to detect the pressures at low temperatures directly. The
volume of the both sides, Vld and Vlu are 0.59 and 0.91 cm3 , respectively.
Before the experiments of superﬂow, we measured the conductance of gas 4 He under 0.01 MPa
at 300 K, and obtained the value of (4.6 ± 0.5) × 10−11 m3 ·s−1 . Under this condition, due to the longmean free path (1.2 µm), a molecular ﬂow passes both the nanochannel and the 100-nm pore. On theassumption that their surfaces are smooth and specular, we calculated the total conductance (Ct ) per
one pore serially connected to the bundle of nanochannels, using the equation
11
1=+.
Ct Cp n Cn

(1)Here, Cp and Cn are the conductances of the 100-nm pore and the nanochannel, respectively. The
measured value is as low as ∼6% of the calculated value. There are several possible factors in lowering
the conductance, such as blocking the edge of the nanochannels by contaminations and lowering
diﬀusivity by a roughness or narrowness of the nanochannel. Recently, the eﬀective diﬀusivity ofvarious porous media has been measured, and the eﬀective diﬀusivity dropped by a factor has beenreported for several sets of adsorbents and porous media under low pressure [4].
The experimental cell was set to the stage, which was anchored to a mixing chamber of a dilution
refrigerator. The temperature was measured by a Ruthenium Oxide thermometer. For the temperature2

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JPS Conf. Proc. 38, 011006 (2023)

ranging from 0.2 to 2.3 K, we measured the ﬂow of liquid 4 He under 0.13 and 0.75 MPa. First, weset the pressures, Pru , Plu , Pld , Prd align to the same, and then closed the all valves. (see Fig. 1 (b))
Next, we lowered only Prd by 0.01 or 0.02 MPa and opened the valve Vd . After that, we observed thevariation of Plu and Pld until they reached equilibrium.

3.

Results and Discussion

Figure 2 (a) shows a typical variation of Plu and Pld above the superﬂuid transition temperature ofbulk liquid 4 He (T λ ) under 0.1 MPa. At 2.40 K (> T λ ), the diﬀerential pressure relaxes exponentiallyand reaches equilibrium smoothly, which indicates a viscous ﬂow. From the observed relaxation, thetotal conductance was obtained as 7.16×10−10 m3 ·s−1 . It is almost 100 times as large as the valueof the no-slip Hagen-Poiseuille ﬂow calculated on the assumption that the viscosity of 4 He in thenanochannel is equal to that in the 100-nm pore (bulk space). Furthermore, it corresponds to thevalue when 32% of the 100-nm pores remain as it is without nanochannels. However, the existence ofso many remnant 100-nm pores is irrational since the conductance of gas 4 He at 300 K is even lower
than the value calculated on the condition that all 100-nm pores contain the bundle of nanochannels.

(a)

(c)
0.132
2.40 KP / MPa

0.13

Plu

0.130
0.128

0.65 K

Pld
0.126

(b)

01000t/s

2000

0.132
2.15 KP / MPa

0.130

1.00 K

Plu / MPa

0.124

0.80 K

1.20 K
1.30 K
1.46 K
1.50 KPlu
0.12

1.60 K

0.128

1.80 K
1.90 K

Pld

0.126

tE

2.10 KtE

0.1240100t/s

200050t/sFig. 2. Variations of Plu (orange) and Pld (black), at (a) 2.40 and (b) 2.15 K under 0.13 MPa. Blue dashedcurves are ﬁtted ones using eq. (2). (c) Variations of Plu at various temperatures under 0.13 MPa. For clarity,
the data are shifted by about 0.01 MPa.

