We observe a thermo-mechanical effect when a chemical potential difference is
created by a temperature difference imposed between two liquid reservoirs
connected to each other through Vycor rods in series with solid hcp 4He. By
creating a temperature difference, $\Delta T$, between the two reservoirs, we
induce a rate-limited growth of a pressure difference between the two
reservoirs, $\Delta P$. In equilibrium $\Delta P {\it vs.} \Delta T$ is in
quantitative agreement with the thermo-mechanical effect in superfluid helium.
These observations confirm that below $\sim$ 600 mK a flux-limited flow exists
through the solid helium.Comment: 4 pages, 4 figure

Hallock, R. B.

Ray, M. W.

English

arXiv

We observe a thermomechanical effect when a chemical-potential difference is created by a temperature difference imposed between two liquid reservoirs connected to each other through Vycor rods in series with solid hcp 4He. By creating a temperature difference, ΔT, between the two reservoirs, we induce a rate-limited growth of a pressure difference between the two reservoirs, ΔP. In equilibrium ΔP vs ΔT is in quantitative agreement with the thermomechanical effect in superfluid helium. These observations confirm that below ∼600 mK a flux-limited flow exists through the solid helium

Ray, M

Hallock, R

ScholarWorks@UMass Amherst

Observation of thermomechanical equilibration in the presence of a solid 4He conduit

This is the pre-published version harvested from ArXiv. The published version is located at http://prb.aps.org/abstract/PRB/v82/i1/e012502We observe a thermomechanical effect when a chemical-potential difference is created by a temperature difference imposed between two liquid reservoirs connected to each other through Vycor rods in series with solid hcp 4He. By creating a temperature difference, ΔT, between the two reservoirs, we induce a rate-limited growth of a pressure difference between the two reservoirs, ΔP. In equilibrium ΔP vs ΔT is in quantitative agreement with the thermomechanical effect in superfluid helium. These observations confirm that below ∼600 mK a flux-limited flow exists through the solid helium

