Analysis of the logarithmic equation of state for materials at high pressures

Physica B 410 (2013) 259–261

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Analysis of the logarithmic equation of state for materials at high pressures

a r t i c l e i n f o

abstract The Hartmann–Haque equation of state widely used in recent literature for polymers, liquids and solids has been demonstrated here to be a simplified version of the generalised logarithmic equation of state ¨ due to Poirier and Tarantola. The identity for the third-order Gruneisen parameter in terms of the pressure derivatives of bulk modulus at infinite pressure has been used to test the validity of the equations of state. The thermodynamics of solids at extreme compression requires a positive and finite ¨ value of the third-order Gruneisen parameter at infinite pressure. It has been found that both the logarithmic equations of state considered here fail to satisfy this constraint. & 2012 Elsevier B.V. All rights reserved.

Keywords: Hartmann–Haque equation of state Pressure derivatives of bulk modulus ¨ Third-order Gruneisen parameter Extreme compression behaviour

1. Introduction

and

On the basis of thermodynamic principles, Hartmann and Haque [1,2] formulated an equation of state (EOS) for studying the pressure–volume–temperature (P–V–T) relationship in case of polymers and liquids. This EOS is written as [1,2]  g    3=2 P V V T þ ¼ ln ð1Þ K0 V 0 V0 T0

A3 ¼

where V0 is the volume V at pressure P¼ 0, K ¼0 is the bulk ¨ modulus at zero-pressure, and g is the Gruneisen parameter assumed to be a constant, independent of compression. T0 is the characteristic temperature of the material. At T ¼0, Eq. (1) gives the pressure–volume relationship  g   P V V ¼ ln ð2Þ K0 V 0 V0 The Hartmann–Haque Eq. (1) has been widely used for polymers and liquids [3–5] and also for solids [6]. In the present paper we demonstrate that Eq. (2) is a simplified version of the generalised Poirier–Tarantola logarithmic EOS [7,8]. To test the validity of the logarithmic EOS, we have used the identity for ¨ the third-order Gruneisen parameter in terms of pressure derivatives of bulk modulus at infinite pressure [9,10]. The thermodynamics of solids at extreme compression [11] requires a positive and finite ¨ value of the third-order Gruneisen parameter at infinite pressure.

2. Generalised logarithmic EOS Poirier and Tarantola [7] have formulated a logarithmic EOS based on the finite strain theory using the Hencky strain in terms of ln (V/V0). This EOS can be written as 2

Px ¼ A1 ln x þ A2 ðln xÞ A3 ðln xÞ

3

ð3Þ

   K0 0 K 0 K 000 þ K 02 0 3K 0 þ3 6

ð6Þ

where K 00 and K 000 are respectively the values of K 0 ¼ dK=dP and 2 K 00 ¼ d K=dP2 , both at zero-pressure. All physically acceptable equations of state must satisfy the Stacey thermodynamic constraint [12,13] according to which K 01 is greater than 5/3. Here K 01 is the value of K 0 ¼ dK=dP in the limit of infinite pressure (P-N) or at extreme compression (V-0). Eq. (3) does not satisfy this constraint as it yields K 01 ¼1. In order to remove this inadequacy, Eq. (3) can be generalised in the same manner in which the Rydberg EOS [14,15] was generalised by Stacey [13]. The generalised logarithmic EOS can be written as follows 0

PxK 1 ¼ a1 ln x þa2 ðln xÞ2 a3 ðln xÞ3

ð7Þ

where a1 ¼ K 0 a2 ¼



ð8Þ

  K0  0 K 0 2K 01 2

ð9Þ

and a3 ¼



  K0  2 K 0 K 000 þ K 00 3K 00 K 01 þ 3K 01 2 6

ð10Þ

Eqs. (9) and (10) reduce to Eqs. (5) and (6), respectively, for K 01 ¼1. The expression for bulk modulus K¼  V(dP/dV)¼ x (dP/dx) is obtained from Eq. (7) as follows:   0 ð11Þ KK 01 P xK 1 ¼ a1 2a2 ln x þ 3a3 ðln xÞ2 Eq. (11) divided by Eq. (7) gives K K 01 ¼ f ðxÞ P

ð12Þ

where x ¼V/V0. A1, A2, and A3 are constants for a given material expressed as follows [11]:

where f(x) is a function of x ¼V/V0 written as follows:

A1 ¼ K 0

ð4Þ

f ðxÞ ¼

ð5Þ

In the limit of extreme compression, Eq. (13) reveals that f(x) tends to zero when x tends to zero. Dividing the numerator and

