thermal contact resistance using laser photothermal method at cryogenic temperatures

Accepted Manuscript Measurement of Thermal diffusivity/Thermal Contact Resistance Using Laser Photothermal Method at Cryogenic Temperatures Bi Dongmei, Chen Huanxin, Liu Shanjian, Shen Limei PII: DOI: Reference:

S1359-4311(16)31332-1 http://dx.doi.org/10.1016/j.applthermaleng.2016.07.188 ATE 8786

To appear in:

Applied Thermal Engineering

Received Date: Revised Date: Accepted Date:

21 April 2016 28 July 2016 28 July 2016

Please cite this article as: B. Dongmei, C. Huanxin, L. Shanjian, S. Limei, Measurement of Thermal diffusivity/ Thermal Contact Resistance Using Laser Photothermal Method at Cryogenic Temperatures, Applied Thermal Engineering (2016), doi: http://dx.doi.org/10.1016/j.applthermaleng.2016.07.188

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Measurement of Thermal diffusivity/Thermal Contact Resistance Using Laser Photothermal Method at Cryogenic Temperatures Bi Dongmei a*, Chen Huanxin b , Liu Shanjiana , Shen Limei b a

School of Agricultural Engineering and Food Science, Shandong University of Technology, Zibo, China

b

School of Energy and Power Engineering，Huazhong University of Science and Technology，Wuhan，China1

ARTIC LE

INFO

ABSTRACT

Article history:

The Laser Photothermal Method (LPM), which is a transient non-contact

Received

technique, was employed to measure the thermal physical parameters of Stainless

Received in revised form

Steel 304 (SS304), Oxygen-Free Copper (OFC) and Aluminum Nitride (AlN)

Accepted

ceramics. The thermal diffusivity of SS304, OFC and AlN ceramic were

Keywords:

measured by the LPM and some explanations for the tendency with the

Laser Photothermal Method

temperature were given. Compared with the reference values, it showed that the

Thermal diffusivity

LPM is reliable in measuring thermal parameters. Following, the thermal contact

Thermal Contact Resistance

resistances (TCRs) of OFC-AlN ceramic, SS304-AlN, SS304-SS304 and AlN

Emprical formula

ceramic were measured in the temperature range from 70 K to 300 K and the contact pressure was on the order of 0.10 MPa. Some explanations were given to make clear the changing trend of TCRs with the increase of temperature. Moreover, some empirical formulas of the TCRs were established. The accuracy of the establishing empirical formulas had been validated by the experiment, and the error was smaller than 9%.

Nomenclature

* Corresponding author. Tel.: 86 05332780092. E-mail address: [email protected] -1-

c d f k L m, n P Q R T

specific heat capacity (J·kg-1·K-1) thickness of specimen (mm) modulation frequency (Hz) thermal conductivity (W·m-1·K-1) thickness of the thin film (mm) constant contact pressure (MPa) the power of the heating laser (W) correlation coefficient contact temperature (K)

Greek Symbols α ρ φ △

thermal diffusivity (m2·s-1) density (kg·m-3) the phase lag (rad) error

Subscripts 1 2

the specimen beside the heating laser the specimen beside the probe laser

1. Introduction

In some special fields, effective heat transfer between solid and solid has become an important issue in instrument design and operation, such as the spacecraft thermal control, high-density electronics cooling, superconductivity and direct cooling refrigerator system, especially those in a vacuum environment. When the solids contact each other, the real contact area is only a small fraction of the nominal contact area in engineering applications [1]. Because of the smaller real contact area, the heat transfer at the interface will be obstructed and the Thermal Contact Resistance (TCR, the reciprocal of thermal contact conductance) holds an important role in here. The microscopic interface between two solids is a three-dimensional layer structure with a thickness of several microns or dozens of nanometers. The TCR of the contact layer has a great influence on the heat transfer in some cases. Therefore, studies on the mechanism of the microscopic heat transfer at the interface are of great significance. Erenow, many theoretical and experimental works on the TCR have been done. Swartz, et al measured the TCR between metal film and the dielectric substrates in the temperature range from 1 to 300 K and established the Diffuse Mismatch Model (DMM). The experimental results agree well with the prediction of the DMM when the temperature is below 30 K [2]. Moreover, when a weak van der waals force is applied on the contact surface, the traditional

