# Measurements of thermal resistance of thin brittle samples under high pressure

D. Djurek, Denis Jerome, C. Weyl

<figure><img src="image_1.png" /></figure>

## To cite this version:

D. Djurek, Denis Jerome, C. Weyl. Measurements of thermal resistance of thin brittle samples under high pressure. Journal of physics E: Scientific instruments, 1982. hal-04998141

HAL Id: hal-04998141

https://hal.science/hal-04998141v1

Submitted on 20 Mar 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_2.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Measurements of thermal resistance of thin brittle samples under high pressure

D Djurek†, D Jérôme‡ and C Weyl‡

† Institute of Physics of the University, Zagreb, Yugoslavia
‡ Laboratoire de Physique des Solides, Université Paris Sud
91405 Orsay, France

Received 20 November 1980

**Abstract.** An accurate method for mesasuring the thermal resistance of small brittle samples of unusual geometry is developed. The method makes use of a thermal potentiometer which compares the temperature gradients in investigated and reference samples. The application of the method to measurement of the thermal resistance of quasi-one-dimensional conductors under normal and high pressures (up to 1.2 GPa (12 kbars)) is described.

## 1. Introduction

Samples of the majority of quasi-one-dimensional (quasi-1D) conductors consist mainly of small needles of unusually high brittleness. The average sample diameter of a well known organic compound tetrathiofulvalinium tetracyanoquinodimethan (TTF-TCNQ) ranges from 10 to 30 µm and measurements of its properties pose a considerable difficulty. Among other physical properties investigated (Ehlers et al 1974), the thermal conductivity of TTF-TCNQ was measured by a modification of the AC calorimetric method (Salamon et al 1975) and calculated from the product of the specific heat and the thermal diffusivity. In this method an unmasked region of the specimen was exposed to light and thermal diffusivity was evaluated by the measurement of temperature oscillations in a masked region. The amplitude of these oscillations is affected by the diffusion of heat from one region to the other and the thermal resistivity of the sample may then be calculated provided the heat capacity of the sample is known from an independent measurement.

A more recent investigation of the quasi-1D conductor NbSe₃ reveals two phase transitions of the Peierls type to a charge-density-wave (CDW) condensate at 146 K and 59 K. The formation of CDW is accompanied by pronounced anomalies in the electrical conductivity (Haen et al 1975), and a well established non-ohmic behaviour due to the properties of depinned CDW also appears. Thermal conductivity data provide a powerful tool for estimating the fraction of condensed electrons in the CDW phase of NbSe₃ (Djurek and Tomić 1981).

The charge-transfer salt tetramethyltetraselenafulvalene hexafluorophosphate (TMTSF)₂PF₆ based on the donor molecule (TMTSF) is the first known organic superconductor and has attracted a considerable interest due to the superconductivity observed at 1.2 K under the pressure of 1.2 GPa (Jérôme et al 1980). At low temperatures the high pressure stabilises the metallic phase which, in effect, displays a high electrical conductivity of the order of 10⁶ Ω⁻¹ m⁻¹ at 20 K. This is an order of magnitude larger than the conductivity in other recently investigated organic conductors.

The measurement of the thermal conductivity of (TMTSF)₂PF₆ under pressure is therefore of considerable importance and this paper is devoted to the problems encountered in high-pressure calorimetric experiments.

## 2. Experimental

The experimental set-up used for measuring the thermal conductivity of NbSe₃ under low pressure and (TMTSF)₂PF₆ under pressure of 1.2 GPa is based on a simple principle of a thermal potentiometer. The advantage of the method lies in the complete removal of the parasitic heat flow through the electrical connections. Additional radiation losses and lateral heat flow in the high-pressure cell may also be eliminated in a standard way. Finally, testing of the high-pressure cell under various conditions of temperature and pressure may be easily performed in a rather reproducible way.

The mounting of the sample in the thermal potentiometer assembly is shown in figure 1(a). The sample s is placed between the heat sink A and the reference sample r bonded to another sink B. Sinks A and B are kept at different temperatures T_A and T_B. The heat flow between the heat sinks A and B is partly channelled through the sample and the reference sample, and the heat current, defined as energy released per unit time, is equivalent in both samples:

$$ \left(\frac{dQ}{dt}\right)_s = \left(\frac{dQ}{dt}\right)_r \quad (1) $$

<figure><img src="image_3.png" /></figure>

<figure><img src="image_4.png" /></figure>

<figure><img src="image_5.png" /><figcaption>**Figure 1.** (a) Schematic diagram of thermal potentiometer for the measurement of thermal conductivity; (b) and (c) show the calibration procedure for the conductance of the thermal link.</figcaption></figure>

Then, depending on the differing thermal conductivities of the sample ($k_s$) and the reference sample ($k_r$), the temperature gradients between them are shared in accordance with the
Fourier definition of the heat flow

$$k_s A_s \frac{T_B - T_C}{X_s} = k_r A_r \frac{T_C - T_A}{X_r} \quad (2)$$

$A_s$, $X_s$ and $A_r$, $X_r$ are cross sections and lengths of the sample and reference sample respectively.

