Measurement of small sample thermal conductivity by parallel thermal conductance technique

Transcription

1 Measurement of small sample thermal conductivity by parallel thermal conductance technique Bartosz M. Zawilski 1, Roy T. Littleton IV 2, and Terry M. Tritt 1, 2 1 Department of Physics and Astronomy 2 Materials Science and Engineering Department Clemson University, Clemson, SC USA In order to measure thermal conductivity of small samples (3.0 x 0.1 x 0.2 mm 3 ) such as pentatellurides and single carbon fibers, we have developed a new technique called parallel thermal conductance (PTC). In this paper we describe the PTC method providing measurements on standards as well as single carbon fibers and preliminary pentatelluride measurements.

2 I. INTRODUCTION Thermal conductivity measurements on relatively small samples are important challenges for the scientific community. For several reasons, including size dependence or perhaps due to natural size limitations, it is important to be able to measure these smaller samples. For relatively big samples we have at our disposal several well established steady-state and nonequilibrium techniques, however, small samples give us more difficulties because of the heat loss importance. Several techniques have been developed for different types of small samples. Indeed, different kind of small samples impose different constraints and experimental conditions: thin wires are geometrically reproducible [ 1 ], thin films require a substrate and a non-equilibrium technique such as pulse [ 2 ] or 3ω method [ 3 ] necessitating a large surface for deposition of a heater and thermometer. Ribbon like samples and fibers are probably the most difficult to measure due to geometrical factors that may vary from sample to sample while dimensions may be limited by the nature of the material. Motivated by the need to measure thermal conductivity of pentatelluride materials (typical dimensions are: 3.0 x 0.1 x 0.2 mm 3 ), which show a promising power factor for thermoelectric use [ 4 ], we have developed a new method or more exactly a new approach to do so. This technique is also well adapted for other ribbon like samples such as carbon fibers (diameter 100 µm). Indeed, the thermal conductivity of these types of samples may be measured by the classic two-wire method. Due to the inability of the sample to support a heater and thermocouple, a sample holder has been developed. First, the characteristics of the sample holder itself must be measured, which determines the base line or background thermal conduction and losses associated with the sample stage. The second step consists of attaching the sample and measuring the new characteristics of the system. By subtraction, the parallel thermal conductance (PTC) is calculated. This conductance is due to the sample, thermal contacts and black body radiation from the sample. The interest of this technique consists of the easiness and rapidity of the measurement. The entire measurement procedure, including the mounting process, cooling time, and 35 power-sweep points between 10 K and room temperature takes approximately 24 hours. Each power-sweep point at a stable temperature, typically consists of 3 stepped heater powers applied in order to determine the slope dp/d[t]). Only one heater, one thermometer, and one differential thermocouple are necessary. This technique is also well adapted for a broad range of samples. Every sample of at least 1 mm length with a thermal conductance of at least one-tenth that of the baseline (K BL 1x10-4 W/K at room temperature) and a thermal contact conductance (K Contact 5x10-3 W/K at room temperature (Fig 5)) are measurable. The first measurement, the base line evaluation, is necessary in order to determine the thermal conductance signal associated with the sample holder, lead wires, and differential thermocouple. This measurement also includes losses due to black body radiation and poor vacuum. Figure 1a) shows the sample holder configuration. a) b) Fig. 1) Sample holder configuration: a) Sample holder alone, b) Sample holder with sample. The second measurement determines the thermal conductance of the previous system in parallel with the sample. By subtraction, assuming that the parallel thermal conductance is limited by the sample and not by the contacts, the thermal conductance of the sample is directly obtained. Naturally, the black body radiations of the sample are included but the mean losses are due to the radiation of the hot surface (heater), which is already included in the base line. Figure 1b) shows the sample holder with attached sample. Thus, the PTC technique remains a very classic method with the well know problems and remedies tied to this kind of measurement. Except for the base line measurement, neither a calibration nor a special sample treatment/preparation is

