The thermal diffusivity of pyrolytic graphite at high temperatures

Co&n,

1972, Vol. IO, pp. 253-257.

Pergamon

Press.

Printed

m Great

Britain

THE THERMAL DIFFUSIVITY OF PYROLYTIC GRAPHITE AT HIGH TEMPERATURES TAKAHO TANAKA and HIROSHIGE SUZUKI National Institute for Researches in Inorganic Materials, Tokyo,Japan (Received 10Mnrch 1971) thermal diffusivity of three pyrolytic graphites was measured using the modulating electron beam method over the temperature range from 1300°C to 1900°C. One of the specimens was as-deposited at 2500X, and the others were the same heattreated at 3000°C after deposition. The mean free path of phonons was estimated by analysis of the experimental data. The mean free path seems to be indepevdent of temperature in the direction perpendicular to the deposition plane; being 4.8 A for the as-deposited specimen, 6.4 and 7.4 A4 for the two heat-treated ones, respectively. The temperature dependence of the mean free path parallel to the deposition plane cannot be expressed in analytical form. However, a monotonic decrease from 91 A at 1330°C: to 56 A at 1980°C is indicated by a tentative approximate calculation.

Abstract-The

1. INTRODUCTION The thermal conduction of pyrolytic graphite because of its high anisotropy is of theoretical interest and of practical importance. The measurements of thermal conductivity of pyrolytic graphite below room temperature were reported by a number of authors[l-61, but data at high temperature are relatively few [7,8]. The present paper deals with measurements of the anisotropic thermal diffusivity of pyrolytic graphite at high temperature. A rough estimate of the mean free path of phonons is obtained from the measured thermal diffusivities on the basis of the theories of Komatsu and Nagamiya [9, IO]. 2. EXPERIMENTAL 2.1. Specimens Measurements were made for three different specimens of pyrolytic graphite. One of them was an as-deposited at 2500°C material (PC-l),

and

the others

were

heat-treated

at

3000°C for 1 hr (PG-2) and for 5 hr (PG-3). Characteristics of these specimens are tabulated in Table 1.

2.2. Experiments The thermal diffusivity measurements were wholly carried out by the modulating electron beam method developed in Cowan [ 111 and Wheeler[l2]. A thin plate specimen mounted in vacuum is heated uniformly by bombardment of the surface with an electron beam. If the intensity of the electron beam is sinusoidally modulated with a frequency v, the phase difference A0 of the temperature variation between the bombarded and the back surface is related to the thermal diffusivity K by an equation [ 121, 2.9 d2v K=F

(1)

where d denotes the thickness of specimen. The apparatus used is schematically shown in Fig. 1. The electric current in the focusing coil is controlled by a low frequency oscillator to modulate the intensity of electron beam. Variation of the surface temperature is taken out as an electric signal by a photo-cell and a low frequency amplifier. The phase difference transformed into time interval 2.53

254

T. TANAKA

and H. SUZUKI

Table 1. Characteristics of pyroiytic graphite specimens

Specimen Deposition temperature f”c1

Heattreatment tempi;;a.;e

Apparent density (glen-4

Layer spacing 69

Electrical resistivity* (Q cm) RI R,

(“G fir) PG-1 PG2 PG-3

2500

30&l 3000,5

3::

2.20 2.25 2.25

3‘35

400 x 10-g 100 - 130X lO-‘j 100 - 130x 10-a

*RI, and R, are respectively the electrical resistivity paraIle1 and ~r~ndi~~~ar plane at room temperature.

Maindeflecting

0.5 0.26 - 0.3 0.26 - 0.3 to deposition

Average temperature was measured by an optical pyrometer corrected on the assump tion that emissivity was O-8 for all specimens.

Electron gun CO

3. RESULTS AND DISCUSSION

system

Fig. 1. Apparatus for thermal diffusivity measurements and the schematic diagram of the photocell circuit. is measured by an electronic counter which is regulated so as to start counting on receiving the signal from the bombarded surface and to stop cm receiving the signal from the other surface. Excremental conditions are shown in Table 2.

