Oscillating nematic aerogel in superfluid 3He
V. V. Dmitriev, M. S. Kutuzov, A. A. Soldatov, E. V. Surovtsev, A. N. Yudin
aa r X i v : . [ c ond - m a t . o t h e r] N ov Oscillating nematic aerogel in superfluid He V. V. Dmitriev +1) , M. S. Kutuzov ∗ , A. A. Soldatov + , E. V. Surovtsev + , and A. N. Yudin + + P. L. Kapitza Institute for Physical Problems of RAS, 119334 Moscow, Russia ∗ Metallurg Engineering Ltd., 11415 Tallinn, Estonia
Submitted November 30, 2020
We present experiments on nematic aerogel oscillating in superfluid He. This aerogel consists of nearlyparallel mullite strands and is attached to a vibrating wire moving along the direction of the strands. Previousnuclear magnetic resonance experiments in He confined in similar aerogel sample have shown that thesuperfluid transition of He in aerogel occurs into the polar phase and the transition temperature ( T ca ) is onlyslightly suppressed with respect to the superfluid transition temperature of bulk He. In present experimentswe observed a change in resonant properties of the vibrating wire at T = T ca and found that below T ca anadditional resonance mode is excited which is coupled to the main resonance.
1. INTRODUCTION
Superfluidity of He in aerogel can be investigatedusing a vibrating wire (VW) resonator immersed inliquid He with an aerogel sample attached to it. Inthis case an appearance of the superfluid fraction of He in aerogel influences resonant properties of the VW.Experiments with aerogel attached to the VW have beendone previously only with silica aerogel [1, 2, 3, 4] wheresuperfluid phases (A-like and B-like) have the sameorder parameters as A and B phases of bulk He. Theseexperiments have allowed to estimate the temperaturedependence of the superfluid fraction in A-like and B-like phases as well as to detect an influence of thesuperfluid flow on the texture of the order parameterin the A-like phase.In this Letter, we present results of experimentswith He in the so-called nematic aerogel using VWresonator. The nematic aerogel is a highly porousstructure consisting of strands with almost parallelorientation [5]. It has been established that in thiscase the superfluid transition occurs into a new phase(polar phase) that does not exist either in bulk He orin He in silica aerogel [6]. The polar phase becomesfavorable due to an essentially anisotropic scattering of He quasiparticles inside the aerogel [7, 8, 9, 10]. Thisphase has a superfluid gap with line of zeroes in theplane perpendicular to the specific direction [11, 12, 13].In nematic aerogel this direction coincides with theaverage direction of the strands, along which the meanfree path of He quasiparticles is maximal [7]. e-mail: [email protected]
2. SAMPLE AND METHODS
We used a sample of mullite nematic aerogel whichhas a form of cuboid with a size along strands ≈ . mmand characteristic transverse sizes ∼ × mm. It wascut from a larger piece of the original sample synthesizedby Metallurg Engineering Ltd so that it has perfectlyflat ends (the planes where strands begin and end):irregularities are about 100 nm. The sample consistsof nearly parallel mullite strands with diameters of ≤ nm (estimated from the scanning transmissionelectron microscope images) and has the overall density ≈ mg/cm . If we assume that the density of mulliteis 3.1 g/cm then the porosity of the sample is 95.2%and the average distance between the strands is 60 nm.Similar mullite sample (which was cut from the sameoriginal piece and was placed in a separate cell of thesame experimental chamber) has been used in nuclearmagnetic resonance (NMR) experiments in He [14, 15]where it was found that the superfluid transition of He in this sample actually occurs into the polar phaseand that the superfluid transition temperature ( T ca ) isonly slightly suppressed with respect to the transitiontemperature ( T c ) of bulk He. It was also found thaton further cooling the second order transition into thepolar-distorted A phase (the PdA phase) occurs, andthat effective mean free paths of He quasiparticles indirections parallel and transverse to the aerogel strandsin the limit of zero temperature are ≈ nm and ≈ nm correspondingly.The present sample was glued using a small amountof Stycast-1266 epoxy resin to a 240 µ m NbTi wire,bent into the shape of an arch with a total hight of10 mm and a distance between the legs of 8 mm asshown in Fig. 1. The stycast was left to thicken untilit was almost set before it was applied to the aerogel1 V. V. Dmitriev, M. S. Kutuzov, A. A. Soldatov, E. V. Surovtsev, A. N. Yudin
V H (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)
I=I sin(2 ft)
