Journal of Low Temperature Physics, VoL 109, Nos. 5/6, 1997

Thermal Response of Superfluid 4He near to an AC Heat Flux

Daniel Murphy and Horst Meyer

Department of Physics, Duke University, Durham, North Carolina 27708-0305, USA

(Received June 12, 1997 by F. Pobell; revised August 13, 1997)

We have measured the thermal response of a superfluid 4He layer near Tz to a time-varying heat flux. In the low-frequency range studied, 1 x 10-3< f< 0.1 Hz, we find the response to be independent of the applied frequency f, a result consistent with the expectation that the measured response is due to the boundary resistance R b alone. These findings differ from those by Olafsen and Behringer. The thermal conductivity cell used by these authors had extraneous surfaces in contact with the superfluid which were eliminated in the cell used for our experiments; we believe these surfaces were responsible for the frequency dependence observed in previous work. Furthermore, we show that AC and DC measurements of R b differ by no more than 1%, and that both exhibit a weak singularity near Tz.

I. INTRODUCTION

Measurements of the transport properties of fluids with large thermal conductivities present unique challenges to the experimentalist. When the thermal resistance of the fluid being studied becomes comparable to ther- mal resistances of parts of the experimental apparatus, it becomes difficult to extract the quantity of interest. For instance, the effective thermal con- ductivity, Ke~, of dilute mixtures of 3He in superfluid 4He diverges as the molar concentration of 3He, X, goes to zero in the regime where counter- flow dominates (i.e. T~> 1.2 K). When a thermal conductivity cell with ther- mometers attached to the endplates is used to measure Keff, the thermal resistance of the fluid layer, Rn, can be smaller than that of the boundary, Rb, between the metal comprisingthe cell and the fluid. The resistances Rn and Rb are only measured in combination, and as we have recently shown, the large correction due to R b led to systematic errors in previous measure- ments of xoff.1

A second example where measurements in high-conductivity fluids are difficult to analyze is the boundary (or Kapitza) resistance of superfluid

801

0022-2291/97/1200-0801512.50/0 �9 1997 Plenum Publishing Corporation

802 D. Murphy and H. Meyer

4He near Tz. The boundary resistance, Rb, is predicted to exhibit a weak divergence as Tx is approachedfl a divergence which was first observed by Duncan, Ahlers, and Steinberg.3 However, in several cells an analysis of the Rb divergence was complicated by a non-linear, or heat-flux dependent, effect which was much larger than the divergent behavior predicted for Rb .n-6 The non-linear effect occurred in these experiments because of the nearly infinite thermal conductivity of superfluid 4He, which led to large heat fluxes in narrow fluid-filled spaces within typical thermal conductivity cells, as is discussed in one of our recent papers. 7

In order to separate the contribution of the bulk fluid from that of the boundary resistance, Behringer proposed measuring the response of a fluid layer to an AC heat flux. 8 In addition, Olafsen and Behringer 9 derived a general expression for the thermal response of a dilute mixture of 3He in superfluid 4He to a heat flux of the form Q(t) = Qo exp(i2nft), and showed that in the limit of zero concentration the low-frequency response of a layer of pure superfluid 4He should be given by

AT(t) = 2QoRb exp( i2zcft), (1)

where AT(t) is the temperature difference across the fluid layer as function of time and f is the frequency of the applied heat flux; AT(t) includes the temperature difference due to the two boundary resistances coupled by the bulk fluid. In deriving this result, it was assumed that the temperature difference due to the metal-fluid boundary resistance is established instan- taneously. Equation 1 implies that 2QoR b, the amplitude of AT(t), should be independent of the applied frequency.

Olafsen and Behringer 1~ measured the response of a nearly pure 4He superfluid layer (with a 3He impurity concentration X ~ 10 -7) to such an AC heat flux. Close to Tz, they found that, contrary to their predictions, the amplitude of the temperature response depended strongly on the applied frequencies for values o f f as low as 10 -4 Hz when [el < 5 x 10 -3, where e ~ ( T - T~)/Tz. In particular, a peak in the amplitude was observed at 0.02Hz. For le{=10 -3 and Q=20#W/cm 2, the amplitude of the response increased from 1.0 cm z K/W to 1.8 cm 2 K/W as the frequency increased from 2 x 10-4 Hz to 0.02 Hz, and decreased to 1.5 cm 2 K/W as f was increased further to 0.3 Hz. Further away from Ta, the measured response was independent of the applied frequency in agreement with their predictions, implying that the anomalous AC response was associated with the superfluid transition. These results were taken to mean that the bound- ary resistance R b becomes dependent on the frequency of the applied heat as the temperature of the fluid layer approaches T~, and it was suggested that the non-linear DC effect mentioned above and the anomalous AC