Measurements of thermal diffusivity in the direction parallel to the deposition plane were made for the heat-treated specimens, PC-2 and PG-3 (Fig. 2). The K vs. T plot: shows considerable scatter because of the difficulties connected with measuring such high diffusivity. The thermal diffusivity perpendicular to the deposition plane (Fig. 3) shows little temperature dependence. The anisotropy ratio KJ& is about 10% where Ka and K, are the thermal diffusivities parallel and perpendicular to the deposition plane, respectively. The differences in the thermal diffusivities between various specimens may be due to their different degrees of graphitization. Taylor[7] reported data on thermal conductivity of pyrolytic graphite up to SOO”K, and Pappis, et ai, [S] up to 22O@‘K, The values of conductivity calculated from

Table 2, Experimental Acceleration voltage Frequency of modufation ~eprodu~ibi~~t~ Specimens size

conditions

150 mA 3 - 4 kV: Total electron beam current 8 x IO-$ Torr O-5 Hz: Pressure of vacuum chamber 2 5% for measurement paralLI to deposition plane 2 1% for measurement perpendicular to deposition plane S$YX 1-5 mm for measurement parallel to deposition plane Sb, X 0.3 mm for measurement perpendicular to deposition plane

THERMAL

DIFFLJSIVITY

OF PYROLYTIC

255

GRAPHITE

into one out-of-plane mode and two in-plane modes along each principal axis. In the relaxation time approximation, the Iattice thermal conductivity K along a principal axis of the crystal lattice is expressed by

K=F

U:(k), Tj(k) . S,(k) = F v,(k) , l,(k) (2)

x Sdkf I

.

1200

1400

1605

l85a

2000

Temperatufe, T Fig. 2. Temperature dependence of the thermal diffusivity of pyrolytic graphite parallel to the deposition plane. OPT-1 ‘5%~2

xffi-3

where uj, T, 1 and S are vector components of the group velocity, relaxation time, mean free path (= VJ) and specific heat, respectively. The summation is taken over all modes of phonons. Each normal mode is designated by a wave vector k and a proper polarization branch j. The thermal diffusivity is related to the conductivity by an equation K

K=z S,(k)

‘i..

1200

14al

1600

1800

2000

Temperotur@, -c Fig. 3. Temperature dependence of the thermal diffusivity of pyrolytic graphite perpendicular to the deposition plane. the present diffusivity data agree, within an order of magnitude, with those obtained by an extrapolation of Taylor’s data to the temperature region of the present study. They are also comparable with those of Pappis, et al., except for K~ of annealed specimen. The Pappis’ data show that K~ is significantly lowered by annealing, which is opposite to our results. In order to discuss the thermal conduction in pyrolytic graphite, it is convenient to assume a layer system composed of elastically coupled planes. These planes may correspond to the deposition planes, in a rough approximation, and may hereafter be called basal planes. We define a-axis lying in the basal plane and c-axis perpendicular to it. In this scheme, three acoustic modes resolve

’

The specific heat of graphite is known to reach the classical Dulong-Petit value at about ZOOO”K, while the phonons effective in thermal conduction have relatively low energy. Thus, in the temperature range considered (from 1600°K to 2200°K) the thermal energy kT is higher than the energy ho of effective phonons. Therefore, the specific heat Sj(k) may be approximated by the Boltzmann constant k. On this basis,

F vj(k)b(k) K=

3nN

(4)

where n and N are the number of atoms per unit cell and the number of unit cells per unit volume of crystal, respectively. The group velocity and the mean free path are assumed to be independent of wave vector k. Then, t.he numerator of equation (4) becomes NC VjZj (j = 1,2,3) for the three acoustic modes and hence

T. TANAKA

256

Kc; 23: 3

Ujlj

(j=

1,2,3).

(5)

The mean free path of phonons at high temperature is mostly determined by the Umklapp processes for which Peierls [13] has given expressions I, = A . exp (B/bT)

T-
1, = B/T

T>tl

8

(64

and H. SUZUKI

too seriously but just as a parameter representing some aspect of heat transport in graphite. For the u-axis conduction, K;, is connected with an out-of-plane mode and with two inplane modes as stated above. If the mean free path 1, is assumed to be the same for all polarization branches j, equation (5) can be rewritten in the same form as equation (7):

W

where 8 is the Debye temperature, and A, b and B are parameters. For thermal conduction along the c-axis, the diffusivity shows little temperature dependence as shown in Fig. 3, which evidently corresponds to the case of T s=-0 in equation (6b). This may be due to the fact that the phonon density of the effective modes is so high that all phonons are scattered while they penetrate one or two basal planes along c-axis. Therefore, the mean free path will be comparable with the spacing of basal planes, and may be taken out of the summation as follows.

