Indian Journal of Pure & Applied Physic); Vol . 37. November 1 999. pp. 8 1 8-822

Analysis of compressibility and thermal expansivity for MgSi03 perovskite

S Meenakshi & B S Sharma

Department of Physics, Institute of Basic Sciences. Khandari. Agra 282 002

Received 3 M arch 1 999; revised 26 July 1 999

An analysis of the compressihil ity of MgSi03 perovskite under high pressure and at room temperature has been presented. The pressure-volume isothermal relationship. isothermal bulk modulus and its pressure derivative have heen calculated at high pressures. down to n compression of fifty percent. using the Birch-Murnaghan third order equation of state ( EOS) [J Geophrs Res 57 ( 1 952) 227J . the Vinet EOS [Phvs Rev B. 35 ( 1 987) 1 945] and the Shnnker EOS LPhysica B. 239 ( 1 997) 337]. The resu lts obtained from the Vinet universal EOS have been used further to investigate the volume thermnl expansion at high pressures nlong isobars for the si l icate perovskite under study with the help of the Shanker equation of thermal expansivity. The calculated values of volume thermal expansion have been found to present good agreement with the recent experimental data.

1 Introduction The si l icate perovskites are important geophysical

minerals found in the lower mantle of the earth I . It is, therefore, desirable to investigate the properties of such minerals under the effect of h igh pressures and h igh temperatures . In the present paper, we study the compress i b i l i ty and t hermal expans i v i ty of MgS iO, perovskites crystal using the three phenomenological equations of state viz. the Birch-Murnaghan third order equation of state2 (EOS) the Vinet EOS3 and the Shanker EOS4. The pressure - volume relationship, isothermal bulk modulus and its pressure derivative at room temperature under the effect of high pressures producing a compression up to fifty percent have been calculated . The results thus obtained are then used to study the temperature dependence of isobaric volume thermal expansion for MgSi03 under h igh pressures.

Anderson and Masuda� have recently developed a thermodynamic method for computing volume thermal expansion versus temperature along isobars for si l icate perovskites . However, their method is quite lengthy and suffers from the uncertainties, regarding the values of thermoelastic parameters used . They have performed calculations in several steps.

The results obtained by Anderson and Masuda, apart from being based on a lengthy method, are sensitive to the values of thermoelastic parameters used . In particular the values of Gruneisen parameter used as input by Anderson and Masuda show large variations of about seventy percent. In the present paper an alternative

thermodynamic method based on the S hanker equation for thermal expansivit/ which is free from the uncertainties in the values of Gruneisen parameter has been used. The present method is much simpler than the method fol lowed by Anderson and Masuda.

2 Analysis of Compressibility at Room Temperature The compressibil ity of a solid along an isotherm can

be studied with the help of pressure-volume re lationship using an i sothermal equation of state. We consider the fol lowing three equations of state:

The Birch-Murnaghan EOS - This EOS derived from the finite strain theor/ is expressed as fol lows :

P = � Ko ( x-7 - x-5 ) [. I + % ( K '0 - 4 ) ( x-? - 1 ) ] . . . ( I )

where x = ( VIVo) 1 /3 Ko and K' 0 are the values of KT and dKT/dP at P = O. The corresponding expressions for isothermal bulk modulus KT and its pressure derivative K'T= dK,./dP obtained from Eq. ( I ) are given below.

Kr = t Ko ( 7 £7 - 5 x-5 ) + � K 0 ( K '0 - 4 ) ( 9 x'i - 1 4 x-7 + 5 x' ) . . . (2)

and

K 'T = d KT = � [ ( K 'n - 4 ) ( 8 1 x9 - 98 x7 + 25 x -5 )

d P 8 Kr

,

;- -

MEENAKSHI & SHARMA: MgSi03 PEROVSKITE 8 1 9

. . . (3)

The Vinet EOS - Using a universal relationship between binding energy and interatomic separation for solids, Vinet et aC lo obtained the following EOS :

P = 3 Knx·2 ( I -x) exp [l1( l -x) ] . . . (4) where 11 = � ( Ko - I ) and x is same as in Eq. ( I ) . Expressions for KT and obtained from Eq. (4) are given below.

