HEAT TRANSFER OF SEVERAL MATERIALS AT CRYOGENIC TEMPERATURE

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1 WeID HEAT TRANSFER OF SEVERAL MATERIALS AT CRYOGENIC TEMPERATURE Takayuki TOMARU 1, Kunihiko KASAHARA 2, Takakazu SHINTOMI 1, Toshikazu SUZUKI 1, Nobuaki SATO 1, Tomiyoshi HARUYAMA 1 and Akira YAMAMOTO 1 1 High Energy Accelerator Research Organization (KEK), 1-1 Oho, Tsukuba, , Japan, Phone: , Fax: , tomaru@post.kek.jp 2 Institute for Cosmic Ray Research (ICRR), University of Tokyo, 5-1-5, Kashiwanoha, Kashiwa, , Japan ABSTRACT We measured the thermal conductivities of sapphire fibers, pure aluminum and copper wires, and a pure graphite sheet at cryogenic temperature. They were investigated as materials of heat conductors used at different temperature ranges for a cryogenic interferometric gravitational wave detector. The thermal conductivities of sapphire fibers (sub millimeter in diameter) were over 10 3 W/m/K at around 40K. Below this temperature, we confirmed that they were limited by the sample diameter. The thermal conductivities of pure aluminum and copper wires were over 10 4 W/m/K at around 10K, and were limited by internal defects in the samples. For the graphite sheet, the thermal conductivity was over 600W/m/K between 200K and 300K. It is effective as a heat conductor to reduce cooling-time of the system. INTRODUCTION Sapphire fibers, pure metals and pure graphite are used as heat conductors in the Large-scale Cryogenic Gravitational wave Telescope (LCGT), which is an optical interferometer in the planning stage aimed at the first detection of gravitational waves (Kuroda et al., 1999). A technical issue is to cool the mirrors to below 20K against heating (about a few watts) while being irradiated by a high-power laser and maintaining vibration-isolation of the mirrors. Figure 1 shows a schematic diagram of the cooling system of the LCGT. Figure 1: Schematic diagram of the cooling system for cryogenic interferometric GW detectors. 1

2 The mirror must be cooled by using only heat conduction of suspension-fiber. Sapphire is a promising material as the mirror-suspension-fiber, since it is effective to reduce the thermal noise, one type of detector noise (Uchiyama et al., 1999; Saulson, 1994). We require a large thermal conductivity of the sapphire-fiber in the temperature range over 10K (Touloukian and Ho, 1970), since the fiber diameter is limited below 1mm to reduce the thermal noise. However, the thermal conductivity can be limited by the sample diameter (size effect) in this temperature range, since the mean free paths of phonons (main heat carrier in crystals) can be over one millimeter at sufficiently low temperature (Rosenberg, 1963). For a heat link between the mirror and the cryostat, and for a heat conductor between the cryocooler and the cryostat, we require a material to be able to conduct a large amount of heat without conduction of vibrations. Pure aluminum and copper are expected to be a good heat link materials, since they have extremely large thermal conductivities below 10K and softness (Touloukian and Ho, 1970). In order to make a soft heat link, it is important whether these metals also have a size effect or not. For cryogenic experiments using conduction-cooling, a lot of initial cooling-time is needed in the temperature range over 100K, since the thermal conductivities of metals are small and the specific heats of materials are large in this temperature range. We can expect to reduce the initial cooling-time of the LCGT by using pure graphite as a part of the heat conductors and radiation shields, which are about a few kilograms and about 100kg, respectively, since it can have large thermal conductivity in the room-temperature range. To investigate the properties of heat conduction, especially the size effect, for the sapphire fiber and pure aluminum and copper wires, we measured their thermal conductivities. We also measured the thermal conductivity of a pure graphite sheet to develop a heat conductor with a short cooling-time. In this paper, we report on the measured thermal conductivities and discussion of their limitation mechanisms. 1 MEASUREMENT METHOD OF THERMAL CONDUCTIVITY Figure 2: Experimental setup for the thermal conductivity measurements. (Longitudinal heat flow method) Figure 2 shows the experimental setup for the thermal conductivity measurements. The base temperature range of the measurement was made by heater-2, since it generated no temperature gradient in the sample at the thermally steady state. Heater-1 generated a flow of heat along the sample. In the thermally steady state, the temperature difference ( DT ) between the two points on the sample was measured by Carbon-Glass-Resistance thermometers or silicon-diode thermometers. The thermal conductivity was calculated by k ª P L DT S, (1) 2

