ORIGINAL PAPER

Analysis of high derivative thermoelastic properties of MgO

P K Singh*

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

Received: 08 April 2010 / Accepted: 19 April 2011 / Published online: 15 May 2012

Abstract: We have studied high derivative thermoelastic properties such as the pressure derivatives of bulk modulus and

the volume dependence of the Gruneisen parameter in case of MgO for a wide range of pressures down to compression

V/V0 = 0.6, and temperatures up to 3,000 K approaching the melting temperature. We have used the isothermal pressure–

volume equation of state (EOS) based on the adapted polynomial expansion of second order (AP2) due to Holzapfel. The

results for the P–V–T relationships and high derivative properties have been obtained using the Holzapfel AP2 EOS. The

pressure derivatives of bulk modulus and volume derivatives of the Gruneisen parameter have been determined using

the free volume theory. A relationship between the pressure derivative of bulk modulus and the ratio of pressure and bulk

modulus has been found to hold good.

Keywords: MgO; Gruneisen parameter; Holzapfel AP2 EOS; Thermoelastic properties

PACS No.: 62.20. fq

1. Introduction

MgO (periclase) is an important ceramic material and

geophysical mineral. In its solid state form, this compound

crystallizes with NaCl type (B1) structure, and does not

exhibit any phase transition for [3,00 GPa [1, 2]. The

melting temperature Tm of MgO is about 3,000 K which is

more than thrice of its Debye temperature hD. Such a wide

range of pressure P and temperature T for MgO makes it a

suitable material to be used as a high-temperature pressure

calibrant for experimental work [3]. Reliable pressure–

volume (P–V) data are required along different isotherms

at high-temperatures for a material to be used as a pressure

scale. A number of equation of state (EOS) models have

been constructed by Dorogokupets and Oganov [2, 4–6] for

determining P–V–T results in case of several solids

including MgO. They have used various equations such as

the Birch EOS [7], Vinet EOS [8] and the Holzapfel EOS

[9]. It has been found by Belonoshko et al. [10] that the

Holzapfel adapted polynomial of second order (AP2) EOS

is consistent with the ab initio molecular dynamics results

and gives a correct Thomas–Fermi limit for materials at

extreme compression [11, 12]. In the present study we have

therefore used the Holzapfel AP2 EOS for determining P–

V–T relationships as well as high derivative thermoelastic

properties of MgO using the EOS parameters correspond-

ing to each isotherm. The high derivative properties to be

investigated in the present study are the bulk modulus K

and its pressure derivatives, K 0 ¼ dK=dP and K 00 ¼d2K=dP2, and the Gruneisen parameters c and

q ¼ dlnc=dlnVð ÞT. The results for P, K and K 0 for MgO

obtained from the Holzapfel AP2 EOS are used to establish

a relationship between K 0 and P/K. The expressions for the

volume dependences of c and q obtained by Shanker and

Singh [13] from the free-volume theory [14] have been

used in the present study.

2. Method of analysis

The Holzapfel AP2 EOS can be written as [9]

P ¼ 3K0x�5 1� xð Þ 1þ c2x 1� xð Þ½ � exp c0 1� xð Þ½ � ð1Þ

where x = (V/V0)1/3, K0 is the zero-pressure value of bulk

modulus K, and

c0 ¼ � ln3K0

PFG0

� �ð2Þ

� 2012 IACS

*Corresponding author, E-mail: pramod0002000@yahoo.com

Indian J Phys (April 2012) 86(4):259–265

DOI 10.1007/s12648-012-0048-8

PFG0 ¼ aFGZ

V0

� �5=3

ð3Þ

and

c2 ¼3

2K 00 � 3� �

� c0 ð4Þ

Here K 00 is the value of pressure derivative of bulk

modulus K 0 ¼ dK=dP at zero-pressure. Eq. (1) provides us

with a correct Thomas–Fermi limit of pressure at infinite

compressions [15]. PFG0 is the pressure of Fermi gas with

a = 0.2337 GPa nm5. For MgO, we have Z, the number of

electrons equal to 20 multiplied by the Avogadro number,

and V0 equal to 11.248 cm3/mol.