-2-

Acoustic Mismatch Model (AMM) can predict well the TCR between Si and Si, Si and Pt [3]. The AMM and DMM are more consistent with their experimental data below a certain temperature. But with the increase of the temperature, the deviation between predicted values and measured value increases. Prasher, et al considered that the scattering near the interface plays a far more dominant role than any other mechanism to the TCR. The Scattering-Mediated Acoustic Mismatch Model (SMAMM) was proposed based on the analysis of the previous models. This model can provide an accurate description of the TCR both at low and high temperatures [4]. To sum up, those models of the TCR have their own application range. Even for the same two materials, once the temperature or the contact pressure changes, the models may no longer be valid. So the experimental measurement is still an important mean to study the mechanism of the TCR. In general, the measurement methods of the TCR can be divided into steady-state and transient-state method. The steady-state method is considered to be a reliable method to measure the TCR. But it takes a relatively long time to come to the steady-state condition for measuring [5]. And the heat flow is more difficult to control in practice. In recent ten years, the photothermal microscopy has been used in measuring the thermal parameters of the object. The technique is with high time resolution and high spatial resolution, such as mirage microscope and thermo-reflectance microscope [6-9]. A modulated or pulsed laser beam is used in the photothermal technique as the heating source. The modulated laser allows the specimen to be thicker compared with the pulsed laser [10]. In this paper, the Laser Photothermal Method (LPM) using the modulated laser was employed to measure the TCR. It can complete the measurement in a short time and can overcome the difficulty in controlling heat flow of the steady-state method. The Stainless Steel 304 (SS304) has a low thermal conductivity and can endure high pressure at a low temperature. Because of the two advantages, SS304 is commonly used in spacecrafts. The cryogenic valves, frameworks and containers of the liquid fuel in spacecraft are mostly made of SS304 [11-13]. So the heat transfer between SS304 and other materials will influence the operating performance of the equipment greatly. In this paper, the thermal diffusivity of SS304, Oxygen-Free Copper (OFC) and Aluminum Nitride (AlN) ceramics -3-

were measured using the LPM firstly. The experimental data is matched well with the reference values. This shows that the LPM is effective and reliable to measure the thermal diffusivity or TCR. Then the TCRs of OFC-AlN ceramic, SS304-AlN ceramic, SS304-SS304 and AlN-AlN ceramic were measured by the LPM. The experimental principle is briefly described in section 2. Then the experimental results and discussions are given in section 3. The main purpose of this paper is to analyze the influence of the temperature and the contact pressure on the TCR. Furthermore, the thermal diffusivity of materials was measured to populate the database. Last the experimental error was analyzed.

2. Experimental principle of LPM

Recently, the photothermal technique has become a research hotspot in the measurement of the thermal parameters because of its high spatial resolution and the non-contact detection technique. The LPM is an application approach of the photothermal technique. The principle of measuring TCR using the LPM is shown in Fig. 1 and the detailed description can be found in reference [14].

Heating Laser Beam Carbon Film

L1 Specimen 1 d Specimen 2 L2 3

4 Gold Film Filter

Probe Laser

Silicon Photodiode Lock-in amplifier Signal Control System

Fig. 1. Schematic of the modulation photothermal method. d is the thickness of the specimen. L1 is the thickness of the carbon film. L2 is the thickness of the gold film. 3 is the incident beam of the probe laser and 4 is the reflected beam

The characteristics of the thermal wave are dependent on the thermal parameters in the homogeneous materials. The TCR and thermal diffusivity of the bulk sample in the two-layer system are the function of the characteristics of the thermal wave. Thermal wave heat transfer was assumed one dimensional in the specimens without radial heat loss. The -4-

thermal wave followed the law of conservation of energy when across the interfaces. Equations (1) and (2) are respectively the formula of the thermal diffusivity and TCR. Details of the sovling process can be found in reference [15] and [16].



[1  TCR 

Where x  d1

 fd 2 2    / 4 

(1)

(k  c) 2 ] tan(  x) (k  c)1

(2)

 f (k  c) 2 [1  tan(  x)]

f f   d2  1 2 4

3. Experimental Results and Discussions

3.1 Measuring cell device In this paper, the influence of the temperature and the contact pressure on the TCR is studied by using the LPM. A spring is used to produce the pressure on the specimens. The contact force could be obtained according to the equation F=μ×S, where μ is the elastic coefficient and S is the compression length. It should be noted that the contact pressure is calculated by the formula, not measured by a force sensor. This is because that the general force sensor on the market cannot meet the experimental requirements, and the applicable temperature range of the force sensors is generally higher than 150 K. Moreover, the size of the force sensors is too large to place in the specimen holder, and the force sensor will block the light of the heating laser or the probe laser. The schematic diagram of the contact pressure is depicted in Fig. 2. Specimen 2