University of Massachusetts AmherstScholarWorks@UMass AmherstPhysics Department Faculty Publication Series Physics2010Observation of thermomechanical equilibration inthe presence of a solid 4He conduitM RayUniversity of Massachusetts - AmherstR HallockUniversity of Massachusetts - Amherst, hallock@physics.umass.eduFollow this and additional works at: https://scholarworks.umass.edu/physics_faculty_pubsPart of the Physics CommonsThis Article is brought to you for free and open access by the Physics at ScholarWorks@UMass Amherst. It has been accepted for inclusion in PhysicsDepartment Faculty Publication Series by an authorized administrator of ScholarWorks@UMass Amherst. For more information, please contactscholarworks@library.umass.edu.Recommended CitationRay, M and Hallock, R, "Observation of thermomechanical equilibration in the presence of a solid 4He conduit" (2010). PhysicsReview B. 1095.Retrieved from https://scholarworks.umass.edu/physics_faculty_pubs/1095arXiv:1005.1022v1  [cond-mat.other]  6 May 2010Observation of thermo-mechanical equilibration in the presence of a solid 4He conduitM.W. Ray and R.B. HallockLaboratory for Low Temperature Physics, Department of Physics,University of Massachusetts, Amherst, MA 01003(Dated: May 7, 2010)We observe a thermo-mechanical effect when a chemical potential difference is created by a tem-perature difference imposed between two liquid reservoirs connected to each other through Vycorrods in series with solid hcp 4He. By creating a temperature difference, ∆T , between the two reser-voirs, we induce a rate-limited growth of a pressure difference between the two reservoirs, ∆P . Inequilibrium ∆P vs.∆T is in quantitative agreement with the thermo-mechanical effect in superfluidhelium. These observations confirm that below ∼ 600 mK a flux-limited flow exists through thesolid helium.PACS numbers: 67.80.-s, 67.80.Mg, 67.40.Hf, 67.90.+zKim and Chan[1, 2], motivated by the work of Ho,Bindloss and Goodkind [3], observed a significant shiftin the resonant period of a torsional oscillator filled withsolid 4He in the vicinity of 100 mK. This shift was inter-preted as due to mass decoupling from the oscillator andit was suggested that this was likely evidence for a su-persolid phase of solid 4He that was first predicted manyyears earlier [4–7]. This interpretation of the presence ofnon-classical rotational inertia (NCRI) continues to becontroversial, but has spawned considerable experimen-tal and theoretical activity[8]. Efforts to cause the flowof solid helium through confined geometries by directlysqueezing the solid lattice have not been successful[9–12]. In an effort to create a DC flux of atoms throughthe solid, we took a different approach and by inject-ing atoms from the superfluid have demonstrated masstransport through the solid[13–15] at temperatures be-low ∼ 600 mK. In the present work we create a chemicalpotential difference between two reservoirs by imposing atemperature difference and observe a rate-limited changeof the thermo-mechanical pressure, which indicates thata pathway percolates the solid.Figure 1 shows the apparatus used for this work, whichis similar to that used previously[13–15]. It consists ofa cylindrical copper chamber, S, pierced by two new 3.0mm dia. epoxy-coated Vycor rods, V1 and V2, whichextend to the horizontal axis of chamber S. The flat endsof V1 and V2 in the cell were coated with epoxy. Withsolid 4He in region S, and the pressure in the Vycor lessthan 37 bar, there is bulk solid 4He in S and superfluidliquid helium in the Vycor rods[16–18]. Atop the Vycorrods are thermometers T1 and T2 and liquid reservoirsR1 and R2 that are fed by capillaries (1 and 2 in fig. 1).The temperatures in the reservoirs are controlled usingheaters, H1 and H2, so that the helium in R1 and R2 re-mains a liquid. A temperature gradient is present acrossthe Vycor; the cell remains at a low temperature. Thepressure of the solid is measured by capacitance straingages [19] C1 and C2 and the pressures of the reservoirsare measured by pressure gauges P1 and P2 located out-side the cryostat. Using our original apparatus[13, 14],a chemical potential difference, ∆µ, could be imposedbetween the Vycor rods by the creation of a pressure dif-ference ∆P = P2 − P1 6= 0 by adding or subtractingatoms from R1 or R2 using lines 1 and 2. For TC < 600mK an imposed abrupt increase in pressure P1 resultedin an increase in P2 accompanied by a rise in the pres-sure C1 and C2 with dP2/dt ≈ constant, independent ofP2− P1.If one has a superleak in a conducting pathway betweenR1 and R2, it should be possible to induce a