The use of this arrangement enables us to compare the thermal resistivities of two samples by measuring the temperature differences between A, B and C. If the thermal resistivity of one of the two samples is known from separate calibration runs, it is possible to measure the thermal resistivity of the unknown one. Depending on the quality of the thermal contacts at C and A it is possible to achieve good accuracy in the measurement. The sample of unknown thermal resistivity may easily be replaced when the reference sample is fixed. Both sinks A and B are copper plates cut as shown in figure 2. The reference sample consists of a differential thermocouple with the cold end at the junction point C and the hot end at sink B. The thermocouples used are standard copper-constantan, chromel-constantan and chromel 0.07% AuFe.

### 3. Zero-pressure regime

The copper wires, as a part of the thermal link, are shown in figure 2 and are labelled with numbers 1, 2 and 4. The constantan wire 3 runs from the cold end at C to the hot end at sink B and then to sink A. In this manner it is possible to measure the temperature difference between two sinks $T_A - T_B$ and the temperature difference on the reference sample $T_B - T_C$. $T_B - T_C$ and $T_C - T_A$ are equivalent to voltage differences in ordinary electric potentiometers. The thermal conductances of the thermocouple $K_r$ and the sample $K_s$ may be expressed in terms of temperature differences measured by differential thermocouples

$$\frac{K_s}{K_r} = \frac{T_B - T_A}{T_B - T_C} - 1. \quad (3)$$

In the zero-pressure regime the thermocouple is prepared by welding copper (50 µm) onto constantan (120 µm). Great care must be paid to the quality of the thermal contacts between the

<figure><img src="image_6.png" /><figcaption>Figure 2. Thermal potentiometer apparatus.</figcaption></figure>

sample and the thermocouple in the case when the resistivity of the sample is low and the contribution of the contact resistivities becomes important. When whisker-type samples are used the thermal contact at C may be prepared by the use of a standard low-temperature Ge varnish. The contacts at the sinks are made by drilling holes in the copper and filling them with molten indium, in order to minimise the temperature difference between the wires and the sink. Before being filled with indium the ends of the thermocouples must be electrically insulated with a thin film of high-temperature varnish. Thermal contact of the sample and the copper sink A is attained by the use of silicone grease in the hole drilled in copper. Thermocouple t is used to measure the temperature of the sink A and thus the temperature of the sample.

The temperature difference $T_B - T_A$ between sinks A and B is maintained in the calorimeter assembly shown in figure 3. The copper plates are fixed together by two copper rods, L, soldered vertically on the main heat sink $S_1$ which is kept at the temperature of the experiment by using the heater $H_1$.

<figure><img src="image_7.png" /><figcaption>Figure 3. Calorimetric cell for the measurement of the thermal conductivity at zero pressure. V, vacuum; O, O ring.</figcaption></figure>

Temperature inhomogeneities within the copper plates may be of considerable importance for the accuracy of measurement, and in order to reduce them it is necessary to keep the copper rods as thin as possible. If the rods are not very thin they contribute to the fluctuation of the mean temperature within the copper plates. In our experiments the thickness of the rods varied from 2 to 3 mm and the highest temperature difference measured within the plates was of the order 0.1 K. Thermocouples on one plate are sunk in indium close to each other and the estimated error due to temperature inhomogeneities within the copper plates is less than one per cent. The upper sink $S_2$ and the heater $H_2$ are used for creating the temperature difference between the sinks. Both heaters, $H_1$ and $H_2$, were wound from manganin wire of diameter 50 µm.

The radiation shielding can R is made of copper and prevents radiation from the outer parts of the calorimeter reaching the sample. The sink assembly with the radiation shield is situated within the copper can, which in turn is in thermal contact with a cryogenic liquid, nitrogen or helium. The whole
can is evacuated to a vacuum higher than about 0.133 mPa (10⁻⁶ Torr). Considerable care must be paid to the heat radiation from the sample to the interior of the calorimeter. The diameter of the Ge varnish droplet at the connection point C is about 0.3 mm. At a temperature of 150 K the radiation power from the droplet is ~10⁻⁷ W when T_B - T_A is about 1 K. For very high ratios of length to cross section the current through the sample may be comparable to the radiation power. In our measurements the length of the sample was about 2 mm and the cross section 0.04 mm². The heat current transported by conduction corresponding to this geometry is about 10⁻⁴ W, being in satisfactory relation to the radiation losses from the droplet and the sample.

