The Kapitza conductance of clean copper surfaces between 0.3 K and 1.3 K

Results are presented of the Kapitza conductance between annealed and electropolished "clean" copper samples and I/quid He4 in the temperature range 0.3 K to 1.3 K A description is given of the apparatus and techniques developed to enable in situ high vacuum treatment of the sample surface, without an excessive heat load on the He3 refrigerator during measurements of the Kap/tza conductance. The experimental results of this work are in excellent agreement with previous results, measured at higher temperatures, for similarly prepared surfaces. The temperature dependence of the measured heat transfer coefficient shows a clear maximum at 1 I~ This new result is in qualitative agreement with the theoretical treatment of the Kapitza conductance in terms of acoustic theory, modified to include phonon attenuation and impedance matching. However the large magnitude of the heat transfer coefficienE suggests that some alternative mechanism may dominate the Kapitza conductance, even to "clean" surfaces.

The Kapitza conductance of clean copper surfaces b e t w e e n 0.3 K and 1.3 K A,W. Pattullo and J.C.A~ Van der Sluijs Key words: cryogenics, Kapitza conductance, copper When a heat flux Q flows across a boundary between two dissimilar materials a temperature discontinuity A T appears at the interface. The thermal boundary conductance or more specifically for interfaces involving liquid helium the Kapitza conductance H k , is defined by; H k = Q/(A AT)

(Wm -2 K - ' )

(1)

whereA is the area of the interface and AT < < T, the absolute temperature. There has been a great deal of work done on the Kapitza problem since it was discovered in 19417 Nevertheless heat transfer between a solid and liquid helium is still poorly understood. In this paper we describe an investigation of the Kapitza conductance between copper and liquid He ~ well below the h transition. The high thermal conductivity of the media and the intimate contact between them makes the Kapitza conductance of this system one of the best defined and most easily measured thermal boundary conductances. For brevity we restrict our comments to the copper to liquid He 4 interface. For more general information about the thermal boundary conductance the reader is referred to a number of review articles, 2-5 and recent papers. 6.7 The fundamental theory of the Kapitza conductance, the acoustic mismatch theory, was developed by Khalatnikov. 8,9 Heat transfer across the interface is described in terms of the transmission and reflection of phonons or acoustic waves at the interface. The fraction of the incident phonons transmitted across the boundary is determined by the mismatch between the acoustic impedances of the media. Only phonons within the He 4 at near normal incidence to the interface may exchange energy with the solid. Phonons incident outside this narrow critical cone are totally internally reflected and make no contribution to H k . An important result of the theory is that the phonon transmission coefficient is temperature independent.

Hence assuming a Debye specific heat (proportional to TJ) the acoustic theory predicts H k am ~ 8 T 3 ( W m -2 K-~). (In calculating this figure, surface modes are assumed 4 to make no contribution to heat transfer). Experimental results are found to lie between H k am and an upper limit H k P rl, the phonon radiation limit? ° H k P n is defined as the Kapitza conductance for 100% p h o n o n transmission. For copper to liquid He 4, H k O r ~ 4240T 3 (Wm -2 K-~). We may now define a heat transfer coefficient ~t for an interface with Kapitza conductance H k as; at

=

H k / H k prl ×

100%

(2)

Hence for a copper to liquid He 4 interface acoustic theory predicts Ofta m -~0.19%. At temperatures below about 0.2 K the results of experiments and the acoustic theory are in general agreement. However at higher temperatures, experimental results show a much larger magnitude (at < 40%), lack of reproducibility both in magnitude and temperature dependence, and large scatter. Some of these points are illustrated in Fig. 1. Bare acoustic theory is clearly inadequate to account for this anomalous behaviour at higher temperatures. A number of attempts have been made to modify the acoustic mismatch theory 24-~7 and to introduce parallel heat transfer mechanisms ~.'.28-31 to explain the large conductance at high temperatures. However these attempts have been severely hampered by the poor reproducibility of experimental results, which cannot be attributed to experimental errors. Instead we must assume that the differing results are from measurements on different physical systems. The most obvious variables are the purity and surface preparation of the sample material. Where known, these variables are listed in the caption to Fig. 1. Work by Alnaimi and van der Sluijs 32 and more recently by Rawlings et al, 22.3° has shown how results may be obtained which reproduce for different copper

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samples and for preparation by different workers. Samples of 5N purity were prepared by vacuum annealing, deep electropolish (> 100 ~ m removed) and finally exposure to 'clean' high vacuum, in situ in the experimental cell. So far, measurements on 'clean' vacuum treated samples have been limited to temperatures above 1.2 K.z2,z3.32 The Kapitza conductance for these carefully prepared surfaces,are found to be consistently lower than previous measurements and t~t is found to increase with decreasing temperature towards 1.2 K, Measurements of the angular dependance of the phonon transmission coefficient at N a F - H e 4 interfaces? 3 confirm the presence of the narrow critical cone predicted by acoustic theory. However as the temperature of incident phonons at the interface rises above 0.2 IL a second channel is seen to dominate heat transfer across the boundary. The mechanism of the second channel is still unknown, However the reduction o f H k / T 3 for samples after vacuum treatment zz.~Uz,34,as suggests that the excess heat is transferred by mechanisms associated with impurities at the interface. Furthermore, for vigorously cleaned

588

surfaces, heat transfer associated with dislocations at the sample surface becomes important, z2.3s The purpose of this work was to extend the measurements on 'clean' surfaces to lower temperatures and to obtain results for H k in the transition region between the high temperature (> 1 K) anomalous behaviour and the almost acoustic behaviour below 0.2 K. In particular we wanted to investigate the hitherto uncertain temperature dependencd of the Kapitza conductance in this regime. Preliminary results of this work have been published? 6 A more complete description of experimental details may be found elsewhere? 7 In the next section brief details are given of a H e 3 refrigerator and the sample high vacuum system. Further on details of the three thermometer systems are given and the sample preparation and experimental technique are described. The steady state or dc measuring method was used to obtain the experimental results of the Kapitza conductance. These results are presented, followed by corrections to the results and a discussion. Finally conclusions are briefly given.

