olivine, pyroxene and dunite yoji kobayashi

J. Phys. Earth, 22, 359-373, 1974

ANISOTROPY OF THERMAL DIFFUSIVITY IN

OLIVINE, PYROXENE AND DUNITE

Yoji KOBAYASHI

Department of Earth Sciences, Faculty of Science,

Kanazaza University, Kanazawa, Japan

(Received May 31, 1974; Revised August 22, 1974)

The thermal diffusivities along three crystallographic orientations aremeasured for olivine and pyroxene up to 1250K. The result shows that theanisotropy amounts to about 70%. The anisotropy is attributed mostly toanisotropy of mean free path of phonon. The anisotropy of thermal con-ductivity and thermal diffusivity in dunite is measured and is discussed onthe basis of preferred orientation and the anisotropy of constituent minerals.

1. Introduction

The anisotropy of thermal conductivity or thermal diffusivity is theoreti-

cally expected in crystals (NYE, 1957; CARSLAW and JAEGER, 1959) and thus

in rocks having the anisotropic structure. Anisotropy of thermal conductivity

was experimentally determined only for a limited number of silicate and oxide

minerals. BIRCH and CLARK (1940) measured the anisotropic thermal con-

ductivity of quartz at high temperature. KANAMORI et al. (1968) showed

that anisotropy of the thermal diffusivity of quartz decreased with increasing

temperature. SASS (1965) reported that feldspars were anisotropic with the

minimum conductivity parallel to [100] and the maximum parallel to [010],

and the amount of anisotropy was about 10% at 25℃. Recently SCHLOESSIN

and DVORAK (1972) measured the anisotropic thermal conductivity of enstatite

at pressures between 19 and 56kbar and at temperature between 300 and

400K. They reported that the amount of anisotropy was about 60% with

the minimum conductivity parallel to [100] and the maximum parallel to

[001].

In the present paper, new measurements of the thermal diffusivities of

olivine, orthopyroxene and clinopyroxene are presented and discussion is

made on (1) a relationship between anisotropy of thermal diffusivity and those

of phonon velocity and mean free path, and (2) the anisotropic thermal dif-

fusivity of dunite.

359

360 Y. KOBAYASHI

2. Experiment

2.1 Specimen

The localities, chemical composition, cell parameters and density of thesingle crystal specimens used here are listed in Tables 1 and 2. The densitiesof the specimens were measured by hydrostatic weighing. The chemical com-

positions of two olivine single crystal specimens and one orthopyroxene single

Table 1. The description of the specimen.

Table 2. Chemical composition of orthopyroxene and clinopyroxene (weight %).

Anisotropy of Thermal Diffusivity in Olivine, Pyroxene and Dunite 361

crystal were determined by an electron probe micro-analyzer. The orienta-tion of crystallographic axes of the specimen was determined by the X-rayoscillation photograph technique. Each specimen was cut into a rectangular

prism 3 to 6mm long, every surface being parallel or perpendicular to crystal-lographic axes. The surfaces of the specimens were polished flat.

The dunite specimen was collected from the Horoman peridotite massin Hokkaido, and its fabric of elastic wave velocities and preferred orientationof olivine were reported by KASAHARA et al. (1968 a, b). Kasahara et al. haveshown that crystallographic a-axes of olivine was concentrated around X-axis

(orientation of major lineation), b-axes showed an imperfect girdle in the YZplane (Y-axis is normal to the foliation), and c-axes an imperfect concentrationaround Z-axis (the direction normal to X- and Y-axes).

The specimens of dunite used for the thermal conductivity measurementwere prepared by cutting the same piece of dunite as the measurement of

elastic wave velocity by Kasahara et al. into the shape of a circular disk.

They were three disks of 25mm in diameter with thickness of 2 to 6mm long.

The normal of the disks is parallel to either coordinates X, Y and Z axes,

which are respectively corresponded to tectonic b, a and c axes of dunite.

The thermal diffusivities are measured on the same specimen of rectangular

prisms, which have a cross sectional area of 2.5-4.0mmmm2 and length of 3 to

5mm along each direction of rectangular coordinates X, Y and Z axes.

