Thermal boundary resistance at interfaces between two dielectrics

Volume 50A, number 4

PHYSICS LETTERS

16 December 1974

THERMAL BOUNDARY RESISTANCE AT INTERFACES BETWEEN TWO DIELECTRICS G. SCHMIDT Institut f~irExperimentelle Kernphysik, Kernforschungszentrum Karisruhe, 75 Karisruhe, Postfach 3640, W. Germany Received 29 October 1974 The thermal boundary resistance (Kapitza resistance) was measured at boundaries between sapphire and epoxy resin at temperatures from 1 to 3 K. The results are discussed in terms of the acoustic mismatch theory including phonon attenuation near the boundary.

The temperature discontinuity occuring at the boundary between two materials, when a heat flux flow across it, is caused by phonon reflexion at that boundary due to the different acoustic properties of the two materials. This concept of acoustic mismatch, first established for boundaries between solids and liquid helium where the effect (Kapitza resistance) was discovered, was extended to solid-solid boundaries by Little [1] and showed an agreement with most experiments within a factor of two [2—4].Previous experiments were all performed at the boundaries between a metal and a dielectric, where the metal electrons play an additional role [4]. Therefore, it seemed to be useful to investigate an interface between two dielectrics. The system sapphire-epoxy is well adapted to test the acoustic mismatch theory because of the very different acoustic properties of these materials. Because of the low thermal conductivity of epoxy, a sandwich method was used: The polished end faces of two synthetic sapphire rods (3 cm long, 0.6 cm diam) were pressed together with a drop of epoxy (CIBA CY 221 + Hardener HY 979) between them. As a spacer a ring of 20 pm Capton-foil was used with an inner diameter of 0.5 cm (sample 1) or 0.55 cm (sample 2). After curing the two sapphire rods were tightly bond together. The measurements were made using a constant heat flux, the temperatures were measured at two points of each sapphire rod with carbon resistors and extrapolated to the end faces. The boundary resistance Rk was calculated to be 2Rk = ~TA/~ d/i~.i~Tisthe temperature difference between the end faces of the sapphire rods, A is the cross-section, Q is the heat flux, d is the thick—

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T• K Fig. 1. Thermal boundary resistance between sapphire and epoxy versus temperature for two samples, as compared to the acoustic mismatch theory [1] (dashed line). The insert shows a calculation based on the attenuated phonon model [6] (see text).

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ness of the epoxy layer and i its known thermal conductivity [3]. It should be mentioned that mean free path effects [5] play no part in this temperature range, since the phonon mean free path in epoxy is small compared with d. The estimated uncertainty of 20% for d/K leads to a systematic error in Rk of 2% at 1 K and 7% at 2 K. The results for the two samples measured are shown in fig. I. They differ somewhat in absolute magnitude and in temperature dependence, which may partly be due to the different fraction of the boundary area 241

Volume 5(A. number 4

I’IIYSICS l.lTTFRS

covered with capton foil. The data were not corrected to take this effect into account. Since the foil is believed to have a lower thermal conductivity than the epoxy, its influence will be greater at higher temperalures which would explain the different temperature dependence. So the second sample seems to he more representive. The graphs given by Little El cannot be used to calculate the theoretical value of the acoustic mismatch theory because they are not extended to the ratios of sound velocities and the ratio of densities for sapphire and epoxy. Therefore, a computer program written by Peterson [61 was used for the calculation, which yielded Rk T3 = 51 cm2K4/W (dashed line in the figure). The sound velocities used for sapphire were c 5 cm/sec and for epoxy 1 = 11.4. c~ = 6.4X i0 3.2 and 2.3 X l0~cm/see, while the densities were 4 and 1 .1 g/cm3. respectively. The agreement in the absolute value is quite good, but the experimental resuits show appreciable deviation from the expected T3 behavior. An improvement of the acoustic mismatch theory by Peterson and Anderson [2,6] takes into account the phonon attenuation near the boundary (attenuated phonon model), which leads to an additional mnechanisni of energy transport by surface waves generated by phonons undergoing total reflextion. A T dependence of Rk can only be expected if the ratio of phonon wave length ?~to phonon mean free path I in the material with the higher sound velocities, in this case sapphire, is independent of temperature. While the

phonon mean free path in the bulk sapphire is limited only by its dimensions. surface waves may he scattered by surface imperfections so that the actual I needed for calculation could be small. If we suppose that the mean free path due to static imperfections is independent of temperature and since X is inversely propor. tional to T (at I K typically 1400 A in sapphire for longitudinal phonons), we get X/I l/T. The insert in 3 against X/l. the hgure shows a calculation of Rk T With X/I ~ lIT. Rk’ T3 would increase with temperature and the deviation of the results from the T~ law could partly be explained. For example. the arbitrary assumption X/I = 0.1 yields Rk ~ T28. Of course. this argumentation cannot quantitatively explain the experimental results since the real / is unknown Further it must be pointed out that no explanation can he given for an increase in Rk’ T3 above the acoustic mismatch theory without attenuation (dashed line), as is the case above 1 .7 K.

References II] WA. little, Can. .1. Phys. 37 (1959)334. 121 RE. Peterson and AC. Anderson. .1. l.ow Temp. Phvs. II (1973) 639.

13] C. Schmidt. Disseratation (KEK-report 2030. Kartsruhe 1974)

[4] B.S Park andY Narahara, J. Phvs Soc Japan 30 (1971 760

[5] 1,.J. Challis and R.A. Sherlock. J. Phys.. (‘3 (1971)) 1193. 61 RE. Peterson, Ph. D. thesis (University of Illinois. Urbana 1973).

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