Experimental apparatus Hes refrigerator A two stage Hea recirculating refrigerator was built to provide the required temperature range. The first stage was a He 4 evaporator used to condense He a gas at 1.2 K. The second stage was a Hea evaporator on which the experimental cell was mounted. Both evaporators were enclosed in an evacuated can immersed in the main He 4 bath at 4.2 K. The cooling power of the fridge was l/,tW at 0.35 K and 100 ~W at 0.43 K. This provided a useful working temperature range from 0.38 K to > 1.3 K with the experimental cell full of liquid He 4. A novel feature of the refrigerator was the design of electronic temperature regulator for the He 3 platform. An electrical heater mounted on the He a evaporator was used to provide fine regulation of the temperature of the evaporator and experimental cell. The heater was driven by a two term (proportional and integral) electronic controller. The controller input was the off-balance error signal from a low power cryobridge, described in the following section. The cryobridge was used to monitor the resistance of one of the resistance thermometers mounted on the He 3 evaporator (220 fZ Speer) or the experimental cell (10 and 47 fl Allen Bradleys). This regulation system was capable of holding the temperature of the He 3 platform constant to within the measuring accuracy of the thermometry. The novel feature of the controller design was a hold facility, whereby once thermal equilibrium was established, the heater power could be kept constant while the bridge was used to monitor other resistors. In this way the one bridge was successfully used for both temperature measurement and control.

Sample vacuum system Previous work has shown that to obtain reproducible results of the Kapitza conductance, high vacuum treatment, ~Uz in particular a vacuum free of oil p u m p contamination n.~s was an essential part of the sample preparation. Therefore we used a getter ion pump, connected through a low flow impedance

CRYOGENICS. NOVEMBER 1983

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pumping line to the experimental cell, to provide the 'clean' vacuum at the sample surface. However due to the high thermal conductivity of superfluid He 4, such a wide bore pumping line would present an intolerable heat load on the He 3 platform when the experimental cell and pumping line were full of He 4, during a measurement of the Kapitza conductance. The simplest solution to the problem was to have two lines leading to the cell. A large bore tube was used to provide the high vacuum at the sample surface, and a long narrow capillary to fill the cell during an experimental run. The complete sample high vacuum system is illustrated schematically in Fig. 2. The large bore tube terminated at a superfluid tight valve. This valve and the experimental cell were connected by demountable indium seals to a drilled mounting block. This in turn was bolted to the He 3 evaporator. The needle valve was closed just prior to a run to prevent the superfluid from the cell forming a film or liquid column in the large bore tube and consequently a large heat leak to the cell. The impedance from the p u m p to the sample surface through the large bore tube and needle valve was ~ l0 s m -3 s less than that through the capillary. After sealing in each new sample and after every experimental run, the system was roughed out using an external rotary and diffusion p u m p set. A cold trap, filled with activated charcoal, prevented contamination of the system by oil backstreaming from the pumps. The dump and dial gauge allowed accurate filling of the cell. The He 4 was taken from the liquid H d bath exhaust to ensure high purity. A small reservoir mounted above the cell was used to accommodate any excess He 4 and so prevent a column of liquid standing in the fill capillary.

CRYOGENICS. NOVEMBER 1983

Experimental cell The cell was of the tube type, in which a known amount of heat was supplied and flowed across the sample to He 4 interface. This produced a temperature difference between the sample and liquid He 4 which could be measured when the system came to thermal equilibrium. An important requirement of the design was that the bulk of the energy supplied to the heater went through the interface. To facilitate this, a hard vacuum was maintained inside the enclosing vacuum can throughout the experiment. A detailed discussion of alternative heat paths will be given later. A cross section through the cell is shown in Fig. 3. As can be seen in Fig. 3, the cell consisted of a thin walled (0.15 mm) stainless steel tube. Protruding from the cell wall were two tags to which small copper blocks were hard soldered. Drillings in the copper blocks allowed carbon resistance thermometers to be mounted. The base of the cell was soldered to a copper block. A flange on the block allowed the cell to be sealed with indium wire to the cell mounting block. The copper sample was in the form of a disc, 3 m m high and 5.6 m m diameter, with a 2 m m diameter hole drilled 2 m m into it to accept the sample thermometer post. The sample was sealed with indium wire into the flange at the top of the cell tube, and secured with three 2 m m diameter stainless steel bolts. A layer of GE7031 varnish ensured good thermal contact between sample and heater block. The carbon resistance thermometers were l0 and 47 13 Allen Bradley resistors. The l0 and 47 N resistors were mounted as pairs on the sample resistor post and on the upper tag of the cell wall. These were used to measure AT across the interface from 0.3 K to 1.3 K. A further l0 l'l resistor was mounted on the lower cell wall tag, to allow thermal gradients in the helium to be measured. The 40 swg varnished constantan leads from the resistors and the cell heater were thermally anchored at the copper base of the cell. Superfluid tight valve. The needle valve was based on a commercially available stainless steel needle (obtained from Oxford Instruments Ltd of Osney Mead, Oxford, UK). A cross section through the valve Constanton heater Copper core Stoinless steel collor Copper resistor p o s t