Thermal Response of Superfluid 4He near T~ to an AC Heat Flux 803

response are related because they appear at approximately the same distance from the transition temperature. 1~

Recently, we have shown ~'7 that the non-linear DC boundary resistance measurements near T~ in pure 4He and the size-dependent xefr results in dilute superfluid mixtures were due to imperfections in previous cell designs. It is possible that Olafsen's and Behringer's AC data were affected by similar experimental problems. Using the same cell as in our recent experiments--a cell in which the amount of surface area in contact with the helium is carefully controlled--we have measured the thermal response of a pure superfluid layer to an AC heat flux; we present our results below.

II. E X P E R I M E N T

The conductivity cell we used in our experiment, which has been described in detail elsewhere, 7 is composed of two cylindrical, high-conduc- tivity copper endplates, separated by a cylindrical, low-conductivity stain- less steel sidewall. The sidewall is joined to the endplates in such a way that no copper surfaces aside from the flat circular ones are in contact with the fluid. The temperature of the top plate of the cell is kept fixed by a Propor- tional-Integral-Differential (PID) controller, while a non-inductively wound wire heater attached to the bottom plate of the cell is used to create a heat flux. The cell was filled with 4He with a 3He molar concentration X[ 3He ] = 2 x 10 -9.

In previous experiments using this cell, DC power was applied via the heater; however, in the present case AC heat fluxes were used, and the manner in which the heat was applied and the boundary resistance extracted requires further description. We have also conducted DC measure- ments for comparison with the AC results. Our AC technique is based on that used by Olafsen and Behringer.l~

To apply an AC power to the bottom of the cell, we supplied an AC voltage of the form V(t)= Vo sin(2n(f/2)t) to the heater using an Hewlett Packard 3325A oscillator. All of the measurements described in this paper used Vo = 5 V, which was applied across a series combination of a 20,000 f2 standard resistor and a 20,000 f2 cell heater, leading to an rms heat flux Qrms = 14.7 pW/cm 2. The applied voltage should then result in a heat flux of the form V2(1 -cos(2nf t ) ) /2R; however, a small DC bias Vbias ~ 0.05 V in our applied AC voltage gave an additional contribution to the applied heat, ~Q = [ V~ias + 2V0 Vbias sin(2n(f/2)t)]/R. Since Vbias is small, the DC

2 contribution Vbias can be ignored, but the AC one is linear in Vbias, and makes a noticeable contribution to the measured temperature response.

804 D. Murphy and H. Meyer

The applied heat flux resulted in a time-dependent temperature response of the cell. The temperature difference across the cell, AT(t), was measured using a differential bridge with germanium thermometers attached to the top and bottom plates; the thermometers have a resolution of 0.3/aK. The excitation voltage for the bridge operation was set at a frequency of 90 Hz, and the output voltage of the bridge was measured with the use of an Ithaco 391A lock-in. The output filter of the lock-in used on the differential bridge was set to 12 dB/octave and a time constant of 1.25 s (i.e. a double-pole low-pass filter with a calibrated time constant for each pole of 1.25 s.) The output filter of the lock-in therefore limited our measurements to heating power frequencies less than 1 Hz. As we will see below, significant corrections to the data are required for lower frequencies as well.

The output of the differential bridge was digitized, along with the voltage and current applied to the heater. The differential bridge trace was converted to AT(t), and the voltage and current traces were multiplied to give the total power P(t) applied to the bottom of the cell. Both AT(t) and P(t) were Fourier transformed, and the ratio of the amplitudes AT/P and the difference in phase between the two signals recorded. If AT/P is due to the boundary resistance alone, then Rb = �89 ATA/P, where A is the cross- sectional area of the cell; the factor of �89 arises from the fact that we measured the resistance of both surfaces in the cell. The resistance measured in this way is R b = R K + R c u , where R/c is the Kapitza resistance and Rcu that of the copper plate between the metal-fluid boundary and the location of the thermometer.

