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Measurements on the Kapitza resistance of very thin superfluid helium films
Physica B 293 (2001) 297}303
Measurements on the Kapitza resistance of very thin super#uid helium "lms A. van der Hoek, H. van Beelen* Kamerlingh Onnes Laboratory, Leiden University, P.O. Box 9506, 2300 RA Leiden, The Netherlands Received 14 March 2000; received in revised form 19 June 2000
Abstract Measurements on the Kapitza resistance of super#uid He "lms on silver substrates at temperatures above 1.0 K are reported. No apparant change in value is found upon decreasing the "lm thickness, all the way down to the Kosterlitz}Thouless transition occurring at thicknesses as small as two to three atomic layers. 2001 Elsevier Science B.V. All rights reserved. Keywords: Kapitza resistance; Thin "lms; He
1. Introduction In 1941, Kapitza observed that when heat #ows from a solid into super#uid He II a temperature discontinuity appears across the boundary [1]. Earlier observations of the phenomenon (1936) had passed without attracting particular attention [2]. Kapitza found that in the range of small #ow rates, the drop in temperature is proportional to the heat #ux. The associated thermal boundary resistance, i.e., the ratio of the temperature drop and the heat #ux per unit contact area, is nowadays known as the Kapitza resistance of the heat-exchanging interface. At the time of its discovery the phenomenon
* Corresponding author. E-mail address: [email protected] (H. van Beelen). Present address: Baan Company, P.O.Box 143, 3770 AC Barneveld, Netherlands.
was attributed to the special character of the super#uid, but it has since become clear that a thermal boundary resistance occurs at the boundary between any two di!erent media. Nowadays, a rather complete quantitative understanding of the measured values of the thermal boundary resistances of di!erent contact surfaces is available. An extensive review of this understanding has been given in 1987 by Swartz and Pohl [3]. In cases that the interface is su$ciently #at, the mechanism behind the heat exchange is described by the &acoustic mismatch model'. This AM-model was "rst introduced in 1952 by Khalatnikov for the boundaries between solids and He II [4]. In the AM-model the heat transfer is due to the exchange of thermal phonons, their transmission through the plane interface being governed by the rules of the acoustics of continuous media. Because of the &acoustic mismatch' between He II and solids, viz., the large di!erences between the sound velocities as
0921-4526/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 0 ) 0 0 5 5 7 - 3
298
A. van der Hoek, H. van Beelen / Physica B 293 (2001) 297}303
well as between the acoustic impedances, the transmission coe$cients of the interfaces are very small. This results in relatively large values of the Kapitza resistances. Furthermore, since at low temperatures a system formed by bulk matter can be viewed as the acoustic analogue of the black-body radiator, the available phonon energy increases with the temperature as ¹. The AM-model therefore predicts the thermal boundary resistance to vary as ¹\. Experimentally it is found that the value of the boundary resistance depends very strongly on the surface preparation of the contact area. Nevertheless, for some carefully polished solid}solid boundaries a fair agreement with the AM-prediction is found. For well-prepared solid-He II boundaries, however, the AM-prediction is only approached for temperatures below a few tenths of a kelvin; towards higher temperatures the Kapitza resistance decreases gradually by up to two or three orders of magnitude below its AM-value. A plausible cause of the deviation is the scattering of high-frequency phonons in the boundary region, which indeed becomes the more important the higher the temperature. The e!ect of phonon scattering on the thermal boundary resistance has "rst been estimated by Swartz in 1987 [3]. Swartz considered the extreme case of completely-di!use, elastic scattering. In that case the probability of either forward or backward scattering is simply proportional to the density of phonon states in the material on either side of the interface. Since in He II this density of states is larger by several orders of magnitude than in solids, the applicability of the &diwuse mismatch model' (DM-model), in whole or in part, can explain the observed stronger decrease of the Kapitza resistance with temperature. The mismatch of the densities of phonon states between di!erent solids is much smaller, which explains the fair agreement with the AM-prediction found for well-prepared solid}solid contact surfaces. From the above models of the thermal boundary resistance of bulk He II one can try to predict the properties of the thermal contact between a very thin He "lm and its substrate. It is known that the excitation spectrum changes signi"cantly with "lm thickness [5], particularly the phonon part at long wavelengths, so it can be expected that the acoustic