2.2 Measurement

In an anisotropic solid, the heat conduction is generally expressed asfollows:

(1)

where Qi is i component of heat flux vector, Kij is thermal conductivity tensor,

and ∂T/∂Xj is component of temperature gradient along Xj axis. For ortho-

rhombic system, Kij is written as follows (NYE, 1957):

where Kij are the three principal thermal conductivities, corresponding respec-

tively to three directions of the crystallographic axes. The thermal diffusivity

kij takes the same matrix form to Kij and the relation of kij to Kij is represented

by kij=Kij/ρC, where ρ is density and C is specific heat. Therefore, we have

only to measure the thermal conductivities along each of three crystallographic

axes of olivine and orthopyroxene to obtain the Kij or kij matrix.

On the other hand, for monoclinic system, Kij is written as follows:

362 Y. KOBAYASHI

Therefore, the measurement of four thermal conductivities along three princi-

pal axes and another direction in XZ plane is necessary in order to determinethe conductivity tensor of clinopyroxene (diopside). However, the authormeasured the thermal diffusivities of clinopyroxene only along a, [010], and

[001] axis. Therefore, K13 was not determined. The thermal diffusivities of olivine, orthopyroxene, clinopyroxene anddunite are measured by the modified Angstrom method developed by KANA-

MORI et al. (1968, 1969). In this method, the thermal diffusivity is calculated

from the measured amplitude decay and the phase lag of periodic temperature

wave. Silver coating paste was used to fix a heater and C-A thermocouple

of 0.1mm thickness with the specimen. The measurements were made in the

atmosphere of argon gas in order to avoid the effects of oxidation. Using three

fused quartz rods of 2.5mm in diameter with lengths 3.98, 4.82 and 7.33mm,

the measuring system was calibrated. The reciprocal of measured thermal

diffusivity of the fused quartz specimen is shown in Fig. 1. The present results

agree well with the previously reported values by KANAMORI et al. (1968)

within a range of ±10%.

The thermal conductivity was measured by the divided-bar method. The

data reduction follows the method given by HORAI (1959). The thermal con-

ductivity of the fused quartz is 3.20×10-3cal/cm・sec・deg at room temperature,

which shows good agreement with the published data (BIRCH and CLARK, 1940;

Fig. 1. Reciprocal of thermal diffusivity of fused quartz.


KINGERY and MCQUARRIE, 1954; KANAMORI et al., 1968). The experimental

error is estimated to be 3%.

3. Result

The thermal diffusivity of dunite at room temperature and 1 atm pressureis shown in Fig. 2. It shows the values of thermal diffusivity plotted againstthe periods of temperature wave. The thermal diffusivity was independentof the periods within the measured ranges. This shows that the effect of lateralside of the specimen does not significantly affect the result.

The thermal diffusivities of single crystal specimens: two olivines, one

Fig. 2. Thermal diffusivity versus period for Horoman dunite.

Fig. 3. Thermal diffusivity of olivine 1 along [100], [010] and [001] directions.

364 Y. KOBAYASHI

orthopyroxene and two clinopyroxenes, were measured in the temperature

range of 300 to 1250K, and the results are shown in Figs. 3, 4, 5, 6, 7 and

in Table 3.

The thermal conductivities of these single crystal specimens are calculatedfrom the diffusivity data using specific heat data (compilation by ROBIE andWALDBAUM, 1968). The thermal resistivities (reciprocal of conductivity) areshown in Figs. 8 and 9.

Fig. 4. Thermal diffusivity of olivine 2 along [001] direction.

Fig. 5. Thermal diffusivity of orthopyroxene along [100], [010] and [001] directions.

Anisotropy of Thermal Diffusivity in Olivine , Pyroxene and Dunite 365

Fig. 6. Thermal diffusivity of clinopyroxene 1 along a-axis, [010] and [001] directions.

Fig. 7. Thermal diffusivity of clinopyroxene 2 along a-axis.

Olivine. Theoretically the thermal resistivity increases linearly with tem-

perature at low temperature. As shown in Fig. 8, the thermal resistivities ofolivine single crystals certainly increase almost linearly with temperature be-low 700K.

Anisotropy of the thermal diffusivity for olivine single crystals is clearlypresent as is inferred from Fig. 3. The thermal diffusivity measured along the[100] direction is the largest and that of [010] is the smallest in the tempera-ture range of 300 to 1250K. BIRCH and CLARK (1940) and KANAMORI et al.

(1968) suggested that the crystal anisotropy of quartz decreased at highertemperatures. Anisotropy of olivine, however, is almost constant withtemperature.

366 Y. KOBAYASHI

Table 3. Results of thermal diffusivity measurements.