A2 ¼



  K0  0 K 0 2 2

0921-4526/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physb.2012.10.023

a1 2a2 ln x þ 3a3 ðln xÞ2 a1 ln x þa2 ðln xÞ2 a3 ðln xÞ3

ð13Þ

260

P.K. Vidyarthi and B.P. Singh / Physica B 410 (2013) 259–261 3

denominator of the right hand side of Eq. (13) by ðln xÞ and then taking the limit x-0, we get ½f ðxÞx-0 ¼ 0

ð14Þ

Eqs. (26) and (27) taken together yield ! K 2 K 000 ¼ K 01 KK 00

ð28Þ

1

Eq. (12) then yields   1 P 0 ¼ K 1 K1

ð15Þ

Eq. (15) represents an identity originally due to Stacey [13,16] satisfied by all such equations of state for which K 01 is greater than zero. Eq. (12) has been found suitable for investigating high pressure derivatives of bulk modulus at infinite pressure [13,15]. Expressions for pressure derivatives of bulk modulus up to thirdorder are obtained by differentiating Eq. (12) with respect to P. Thus we find   K K 0 K0 ¼ xf ðxÞ ð16Þ P P     K K 2K 0 00 KK 00 þ K 0  K0 ¼ xf ðxÞ þ x2 f ðxÞ P P P

ð17Þ

"       # K K K 6K K 2 K 2 K 000 þ4KK 00 K 0  þ K 02 K 0   K0 P P P P P 0

00

000

ð18Þ ¼ ½xf ðxÞ þ3x2 f ðxÞ þ x3 f ðxÞ    0 00 2 000 3 where f xÞ ¼ df ðxÞ=dx,f xÞ ¼ d f ðxÞ=dx2 and f xÞ ¼ d f ðxÞ=dx3 : At extreme compression (x-0), Eq. (13) on differentiation and taking the limit yields the following results: 0

½xf ðxÞx-0 ¼ 0

Eq. (28) is a result based on the generalised logarithmic EOS (Eqs. (7) and (13)).

3. Hartmann–Haque EOS Eq. (2) due to Hartmann and Haque [1,2] is examined here by ¨ determining high derivative properties. The Gruneisen parameter g in Eq. (2) should be replaced more appropriately by K 01 . By differentiating Eq. (2) with respect to V, we get    g dP V ¼ K0 K ¼ V þ gP ð29Þ dV V0 Eq. (29) reveals that K ¼K0 at P¼0 (V¼V0), and K becomes infinite at infinite pressure and extreme compression (V-0). Eq. (29) can be written as follows with the help of Eq. (2):    1 K V ¼ ln þg ð30Þ P V0 In the limit V-0, the first term on the right hand side of Eq. (30) becomes zero, so that Eqs. (15) and (30) give   K ¼ g ¼ K 01 ð31Þ P 1 Also, on differentiating Eq. (29) with respect to P, we get

ð19Þ

K0 ¼

00

½x2 f ðxÞx-0 ¼ 0

ð20Þ

 dK g ¼ gþ KgP dP K

At infinite pressure, Eq. (32) yields g, and at P¼0 we get g ¼ K 00 =2. Thus K 01 ¼ K 00 =2. Eq. (2) can now be written as

and 000

½x3 f ðxÞx-0 ¼ 0

ð21Þ 0

Eqs. (19) and (21) when substituted in Eqs. (16)–(18) give the following results at infinite pressure, represented by the subscript 1.   K ¼0 ð22Þ K 0 P 1 ½KK 00 1 ¼ 0

ð23Þ

and ½K 2 K 000 1 ¼ 0

ð24Þ

The quantities appearing in Eqs. (22)–(24) become zero at infinite pressure, but their ratios remain finite. Dividing Eq. (17) by Eq.(16), we get   00 KK 00 K xf ðxÞ  0  þ K 0 2 ¼ 1 0 ð25Þ P K K=P f ðxÞ Using the expression for f(x) given by Eq. (13) we find that the right hand side of Eq. (25) becomes zero at infinite pressure. With the help of the identify represented by Eq. (15), Eq. (25) yields at infinite pressure " # 1 K 2 K 00  0 ¼ K 01 ð26Þ K 1 KK 0 P

PxK 1 ¼ K 0 ln

000

K K KK 00

! ¼ 2K 01  1

1 K 01

2

00

K K KK 0 P

! ð27Þ 1

V V0



ð33Þ

and 2

Higher pressure derivatives of bulk modulus at infinite pressure satisfy the following identity [9,10]:



This is the appropriate form of the Hartmann–Haque EOS [1,2] describing the pressure–volume data for polymers, liquids and solids [3–6] with the help of fitted parameters. For investigating the thermoelastic properties of materials at high pressures, an EOS must satisfy the thermodynamic constraints at infinite pressure. The properties extrapolated to infinite pressure or to extreme compression by considering a material to remain in the same phase have a powerful and fine control on the properties to be determined in the finite range of pressures. The calculation of high derivative properties such as the pressure derivatives of bulk modulus with the help of an EOS is much more crucial than the P–V data determined by adjusting the EOS parameters. The P–V relationship is based on the first derivative of interatomic potential energy, whereas the bulk modulus and its pressure derivatives up to third order depend on the derivatives of potential energy up to fifth order [11]. Expressions for the high pressure derivatives of bulk modulus are obtained from Eq. (32) with g ¼ K 01 as follows:     KK 00 ¼ K 01 ½ K 0 K 01 K 0 1K 01 P=K  ð34Þ