3 where P is the applied power of the heater-1, L is the length between the thermometers, and S is the cross section of the sample. This measurement method is called the longitudinal heat flow method. We measured the thermal conductivities of sapphire fibers, pure aluminum and copper wires and a graphite sheet by using this measurement method. 2 THERMAL CONDUCTIVITY OF SAPPHIRE FIBERS 2.1 Diameter-Dependence of Thermal Conductivity of Sapphire Fiber To investigate the size effect of sapphire-fiber and its temperature range, we measured the thermal conductivities of sapphire fibers with different diameters of 390mm, 250mm and 160mm. The samples were produced by Photran LLC using the Edge defined Film-fed Growth (EFG) method. The c-axis was along the fiber axis. The thermal conductivity was measured between 4K and 100K. Figure 3 (a) shows the measured thermal conductivities of sapphire fibers. We found peaks of the thermal conductivities at around 40K. Below this temperature, we observed differences of the thermal conductivities. The thermal conductivity of sapphire was almost proportional to the sample diameter in this temperature range. Therefore, we concluded that the thermal conductivity of sapphire fiber with sub millimeter in diameter was limited by the size effect below 40K (Tomaru et al. 2002). f390mm f250mm f160mm Figure 3: (a): Measured thermal conductivities of sapphire fibers with different diameters, (b): thermal conductivities of f250mm samples with and without scratched surface. When we applied the measured thermal conductivity of sapphire fiber to the suspension-fiber of the LCGT, the maximum heat transfer from the 20K mirror to 10K intermediate-mass of the pendulum was 340mW. This value is not sufficient for the present estimated heat generation in the mirrors. Therefore, we concluded that we must reduce the heat generation in the mirror and devise some enhancement of the mean free path of the phonons. 2.2 Phonon Reflection Effect The mean free paths of the phonons can be possibly enhanced by using phonon reflection on the side surface of the sample (Rosenberg, 1963). To investigate the existence of such a phonon reflection effect, we measured the thermal conductivity of sapphire fiber with f250mm after scratching the sample surface by #8000 diamond paste. Figure 3 (b) shows a comparison of the measured results before and after scratching the surface. Since this data show that thermal conductivity of the scratched sample was reduced to half from that of a non-scratched sample at 5K, we concluded that phonons were reflected at least twice on the original surface of the sample. Therefore, we can possibly enhance the thermal conductivity of the sapphire fiber by polishing the surface with higher quality. A group in USA is studying how to enhance the phonon reflectivity in sapphire fibers (Hall, 2003). 3

4 3 THERMAL CONDUCTIVITIES OF PURE ALUMINUM AND COPPER WIRES Since it is expected that the thermal conductivities of pure aluminum and copper are very large around 10K, we measured the thermal conductivities of these metals and investigated their limitation mechanism. The samples used in the measurement were as follows: Aluminum wires with % purity, and with f1.99mm, f1.00mm, f0.50mm and f0.20mm in diameter, Copper wires with % purity, and with f1.00mm and f0.20mm in diameter. These samples were annealed at 500 during one hour in a vacuum below 10-3 Pa. Figure 4 shows the measured thermal conductivities for these wires (Kasahara et al., 2003). For all samples, the thermal conductivities were larger than 10 4 W/m/K below 10K. Although the thermal conductivities scattered below 10K, we did not find any correlation between the thermal conductivity and the sample diameter. Aluminum Copper Figure 4: Measured thermal conductivities for pure aluminum and copper wires. Table 1: Measured RRRs and thermal conductivities at 5K for aluminum and the copper wires. Sample Aluminum Copper f0.20mm f0.50mm f1.00mm f1.99mm f0.20mm f1.00mm RRR Another cause of their thermal conductivity limitation is internal defects in the samples. Since electrons mainly contribute to the thermal conductivity of metals, one way to evaluate the effect of internal defects is to measure the residual resistance ratio ( RRR) of electricity, which is defined as RRR = R 300K R 4.2K, (2) where, R 300K and R 4.2K are the electrical resistances at 300K and at 4.2K, respectively. The electrical resistances were measured by the four-wire method. Table 1 gives the measured RRRs. The measured thermal conductivities were fitted to the following equations: 1 k (Al ) = T RRR T, (3) 1 k (Cu) = T RRR T, (4) by using measured RRRs. The temperature dependence in these equations was derived from the theory of thermal conductivity (Rosenberg, 1963). Therefore, we concluded that the thermal conductivities of pure aluminum and 4