mismatch between a thin "lm and its substrate will be quite di!erent from that of the bulk. A distinct crossover from bulk- to "lm-behaviour might thus be observable in the Kapitza resistance of "lms on cleaved crystal surfaces or on well-polished substrates at temperatures smaller than a few tenths of a kelvin, cases in which the AM-model describes the bulk behaviour pretty well. However, for unpolished substrates or at higher temperatures diffuse phonon scattering at the interface will become more and more dominant, so that the crossover behaviour may be expected to become less evident. According to the DM-model the phonon transmission from solids into He II is almost 100%, so that only very dramatic changes of the density of states of the excitations in the "lms will a!ect the value of the Kapitza resistance signi"cantly. In Ref. [3] a similar argument is used to explain the observed insensitivity to the pressure of the Kapitza resistance in the case of He [6]. Experimentally, the situation is not yet completely clear; the measurement of the Kapitza resistance of very thin "lms is by no means straightforward. At present, only some data for temperatures above 0.3 K are available. In 1974 Long et al. [7,8] reported measurements, carried out at ¹"1.1 K, on the re#ection of sub-microsecond pulses of high-frequency phonons (with black-body temperature of about 10 K), from the solid side of the interface between He and a cleaved sodium#uoride crystal. The data resolved the re#ections of the longitudinal and transverse phonons, and demonstrated the occurrence of mode conversion as well as of the excitation of surface waves. On lowering the vapour pressure it was found that the re#ection pattern only starts to change at helium coverages thinner than a few atomic layers. From the analysis of their data the authors concluded that the mechanism, responsible for the transfer of energy into the helium side, is the desorption of helium atoms from the second, solid, helium layer, mediated by the excitation of surface waves. In 1981, Taborek et al. [9] also studied the transmission of narrow heat pulses into helium "lms as a function of "lm coverage. They observed that the transmission of sub-microsecond pulses already attains its bulk value at "lm coverages of only three atomic layers. From this observation they
A. van der Hoek, H. van Beelen / Physica B 293 (2001) 297}303
concluded that the Kapitza resistance does not change with "lm thickness down to such small coverages. A revival of interest in the problem occurred in 1992, when Mester et al. [10] reported the observation of a dramatic increase of the Kapitza resistance, by up to three orders of magnitude, for "lms thinner than four atomic layers; they had measured the temperature rise of a bolometer from which "lm liquid was steadily evaporated by a constant heat input. A subsequent renewed interpretation of Mester's measuring results by Wyatt [11], however, demonstrated that no change at all had occurred. Such a conclusion was also drawn in 1993 by Eggenkamp et al. [12], from their measurement of the heat-pulse di!usion along a glass substrate coated by a He "lm of 2.5 atomic layers coverage. Finally, the absence of a dramatic increase of the Kapitza resistance with decreasing "lm thickness down to three atomic layers was also con"rmed by van Beelen et al. [13], but a far more gradual increase towards thinner "lms was suggested by the measurements. In the latter experiment, heat is supplied to the "lm at a steady rate by the latent heat of vapour condensation. The temperature rise due to the removal of this heat through the glass substrate to the surrounding He II bath is measured, while the condensed mass is removed by a steady super#uid mass#ow through the "lm back to a heater, by which it is evaporated again. We have repeated the experiment using an improved device with a welllocalized heat exchange. As we have already brie#y reported (in 1996) [14], we could demonstrate that the observed apparent gradual increase was an artefact of the old device, essentially caused by the shrinking of the heat-exchanging surface area with decreasing "lm thickness. In the present article, a full account of these investigations is presented.
2. The measuring cell for the study of the Kapitza resistance The measurements form part of the investigations on the #ow properties of very thin He "lms reported in Ref. [15]. The Kapitza resistances of the helium "lms are studied on "lms covering the inner wall of the all metal device, sketched in Fig. 1.
299
The device is immersed in the main He bath. It is provided with a well-localized, highly conductive (silver) heat exchanger with the bath. The net thermal conductance per unit area of the exchanger i is composed of the Kapitza conductances i (d, ¹) and i (¹) of the two interfaces between ) "lm and silver and between silver and bulk liquid, respectively, connected in series with the conductance of the silver j /d (with d the thickness and j the thermal conductivity), so that 1 1 d 1 " # # . (1) i j i i ) Since, of the three contributions to i , only i may vary with the "lm thickness d, measurements of the thermal resistance of the exchanger can thus be used in principle to investigate the possible variation of the Kapitza conductance i with d. We used the silver heat exchangers sketched in Fig. 2. Three di!erent sizes were used, as listed in Table 1. For these heat exchangers, machined out of high-purity silver, the surface contributions fully dominate their total thermal resistance. The net thermal resistance of the heat exchangers is thus 1 1 R " # . (2) i A iA ) The He "lm covers the inside walls of the measuring cell, and in fact also of the &"ll tube' FT, up to some height above the He bath level (see Fig. 1). In order to vary the "lm thickness, discrete amounts of He gas are added to or removed from the interior of the cell via the "ll tube. A pumping tube, not shown in the "gure, is connected to the vacuum can. Both "ll tube and pumping tube lead through the top of the cryostat to a gas-handling system at room temperature. In order to suppress thermal radiation from the top of the cryostat, both tubes are provided with radiation barriers, formed
With an estimated thermal conductivity of the high-purity silver better than 10 W m\ K\ and a wall thickness of about 1 mm, the middle term of Eq. (1) is smaller than 10\ K m W\, much smaller than the 10\ K m W\ of the other two contributions.
300
A. van der Hoek, H. van Beelen / Physica B 293 (2001) 297}303 Table 1 The dimensions of the three silver heat exchangers
Fig. 1. Schematic drawing of the experimental arrangement used for the study of the Kapitza resistance between "lm and substrate.
