Simultaneous measurement of thermal diffusivity, heat capacity, and thermal conductivity by Fourier transform thermal analysis

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1 High Temperatures ^ High Pressures, 2001, volume 33, pages 387 ^ ECTP Proceedings pages 955 ^ 963 DOI: /htwu377 Simultaneous measurement of thermal diffusivity, heat capacity, and thermal conductivity by Fourier transform thermal analysis Junko Morikawa, Toshimasa Hashimoto Tokyo Institute of Technology, O-okayama, Meguro-ku, Tokyo , Japan; fax: ; toshimas@o.cc.titech.ac.jp, jmorikaw@o.cc.titech.ac.jp Akikazu Maesono Ulvac Sinku-Riko Inc., Hakusan, Midori-ku, Yokohama , Japan; fax: ; maesono@ulvac-riko.co.jp Presented at the 15th European Conference on Thermophysical Properties, Wu«rzburg, Germany, 5 ^ 9 September 1999 Abstract. The Fourier analysis of higher-order harmonics of a temperature wave provides a new technique for simultaneous measurement of thermal properties of thin films as a function of frequency under temperature scanning. The mathematical formula for the propagation of a square wave is examined first in comparison with that of a sine wave and the principle of Fourier transform thermal analysis is validated experimentally. Results on thermoplastics are shown and the glass transition and crystallisation are discussed. The features are summarised as follows: it employs a thin high-sensitivity sensor with negligible heat capacity; the actual temperature wave that is propagating in the specimen can be directly observed; the thermal properties of thin film specimens (thickness of the order of micrometres) can be determined nondestructively at a single time measurement. The principle applied to the technique, called `Fourier transform thermal analysis', makes it possible to determine simultaneously thermal diffusivity, heat capacity per unit volume, and thermal conductivity as a function of frequency and temperature. 1 Introduction Temperature modulation techniques have been well established (Parker et al 1961; Sullivan and Seidel 1968; Rosencwaig and Gersho 1976; Adams and Kirkbright 1977; Birge and Nagel 1985; Hashimoto et al 1990; Reading et al 1992, 1993; Hashimoto and Tsuji 1992, 1993; Morikawa and Hashimoto 1998) and utilised to obtain thermal properties during the last three decades. The significance of dynamic thermal properties has been recognised especially for the imaginary part of the complex thermal property. However, at this stage a simultaneous measurement of different kinds of thermal properties, such as heat capacity and thermal diffusivity, at plural frequencies under temperature scanning, is necessary for further study. The present work shows that the observation of higher-order harmonics of the temperature wave propagating in a specimen begins to solve these problems. The higher-order harmonics of the temperature wave are successfully detected by inputting a square pulse train with a variable duty factor. The generated temperature wave is measured with a sensor located at a certain distance in the thickness direction. Fourier transform thermal analysis has been applied to glass transition and crystallisation of thermoplastics with simultaneous measurement of thermal diffusivity, heat capacity per volume, and thermal conductivity as a function of frequency and temperature in single time measurement. 2 Principle In order to analyse the propagation of the temperature wave, a one-dimensional heat conduction model is assumed (Carslaw and Jaeger 1946), as depicted in figure 1. The film specimen with thickness d is attached to substrates at x ˆ 0 and x ˆ d. The thermal