We use Eqs. (1–4) for determining isothermal pressure–

volume (P–V) relationships for MgO at selected tempera-

tures in the range 300–3,000 K. This is a very broad range

of temperature. The maximum temperature considered here

i.e., T = 3,000 K is more than thrice of hD, the Debye

temperature for MgO [16]. Input values of EOS parameters

along different isotherms are taken from the literature [1, 2,

17, 18] and given in Table 1 along with the values of PFG0,

c0 and c2 calculated from Eqs. (2–4). Values of input

parameters are temperature-dependent, and we have used

appropriate values corresponding to each temperature. For

other equations of state, this method has been used suc-

cessfully by earlier workers [17, 19, 20]. In addition to P–V

isotherms we have also calculated higher derivative ther-

moelastic properties such as the bulk modulus and its

pressure derivatives, first and second Gruneisen parameters

c and q. The expressions for the bulk modulus K and its

pressure derivatives K 0 ¼ dK=dP and K 00 ¼ d2K=dP2 are

obtained using the following relationships

K ¼ �VdP

dV

� �¼ � x

3

dP

dx

� �ð5Þ

K 0 ¼ � x

3

dK

dx

� �ð6Þ

KK 00 ¼ x2

9K

d2K

dx2

� �� K 0 K 0 þ 1

3

� �ð7Þ

where

dK

dx¼ � x

3

d2P

dx2

� �� 1

3

dP

dx

� �ð8Þ

and

d2K

dx2¼ � x

3

d3P

dx3

� �� 2

3

d2P

dx2

� �ð9Þ

The expression for pressure P, as a function of x, is

given by Eq. (1), the Holzapfel AP2 EOS. The pressure

derivatives of bulk modulus are related to the Gruneisen

parameters c and its volume derivative q through the

following relationships based on the free-volume theory

[13, 14]:

c ¼ K 0

2� 1

6� e ð10Þ

where

e ¼ f K � K 0Pð Þ3K � 2fPð Þ ð11Þ

The parameter f takes different values for different

formulations. Thus f = 0 for Slater’s formula [21], f = 1

for the Dugdale–MacDonald formula [22], and f = 2 for

the Vashchenko–Zubarev formula [14]. Value of f for a

given material such as MgO can also be determined by

taking c = c0 = 1.54 [16], at P = 0. Thus we have

c0 ¼K 002� 1

6� f

3ð12Þ

giving f = 1.105 for K 00 ¼ 4:15: Equation (12) is obtained

from Eqs. (10) and (11) at P = 0, c = c0, and K 0 ¼ K 00. The

second Gruneisen parameter q is obtained by

differentiating Eq. (10) as follows [13]

q ¼ V

cdcdV

� �T

¼ 1

c�KK 00

2þ K

dedP

� �ð13Þ

where

KdedP¼ � fKK 00Pþ eK 3K 0 � 2fð Þ½ �

3K � 2fPð Þ ð14Þ

We have used Eqs. (10–14) for determining c and q at

different compressions along isotherms at selected

temperatures.

Table 1 Values of input

parameters [16–18] for MgO

used in the Holzapfel AP2 EOS

T (K) Vo = V(T,0) (cm3/mol) Ko (GPa) K0o PFG0 (GPa) c0 c2

300 11.248 162 4.15 2,626 1.69 0.038

500 11.330 156 4.21 2,594 1.71 0.102

1,000 11.567 141 4.36 2,506 1.78 0.261

1,500 11.841 126 4.53 2,410 1.85 0.442

2,000 12.146 109 4.74 2,310 1.96 0.655

2,500 12.486 94.0 4.95 2,206 2.06 0.868

3,000 12.860 79.0 5.16 2,100 2.18 1.058

260 P. K. Singh

3. Results and discussions

The values of P, K, K 0 and KK 00 with the change in, versus

compression down to V/V0 = 0.6, for MgO, obtained with the

help of Eqs. (1–9), using the input data given in Table 1, along

different isotherms, are reported in Tables 2, 3, 4 and 5. There is

good agreement between the results obtained in the present

study and the pressure, volume, bulk modulus data for MgO

reported in the literature [1, 2, 17, 18, 23, 24]. We study the high

derivative properties using the results for P, K and K 0 given in

Tables 2, 3 and 4. We have plotted 1/K 0 versus P/K in Fig. 1.

These plots based on the Holzapfel AP2 EOS (Eq. 1) satisfy the

following relationship proposed by Shanker et al. [25]:

1

K 0¼ Aþ B

P

K

� �þ C

P

K

� �2

ð15Þ

where A = 1/K 00, B ¼ �K0K 000=K 020 , and C ¼ K 01=K 020� �

K0K 000 þ K 00ðK 00 � K 01Þ�

. Values of A, B and C thus

determined are found to be temperature-dependent, and

given in Table 6. The validity of Eq. (15) has been discussed

recently by Shanker et al. [25, 26] and by Kushwah and

Bharadwaj [27]. Equation (15) gives the following expression

for the second pressure derivative of bulk modulus

KK 00 ¼ �K 02 Bþ 2CP

K

� �1� P

KK 0

� �ð16Þ

Values of KK 00 determined from Eq.(16) using the val-

ues of the parameters A, B and C, as given in Table 6,

fairly agrees with the corresponding values of KK 00 calcu-

lated from the Holzapfel AP2 EOS (Table 5). Eqs. (15) and

(16) are very convenient for determining pressure deriva-

tives of bulk modulus.