Specimen 1

Specimen holder

Bolt

Probe laser beam

Heating laser beam Hole 1 Ring Hole 2

Refrigerator cold head 2

Spring

Fig. 2. Schematic diagram of the contact pressure -5-

In order to ensure the contact pressure distributed on the surface of the specimen uniformly. A ring is placed between the specimen 1 and the spring. The material of the ring is Poly-Tetra-Fluoro-Ethylene (PTFE) which has low thermal conductivity, so the existence of the ring does not affect the calculation of heat transfer. Vacuum Chamber

Second-stage Radiation Shield First-stage Radiation Shield Specimen1

Specimen 2

Thermocouple 1 2 3 4

Refrigerator Cold Head 2

Refrigerator Cold Head 1

5

Concentrator

6

GM Refrigerator

Data Acquisition System Inlet

Outlet Valve Vacuum Pump

Fig. 3. Schematic diagram of LPM experiment rig The schematic experimental setup of the LPM is shown in Fig. 3. The experiments were conducted in a vacuum chamber, and the vacuum was better than 0.1 Pa. The specimen holder was mounted on the refrigerator cold head 2. Two stage radiation shields were placed on the two refrigerator cold heads respectively. The shields can reduce the affection of the external radiation during the experiment. The difference of the temperature in two radiation shields is smaller than 3 K. This shows that the environmental temperature can be controlled in an exact manner. Some thermocouples were placed in order to measure the experiment temperature. Note that two thermocouples of the Ni-Cr/Cu+0.13% Fe (at) were placed to detect the temperature of the specimen 1, 2 respectively. Meanwhile a Cu/Cu-Ni thermocouple was placed to detect the temperature of the specimen holder. In addition，the refrigerator cold head 2 was equipped with a thermocouple of Ni-Cr/Cu+0.13% Fe (at). The temperature of first-stage and second -stage radiation shields were tested using a Cu/Cu-Ni thermocouple respectively. -6-

As it is known, the applicable temperature the Ni-Cr/Cu+0.13% Fe (at) thermocouple is in the range of 1 K to 275 K and the time delay is shorter than 0.1 s. The Cu/Cu-Ni thermocouple covers the temperature range from 70 K to 600 K. So the Ni-Cr/Cu+0.13% Fe (at) thermocouple is much more expensive than the Cu/Cu-Ni thermocouple. Therefore, the Ni-Cr/Cu+0.13% Fe (at) thermocouples were just used for measuring the lower temperature. In order to achieve high precision, the thermocouples were calibrated in ice-water mixture, which temperature was used as the reference point for measuring. The values of each temperature measuring point were recorded by the data-acquisition system, and the monitor is Keithley 2700. The reflected signal of the probe laser was received by a silicon photodiode, which was used to transform the optical signals to the electrical signal. Then a Lock-In Amplifier (LIA) of SR830 would receive the electrical signals, as depicted in Fig. 1. The phase lag and the amplitude of the signal was displayed on the LIA. Fig. 4 shows the LPM test rig.

Fig. 4 The LPM test rig

3.2 Experimental results and discussion The formulas of the thermal parameters were derived basing on that the specimen had been in the thermal equilibrium. For example, the specimen was considered working in the thermal equilibrium state when the thermal contact conductance did not vary more than 1%in 1 h, and we call this measurement method as steady-state method -7-

[17]. In other words, when the fluctuation of the temperature was smaller than 1 K, the specimen was considered reaching the thermal equilibrium. The LPM bases on that the reflectance of solid surface obeys the linear relationship with the temperature fluctuation. The temperature variation of the surface can be obtained by analyzing the information of the reflected laser beam. The LIA detected the phase lag and the modulation frequency of the reflected laser signals. Then the thermal diffusivity and the TCR can be calculated according to Eqs. (1) and (2) respectively. Under experimental conditions, when the modulation frequency was higher than 600 Hz, the relation between the phase lag and the modulation frequency was linear, φ=0.0001f+0.6804. For convenience, the value of the modulation frequency should be larger than 600 Hz when the LPM was employed to measure the thermal parameters. The specimens were done some pre-processing to improve the precision of measurements. The specimens were processed into thin circular pieces by wire-electrode cutting technology. The diameters of the specimens were all 10 mm. The surface of the specimens was polished, and the roughness was on the order of 0.1 μm. The specifications of the specimens are listed in table 1. Meanwhile in order to absorb the power of the heating laser immediately, the heating surface of the specimen 1 was coated with carbon film about 40 nm thick by the magnetron sputtering technology. The detection surface of the specimen 2 was deposited with a gold film about 0.3 μm thick by the evaporation technology to increase the signal-to-noise ratio. The specimens are shown in Fig.5.

