Effect of boundary condition on Kapitza resistance between superfluid 3 He-B and sintered metal
S. Autti, A.M. Guénault, R.P. Haley, A. Jennings, G.R. Pickett, R. Schanen, A.A. Soldatov, V. Tsepelin, J. Vonka, D.E. Zmeev
EEffect of boundary condition on Kapitza resistance between superfluid He-B andsintered metal
S. Autti, A. M. Gu´enault, ∗ A. Jennings, † R. P. Haley, G. R. Pickett,R. Schanen, V. Tsepelin, J. Vonka, ‡ and D. E. Zmeev Department of Physics, Lancaster University, Lancaster, LA1 4YB, United Kingdom
A. A. Soldatov
P. L. Kapitza Institute for Physical Problems of RAS, 119334 Moscow, Russia (Dated: May 13, 2020)Understanding the temperature dependence of thermal boundary resistance, or Kapitza resistance,between liquid helium and sintered metal has posed a problem in low temperature physics fordecades. In the ballistic regime of superfluid He-B, we find the Kapitza resistance can be describedvia scattering of thermal excitations (quasiparticles) with a macroscopic geometric area, rather thanthe sintered metal’s microscopic area. We estimate that a quasiparticle needs on the order of 1000collisions to successfully thermalise with the sinter. Finally, we find that the Kapitza resistanceis approximately doubled with the addition of two mono-layers of solid He on the sinter surface,which we attribute to an extra magnetic channel of heat transfer being closed as the non-magneticsolid He replaces the magnetic solid He.
I. INTRODUCTION
The thermal boundary resistance, or Kapitza resis-tance, R K between liquid helium and sintered metal is acommon limiting factor in the pursuit of ultra-low tem-peratures. As temperature decreases the thermal bound-ary resistance R K increases, reducing the cooling powerattainable in dilution refrigerators. According to acousticmismatch theory, phonon-phonon transfer should yield aboundary resistance proportional to inverse temperaturecubed, R K ∝ T − [1]. A common way of decreasing theKapitza resistance between liquid helium and metal andincreasing refrigerator performance is to increase the sur-face area using sintered metal powders. However, theproblem of Kapitza resistance between sintered metaland liquid He is still yet poorly understood despite thelarge scale use of dilution refrigerators, in part due tosintered metals’ complicated geometry [1]. At very lowtemperatures, T ≤
10 mK, the boundary resistance de-viates significantly from the prediction of acoustic mis-match theory. A lower than expected Kapitza resistanceis observed in general, and with a temperature depen-dence of R K ∝ T − . In superfluid He-B at ultra-lowtemperatures (about 0 . T c or 0 . ∗ Deceased † [email protected] ‡ Current address: Paul Scherrer Institute, Forschungsstrasse 111,5232 Villigen PSI, Switzerland effects of solid helium at low temperatures as a few lay-ers of solid helium will form on the surfaces due to vander Waal’s forces [5, 6]. It is thought that the magneticproperties of He could play a large role in the smallerthan expected Kapitza resistance, by the creation of amagnetic channel of energy exchange with magnetic im-purities in the sintered metal [7, 8].Recently, it has become a popular technique in ultra-low temperature physics to pre-plate the surfaces of theexperimental cell and objects with solid He. He pref-erentially adsorbs to surfaces over He due to its lowerzero-point energy. Control over the number of He lay-ers allows fine tuning of the surface scattering specular-ity. This technique has been used for the stabilizationof the newly discovered polar phase in nematic aerogels[9] and the suppression of the B phase in thin slab ge-ometries when the scattering at the slab surface becomescompletely specular [5, 6]. Importantly, solid He is alsonon-magnetic. In the ballistic regime of the B phase,heat transfer should be dominated by quasiparticle col-lisions with the surface. The difference in magnetic andnon-magnetic quasiparticle scattering has been found tohave a significant effect on the stabilisation of the Aphase [10] and of the polar phase [11] in anisotropic me-dia. It is therefore likely that the magnetic propertiesof solid He are an important factor in the Kapitza re-sistance. Hence pre-plating surfaces with non-magnetic He should change the effective Kapitza resistance, animportant consideration for future experiments at ultra-low temperatures.In this work we demonstrate the difference in Kapitzaresistance between superfluid He-B and sintered silverwith and without pre-plating of the surface with 2 layersof He. We compare the measured Kapitza resistance toearlier work [2, 12] and find very good agreement withthe reported exponential temperature dependence. Ourobserved difference in Kapitza resistance between solid a r X i v : . [ c ond - m a t . o t h e r] M a y He and solid He coverage provides evidence for the roleof the magnetic channel in decreasing the Kapitza resis-tance between metal and liquid He. We use a simplemodel to calculate the increase in probability of a quasi-particle collision with sintered metal eventually result-ing in quasiparticle recombination. We find that sintercovered with He offers about double the probability ofrecombination. Furthermore, the simple model seems tosuggest a larger geometric scattering area – that is, thearea of sintered plate surfaces that face the experimen-tal volume containing the quasiparticle source with noother sintered surfaces between it and the volume – isvery important to the lowering of Kapitza resistance atultra-low temperatures. The total (interfacial) area ofall plates or the microscopic sinter area – the area of thesinter’s sponge-like geometry – do not provide consistentresults between different measurements.