pressuredifference P1 − P2 by the imposition of a temperaturedifference T 1 − T 2. In liquid helium, this is the wellknown fountain effect (or thermo-mechanical effect)[20].Quantitatively, the fountain effect relates a temperaturedifference, ∆T = Tb − Ta, between two liquid reservoirsconnected by a superleak to a fountain pressure differ-ence, ∆Pf , by the relation∆Pf =∫ TbTaρSdT, (1)1 23R 1 R 2V 1 V 2SC 1 C 2H 1 H 2T 1 T 2T CP 1 P 2FIG. 1. Schematic diagram of the apparatus used to searchfor flow through solid 4He. Superfluid helium in Vycor allowsfor an interface between the solid and superfluid at pressuresgreater than the bulk melting pressure of solid helium.2where ρ is the liquid density and S is the entropy.The apparatus shown in fig. 1 can be used to utilizethe fountain effect in the presence of solid helium. Withsolid helium in region S, we use H1 or H2 to vary T 1or T 2, create a chemical potential difference between thereservoirs, R1 and R2, and then measure the resultingpressures P1 and P2 in the reservoirs. If a conductingpathway through the solid is present, we should be ableto observe a pressure-temperature relationship given byeq. 1. With liquid helium in the cell at P ≈ 24 bar,TC = 250 mK, and T 1 6= T 2 near 1.6 K we confirmthat this fountain pressure can be created in quantitativeagreement with eq. 1. In the present report we show thatthis is also true with solid 4He in the cell.All of our solid helium samples were grown in the hcpregion of the phase diagram from the superfluid at con-stant temperature using commercial grade 4He (∼300ppb 3He concentration). To fill the cell initially, thehelium is introduced through line 3 and does not passthrough the Vycor rods. To grow the solid at constanttemperature, we begin with the pressure in region S be-low the bulk melting curve for 4He at low temperatureand then add atoms to the solid through lines 1 and 2,which lead to the Vycor. Because these fill lines remainopen we can continue to add atoms to the solid at pres-sures greater than the bulk freezing pressure of the he-lium, and grow our solid sample[15].Figure 2a shows data from a solid sample denoted FSat P = 26.1 bar, TC = 304 mK. In the case shown,thermal energy was supplied to H2, which caused T 2 tochange to a larger value while T 1 was held constant. ∆Tstabilized within 2 min. The result, evident in the figure,was an approximately linear increase (decrease) in P2(P1) to a new stable value with an increase in T 2. Forthe parameters here, one would expect using eq.1 that att ∼ 20 min P2− P1 = 0.056 bar; 0.052 bar is measured.In the vicinity of t ∼ 35 min, P2−P1 is predicted to be0.074 bar and 0.075 bar is measured. The rate of changeof, for example, P2 following the change in T 2 near t= 29 min was dP2/dt ≈ 2.0 mbar/min with ∆P stablein δt ≈ 6 min. With liquid helium in the apparatus at250 mK and P ≈ 24.4 bar a similar ∆T created a similar∆P , but with the result dP2/dt ≈ 22 mbar/min., a muchfaster rate, which indicates clearly that the epoxy-coatedVycor does not restrict the flux; the presence of the solidsample limits the flux.Another solid sample shown in Figure 2b, FT, pro-duced a constant flux similar to sample FS withdP2/dt ≈ 2.2 mbar/min. Two conclusions from sampleslike FS and FT are apparent: (1) a change in the tem-perature difference T 2− T 1 results in a flux of atoms toproduce a change in the pressure difference P2−P1 thatis in agreement with expectations based on the fountaineffect and (2) the change in P2− P1 with time is linear,which suggests that there is a critical flux imposed by thepresence of the solid helium, which is a characteristic of0 10 20 30 40 5026.0826.1026.1226.1426.1626.181.561.601.641.6810 20 30 40 5026.0626.0826.1026.1226.141.561.601.641.68Pressure (bar)time (minutes)P1T1T2b P2 TVycor (K)T1P1T2P2apressure (bar)time (minutes) TVycor (K)FIG. 2. (color online) (a) Sample FS, TC = 304 mK. Ap-plication of thermal energy to H2, results in a change in T2,with T1 held constant. T2 is stabilized to its new value in≈ 2 min. The resulting change in P2 − P1 stabilizes in ≈ 6min, with changes in P1 and P2 linear in time. The rapidsmall oscillations at the start of changes in T2 result from theelectronic temperature stabilization. (b) Sample FT, TC =304 mK. A small overall long-term systematic drift has beenremoved from the pressure data for sample FT.the flow of a superfluid limited by a critical velocity.Figure 3a shows data from sample FI, grown at P =26.7 bar and studied at TC = 250 mK. Changes in ∆Tfor this sample produced changes in ∆P similar to thoseseen in samples FS and FT. Also shown here (and seenin