Recently, an enormous enhancement of water ﬂow in carbon nanotube (CNT) compared withthat of the no-slip Hagen-Poiseuille ﬂow has been reported by many groups [5–10]. For 7-nm multiwalled CNT membrane, 4-5 orders of magnitude enhancement were reported [5], and for 1.3-2.0 nm,
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enhancement factors were 560-8400 [6]. Since the ﬁrst observation [5], various causes have beenstudied. The eﬀect of conﬁnement has been well-researched, and it is understood that below a certain
size, a slip, i.e., low resistance to liquid ﬂow, occurs. Considering that the interaction between 4 Heand SiO2 (the material of the nanochannel) is smaller than that between H2 O- and C, a slip may occur
in the nanochannel.
Figure 2 (b) shows the variation of Plu and Pld at 2.15 K, which is just below T λ . Plu shows abreak when it reaches the same value as Pld . It means that there exists a ﬁnite ﬂow rate independentof the diﬀerential pressure, which seems to be caused by a superﬂow. The behavior just below T λ wasagainst expectation since in other experiments using a quartz tuning fork, a superﬂuid response in thenanochannel was not observed down to ∼1.6 K under unsaturated vapor pressure [11]. In addition,
the value of 1.6 K seems to be consistent with the previous torsional oscillator measurements for 4 He
conﬁned in nanochannels, where the superﬂuid response appears at 0.89 K in a 2.8-nm channel andat 1.80 K in a 4.7-nm channel [12]. We will discuss the observed strange behavior later.
We measured the relaxation of Plu for various temperatures below T λ as shown in Fig. 2 (c).
For all temperatures, the break at which Plu reaches the same value as Pld was observed clearly. Theelapsed time (tE ) from the start of ﬂow to the break of Plu is about 50 s at 2.10 K. ∆t decreases
gradually with decreasing temperature. Then, it shows a drop below 1.50 K and becomes almost
constant below 1.30 K. For all temperatures, the time dependence of Plu is always slightly convex
downward. It means that in addition to the superﬂow whose velocity is independent of the diﬀerentialpressure, there exists the ﬂow whose conductance depends on the diﬀerential pressure, like a viscous
ﬂow. Thus, we assume that the viscous ﬂow and the superﬂow coexist and adopt the model of thevariation of Plu as,[]dPlu (t) = −Cv (Plu (t) − Pld (t)) − γ s dt.
(2)Here, Cv is the conductance of viscous ﬂow, and γ s is the rate of pressure variation due to superﬂow.
Putting the origin of the viscous ﬂow aside, we ﬁtted the data by computing eq. (2) using Cv and γ s
as ﬁtting parameters. The data were well ﬁtted as shown as a blue dashed curve in Fig. 2 (a) and (b).
(a)(b)
2000.75 MPa

Cv (10-8 m3 s-1)

0.75 MPa

gs (Pa㺃s-1)

To
1000.13 MPa

640.13 MPa

2001002T (K)
Fig. 3.

12T (K)

Temperature dependence of (a) γ s and (b) Cv under 0.13 and 0.75 MPa.

The obtained Cv and γ s are summarized as a function of temperature in Fig. 3 (a) and (b), respectively. With decreasing temperature, γ s rises at around T λ and then shows the second rise at a certain4

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JPS Conf. Proc. 38, 011006 (2023)