(a) AlN ceramic

(b) AlN ceramic with the carbon film

(c) OFC with the gold film

(d) SS 304

Fig.5 . Specimens

In the temperature range of 70-300 K, the thermal diffusivity of single specimen was measured by the LPM based on two purposes: ⑴assess the reliability and validity of the LPM; ⑵ provide the thermal diffusivity of materials for cryogenic applications. Firstly, the thermal diffusivity of OFC was measured. Secondly, the thermal diffusivity of SS304 was measured and the fitting formula was obtained. -8-

The thermal diffusivity of OFC is shown in Fig. 6. The experimental data and the reference values [18] are shown in the figure for comparison. We can see that the experimental data were in consistency with the reference values at different temperatures. This shows that the technique is reliable in measuring thermal diffusivity using the LPM. The average relative error between the experimental data and reference values is 6.7%. 0.01

28 Reference values

Experimental data

Relative error

14

0.001

7

Relative error/%

Thermal diffusivity/m2 ·s -1

21

0

-7

0.0001

-14 20

60

100

140

180

220

260

Temperature/K

Fig. 6. Thermal diffusivity of OFC

For SS304, the reason that the thermal diffusivity decreases with the temperature is different from other two materials. The thermal conductivity of SS304 increases slightly with the increase of temperature. So the thermal diffusivity is mainly associated with the specific heat capacity and the density. The density of SS304 decreases with the increase of temperature, while the specific heat capacity increases in the meantime. At the low temperature, the change of the specific heat capacity is more remarkable than that of the density. But the specific heat capacity will change little when the temperature is higher than a certain value (about 250 K), while the density is continuing to decline. So the thermal diffusivity of SS304 decreased with the increase of temperature and the change of the thermal diffusivity was obvious at the lower temperature. But at the higher temperature, the thermal diffusivity has the tendency of increasing gradually. log10（α）=1.1524[log10（T）]2-5.4447[log10（T）]+0.9734

-9-

(3)

The experimental data of thermal diffusivity of SS 304 is shown in Fig. 7 (a). The experimental data and the reference values [19] are shown in Fig. 7 (b) for comparison. In the temperature range of 30-260 K, the thermal diffusivity of SS 304 decreased with the increase of the temperature. From 30 K to 50 K the thermal diffusivity of SS 304 changed fiercely. The Cryogenics Technologies Group measured the thermal conductivity and the specific heat of SS 304 [20], and results showed that the thermal conductivity and the specific heat of SS 304 followed

1.00E-05 Experimental data Thermal diffusivity/m 2 ·s-1

Thermal diffusivity/m 2 ·s-1

2.50E-05

2.00E-05 1.50E-05

1.00E-05 5.00E-06

Reference values

8.00E-06

6.00E-06 4.00E-06 2.00E-06

0.00E+00

0.00E+00 25

65

105

145 185 Temperature/K

225

265

50

(a)

100

150 200 Temperature/K

250

300

(b) Fig. 7 Thermal diffusivity of SS 304

the exponential law of 10m. According to the defined expression and the density of SS 304 remains the same, α=k/ρc, the thermal diffusivity of SS 304 followed the exponential law of 10n. m and n are the functions of log10T. So the logarithm of the thermal diffusivity of SS 304 is the polynomial function of the logarithm of the temperature and the simulation curve was shown in Fig. 8. Eq. (3) is the fitting formula he logarithmic of the thermal diffusivity of SS 304 and the logarithm of the temperature. The fitting relative coefficient was 0.9426. The density is only a weak function of temperature, so the temperature dependence of the thermal diffusivity is determined by the thermal conductivity and the specific heat.

- 10 -

log10 (T) -2 -2.5 1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

-3

log10 (α)

-3.5 -4

Experimental data Simulation curve

-4.5 -5 -5.5 -6 -6.5 -7

Fig. 8 Thermal diffusivity of SS304. The x-axis represents the logarithmic of the temperature, and the y-axis represents the logarithmic of the thermal diffusivity of SS 304. ◆represents the experimental data, and the solid line represents the simulation curve.