3 required. The mounting process is fast and easy due to the fact that the thermocouple is mounted on the sample holder and that the baseline magnitude is only associated with the sample holder which may be reused several times. II. BACKGROUND Several methods have been successfully used in order to measure the thermal conductivity of ribbon like samples. A thermal potentiometer has been developed by Piraux et. al. [ 5 ] for carbon fibers. This technique provides direct thermal conductivity measurements of single carbon fibers and is applicable for other samples such as pentatellurides. However, this technique implies a relatively difficult mounting process as well as a complicated measurement procedure that must require significant amount of time. A variant of the 3ω method or self-heating technique [ 6 ] has been successfully used for carbon nanotubes. This technique requires a good knowledge of the sample resistance versus temperature, which should change enough to make the thermometer (the sample itself) sensitive to temperature variation. In the pentatellurides, due to the relative high instability of the resistivity under 250 K and contrarily relative flatness in temperature below [ 7 ] these conditions are not satisfied. III. METHOD As we reported above, the main advantages of this technique are simplicity and speed of each measurement. For these reasons we use an APD closed cycle cryo-cooler, which provides fast cooling (1 hour 35 minutes from room temperature to 10 K) which does not require any cryogenic fluid refilling. The electronics devices used for our method are relatively generic: temperature controller, current sourcemeter, and voltmeter. All electronic equipment is entirely under automatic computer control, which plays the central role. Figure 2) present the scheme of our set-up. The design of the sample holder is extremely important in order to insure several characteristics making the technique useful. The main characteristic of the design is to ensure a thermal conductance as low as possible in order to allow base line subtraction from the total thermal conductance (i.e. sample holder and the sample in parallel) conserving a good precision. For the same reasons, in order to avoid abusive black body radiation, the surface of the heating source should be minimized. Fig 3) shows a brief history of the base line improvement. However, the sample holder also needs to be able to fulfill its mechanical role to support the heater and sample. Also, the absolute heat capacity of the heater head should be minimized to avoid excessive relaxation time. Fig. 2) Scheme of the experimental set-up Thermal Conductance (W/K) Fig. 3) Base line improvement, as we can see, sample holder design progress allows us to decrease the base line over one order of magnitude. A. Sample holder description. For good accuracy of the measurement we should respect some limitations. The sample thermal conductance magnitude should be at least of the order of one tenth of the sample holder thermal conductance (base line). Since this technique is a two-probe measurement, the thermal resistance of the sample should be much larger than the thermal resistance of the contacts (Cf. Fig. 5).

4 10-2 Instrument Configuration Thermal Conductance (W/K) Non Tempeture T Stable? Yes Applay Power P Fig. 4) Thermal conductance versus temperature: Base line (opened diamonds), typical thermal conductance of sample holder with sample (full circles), and contact thermal conductance (opened squares). This implies that the thermal conductance of the sample be smaller than that of the contacts. Fig. 4) shows a typical thermal conductivity of the base line, sample holder with a sample (pentatelluride) and thermal contacts. Non Thermal Gradient ²T Stable? Yes Point Acquisition ²T Write to File P=PxPstep Non ²T Significative? Fig. 5) Scheme of thermal contacts configuration with corresponding formula. Yes B. Program description. As mentioned earlier, the acquisition program plays a central role. Indeed, a fast measurement implies a minimum of operations while accuracy implies a valid range for each parameter such as temperature stability, thermal gradient stability, heater current, etc. In order to incorporate speed with accuracy, the acquisition program should be able to determine, judiciously, each parameter and, if possible, in advance. The organization chart (Fig. 6) shows the procedure adopted to acquire highly reliable data. When the base temperature becomes stable the program checks the thermocouple stability. T=T+Tstep Non T = End Temperature? Yes Fig. 6) Program organization chart. End The time necessary for the differential thermocouple temperature to become stable depends on the relaxation time (which evolves with temperature as a function of the