Gruneisen parameter c is a quantity of central impor-

tance [16] giving the following relationship between ther-

mal and elastic properties of solids

c ¼ aKT V

CV¼ aKSV

CPð17Þ

where KT is the isothermal bulk modulus, KS the isentropic

bulk modulus, CV and CP are the specific heats at constant

volume and constant pressure, respectively. We have

determined c and its volume derivative q for MgO along

different isotherms with the help of Eqs. (10–14) using the

results for P, K, K 0 and KK 00 given in Tables 2, 3, 4 and 5.

We have, at first, determined e and Kde/dP using Eqs. (11)

and (14) at different compressions and temperatures. The

results are given in Tables 7 and 8. Variations in the values

of c as well as q are quite significant (Figs. 2, 3). c and q

both increase with the increase in temperature, and

decrease with the increase in pressure. The results for c and

q are obtained in the present study using the free volume Ta

ble

2P

ress

ure

–v

olu

me

rela

tio

nsh

ips

for

Mg

Oal

on

gd

iffe

ren

tis

oth

erm

sat

sele

cted

tem

per

atu

res

T.

Th

ere

sult

sb

ased

on

mo

lecu

lar

dy

nam

ics

sim

ula

tio

n[1

0,

36]

are

giv

enw

ith

in

par

enth

eses 30

0K

50

0K

1,0

00

K1

,50

0K

2,0

00

K2

,50

0K

3,0

00

K

V(T

,P)

V(T

,0)

V(T

,P)

(cm

3/m

ol)

P(G

Pa)

V(T

,P)

(cm

3/m

ol)

P(G

Pa)

V(T

,P)

(cm

3/m

ol)

P(G

Pa)

V(T

,P)

(cm

3/m

ol)

P(G

Pa)

V(T

,P)

(cm

3/m

ol)

P(G

Pa)

V(T

,P)

(cm

3/m

ol)

P(G

Pa)

V(T

,P)

(cm

3/m

ol)

P(G

Pa)

11

1.2

48

01

1.3

30

01

1.5

67

01

1.8

41

01

2.1

46

01

2.4

86

01

2.8

60

0

0.9

51

0.6

86

9.2

4

(9.2

2)

10

.76

48

.91

(8.9

3)

10

.98

98

.09

(8.1

0)

11

.24

97

.26

(7.2

4)

11

.53

96

.31

(6.3

5)

11

.86

25

.47

(5.4

4)

12

.21

74

.62

(4.5

8)

0.9

01

0.1

23

21

.2 (21

.2)

10

.19

72

0.5 (2

0.4

)

10

.41

01

8.7 (1

8.8

)

10

.65

71

6.8 (1

6.8

)

10

.93

11

4.7 (1

4.8

)

11

.23

71

2.8 (1

2.7

)

11

.57

41

0.9 (1

0.8

)

0.8

59

.56

13

6.8 (3

6.7

)

9.6

31

35

.6 (35

.6)

9.8

32

32

.5 (32

.4)

10

.06

52

9.5 (2

9.6

)

10

.32

42

5.9 (2

5.9

)

10

.61

32

2.7 (2

2.6

)

10

.93

11

9.4 (1

9.2

)

0.8

08

.99

85

7.2 (5

7.1

)

9.0

64

55

.4 (55

.5)

9.2

54

50

.8 (50

.7)

9.4

73

46

.2 (46

.3)

9.7

17

40

.8 (40

.9)

9.9

89

35

.9 (35

.8)

10

.28

83

0.8 (3

0.5

)

0.7

58

.43

68

4.0

8.4

98

81

.58

.67

57

5.1

8.8

81

68

.69

.11

06

0.9

9.3

65

53

.99

.64

54

6.5

0.7

07

.87

41

20

7.9

31

11

68

.09

71

08

8.2

89

98

.78

.50

28

8.1

8.7

40

78

.49

.00

26

8.0

0.6

57

.31

11

68

7.3

65

16

37

.51

91

52

7.6

97

14

07

.89

51

25

8.1

16

11

28

.35

99

7.7

0.6

06

.74

92

33

6.7

98

22

86

.94

02

12

7.1

05

19

67

.28

81

77

7.4

92

15

97

.71

61

39

Th

ere

sult

sb

ased

on

mo

lecu

lar

dy

nam

ics

sim

ula

tio

n[1

0,

36

]ar

eg

iven

wit

hin

par

enth

eses

Analysis of high derivative thermoelastic 261

formulation derived from the fundamental relationship

between thermal pressure and thermal energy [16], and

using the values of pressure derivatives of bulk modulus

determined from the Holzapfel AP2 EOS (Eq. 1).