3.2.2 TCR of Two-layer system In this paper, the TCRs of OFC-AlN ceramic, SS304-AlN ceramic, SS304-SS304, AlN-AlN ceramic were measured by the LPM in the temperature range from 70 K to 300 K. Then the comparison between the TCRs of OFC-AlN ceramic, SS304-AlN ceramic and AlN-AlN ceramic was done. Meanwhile the thermal conductivity, specific heat and density of OFC, AlN ceramic, SS304 were listed in Tab. 2. Because the cost of gliding is high, so in the experiments of measuring the TCRs between OFC and AlN ceramic, between SS304 and SS304, between AlN-AlN ceramic, the AlN ceramic was placed beside the probe laser. In the temperature range of 70-300 K, the TCR of OFC-AlN ceramic are depicted in Figs. 9 and 10 from 0.20 MPa to 0.57 MPa. The TCR decreased with the increase of the temperature and the contact pressure. The maximum value was 1.05×10-5 m2·K·W-1, while the minimum value was 1.21×10-6 m2·K·W-1. The TCR and the temperature followed an exponential relationship. And the fitting formulas are listed in Tab. 3 and the correlation coefficients were better than 0.94. So the fitting curve agrees well with the experimental data. And the TCR followed a linear relationship with the contact pressure, as listed in Tab. 4 and the correlation coefficients were better than 0.94. So the empirical formula could well descript the relationship between the TCR and the contact pressure.

- 11 -

1.20E-05 0.20MP a

1.00E-05

0.34MP a

TCR/m 2 ·K·W -1

0.42MP a

8.00E-06

0.48MP a 0.57MP a

6.00E-06 4.00E-06 2.00E-06 0.00E+00 0

50

100

150

200

250

300

350

T emperature/K

Fig. 9 Variation of TCR of OFC-AlN ceramic with the temperature

As seen in Fig. 9, the TCR of OFC-AlN ceramic decreased with the increase of temperature. For calculation of the TCR the thermal conductivity and specific heat capacity are unknown, and they are extracted in reference [21]. When the temperature was lower than 100 K, the TCR would change fiercely with the temperature. The similar phenomenon occurred in another system of SS304-OFC [14]. In general, the temperature, contact pressure, surface morphology and surface deformation influence the heat flow of different contacting materials. To a large extent, the TCR is related to the thermal conductivity of the bilateral materials. The heat conduction in solids mainly depends on the electrons and phonons. Among them, the heat transfer in metals mainly by electron motion. To insulators or semiconductors, the phonons are considered to be the main hot-carriers. At lower temperatures, for OFC and AlN ceramic, increasing the temperature can enhance the heat transfer because of the improved electron motion. The thermal conductivity of OFC decreases with the increase of temperature when the temperature is above about 25 K. Especially, the value is sharply reduced in the temperature range from 25 K to 70 K [22]. Meanwhile, the thermal conductivity of SS304 is increased with the increase of temperature. But the thermal conductivity of OFC is far higher than that of SS304 in the whole experimental temperature range. Therefore, the value of TCR between OFC and AlN ceramic is mainly influenced by the characteristics of SS304. This is the reason that the TCR of OFC-AlN ceramic decreases with the temperature. AlN ceramic is a kind of electrical insulator, so the heat is mainly transported by phonons. The thermal conductivity of AlN ceramic is higher than that of other common electrical insulation materials. The theoretical value of its thermal - 12 -

conductivity can achieve 320 W/ (m·K) at room temperature [23]. But because of the influence of the impurities and lattice defects, the actual value of thermal conductivity is less than the theoretical value. Among these reasons, the thermal diffusivity of AlN ceramic is significantly decreased by the addition of SO2 [24]. For AlN ceramic, the thermal conductivity is mainly influenced by the phonon mean free path. At the lower temperature (< 20 K), the influence of phonon scattering on the mean free path is weak. The thermal conductivity of AlN ceramic with the change of temperature is mainly corresponded to phonon heat capacity. So the value of the thermal conductivity follows a cubic relationship with the temperature then. When the temperature is higher than Debye temperature, the phonon mean free path is proportional to the multiplicative inverse of the temperature. The thermal conductivity of AlN ceramic decreases with the increase of temperature at the higher temperature. For the same reason, the thermal conductivity of AlN ceramic is far higher than that of SS304 in the whole experiment temperature range. Fig. 10 shows the variation of the TCR between OFC and AlN ceramic at the temperature range from 71 K to 290 K. The TCR decreased with the increase of the contact pressure. The experimental results showed that the relationship between the TCR and contact pressure was nearly linear. The reason that the TCR appeared is the imperfect contact between materials. Increasing the contact pressure can enlarge the actual contact area, which causes the TCR to decrease. 1.20E-05 71K

