The Measurement and Uncertainty Analysis of Thermal Resistance in Cryogenic Temperature Sensor Installation

The choice of the appropriate installation method plays an important role for accurate temperature measurement. In the cryogenic and high vacuum environment, due to poor contact between the cryogenic temperature sensor and the surroundings that the sensor is installed and intended to measure, the self-heating from sensor measuring current brings about temperature difference and creates a potential temperature measurement error. The self-heating temperature difference is directly proportional to the thermal resistance for a mounted sensor, which means that lower installation thermal resistance of sensors is advantageous to obtain better measurement results. In this paper, a measurement model for the installation thermal resistance of sensor is built in terms of two currents method which is always used to measure self-heating effect. A cryostat that can provide variable temperature in the accurate temperature measurement and control experiments is designed and manufactured. This cryostat can reach 3K in a few hours and the sample temperature can reach as high as 20 K. Based on the experimental results, the measurement uncertainty of the thermal resistance are also analyzed and calculated. To obtain the best measurement results in our cryostat, the thermal resistances of sensors with two installation methods are measured and compared.


Introduction
Operation of resistance thermometer requires dissipation of power in the sensor. The heat flux generated by the measurement current creates the temperature difference between the sensor and the environment is intended to measure, which is no doubt to produce temperature measurement error or uncertainty. The self-heating temperature difference relates to the thermal resistance between the sensor and its surroundings. Thermal resistance is a significant factor in temperature measurement uncertainty. How to accurately measure the thermal resistance between the sensor and the environment is an important problem in high precise temperature measurement. Thermal resistance is found to depend on temperature and details of sensor mounting. Apiezon grease and Varnish are always used to mount thermometer sensors in low temperature, because they have high thermal conductivity in low temperature. Details of construction unknown to a user can strongly affect the thermal resistance. It is difficult to calculate an effective thermal resistance for a mounted temperature sensor. Many scholars have done research on the thermal resistances of the cryogenic temperature sensors. Rusby [1] studied the thermal resistances of a standards grade rhodium-iron sensor from 0.4 to 280K; Schoepe [2] calculated the thermal resistances of a thick film resistor from 0.4 to 280K. Thermal resistances were measured at cryogenic temperatures from 1 to 300K on several commercially available temperature sensors by Holmes and Courts [3]. In this paper, the thermal resistances of the Cernox temperature sensor that mounted by two different methods (Apiezon N Grease, VGE-7031 varnish) from 4.2 to 14K are calculated. The measurement uncertainty of the thermal resistance is also analyzed. A cryostat cooled by Gifford-McMahon (GM) cryocooler that can provide variable temperature in the accurate temperature measurement and control experiments is designed and manufactured. A thermal damper made of PTFE is inserted between the second cold head and the sample to decrease the temperature oscillation of the sample caused by the second cold head [4].

Theoretical background
When the resistance R of the Cernox sensor is measured, the measurement current I dissipates power P: Heat dissipation in the sensor causes a heat flux that generates a temperature gradient from the sensor to the mounting surface. The Cernox sensor therefore measures a higher temperature T instead of the actual temperature T 0 , which would be measured without the self-heating effect [6]. The temperature difference T  can therefore be calculated as: Where t R is the thermal resistance between the sensor and its environment, and I is the measurement current, R is the resistance of the temperature sensor at the measurement current. The self-heating value can also be calculated in terms of resistance Where 0 X and 1 X are the resistance ratios of zero-current and current 1 I separately.
Where s is the sensitivity of the Cernox thermometer at the measurement temperature: The resistance ratio of zero-current is unable to be measured. For the highest accuracy measurements, corrections should be applied by measuring at two currents, 1 I and 2 I , and extrapolating to zero current. In addition Substituting Eq.(5) to Eq.(3) yields: Therefore, the thermal resistance t R of the temperature sensor and its mounting environment is: Differentiation of Eq. (7) leads to the propagation-of-error equation: I  dI  dI  dX  dX  I I  I I  I  I  dR  I I  I I  sX sR dX R X ds As a consequence, the uncertainty due to the measured effective thermal resistance can be expressed as: Where ( ) u X and ( ) u I are the uncertainties due to the measurement of resistance ratios and excitation current respectively, while ( ) u s is the uncertainty due to the fitting of temperature sensitivity. The uncertainties 1 ( ) u I and 2 ( ) u I can be obtained from calibration specification of Fluke 1594A, as seen in Table 1, while ( ) u s is the uncertainty due to the fitting of temperature sensitivity. It can be seen clearly that choose the 1:√2 current ratios to calculate the thermal resistance and its uncertainty is most convenient. The thermal resistance of the Cernox sensor we calculated above consists of two parts: one is the internal thermal resistances, which is related to the structure and materials of the thermometer itself, and the other is the interfacial boundary thermal resistance of which we are concerned.