It is found that c decreases with the increase in pressure,

i.e. with the decrease in volume along different isotherms.

Earlier workers used the following relationship [16]

cc0

¼ V

V0

� �q

ð18Þ

Eq. (18) is valid only when q is assumed to remain

constant. However, it has been found that q does not

remain constant, but changes significantly with volume

along different isotherms (Fig. 3). It may be more

appropriate to write [28]

q

q0

¼ V

V0

� �k

ð19Þ

To examine the validity of Eq. (19), we have plotted

ln(q/q0) versus ln(V/V0) in Fig. 4 using the results obtained

from Eqs. (10–14). The nature of these plots reveals that kin Eq. (19) does not remain constant. The slope of the plots

in Fig. 4 decreases with the increase in compression along

different isotherms leading to the conclusion that k changes

Table 3 Values of bulk

modulus K(GPa) for MgO at

different compressions and

temperatures

V(T,P)

V(T,0)

Bulk modulus K(GPa)

300 K 500 K 1,000 K 1,500 K 2,000 K 2,500 K 3,000 K

1 162 156 141 126 109 94 79

0.95 199 192 175 158 138 120 102

0.90 245 237 217 197 174 153 131

0.85 302 293 270 246 219 194 167

0.80 373 362 336 308 276 246 213

0.75 462 450 419 387 349 313 273

0.70 576 563 526 489 443 400 351

0.65 725 709 666 621 566 514 455

0.60 920 902 851 797 731 667 593

Table 4 Values of pressure

derivative of bulk modulus (K0)for MgO at different

compressions and temperatures

V(T,P)

V(T,0)

Pressure derivative of bulk modulus K0

300 K 500 K 1,000 K 1,500 K 2,000 K 2,500 K 3,000 K

1 4.15 4.21 4.36 4.53 4.74 4.95 5.16

0.95 3.92 3.98 4.10 4.25 4.42 4.59 4.77

0.90 3.73 3.77 3.88 4.00 4.15 4.30 4.44

0.85 3.56 3.60 3.69 3.80 3.92 4.05 4.17

0.80 3.41 3.44 3.52 3.61 3.72 3.83 3.94

0.75 3.27 3.30 3.37 3.45 3.55 3.64 3.74

0.70 3.15 3.17 3.24 3.31 3.39 3.47 3.56

0.65 3.04 3.06 3.12 3.18 3.25 3.32 3.40

0.60 2.92 2.95 3.00 3.06 3.12 3.19 3.25

Table 5 Values of second

pressure derivatives of bulk

modulus for MgO at different

compressions and temperatures

V(T,P)

V(T,0)

Second pressure derivatives of bulk modulus KK00

300 K 500 K 1,000 K 1,500 K 2,000 K 2,500 K 3,000 K

1 -4.88 -5.08 -5.60 -6.23 -7.04 -7.90 -8.79

0.95 -3.98 -4.13 -4.51 -4.97 -5.55 -6.15 -6.75

0.90 -3.28 -3.39 -3.69 -4.02 -4.45 -4.88 -5.31

0.85 -2.73 -2.82 -3.04 -3.30 -3.62 -3.94 -4.25

0.80 -2.29 -2.36 -2.54 -2.73 -2.98 -3.22 -3.45

0.75 -1.93 -1.99 -2.13 -2.29 -2.47 -2.66 -2.84

0.70 -1.64 -1.69 -1.80 -1.92 -2.07 -2.22 -2.36

0.65 -1.40 -1.43 -1.53 -1.63 -1.75 -1.86 -1.97

0.60 -1.19 -1.22 -1.30 -1.38 -1.48 -1.57 -1.66

262 P. K. Singh

with the increase in pressure. In fact, k the third order

Gruneisen parameter is defined as [13, 28]

k ¼ d ln q

d ln V

� �T

ð20Þ

Eq. (20) yields Eq. (19) only when k is constant, and in

that case the plots given in Fig. 4 should be straight lines.

But this is not the case, it has been found [13] that kdecreases with the increase in pressure and approaches a

finite value k?[ 0 in the limit of infinite pressure [29–31].