The calibration of the thermal link was performed by the use of copper wire at the position of the sample investigated; the whole assembly is shown schematically in figures 1(b) and (c). Copper wire (full line) and constantan (broken line) are glued together at the junction point C with copper wire of equal length and cross section spanned between A and C. The temperature dependence of the different T_B - T_A and T_B - T_C was then measured. Thus we have

$$ \frac{K + K_0}{K} = \left( \frac{T_B - T_A}{T_B - T_C} \right) - 1 = 1 + \frac{K_0}{K} \quad (4) $$

where K and K₀ are thermal conductances of the copper and constantan wire respectively.

In addition the calibration was performed as is shown in figure 1(c). The thermocouple wires (copper and constantan) were of the same length as in the former case. Thus we have two additional copper wires of the same length connecting the heater and the current source. Then under steady-state conditions we have

$$ \frac{I^2 R}{T - T_0} = 3K + K_0 \quad (5) $$

where T - T₀ is the temperature difference between the heater and the heat sink. Thus it is possible to evaluate the conductances K and K₀ from the expressions (4) and (5).

The method was tested by the measurement of the thermal conductivity of copper and constantan at low temperatures (figure 4(b)). A satisfactory agreement is obtained with the data found in the literature (American Institute of Physics Handbook 1972). The main error in the comparison of the measured thermal conductivity with the values found in the literature is connected with difficulties in the determination of the true geometry.

The method described was used for the measurement of the thermal resistance of NbSe₃ in the phase transition region (figure 4(a)). In the temperature region around 145 K a pronounced anomaly in thermal resistance was found (Djurek et al 1981) and interpreted in terms of a structural instability caused by Peierls distortion. No anomaly has been observed at the lower transition. This result contradicts the generally accepted belief that most of the electrons are condensed at the lower transition. Therefore it is more probable that the order parameter at the lower transition is only supporting the order parameter vanishing at 146 K.

### 4. High-pressure regime

The high-pressure cell for measuring the thermal conductivity is made of two copper sinks (A and B in figure 5) thermally and mechanically connected by two copper rods, L. The upper sink, B, is subjected to the Joule power source comprising a 10 Ω manganin heater, H. To establish the thermal link, r, chromel wire of 0.3 mm in diameter is suspended between the upper sink and the sample, s. The thermal contact between the sample and

<figure><img src="image_8.png" /><figcaption>Figure 4. The thermal conductivity of (a) the quasi-1D conductor NbSe₃ and (b) copper.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Figure 5. High-pressure cell for measuring thermal conductivity by the thermal potentiometer method.</figcaption></figure>
<figure><img src="image_10.png" /><figcaption>**Figure 6.** Thermal conductivity of isopentane under a pressure of 1.2 GPa. The inset curve shows the dependence of the thermal conductivity of isopentane on pressure at 300 K. The inset diagram shows the experimental set-up for measuring the thermal conductivity of isopentane.</figcaption></figure>

the lower sink is made of polyurethane immersed in a hole 0.4 mm in diameter. Constantan wire of 15 µm diameter, as a part of the differential thermocouple, is attached in the same way to the upper sink. In order to account for the temperature difference $T_B - T_A$, a differential thermocouple, E, is spanned between both sinks. Isopentane has been used as a pressure-transmitting medium, its solidification temperature being at 210 K for a pressure of 1.2 GPa. The main difficulty in the measurements of thermal conductance arise with the removal of the side heat flow in isopentane. The lateral heat current competes with the current transported by conduction in the sample and the thermal link. Certain optimisation of the sample geometry and high-pressure cell is needed to minimise the lateral

heat flow and accordingly produce the correct interpretation of the data. The sample together with the thermal link and the high-pressure cell form an assembly of cylindrical symmetry, and the heat current from the interior to the side wall of the cell may be expressed as

$$Q_{-} = \frac{2\pi k_{iso}}{\ln R_{2}/R_{1}} \int \frac{\Delta T}{X} dX \quad (6)$$

with the integration being over the sample and the thermal link. Here $k_{iso}$ stands for the thermal conductivity of isopentane and $\Delta T$ is a temperature difference between the sample (diameter $R_1$) and the side wall (diameter $R_2$).