Fig. 1. Signal measurement circuit of a VW immersedin liquid He in external steady magnetic field H . Thestrands of nematic aerogel glued to the wire are orientedalong the oscillations to prevent the aerogel from soaking up the glue. Thewire is mounted in one of the cells of our experimentalchamber. The experiments were carried out at pressures7.1, 15.4, and 29.3 bar and in magnetic fields 305–1650 Oe. Necessary temperatures were obtained by anuclear demagnetization cryostat and measured by aquartz tuning fork calibrated by Leggett frequencymeasurements in bulk He-B and He-A in separateNMR experiments. To stabilize the polar phase innematic aerogel [16], the samples were preplated with & . atomic layers of He.A measurement procedure of the aerogel resonator isthe same as in the case of a conventional wire resonator[17]. The mechanical flapping resonance of the wire isexcited by the Lorentz force on an alternating currentwith amplitude I (from 0.05 mA to 0.5 mA in ourexperiments), passing through the wire in a steadymagnetic field. In liquid He the maximum velocityof the wire at such currents in the used range oftemperatures did not exceed 0.1 mm/s. The motion ofthe wire generates a Faraday voltage which is amplifiedby a room-temperature step-up transformer 1:30 andmeasured with a lock-in amplifier. In-phase (dispersion)and quadrature (absorption) signals are joint fitted toLorentz curves in order to extract a resonance frequency f a and a full width at half-maximum of the absorption(a resonance width) of the signal ∆ f a . For our resonator,the resonance frequency in vacuum ( f ) is 752 Hz.
3. THEORETICAL MODEL In He the resonance frequency of the VW resonatoris inversely proportional to the square root of theeffective mass ( M ) which is oscillating. We use a simple model in which we neglect effects of He flow aroundthe wire. We also do not consider effects due to afinite mean free path of He quasiparticles in bulk Hebecause our measurements have been carried out at
T > . T c where these effects can be neglected. Then M has five contributions [3, 18, 19, 20]: (i) the mass ofthe oscillating part of the wire and mass of the emptyaerogel which sum ( m ) defines f , (ii) the mass of thenormal-fluid fraction entrained in the aerogel ( m n = ρ an V ), (iii) the effective mass of the superfluid flow( m sf ), (iv) the effective mass of the normal componentpotential backflow ( m nf = αρ n V ), and (v) the effectivemass m v which is carried by the body due to viscosityof the normal component of liquid He. Here V is thevolume of He in aerogel, ρ an and ρ n are the densitiesof normal components of He in aerogel and in bulk He, and α ∼ is a geometrical factor ( α = 0 . fora sphere and α = 1 for a cylinder oscillating alongthe direction normal to its axis). Then the expectedresonant frequency equals f a = f m m + m n + m sf + m nf + m v , (1)The effective mass of the superfluid flow for anellipsoidal sample moving along a principal axis canbe found using analogy with a dielectric sample in anelectric field [19]: m sf = αV ( ρ s − ρ as ) ρ s + αρ as , (2)where ρ as and ρ s are the densities of superfluidcomponents of He in aerogel and in bulk He. Itshould be noted that Eq. (2) is obtained in the caseof isotropic superfluid density tensors of the phases inthe problem. The latter is valid only for the B phaseof superfluid He. The superfluid density tensor of thepolar phase (or the PdA phase) is anisotropic. As aresult, the mass of the superfluid flow depends on theangle between the intrinsic anisotropy axis of aerogeland the direction of motion of the sample. In our casethis angle is zero and ρ as is the superfluid density of thepolar phase (or the PdA phase) along the strands.The mass m v is related to the inertial part of theviscous drag force and can be estimated as ρ n Sδ , where δ is the viscous penetration depth and S is the surfacearea of the body. For a sphere with diameter D the exactresult is m v = 3 D r πρ n ηf , (3)where η is the shear viscosity and f is the frequencyof oscillations [21]. The dissipative part of the viscousdrag force determines the resonance width. If δ is much scillating nematic aerogel in superfluid He D/ then in resonance m v ≈ ρ n V ∆ f a /f a .Therefore, for the case of a general shape of the sample,the resonant frequency at T > T ca is expected to begiven by f a = f m m + ρV (1 + α ) + βρ n V ∆ f a /f a , (4)where β is a geometrical factor ( β = 1 for a sphere) and ρ is the density of He. The resonant frequency f n innormal He in the limit of η → (that is ∆ f a → )satisfies the following condition: f n − f = (1 + α ) Vm f ρ. (5)If ∆ f a ≪ f a then at T > T ca f a = f n − β ∆ f a . (6)In the above reasoning, we have assumed that thenormal and superfluid components of He are separatelyincompressible. As it is shown in Ref. [22] in superfluid He the compressibility affects Stokes parameters ( C and C ′ in notations of Ref. [22]) that affects m v (dueto change of C ) and ∆ f a (due to change of C ′ ).Fortunately, these changes are not large and C variesalmost linearly with C ′ [22]. Therefore, in the firstapproximation, in superfluid He m v also should beproportional to ∆ f a .