4. Conclusions

The thermoelastic properties of MgO have been studied

over a wide range of pressure and temperature with the

help of the Holzapfel AP2 EOS. The available P–V–T

results for MgO based on the molecular dynamics simu-

lation [10, 36] are included in Table 2. The EOS has been

found to be compatible with the ab initio molecular

dynamics results [10], and consistent with the boundary

conditions at zero-pressure and infinite pressure based on

the Thomas–Fermi model [29]. Thus the Holzapfel AP2

EOS is physically more sound than the pressure–volume

relationships used by previous workers [32–35, 37–41].

MgO

300K500K

1000K1500K

2000K2500K

3000K

0.100

0.150

0.200

0.250

0.300

0.350

0.400

0 0.05 0.1 0.15 0.2 0.25 0.3

P/K

1/K

'

Fig. 1 Plots of 1/K0 Vs. P/K for

MgO along different isotherms

Table 6 Values of the parameters for MgO appearing in Eq. (15)

T (K) A B C

300 0.241 0.283 0.525

500 0.238 0.287 0.529

1,000 0.229 0.295 0.538

1,500 0.221 0.303 0.548

2,000 0.211 0.313 0.558

2,500 0.202 0.323 0.568

3,000 0.194 0.330 0.578

Table 7 Values of the

parameter e for MgO based on

Eq. (11)

V(T,P)

V(T,0)

e

300 K 500 K 1,000 K 1,500 K 2,000 K 2,500 K 3,000 K

1 0.368 0.368 0.368 0.368 0.368 0.368 0.368

0.95 0.312 0.311 0.309 0.307 0.304 0.301 0.299

0.90 0.266 0.265 0.262 0.259 0.255 0.251 0.247

0.85 0.229 0.228 0.224 0.221 0.216 0.212 0.208

0.80 0.198 0.197 0.193 0.190 0.186 0.182 0.178

0.75 0.172 0.171 0.168 0.165 0.161 0.157 0.154

0.70 0.151 0.149 0.147 0.144 0.140 0.137 0.134

0.65 0.132 0.131 0.129 0.126 0.123 0.121 0.118

0.60 0.116 0.115 0.114 0.112 0.109 0.107 0.105

Analysis of high derivative thermoelastic 263

Table 8 Values of K(de/dP) for

MgO based on Eq. (14)V(T,P)

V(T,0)

Kde/dP

300 K 500 K 1,000 K 1,500 K 2,000 K 2,500 K 3,000 K

1 -1.26 -1.28 -1.34 -1.40 -1.48 -1.55 -1.63

0.95 -0.96 -0.97 -1.00 -1.03 -1.06 -1.10 -1.13

0.90 -0.74 -0.74 -0.76 -0.77 -0.78 -0.79 -0.80

0.85 -0.58 -0.58 -0.58 -0.58 -0.58 -0.58 -0.58

0.80 -0.45 -0.45 -0.45 -0.44 -0.44 -0.44 -0.43

0.75 -0.36 -0.35 -0.35 -0.34 -0.34 -0.33 -0.32

0.70 -0.28 -0.28 -0.27 -0.27 -0.26 -0.25 -0.25

0.65 -0.22 -0.22 -0.21 -0.21 -0.20 -0.19 -0.19

0.60 -0.18 -0.17 -0.17 -0.16 -0.16 -0.15 -0.15

MgO

300K500K

1000K

1500K

2000K

2500K

3000K

0.50

0.70

0.90

1.10

1.30

1.50

1.70

1.90

2.10

2.30

0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05

V/V0

γγ

Fig. 2 Volume dependence of cfor MgO along different

isotherms

MgO

300K500K

1000K

1500K

2000K

2500K

3000K

0.15

0.30

0.45

0.60

0.75

0.90

1.05

1.20

1.35

1.50

0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05

V/V0

q

Fig. 3 Volume dependence of

q for MgO along different

isotherms

264 P. K. Singh

A relationship between the reciprocal K 0 (1/K 0) and the

ratio (P/K), which is quadratic in P/K (Eq. 15), has been

found to hold good, and used for determining the second

pressure derivatives of bulk modulus. The thermoelastic

behaviour has been studied in terms of the Gruneisen

parameter c and its volume derivatives using the EOS

results in the free-volume theory. The results obtained in

the present study are useful particularly in view of the fact

that MgO is an important high-temperature pressure cali-

brant for experimental work [3]. The results obtained in the

present study reveal that the Gruneisen parameter and its

volume derivatives change significantly with the change in

volume due to the variations in pressure and temperature.

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MgO

300K500K1000K1500K

2000K

2500K3000K

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55

-ln(V/V0)

-ln(q

/q0)

Fig. 4 Plots of ln(q/q0)

Vs. ln(V/V0) for MgO along

different isotherms

Analysis of high derivative thermoelastic 265