KT = Kill x·2 [ I +(l1x+ 1 )( I -x)] exp[l1( 1 - x)] . . . (5) and

K ' 1 [ x ( I - 11 ) + 2 11 x2 2] T = - + l1 x + 3 1 + ( l1 x + I ) ( I - x ) . . . (6)

The Shanker EOS - Shanker et al. 4 have obtained an EOS using a specific form for volume dependence of the short-range force constant defined in terms of interatomic potentials I I . This EOS is obtained in the fol lowing form:

K ( V /Vo l/3 [ ( I 2 ) I P = 0 t . · 1 - t + t2 !exp ( t y ) - I : + Y (- I + Y - � )exp t y '] . . . (7)

where y = I -VIVo and t = K'o-(8/3). Expressions for KT and K'T obtained from Eq. (7) are given below:

4/3 3 KT = Ko [� J 1 1 - [ 1 -:" J ) exp ( t y ) + � p

. . . (8)

and

K ' = ± + (' 1 _ i£, ) [l + � {. t + ( I + 2 y ) } ] T 3 3 Kr 3 Vo ( I + y + / )

. . . (9)

Using Eq. ( 1 -9) we have calculated P, KT and K'T for MgSi03 perovskite. The results thus obtained have been used to study the volume thermal expansion as a function of temperature along isobars at P = 0 and at h igher pressures.

3 Analysis of Thermal Expansivity at Higher Pressures

The thermal expansivity of a sol id can be studied on the basis of an equation of state which takes into account the effect of temperature in terms of thermal pressure (P1h) . Such an EOS can be written as fol lows I 1 . 1 2 :

d V P = - d V + P1h ( T ) . . . ( 1 0)

Here the pressure P is a function of volume and temperature both, V the lattice potential energy is the function of volume Valone, and the thermal pressure P1h, is considered to be a function of temperature only . Considering the sol id at T = To, the room temperature, and under pressure P, we can expand the potential energy V in powers of volume change () = VCT, P) - VeTo, P) using Tay lor' s series expansion. Thus we can write the fol lowing expression truncated beyond the cubic term'):

V = U + V i () + � U ii 82 + �U iii 8' o 0 2 0 6 0 . . . ( I I ) where Vo is the value of V at 0 = 0, i .e . at vcr.p) = V(To,P) . The first-, second-, and third- order volume derivatives of V al l taken at 8 = 0 are represented by V ;) , V � and V �i i respectively. The volume derivati ve of Eq. ( I I ) yields the fol lowing expression

� � = V 0 + V � 8 + � U �i 82 . . . ( 1 2) The equi l ibrium condition at T = To and pressure P is

obtained from Eq. ( 1 0) as fol lows: V '0 = P1h ( To ) - P . . . ( 1 3) Using Eq. ( 1 3) i n Eq. ( 1 2) we get P + � � - P1h (To ) = V � 8 + � V �i 02 . . . ( 1 4) Eqs ( 1 0) and ( 1 4) then y ield

i i I ii i 2 /::;'Pth = Pth ( T ) - Pth (To ) = V 0 0 + '2 V 0 0 . . . ( 1 5) Eg. ( 1 5) represents the relationship between thermal

pressure and 8 the i sobaric volume expansion with temperature. Values of V � and V � can be obtained from the expressions for i sothermal bulk modulus and its pressure derivative. Assuming that the thermal pressure depends on temperature only and that it is independent of volume at constant temperature, Shanker and Kushwah " obtained the following relations:

V i i = K ( To , P ) o V ( To , P )

and

V i i i = _ K( To , P ) [ K ' ( To , P ) + I ] o ---------------7------[ V ( To , P ) r

. . . ( 1 6)

. . . ( 1 7) where K(To, P) and K(To, P) are the values of isothermal bulk modulus and its pressure derivative respectively. Inserting the values of V �i and V :ii from Eqs ( 1 6) and ( 1 7) in Eq. ( 1 5) , and then solving it for 0, Shanker et

820 INDIAN J PURE APPL PHYS, VOL 37, NOVEMBER 1 999

at. 9. 1 3 obtained the fol lowing relationship for the isobaric volume expansion with temperature.