Fig. 2. The silver heat exchangers used for the measurements of the Kapitza resistance.
by a number of brass plates inserted halfway through the walls of the tubes at di!erent heights above the vacuum can. Moreover, the radiation barriers in the cold section of the "ll tube serve at the same time as sound breakers, in order to prevent the spontaneous appearance of thermally driven &organ-pipe like' sound oscillations in the vapour column, observed in our earlier experiments [16]. A steady circulation of helium is set up in the essential part of the device, formed by heater Q and thermometer T inside the stainless-steel
No.
A
S1 S2 S3
15.71 4.92 0.83
(cm)
A
(cm)
13.29 3.61 1.41
vacuum chamber and the (silver) heat exchanger in contact with the surrounding He bath. Unwanted #ow of heat, by circulation of He, to and from the "ll tube is e!ectively suppressed by the two narrow constrictions (d +0.5 mm) shown in the "gure, particularly at the lowest temperature where the vapour density becomes small and the vapour #ow resistance high. The upper one, in combination with the bellows heat exchanger (see Fig. 1), minimizes the input of extra heat from the top of the cryostat; the lower one reduces the heat leak away from Q . The core measurements concern the variation of *¹ ,¹ !¹ with the heater power QQ sup plied by Q , where *¹ is measured di!erentially with respect to the bath thermometer T . The ther mal resistance R ,*¹ /QQ is formed by the con tribution from the #ow resistance along the "lm path between T and the heat-exchanging surface, in series with the thermal resistance R of the (silver) heat exchanger itself. Since, in the super#uid phase, the "lm#ow resistance decreases rapidly for decreasing QQ (according to a powerlaw with expo nent larger than two [17]), the constant thermal resistance of the heat exchanger becomes the dominant contribution at small QQ and can be deduced from the data. A correction for the residual heat #ow along the lower constriction is applied, for which purpose this heat #ow is determined by means of both heaters Q and Q and the addi tional thermometers T , T and T (see the appen dix). Owing to the fact that, at the lower bath temperatures, the super#uid transition already takes place in "lms of only about two atomic layers thickness, the e!ect of the "lm thickness on the value of the thermal boundary resistance could be investigated down to such very thin "lms in this way. Because of the decrease in the resolution of the measurement with increasing temperature, most of
A. van der Hoek, H. van Beelen / Physica B 293 (2001) 297}303
301
our measurements have been carried out at the lowest temperatures attainable in our cryostat.
3. Experimental results In the double-logarithmic plot of *¹ versus QQ in Fig. 3, a representative set of measuring results is presented, obtained at ¹ "1.107 K with S3, the smallest of our heat exchangers (cf. Table 1). Series of datapoints are shown for various values of the `glass-equivalent xlm thicknessa dH [15] in the range just beyond the Kosterlitz}Thouless transition, which is found to occur at about dH "2.33 )2 a.l. for this value of ¹ . For each value of dH two curves are presented, one connecting the ten datapoints measured at almost equal spacings along the ordinate for decreasing values of the heater power QQ , the other connecting the datapoints measured at the same 10 values of QQ but in the reverse order. Each datapoint corresponds to a thermal resistance R ,*¹ /QQ , where *¹ is composed of the temperature drop *¹ along the "lm between T and the heat-exchanging "lm area and the drop *¹ across the heat exchanger. For given dH, R is dominated by the contribution from *¹ at large values of the heater power QQ , but, because of the steep decrease of the "lm#ow resistance with decreasing mass#ow rate [17], the constant contribution from the heat exchanger R should be come dominant at small values of QQ . That this is what is actually observed is demonstrated by the curves in Fig. 3; they all tend to approach a common asymptote with slope one for small QQ . All curves having the same asymptote signi"es that R is independent of dH. Finally, in view of Eq. (2), it can be concluded that also the thermal-boundary conductivity i is independent of the "lm thickness, even for d approaching the Kosterlitz}Thouless value, and
This is the value of the thickness, as calculated from the vapour pressure using the well-known Frenkel}Halsey}Hill procedure with the VanderWaals constant of the substrate given the value usually used for glass, c "2.62;10\ m s\, by 5 lack of knowledge of the value on silver.
Fig. 3. The measured ¹ !¹ versus QQ , determined by the superposition of the Kapitza resistance and the "lm #ow, for a range of dH just beyond the Kosterlitz}Thouless transition at dH +2.33 a.l. The tendency to approach a common asymptote )2 with slope one at small QQ suggests that the Kapitza resistance does not depend on d.
should thus remain equal to the bulk value i . This ) conclusion is corroborated by similar series of measurements carried out at some other values of the bath temperature and with the other heat exchangers listed in Table 1. In the appendix the way in which the value of the Kapitza conductivity is derived from the measurements is discussed. Although the measuring results on i were found to be well reproducible, not only ) during each measuring day but also from week to week and even after replacing one heat exchanger by another, we do not consider the experimental values compiled in the last column of Table 2 to correspond to `thea speci"c value of the Kapitza conductivity of the boundary between liquid He and silver; we are aware of the fact that our surfaces may very well have been contaminated, even though we took care to keep them clean. This observation does not a!ect the main conclusion of our work, however, i.e., the Kapitza resistance is largely independent of the thickness of the helium "lm.