II. EXPERIMENTAL DETAILS
Our experimental apparatus is a Lancaster-style nestednuclear demagnetization cell attached to a custom-builtdilution refrigerator (shown in Fig. 1). The inner cell con-tains 8 copper plates of thickness 1 . . anda filling factor of 0.5 [13]. From the total surface area,which is more or less equal to the total microscopic sintersurface area, we can calculate the amount of gas neededto pre-plate all surfaces with 2 layers of solid He [14].This is accomplished by inserting a small amount of Hegas before filling the cell with He to saturated vapourpressure. After demagnetizing to the required magneticfield (or temperature) the copper plates act as a thermalbath.Inside the cell we have a set of mechanical probes: Arelatively large moving wire with a diameter of 135 µ m,a small vibrating loop wire with a diameter of 4 . µ mand a small quartz tuning fork. The damping in ballistic He-B measured via mechanical resonance widths ∆ f ofthe vibrating loop and tuning fork is determined by thenumber of quasiparticle collisions, and hence the numberdensity of quasiparticles in the superfluid. In the ballisticregime the quasiparticle density is exponential with tem-perature, thus vibrating resonators can be very sensitivethermometers [15]. The large wire can be moved at con-stant velocity with a direct current in a magnetic fieldwhich generates a Lorentz force, which we term a DCpulse [16]. Doing so heats the experimental cell whichincreases the number of quasiparticles above the thermalequilibrium density [17]. Hence monitoring the width re-sponse of the vibrating loop and tuning fork during aDC pulse allows us to detect the heating and subsequentcooldown. Note, that the actual temperature change issmall due to the exponential dependence. Wire thermometer
Large WireTuning Fork
ThermometerInner cell sinter platesTail piece
FIG. 1. The experimental cell. The inner cell contains 800 . . . A ge-ometric 2 cm × × III. RESULTS
Initially, the thermometer is resonating with a base res-onance width of ∆ f base and the bulk quasiparticle popu-lation is in thermal equilibrium with the walls and sinter.A DC pulse is started, which heats the cell and in-creases the bulk quasiparticle population, resulting in arapid increase in thermometer width up to a peak value.The quasiparticles excited begin colliding with surfaceswhich is dominated by the silver sinter. These collisionsare either near-elastic or result in the quasiparticles los-ing enough energy to successfully recombine as a Cooperpair, seen as an exponential decay with time constant τ b until the quasiparticles have reached thermal equilibriumwith the surroundings, at which point the thermometerwidth is now close to the base width. Figure 2 showsa typical response. We can model the thermometer re-sponse as [18]∆ f = ∆ f base + H τ b τ b + τ w (cid:16) e − t/τ b − e − t/τ w (cid:17) . (1)Here τ w is the response time of the thermometer, whichis limited by the resonance width of the thermometer andis approximately equal to 1 /π ∆ f base . H is a constant de-scribing the amplitude of the width response. The decayconstant τ b is governed by the effective boundary resis-tance R K , area A and the heat capacity C B of superfluid He-B [3]
Cell Experimental Volume (cm ) Sinter Area Sinter Mass (g)Interfacial (m ) Microscopic (m ) Geometric (cm )This work 8.6 0.21 79.68 36 96Castelijns et al. [2] 1 0.011 23.7 3 28.5Carney et al. [12] 1 0.169 40.7 3 49TABLE I. Experimental cells for each measurement. The interfacial area is the area of all plate faces covered in sinter andmicroscopic area is the area of the sintered powder’s sponge-like surface. Geometric area is the sintered plate surfaces that facethe experimental volume containing the quasiparticle source with no other sintered surfaces between it and the volume. Theexperimental volume quoted for this work excludes the tail piece, in which a small hole limited heat flow into the main volume.However, the tail piece volume is not large and its inclusion would not change the data significantly. − Time t ( s ) . . . . . T e m p e r a t u r e C h a n g e ( µ K ) He ( T = 193 µK) He ( T = 197 µK) FIG. 2. Comparison of the response of the vibrating loopwire with diameter 4 . µ m for pure He and two layers of solid He coverage on the cell surface. The change in resonancewidth is converted into a change in temperature. Solid linesrepresent the fits from Eq. (1). AR K = τ b C B . (2)The heat capacity of He-B in the ballistic regimeis determined by the number of quasiparticles and istherefore exponentially dependent on temperature C B ∝ exp ( − ∆ B /k B T ). ∆ B is the superfluid energy gap of He-B and k B is the Boltzmann constant.Figure 2 shows the difference in response for the vibrat-ing loop thermometer between the cell filled with pure He and the same cell with 2 layers of He pre-plating.The decay constant extracted for 2 layers of