V ( T , P )

V ( To , P )

I I I I I') 1 + I - i 1 - 2 [ K ( To , P ) + 1 J .1Pth / K ( To , P ) , -I K I ( To , P ) + I ]

. . . ( 1 8) For calculating isobaric volume expansion with tem

perature VeT. P)IV(To, P), we need values of K(To, P) and K' (To, P). These are calcu lated with the help of the equations of state. Values of LlPlh required in Eq. ( 1 8) are obtained from the fol lowing relationship

LlP'h = lu(e, O)J r Kce, 0) ] (T- To) . . . ( 1 9) where (0., 0) and K(e, O) are respectively the thermal expansiv ity and isothermal bulk modulus at T = e, the Debye temperature and at P = O. The val idity of Eq. ( 1 9) in case of MgSlO, perovskite has been discussed by Anderson and Masuda5 . They have reported the value of the product aK at T = e and P = 0 to be 6.92 X 1 0-3 GPa K I . We use the same value in Eq. ( 1 9) to calculate LlP'h at different temperatures.

4 Results and Discussions First, we have calculated pressure P, isothermal bulk

modulus K T and K'T = dK T/dP as a function of compression for MgSiO, perovskite at T = To = 300 K using Eqs .

( 1 -9) based on the Birch-Murnaghan EOS, the Vinet EOS and the Shanker EOS . The input data used are the

zero pressure values, Ko = 26 1 GPa and K'n = 4, taken from Wang et a/7. The results are reported in Table I . We note from this table that the resu lts obtained from the Vinet universal EOS and the Shanker EOS are in close agreement with each other up to the max imum compression VeTo, P)/V(To, 0) = 0.5 considered in the present study . The range of compression considered here is sufficiently large as it corresponds to a pressure of nearly 650 GPa which is about 2.5 times larger than the bul k modulus (26 1 GPa) for the s i l icate perovskite under study. The resul ts obtained from the Birch-Murnaghan EOS deviate significantly at higher compressions for VeT, P)/V (To, 0) < 0.7. The results based on the Vinet universal EOS should be considered to be most rel iable in view of the rigorous test presented recently by Hama and SUito '4.

Using the values of K(To, P) and K'(To, P) calculated

from the Vinet EOS and LlP'h determined from Eq. ( 1 9), we have calculated VeT, P)IV( To, P) with the help ofEq. ( 1 8) for MgSiO, perovskite up to a temperature of 3000 K along isobars at different pressures lip to 649 GPa (Table 2) . Values at simultaneously e levated pressures and temperatures are then obtained using the fol lowing relationship:

Tahle I - Values of pressure P. isothermal bul k modulus KT and its pressure derivative K'T for MgSiO) perovskite as a function of VIVo calculated from (a) the B irch-Murnaghan third order EOS (b) the Vinet EOS and (c) the Shanker EOS at T = To = 300 i< VIVo Pressure (P) Kr K'T

(a) (h ) (c) (a) (h) (c) (a) (h) (e) 1 .00 () 0 0 26 1 26 1 26 1 4.0 4.0 4,() O.l)5 1 4.9 1 4,X 1 4. X 3 1 9 3 1 11 3 1 11 H2 3 ,74 3 ,76 O.l)() .B,l) 33 ,X 33.X 390 3X7 3X7 3.66 3,52 3 ,54 (l.l�5 58.7 5!U 50 479 47 1 472 3.52 3,33 3 ,34 O.XO 9 1 . 1 89.9 9(),O 59 1 573 574 3.40 3, 1 5 3 , 1 6

0 75 1 33 1 3 1 1 3 1 733 699 700 3 ,29 2.99 2.98 0,70 1 90 1 84 1 84 9 1 7 855 854 3. 1 9 2 .85 2.82 (),65 267 255 255 1 1 58 1 05 1 1 046 3 . 1 0 2.7 1 2,67 0 6( ) 372 34l) 34H 1 480 1 299 1 284 3 ,02 2.59 2,53 0,55 5 1 Y 475 472 1 9 1 8 1 6 1 9 1 5H5 2.Y4 , 2.47 2.40 O.5() 730 MY 642 2532 2037 1 972 2.87 2 ,35 2.27

y-'

Y'

�

� �

r � r

t

....

