Universal for elastic ; MgSiO3 perovskite is an example

József Garai

Department of Earth Sciences, Florida International University, Miami, FL 33199, USA

Abstract

Universal (P-V-T) equation of state is derived from theoretical considerations. Correlation coefficients, root-main-square-deviations, and Akaike Information Criterias are used to evaluate the fitting to the experiments of perovskite 0-109 GPa and 293-2000 K. The proposed equation remains valid through the entire and range and has superior fitting parameters in comparison to the Birch-Murnaghan, Vinet, and Roy & Roy equation of states.

Contents

1. Introduction

2. The equation of state

2.1 Thermal EoS

2.2 The temperature effect on the bulk modulus

2.3 Isothermal EoS

2.3.1 Finite-strain EoS

2.3.2 Inter-atomic potential EoSs

2.3.3 Empirical EoSs

3. Fundamental components of the in

4. New description for the pressure-temperature-volume relationship

4.1 The affect of pressure and temperature

5 Testing the EoS to experiments of perovskite

5.1 Fitting criterias

5.2 Fitting parameters

6. Conclusions

Acknowledgement

References

- 2 - 1. Introduction

The relationships among the pressure, the volume, and the temperature are described by the

Equation of State (EoS). The volume-temperature relationship is described by the definition of

the volume coefficient of expansion [ αV ]

1  ∂V  α V p ≡   (1) V  ∂T p

The relationship between the pressure and the volume is given by the isothermal bulk

modulus [BT ]

 ∂p  BT ≡ −V  . (2)  ∂V T

For the validity of equation (2) it is assumed that the solid is homogeneous, isotropic, non- viscous and has linear . It is also assumed that the stresses are isotropic; therefore, the

principal stresses can be identified as the pressure p = σ1 = σ2 = σ3 . The schematic relationships between the thermodynamic quantities are shown on Fig. 1-a.

Experiments show that both the volume coefficient of expansion and the isothermal bulk modulus are pressure and temperature dependent; therefore, it is necessary to know the derivatives of these parameters.

 ∂αV   ∂αV   ∂BT   ∂BT    ;   ;   ;   (3)  ∂T p  ∂p T  ∂T p  ∂p T

A universal EoS must cover the entire pressure and temperature range; therefore, it is necessary to incorporate all of the derivatives of the volume coefficient of expansion and the isothermal bulk modulus. There is no single expression known for universal (P-V-T) EoS (MacDonald,

- 3 - 1965; Baonza et al., 1996). An attempt is made here to derive and test the first universal EoS for elastic solids.

2. The equation of state

In order to overcome the complexity of the EoS, the common practice is that the temperature of the substance is raised first and then the substance is compressed along the isotherm of interest

(Duffy and Wang, 1998; Angel, 2000). The relevant equations are called the thermal and the isothermal equation of state (EoS) respectively.

The thermal EoS is used to calculate the volume at atmospheric pressure and temperature T

[]V0,T . It is also necessary to know the temperature affect on the bulk modulus []B0 ()T . Using the values of the volume and the bulk modulus at the corresponding temperature the isothermal

EoS calculates the affect of pressure by using the first and the second derivates of the bulk

 ∂B   ∂ 2 B  modulus,   and   at the given temperature.    2   ∂p T  ∂p T

The simplest complete thermodynamic description of a single component solid then requires a minimum of four parameters:

 ∂B   ∂B  ; B T ; ;   α V 0 ()0     (4)  ∂T p=0  ∂p T a more precise description would require six parameters

 ∂α   ∂B   ∂B   ∂ 2 B  α ; V ; B T ; ;   ; and   . V   0 ()0      2  (5)  ∂T p=0  ∂T p=0  ∂p T  ∂p T

- 4 - 2.1 Thermal EoS

The simplest thermal equation of state is derived by the integration of the thermodynamic definition of the volume coefficient of Eq.(1)

T α ()T dT ∫ Vp (6) T0 V0 ()T = V0 (T0 )e .

If a wider temperature range is considered then the temperature dependence of the volume coefficient of thermal expansion should be known. Knowing the first derivative of this parameter allows one to calculate the high temperature values:

 ∂α V  α V p ()T = α V p (T0 )+   ()T − T0 . (7)  ∂T p

The thermodynamic Gruneisen-Anderson parameter [δT ] is defined as (Anderson, 1987):

 ∂ln K T  1  ∂ln K T  δT =   = −   (8)  ∂lnρ p α V  ∂T p

Assuming that the solid at higher follows classical behavior, then the product of

α K is constant and the Gruneisen-Anderson parameter is independent of temperature, Vp T

Anderson et al. (1992) and Shanker (1993) proposed the following isobaric EoS:

1 − V = V 1− α δ T − T δ0 (9) 0 []V0 0 ()0

where the subscript zero values of the parameters refers to the initial temperature of T.0

Assuming that the product of α K is constant and the Gruneisen-Anderson parameter Vp T changes linearly with the volume, the following EoS has been proposed by Kumar (2002) and

Kushwah et al. (1996)

- 5 -  1  V= V0 1− ln[]1− α V A()T − T0  (10)  A 0 

where A = δ0 +1.

