Analysis of thermal expansivity of periclase (MgO) at high temperatures

Physica B 225 (1996) 283-287

ELSEVIER

Analysis of thermal expansivity of periclase (MgO) at high temperatures S.S. Kushwah*, J. Shanker Department of Physics, Institute of Basic Sciences, Khandari, Agra, India

Received 20 November 1995; revised 19 March 1996

Abstract

The thermal expansivity of MgO has been studied in the temperature range (300-1800 K) using various thermodynamic relationships reported in the recent literature. The volume-temperature relationship and thermal expansion coefficient as a function of temperature are predicted and compared with each other. It is found that among the various equations used the Suzuki equation presents the best agreement with the experimental data.

1. Introduction

The volume-temperature relationship and thermal expansivity data are required for investigating the equation of state and predicting the compression data of solids at high temperatures [1,2]. There are certain difficulties associated with the measurements of volume thermal expansion coefficient ~ at high temperatures which lead to considerable uncertainties in the experimental values [3,4]. It has been emphasised [4] that most serious error in the calculation of thermodynamic functions arises from the uncertainty of thermal expansivity at high temperatures. It is, therefore, desirable to develop semi-empirical and semi-phenomenological models for predicting V ( T ) and ~(T). Several important thermodynamic relationships have been reported in the literature [5-10] which can be used for developing such models. In the present paper we compare various thermodynamic relationships to determine V ( T ) and *Corresponding author. Permanent Address: Rishi Galav College, Morena, MP, India.

ct(T) for MgO. The method of analysis is given in Section 2. Results are discussed and compared with available experimental data in Section 3.

2. Method of analysis

The original Murnaghan's equation of state was formulated [11] to predict P - V data for a solid at a fixed temperature. It did not contain any terms showing the variation of volume with temperature. More recently, attempts have been made [2, 12-14] to incorporate the thermal effects so as to estimate the volumes at simultaneously elevated temperatures and pressures. According to Akaogi and Navrotsky [12] we can write V

--=1

Vo

+ ~o(T -- To) + ~ (T -- To) 2

(1)

[~o + 2 a ; ( T -- To)],

(2)

and :~ =

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284

S.S. Kushwah, J. Shanker / Physica B 225 (1996) 283-287

where ao Js the thermal expansion coefficient, Vo the volume, and a~ the temperature derivative of a at T = To and P = 0. It should be mentioned that Eqs. (1) and (2) are consistent with each other, and are based on the standard definition of ~ given as

1<)

Fei and Saxena [13] have used an expression which can correctly be expressed as follows: V --=l+%(T-To)+~ao(T-To) Vo -

A%(T

-

1 ,

]

To) -1,

[co + e'o(T - To) + Aao(r - ro)-2].

Guillermet and Gustafson [14] considered an exponential dependence of V/Vo on temperature given below:

~=~O+K~ ~

I(dK [Ko ldK r +

[ ~;

' (aKVl.J ( T -

2 , KoKo \ a T ]

~ ; ( T - To)21 ~ - .

=exp ~o(r-ro)+

(6)

Eqs. (3) and (6) yield a linear dependence of ~ on temperature as follows: = eo + a~(T - To).

v

[

roll-,,.~

i+

\aTj\

Ko +

To).

(9)

= ~o[1 - ~O6T(T -- To)] -1,

(10)

where fiT is the Anderson-Gruneisen parameter defined as

1<)

(11)

Eq. (10) has been derived by integrating the following relationship and taking ~r to remain constant, =~T. V1 ( d ~ )-f,

(12)

(7)

Plymate and Stout [2] have presented a more involved expression for the temperature dependence of V/Vo by taking into account the variation of isothermal bulk modulus (K) with temperature and pressure. The expression obtained by Plymate and Stout is given as

Vo =

-'

In deriving Eqs. (8) and (9), the bulk modulus has been assumed to vary linearly with temperature. This implies that (dK/dT) is constant and its value is taken at room temperature and atmospheric pressure. An expression for the temperature dependence of ~ has been obtained by Anderson et al. [9] which can be expressed as follows:

aK d-T v" [

(8)

where (dK/dT) is the temperature derivative at constant pressure. Ko and K~ are the values of the bulk modulus and its first pressure derivative at atmospheric pressure. The expression of a corresponding to Eq. (8) is obtained as follows:

(4)

(5)

V

-

2

where A is a constant having the units of T 2 and value equal to i. We have introduced the constant A in the last term to make it consistent with the other terms in Eq. (4) with respect to units. An expression of • corresponding to Eq. (4) is then obtained as follows: =

-- \ d T

/J

\dr] goK'o 1 (T -- To)

The expression for the temperature dependence of V/Vo corresponding to Eq. (10) is then obtained as V = Vo

--

[1 -

O~06T(T -- T o ) ] -1/~T

(13)

Anderson et al. [9] have pointed out that Eq. (10) is a high-temperature approximation. It only works above the Debye temperature 0. Therefore, To in Eq. (10) must be about 1000K, and ao should be taken as the experimental • measured at 1000K. We use Eqs. (10) and (13) for estimating e and V/Vo in the temperature range 1000-1800K.