2 388 J Morikawa, T Hashimoto, A Maesono 15 ECTP Proceedings page 1342 heater sensor substrate specimen substrate flow vertical to the thickness direction is neglected because the area of the gold-sputtered sensor, 1 4 mm 2, is much larger than the thickness of the specimen, d 50 mm, which is described in detail by Morikawa et al (1995). The one-dimensional heat diffusion equation is: qt qt ˆ a q2 T qx, 0 < x < d, 2 qt qt ˆ a q 2 T s, x < 0 and d < x, (1) 2 qx where T is modulated temperature, t is time, a is the thermal diffusivity of the specimen, and a s is that of the substrate. The thermal wave in the substrate is assumed to decay to become negligible, because the substrate is thick enough (Pyrex, 2 mm). It is assumed that the substrate is infinite, so the temperature oscillations at infinity are: T x! 1, t ˆ0, T x! 1, t ˆ0. (2) The boundary conditions at x ˆ 0 and x ˆ d are: T x! 0, t ˆT x! 0, t, T x! d 0, t ˆT x! d 0, t. (3) It is assumed that there is no contact resistance between the gold-sputtered layer (sensor and heater) and the specimen or the substrate. The sputtered layer is much thinner ( 100 Ð) than the specimen thickness d, so the temperature is assumed continuous at the boundary of the specimen and the substrate. The thermal balances at the interface of the specimen and the substrate are: qt l s l qt ˆ j t ˆj qx qx 0 exp iot, l qt qx x! 0 x!d 0 x ˆ 0 qt l s qx x ˆ d x! 0 x!d 0 x Figure 1. Schematic diagram of a mathematical model: specimen, heater (at x ˆ 0), sensor (at x ˆ d ), and an infinite substrate. ˆ 0, (4) where l and l s are the thermal conductivity of the specimen and the substrate, respectively, and j(t) is a periodic heat flux. When a temperature wave is generated on the front surface (x ˆ 0), it propagates in the thickness direction and is detected on the rear surface (x ˆ d ). The temperature oscillation at the rear surface, T(d, t), is obtained by solving

3 Simultaneous measurement by Fourier transform thermal analysis ECTP Proceedings page 1343 the one-dimensional thermal diffusion equation, with boundary conditions (2) ^ (4). This is described as follows (Hatta et al 1988; Kato et al 1993; Morikawa et al 1995): T d, t ˆ f j 0 exp iot = 1 i Šg exp i 1 kd Š f lk l s k s 2 lk l s k s 2 exp 2 i 1 kd Šg=2lk, (5) where k is (o=2a), and subscript s means the plate substrate. When a square pulse train with constant angular frequency o and pulse height V 0 is input to a heater, a temperature wave consisting of multiple frequencies is generated by AC Joule heating. The input voltage, V(t), is described as a function of time, as follows: V t ˆ V0, 2mp < ot < 2 m a p, (6) 0, 2 m a p < ot < 2 m 1 p, where a (0 < a < 1) is a duty factor and m is a natural number. The Fourier transform of V(t) is derived as follows: ( V t ˆV 0 a X1 n ˆ 1 n 6ˆ 0 sin anp np ) exp i not anp Š, (7) where n is an integer. If kd 4 1 or kl k s l s 0, and assuming a linear combination of each harmonic, T(d, t) is derived by solving the one-dimensional thermal diffusion equation with boundary conditions described above, as follows: 2ca T d, t ˆ ca c s as j 2 0 exp i not X 1 n ˆ 1 n6ˆ0 sin anp np 1 d p anp 2a 4 exp d 2a, (8) where c is heat capacity per unit volume. The phase delay (Dy n ) and the amplitude (A n ) of each nth order harmonic are derived as follows: Dy n ˆ d p anp, (9) 2a 4 and 2ca A n ˆ B sin anp 1 ca c s as 2 np exp d, (10) 2a where B is defined as B ˆ Aj 0, and A is a conversion factor of temperature and voltage which is defined in equation (11) of Morikawa et al (1995). If B is previously obtained as a function of temperature by the use of a standard material, such as Pyrex 7740, then c is obtained from the amplitude of the detected signal. It is noteworthy that the phase shift Dy directly gives the thermal diffusivity a without any standard material. Thermal conductivity l is calculated from the relationship l ˆ ac. As a consequence, a, c, and l are simultaneously obtained as a function of temperature. 3 Experimental Figure 2a shows the schematic diagram of the measurement system. The detail is described by Maesono et al (2001). The basic measurement system consists of a function synthesiser (NF Electronic Instruments, NF1946, Yokohama, Japan), a digital lock-in amplifier (Stanford Research, SR830, California, USA), a temperature controller (HPC7000), a specimen holder on a hot stage, and a personal computer. It is possible to input a square pulse train with frequency f ˆ 0:1 Hz ^ 100 khz and duty factor