Fig. 8. Thermal resistivities of olivine 1 and 2. Dashed line in from data of

KANAMORI at el. (1968).

The value of thermal diffusivity measured along the [001] direction of(Mg91.6Fe8.4)2SiO4 is larger than that of (Mg87.4Fe12.6)2SiO4. The dependence ofthe thermal diffusivity on the fayalite content is consistent with that of thermalconductivity obtained by HORAI and SIMMONS (1969).

Orthopyroxene (Bronzite). As shown in Figs. 5 and 9, the thermal dif-


Fig. 9. Thermal resistivities of orthopyroxene and clinopyroxene 1 and 2.

fusivities decrease with increasing temperature and the thermal resistivities

increase linearly with increasing temperature in the temperature range of 300

to 800K.

Anisotropy of the thermal diffusivity is present as in the case of olivine.The value along [001] direction is the largest and the value along [100] direc-tion is as large as that along [010] direction. According to SCHLOESSIN andDVORAK (1972), the value of thermal conductivity of enstatite along [001]direction is the largest, and the value along [100] direction is the slightly smallerthan that along [010] direction. Considering the experimental errors of the

both data sets, we think the present result on the anisotropy of thermal dif-fusivity of the bronzite agrees with that of thermal conductivity of enstatiteobtained by Schloessin and Dovorak.

Clinopyroxene (Diopside). Two specimens of clinopyroxene with slightlydifferent chemical composition show almost identical dependence of thermalconductivity on temperature. As shown in Figs. 6 and 7, the thermal dif-fusivities decrease with increasing temperature at temperatures lower than

600K, however, they are almost constant at temperatures higher than 600K.As shown in Fig. 9, the thermal resistivities of clinopyroxenes also show alinear dependence on temperature at lower temperatures.

Anisotropy of the thermal diffusivity also exists clearly as in the case of olivine and orthopyroxene. The value along [001] direction is the largest inthe temperature range of 300 to 1250K. The value along a-axis is same as

that along [010] direction.

368 Y. KOBAYASHI

The diffusivities of clinopyroxene are also comparable with those oforthopyroxene.

Dunite. Anisotropy of the thermal diffusivity exists clearly. As shownin Fig. 2, the dunite has the largest value along X-direction and the smallest

in the Y-direction. The thermal conductivities were also measured at room

temperature and 1 atm pressure. Anisotropy of the thermal conductivity

showed the tendency similar to that of thermal diffusivity.

4. Discussion

A linear relation between lattice thermal conductivity Kp (or lattice ther-mal diffusivity kp) and elastic wave velocity is anticipated from Gas kinetic

theory,

(4)

where λ, Vm, C and ρ are the mean free path of phonon, phonon mean velocity,

specific heat and density, respectively. The phonon mean velocity Vm in an

isotropic media is identified with the mean elastic or acoustic wave velocity

given by

(5)

where Vp and Vs are the compressional and shear wave velocities, respectively.

DUGDALE and MACDONALD (1955) have suggested that reasonable values of

lattice thermal conductivity Kp are empirically obtained for alkali halides, if

the mean free path λ of phonon is given by

(6)

where A0 is the lattice parameter, α is the thermal expansion coefficient, γ is

Gruneisen parameter and T is absolute temperature.

Equation (4) shows that the thermal conduction is directly related to the

speed Vm and the efficiency λ of phonon transfer. In an anisotropic medium

both Vm and λ should be dependent on orientation. Therefore, the present

author is extending Kp to the orientation-dependent quantity Kpi in terms of

orientation-dependent Vmi and λi as follows;

(7)

where i denotes the orientation.

The orientation-dependent mean acoustic wave velocity Vmi is simplygiven by

(8)

where Vpi is the compressional wave velocity, VsIi and VsIIi are two shear

wave velocities propagating along the orientation i. A simple extension of λ


in (6) to anisotropic quantity involves a little superficial procedure. A0 in (6)is a basic unit length in the periodic crystal structure, which makes an interac-

tion with phonon propagation in a periodic lattice. In an anisotropic lattice,

A0 in (6) is corresponded to a wave length of lattice periodicity measured along

the orientation of phonon propagation. Therefore A0 is replaced by a lattice

constant A0i along the orientation i. The αγT is a parameter showing the

degree of anharmonicity of the lattice property which makes interaction of

phonon-phonon interaction to reduce the efficiency of phonon propagation,

and in an anisotropic material it is an orientation-dependent quantity. Then

an extension of (6) is written by

(9)

where αi and γi are linear thermal expansion coefficient and Gruneisen constant

along i-axis. The value of γi in (9) is evaluated by the two different ways;

Table 4. Physical constants of olivine (Fo91.6Fa8.4) at room temperature.