Acknowledgements The authors are grateful to Lex Reesink for his expert help in the preparation of the manuscript.
A. van der Hoek, H. van Beelen / Physica B 293 (2001) 297}303
302
Table 2 The measured and corrected values of the thermal resistances for various heat exchangers No.
¹ (K)
a (K W\)
b (K W\)
c (K W\)
R (K W\)
R (K W\)
R (K W\)
i ) (W K\ m\)
S1 S2 S3 SS
1.065 1.074 1.102 1.103
1.2 4.0 13 50
0.07 0.2 0.6 2
4.7 4.8 5.0 4.4
1.2 4.2 15 90
5.0 5.0 5.2 4.5
81 96 108 108
1.1;10 1.1;10 1.3;10
Appendix The common asymptote in Fig. 3 corresponds to a value of the thermal resistance of the heat exchanger R "14.7 K/W. With the dimensions of S3 given in Table 1, Eq. (2) yields for ¹ "1.10 K the bulk value i "1.3;10 W m\ K\, in ) fair agreement with values for unpolished surfaces reported in literature, in between the predictions from the acoustic}mismatch model and the diffuse-mismatch model (cf. Fig. 1 of Ref. [3]). Furthermore, by employing also the second heater Q and thermometer T , it could be veri"ed that the corrections due to the heatleak across the narrow constriction remain almost negligibly small for these thin "lms at these low temperatures. This is illustrated by the example presented in Fig. 4. It shows the curves for *¹ (QQ ), *¹ (QQ ), *¹ (QQ ) and *¹ (QQ ), measured at ¹ "1.102 K for heat exchanger S3, in a double-logarithmic plot. Evidently, also the datapoints *¹ (QQ ) ap proach an asymptote for decreasing QQ , which is found to be independent of the thickness of the "lm and corresponds essentially to the thermal resistance R of the heat exchange between the "ll tube and the bath (cf. Fig. 1). The common asymptote expected for the other two series of data, has clearly not been reached within the measuring window presented in the "gure; since the temperature di!erences on these asymptotes are proportional to the heat#ow across the constriction, this data shows that this heatleak does not amount to more than a few percent of the heat input. In the regime of the asymptotes, where the thermal resistances R and R , as well as R of the constriction, have become independent of the heat-
Fig. 4. The measured ¹ !¹ versus QQ , for a range of dH.
From these measurements the Kapitza conductivity, corrected for the heat leak, is computed.
#ow, a more quantitative analysis of the data in Fig. 4 can easily be made. In this regime the contributions of the two heaters to the temperature di!erences simply add, so that: *¹ "aQQ #bQQ ,
(3)
*¹ "bQQ #cQQ .
(4)
Constants a}c clearly correspond to the respective positions of the three dashed asymptotes in Fig. 4. As is easily veri"ed, in terms of R , R and R , a, b and c equal, respectively, (R #R )R , a" R #R #R (R #R )R . c" R #R #R
R R b" , R #R #R (5)
A. van der Hoek, H. van Beelen / Physica B 293 (2001) 297}303
The inversion of these relations yields for the thermal resistance values in terms of the measured quantities: ac!b ac!b ac!b R " , R " , R " . (6) c!b a!b b In Table 2 results, obtained for the three silver heat exchangers of Table 1 have been compiled, together with results obtained for a stainless-steel heat exchanger SS, formed by a disc of 20 mm diameter with a wall thickness of 5 mm. As for the run of Fig. 4, the values of b in the table must be considered to be upper bounds. In the last column, the values of the Kapitza conductivity i of the ) bulk liquid is given, as it is deduced from the data on R "R and the dimensions given in Table 1 according to Eq. (2). Their values of about 1.2; 10 W m\ K\ are very well compatible with the values reported in literature [3]. Also the value of R of the heat exchanger SS, which is dominated by the resistance of the wall, agrees well with the value of 96 K W\ calculated from the thermal conductivity j "0.15(¹/K) W m\ K\. Finally, the values of R of about 100 K W\ are very well compatible with the estimated value of the Poiseuille resistance of the vapour #ow through the constriction, which itself strongly depends on the vapour density, and thus on the "lm thickness and the temperature.