4 390 J Morikawa, T Hashimoto, A Maesono 15 ECTP Proceedings page 1344 temperature controller 8C lock-in amplifier heater specimen signal input ref. in sensor heater cell R function synthesiser E GP IB GP IB RS-232C (a) PC sputtered gold layer substrate (Pyrex) lead to a sensor specimen lead to a heater spacer (b) Figure 2. (a) Schematic diagram of the measuring system for Fourier transform thermal analysis. (b) An example of a measurement specimen sputtered with gold sensor and heater. a ˆ 0:05 0:5 to the heater for the temperature modulation. The generated temperature wave was selected to be less than 0.1K on the front surface, and less than 10 mk on the rear surface. A 0:2Kmin 1 rate heating and cooling was used in order to minimise the temperature distribution in the specimen. A fast Fourier transform (FFT) analyser (Ono Sokki, CF360, Yokohama, Japan) was used to observe the waveform and the Fourier spectrum of the temperature wave on the sensor. The film specimens used in this study were borosilicate glass (Matsunami micro coverglass, Osaka, Japan) and poly(ethylene terephthalate) (PET) (Mitsui Chemical Co. Ltd, Tokyo, Japan ). Thin gold resistors controlled at about 50 O with rectangular areas of 1 4 mm 2 were sputtered on both surfaces of the specimen films, one as a heater

5 Simultaneous measurement by Fourier transform thermal analysis ECTP Proceedings page 1345 and the other as a sensor with negligible heat capacity (figure 2b). Because of the direct sputtering, good thermal contact between the specimen and the resistors was obtained. The specimen was inserted between substrate plates of Pyrex 7740, with a spacer to maintain a constant thickness, d. The precision of the measurement of Dy n or A n is less than 0.5% if the magnitude of the signal is above 0.5 mv. The standard deviation of the average value of thermal diffusivity is less than 2% including the error of the thickness measurement, and less than 5% for heat capacity per unit volume and thermal conductivity. 4 Results and discussion Figure 3 shows a typical waveform in borosilicate glass (78 mm), detected on the sensor, with fundamental frequency f ˆ 23 Hz (kd ˆ 1), duty factor a ˆ 0:5 and The signal is pre-amplified (gain ˆ 100) for input into the FFT analyser. The waveform varies with kd and it appears like a triangular wave when kd ˆ 1 in figure 3. The Fourier transform of temperature wave in figure 3 is shown in figure 4, in the frequency range up to 200 Hz. At frequencies of an integral multiple of the fundamental frequency, harmonics are observed. According to equation (10), no signal can be observed when sin (anp) is zero. Then in figure 4b, sin (np=4) becomes zero when n is a multiple of 4, so harmonics of numbers which are a multiple of 4 are not allowed. In order to improve the signal-to-noise ratio of the harmonic signals, a digital lock-in amplifier is required.the fundamental frequency was scanned from 2 Hz to 400 Hz in steps of 5 Hz and each component of harmonic signal, phase delay or amplitude, was detected, respectively. Figure 5 shows the relationship between the phase delay (Dy n ) and the square root of angular frequency of each harmonic. The phase delay, Dy n, shows good linearity as predicted in equation (9). In the case of a ˆ 0:125, seven split straight lines are observed, with a phase shift of p=8 between the adjacent harmonics. Each line T=arb. units a ˆ 0:5 a ˆ 0: t s t s (a) (b) Figure 3. Waveforms on the sensor in borosilicate glass, d ˆ 78 mm. The substrate is Pyrex 7740, the duty factor a is 0.5 or 0.25, and the fundamental frequency f is 23 Hz. 10 log [100T ac (d, t)=t ac (0, t)] 30 a ˆ 0: n ˆ 1 a ˆ 0:25 n ˆ n ˆ n ˆ 3 n ˆ n ˆ 5 n ˆ 5 80 n ˆ 6 80 n ˆ 7 n ˆ f Hz f Hz (a) (b) Figure 4. Fourier transform of waveform in borosilicate glass detected on the sensor. The frequency of the temperature wave is 23 Hz and the duty factor a is 0.5 or The nth order harmonic is not observed when sin (anp) ˆ 0.