1

2

ð32Þ K 01 ¼

000

K K ¼ KK

00

!   2 K 01 P K0P 0 K þ K 0 K 01 2 1 K K

ð35Þ

Eqs. (34) and (35) reveal that KK 00 and K 2 K 000 both become zero at infinite pressure. At P¼0, we get K 0 K 000 ¼ K 01 2

ð36Þ

P.K. Vidyarthi and B.P. Singh / Physica B 410 (2013) 259–261

and K 20 K 000 0

¼ 4K 01 3

ð37Þ

It should be emphasised here that the Hartmann–Haque EOS is obtained from the generalised logarithmic Equation (7) by taking a2 and a3, both equal to zero. Eq. (9) gives K 01 ¼ K 00 =2 when a2 ¼0. When we put K 01 ¼ K 00 =2 and a3 ¼0 in Eq. (10), we get Eq. (36) based on the Hartmann–Haque EOS. Eq. (35) yields the following expression for the higher pressure derivatives of bulk modulus at infinite pressure: " #  K 2 K 000 1K 0 P=K ¼ K 01 3 ð38Þ 00 00 KK KK 1 1

Eq. (38) based on the Hartmann–Haque EOS with the help of the identity, Eq. (27) yields the same expression as given by Eq. (28) based on the generalised logarithmic EOS.

4. Discussions and conclusions The Hartmann–Haque EOS most widely used in recent literature [1–6] for polymers, liquids and solids has been demonstrated here to be a simplified version of the generalised logarithmic EOS. Both EOS, Eq. (7) as well as Eq. (33), satisfy the basic  00identity,  Eq.  (15),  due to Stacey [16], and also reveal that KK 1 and 2 000 K K become zero. The most critical test of an EOS can be 1 ¨ performed using the identity for the third-order Gruneisen parameter l1 at extreme compression [9,10]: ! K 2 K 000 0 l1 ¼ K 1  ð39Þ KK 00 1

where   d ln q l1 ¼ d ln V 1 q1 ¼



d ln g d ln V

 ð41Þ 1

and

g1 ¼

K 01 1  6 2

recognisable defect in the theories considered, but that may be related to the lambda-infinity problem. Finally it should be mentioned that the Stacey reciprocal K-primed EOS [11,12] yields the following expression [9]: " # ! K 2 K 000 K 02 0 1 ¼  K þ ð43Þ 1 KK 00 K 00 1

On comparing Eq. (43) and Eq. (39), we get

l1 ¼

ð42Þ

Eq. (42) was found by Stacey and Davis [11] to have the status of an identity. The Hartmann–Haque EOS and the generalised logarithmic EOS, both, have been found to yield Eq. (28). A comparison of Eqs. (28) and (39) gives l1 ¼ 0 This is the common result based on both EOS. However, according to the thermodynamic constraint [11,13], l1 must be positive and finite. If l1 ¼ 0, then q1 must be positive and finite, and also g1 must be zero. But this is not physically acceptable because g1 is always positive and finite, Eq. (42), and q1 tends to zero. Thus the result, l1 ¼ 0, based on the two EOS is not consistent with the thermodynamic constraints [10,11]. In the Hartmann–Haque EOS, gamma is assumed to be a constant, independent of compression. Then the conflict with gamma-infinity would be a

K 02 1 K 00

ð44Þ

Eq. (44) gives a positive and finite value forl1 . It is also revealed from Eq. (44) that l1 is a material-dependent parameter in the same sense as K 00 and K 01 depend on the material. The Kushwah logarithmic EOS [17], based on the parameters [18] derived from the Stacey reciprocal K-primed EOS, gives l1 ¼ 1, a 0 positive and finite value. This EOS [17,18] is an expansion of PxK 1 in powers of ln(2 x) in place of lnx in Eq. (7).

Acknowledgements Thanks are due to both the reviewers for their helpful comments. Thanks are also due to Professor Jai Shanker for some useful discussions. References [1] [2] [3] [4] [5]

ð40Þ

261

[6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]

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P.K. Vidyarthi, B.P. Singh n Department of Physics, Institute of Basic Sciences, Dr. B.R. Ambedkar University, Khandari Campus, Agra, India E-mail addresses: [email protected], [email protected] (B.P. Singh) Received 6 August 2012 Available online 2 November 2012

n

Corresponding author. Tel.: þ91 9837019242.