Isopentane is an organic liquid with moderately low thermal

<figure><img src="image_11.png" /><figcaption>**Figure 7.** Low temperature data for the ratio $R_1/R_2 = (T_C - T_A)/(T_B - T_C)$ at 1.2 GPa for chromel-constantan (A). The same ratio obtained in vacuum is shown as a full curve; the broken curve is the ratio of thermal conductances of chromel and constantan obtained in independent experiments and in the literature. Chromel-chromel data (B) are evaluated in order to estimate the contribution of contacts at point C and sinks.</figcaption></figure>
Thermal resistance of thin brittle samples

conductivity (0.217 mW mm⁻¹ K⁻¹ at room temperature) and when ordinary metal samples are investigated the lateral heat flow may be neglected. However, to account for possible modifications of the simple expression (3) when low-conductivity samples are used, very careful tests in various conditions need to be performed. Firstly the thermal conductivity of isopentane under high pressure must be known over the whole temperature range considered. It turns out to be necessary to know the thermal data from independent measurements of the thermal conductivity of isopentane, and these have been carried out separately. For that purpose the cylindrical cell has been used as shown inset in figure 6. In that case one can make use of the expression (6) for the evaluation of the thermal conductivity. The Joule power delivered to the heater coil 4 wound about the copper rod 3 is balanced by the lateral heat flow in isopentane from the heater to the side wall 2. The temperature difference between the copper rod (diameter R₁) and side wall (diameter R₂) is constant along the cell and is measured by means of the differential chromel-constantan thermocouple (20 µm in diameter). The thermal conductivity of isopentane is strongly pressure dependent and at 1.2 GPa it exhibits a moderate increase with decreasing temperature (main curve of figure 6). A decrease of thermal conductivity with decreasing temperature occurs below 5 K and this may be attributed to the size effect in solid isopentane. Owing to the moderately slow variation of the thermal conductivity with temperature, isopentane is a good material for measuring thermal conductance.

The protection of the high-pressure cell by means of a Teflon shield (T in figure 5) provides an appreciable reduction of the lateral heat flow. The top side is in good thermal contact with the upper sink which gives rise to the formation of a temperature gradient in the side wall. The temperature difference ΔT is then appreciably reduced (the mean value of ΔT is about 0.1 K) leading to the reduction of lateral heat flow.

The modification of the ratio (3) due to the contribution of the lateral heat flow was estimated in several different ways by replacing the sample with other samples of known conductance and equivalent geometry. In the first experiment constantan was located between the thermal link and the sink A. The dimensions of the constantan were 0.04 mm² × 1.3 mm and of the chromel 0.09 mm² × 2.7 mm.

An additional measurement of the thermal conductivity of chromel was performed by investigating the temperature difference along a chromel wire subjected to Joule heating at one end. From the thermal conductivity obtained by this conventional method the thermal conductance was calculated for the above geometry and compared with the conductance of constatan found in the literature (White 1968). The broken curve in figure 7 is the ratio of the contributions of the two conductances. When compared with the data obtained in vacuum certain discrepancy is evident. This may be attributed to the nonlinear contribution of contacts in the construction of the thermal potentiometer where the thermal resistance of contacts is not negligible in comparison with the thermal resistance of the sample and the thermal link. In order to estimate possible errors due to the contacts, the constantan from the thermal potentiometer assembly was replaced by chromel (0.3 mm in diameter). The ratio T<sub>B</sub> - T<sub>A</sub>/T<sub>B</sub> - T<sub>C</sub> for the 'chromel-chromel' set-up is also shown in figure 7. Instead of a constant value over a broad temperature range a moderate variation occurs and the true ratio, 1.48, is finally approached at liquid helium temperatures when the relative contribution of the contact resistance is negligible in comparison with the resistance of chromel.

In conclusion, we have developed a thermal potentiometer method for measuring thermal resistance at zero and high

pressure. Lateral heat flow in the high-pressure cell is shown to influence tolerably the accuracy of the measurement and the reproducibility of the data.

The authors are indebted to A Andrieux for valuable collaboration and B Gumhalter for critical reading of the manuscript. One of us (DD) expresses gratitude to Professor J Friedel for the support during his stay in Orsay.

References

American Institute of Physics 1972 *American Institute of Physics Handbook* 3rd edn pp 4–155

Djurek D and Tomić S 1981 Thermal conductivity of Nb<sub>3</sub><sup>5</sup> *Phys. Lett.* **85A** 155–6

Ehlers I, Hepp K and Weidenmüller H A (ed.) (1974 GPS Summer School Proceedings on One-Dimensional Conductors (Berlin, Heidelberg, New York: Springer)

Haen P, Monceau P, Tissier B, Waysand G, Meerschaut A, Molinie P and Rouxel I 1975

Proc. 14th Int. Conf. on Low Temperature Physics, Otaniani, Finland 5 pp 445–8

Jérôme D, Mazaud A, Ribault M and Bechgaard K 1980 Superconductivity in a synthetic organic superconductor (TMTSF)<sub>2</sub>PF<sub>6</sub>

*J. Physique Lett.* **41** 95–8

Salamon M B, Bray I W, De Pasquali G, Craven R A, Stucky G and Schultz A 1975 Thermal conductivity of TTF-TCNQ near the metal-insulator transition

*Phys. Rev. B* **11** 619–22

White G K 1968 *Experimental Techniques in Low-Temperature Physics* (Oxford: Clarendon)