Thermal EoS have been suggested by Akaogi and Navrotsky (1984; 1985), assuming that the thermal expansion is quadratic in the temperature, and independent of pressure

V = V 1+ α T − T + α ' T − T 2 , p 0 [ V0 ()()0 V0 0 ] (11) where α ' is the temperature derivative of α at temperature T = T . Taking into consideration V0 V 0 the affect of the pressure the equation can be written as:

−1 ' K'  B ()p − p  0 2 V = V 1+ 0 0 1+ α T − T + α ' T − T , (12) p,T 0   []V0 ()()0 V0 0  B0 

Fei and Saxena (1986) revised the quadratic relationship of Eq. (12) and proposed the following empirical expression:

 1 ' 2 −1  V.p = V0 1+ α V ()T − T0 + α V ()T − T0 − α V ()T − T0 (13)  0 2 0 0 

Assuming linear change as a function of temperature in the volume coefficient of thermal expansion to the following expression:

1 ' 2 αV ()T−T0 + αV ()T−T0 0 2 0 (14) Vp = V0e

The proposed general expression of Eq. (14) for bcc is:

−1 ' 1 K0 α ()T−T + α' ()T−T 2  '   ∂B   p − p  V0 0 V0 0 V = V 1+ B B − T − T 0 e 2 (15) p 0  0  0   ()0  2     ∂T 0  B0 

An empirical expression has been given by Plymate and Stout (1989)

- 6 - 1    2  2  −  ∂BT  1  '  ∂BT  1  ()T−T0  ' αV +  ()T−T0 + αV +  B0  0  ∂T  '   0  ∂T  2 '  2   ∂B   T − T   0 B0B0   0 B B0   V = V 1+ T  0  e  0  0       (16)   ∂T 0  B0    

2.2 The temperature effect on the bulk modulus

The temperature has an effect not only on the volume and the volume coefficient of thermal expansion but on the bulk modulus as well. In order to use the isothermal EoS it is necessary to know the value of the bulk modulus at the temperature of interest, which can be obtained from

 ∂BT  BT0 ()T = BT0 (T0 )+   ()T − T0 . (17)  ∂T p

Theoretically the temperature dependence of the elastic constants can be determined as the sum of the anharmonic terms (Kittel, 1968; Levy, 1986). At sufficiently low temperatures the elastic constant should vary as T4 (Born & Huag, 1956, p 437). Contrary to this suggestion some metallic substances have been found to show a T2 rather than a T4 dependence at low temperatures (Alers, 1961; Chang. & Graham, 1966). There is no general prediction for higher temperatures. Experiments on , conducted at higher than room temperature, show a linear relationship between the bulk modulus and the temperature (Wachtman, 1959).

The third law of requires that the derivative of any elastic constant with respect to the temperature must approach zero as the temperature approaches absolute zero.

Combining this criterion with the observed linear relationship at higher temperatures, Wachtman et al. (1961) suggested an equation in the form of

 T   − 0   T  (18) B.= B0 − b1Te

- 7 - where K0 is the bulk modulus at absolute zero, and b1 and T0 are arbitrary constants.

Theoretical justification for the Wachtman’s Equation was suggested by Anderson (1966).

Based on shock-wave and static-compression measurements on , a linear relationship between the logarithm of the bulk modulus and the specific volume has been detected for metals

(Grover et al., 1973):

 ∆V  ln BT = ln B0 + α  , (19)  V  where α is a constant depending on the . The linear correlation is valid up to 40% volume change. Using this linear correlation Jacobs and Oonk (2000) proposed a new equation of state. They rewrite equation (19) as

 B0 (T)  V 0 (T) = V 0 (T ) + b ln  , m m 0  0  (20)  B (T0 ) 

0 where Vm denotes , T0 the reference temperature and the superscript “0” refers to standard pressure (1 bar). Equation (20) successfully reproduces the available experimental data

for MgO, Mg 2SiO 4 , and Fe2SiO 4 (Jacobs and Oonk, 2000; Jacobs et al. 2001; Jacobs and

Oonk, 2001).