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in the experimental cell. The resistance thermometers were calibrated against the susceptibility thermometer to the lowest temperatures used. Vapour pressure thermometry. The values of vapour pressure against temperature tabulated by Sydoriak et al? 9 may be fitted by an empirical relation. 4° This relation was used to facilitate computer processing of the experimental results. A drilling in the cell mounting block (see Fig. 2) formed the sensing bulb for the vapour pressure thermometer. The bulb was packed with copper wire to improve thermal contact to the He 3. A thin walled stainless steel tube coupled the bulb to a dual mercuryoil manometer at room temperature. The tube was heat sunk to the 1 K pot for radiation trapping. The thermometer was not used below 0.7 K to avoid significant thermomolecular pressure corrections (> 1%)..1 Moreover to avoid overfilling the sensing bulb at the lowest temperatures, the quantity of gas, and hence the thermometers upper measuring limit was restricted to 2 K. The manometer could be read to an accuracy of ___ 0.25 m m using a cathetometer. The overall measurement error was + 4 m K at 0.7 K, becoming less at higher temperatures. tJaramagnetic susceptibility thermometer The magnetic susceptibility Of the paramagnetic salt C M N obeys Curie's law down to about 7 mlC A convenient way to measure the susceptibility, and hence the temperature, is to convert the susceptibility variation to a frequency variation. This was done by filling a small coil with CMN. The coil was connected in parallel with a fixed capacitor to form the tuned circuit for a low power, back diode oscillator? ~ The circuit diagram is shown in Fig. 5. The frequency f, of the oscillator is related to the absolute temperature T by,43

Cross-section through the superfluid tight needle valve

(fo - .])/fo is shown in Fig. 4. The valve was designed to be operated whilst at room temperature and with the vacuum can which enclosed the He 3 and He 4 evaporators removed. The valve casing was hexagonal to allow a spanner to be used to hold it, while the top nut was screwed up or down, so opening or closing the valve. A double 'O' ring seal prevented atmospheric air entering the high vacuum parts of the valve during room temperature manipulations. After opening or closing the valve, a brass cap could be sealed over the top of the valve to allow a cold leak test of the cell (valve open), or to allow the large bore tube to be monitored for leaks through the valve (closed) during an experimental run. The base of the valve was soldered into a copper block which had an indium seal flange to connect it to the cell mounting b l o c k

Thermometry The three separate thermometer systems are briefly described in turn in the following subsections. A He 3 vapour pressure thermometer provided the primary thermometry from 2 K down to about 0.7 IC Hence all absolute temperatures were calculated from the 1962 He 3 scale, tabulated by Sydoriak et al? 9 Calibrated against the vapour pressure thermometer was a cerous magnesium nitrate (CMN) paramagnetic susceptibility thermometer. This provided the absolute thermometry over the entire temperature range of the experiment. Finally carbon resistance thermometers were used to measure thermal gradients

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where f0 is the frequency at 1/T = 0, B depends on the Curie constant of the C M N a n d j is the coil filling factor. The most critical component of the thermometer was the coil, based on a design by Hartley et al. ~4 A slurry of powdered C M N and glycerine was pressed around copper wires and the whole enclosed in a nylon former. A threaded copper stud, hard soldered to the copper wires, was used to attach the coil to the cell mounting block on the refrigerator. Around the nylon former was wound the 120 turn coil of 38 swg copper wire. The C M N was used in powder form to improve the thermal contact between the salt and the cell mounting block. Hence the value ofjB was found by calibrating the thermometer against the He 3 vapour thermometer (0.7 K < T < 2 K) and the boiling point of He ~ in the main helium bath (4.2 K). The calibration also confi~rmed the excellent thermal contact between the C M N and vapour pressure thermometers, and the cell mounting b l o c k The constants in (3) were found from a least squares fit to the calibration data. A small but significant change in B was found to occur at about 1 K. This anomaly may have been due to some magnetic transition in the vicinity of the C M N coil. However the cause has yet to be fully investigated. Allowing for the change in B at 1 K, the standard deviation of the calibration points was 1.3%, giving a predicted error of + 15 m K at 0.35 IC Although f0 had to be found for every experimental run using the 4.2 K

CRYOGENICS. NOVEMBER 1983

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fixed point, subsequent recalibration confirmed the constancy offojB with time. Resistance thermometry. Due to their small size and high sensitivity, carbon resistors were used to measure thermal gradients in the experimental cell. Mounted together in pairs as illustrated in Fig. 3, 10 fl (Ri) and 47 fl (Ri') Allen Bradley resistors were used to cover the temperature ranges 0.3 K < T < 1.2 K and T > 1.2 K respectively. By using different nominal resistor values, sensitivity was maintained at the higher temperatures and the measurement of excessively high resistance avoided. The resistance values were measured to an accuracy of 0.1%, using an ac Wheatstone bridge (cryobridge type $72, supplied by the Low Temperature Dept, Nuclear Physics Institute, Czech Academy Science, Prague). The cryo-bridge was of the three terminal type, eliminating the measurement of lead resistance. The m i n i m u m sensor excitation voltage was 20/,tV giving a power dissipation of 10- ~ W for a 10 fl resistor at 0.35 K. With suitable switching, the bridge was also used to measure differences between resistors; DR1 ( = R I - R 2 ) , D R I ' ( = R I ' - R 2 ' ) and DR2 (=R2-R3), (see Fig. 3). Hence care was taken to match the characteristics of resistance thermometers of the same nominal value. The difference measurements were used to improve the precision and reduce the effects of temperature drift upon the measurement of A T at the copper to liquid helium interface. All electrical leads into the refrigerator were filtered against rf interference. During each experimental run, the resistors were calibrated against the C M N thermometer throughout the temperature range used. A polynomial of the form, l/T