370 Y. KOBAYASHI

thermal and acoustic. The thermal γi is given by

(10)

where βi is linear compressibility along i-axis. Equation (10) is a modification

of Gruneisen's relation. The acoustic γi is given by an extension of BARRON'S

formula (1957);

(11)

The equation; λi=3kpi/Vmi derived from (7) is used to determine the

phonon mean free path from the measurements of kpi. Although λi changes

with temperature, λiT does not change as suggested by (6) or (9). Therefore,

λiT obtained from (7) is compared with those given by (9) with (10) and (9)

with (11) in Tables 3 and 4. The theoretical prediction of λiT by (9) gives

larger values than that experimentally determined by two orders of magnitude

Table 5. Physical constants of orthopyroxene at room temperature.


in the present silicate minerals. This shows that Dugdale and MacDonald

formula (6) or (9) which is given for a simple cubic insulator, cannot be ap-

plied to estimating the numerical values of λ of complicated silicate minerals,

but is limited to relate the change of kp with the other factors such as pressure

or temperature as discussed by FUJISAWA et al. (1968). Another important

point to be noted is that the theory does not explain the right order of relative

magnitudes among three kpi along the different orientations in one crystal,

although the large anisotropy is predicted. These situations seem to suggest

that some important factor is out of consideration not only in (9) but also in

(6).

In order to know which factor is most significantly affecting the aniso-

tropic thermal conduction, the anisotropies of kpi, Vmi and λi at room tem-

perature are compared also in Tables 4 and 5. The difference of kpi and λi

along different orientations amounts to factor of 2, while the anisotropy of

Vmi is less than 10%. This suggests that the most fraction of the anisotropic

thermal conduction is attributed to the anisotropy of phonon mean free path.

Several physical constants of the dunite are calculated from the data

obtained and are shown in Table 6. The thermal diffusivity of the dunite is

compared with that of olivine, a major constituent (95%) of the present dunite.

The mean thermal diffusivity of the dunite 15.6×10-3cmcm2/sec is almost

Table 6. Physical constants of Horoman dunite at room temperature.

372 Y. KOBAYASHI

the same to 16.5×10-3 of olivine. The anisotropy of the thermal diffusivity

(or thermal conductivity) is also well correlated with the fabric structure of

the dunite and the properties of single crystal olivine. The anisotropy of the

thermal diffusivity in the dunite is 48% with the maximum value along X-

direction and the minimum value along Y-direction. The fabric diagram of

olivine in the present dunite shows that a-axis (maximum diffusivity) and b-

axis (minimum diffusivity) of olivine are most frequently oriented respectively

along X-and Y-directions of the dunite.

The reduction of anisotropy of compressional wave velocity by aggrega-tion of olivine is 1/2 (22% of olivine single crystal to 11% of the presentdunite) in the present case. By assuming the same reduction rate is appliedto the thermal diffusivity, the anisotropy of diffusivity is obtained as 34%

(1/2 of the single crystal value of 68%), which is approximately coincidingwith 48% for the present dunite specimen. Therefore, the thermal diffusivityanisotropy in the olivine rocks due to the preferred orientation of olivine isexpected to be 2-3 times of the volocity anisotropy of the compressional wavevelocity.

The significant preferred orientation of olivine and the resulting velocityanisotropy in peridotitic rocks are very common both for nodule and Alpine

type intrusion (KUMAZAWA et al., 1971). In the East Pacific Ocean the presenceof velocity anisotropy of 10% has been reported by RAITT et al. (1969) and

MORRIS et al. (1969). Therefore, the anisotropy of the thermal conductivityin the upper mantle is expected to exist. The expected magnitude of the con-ductivity anisotropy is 30-50% from the discussions in the above. Such alarge anisotropy of the thermal conductivity in the upper mantle may signifi-cantly affect the interpretation of the thermal regime in the upper mantle.

The author owes to Dr. Mineo Kumazawa of Nagoya University for his advice andcritical reading the manuscript of this paper. The author is indebted to Dr. Y . Kono for hisvaluable discussion and to Prof. Y. Kaseno for his continuing support throughout the presentwork.