By examining the power spectra of AT and P we found that the fre- quency f of the response AT(t) was the same as that of the applied power. Furthermore, higher-order harmonics o f f were not present in the Fourier transform of the measured response, suggesting that non-linear processes were not observed. (The sub-harmonic component present in the applied power, due to the small DC bias voltage, led to a sub-harmonic in the measured response AT(t) at a frequency f/2, but did not affect our results.)

The temperature control of the top plate posed a technical problem for f > 0.1 Hz. Above this frequency, the temperature of the top plate began to oscillate at the frequency of the applied heat because the heating power applied to the bottom of the cell was an appreciable fraction of the power applied to control the temperature of the top plate. Also, the large thermal diffusivity of the superfluid helium layer and the copper endplates meant that the heat applied to the bottom of the cell was transmitted very quickly to the control thermometer. The thermal relaxation time of the cell, including the 1 mm fluid layer, is determined by the fluid heat capacity and the boundary resistance and is approximately 1 s. Our use of differential thermometry

Thermal Response of Superfluid 4He near T~ to an AC Heat Flux 805

eliminated most of the above mentioned effect in our data, but the remain- ing temperature oscillations of the top plate restricted our measurements to frequencies less than 0.1 Hz.

III. RESULTS

The observed amplitude of the boundary resistance R b and its phase angle | are plotted in Fig. l(a) and Fig. 2(a) versus the f requencyf of the applied power for several reduced temperatures of the top plate, gtop- The solid lines on the figures indicate the expected response, calculated from the value of R b measured using DC heat fluxes at le] = 1.5 • 10-3 and the nominal characteristics of the output filter of the lock-in. The equations describing these curves are given by

Rb(f = O) Rb(f) ---- 1 + (2gfz) 2 (2)

and

tan-1 \(1 7 4gf _'~ | - (2gfv)2J '

(3)

where Rb( f = 0) is the DC limit of the boundary resistance and z is the time constant of the double-pole output filter of the lock-in. The calculated curves reflect the effect of a lock-in on the frequency response of an ideal system, and do not take into account the response of the cell or of the tem- perature controller. As a result, the curves are only valid when f < 0.1 Hz.

For a given etop, the measured R b is found to be independent o f f to within the experimental scatter when f < 0.02 Hz. Above this frequency, the measured response drops with increasing frequency due to the output filter on the differential bridge's lock-in. It can be seen that there is good agree- ment between the measured and the predicted response, Eq. 2. Hence the amplitude of Rb, divided by Eq. 2, represents the corrected response and these data, taken at several values of letop[ are plotted in Fig. l(b). (Note the expanded vertical scale in Fig. l(b)).

In Fig. 2(b) it can be seen that the phase angle | of the thermal reponse, after correcting for that from the lock-in, is nearly zero at all frequencies measured in our experiment. (Note the expanded vertical scale in Fig. 2(b).) This zero-valued phase shift indicates that the time taken to establish the temperature difference across the fluid layer as well as the boundaries is less than 10 s. Therefore, our results suggest that any dynamics in the boundary layer itself occur on timescales shorter than 10 s.

806 D. Murphy and H. Meyer

a) 2.4

2.2

2.0

1.8

--" 1.6

1.4

1.2

1.0

. , . . . . . . . . i . . . . i i 1 i i | i | . . . . . . . . . i . . . . i i i i i i i ~

: Original Data

J t

" IEtop I ~ ~ "J

�9 4.5x 10 -s " ~ - ~ 7 5 x 1 0 -5 ~ -~

o 1 5 x 1 0 ' X 3 - ~ 7 5 x 1 0 ' \-]

o 1.5 x 10 3 " x ~

b) . . . . . . . . i . . . . t i i i i i i ] . . . . . . . . . i . . . . | 1 ! i | i |

2.2 Corrected for Filter ,

2.0 ~ o i M

"% 1.81

1.6

. . . . . . . . I . . . . i

2 3 0.001

0

[3

o

o +

I~top I

4.5 x

7.5 x

1.5 x

7.5 x

10 .5

l f f 5

l f f 4

l f f 4

o 1.5 x 10 -3

+ 7.5 x 10 .3 i I i i i i | . . . . . . . . . i . . . . i i ! t r i

4 5 6 7 8 2 3 4 5 6 7 8 0.01 0.1

f (Hz)

Fig. 1. (a) The amplitude ]RbJ plotted ve r su s f for several reduced temperatures of the top plate, letop [. The solid line is the normalized predicted frequency response of the lock-in's output filter (See text). (b) The amplitude IRbl, corrected for the lock-in frequency response, plotted versus f for several values of I~topl.