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CRYOGENICS. NOVEMBER 1983

ensured by soldering one resistor lead to the copper mounting tag (see Fig. 3), coating the resistor body with vacuum grease and inserting it into the drilling in the tag. At no stage during the experiment was any evidence of Joule heating of the thermometers seen. Over three runs, the resistance thermometer on the copper sample (R1) gave results reproducible to 8 mK~ and consistent with similar literature data? 5

Experimental

technique

In this section we describe in some detail the two most crucial techniques; sample preparation and the measurement of the Kapitza conductance. Sample preparation. The samples were produced from a bar of 99.999% pure polycrystalline copper. They were machined to form discs of 5.6 m m diameter and 3 m m thick. A 2 m m diameter hole was drilled 2 m m into the centre of one face to accept the sample thermometer post (see Fig. 3). The machined samples were annealed under vacuum at 950°C for 11 h and then cooled over a further 13 h. Prior to electropolishing, the annealed samples were hand polished using 400 grade emery and finally 'crocus' paper. The samples were electropolished in an aqueous solution (sg~ 1.53) of orthophosphoric acid. Immediately after electropolishing, each sample was washed in distilled water and mounted into the experimental cell. The cell was then sealed to the cell mounting block on the He a refrigerator and the whole sample high vacuum system evacuated. During evacuation precautions were taken against contamination, as discussed previously. The mounting process took betweem 15 and 30 minutes. With the valve at the base of the wide bore tube open (see Fig. 2), the ion p u m p was left to p u m p on the sample vacuum system for a number of days. The final vacuum, measured at the ion p u m p was typically 3 × l0 -~ torr. From the pumping speed of the pump,

591

and the impedance of the large bore line, we estimate that the pressure at the sample surface was less than 3 × 10-4 torr. Prior to a measuring run, the helium tight valve was closed and the vacuum can and dewars assembled. The ion p u m p remained pumping on the cell fill capillary and the dump. The cell was cooled to 4.2 K and a measured quantity of pure He 4 gas, from the top of the cryostaL slowly introduced down the fill capillary. Once the cell was full, it was cooled further and measurements of i l k taken. After the experiment, the cell was evacuated and allowed to warm up to room temperature. The helium tight valve was reopened and the ion p u m p allowed to pump the sample surface to high vacuum again. After a further few days pumping, preparations were made for the next experimental run on the sample. Kapitza conductance measurement. According to (1), we must measureA, (2 and AT to allow Hk to be calculated. We must also know the absolute temperature T at which the measurements were taken. The interface area A, was taken to be equal to 15.9 m m 2, the cross section area of the cell tube. The rest of the sample surface was assumed to be covered with indium seal. While this assumption may have slightly altered the calculated magnitude of ilk, it would have no effect upon the temperature dependence of the results. With the cell full of liquid He 4, the heat flux across the interface (2, was assumed to be equal to the electrical power supplied to the cell heater. The validity of this assumption, together with probable corrections are discussed later. A standard four wire technique was used to measure the power dissipated in the cell heater. Two wires carried the current to the heater coil and a further two wires were used to measure the potential drop across it. The voltage drop across a standard resistor in series with one current lead enabled the current to be calculated. A small correction 46 was made for energy disspated in the current leads between the heater and the heat sink at the base of the cell. The magnitude of the power supplied to the sample ranged between 30-400/tW at 1.3 K to 0.5-3/xW at 0.35 K= The electronic temperature regulator was used to stabilise the temperature of the He 4 in the cell using R2 (or R2' depending on the temperature) as sensor. It was assumed that for zero power to the cell heater, the sample, liquid He 4 and the C M N thermometer were at the same temperature, allowing calibration of the resistors using the C M N thermometer. The power was switched on to the cell heater and the temperature regulator was left monitoring R2 for a few minutes to allow it to compensate for the additional heat load. Once a steady state was established, data for at least five values of (~ for each temperature of the He 3 platform were taken, with repeated checks of zero power points. The data obtained was the change in resistance of the sample thermometer for variation in Q at each measuring temperature. The temperature difference across the interface AT, was interpolated solely from the calibration polynomial of R1. This polynomial, (4), was obtained from the zero power measurements at each temperature during the experimental run. AT was always less than 4.4% of T, the absolute temperature. Hk was calculated from the gradient of a least squares fit to the (2 versus AT data (see (1)) for each