303
References [1] P.L. Kapitza, J. Phys. USSR 4 (1941) 181. [2] W.H. Keesom, A.D. Keesom, Physica 3 (1936) 359. [3] E.T. Swartz, R.O. Pohl, Rev. Mod. Phys. 61 (1989) 605. [4] I.M. Khalatnikov, Zh. Eksperim. Teor. Fiz. 22 (1952) 687. [5] H.J. Lauter, H. Godfrin, V.L.P. Frank, P. Leiderer, in: A.F.G. Wyatt, H. Lauter (Eds.), Excitations in Two-Dimensional and Three-Dimensional Quantum Fluids, Plenum Press, New York, 1991, p. 419. [6] A.C. Anderson, J.I. Connolly, J.C. Wheatley, Phys. Rev. 135 (1964) 910A. [7] A.R. Long, R.A. Sherlock, A.F.G. Wyatt, J. Low Temp. Phys. 15 (1974) 523. [8] A.R. Long, J. Low Temp. Phys. 17 (1974) 7. [9] P. Taborek, M. Sinvani, M. Weimer, D. Goodstein, J. Phys. C6 (1981) 825. [10] J.C. Mester, E.S. Meyer, M.W. Reynolds, T.E. Huber, I.F. Silvera, Phys. Rev. Lett. 68 (1992) 3068. [11] A.F.G. Wyatt, Phys. Rev. Lett. 69 (1992) 1785. [12] M.E.W. Eggenkamp, H. Khalil, J.P. Laheurte, J.C. Noiray, J.P. Romagnan, Europhys. Lett. 21 (1993) 587. [13] H. van Beelen, R.W.A. van de Laar, A. van der Hoek, Physica B 194}196 (1994) 669. [14] A. van der Hoek, H. van Beelen, Czech. J. Phys. 46 (Suppl. S1) (1996) 415. [15] A. van der Hoek, PhD Thesis, Leiden University, 2000. [16] R.W.A. van de Laar, A. van der Hoek, H. van Beelen, Physica B 216 (1995) 24. [17] V. Ambegaokar, B.I. Halperin, D.R. Nelson, E.D. Siggia, Phys. Rev. B 21 (1980) 1806.
Measurements on the Kapitza resistance of very thin super#uid helium "lms A. van der Hoek, H. van Beelen* Kamerlingh Onnes Laboratory, Leiden University, P.O. Box 9506, 2300 RA Leiden, The Netherlands Received 14 March 2000; received in revised form 19 June 2000
Abstract Measurements on the Kapitza resistance of super#uid He "lms on silver substrates at temperatures above 1.0 K are reported. No apparant change in value is found upon decreasing the "lm thickness, all the way down to the Kosterlitz}Thouless transition occurring at thicknesses as small as two to three atomic layers. 2001 Elsevier Science B.V. All rights reserved. Keywords: Kapitza resistance; Thin "lms; He
1. Introduction In 1941, Kapitza observed that when heat #ows from a solid into super#uid He II a temperature discontinuity appears across the boundary [1]. Earlier observations of the phenomenon (1936) had passed without attracting particular attention [2]. Kapitza found that in the range of small #ow rates, the drop in temperature is proportional to the heat #ux. The associated thermal boundary resistance, i.e., the ratio of the temperature drop and the heat #ux per unit contact area, is nowadays known as the Kapitza resistance of the heat-exchanging interface. At the time of its discovery the phenomenon
* Corresponding author. E-mail address: [email protected] (H. van Beelen). Present address: Baan Company, P.O.Box 143, 3770 AC Barneveld, Netherlands.
was attributed to the special character of the super#uid, but it has since become clear that a thermal boundary resistance occurs at the boundary between any two di!erent media. Nowadays, a rather complete quantitative understanding of the measured values of the thermal boundary resistances of di!erent contact surfaces is available. An extensive review of this understanding has been given in 1987 by Swartz and Pohl [3]. In cases that the interface is su$ciently #at, the mechanism behind the heat exchange is described by the &acoustic mismatch model'. This AM-model was "rst introduced in 1952 by Khalatnikov for the boundaries between solids and He II [4]. In the AM-model the heat transfer is due to the exchange of thermal phonons, their transmission through the plane interface being governed by the rules of the acoustics of continuous media. Because of the &acoustic mismatch' between He II and solids, viz., the large di!erences between the sound velocities as
0921-4526/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 0 ) 0 0 5 5 7 - 3
298
A. van der Hoek, H. van Beelen / Physica B 293 (2001) 297}303
well as between the acoustic impedances, the transmission coe$cients of the interfaces are very small. This results in relatively large values of the Kapitza resistances. Furthermore, since at low temperatures a system formed by bulk matter can be viewed as the acoustic analogue of the black-body radiator, the available phonon energy increases with the temperature as ¹. The AM-model therefore predicts the thermal boundary resistance to vary as ¹\. Experimentally it is found that the value of the boundary resistance depends very strongly on the surface preparation of the contact area. Nevertheless, for some carefully polished solid}solid boundaries a fair agreement with the AM-prediction is found. For well-prepared solid-He II boundaries, however, the AM-prediction is only approached for temperatures below a few tenths of a kelvin; towards higher temperatures the Kapitza resistance decreases gradually by up to two or three orders of magnitude below its AM-value. A plausible cause of the deviation is the scattering of high-frequency phonons in the boundary region, which indeed becomes the more important the higher the temperature. The e!ect of phonon scattering on the thermal boundary resistance has "rst been estimated by Swartz in 1987 [3]. Swartz considered the extreme case of completely-di!use, elastic scattering. In that case the probability of either forward or backward scattering is simply proportional to the density of phonon states in the material on either side of the interface. Since in He II this density of states is larger by several orders of magnitude than in solids, the applicability of the &diwuse mismatch model' (DM-model), in whole or in part, can explain the observed stronger decrease of the Kapitza resistance with temperature. The mismatch of the densities of phonon states between di!erent solids is much smaller, which explains the fair agreement with the AM-prediction found for well-prepared solid}solid contact surfaces. From the above models of the thermal boundary resistance of bulk He II one can try to predict the properties of the thermal contact between a very thin He "lm and its substrate. It is known that the excitation spectrum changes signi"cantly with "lm thickness [5], particularly the phonon part at long wavelengths, so it can be expected that the acoustic