S.S. Kushwah, J. Shanker / Physica B 225 (1996) 283-287

Geophysicists need to know the behaviour of oxides at very high temperatures, and further, one of the favourite geophysical materials is MgO which is under present study. The Suzuki theory of thermal expansivity has received a lot of exposure in the geophysical literature 1-15-17]. The Suzuki equation is expressed as V

[1 + 2 k - ( 1

Relationships (1)-(9) are used to calculate V/Vo and s as a function of temperature in the range 300-1800K for periclase (MgO). The input data used in calculations are taken from Xia and Xiao [4] and Anderson et al. 1-18]. These are So = 31.2x 10-6K -1, Ko = 162GPa, K~ = 4.2, fT = 6.5 and s~ = 8 . 5 7 x 1 0 - 9 K -2. For determining the value ofdK/dT we have used Eq. (11) taking s = So and K = Ko. The value of dK/dT thus determined corresponds to room temperature and atmospheric pressure. It has been shown by Plymate and Stout [2] and by Xia and Xiao 1-4] that (dK/dT) thus estimated can be taken as constant. The results obtained from Eqs. (1)-(9) are compared with available experimental data 1-18] in Fig. 1 for the temperature dependence of V/Vo and in Fig. 2 for the temperature dependence of s. It is found that the values calculated from Eqs. (1)-(9) deviate significantly from the experimental data. The deviations are more pronounced for the temperature dependence of s (Fig. 2). Eqs. (10) and (13) are used in the temperature range 1000-1800K for the reason already discussed. The results agree closely with the experimental data (Figs. 1 and 2). In calculations we have

(e)

MgO

[d) (e)

1.05

T

1.04

1.03

(14)

3. Results and discussion

(f) (b)

1.06

--4kETH/Q)I/2](Vo/V(To)) 2kav

where the symbols are well-known 1,15]. This equation has a solid theoretical background. It is derived from the Mie-Gruneisen equation of state, which, in turn, is a special case of the anharmonic approximation of lattice dynamics. On the basis of Eq. (14) we study the thermal expansivity as a function of temperature.

(a)

1.07

>o

V(To)

285

1.02

1.01

1.00 30O

600

, 1200

900

, 1500

1800

TEMPERATURE (K)

Fig. 1. Plots of the volume ratio V/Vo versus temperature for MgO: (a) experimental [3]; (b) calculated from Eq. (1); (c) calculated from Eq. (4); (d) calculated from Eq. (6); (e) calculated from Eq. (8); (f) calculated from Eq. (14); and circles represent the values calculated from Eq. (13).

55

;b) a) Le) :f)

5O EXPERIMENT ~gO

,d)

-C,

,{c)

35

30

25

300

I

1

I

I

600

900

1200

1500

T ENPERATURE (K)

1800 :>

Fig. 2. Plots of the volume thermal expansion coefficient ~ versus temperature for MgO: (a) experimental I-3]; (b) calculated from Eq. (2); (c) calculated from Eq. (5); (d) calculated from Eq. (7); (e) calculated from Eq. (9); (f) calculated from numerical differentiation of Eq. (14); and circles represent the values calculated from Eq. (10).

S.S. Kushwah, J. Shanker/ Physica B 225 (1996) 283-287

286

taken Cto = 44.7x 10-6K -1 at To = 1000K and 6T = 4.8. This value of 6T is in the range of values 5.26--4.66 reported by Anderson et al. [9]. Thus, Eqs. (10) and (13) hold good in the high-temperature approximation. The Suzuki equation (14) provides an important and useful method for estimating the temperature dependence of thermal expansivity in the entire temperature range. V/V(To) is calculated from Eq. (14). The analytical expression for ~ found by differentiation of Eq. (14) is quite complicated. We have therefore used the method of numerical differentiation [19] for estimating ~(T). The input data used in calculations are Ko = 162GPa, K~ = 4.13, 0 = 945K, 7 = 1.54 and Vo = ll.24cm3mo1-1, all at P = 0. The results based on the Suzuki equation present close agreement with experimental data (Figs. 1 and 2). One of the advantages of the Suzuki equation is that it follows the curvature of the data quite well, so that the d:t/dT predicted from this equation is an exact fit to the d~/dT of the experiment. More recently, an attempt has been made by Xia and Xiao [4] to fit the experimental data on ~(T) for MgO in terms of an empirical relationship given as ~t