TCR/m2 ·K·W -1

1.00E-05

85K 100K

8.00E-06

120K 150K

6.00E-06

200K

4.00E-06

250K 290K

2.00E-06 0.00E+00

0

0.1

0.2

0.3

0.4

0.5

0.6

Loading Pressure/MPa Fig. 10 Variation of TCR of OFC-AlN ceramic with the contact pressure

Fig. 11 shows the experimental data of the TCR of SS304-AlN ceramic at the contact pressure of 0.35 MPa. The - 13 -

SS-AlN ceramic TCR decreased with the increase of the temperature. The experimental data were fitted. The relationship between TCR and temperature obeyed an exponentiation relation with R=0.0014T-0.9479, and the correlation coefficient was 0.9873. So the empirical formulas could well descript the experimental data. 3.00E-05 Experimental data Fitting curve

2.00E-05

2

TCR/m ·K·W

-1

2.50E-05

1.50E-05

1.00E-05

5.00E-06 50

100

150

200

250

300

Temperature/K Fig. 11. Variation of the SS304-AlN ceramic TCR with the temperature (0.35 MPa)

The TCR of SS304-SS304 is shown in Fig. 12 and the contact pressure is 0.92 MPa. In the temperature range of 76-263 K, the TCR decreased with the increasing of the temperature with R=0.4041T-1.5391. And the correlation coefficient was 0.9667. So the fitting curve agreed well the experimental data.

2

TCR/m ·K·W

-1

4.00E-04 3.50E-04

Experimental data

3.00E-04

Fitting curve

2.50E-04 2.00E-04 1.50E-04 1.00E-04 5.00E-05 0.00E+00 50

100

150

200

250

Temperature/K

Fig. 12. Variation of the SS304-SS304 TCR with the temperature (1.02 MPa)

The TCR of AlN-AlN ceramic is shown in Fig. 13 and the contact pressure is 0.51 MPa and 0.76 MPa. From 70 K to 240 K, the TCR firstly decreased and then increased with increasing of the temperature. The TCR reached a - 14 -

minimum value at 140-157 K. 1.30E-04 1.20E-04 0.51MPa

TCR/m2·K·W-1

1.10E-04

0.76MPa

1.00E-04 9.00E-05 8.00E-05 7.00E-05 6.00E-05 5.00E-05 4.00E-05 40

90

140

190

240

Temperature/K Fig. 13. Variation of the AlN-AlN ceramic TCR with the temperature

3.2.3 Experimental error As shown in Eq. (2), the error in TCR determined by the shift of phase lag can not be negligible. Take the derivative of R：   C 2  2 1 +  s e c   C 1   R'= 2  f  C 2  1 - t an   -

R = 

(4)

Eq. (3) multiplied by1/R：

R =

sec2  - 

tan  -  1- tan  -  

=

R R 

Then：

 R  =

R = R R

(5)

Eq. (5) is the measurement error of TCR induced by the shift of phase lag. Similarly, the measurement error of thermal diffusivity induced by the shift of phase lag can be derived.

   =





=

(6)

The followings are the measurement errors of instruments: (1) Error in temperature measurement: The Ni-Cr/Cu+0.13% Fe (at) thermocouple and Cu/Cu-Ni thermocouple were employed. The diameters of thermocouples are 0.2 mm and the measurement accuracy is 0.1 K. So the Error in - 15 -

temperature measurement is as Eq. (7). T =

T T

(7)

(2) The caliper was employed to measure the thickness of the specimen and the measurement accuracy is 0.02 mm. L =

L L

(8)

The above errors are not related, so the transfer functions of all errors are 1. According to the principle of error propagation [25] the experimental error can be derived.  = T2 +2R +2 +2L

(9)

4. Conclusions

In this paper, the LPM which can complete the measurement in a short time was employed to measure the thermal diffusivity of SS304, OFC and AlN ceramic. Compared with the reference values, it showed that the LPM is reliable in measuring thermal parameters. Then the TCRs of OFC-AlN ceramic, SS304-AlN ceramic, SS304-SS304 and AlN ceramic were measured in the temperature range from 70 K to 300 K. For the changing trend of TCRs with the increase of temperature, some empirical formulas of the TCRs were established to fit the experimental results, and the correlation coefficients were all higher than 0.91. This shows that the formulas agreed well with the experimental data in the whole temperature range. The experimental error consisted of the function error and the measurement errors of instruments.