mismatch between a thin "lm and its substrate will be quite di!erent from that of the bulk. A distinct crossover from bulk- to "lm-behaviour might thus be observable in the Kapitza resistance of "lms on cleaved crystal surfaces or on well-polished substrates at temperatures smaller than a few tenths of a kelvin, cases in which the AM-model describes the bulk behaviour pretty well. However, for unpolished substrates or at higher temperatures diffuse phonon scattering at the interface will become more and more dominant, so that the crossover behaviour may be expected to become less evident. According to the DM-model the phonon transmission from solids into He II is almost 100%, so that only very dramatic changes of the density of states of the excitations in the "lms will a!ect the value of the Kapitza resistance signi"cantly. In Ref. [3] a similar argument is used to explain the observed insensitivity to the pressure of the Kapitza resistance in the case of He [6]. Experimentally, the situation is not yet completely clear; the measurement of the Kapitza resistance of very thin "lms is by no means straightforward. At present, only some data for temperatures above 0.3 K are available. In 1974 Long et al. [7,8] reported measurements, carried out at ¹"1.1 K, on the re#ection of sub-microsecond pulses of high-frequency phonons (with black-body temperature of about 10 K), from the solid side of the interface between He and a cleaved sodium#uoride crystal. The data resolved the re#ections of the longitudinal and transverse phonons, and demonstrated the occurrence of mode conversion as well as of the excitation of surface waves. On lowering the vapour pressure it was found that the re#ection pattern only starts to change at helium coverages thinner than a few atomic layers. From the analysis of their data the authors concluded that the mechanism, responsible for the transfer of energy into the helium side, is the desorption of helium atoms from the second, solid, helium layer, mediated by the excitation of surface waves. In 1981, Taborek et al. [9] also studied the transmission of narrow heat pulses into helium "lms as a function of "lm coverage. They observed that the transmission of sub-microsecond pulses already attains its bulk value at "lm coverages of only three atomic layers. From this observation they
A. van der Hoek, H. van Beelen / Physica B 293 (2001) 297}303
concluded that the Kapitza resistance does not change with "lm thickness down to such small coverages. A revival of interest in the problem occurred in 1992, when Mester et al. [10] reported the observation of a dramatic increase of the Kapitza resistance, by up to three orders of magnitude, for "lms thinner than four atomic layers; they had measured the temperature rise of a bolometer from which "lm liquid was steadily evaporated by a constant heat input. A subsequent renewed interpretation of Mester's measuring results by Wyatt [11], however, demonstrated that no change at all had occurred. Such a conclusion was also drawn in 1993 by Eggenkamp et al. [12], from their measurement of the heat-pulse di!usion along a glass substrate coated by a He "lm of 2.5 atomic layers coverage. Finally, the absence of a dramatic increase of the Kapitza resistance with decreasing "lm thickness down to three atomic layers was also con"rmed by van Beelen et al. [13], but a far more gradual increase towards thinner "lms was suggested by the measurements. In the latter experiment, heat is supplied to the "lm at a steady rate by the latent heat of vapour condensation. The temperature rise due to the removal of this heat through the glass substrate to the surrounding He II bath is measured, while the condensed mass is removed by a steady super#uid mass#ow through the "lm back to a heater, by which it is evaporated again. We have repeated the experiment using an improved device with a welllocalized heat exchange. As we have already brie#y reported (in 1996) [14], we could demonstrate that the observed apparent gradual increase was an artefact of the old device, essentially caused by the shrinking of the heat-exchanging surface area with decreasing "lm thickness. In the present article, a full account of these investigations is presented.
2. The measuring cell for the study of the Kapitza resistance The measurements form part of the investigations on the #ow properties of very thin He "lms reported in Ref. [15]. The Kapitza resistances of the helium "lms are studied on "lms covering the inner wall of the all metal device, sketched in Fig. 1.
299
The device is immersed in the main He bath. It is provided with a well-localized, highly conductive (silver) heat exchanger with the bath. The net thermal conductance per unit area of the exchanger i is composed of the Kapitza conductances i (d, ¹) and i (¹) of the two interfaces between ) "lm and silver and between silver and bulk liquid, respectively, connected in series with the conductance of the silver j /d (with d the thickness and j the thermal conductivity), so that 1 1 d 1 " # # . (1) i j i i ) Since, of the three contributions to i , only i may vary with the "lm thickness d, measurements of the thermal resistance of the exchanger can thus be used in principle to investigate the possible variation of the Kapitza conductance i with d. We used the silver heat exchangers sketched in Fig. 2. Three di!erent sizes were used, as listed in Table 1. For these heat exchangers, machined out of high-purity silver, the surface contributions fully dominate their total thermal resistance. The net thermal resistance of the heat exchangers is thus 1 1 R " # . (2) i A iA ) The He "lm covers the inside walls of the measuring cell, and in fact also of the &"ll tube' FT, up to some height above the He bath level (see Fig. 1). In order to vary the "lm thickness, discrete amounts of He gas are added to or removed from the interior of the cell via the "ll tube. A pumping tube, not shown in the "gure, is connected to the vacuum can. Both "ll tube and pumping tube lead through the top of the cryostat to a gas-handling system at room temperature. In order to suppress thermal radiation from the top of the cryostat, both tubes are provided with radiation barriers, formed
With an estimated thermal conductivity of the high-purity silver better than 10 W m\ K\ and a wall thickness of about 1 mm, the middle term of Eq. (1) is smaller than 10\ K m W\, much smaller than the 10\ K m W\ of the other two contributions.