592

temperature p o i n t This procedure had the advantages that: systematic calibration errors of R1 could be removed; any non-linear behaviouP 7-49 could be detected; and the standard deviation of the experimental points at each temperature gave an estimate of the error in Hk. Finally Hk/T a was calculated. To the standard deviation in Hk was added estimates of the errors in T, Q, and A to give the error bars for the results presented in the next section. Throughout our calculations, we assume that AT was equal to the temperature difference between Rl and R2. This was justified because the temperature gradient in the copper sample was calculated to be negligible. Similarly a negligible gradient in the liquid He 4 was confirmed by m e a s u r e m e n t This latter measurement also confirmed that R2 and R3 were in good thermal contact with the liquid He 4 inside the cell. s°

Experimental results The experimental results are presented as H k / T 3 versus temperature to remove the temperature dependence of the incident phonon energy density. Results are presented for two samples of polycrystalline copper. Sample 1 had 152/tm removed from the surface by electropolishing. The sample surface was pumped on for a total of 29 days before the first measuring run. The vacuum finally achieved was 3 x 10-~ torr at the ion pump, corresponding to an estimated pressure of better than 3 × 10-4 tort at the sample surface. Only the results from three of the five runs on sample l are presented in Fig. 6. Although the results of the other two runs agreed well with those illustrated in Fig. 6, they have been omitted because insufficient power steps were taken at each temperature to allow an estimate of the measurement errors. As can be seen from Fig. 6, there is excellent agreement between the results from the separate experimental runs. The curve in Fig. 6 is a weighted least squares polynomial fit to the experimental data points.

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CRYOGENICS. NOVEMBER 1983

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additional experimental runs were not possible and so we consider the data for sample 2 to be less accurate than that for sample 1. However as the general behaviour of H k / T 3 for sample 2 is in excellent agreement with that observed for sample 1, it provides strong supportive evidence and is therefore included. From the two samples examined` it is clear that the values of H k / T 3for 'clean' copper samples exhibit a m a x i m u m at about 1 K. In the next section, this behaviour is compared with results reported by other workers in this laboratory and elsewhere. Neither sample showed non-linear behaviour. This was not surprising considering the fairly low heat flux densities used here and the largely unknown dependence of Qc upon the surface preparation.

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250 I I I I I I I I I I I 0.4 0.5 06 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 15 Temperature, K Fig. 7 Experimentalresults for sample2 The linear dependence of AT on Q (1) is reported47 to break down above some critical heat flux density Qc (Wm-2), for a number of metal-liquid helium interfaces. In particular, work on copper samples? 8,49 prepared in a simi!ar manner to those reported here, has shown that Qc is surface dependent and lies between 18 and 70 Wm -2 for 1.4 K < T < 2.1 K. During the measurements on sample 1, the fit of the results to (1) was checked over a range of applied powers. Nine separate powers were applied at an ambient temperature of 1.01 K. The range of power was limited by resistance thermometer sensitivity at low powers (< 33 p,W) and the refrigerator cooling power at high powers (> 149/.tW). These heater powers corresponded to heat flux densities of 2.1 Wm -2 to 9.4 Wm -2 respectively. No significant deviation from linearity was observed. This was also the case even at the high heat flux densities (25 Wm -2) at 1.3 K. Sample 2 had 61 p.m removed from the surface by electropolishing. The sample surface was pumped on for 15 days to an estimated pressure of less than 6 x 10-4 torT, (6 x 10-7 torr measured at the ion pump). Unfortunately the investigation was prematurely curtailed by a superfluid leak. Hence it was only possible to do one experimental run on sample 2. The results are shown in Fig. 7. A polynomial fit to the data points for sample 2 was not possible due to the limited number and accuracy of the results. As can be seen, although the shape of the curve for sample 2 is similar to that for sample 1, the magnitude o f H k / T 3 is considerably greater for the second sample. Possible explanations for this increased heat transfer will be put forward. Again non-linear behaviour was not detected, in this case for heat flux densities up to 57 Wm -2 at 1.4 K. Previous workers z2 show that over a period of vacuum treatment, the magnitude of Hk/T3reduces gradually to some base value. For sample 1, the excellent agreement between the results from consecutive runs showed that these results had indeed reached their base value. The base value for the results for sample 2 could not be established unequivocally as

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Heat sink b Fig. 8 Alternativeheat paths in the experimental cell: a - simple case of heat flow down cell wall and into He4;and b - includingthe heater clamping bolts and indium seal and associated flange