Acknowledgements

The research was sponsored by China National Natural Science Fund (No.51406108).

References

[1] J. H. Zhao, S. Nagao, Z. L. Zhang, Loading and unloading of a spherical contact: From elastic to elastic-perfectly plastic materials, Int. J. Mech. Sci. 56(2012) 70-76. [2] E. T. Swartz, R. O. Pohl, Thermal resistance at interfaces, Appl. Phys. Lett. 51(1987) 2200-2202. [3] R. S. Prasher, X. J. Hu, Y. Chalopin, N. Mingo, K. Lofgreen, S. Volz, F. Cleri, P. Keblinski, Turning carbon - 16 -

nanotubes from exceptional heat conductors into insulators, Phys. Rev. Lett. 102(2009) 105901-105905. [4] F. Gong1, Y. S. Tam; S. T. Nguyen; H. M. Duong , Prediction of thermal resistances and heat conduction of carbon nanotube aerogels in various permeated gases, Chem. Phys. Lett. 627(2015) 116-120. [5] Z. G. Jiang, Y. Z. Lia, W. G. Le, H. Tan, An improved thermal contact resistance model for pressed contacts and its application analysis of bonded joints, Int. J. Heat Mass Tran. 61(2014) 133-142. [6] A. R. Warrier, K.G. Deepa, T. Sebastian, C. S. Kartha, K. P. Vijayakumar, Non-destructive evaluation of carrier transport properties in CuInS2 and CuInSe2 thin films using photothermal deflection technique, Thin Solid Films 518(2010) 1767-1773. [7] F. Agresti, A. Ferrario, S. Boldrini, A. Miozzo, F. Montagner, S. Barison, C. Pagura, M. Fabrizio, Temperature controlled photoacoustic device for thermal diffusivity measurements of liquids and nanofluids, Thermochimica Acta 619(2015) 48-52. [8] V. Eremenko, V. Sirenko, V. Ibulaev, J. Bartolomé, A. Arauzo, G. Reményi, Heat capacity, thermal expansion and pressure derivative of critical temperature at the superconducting and charge density wave (CDW) transitions in NbSe2, Phys. C 469(2009) 259-264. [9] P. Korpiun, R. Osiander, 3 rd edn. (Springer, Heidelberg, 1992) [10] D. Almond, P. Patel, Photothermal Science and Technique. (Great Britain by St Edmundsbury Press, London, 1996) [11] M. Talha, C.K. Behera, O.P. Sinha, Effect of nitrogen and cold working on structural and mechanical behavior of Ni-free nitrogen containing austenitic stainless steels for biomedical applications, Mat Sci Eng C 47(2015) 196-203. [12] S. T. Wen, Y. Tan, S. Shi, W. Dong, D. C. Jiang, J. Liao, Z. Zhu, Thermal contact resistance between the surfaces of silicon and copper crucible during electron beam melting, Int J Therm Sci 73(2013) 37-43. [13] S. C. Somé, D. Delaunay, J. Faraj, J. L. Bailleul, N. Boyard, S. Quilliet, Modeling of the thermal contact resistance time evolution at polymeremold interface during injection molding: Effect of polymers' solidification, - 17 -

Appl. Therm. Eng. 84(2015) 150-157. [14] D. M. Bi, H. X. Chen, Y. Tian, Influences of temperature and contact pressure on thermal contact resistance at interfaces at cryogenic temperatures, Cryogenics 52(2012) 403-409. [15] Y. Ohsone, G. Wu, D. Dryden, Optical measurement of thermal contact conductance between wafer-like thin solid samples, ASME 121(1999) 954-961. [16] V. Suarez, J. Hernández Wong, U. Nogal, A. Calderón, J.B. Rojas-Trigos, A.G. Juárez, E. Marín, Study of the heat transfer in solids using infrared photothermal radiometry and simulation by COMSOL Multiphysics, Appl. Radiat Isotopes 83(2014) 260-263. [17] C. Ding, R. S. Wang, Thermal contact conductance of stainless steel-GFRP interface under vacuum environment, Exp. Therm. Fluid Sci. 42(2012) 1-5. [18] H. Morkoç, Handbook of Nitride Semiconductors and Devices, Materials Properties, Physics and Growth, (2008) [19] V.S. Arpaci, S.H. Kao, A. Selamet, Introduction to Heat Transfer, Prentice-Hall, NJ, 1999, pp. 579-581. [20]

Cryogenics

Technologies

Group.