Experiment setup
In order to obtain the cryogenic temperature, a new cryostat was designed [5], as shown in Fig.1. This cryostat had a simple structure that consists of a two-stage GM cryocooler (Sumitomo (SHI) Cryogenics of America, Inc. RDK-415D), cryostat wall, radiation shield, thermal damper, sample holder, Cernox temperature sensors, temperature controller (Lakeshore,340) and other measuring instruments (Fluke 1594A). All the measurement devices used for acquiring data were connected by IEEE-488 cables and controlled by a personal computer using a program written by LabView software.  All measurements were performed with a DC bridge Fluke 1594A (uncertainty 0.8ppm) in combination with a 10000ohm standard resistor (uncertainty 1.5ppm) placed in a temperaturecontrolled bath. Thermal resistances of two different mounting methods (VGE-7031 Varnish, Apiezon N Grease) were measured at cryogenic temperatures (4.2K 、 6K 、 8K 、 10K 、 14K) on Cernox temperature sensor (Cernox-1050-SD SN 70210) by two-current method. The measurement uncertainty of the thermal resistance is also analyzed and calculated by the present theory. It is worth mentioning that the increase in the sensor excitation current will result in a decreasing temperature measurement standard deviation. A larger measurement current will also lead to more obvious temperature difference between the two measurement currents 1 I and 2 I . Nevertheless, a larger excitation current will dissipate more power in the temperature sensor, raising its temperature above the mounted environment. It is a significant problem to choose an appropriate measurement current to balance the standard deviation and the self-heating effect. In our experiment, we choose 35 A  as the measurement current for 4.2K and 6K, and 65 A  for 8K, 10K and 14K.

Results and discussion
The temperature measurement results of the Cernox thermometer (X70210) measured by two-current method was shown in Figure 2. The temperature 1 . It can be seen clearly that the temperature fluctuation is less than 2mK at all temperatures. The thermal resistances can be calculated by the measurement results of the Figure 2. The thermal resistance is shown as a function of temperature for the Cernox sensors in Table 2 and Figure 3. The thermal resistances of the Cernox thermometer (X70210) installed by VGE-7031 Varnish and Apiezon N-Grease were measured. The relative difference of the thermal resistance mounted by these two methods is less than 2%. In other words, the temperature uncertainty of the two mounting methods caused by self-heating is roughly equivalent. The thermal resistances of the Cernox thermometer (X70208) mounted by N-Grease were also measured. The thermal resistances of these two thermometers have the same order of magnitude.   The uncertainty of the thermal resistances (X70210) at different temperatures installed by Varnish was analyzed in Table 3. It is remarkable that the uncertainty of the thermal resistance decreasing with the 6

ICECICMC
IOP Publishing IOP Conf. Series: Materials Science and Engineering 171 (2017) 012126 doi:10.1088/1757-899X/171/1/012126 increasing of the temperature, while the relative uncertainty of the thermal resistance for different temperatures are almost equal. The uncertainty of the thermal resistance at all temperature points are less than 3%, it means that the calculation results of the thermal resistance is credible.

Conclusions
Cernox thermometer self-heating is a significant factor in high accuracy cryogenic temperature measurement that cannot be eliminated. The temperature difference caused by self-heating is depended on the thermal resistance of the sensor and its environment. In this paper, the thermal resistance of the Cernox temperature sensor and its surroundings from 4.2K to 14K is calculated by two-current method. The results show that the thermal resistances of the two mounting methods (VGE-7031 Varnish and Apiezon N Grease) are roughly equivalent. It can be observed that with the increasing of the temperature, the effective thermal resistances are gradually decreased. The uncertainty of the thermal resistance is also analyzed in this paper. The uncertainty of the thermal resistance decreased with the increasing of the temperature, while the relative uncertainty for different temperature is equal and less than 3%.