samples FS and FT) are changes in C1 and C2 in-duced by the changes in ∆T . Note that with T 1 fixed,a change in P2 and P1 is induced by a change in T 2;atoms are supplied to R2 from R1, but also from thesolid. This may be a manifestation of the isochoric com-pressibility of the solid[15, 21]. Changes in C1 and C2maintain the fountain effect along the Vycor between thecell and R1 and R2. Figure 3b shows data from sam-ple FM, which was grown and studied at P = 26.5 bar,TC = 700 mK. Here changes to T 2 produced very smallchanges in P2, and no corresponding change in P1. Also,note that the small changes in P2 due to changes in T 2are opposite what would be expected for a fountain ef-fect. These small pressure changes are likely due to asmall density change of the helium in the reservoir as30 10 20 30 40 5026.526.626.70 10 20 30 40 5026.126.226.326.426.526.61.561.601.641.681.561.601.641.68C1C2T1T2P1Pressure (bar)time (min)P2aT1T2C1C2P2P1Pressure (bar)time (min)b TVycor (K) TVycor (K)FIG. 3. (color online) (a) Sample FI, TC = 250 mK. Flow wasinduced through the solid by changing T2. (b) Sample FM,TC = 700 mK. Changes to T2 resulted in no correspondingchanges in P1, P2, C1 or C2.the temperature in the reservoir changes. In fig. 3b T 2changed from 1.617 K to 1.641 K. Over this temperaturerange, the density changes[22] by ∆ρ/ρ ≈ 8.5 × 10−4.Also, there were no changes in the cell pressure, such asthose that were observed in fig. 3a. Thus, at TC = 700mK we were unable to induce a mass flow through thesolid region by changing T 1 or T 2. Whatever was con-ducting the flow in sample FI at 250 mK seen in fig. 3adid not do so for sample FM at 700 mK. This observationis consistent with our previous results [14] in which wewere unable to induce mass flow across the solid whenwe changed the pressure in one fill line for TC & 550mK.Finally, this sample created at 700 mK was cooled toTC = 300 mK, and a change in ∆T did not induce achange in ∆P between the two reservoirs. This is alsoconsistent with our previous observations, which showedthat samples created at higher temperatures don’t flowwhen cooled undisturbed to lower temperatures. Freshsamples created at low temperatures allow a mass flow[14].In addition to the fountain pressure that can be mea-sured between the two reservoirs when a temperature dif--80 -40 0 40 80 120-4004080120TC = 250 mK Liquid data Solid, vary T2  Solid, vary T1 Calculation - liquid Calculation - solid  P V (mbar)TV (mK)FIG. 4. (Color online) Measured pressure difference betweenR1 and R2, ∆PV = P2−P1 at TC = 250 mK for a tempera-ture difference, ∆TV = T2− T1 between the reservoirs. Thelines show eq. 1 for P = 24.73 bar and P = 26.75 bar, whichis the pressure in the reservoirs for the liquid data and soliddata (sample FI) respectively.ference is imposed between them, there is a fountain pres-sure between the helium in the Vycor rods in the cell andthe respective individual reservoirs due to the tempera-ture difference between the cell and the reservoirs. Thisremains, of course, even for the case when T 1 = T 2.When we measure the pressure between the cell and thereservoirs, δP i = Pi−Pcell vs. δT i = T i−Tcell, i = 1, 2,we find quantitative agreement with the fountain effectfor liquid helium in the cell at 24.34 bar and TC = 250mK for reservoir temperatures we have studied (1.52 K< T < 1.64 K). With solid in the cell at 250 mK, a givensolid sample shows the same dependence of δP i on δT ias that with liquid in the cell, but with a modest sampleto sample offset consistent with an inability to preciselyknow the sample pressure due to the presence of modeststable gradients[14, 15] that are sometimes observed withsolid in the cell.As long as there is a path for helium to flow throughthe solid any change in the chemical potential in onereservoir will be communicated to the other reservoir viathese flow paths until the chemical potentials are equalin both reservoirs. In other words, if T 1 = T 2 thenP1 = P2 regardless of the presence of a pressure gradientin the solid lattice that might be recorded by C1 or C2as long as there are flow paths through the solid. Thus,the pressure difference between the R1 and R2 inducedby T 1− T 2 6= 0 is given by eq.1.Figure 4 shows the pressure difference between the twoliquid reservoirs, ∆PV = P2 − P1 vs. the temperaturedifference between the two reservoirs, ∆TV = T 2 − T 14for sample FI at TC = 250 mK, PC ≈ 26.58 bar andfor liquid in the cell at TC = 250 mK, PC = 24.34 baralong with the expectation based on eq. 1 for reservoirpressures P1, P2 = 24.73 bar (the liquid data) and forP1, P2 = 26.75 bar (solid