Material

Properties:

304

Stainless

(UNS

S30400).

http://cryogenics.nist.gov/MPropsMAY/materialproperties.htm. [21] G. B. Chen, 1st edn. (Zhejiang University Press, Zhejiang, 1998) [22] R. Berman, D. K. C. MacDonald, in Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, London, 1952, edited by the Royal Society, p.122 [23] V. Gedama, A. Pansaria, A. K. Sinhab, B. K. Sahoo, The effect of macroscopic polarization on intrinsic and extrinsic thermal conductivities of AlN, J. Phys. Chem. Solids 78(2015) 59-64. [24] Y. Zhou, H. Hyuga, D. Kusano1, Yu-ichi Yoshizawa, T. Ohji, K. Hirao, Development of high-thermal-conductivity silicon nitride ceramics, J. As. Cer. Soc. 3(2015) 221-229. [25] J. R. Taylor, An introduction to error analysis: The study of uncertainties in physical measurements, University Science Books 1997, United States.

- 18 -

Table(s)

Table 1 Specifications of the specimens Category

OFC

Thickness

Surface roughness

/mm

of both parallel planes

0.26

0.8 μm /0.3 μm

Composition

99.95% Cu N>32.5-33;

AlN

0.3

0.6 μm /0.3 μm

O<1.0-1.5; C<0.1

SS304

0.6

0.5 μm /0.4 μm

OCr18Ni9

Table 2 Thermal parameters of the specimens

比热/(J·kg-1)

温度

热导率/(W·m-1·K-1)

OFC

AlN 陶瓷

SS 304

OFC

AlN 陶瓷

SS 304

OFC

AlN 陶瓷

SS 304

30

38

59

40

5.4

103

1300

111

6.52

40

50

70

70

10.13

167

1150

155.6

7.4

50

55

77

100

22.11

191.5

1000

174.2

7.89

59

65

81

127

60.33

205

865

211.4

8.17

70

70

85

161

58.021

217

700

230

8.45

77

75

90

183

69.99

232

595

248.6

8.8

81

77

100

196

74.78

262

545

256

9.4

85

80

110

210

82.37

280

475

265

9.8

90

85

120

226

96

305

500

284

10

100

90

148

254

110.09

360.8

450

300

10.76

110

100

150

254

139.74

364

450

320

10.8

120

110

153

290

170.91

368.2

440

335

10.89

148

120

195

321

203.31

413.5

426

341

12.15

150

130

200

323

333.4

417

425

344

12.3

153

150

204

331.7

333.41

419.8

423.5

350

12.42

195

170

208

354

374.85

422.6

417.5

330

12.54

198

183

249

355

459.55

455.2

417.2

317

13.58

200

190

250

356

465.09

456

417

310

13.6

204

200

282

358

473.01

456

416.6

300

14.55

208

210

285

359

489.19

456

416.2

290

14.62

250

221

290

376

525.06

456

410

279

14.73

282

231

294

381.5

557.66

456

403.7

269

14.83

290

250

383.3

619.61

404.9

244

294

290

384.3

720.92

409.6

200

Table 3 Empirical formula of the TCR Contact pressure

Empirical formula

Correlation coefficient

0.20 MPa

R=0.0001T-0.5425

0.9473

0.34 MPa

R=0.0001T-0.5603

0.9804

0.42 MPa

R=0.0002T-0.7903

0.9820

0.48 MPa

R=0.0002T-0.7545

0.9780

0.57 MPa

R=0.0002T-0.8826

0.9457

Table 4 Fitting expressions of TCR of OFC-AlN ceramic Temperature

Empirical formula

Correlation coefficient

71 K

R=10-5(－P+1)

0.9877

85 K

R=10-5(－P+1）

0.9837

100 K

R=10-5(－P+1）

0.9508

120 K

R=10-5(－P+1）

0.9415

150 K

R=10-5(－P+0.9）

0.9228

200 K

R=10-5(－P+0.7）

0.9083

290 K

R=10-5(－P+0.6）

0.9452

Highlights

1. Thermal diffusivity of oxygen-free copper and stainless steel was measured by the LPM. 2. Thermal contact resistance between thin films was measured by the LPM. 3. Empirical formulas of thermal contact resistance are given. 4. The experimental error was analyzed.

- 19 -