6 392 J Morikawa, T Hashimoto, A Maesono 15 ECTP Proceedings page a ˆ 0:125 n ˆ 1,2,3,4,5,6,7 2 3 sinusoidal wave (Dy anp) rad 4 5 n ˆ (no) rad s Figure 5. Phase delay of each harmonic, Dy n, plotted against angular frequency, o n, of a temperature wave in borosilicate glass (80 mm). The phase shift, anp, can be observed between the adjacent harmonics. Duty factor a ˆ 0:125. The phase delay for the sinusoidal wave is also shown. corresponds, respectively, to the phase delay of the fundamental frequency, second harmonic, third, etc, up to seventh. The phase delay for the sine wave is also shown in figure 5 as a reference. The identical slope was also observed when the amplitude of each harmonic corrected by Fourier coefficient, as described in equations (11) and (12), was plotted against square root of angular frequency: An o ln V rad s ˆ d ln 2a F n F n ˆ sin anp np 2ca ca c s a s 2 V ln B=V WK 1 m 2, (11) 1. (12) jnj To confirm equations (9) and (10), these data were replotted as Dy anp,orln(a n o =F n ) versus (o n ) in figure 6, where linear master plots with identical slope were obtained. The slope leads to a thermal diffusivity of borosilicate glass, a ˆ 4: m 2 s 1 at 25 8C, with an accuracy within 0.1%. If ln (B) is obtained from the standard material, Pyrex 7740, where c s a s of the substrate is already known (Japan Society forthermophysical Properties 1990), the value of ca of borosilicate glass is 1: Jm 2 K 1 s. It was also confirmed that identical slopes as in figure 6 were obtained by varying the harmonic-order number at a fixed fundamental frequency. To obtain the thermal effusivity, ca, another method was also examined. When the heat flux j 0 increases, the amplitude A n increases following the rule described in equation (10). In figure 7, A n is plotted against j 0, when a ˆ 0:125, f ˆ 23 Hz, and n ˆ 1 7

7 Simultaneous measurement by Fourier transform thermal analysis ECTP Proceedings page a ˆ 0:125 n ˆ 1,2,3,4,5,6,7 2 Dy anp Fn ) Dy anp, ln (A n o n ln (A n o n =F n ) rad s o n Figure 6. Master plots of phase delay, Dy n, and amplitude, A n, of each harmonic in borosilicate glass based on equations (9) and (10). F n is defined in equation (12). 12 a ˆ 0:125 kd ˆ 1(f ˆ 23 Hz) 10 n ˆ 1 8 A n mv 6 4 n ˆ 2 2 n ˆ 3 n ˆ j 0 Wmm 2 Figure 7. The relationship between amplitude, A n, and heat flux, j 0, of each harmonic in borosilicate glass. The fundamental frequency is 23 Hz, kd ˆ 1, a ˆ 0:125, and n ˆ 1 7.