He is ap-proximately double that of pure He. As clearly shownin Fig. 3, the thermalization time τ b is independent oftemperature for both types of coverage, with τ b being 0 . ± . He compared to 1 . ± . He. Similar results were obtained for the quartztuning fork. The time constant was found to not varywith the wire velocity of the DC pulse and is thus in-dependent of the energy of the DC pulse from 1 pJ to50 pJ (wire velocities of 2 mm s − to 50 mm s − ). FromEq. (2) this demonstrates an exponential temperature de-pendence on Kapitza resistance in both cases, and thatpre-plating with solid He results in roughly twice theKapitza resistance for a given temperature.Another possible contribution to the heat capacity isthat of the solid He on the walls in a magnetic field.Elbs et al. [19] measured the surface contribution to theoverall heat capacity for a much smaller container of Heand found that at higher temperatures around 0 .
16 mKto 0 .
19 mK in similar magnetic field to our measurementsthe heat capacity of solid and liquid helium were roughlyequal. For our much larger volume the heat capacity ofsuperfluid He-B is much larger than the contribution toheat capacity from any solid layers, thus we neglect thisheat capacity.The described bolometry method can only be appliedin the ballistic regime of the B phase and cannot be usedin the A phase. Therefore, we were unable to make asimilar measurement in the A phase. We expect Kapitzaresistance in the A phase to be different than in the Bphase for at least two reasons: the heat capacity has adifferent form of temperature dependence and the prop-agation of quasiparticles is significantly different [20].
IV. DISCUSSION
The exponential temperature dependence is entirelyconsistent with earlier work by Parpia [3] and at Lan-caster by Castelijns et al. and Carney et al. [2, 12]. τ b is also consistent with the time constants expectedby Castelijns et al. , however, they observed an actualtime constant about ten times larger than expected. Inthat work an electrical heater was used to create heatingbursts rather than our mechanical method, which likelycreated a bubble of normal fluid around the heater anddramatically changed the time constant [2]. The advan-
140 160 180 200 220 240 260
Temperature ( µK ) . . . . . . τ b ( s ) He He FIG. 3. Comparison of quasiparticle relaxation time con-stant τ b as measured by the thermometers. The average τ b is0 . ± . . ± . He and He coverage, respec-tively. As can be seen there is no discernible variation overtemperature, which is consistent with an exponential depen-dence of R K . There was also no variation with velocity of thewire in the DC pulse. tage of our method is we see the direct creation and re-laxation of quasiparticles as the wire moving at supercrit-ical velocities does not destroy the superfluid state [17].Therefore we see no such problem as possible regions ofnormal fluid being formed around a heater and we ob-serve purely the dynamics of the quasiparticles, which inthe ballistic regime dominate the heat transport proper-ties of the superfluid. For He coverage we also observeda similar magnetic field dependence as measured by Os-heroff and Richardson [21] with pure He (about a 20%increase in τ b as the field was doubled from 63 mT to125 mT).Consider a superfluid quasiparticle excited by the DCpulse. The quasiparticle travels with group velocity v g which is approximately 20 m s − at these temperaturesand pressure. Due to the superfluid energy gap, whenthe quasiparticle scatters with a wall it has a chance toeither scatter elastically or lose energy ∆ B and recombineinto a Cooper pair. The recombination results in a re-duction of the overall superfluid temperature. However,losing enough energy for recombination after scatteringwith cell walls is extremely unlikely and quasiparticle-quasiparticle interactions can be neglected in the bal-listic regime, hence only scattering with sinter matters.Note that the coherence length of the superfluid pairs isaround 80 nm at zero pressure [22]. With the pore sizein the sinter roughly equal to the coherence length, thesinter could appear as a wall of metal and normal fluidto any approaching quasiparticle. If we consider a quasiparticle moving in a box withsides of length d we can estimate the number of colli-sions needed for it to eventually recombine. The timeit takes to traverse from one wall to another is roughly t col = d/v g and there are six walls, of which only one isthermally conductive due to the sintered copper plates.The average time taken for there to be no more collisionsis approximately τ b . Therefore the probability of collisionsuccessfully leading to recombination is P = t col τ b . Thisgives an estimate on the order of 1000 collisions neededfor a quasiparticle to lose its energy with the sinter andrecombine. And, as we demonstrate, a surface coveredwith solid He reduces the amount of