- 0to = B l n T/To,

(15)

5.50 5.00

T

4.50

A

'o "- ~ 0 ::¢ ~>

350

MgO I

300

I

0.(,0

000 (a)

I

I

080 1.20 In(kl

I

1.00 •

2.O0

18o

'o ~

g "~

14.0

12D NaCl

where B is treated as a constant. Xia and Xiao estimated ~ as a function of temperature for MgO with the help of Eq. (15) using the value of B calculated from the experimental data on the pressure dependence of specific heat Cp in the range of elastic Debye temperature [4]. In order to demonstrate the validity of the relationship (15) for MgO, we have plotted the experimental data represented by dots in Fig. 3(a). It is seen from the figure that the linear relationship holds to a good approximation. However, this may not hold good for other materials. For example, in case of NaCI the experimental data are available [20], and we have plotted them in Fig. 3(b). In this case significant deviations are noted and the experimental data reveal a nonlinear relationship. Finally, it should be mentioned that a direct relationship between ~/~o and V/Vo has also been

10.0

i

000

020

(b)

J

i

0.~ 0.60 In (T_.). T,,

I~

i

0 >

tOO

Fig. 3. (a) Plot of ~V/Vo versus ln(T/To) for MgO; dots represent the experimental data, and (b) plot of ~V/Vo versus ln(T/To) for NaC1.

reported in the literature [5] which can be expressed as

So

\VoJ

'

(16)

This relationship is based on the two assumptions viz., 6r remains constant and the product ~Kr is also constant. We have assessed the validity of Eq. (16) for the present case of MgO by plotting

S.S. Kushwah, ,L. Shanker / Physica B 225 (1996) 283-287

287

References 050 0.40

0.3O 0.20

Mg0

¢::;

0.10

0.00

I

I

0010

0020

I

0030

tn v • Vo

I

I

QOt~O 0050

I

I

0.060 0070

>

Fig. 4. Plot of ln(~/c~0) versus ln(T/To) for MgO.

ln(~/~o) and ln(V/Vo) based on experimental data (Fig. 4). Eq. (16) would imply the plot to be a straight line. However, significantly large deviations from the linearity are found. Thus, the assumptions used in Eq. (16) may not be true strictly.

Acknowledgements We are thankful to the Referee for his valuable comments and for providing the data and other relevant information on the Suzuki equation. The financial support received from the UGC, Bhopal (MP), India is gratefully acknowledged.

[1] R. Boehler and G.C. Kennedy, J. Phys. Chem. Sol. 41 (1980) 517, 1019. [2] T.G. Plymate and J.H. Stout, J. Geophys. Res. B 94 (1989) 9477. [3] O.L. Anderson and K. Zou, Phys. Chem. Min. 16 (1989) 642. [4] X. Xia and J.K. Xiao, J. Phys. Chem. Sol. 54 (1993) 629. [5] O.L. Anderson, J. Geophys. Res. 72 (1967) 3661. [6] I. Suzuki, J. Phys. Earth 23 (1975) 145. [7] I. Suzuki, S. Okajima and K. Seya, J. Phys. Earth 27 (1979) 63. [8] A. Dhoble and M.P. Verma, Phys. Star. Sol. B 133 (1986) 491; 136 (1986) 497. [9] O.L. Anderson, D.G. Isaak and H. Oda, Rev. Geophys. 30 (1992) 57. [10] J. Shanker and M. Kumar, Phys. Stat. Sol. B 179 (1993) 351. [11] F.D. Murnaghan, Proc. Natl. Acad. Sci. 30 (1944) 244. [12] M. Akaogi and A. Navrotsky, Phys. Earth Planet Inter. 36 (1984) 124. [13] Y. Fei and S.K. Saxena, Phys. Chem. Min. 13 (1986) 311. [14] A.F. Guillermet and P. Gustafson, High Temperatures High Pressures 16 (1985) 591. [15] O.L. Anderson, Equations of State of Solids for Geophysics and Ceramic Science (Oxford University Press, Oxford, 1995). [16] I. Suzuki, E. Ohtani and M. Kumazawa, J. Phys. Earth 28 (1980) 273. [17] I. Suzuki, H. Takei and O.L. Anderson, Proc. 8th Internat. Thermal Expansion Symp. ed. T. Hahn (Plenum, New York, 1984) pp. 79-88. [18] O.L. Anderson, D.G. Isaak and H. Oda, J. Geophys. Res. 96 (1991) 18037. [19] D. Potter, Computational Physics (Wiley, New York, 1980). [20] S. Yamamoto, I. Ohno and O.L. Anderson, J. Phys. Chem. Sol. 48 (1987) 143. -