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A. van der Hoek, H. van Beelen / Physica B 293 (2001) 297}303 Table 1 The dimensions of the three silver heat exchangers
Fig. 1. Schematic drawing of the experimental arrangement used for the study of the Kapitza resistance between "lm and substrate.
Fig. 2. The silver heat exchangers used for the measurements of the Kapitza resistance.
by a number of brass plates inserted halfway through the walls of the tubes at di!erent heights above the vacuum can. Moreover, the radiation barriers in the cold section of the "ll tube serve at the same time as sound breakers, in order to prevent the spontaneous appearance of thermally driven &organ-pipe like' sound oscillations in the vapour column, observed in our earlier experiments [16]. A steady circulation of helium is set up in the essential part of the device, formed by heater Q and thermometer T inside the stainless-steel
No.
A
S1 S2 S3
15.71 4.92 0.83
(cm)
A
(cm)
13.29 3.61 1.41
vacuum chamber and the (silver) heat exchanger in contact with the surrounding He bath. Unwanted #ow of heat, by circulation of He, to and from the "ll tube is e!ectively suppressed by the two narrow constrictions (d +0.5 mm) shown in the "gure, particularly at the lowest temperature where the vapour density becomes small and the vapour #ow resistance high. The upper one, in combination with the bellows heat exchanger (see Fig. 1), minimizes the input of extra heat from the top of the cryostat; the lower one reduces the heat leak away from Q . The core measurements concern the variation of *¹ ,¹ !¹ with the heater power QQ sup plied by Q , where *¹ is measured di!erentially with respect to the bath thermometer T . The ther mal resistance R ,*¹ /QQ is formed by the con tribution from the #ow resistance along the "lm path between T and the heat-exchanging surface, in series with the thermal resistance R of the (silver) heat exchanger itself. Since, in the super#uid phase, the "lm#ow resistance decreases rapidly for decreasing QQ (according to a powerlaw with expo nent larger than two [17]), the constant thermal resistance of the heat exchanger becomes the dominant contribution at small QQ and can be deduced from the data. A correction for the residual heat #ow along the lower constriction is applied, for which purpose this heat #ow is determined by means of both heaters Q and Q and the addi tional thermometers T , T and T (see the appen dix). Owing to the fact that, at the lower bath temperatures, the super#uid transition already takes place in "lms of only about two atomic layers thickness, the e!ect of the "lm thickness on the value of the thermal boundary resistance could be investigated down to such very thin "lms in this way. Because of the decrease in the resolution of the measurement with increasing temperature, most of
A. van der Hoek, H. van Beelen / Physica B 293 (2001) 297}303
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our measurements have been carried out at the lowest temperatures attainable in our cryostat.
3. Experimental results In the double-logarithmic plot of *¹ versus QQ in Fig. 3, a representative set of measuring results is presented, obtained at ¹ "1.107 K with S3, the smallest of our heat exchangers (cf. Table 1). Series of datapoints are shown for various values of the `glass-equivalent xlm thicknessa dH [15] in the range just beyond the Kosterlitz}Thouless transition, which is found to occur at about dH "2.33 )2 a.l. for this value of ¹ . For each value of dH two curves are presented, one connecting the ten datapoints measured at almost equal spacings along the ordinate for decreasing values of the heater power QQ , the other connecting the datapoints measured at the same 10 values of QQ but in the reverse order. Each datapoint corresponds to a thermal resistance R ,*¹ /QQ , where *¹ is composed of the temperature drop *¹ along the "lm between T and the heat-exchanging "lm area and the drop *¹ across the heat exchanger. For given dH, R is dominated by the contribution from *¹ at large values of the heater power QQ , but, because of the steep decrease of the "lm#ow resistance with decreasing mass#ow rate [17], the constant contribution from the heat exchanger R should be come dominant at small values of QQ . That this is what is actually observed is demonstrated by the curves in Fig. 3; they all tend to approach a common asymptote with slope one for small QQ . All curves having the same asymptote signi"es that R is independent of dH. Finally, in view of Eq. (2), it can be concluded that also the thermal-boundary conductivity i is independent of the "lm thickness, even for d approaching the Kosterlitz}Thouless value, and
This is the value of the thickness, as calculated from the vapour pressure using the well-known Frenkel}Halsey}Hill procedure with the VanderWaals constant of the substrate given the value usually used for glass, c "2.62;10\ m s\, by 5 lack of knowledge of the value on silver.