..

MEENAKSHI & SHARMA: MgSi03 PEROVSKITE 82 1

Table 2 - Values of isobaric volume expansion VeT. P)/v(To, P) for MgSi03 perovskite calculated from Eg. ( 1 8) usi ng the data on K(To. P) and K'(To. P) predicted from the Vinet EOS. At T = To = 300 K , VeT, P)/V(To, P) = I

VeT. P)/ VeTo. P)

Pressure T = 500K T = 1 000K T = 1 500K T = 2000K T = 2500K T = 3000K

() 1 .00537 1 .0 1 95 1 1 .034R5 1 .05 1 77 1 .07089 1 .09339 1 4.H I .00440 1 .0 1 583 1 .02797 1 .04097 1 .05506 1 .07055

33 .H 1 .00361 1 .0 1 289 1 .02261 1 .03283 1 .04364 1 .055 I 5

58.3 1 .00296 1 .0 1 052 1 .0 1 836 1 .02650 1 .03497 1 .04383

89.9 1 .00243 1 .0086 1 1 .0 1 496 1 .02 1 49 1 .02822 1 .035 1 7

1 3 1 1 .00 1 99 1 .00703 1 .0 1 2 1 7 1 .0 1 744 1 .02282 1 .02833

1 84 1 .00 1 62 1 .00573 1 .00990 1 .0 1 4 1 4 1 .0 1 846 1 .02286

255 1 .00 1 32 1 .00465 1 .00802 1 .0 1 1 44 1 .0 1 489 1 .0 1 84 1

349 f .()O I 07 I .O(l375 1 .00647 1 .0092 1 1 .0 1 1 98 1 .0 1 47H

475 1 .00085 1 .00301 1 .005 1 7 1 .00736 1 .00956 1 .0 1 1 78

649 1 .00068 1 .00239 1 .004 1 0 1 .00583 1 .00757 1 .00932

Table � - Values of V(T, P)/V(To,O) for MgSi03 perovskite calculated using Eg. (20) taking the values of VeTo, P )/v(To, 0 ) derived from the Vinet EOS and the values of V(T. P)/V(To, P) from Eg. ( 1 8)

P VeT, P)/ VeTo, P)

(GPa) 300K 500K

0 1 .00 1 .00537

1 4.R 0.95 0.954 1 8

33.8 0.90 0.90325

5R.3 0.85 0.85252

89.9 0.80 0.80 1 94

1 3 1 0.75 0.75 1 49

1 84 0.70 0.70 1 1 3

255 0.65 0.65086

349 0.60 0.600642

475 0.55 0.55047

649 0.50 0.50034

V e T , ? ) V ( T o , P ) V ( T , P ) =

V ( To , 0 ) V ( To , 0 ) V ( To , P )

l OOOK

1 .0 1 95 1

0.96504

0.9 1 1 60

0.85894

0.80689

0.75527

0.7040 1

0.65302

0.60225

0.55 1 66

0.50 1 1 9

. . . (20)