4. EXPERIMENTS IN NORMAL HE In our experiments δ is essentially less than thecharacteristic sizes of the aerogel sample: even near T c at P = 29 . bar δ ≈ . mm, so the observed resonanceproperties of our VW at T > T c are well describedby the theoretical model. In Fig. 2 by open symbolswe show dependencies of the resonant frequency onthe resonance width measured at different pressures at T > T c . It is seen that these dependencies agree withEq. (6). The solid lines in Fig. 2 are the best linear fitsto the data which allow us to determine f n and β . Weobtain that at all pressures β is close to 1, that is tothe value expected for a sphere. The obtained valuesof f n also agree with Eq. (5) (see the inset to Fig. 2):the slope of the line in the inset is . × − cm s /gwhile the value of the slope calculated from Eq. (5)(with α = 0 . and estimated value of m ≈ . mg)is . × − cm s /g. We note that the dependenceof f a on ∆ f a remains the same also at T ca < T < T c (filled symbols in Fig. 2). It means that the influence ofcompressibility of normal and superfluid components is
40 60 80 100 120 140 160 180 200 220480500520540560 29.3 bar 15.4 bar f a ( H z ) f a (Hz) f - - f - ( k H z - ) (g/cm ) Fig. 2. The resonant frequency versus the resonancewidth measured at 29.3 bar (circles), at 15.4 bar(triangles), and at 7.1 bar (squares). Open symbolscorrespond to measurements in normal He (
T > T c ),filled symbols have been obtained at T ca < T < T c .Solid lines are fits to the data at T > T c using Eq. (6)(squares: β = 1 . , triangles: β = 0 . , circles: β = 0 . ). I = 0 . mA, H = 1650 Oe. Inset: Thedependence of f n on ρ determined from linear fits shownin the main panel. Solid line is the best fit according toEq. (5). not essential. Our experience with bare VWs resonatorsshow that this dependence follows Eq. (6) down to T ∼ . T c .