REFERENCES

BARRON, T.H.K., Gruneisen parameters for the equation of state of solids, Ann. Phys. N.Y., 1, 77-90, 1957.BIRCH, F. and H. CLARK, The thermal conductivity of rocks and its dependence upon tem-

perature and composition, Am. J. Sci., 238-529, 530-613, 1940.CARSLAW, H.S. and J.C. JAEGER, Conduction of Heat in Solids, 2nd edition, PP. 38-49, Oxford University Press, London, 1959.DUGDALE, J.S. and D.K.C. MACDONALD, Lattice thermal conductivity, phys. Rev., 98, 1751-1752, 1955.FUJISAWA, H., N. FUJII, H. MIZUTANI, H. KANAMORI, and S. AKIMOTO, Thermal diffusivity

of Mg2SiO4, Fe2SiO4, and NaCl at high pressures and temperatures, J. Geophys. Res., 73, 4727-4733, 1968.


HORAI, K., Studies of the thermal state of the Earth. The third paper; Terrestrial heat flow

at Hitachi, Ibaraki Prefecture, Japan, Bull. Earthq. Res. Inst. Tokyo Univ., 37, 571-591,

1959.

HORAI, K. and G. SIMMONS, Thermal conductivity of rock-forming minerals, Earth Planet.

Sci. Lett., 6, 359-368, 1969.

KANAMORI, H., N. FUJII, and H. MIZUTANI, Thermal diffusivity measurement of rock-forming

minerals from 300° to 1100°K, J. Geophys. Res., 73, 595-605, 1968.

KANAMORI, H., H. MIZUTANI, and H. FUJII, Method of thermal diffusivity measurement, J.

Phys. Earth, 17, 43-52, 1969.

KASAHARA, J., I. SUZUKI, M. KUMAZAWA, Y. KOBAYASHI, and K. IIDA, Anisotropism of P

wave in dunite, Zisin (J. Seismol. Soc. Japan), 21, 222-228, 1968a (in Japanese with English

abstract).

KASAHARA, J., I. SUZUKI, M. KUMAZAWA, and K. IIDA, Anisotropism of S Wave in dunite,

Zisin (J. Seismol. Soc. Japan), 21, 229-236, 1968b (in Japanese with English abstract).

KINGERY, W.D. and M.C. MCQUARRIE, Thermal conductivity: 1, Concepts of measurement

and factors affecting thermal conductivity of ceramic materials, J. Am. Ceram. Soc., 37,

67-72, 1954.

KUMAZAWA, M. and O.L. ANDERSON, Elastic moduli, Pressure derivatives, and temperature

derivatives of single-crystal olivine and single crystal forsterite, J. Geophys. Res., 74,

5961-5972, 1969.

KUMAZAWA, M., The elastic constants of single-crystal orthopyroxene, J. Ceophys. Res., 74,

5973-5980, 1969.

KUMAZAWA, M., H. HELMSTAEDT, and K. MASAKI, Elastic properties of eclogite xenoliths

from diatremes of the east Colorado Plateau and their implication to the upper mantle

structure, J. Geophys. Res., 76, 1231-1247, 1971.

MORRIS, G.B., R.W. RAITT, and G.G. SHOR, Velocity anisotropy and delay-time maps of

the mantle near Hawaii, J. Geophys. Res., 73, 4300-4316, 1969.

NYE, J.F., Physical Properties of Crystals, pp. 195-214, Oxford University Press, London, 1957.

RAITT, R.W., G.G. SHOR, T.J.G. FRANCIS, and G.B. MORRIS, Anisotropy of the Pacific

upper mantle, J. Geophys. Res., 74, 3095-3109, 1969.

ROBIE, R.A. and D.R. WALDBAUM, Thermodynamic properties of minerals and related sub-

stances at 298.15 (25.0℃) and one atmosphere (1.013bar) pressure and at high tempera-

ture, Bull. Geolog. Survey, 1259, 256, 1968.

SASS, J.H., The thermal conductivity of fifteen feldspar specimens, J. Geophys. Res., 70, 4064-

4065, 1965.

SCHLOESSIN, H.H. and Z. DVORAK, Anisotropic lattice thermal conductivity in enstatite as a

function of pressure and temperature, Geophys. J.R. Soc., 27, 499-516, 1972.

olivine, pyroxene and dunite yoji kobayashi

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