In Eg . ( 20) we have taken the values of VeTo,

P)/V(To, 0) calcu lated from the Vinet EOS and the val

ues of VeT, P)/V(To, P) from Eg . ( 1 8) . The values of

VCT, P)/V( To, 0) resulting from Eg. (20) are reported in

Table 3 . The volume expansion is about ten percent

when the temperature is increased from 300 to 3000 K

1 500K 2000K 2500K 3000K

1 .03485 1 .05 1 77 1 .07089 1 .09339

0.97657 0.98892 1 .0023 1 1 .0 1 702

0.92035 0.93687 0.94955 0.96349

0.8656 1 0.87252 0.87972 0.88726

0.8 1 1 97 0.8 1 7 1 9 0.82258 0.828 1 4

0.759 1 3 0.76308 0.767 1 1 5 0.77 1 25

0.70693 0.70989 0.7 1 292 0.7 1 600

0.655 2 1 0.65744 0.65968 0.66 1 97

0.60388 0.60553 0.607 1 9 O.608R7

0.55284 0.55405 0.55526 0.55648

0.50205 0.50292 0.50379 , 0.50466

at P = O. The rate of expansion becomes smal ler as the pressure is increased . At the h ighest pressure and temperature considered in the present study the volume expansion is about one percent only. The experimental and theoretical studies as reviewed by Anderson I also reveal that the thermal expansivity of solids decreases substantially at higher pressures. In order to present a direct comparison between theory and experiment, we use the experimental data reported by Funamori and

R22 INDIAN J PURE APPL PHYS. VOL 37. NOVEMBER 1 999

Table 4 - Data predicted from the V inet universal equation of state for MgSiO� perovskitc

P

(GPa)

25

36

V ( T o. P )

V ( Til . 0 )

O.92 1 R

0.895 1

K (To. P) (GPa)

356

395

K·(To. Pl

3 .62

3 .50

Table 5 - Comparison of the calculated and experimental values of VeT. P)/V(To. 0) for MgSiO, perovskite at 25 GPa and 36 GPa. (a) calculated in the present study as described in the text, and (b) experimental values at 25 GPa from Funamori et al. 15 and at 36 GPa from Funamori and Yagi8

Temp. T(K)

300

400

500

600

70{)

ROO

900

1 000

1 1 00

1 200

1 300

1 400

1 500

1 600

1 700

I ROO

1 900

2000

P = 25GPa

VITo P)/V(To. 0)

(a) (h) 1 5

0.9 2 1 R

0.92 1 &

0.9236

0.9254

0.9273

0.929 1

0.93 1 0

0.9329

0.93411

0.9367

0.9386

0.9406

0.9426

0.9446

0.9466

0.94R7

0.950&

0.9529

0.9550

± 0.0020

0.9225

0.9235

0 9250

0.9266

0.92&3

0.9302

0.932 1

0.934 1

0.936 1

0.9382

0.9404

0.9425

0 9447

0.9469

0.9492

0.95 1 4

0.9535

±O.0035

P = 36GPa VeT, P)/V(To, 0)

(a) (b)8

0.895 1

0. 8967

0.8983

0. R999

0.90 1 5

0.903 1

0.9047

0.9064

0.908 1

0.9098

0.9 1 1 5

0.9 1 32

0.9 1 49

0.9 1 67

0. 9 1 &4

0.9202

0.9220

0.9238

0.8954

± 0.00 1 5

0.8966

0.898 1

0.8997

0.90 1 2

0.9028

0.9046

0.9062

0.9077

0.9092

0.9 1 08

0.9 1 23

0.9 1 39

0.9 1 57

0.9 1 75

0.9 1 94

0.92 1 2

±O.OO 1 5

Yagi8 at 36 GPa, and by Funamori et atl 5 at 2 5 GPa. First we calculate VeTo, P)/V(To, 0), K(To P ) and K' (To, P )

at P = 25 GPa and 36 GPa using the expressions(4.6) based on the Vinet universal EOS . These values are gi ven in

Table 4. Values of VeT, P)/v(To, P) are then calculated with the help of Eg. ( 1 8) . Final ly the values of volume expansion along i sobars at P = 25 and 36 GPa are obtained using Eg. (20) up to a temperature of 2000 K.

The results are compared with the experimental data in

Table 5 . It IS found that the calculated values of VeT, P)/V(To, 0) agree with the experimental values

with in the uncertainties of measurements. This supports

the val idity of the method used in the present study .

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ceramic science, (Oxford University Press, New York) .

1 995.

2

3

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6

7

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Suzuki I, Okajima S & Seya K. J Ph),,, Earth . 27 ( 1 979) 63. Wang Y , Weidner D L & Liebermann R C, Ph),s Earth Planel Inter, 83 ( 1 994) 1 3.

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78.

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