5. EXPERIMENTS IN SUPERFLUID HE In Fig. 3 we show temperature dependencies of theresonance frequency and width measured at 29.3 bar. Oncooling in normal He the resonance width is increasingand the frequency is decreasing due to the Fermi-liquidbehavior of the viscosity ∝ /T corresponding to theincrease of m v . Then a rapid decrease of the width (arise of the frequency) is observed indicating a superfluidtransition in bulk He at T = T c . On further cooling,the second resonance appears (filled triangles in Fig. 3)accompanied by the spike in the width of the mainresonance. This additional resonance mode appearsjust below the superfluid transition temperature of He in the sample used in NMR experiments [14, 15].Therefore, we conclude that this resonance is due tothe superfluid transition of He into the polar phase inthe oscillating sample which occurs at T ca ≈ . T c .Although we have not been able to observe a clearresonance peak at frequencies lower than 470 Hz, we V. V. Dmitriev, M. S. Kutuzov, A. A. Soldatov, E. V. Surovtsev, A. N. Yudin T c R e s on a n ce fr e qu e n c y ( H z ) T/T c T AB T ca f a ( H z ) Fig. 3. Temperature dependencies of the resonancewidth of the main resonance (open circles) and ofthe frequencies of the main (filled circles) and thesecond (filled triangles) resonances. P = 29 . bar, I =0 . mA, H = 1650 Oe. Arrows mark T ca , T c , and ABtransition in bulk He at T = T AB . assume that on cooling from T = T ca the frequencyof the second mode rapidly grows from 0 and slightlylower T ca becomes close to the frequency of themain resonance resulting in an interaction (repulsion)between these modes. It is illustrated by Fig. 4(a) wherewe show the evolution of the VW absorption signalduring a very slow passage through T ca . It is seen thatjust below T ca there are two resonance peaks. Thetemperature dependence of the resonant frequenciesnear T ca is shown in Fig. 4(b) that demonstrates therepulsion of two resonance modes. For clarity, below T ca we continue to call as the main resonance the mode witha smaller frequency. As it is seen from Fig. 5, on coolingthe resonance frequency of another (second) mode ( f a )is increasing up to about 1600 Hz at T = 0 . T c . Similarbehavior of the second mode was observed at lowerpressures.We suppose that the second mode is an analog of aso-called slow sound mode observed previously in silicaaerogel immersed in superfluid helium [23, 24, 25]. Thepoint is that in aerogel the normal fluid component isclamped to the matrix, since δ exceeds the characteristicseparation of the strands. However, the skeleton ofaerogel is elastic and the normal component can movetogether with the strands. Therefore, the superfluidcomponent and the combined normal fluid and aerogelmatrix can move in opposite directions, resulting in asecond-sound-like mode [23] which resonant frequencygrows from 0 on cooling from T ca . In superfluid Hein silica aerogel such resonance mode was observed in
300 400 500 600 700
T = 0.985T c V W a b s o r p ti on Frequency (Hz)
T = 0.992T c
500 nV(a) (b) R e s on a n ce fr e qu e n c y ( H z ) T/T c T ca Fig. 4. (a) Temperature evolution of the VW absorptionsignal on slow warming from T ≈ . T c to T ≈ . T c . For better view, the absorptionlines are successively shifted upward with increasingtemperature. Thick (red) and thin (blue) linescorrespond to T > T ca and T < T ca respectively. (b)Two branches of the wire resonance versus temperaturenear T ca obtained by fitting the lines in panel (a) with asum of two Lorentz peaks. P = 29 . bar, T ca ≈ . T c , I = 0 . mA, H = 1650 Oe. the low-frequency sound measurements [24, 25]. We aredealing with a highly anisotropic aerogel which is softin the direction normal to the strands but is rigid in thedirection along the strand. Therefore, in our case theslow mode should correspond to periodic deformationsof the sample in the direction normal to the strands.We note that we detect motions of the wire, but we scillating nematic aerogel in superfluid He T AB T AB f a ( H z ) T/T c T AB f a ( H z ) T/T c Fig. 5. The resonance frequency and the resonancewidth (inset) of the slow sound mode in nematic aerogelversus temperature measured at P = 29 . bar (circles, T ca ≈ . T c ), P = 15 . bar (squares, T ca ≈ . T c ),and P = 7 . bar (triangles, T ca ≈ . T c ). The givensuperfluid transition temperatures are nearly the sameas measured in NMR experiments [14] with the similarsample. I = 0 . mA, H = 1650 Oe. can excite and detect the slow mode in aerogel, evenif its resonance frequency is far from the original VWmechanical resonance. It means that even well below T ca this second resonance is strong enough to affect the wireoscillations.If we neglect corrections due to that our sample isnot an ellipsoid, then, the results of measurements of thefrequency of the main resonance mode can be used toestimate the superfluid fraction ρ as /ρ of He inside theaerogel. For this purpose at