Fig. 3. The measured ¹ !¹ versus QQ , determined by the superposition of the Kapitza resistance and the "lm #ow, for a range of dH just beyond the Kosterlitz}Thouless transition at dH +2.33 a.l. The tendency to approach a common asymptote )2 with slope one at small QQ suggests that the Kapitza resistance does not depend on d.
should thus remain equal to the bulk value i . This ) conclusion is corroborated by similar series of measurements carried out at some other values of the bath temperature and with the other heat exchangers listed in Table 1. In the appendix the way in which the value of the Kapitza conductivity is derived from the measurements is discussed. Although the measuring results on i were found to be well reproducible, not only ) during each measuring day but also from week to week and even after replacing one heat exchanger by another, we do not consider the experimental values compiled in the last column of Table 2 to correspond to `thea speci"c value of the Kapitza conductivity of the boundary between liquid He and silver; we are aware of the fact that our surfaces may very well have been contaminated, even though we took care to keep them clean. This observation does not a!ect the main conclusion of our work, however, i.e., the Kapitza resistance is largely independent of the thickness of the helium "lm.
Acknowledgements The authors are grateful to Lex Reesink for his expert help in the preparation of the manuscript.
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Table 2 The measured and corrected values of the thermal resistances for various heat exchangers No.
¹ (K)
a (K W\)
b (K W\)
c (K W\)
R (K W\)
R (K W\)
R (K W\)
i ) (W K\ m\)
S1 S2 S3 SS
1.065 1.074 1.102 1.103
1.2 4.0 13 50
0.07 0.2 0.6 2
4.7 4.8 5.0 4.4
1.2 4.2 15 90
5.0 5.0 5.2 4.5
81 96 108 108
1.1;10 1.1;10 1.3;10
Appendix The common asymptote in Fig. 3 corresponds to a value of the thermal resistance of the heat exchanger R "14.7 K/W. With the dimensions of S3 given in Table 1, Eq. (2) yields for ¹ "1.10 K the bulk value i "1.3;10 W m\ K\, in ) fair agreement with values for unpolished surfaces reported in literature, in between the predictions from the acoustic}mismatch model and the diffuse-mismatch model (cf. Fig. 1 of Ref. [3]). Furthermore, by employing also the second heater Q and thermometer T , it could be veri"ed that the corrections due to the heatleak across the narrow constriction remain almost negligibly small for these thin "lms at these low temperatures. This is illustrated by the example presented in Fig. 4. It shows the curves for *¹ (QQ ), *¹ (QQ ), *¹ (QQ ) and *¹ (QQ ), measured at ¹ "1.102 K for heat exchanger S3, in a double-logarithmic plot. Evidently, also the datapoints *¹ (QQ ) ap proach an asymptote for decreasing QQ , which is found to be independent of the thickness of the "lm and corresponds essentially to the thermal resistance R of the heat exchange between the "ll tube and the bath (cf. Fig. 1). The common asymptote expected for the other two series of data, has clearly not been reached within the measuring window presented in the "gure; since the temperature di!erences on these asymptotes are proportional to the heat#ow across the constriction, this data shows that this heatleak does not amount to more than a few percent of the heat input. In the regime of the asymptotes, where the thermal resistances R and R , as well as R of the constriction, have become independent of the heat-
Fig. 4. The measured ¹ !¹ versus QQ , for a range of dH.
From these measurements the Kapitza conductivity, corrected for the heat leak, is computed.
#ow, a more quantitative analysis of the data in Fig. 4 can easily be made. In this regime the contributions of the two heaters to the temperature di!erences simply add, so that: *¹ "aQQ #bQQ ,
(3)
*¹ "bQQ #cQQ .
(4)
Constants a}c clearly correspond to the respective positions of the three dashed asymptotes in Fig. 4. As is easily veri"ed, in terms of R , R and R , a, b and c equal, respectively, (R #R )R , a" R #R #R (R #R )R . c" R #R #R
R R b" , R #R #R (5)
A. van der Hoek, H. van Beelen / Physica B 293 (2001) 297}303
The inversion of these relations yields for the thermal resistance values in terms of the measured quantities: ac!b ac!b ac!b R " , R " , R " . (6) c!b a!b b In Table 2 results, obtained for the three silver heat exchangers of Table 1 have been compiled, together with results obtained for a stainless-steel heat exchanger SS, formed by a disc of 20 mm diameter with a wall thickness of 5 mm. As for the run of Fig. 4, the values of b in the table must be considered to be upper bounds. In the last column, the values of the Kapitza conductivity i of the ) bulk liquid is given, as it is deduced from the data on R "R and the dimensions given in Table 1 according to Eq. (2). Their values of about 1.2; 10 W m\ K\ are very well compatible with the values reported in literature [3]. Also the value of R of the heat exchanger SS, which is dominated by the resistance of the wall, agrees well with the value of 96 K W\ calculated from the thermal conductivity j "0.15(¹/K) W m\ K\. Finally, the values of R of about 100 K W\ are very well compatible with the estimated value of the Poiseuille resistance of the vapour #ow through the constriction, which itself strongly depends on the vapour density, and thus on the "lm thickness and the temperature.
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