5 diffusivity) of the system due to a small thermal conductance. If both, temperature and the gradient are stable a small power is applied and increased until the measured thermal gradient T reach a suitable value. For the first/ lowest temperature evaluated, the thermal conductance of the system must be determined so the first applied power should to be enough small in order to avoid any damage. For subsequent temperatures, (if not too far from the previous one) the first applied power may be determined according to the thermal conductance measured previously. Thermal Conductivity (W/K) Fig. 7) Thermal conductance of the contacts versus temperature: first measurement (opened squares), second measurement (crosses). IV. RESULTS A. Base line. a) b) Validation of this system, as with apparatus, requires evaluation of suitable standards. The thermal conductivity of standards as well as the reproducibility of the thermal contacts and base line stability was evaluated. In order to measure the thermal contact, a 0.83 diameter copper wire was selected as a sample. The thermal conductance of this sample is very high between 10 K and room temperature; thus the conductivity limitation is imposed by the thermal contacts (see Fig. 5). Different thermal contacts have been tested (silver paint, nickel paint, stycast, eventually copper-filed apiezon grease) and determined found the best results with silver paint in term of quotient: (quality) 2 Reproducibility. The Fig. 7 shows thermal Difficulty conductivity of the contacts measured with copper sample. Fig. 8) Standards thermal conductivity versus temperature: a) pyroceram, and b) stainless steel #1461. Standard data (closed circles), PTC measurements (opened circles). B. Standards and first measurements. Standards such as stainless steel (Fig. 8a) and pyroceram (Fig. 8b) were selected and measured. It is important to keep in mind that these measurements concern very small samples, which may provoke an apparition of size effect and may induce thermal transport that is size dependent. Indeed, one of the stainless steel (#302) thin wire (5 mill diameter) gave us invariably a thermal conductivity twice higher than the conductivity of the usual (bigger) samples (Fig 9). After

6 verification, it was concluded that these disturbing measurements do not result from any experimental error. Note that any contact problem would only decrease the measured parallel conductance and the deduced sample conductivity Fig. 9) Thermal conductivity of stainless steel #302: standard data (full circles), PTC measurements of thin wire (opened circles). With this technique we measured, for the first time, the thermal conductivity of a pentatelluride in order to determine thermoelectric figure of merit for these materials (Fig 10a). A detailed study of these materials will be reported elsewhere [ 8 ]. For illustrating the different possibility of our technique we also report the measurement of a single carbon fiber compared with the measurements of B. Nysten et al. [ 9 ] Using the thermal potentiometer (Fig 10b). V. CONCLUSION We describe a very simple and fast technique, which allow us to measure, for the first time, the thermal conductivity of small samples such as pentatellurides. This technique is applicable for a broad range of materials of low thermal conductance, which is generally tied with small cross-area with respect to length. ACKNOWLEDGMENT We acknowledge A. Pope to allow us to use initially her experimental facilities. a) b) Fig. 10) Thermal conductivity versus temperature: a) pentatelluride ZrTe 5 (opened circles), our data are not compared because the pentatellurides have been measured for the first time, b) single carbon fiber: Data taken from reference X (closed circles), and PTC measurement (opened circles). 1 H. Weinstock, C.W. Thompson, Jr and W.C. Overton, Cryogenics 21, 118 (1981). 2 R.L. Filler, and P. Lindelfeld, Rev. Sci. Instrum. 46, 439 (1975) 3 D. G. Cahili, Rev. Sci. Instrum. 61, p 802 (1990). 4 R. T. Littleton IV, T. M. Tritt, J. W. Kolis and D. Ketchum, Phys. Rev. B. 60, (1999). 5 L. Piraux, J.-P. Issi, and P. Coopmans, Measurement 5,2 (1987). 6 W. Yi, L. Lu, Z. Dian-lin, Z.W. Pan, and S.S. Xie, Phys. Rev. B 5, R9015 (1999).

7 7 R. T. Littleton IV, T. M. Tritt, C. R. Feger, J. Kolis, M. L. Wilson, M. Marone, J. Payne, D. Verebeli, and F. Levy, Appl. Phys. Lett. 72, 2056 (1998). 8 B.M. Zawilski, R.T. Littleton IV, T.M. Tritt, in progress. 9 B. Nysten, L. Piraux, and J.-P. Issi, J. Phys. D: Appl. Phys. 18, 1307 (1985).

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