8 394 J Morikawa, T Hashimoto, A Maesono 15 ECTP Proceedings page 1348 for borosilicate glass, and the slope is a function of ca and c s a s. When the thermal effusivity of the reference material is already known, ca of the specimen can be obtained. When A n was corrected with respect to the Fourier coefficient, the master plot was also successfully obtained. Finally, it is possible to utilise this technique for the application of thermal analysis to temperature scanning. Equations (9) and (10) are rewritten as follows: nod 2 a ˆ 2 Dy n p4 2, (13) anp ca ˆf2D 12 D D 1 Š gc s a s, (14) where D ˆ exp n kd o s A F n s n exp ns k s d s o A n F n s. (15) Here Dy n and A n are detected as a function of temperature, and a and c are obtained as a function of temperature either on heating or cooling. Figure 8 shows the results for the simultaneous measurements of a, c, and l of PET under heating of 0:2 8C min 1. The glass transition and cold crystallisation are clearly observed l Wm 1 K a m 2 s 1 (a) c Jm 3 K 1 (b) T 8C (c) T 8C Figure 8. The simultaneous measurement of thermal diffusivity a, heat capacity per unit volume c, and thermal conductivity l, of PET under heating of 0:2 8C min 1 in a single time measurement. The detected frequencies (indicated) are 28, 56, 84, and 112 Hz when a ˆ 0:125, n ˆ 1 4, and d ˆ 44 mm. 5 Summary A proposed method designated `Fourier transform thermal analysis' has been experimentally verified to show that it is possible to obtain the frequency-dependent thermal properties from a single time ^ temperature scan. The detected harmonics were well fitted in the mathematical rule derived from the thermal diffusion equation. The principle was

9 Simultaneous measurement by Fourier transform thermal analysis ECTP Proceedings page 1349 also applied to the measurement on temperature scanning and the phase transitions of a thermoplastic were successfully detected. This technique will be utilised for research on nonreversible chemical reactions and phase transitions. The results indicate that the observed signals are exactly the harmonics of the temperature wave, propagating in the specimen, not caused by some electronic noise source. This measurement system is unique in that the sensor is situated at a distance d from the heater position, and the Fourier spectrum was detected successfully, leading to a frequency-dependent thermal property under temperature scanning. Both the frequency and the temperature dependence of a, c, and l were simultaneously obtained by a single time measurement on a small amount of valuable specimen during phase transitions. References Adams M J, Kirkbright G F, 1977 Analyst ^ 682 Birge N O, Nagel S R, 1985 Phys. Rev. Lett ^ 2677 Carslaw H S, Jaeger J C, 1946 Conduction of Heat in Solids (Oxford: Oxford University Press) Hashimoto T, Tsuji T, 1992, presented at the Tenth International Congress on Thermal Analysis, Hatfield, UK, 24 ^ 28 August Hashimoto T, Tsuji T, 1993 J. Therm. Anal ^ 726 Hashimoto T, Hagiwara A, Miyamoto A, 1990 Thermochim. Acta ^ 324 Hatta I, Sakakibara K J, Suzuki J, Yao H, 1988 Jpn. J. Appl. Phys ^ 2159 Japan Society for Thermophysical Properties, 1990 Thermophysical Properties Handbook (Tokyo: Yokendo) Kato R, Maesono A, Hatta I, 1993 Jpn. J. Appl. Phys ^ 3655 Maesono A, Takasaki Y, Maeda Y, Tye R P, Morikawa J, Hashimoto T, 2001 High Temp. ^ High Press. 33 (in press) Morikawa J, Hashimoto T, 1998 Jpn. J. Appl. Phys. 37 L1484 ^ L1487 Morikawa J, Tan J, Hashimoto T, 1995 Polymer ^ 4443 Parker W J, Jenkins R J, Butler C P, Abbott G L, 1961 J. Appl. Phys ^1687 Reading M, Elliott D, Hill V L, 1992, presented at the Tenth International Congress on Thermal Analysis, Hatfield, UK, 24 ^ 28 August Reading M, Elliott D, Hill V L, 1993 J. Therm. Anal ^955 Rosencwaig A, Gersho A, 1976 J. Appl. Phys ^ 69 Sullivan P F, Seidel G, 1968 Phys. Rev ^ 685

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