data). The data superimpose,and agree well with eq. 1. Data from samples FS andFT show similar, good agreement with eq. 1.The detailed nature of these flow paths is still notknown although there have been several predictions. Forexample, grain boundaries[23] and dislocations[24] havebeen shown in simulations to support mass flow, as hasa glassy, or “superglass” phase of solid 4He [25–27]. Wehave previously reported on quantitative aspects of someof these scenarios, and how they relate to our earlier ob-servations [13, 14]. Another possibility is the presenceof superfluid cores along edge dislocations as proposedby Soyler et al[21]. This proposal explains two aspectsof our experiments: (1) the flux of atoms though solidhelium below a characteristic temperature and (2) thegrowth of the density of the solid at constant volume inthe presence of a chemical potential gradient. It has beensuggested that the flow might be due to liquid channelswhich could exist in the solid[28], but as we have notedpreviously[13, 29], this scenario seems to be inconsistentwith a number of our observations; in particular, it seemsinconsistent with the lack of a mass flow under any cir-cumstances for solids at temperatures above TC & 550mK. While it is not possible for us to identify the specificprocess that leads to the mass flux that we observe, thepresence of a fountain effect with solid helium in the celland linear changes in ∆P with an imposed change in ∆Tallows the conclusion that a flux-limited pathway perco-lates the solid. It is not possible to identify the precisebottleneck that limits the flow; it could be the pathwaysin the solid or it could be at the interface between thesolid and the Vycor.Unclear at present is the connection of this work to thenumerous experiments that have been heavily focused onthe behavior of solid helium in the temperature rangebelow ≈ 100 mK. There may well be no connection.In summary, we have induced a mass flow through solid4He by creating a chemical potential difference (by meansof the imposition of a temperature difference) betweentwo liquid reservoirs that are connected by Vycor rodsin series with solid 4He. When the temperature in onereservoir is changed, we observe a corresponding changeof the pressure, which agrees with predictions based onthe fountain effect. In order for this to happen, mass wassupplied through the cell that contained the solid heliumsample and this happened with a rate-limited flux. At700 mK, equilibration is not achieved, which is consis-tent with our previous results. Our observations confirmthat the relevant quantity that induces flow in our exper-iments is the chemical potential.We thank N. Prokofev and B. Svistunov for numerousdiscussions. This work was supported by NSF DMR 07-57701 and 08-55954,.[1] E. Kim and M. Chan, Nature, 427, 225 (2004).[2] E. Kim and M. Chan, Science, 305, 1941 (2004).[3] P. Ho, I. Bindloss, and J. Goodkind, J. Low. Temp.Phys., 109, 409 (1997).[4] O. Penrose and L. Onsager, Phys. Rev., 104, 576 (1956).[5] A. Andreev and I. Lifshitz, Sov. Phys. JETP, 29, 1107(1969).[6] G. V. Chester, Phys. Rev. A, 2, 256 (1970).[7] A. J. Leggett, Phys. Rev. Lett., 25, 1543 (1970).[8] S. Balibar and F. Caupin, J. Phys.: Condens. Matter,20, 173201 (2008).[9] D. S. Greywall, Phys. Rev. B, 16, 1291 (1977).[10] J. Day, T. Herman, and J. Beamish, Phys. Rev. Lett.,95, 035301 (2005).[11] J. Day and J. Beamish, Phys. Rev. Lett., 96, 105304(2006).[12] A. S. C. Rittner, W. Choi, E. J. Mueller, and J. D.Reppy, Phys. Rev. B, 80, 224516 (2009).[13] M. W. Ray and R. B. Hallock, Phys. Rev. Lett., 100,235301 (2008).[14] M. W. Ray and R. B. Hallock, Phys. Rev. B, 79, 224302(2009).[15] M. W. Ray and R. B. Hallock, arXiv:0908.2591v2 [cond-mat.other] (2010).[16] J. R. Beamish, A. Hikata, L. Tell, and C. Elbaum, Phys.Rev. Lett., 50, 425 (1983).[17] C. Lie-zhao, D. F. Brewer, C. Girit, E. N. Smith, andJ. D. Reppy, Phys. Rev. B, 33, 106 (1986).[18] E. Adams, Y. Tang, K. Uhlig, and G. Haas, J. LowTemp. Phys., 66, 85 (1987).[19] G. C. Straty and E. E. Adams, Rev. Sci Inst., 40, 1393(1969).[20] J. Wilks, Properties of Liquid and Solid Helium (OxfordUniversity Press, 1967).[21] S¸ G. So¨yler, A. B. Kuklov, L. Pollet, N. V. Prokof’ev,and B. V. Svistunov, Phys. Rev. Lett., 103, 175301(2009).[22] J. Maynard, Phys. Rev. B, 14, 3868 (1976).[23] L. Pollet, M. Boninsegni, A. B. Kuklov, N. V. Prokof’ev,B. V. Svistunov, and M. Troyer, Phys. Rev. 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Observation of thermo-mechanical equilibration in the presence of a solid 4He conduit

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