temperature (T o ). T o is 1.5 K under 0.13 MPa, and lowered to 1.4 K under 0.75 MPa. Since these
are close to the temperature at which the superﬂuid response was observed by a tuning fork underunsaturated vapor pressure, the rise at T o can be attributed to the appearance of superﬂow throughthe nanochannel [11]. As is clear from Fig. 3 (b), Cv also shows a two-step increase in a similar wayto that of γ s . First, Cv rises at around T λ , and then rises again at a little higher (lower) temperaturethan T o under 0.13 (0.75) MPa. A possible origin of the co-movement of γ s and Cv is mutual friction
between the superﬂuid and the inert solid layer in the nanochannel. The frictional force is expected
to increase with increasing superﬂuid velocity, resulting in an increase in the viscous ﬂow.
Why does the γ s term appear just below T λ ? Duc et al. also observed the superﬂow throughthe single nanohole just below T λ and explained that the total conductance of the nanohole seriallyconnected to the source reservoir was strongly inﬂuenced by superﬂuidity in the reservoir [2]. In thepresent experiments, the bundle of nanochannels is serially connected to the 100-nm pore. Besides,
the observed total conductance at 2.40 K (> T λ ) demonstrates that the magnitude of Cp is in the sameorder of n Cn , suggesting that Cp can signiﬁcantly vary Ct .
Finally, we discuss the superﬂuid velocity through the nanochannel. The mass velocity per oneγsdρnanochannel Qms due to superﬂow is related to γ s via Qms = Nn· dP· Vlu . The superﬂuid velocity
Qms
through the nanochannel is derived from the equation of v s =. Here, ac is the radius of
πa2c ρ sc
the nanochannel, and ρ sc is the superﬂuid density in the nanochannel. Although the temperaturedependence of ρ sc is unknown, ρ sc can be approximated by the density of 4 He below about 1 K, whereγ s is almost constant. The assumption that all of the nanochannels are open gives us the minimum
value of the superﬂow velocity. The value obtained from the diﬀerence between the γ s at 1.5 and
0.6 K was in the order of 0.01 m·s−1 . The gas ﬂow measurements suggest the possibility that only 6%of nanochannels are open. Even if only 6% of the nanochannels are open, the superﬂuid velocity is as
low as 0.17 m·s−1 , which is two orders of magnitude lower than the value observed for a single 6.3nm hole. To evaluate the critical velocity of 4 He in the nanochannel, further research on temperaturedependence of superﬂuid density or of superﬂuid velocity is needed. However, the order of criticalvelocity is certainly small compared with the value for larger holes, which supports the theoreticalprediction that the critical velocity is reduced by narrowing the channel because of lowering theenergy barrier for vortex line nucleation.

4.

Summary

We have observed the superﬂow of 4 He in nanochannels by means of the bundle of highly oriented channels synthesized in a porous alumina membrane. Under 0.13 MPa, the mass velocity ofsuperﬂow shows a small rise at around 1.5 K, which can be attributed to the superﬂow appearance inthe nanochannel. Under 0.75 MPa, this temperature of superﬂow appearance is lowered slightly. Thesuperﬂuid velocity evaluated from the mass velocity is at most in the order of 10−1 m· s−1 , which isnearly two digits smaller than the value which was previously reported for a single 6.3-nm hole.
References[1] A. Del Masestro and B. Rosenow, Phys. Rev. B 95, 140507(R) (2017).
[2] P. Duc, M. Savard, M. Petrescu, B. Rosenow, A. Del Maestro, and G. Gervais, Sci. Adv. 1, e1400222 (2015).
[3] A. Yamaguchi, F. Uejo, T. Yoda, T. Uchida, Y. Tanamura, T. Yamashita, and N. Teramae, Nat. Phys. 3, 337(2004).
[4] R. Valiullin, J. Kärger, and R. Gläser, Phys. Chem. Chem. Phys. 11, 2833 (2009)[5] M. Majumder, N. Chopra, R. Andrews, and B. J. Hinds, Nature 438, 7064 (2005).
[6] J. K. Holt, H. G. Park, Y. Wang, M. Stadermann, A. B. Artyukhin, C. P. Grigoropoulos, A. Noy, and O.
Bakajin, Science 312, 1034 (2006).
[7] P. Zeigermann, J. Kärger, and R. Valiullin, Microporous Mezoporous Mater. 178 , 84 (2013).
5

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JPS Conf. Proc. 38, 011006 (2023)[8] K. Ritos, D. Mattica, F. Calabro, and J. M. Reese, J. Chem. Phys. 140, 014702 (2014).
[9] S. Casanova, M. K. Borg, Y. M. J. Chew, and D. Mattia, ACS Appl. Mater. Interfaces 11, 1689 (2019).
[10] A. Fan, Y. Hu, Y. Zhang, W. Ma, and X. Zhang, Nano Futures 5, 015003 (2021).
[11] A. Kaneko, J. Taniguchi, M. Suzuki, and M. Hieda, JPS Conf. Proc., in press.
[12] J. Taniguchi, Y. Aoki, and M. Suzuki, Phys. Rev. B 82, 104509 (2010).

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