Thermal Response of Superfluid 4He near Tz to an AC Heat Flux 807

a) 0.0

-0 .2

-0 .4

r a ~

= -0.6 . , m , ~

-0.8

| _1.o I

-1 .2

-1 .4

! . . . . . . . . . . i i i ~ + i J [ . . . . . . . . . i . . . . i | i i | i ~ i na l Data

letop [ N �9 4.5 x l 0 -5 N r 7.5 x 10 -5 N k

D 1.5 x l 0 4 \

tX 7.5 x 10 -4 O k

o 1.5 x 10 3 k + 7.5 x 10 -3 1

. . . . . . . . l . . . . I I m J ~ II I J . . . . . . . . . I . . . . i i i I i I i "

~.Xv) 0 .00 . i . . . . . . n . . . . . . . . . . . J . . . . . . . . . . . . . . . . . . . . . i~ C o r r e c t e d f o r F i l t e r

-0.05

,----, -0.10 L let~ �9 4.5 x 104

r ,~ 7.5 x 10 .5

"~ - 0 . 1 5 /:1 1.5 x 10 -4 + ~ 7.5 x 10 .4 o

- 0 . 2 0 o 1.5 x 10 -3 + 7.5 x 10 -3 �9

-0 .25

-o.3o . . . . . . . . ~ . . . . ~ :~ ; ; ~ " . . . . . . . . . . . . . . 2 ~ ~ ; ; ~ ' 0.001 0.01 0.1

f (Hz)

Fig. 2. (a) The phase angle O plotted versus f for several reduced temperatures of the top plate, ]etopl. The solid line is the predicted frequency response of the lock-in's output filter (See text). (b) The phase angle O corrected for the lock-in frequency response, plotted versus f for several values of [etop ].

808 D. Murphy and H. Meyer

Higher frequency measurements of the boundary resistance proved dif- ficult in our experiment because the large heat fluxes applied to the bottom plate propagate to the temperature-regulated top plate in a time shorter than 1 s. The temperature controller must therefore respond at a faster rate than that of the large AC heat flux, and fails to do so for frequencies above 0.2 Hz. The effects of oscillations of the top-plate temperature are observed at lower frequencies as well, leading to the slight increase in Rb with f seen in Fig. l(b) after correction for the lock-in filter.

As letopl is reduced from 10-3, the amplitude of the response increases slightly at all frequencies. This increase is consistent with the divergent R b

measured in our recent DC experiments. 7 The quantitative comparison of the AC measurements to the DC ones becomes uncertain at reduced tem- peratures comparable with the rms amplitude of fluid temperature oscilla- tion, calculated to be fie= +2 x 10 -5 for Q~s= 15/~W/cm z. Therefore, a direct comparison of the AC data with the DC data must take into account the oscillation of the temperature of the fluid. This is because Je[ was well- defined in the DC experiments, while in the AC measurements it varies periodically with time; one therefore can assign only a time-averaged Ig[ to the AC results, defined by

e= etop+ (Tbo,,o.,- T~op)/2T~, (4)

where Tbottom is the time-averaged temperature of the bottom plate sub- jected to the periodic heat flux of frequency f Also, when fie ,,~ e the dependence of R b o n [gl means that non-linear effects in the response may be evident because R b will vary during each cycle of the applied heat flux. A non-linear effect would presumably result in higher order harmonics of f i n the Fourier transform of AT(t); the absence of these in the measured response is taken as evidence that non-linear effects were not significant in our reduced temperature range.

Figure 3 shows Rb plotted versus [g[ for frequencies f < 2 x 10-2 Hz, where corrections due to the instrumentation are negligible. These results are compared with those obtained by the standard technique of a DC heat flux, carried out during the same cooldown, and also with previous measurements. 7 (The DC measurements in both instances are for Q= 28 pW/cm2.). Although the AC data are systematically 1% larger than the DC results, the agreement is very satisfactory and all sets of data show the same weak increase of R b as [e[ decreases. We also mention that no dependence of R b o n Q was found in the present DC measurements, which confirms the results in Ref. 7. Since the emphasis of this work was a comparison of the AC and DC methods for measuring R b and an exami- nation of the behavior reported by Olafsen and Behringer, we will not

T h e r m a l R e s p o n s e o f S u p e r f l u i d 4 H e n e a r T x t o a n A C H e a t F l u x 8 0 9

t'q

2.05 . . . . .