593

Discussion We divide the discussion into three sections. In the first we give a critical appraisal of the experimental cell design, including corrections for heat flow down the cell wall in parallel with that across the sample-He 4 interface. This is followed by a comparison between the experimental results reported here and previous experimental work. Finally we compare our results with current theoretical models of Hk. The experimental cell and parallel heat flow. The results presented in the last section were calculated using the assumption that all the energy supplied to the cell heater flowed across the copper sample-He 4 interface. We have already seen that there were negligible thermal gradients in the sample and the He 4, and that the thermometers were in good thermal contact with the sample surface and the He ~ in the cell. It was also assumed that all other thermal gradients were negligible in comparison with AT, the gradient at the sample-He 4 interface. While these assumptions were valid for low Hk interfaces above 1 K, they become increasingly less representative at lower temperatures or higher H k values. In fact the experimental results represent more closely the total cell conductance sl rather than H k at the sample-He ~ interface. To calculate Hk, we must examine the thermal properties of the experimental cell and correct the experimental results for parallel heat flow. We show schematically in Figs 8a and 8b the thermal paths in both a simple and more complicated representation of the experimental cell. For both models we make the reasonable assumptions that the heater (source) was isothermal, as was the cell base and liquid He 4 (sink). The behaviour of the simple model (Fig. 8a) has been calculated by both analyticaP 2 and numerical methods? ° A computer model, described elsewhere, 5s was used to calculate the total cell conductance. Thermal conductivities were obtained from the literature. ~53 For a first approximation, the experimental values of i l k for sample 1 from Fig. 6 were used. The total cell conductance so obtained was compared with the experimental results (again from Fig. 6). Generally the results differed and so the value of i l k for the sample-He ~ interface was altered and the program rerun. This iterative process was repeated until the calculated cell conductance was within 0.1% of the measured conductance. The values o f H k / T 3 corrected for heat flow down the cell wall and into the He 4 are shown in Fig. 9, curve 2. Also shown is the experimental curve of Fig. 6 (curve 1). The simple model does not include the heater clamping bolts nor the indium seal and associated flange. Hence these were subsequently included in a more comprehensive model, illustrated in Fig. 8b (cf Fig. 3). Again using data from the literature 16,s3.54 the second model was used to calculate the total cell conductance T = 0.4 K. However the calculated cell conductance was strongly influenced by the sampleflange thermal conductance. Even for the limiting case of i l k -- 0 at the sample-He 4 interface, the calculated cell conductance was still about double the experimental result. This large discrepancy was not surprising when we consider that there were a number of unknown contact resistances which were estimated. These unknowns could not be easily measured experimentally because the usual measurement, of the empty cell con-

594

1500 140C

..o...,'"'"" 2

130C 1200

• •...,......,°

',

900 Y

E

800

\ 600 500

Sample 1

400 300 200 100

0.5

0.6

0.7

0.8 0.9 1.0 Temperature, K

1.1

1.2

13

1.4

Fig. 9 Corrections to the results for both samples for parallel heat flow: curves 1 - experimental results uncorrected; 2 - correction for heat paths in simple model; and 3 - correction for most pessimistic case

ductance (with or without a nylon d u m m y sample), is completely dominated by the cell wall conductance. Taking the most pessimistic view, that at 0.4 K Hk = H k am for the sample-He 4 interface, we could reduce the estimate of the copper-flange conductance (A-A' in Fig. 8b) to give agreement between the model and the experimental result for the total cell conductance. Using these new estimates of the contact resistances, we then calculated the corrected values of H k / T ~ in a similar m a n n e r to that described previously for the simple model. These results represented the lower limit o f H k / T J and are shown in Fig. 9. curve 3. In view of the pessimistic assumptions used to obtain this lower limit, we would expect the true value of H k / T ~ to lie slightly below curve 2, obtained from the simple model. These same techniques were also applied to the experimental results for sample 2. However as the magnitude o f H k / T ~ was greater for sample 2, and the estimates of contact resistances from the analysis of the sample 1 data could be used (same cell!), the corrections to the results for sample 2 were significantly less. These results are also shown in Fig. 9. As can be seen, the data from both samples still exhibit a peak at about 1 K. Comparison with previous experimental work. In Fig. 10, the experimental results of this work are shown, together with the earlier results for copper-He 4 from Fig. 1. The results presented for samples 1 and 2 are those obtained using the simpler cell model, which we consider to give the best representation of the experimental results. As can be seen, the results for both samples seem to follow the general trend of earlier measurements, with H k / T 3 falling rapidly towards acoustic mismatch values below 1 IC However

CRYOGENICS. NOVEMBER 1983

Phonon

radiation

limit

1OO

50 8 Sample 2

1000!

7

500

10

~t 5

ff

E

m"

"

6~

a

.2

mple I

100 :t:

50

1

0.5

10

Acoustic

I i I I O.1

mismatch

I

1

I

I I I , hi 0.5 1.O Temperature, K

I

Fig. 1 0 Comparison of experimental results of this work with previous experimental and theoretical 15 results for the copper-liquid He 4 interface. 1 - a n n e a l e d and mechanically polished sample; 1 t 2 - low purity sample; t2 3 - surface treatment not specified; 13 4 - annealed and electropolished sample; i4 6 - low purity sample; is 7 - low purity electropolished sample;17 8 - annealed electropolished sample;18 9 - electropolished sample; 19 1 0 - annealed and electropolished low purity sample; 20 1 1 - annealed electropolished sample; 21 1 2 - anealed, electropolished and vacuum treated sample 22 - experimental results of this work; . . . theoretical curve for the desorption model 15

where d was the electropolish depth in g m and a0 represented the energy transmission coefficient for the bulk material. A best fit was obtained for a0 = 5% and ai = 35% and 20% for measurements at 1.5 K and 1.8 K respectively. On the basis of these results we should expect the transmission coefficient for sample 2 (d = 61/xm) to be a factor 1.8 to 2.3 greater than that for sample 1 (d = 152/xm). At 1 14, we actually found th at H k / T 3 for sample 2 was 2.7 times greater than for sample 1. This is a reasonable agreement, considering that the results for sample 2 may not have reached their base value (see previous section), and some of the large variations in values of Hk/T3seen before (Fig. 1). Hence we conclude that the increased values of ilk~T3 for sample 2 were due to the shallower electropolish and hence to an enhanced dislocation associated heat transfer mechanism.22, 35 In Fig. 11, the results of sample 1 are shown on a larger scale, together with results for similarly prepared, deeply electropolished samples. Also included in Fig. 1 1 is a theoretical curve, to be discussed in the next section. The excellent agreement between the magnitude and temperature dependence of the results for sample 1 and previous results 22,23 for 'clean" samples can clearly be seen. Here we use the term 'clean' to mean samples with low surface fault densities. Obviously these surfaces are not ideal. The agreement between the results from different 'clean' samples suggests that the m a x i m u m is a feature of the residual energy transmission coefficient, after the impurity based heat transfer had been largely removed. The physical explanation of this residual heat transfer is still controversial. Comparison with theoretical results. Obviously from Fig. 9, neither the measured temperature dependence nor the magnitude o f H k / T 3 for samples 1 and 2 agree with the predictions of the acoustic theory. 8.9 We next look for agreement between the experimental results presented here and the predictions of the modified acoustic models proposed by various workers, z4-27 Only one such modeW predicts results similar to those