collisions by half.A similar argument for the importance of the geometricconditions taking precedence has been made earlier, withan estimate being hundreds of collisions needed [12].The difference between the two types of coverage, webelieve, is due to different magnetic properties of thetwo. It has long been thought that a magnetic chan-nel for transfer of heat is key to explaining the observeddifference in Kapitza resistance between experiment andtheory [23]. Further evidence was added by K¨onig et al. ,who saw a large difference in the performance of differentbrands of silver powder [7, 8, 24]. They concluded thatUlvac powder has a larger content of magnetic impuritiesand a lower Kapitza resistance at millikelvin tempera-tures. Incidentally, we, Osheroff and Richardson [21],and Castelijns et al. [2] all used the 70 nm Ulvac powder,which has the highest magnetic impurity content of allbrands and sizes measured by K¨onig et al .While taking the surface of a sinter to be a flat piecemay be overly simplistic, we can compare our valuesof thermal boundary resistance to those obtained byCastelijns et al. and Carney et al. using the same sil-ver powder and sintering technique. We fit zero pressuredata from Castelijns et al. with a function that uses theBCS gap ∆ B = 1 . k B T c rather than the original func-tion which leaves the energy gap as a free parameter. Thesame method was used for fitting the data from Carney et al. We take A in Eq. (2) as the geometric surface (seeTable I) and find that this gives good agreement with ourresults, as shown in Fig. 4, unlike the other possible val-ues of A given in Table I. What is interesting here is thatin both experiments by Carney et al. and Castelijns etal. the exact same cell geometry was used, but the ratiobetween the measured boundary resistances at the lowesttemperatures was about 2.5. Our data points lie closerto the results of Castelijns et al. , or between the two linesif the tail piece volume is included in the calculation ofthe heat capacity in Eq. (2).The difference between the results of Castelijns etal. and Carney et al. was explained in terms of thequasiparticle-scattering model by the difference in thenumber of sinter plates. In Castelijns et al. the cell hadtwelve 1 mm thick plates, and a square was cut out ofseven to form a 1 cm box. In Carney et al. , ninety-twoplates of 0 . FIG. 4. The effective thermal boundary resistance R K multi-plied by the “geometric” sinter scattering area A for quasipar-ticles created by the wire as a function of inverse temperature. T c = 929 µ K is the critical temperature of superfluid He atzero pressure. The gold and red lines are fits for data fromRefs. [2] and [12], respectively, and converted into thermalboundary resistance. For comparison, our data shown is onlyin pure He which lies mostly on the gold line, or between thetwo lines if the tailpiece volume is included. combination. In our cell, there is a combination of boththick and thin plates.
V. CONCLUSIONS
We have demonstrated that in the ballistic regime thethermal boundary resistance between sintered metal and He is dominated by the collisions of quasiparticles withsinter walls. The probability of an eventually recombin-ing collision can be calculated using the thermal timeconstant. The probability is surprisingly very low, onthe order of one in a thousand. Multiplying by the geo-metric surface area of the sinter quasiparticles can collidewith we find agreement with previous results. To increasethe thermal boundary conductivity it is therefore impor-tant to increase the frontal surface area the particles cancollide with in the experimental volume.When designing cells for ultra-low temperature exper-iments, increasing the geometric surface in contact withthe He is an important factor. Future designs shouldaim to increase the contact area, possibly by using ridge-like protrusions.It is also important to consider the magnetic propertiesof the surface the quasiparticles scatter with. Coveringthe surfaces with two layers of non-magnetic solid Hedoubles the thermal boundary resistance. This providesfurther evidence of a magnetic energy transfer channelbetween liquid He and metals to explain deviations inKapitza resistance from theory. A well-developed theoryexplaining the Kapitza resistance at ultra-low tempera-tures should include the magnetic properties and futuresinter designs should aim at utilizing magnetic impuri-ties.
ACKNOWLEDGMENTS
We acknowledge M.G. Ward and A. Stokes for their ex-cellent technical support. This work was funded by UKEPSRC (grant No. EP/P024203/1) and EU H2020 Eu-ropean Microkelvin Platform (Grant Agreement 824109).S.A. akcnowledges support from the Jenny and Antti Wi-huri Foundation. [1] T. Nakayama, Chapter 3: Kapitza thermal boundary re-sistance and interactions of helium quasiparticles withsurfaces, in
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