0 2.00

1.95

1"90 I

1.85 t . . . . 2

10 .5

s , , , i l l | . . . .

0 I

0 O

0 0

I 1 % o

| | i i i f | I . . . . , , , , l , , w

�9 AC data o Current DC data o Previous DC data

8

8 o

i i i i i s s l . . . , i i 1 1 J i I I [ . . . . I

4 6 8 2 4 6 8 2

10 .4 10 -3

O

t ! i , i l l 4 6

10 .2

Fig. 3. The amplitude R b plotted versus [g[, comparing AC and DC measurements. All DC data are for Q=28pW/cm 2, while for AC measurements, Qrm~= 14.7 ,ttW/cm 2. The "present" DC data were taken during the same cooldown, while the "previous" ones are those of Ref. [7].

compare our new data with predictions and previous experiments, but refer to Ref. [7] .

The data in Figs. l(b) and 2(b) can be compared with those of Figs. 2(a) and 2(b) in the paper by Olafsen and Behringer, ~~ which also have been corrected for the frequency-dependent response of the instrumenta- tion. While our results are independent o f f , those of Ref. [10] show a variation with frequency on the order of 50 %. Although frequencies as low as I0 -a Hz were used in Ref. [10] while our lowest frequency was 1.2 • t0 -3 Hz, most of the amplitude variation in their response occurred over the frequency range covered in our experiment. The absence of a similar structure in our results suggests that the results of Ref. [10] were affected by factors absent in the present work. We have shown in an earlier paper that the anomalous DC results near T~ were due to gaps between the endplates and the sidewall in the thermal conductivity ceils used; 7 such gaps were present in the cell used in Ref. [ 10]i and are probably respon- sible for the results they observed. There is also the possibility that the non- zero, albeit small, 3He impurity present in their helium sample affected their data.

810 D. Murphy and H. Meyer

IV. C O N C L U S I O N

We have measured the frequency response of the boundary resistance R b in superfluid 4He near Ta over the frequency range 1.2• 1 0 - 3 < f < 0.1 Hz, and we find that both the amplitude and phase are independent of f below 2 x 10 -2 Hz where instrumental effects are negligible. The values for Rb derived from our AC measurements agree well with those measured using DC heats. The results of Olafsen and Behringer, where the amplitude and phase were found to depend upon f, were presumably affected by the presence of gaps between the endplates and sidewalls in their experimental cell; similar gaps in other experimental cells led to anomalous results in DC boundary resistance measurements near T~. 7 These results in the superfluid phase should not be construed to imply that those in the normal phase H are invalid, because the situation is quite different there.

Our experimental procedure can be extended to cover higher frequen- cies by increasing the thermal mass of the upper plate through strong ther- mal coupling with a large thermal mass, such as a liquid helium bath, which will dampen the top plate thermal oscillations. Decreasing the time constant of the temperature bridge and most of all decreasing the thickness of the fluid layer will permit measurements at frequencies greater than 0.I Hz.

A C K N O W L E D G M E N T S

We have greatly benefited from help and discussions with F. Zhong. We are also indebted to J.S. Olafsen and R.P. Behringer for discussions and helpful comments on a draft of this paper. The work has been supported by grant NAG3-1838 of the NASA.

R E F E R E N C E S

1. D. Murphy and H. Meyer, J. Low Temp. Phys. 107, 177 (1997). 2. D. Frank and V. Dohm, Phys. Rev. Lett. 62, 1864 (1989); Z. Phys. B 84, 443 (1991) and

references therein. 3. R. V. Duncan, G. Ahlers, and V. Steinberg, Phys. Rev. Lett. 58, 377 (1987). 4. F. Zhong, J. Tuttle, and H. Meyer, J. Low Temp. Phys. 79, 9 (1990). 5. R. V. Duncan and G. Ahlers, Phys. Rev. B 43, 7707 (1991). 6. D. Murphy and H. Meyer, J. Low Temp. Phys. 97, 489 (1994). 7. D. Murphy and H. Meyer, J. Low Temp. Phys. 105, 185 (1996). 8. R. P. Behringer, J. Low Temp. Phys. 81, 1 (1990). 9. J. S. Olafsen and R. P. Behringer, J. Low Temp. Phys. (to appear).

10. J. S. Olafsen and R. P. Behringer, Phys. Rev. B 52, 61 (1995). 11. J. S. Olafsen and R. P. Behringer, J. Low Temp. Phys. 106, 673 (1997).