5 copper wires were limited by the internal defects. 1 By using these equations, we could calculate the thermal conductivities of pure aluminum and copper wires by only measuring RRR. Since the thermal conductivities of pure aluminum and copper wires did not have a size effect, the spring constant of the heat link can be reduced by using many thin wires while keeping the cross section of the heat link. This issue is discussed in Appendix A. 4 THERMAL CONDUCTIVITY OF PURE GRAPHITE Figure 5: Measured thermal conductivity of a PGS graphite sheet We measured the thermal conductivity of a PGS graphite sheet, which had 99.9% purity and 0.1mm in thickness, manufactured by Matsushita Electric Components Co. Ltd. with industrial quantities (PGS graphite sheet online catalog, 2003). The large thermal conductivity of this graphite at room temperature has been reported. Figure 5 shows the measured thermal conductivity of the PGS graphite sheet above 77K. It had a thermal conductivity of 670W/m/K at 200K, which is about three-times larger than that of aluminum and about twice larger than that of copper. Therefore, graphite is effective as a heat conductor at over the 100K-temperature range, and the cooling-time can be reduced a few times by using graphite. In several reports (Touloukian and Ho, 1970), the thermal conductivities of some pyrolytic graphite, which is highly crystallized graphite, are reported as being one order of magnitude larger than our data. Therefore, we can enhance the thermal conductivity of the PGS graphite sheet by increasing the crystallization by annealing. We are preparing an experiment using an the annealed PGS graphite sheet. CONCLUSION We measured the thermal conductivities of mono-crystalline sapphire fibers, pure aluminum and copper wires, and a pure graphite sheet to develop a cooling technique of the LCGT. The thermal conductivity of sapphire fibers with sub millimeter in diameter was over 10 3 W/m/K around 40K, and proportional to the fiber diameter below this temperature. We also pointed out that we can possibly enhance the thermal conductivity by increasing the phonon reflectivity by polishing the surface to high quality. In thermal conductivity measurements of the pure aluminum and copper wires, we did not observe any size effect, but did observe the effect of internal defects, evaluated by RRR measurements. We confirmed that the PGS graphite sheet had large thermal conductivity over the 100K-temperature range, and it is effective as a part of the heat conductor to reduce the initial cooling-time. The thermal conductivity can be increased more by annealing. We are now studying this issue. 1 The reason why thin wires have small RRR is because impurities could be introduced in the process of the wire drawings. Although the RRR of the copper seems to depend on the sample size, we concluded that it is not size effect, since the mean free path of the electrons estimated from the Wiedemann-Franz law is much smaller than the sample size. 5

6 APPENDIX A. Development of a Soft Heat Link Since the thermal conductivities of pure metals had no size effect, we can reduce the spring constant of the heat link by using many thin wires while keeping its heat transfer, because the spring constant ( k ) of the heat link with the number of n and diameter of each wire of d is described as k = n K E d 4 64 l µ P 2 3 n, (6) where, l is the length of the heat link, E is Young s modulus of the wire and K is a factor related to the spring shape. P is the heat power able to be transferred along the heat link, which is proportional to n d 2. To investigate the effectiveness using many thin wires, we made a stranded cable and compared its spring constant with that of a single wire while keeping their cross sections by measuring their resonant frequencies. Since aluminum has a smaller Young s modulus than copper and they have almost the same thermal conductivities, we used aluminum as the material of the cable. The purities of the samples were about 99.99%. Table 2: Measured spring constants of a single wire and a stranded cable of aluminum, normalized by the value of the f0.5mm single wire. f0.5mm Single Wire f0.1mm25 Stranded Cable Relative Spring Constant Table 2 gives the measured results. The spring constant of the stranded cable was about one order of magnitude smaller than that of the single wire. Therefore, we confirmed that the stranded pure aluminum cable was effective as a soft heat link. Kuroda K. et al., 1999, Int. J. Mod. Phys. D 8, p.557. Uchiyama T. et al., 1999, Phys. Lett. A 261, p.5. REFERENCES Saulson P. R., 1994, Fundamentals of Interferometric Gravitational Wave Detectors, World Scientific, Singapore Touloukian Y. S. and Ho C. Y., 1970, Thermophysical Properties of Matter, Vol. 2, Thermal Conductivity - Nonmetallic Solids, The TPRC Data Series, IFI/PLENUM New York-Washington. Rosenberg H. M., 1963, Low temperature solid state physics: some selected topics, Oxford : Clarendon Press Touloukian Y. S. and Ho C. Y., 1970, Thermophysical Properties of Matter, Vol. 1, Thermal Conductivity -Metallic Solids, The TPRC Data Series, IFI/PLENUM New York-Washington. Tomaru T., 2001, Phys. Lett. A 283, p.80. Hall M., 2003, Private Communication Kahahara K., 2003, Under preparing manuscript to submit to Cryogenics. PGS graphite sheet online catalog, 2003, 6