T < T ca we can subtractcontribution of m v into f a , using the dependence of f a = f a (∆ f a ) measured at T > T c . If we denote as ˜ f a the result of the subtraction then using Eq. (1) without m v we obtain the following equation, which allows toestimate ρ as /ρ : ρ s ρ as (1 + α ) ρ ( ρ s + αρ as ) = 1 − ( f / ˜ f a ) − f /f n ) − . (7)However, the existence of the second resonance makesthe estimation impossible. The point is that even wellbelow T ca the interaction between two resonant modesseems to be essential and their frequencies remaincoupled. Fig. 6 illustrates the influence of the secondmode on the frequency of the main resonance. If weexclude data points just below T ca (where frequencies ofresonance modes are too close to each other) then thereis a jump-like decrease in the main resonance frequencydue to the appearance of the second mode. At T =
60 70 80 90 100 110 120 130 140500510520530540 T=0.975 T c T=0.83 T c T=1.7 T c T AB T ca =0.985 T c f a ( H z ) f a (Hz) T c Fig. 6. The frequency of the main mode versus theresonance width measured at P = 15 . bar from . T c to . T c . Open triangles correspond to measurementsin normal He (
T > T c ), filled triangles are the data inthe range of T ca < T < T c , filled circles are the datain the range of . T c < T < T ca . T ca ≈ . T c . Thedata points in the range of . T c < T < T ca wherefrequencies and intensities of the resonance modes areclose to each other are not shown. . T c the frequency of the main resonance is ≈ Hzand it is by 15 Hz smaller than at T = T ca , despite thefact that the frequency of the second mode is alreadymuch higher ( ≈ Hz). On further cooling, thefrequency of the second mode continues to change, andwe cannot distinguish contributions into f a from ρ as andfrom the influence of the second mode. Unfortunately,the theoretical model of the slow mode in He in aerogeldescribed in Refs. [23, 24] is not applicable to ourstrongly anisotropic sample and further developmentof the theory is necessary for treatment of our results.Worthy to mark that the second order transition fromthe polar phase into the PdA phase should not influencethe slope of the temperature dependence of ρ as measuredalong the aerogel strands [26]. As it follows from theNMR experiments [14, 15], the transition into the PdAphase in our sample should occur at T = 0 . T c (at29.3 bar) and at T = 0 . T c (at 15.4 bar), and at thesetemperatures we see no any specific features in thedependencies shown in Figs. 3 and 5.We note that Eq. (7) at ρ s = ρ differs from theequation used in Refs. [1, 2, 3, 4] for determinationof ρ as /ρ of He in silica aerogel. Using Eq. (7) weobtain that values of ρ as /ρ are about 1.3–2 times smaller(depending on temperature and α ) than that reportedin Refs. [1, 2, 3, 4]. V. V. Dmitriev, M. S. Kutuzov, A. A. Soldatov, E. V. Surovtsev, A. N. Yudin
6. CONCLUSIONS
Using the aerogel wire techniques, earlier usedto investigate superfluidity of He in isotropic silicaaerogels, we have observed a superfluid transition of Hein nematic aerogel accompanied by appearance of thesecond (slow sound) mode inside the aerogel sample.Resonance frequencies and widths of both the mainand slow sound modes are measured in a wide range oftemperatures. We think that a proper theoretical modelof the slow mode in nematic aerogel might allow toestimate a superfluid density fraction inside our sample.Our results are promising for experiments onsearching for the beta phase in nematic aerogel [26, 27],a new superfluid phase of He that should appearin a strong magnetic field right below T ca . The betaphase must exist in a narrow temperature regionclose to T ca (proportional to the value of magneticfield), and on cooling from the beta phase a transitionto the distorted beta phase (which is continuouslytransformed to the polar phase on further cooling)should be observed as a kink on a superfluid fractionversus temperature plot [26]. The latter can be seenin the resonant frequencies in VW experiments. Inpresent experiments the maximal magnetic field whichwe were able to apply is 1650 Oe. In this field the rangeof existence of the beta phase is expected to be verysmall (about 0.005 T c ) [26]. Unfortunately, this range oftemperatures is nearly the same as the range, whereinthe interaction of the observed two resonance modesis rather strong and the frequency of the slow soundmode resonance is rapidly changing. This, togetherwith experimental errors in determination of resonancefrequencies, has prevented us from detecting any clearkink on temperature dependencies of f a and f a .We are grateful to V.I. Marchenko for usefuldiscussions. FUNDING
This work was supported by the Russian ScienceFoundation (project no. 18-12-00384).
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