500

at the higher end of the temperature range, the results presented here differ considerably from previous work. The temperature dependence o f H k / T 3 for interfaces between 'clean' copper and H k / T 3 shows a significant m a x i m u m at approximately 1 K. Furthermore for sample 1, the magnitude o f H k / T 3is lower than the majority of other experimental data, over the c o m m o n temperature range. This together with the small scatter of the sample 1 results confirms the behaviour for a 'clean' surface. 22 The significant difference between the surface preparation of sample 1 to that of sample 2 was the much shallower electropolish applied to the latter. Previous work on similarly prepared metal surfaces 22,35 has shown that a correlation exists between the electropolish depth (and hence the dislocation density at the sample surface), and the magnitude of Hk/T3(or at). Results from a number of copper samples of differing electropolish depth were fitted to a curve of the form; 22 a t

=

a 0

+

a I

exp (-d/50)

(%)

CRYOGENICS. NOVEMBER 1983

(5)

400

300 E

m

2oc

10C

I 0.4

l

l 06

I

01.8

I

I 10

I

l 1.2

I

I 14

I

I 1.6

I

I I 8

Temperature, K Fig. 11 Comparison of the experimental results for sample 1 (this work), with results for similarly prepared copper samples and with the predictions of the modified acoustic theory, x - sample annealed, electropolished to a depth of 1 53 ~.m and vacuum treated; 22 • - sample electropolished, ion bombarded, annealed and UHV treated; 23 - - sample 1 (1 52/,¢rn electropolish} this work; - - - theoretical curve for the modified acoustic theory27

595

found for samples 1 and 2. The model incorporates into the bare acoustic model the effects of impedance matching, 24 and phonon attenuation within the solid. 25.26 Impedance matching between the solid and the liquid He 4 is caused by a dense layer of He ~ adjacent to the solid surface. 24 The increase in He 4 density is produced by the Van der Waals attractive force between the He 4 atoms and the solid. The effects of phonon attenuation (without being specific about a mechanism) are included in the treatment by replacing the phonon wavevector K in the solid, by a complex wavevector containing an attenuation coefficient V;2s,26 K =- k(l + iV)

(6)

where the frequency (to) dependent, dimensionless damping parameter V is given by V(to) = 1/tot(to) < 1 and r(to) is the mean p h o n o n relaxation time for the relevant scattering process. For this particular model, 27 the damping parameter was assumed to be frequency independent. Hence the temperature dependence predicted by the model, in particular the position of the m a x i m u m in H k / T J, is determined by the m a x i m u m density and the structure of the dense layer. The damping parameter simply gives most of the increased magnitude o f H k / T 3 over bare acoustic theory. Reasonable agreement between theory and the experimental results for sample 1 is obtained for a m a x i m u m density of 1.3 times the bulk liquid He 4 density and a damping coefficient, V - - 0.36. The theoretical curve is shown in Fig, 11. A plausible phonon attenuation mechanism, giving a frequency independent damping coefficient, is the scattering of phonons at bulk dislocations. 2s For this mechanism, the dislocation density g, is related to the attenuation factor V by 25 V =

ory2b2

(7)

where y is the Gruneisen constant and b the Burgers vector. Insertion of physically realistic values in (7) gives for sample 1 a dislocation density ~ 1018 m -2. The results for both samples show a peak at about the same temperature, as would be expected from the model. Similarly the increase o f H k / T ~for increasing dislocation density, as shown by these and previous 22.35 experimental results, is also as expected. Therefore it is tempting to conclude that for temperatures around 1 K, agreement between experimental and theoretical results for H k to liquid helium has been found. 36 However there are a number of criticisms which bring this agreement into question. For annealed crystals, the dislocation density is 101° m -2. Conversely, the dislocation density deduced for sample 1 implies a highly work hardened surface? 5 Obviously the discrepancy is even greater for sample 2. Stronger scattering processes z5 are unlikely as they would alter the temperature dependence of H k / T 3 predicted by the model. Some other, as yet unknown mechanism may be responsible for the temperature independent phonon attenuation within the copper. This possibility is discussed below. An effect of introducing phonon attenuation into the acoustic treatment is that phonons in the He 4 incident at the interface at increased angles of incidence can contribute to heat transfer. Hence the