In c-direction the out-of-plane mode is longitudinal and the others are degenerate transverse in-plane modes. For as-deposited specimen the group velocities of those modes are presumed to be 3.7 X lo5 cm/set and 0.56 x lo5cm/set [9, 141 respectively, and for the annealed 4.1 X lo5 cmlsec and 0-56X lo5 cm/sec[9,15] respectively. By using these values of n, the mean free path 1, for PG-1 is computed to be 4.8 h;, and for PG-2 and PCS-3 to be 6-4 and 7.4 A, respectively. This result is in qualitative agreement with the preceding considerations. However, since the concept of phonons is defined on basis of weak interaction, it may not be appropriate enough to be used for such strong interaction. Therefore. I,- mavI be taken not

The Debye temperature for u-axis vibrational modes is about 25OO”K, and hence the temperature dependence of the mean free path cannot be expressed by such an analytical form as equation (6) in this case. In u-direction, the acoustic modes are in-plane longitudinal, in-plane transverse and outof-plane transverse, and their group velocities are 2-10 X lo6 cmisec, 1.23 X lo6 cm/set and 0.4 X 10” cm/set, respectively [9, lo]. It should be noted that there is a possibility of overestimation coming from the use of these group velocities for vj in equation (8). The numerical values of mean free path calculated in this way are shown in Fig. 4. They are smaller than the values obtained by extrapolation using the relation 1, = 9.0 X

L 4.4

47

50 Reciprocal

Fig. 4. Temperature

5.3

56 temperature,

59

62

6-5

xd7”K

dependence of the phonon mean free path along u-axis, as calculated by use of the equation (8) of the text.

THERMAL

DIFFLJSIVITY

lo-? exp (812-20 T) given by Taylor [7] for the temperature range T < 812. This may be explained as follows: at temperature exceeding @/2, the density of phonons whose wave vectors are large enough to bring about the Umklapp process increases so remarkably as to shorten the mean free path 1,. One may note that there are some diRerences in the c-axis mean free path for the different specimens PG-1, PG-2 and PG-3, whereas no appreciable difference in the thermal diffusivity along u-axis is found between PG-2 and PG-3. These observations can be explained by the turbostratic structure of pyrolytic graphite. In heat-treatment, the turbostratic structure is modified so as to exhibit a more tri-dimensional stacking order, and then the phonon energy will increase because of the increase in the interaction between the basal planes. This increases the group velocity and decreases the phonon density, leading to an increase of diffusivity. On the other hand, the structure of each plane is little influenced by l~eat-treatment time, and the mean free path of phonons along a-axis at such high temperature is much shorter than the crystalline size. Therefore, the thermal conduction along c-axis is affected by the degree of graphitization, while the u-axis conduction remains much the same. One should note that the Pappis’s case cannot be explained by this consideration. 5. CONCLUSION

Based on the phonon transport scheme, an analysis has been made of the thermal conduction in pyrolytic graphites at high temperature with reference to their aniso-

OF PYROLYTIC

GRAPHITE

257

tropy and to the effect of heat-treatment. The estimated mean free paths along c-axis are found to be comparable with the c-axis lattice spacing. For the n-axis conduction the estimated mean free paths are shorter than those extrapolated from ‘Taylor’s relation. The results are qualitatively consistent with considerations theoretical of the phonon scattering processes along the principal axes of pyrolytic graphite. Acknowledgements-The authors would like to express their appreciation to Dr. T. Iwata of Japan Atomic Energy Research Institute for supplying the pyrolytic graphite specimens. Thanks are also due to Mr. T. Ishii, Dr. M. Iwata and Mr. Y. Uemura for their valuable advice. REFERENCES 1. Slack G. A., Phys. Rev. 127,694 (1962). 2. Hooker C. N., Ubbelohde A. R. and Young D. A., Proc. Roy. Sac. A 276,83 (1965). 3. Hooker C. N., Ubbelohde A. R. and Young D. A., Proc. Hay. Sot. A 284, 17 ( 1965). 4. Holland M. G. and Klein C. A., R&f. rtm. Phys. Sot. 7, 191 (1962). 5. Holland M. G., Klein C. A. and Straub W. D., J. Phys. G&m. Solids 27,903 (1966). 6. Smith A. W. and Rasor N. S., Phy.~. KPV. 104, 885 (1956). 7. Taylor R., Phil. Mug. 13, 157 (1966). 8. Pappis J. and Blum S. L., ,/, ‘4m. Cernm. Sot. 44,592 (1961). 9. Komatsu K. and Nagamiya T., ,j_ Phps. Sot. Jafxzn 6,438 (1951). 10. Komatsu K.,J. Phys. S~r.~~~~~, 10,346 (1955). II. Cowan R. D., J. Appl. Phys. 32,1363 (1963). 12. Wheeler M.J., &-it. J. Ap$. Whys. 16,365 (1964). 13. Peierls R., Ann. Phys. 3, 1055 (1929). 14. Klein C. A. and Holland M. G., Phy,~. Rev. 136,575 (1964). 15. Dolling G. and Brockhouse B. N.. Phys. Ret]. 128, 1120 (1962).