596

modified theory predicts the widening of the critical cone of the bare acoustic mismatch treatment. However as we have already mentioned, measurements of the angular variation of the phonon transmission coefficient between two dielectrics, reveal the existence of a background channel, and confirm the presence of a narrow critical cone. This latter point implies that V is very much smaller than 0.36 at temperatures around 1 K, possibly even zero. 33 Therefore if, as seems likely, the angular distribution of phonons emitted from 'clean' metal surfaces is similar to that from dielectric crystals, phonon attenuation, z5.2~ or models incorporating this treatment of heat transfer, 27 predict results which are inconsistent with experimental observation? 3 Furthermore even for the m a x i m u m possible value of the damping parameter, ie V = 1, the energy transmission coefficient a t is still only 12%. From Fig, 1, we saw that experimental values of t~t can be up to 40%. Hence we have seen that the temperature dependence of H k / T 3for the experimental results presented here is similar to the predictions of the modified acoustic theory. 27 However a number of other requirements of the theory conflict with experimental observation. Therefore we must consider alternative heat transfer mechanisms, acting in parallel with the bare or slightly enhanced acoustic mismatch mechanism. One such mechanism is the single particle excitation model? s in which individual He 4 atoms desorb from the solid surface. However the temperature dependence shown by the results presented here are contrary to the predictions for such a desorption mechanism (illustrated in Fig. 10), or for any other mechanism which has a fairly well defined threshold e n e r g y . 56

As we have already seen, there is strong evidence '2.23.32.34,35 to suggest that impurities play an important role in heat transfer across an interface. Moreover a number of treatments of impurity based heat transfer predict a m a x i m u m in the H k / T 3versus temperature curve, n-3° The dependence of H k / T 3upon dislocation density at the surfacC TM may be explained through this approach if dislocations are considered to be preferred sites for the adsorption of impurity atoms. 22 However there is doubt as to whether these mechanisms are capable of predicting a sufficiently large heat transfer. 37 An alternative model, in which impurities play a dominant role, is the attenuation of loaded Rayleigh waves. 31 In this approach, it is assumed that impurities or dislocations act as generators of interface waves, which subsequently decay by energy loss into the liquid helium. Transmission coefficients a t up to 40% are shown to be possible, comparable to the higher experimental values of ott. The model also gives an explanation for the diffuse signals seen in recent phonon reflection experiments 57 and the disappearance of these signals when He 4 is deposited onto the solid surface. However the one dimensional treatment used so far does not predict a m a x i m u m in H k / T 3 nor can it predict the correct angular dependence for the phonon transmission coefficient. 33 A recent study sa has shown that Rayleigh waves travelling along a disturbed surface produce a wide angular distribution of phonons emitted into the helium. This is distinct from the case for an ideal surface, when phonons are only

CRYOGENICS. NOVEMBER 1983

emitted at the Rayleigh angle. 59 A study o f the two dimensional treatment should be important. Further comparisons between theory and previous experimental work may be found in the literature? We conclude that although acoustic mismatch theory m a y apply to all interfaces, a second m e c h a n i s m acts in parallel and dominates heat transfer even to carefully prepared surfaces. The nature of the second m e c h a n i s m is still controversial. For surfaces not carefully prepared, impurities at the interface are expected to play a d o m i n a n t role.

Conclusions We summarize the conclusions reached in this paper as follows: The experimental results presented here correlate very well with previous, higher temperature data, for interfaces between liquid He 4 and 'clean' copper surfaces, observed by Rawlings et a122 and Snyder. 23 We find that in the transition region between the almost acoustic behaviour at low temperatures, and the a n o m a l o u s behaviour towards 2 K, the temperature dependence o f H k / T 3 for interfaces between liquid He 4 and "clean' copper shows a m a x i m u m at about 1 K. This behaviour is characteristic o f ' c l e a n ' surfaces and has not been observed before. The position o f the m a x i m u m does not seem to depend on the surface fault density. The magnitude of Hk/T~does depend on the dislocation density, in agreement with previous work at higher temperatures) TM In view of the sample preparation, we suggest that the experimental results presented here represent the underlying trend o f i l k , once the impurity based heat transfer has been largely removed. The values of H k / T 3measured at the lowest temperatures investigated here, are in reasonable agreement with previous work for less ideal sample surfaces? 4,2° This confirms that the second m e c h a n i s m becomes inoperative at low temperatures. Although the temperature dependence o f the experimental results is in reasonable agreement with the acoustic mismatch theory, modified to include impedance matching and p h o n o n attenuation, the theory c a n n o t account for the magnitude o f the experimental results. So far, attempts to find a correlation between H k and the bulk properties of the media forming the interface have been unsuccessful? .4 Possibly the position and magnitude of the peak for 'clean' surfaces m a y shed some light on the hitherto unresolved nature of the excess heat transfer m e c h a n i s m ? 1

Authors The authors are from the School of Physical and Molecular Science, University College of North Wales, Bangor, Gwynedd, LL57 2UW, UK. Paper received 6 July 1983. The authors express their gratitude to the Science and Engineering Research Council for support for this work. One of the authors (AWP) thanks the Science and Engineering Research Council for a studentship. The authors would like to thank D r F. Moss of the University o f Missouri, St Louis, Missouri, USA for assitance with obtaining back diodes, and D r A.M. G u e n a u l t o f the University of Lancaster, U K together with m a n y others, for m u c h stimulating discussion. Also this study was only made possible by the support of the technical staff of this and the computing department at Bangor.

CRYOGENICS. NOVEMBER 1983

It is with regret that we must a n n o u n c e the death of Dr J.C.A. van der Sluijs, the co-author o f this paper, w h o died in July 1983 having been taken seriously ill earlier this year. Dr A.W. Patullo expressed the wish that this paper should still be published,

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54 55

CRYOGENICS. NOVEMBER 1983