Assuming that the product of the volume coefficient thermal expansion and the bulk modulus is constant at temperatures higher than the Debye temperature analytical for the temperature dependence of the bulk modulus was derived (Garai and Laugier, 2006).

T − δTαVdT ∫T=0 (21) BT = e K T=0 .

where δT is the isothermal Anderson-Grüneisen parameter given by:

- 8 - 1  ∂BT  δT = −   . (22) α B ∂T Vp T  p

Equation (21) was able to mimic experiments with high accuracy for the investigated substances.

2.3 Isothermal EoS

The determined values of the volume and the bulk modulus at temperature T can be used as initial parameters for an isothermal EoS. The isothermal equation of states follow finite strain, interatomic potential, or empirical approach.

2.3.1 Finite-strain EoS The Birch-Murnaghan EoS (Birch, 1947; Murnaghen, 1937, 1944) assumes that the strain of a solid can be expressed as a Taylor series in the finite Eulerian

strain, fE . Expansion to fourth order in the strain yields an EoS:

5  3 3  35 2  p= 3B f 1+ 2f 2 1+ B'−4 f + B B"+ B'−4 B'−3 + f 0 E ()E  ()E  0 ()() E  (23)  2 2  9  

where fE is

2  V  3  0  −1 V (24) f =   . E 2

The Birch-Murnaghan equation (23) is the most widely used isothermal EoS.

Quite recently Sushil et al. (2004) used n=1 instead of the n=2 in the Eulerian strain measure

n  V  3  0  −1 1 3 (25)  V   V0  f E = =   −1 n  V  and using the method of Stacey (2001) proposed a modified three-parameter Eulerian strain EoS,

- 9 - 4 5 7 − − − 9  3 3 −2 3  p.= B0 − A1x + A 2 x − A 3 x + A 4 x  (26) 2   where

V " ' 2 26 x = , A1 = B0 B0 + ()B0 − 3 + V0 9

" ' ' 66 A 2 = 3B0 B0 + ()()B0 − 3 3B0 −8 + 9 60 A = 3B B" + ()()B' − 3 3B' − 7 + 3 0 0 0 0 9 20 and A = B B" + ()()B' − 3 B' − 2 + 4 0 0 0 0 9

The authors claimed that their modified Eulerian strain EoS is more rapidly convergent than the

Birch-Murnaghan EoS.

2.3.2 Inter-atomic potential EoSs The theoretical base for the interatomic potential EoS lays in the thermodynamic relationship

 ∂p  ∂U p = T  −   (27) ∂T V ∂V T,m where m stands for a mol quantity. Assuming constant temperature, as the case for isothermal

EoSs, the first term can be neglected. Approaching the second term, the so-called , with the volume derivative of the biding energy allows determining the pressure- volume relationship. The resulting EoS contains three parameters, the zero pressure values of the molar volume, the isothermal bulk modulus, and the pressure derivative of the bulk modulus.

Using the potential function proposed by Mie and extended by Grunesisen (Poirier, 1991 p.

37; Partington, 1957)

- 10 - A B A B U.r = − − = − + () m n  m   n  (28) r r     V  3  V  3  where r is the interatomic spacing and A, B, m, and n are constants (not necessarily integers) the

P-V equation of state can be written as (Anderson, 1989, p. 83):

 m+3   n+3        3K ()0  V  3   V  3  p.= T  o  −  o   (29) m − n  V   V    

The so-called universal EoS derived by Rose from a general inter-atomic potential (Rose,

1984) which was promoted by (Vinet, 1987-a, -b) is also commonly used:

 3   ()()B'−1 1−fV  1− f V  2  p= 3K 0 2 e (30) f V where

1  V 3 f =   . (31) V    V0 

The Vinet EoS gives very accurate results for simple solids at very high pressure.

Some authors (Parsafar and Mason, 1994; Campbell and Heintz, 1991) pointed out that there is a restriction on Eq. (30) when it is applied to high-pressure phase solids under low pressure

conditions. The use of p = 0 and V = V0 is sometimes arbitrary since the high pressure phase might not exist under this condition. In order to overcome on this problem Fang, 1998 suggested modifying the original Vinet equation (30) by introducing an additional parameter. In this modified equation it was assumed that the isothermal bulk modulus varies linearly with the pressure.

- 11 - Precise knowledge of the interatomic forces in the -free state and their variation with pressure and temperature would allow calculating all the thermodynamic properties. The lack of such knowledge has resulted in many two and three-parameters empirical EoSs.

2.3.3 Empirical EoSs Empirical EoSs can be divided into two major groups. One uses the original Eulerian strain or Interatomic potential EoSs and refines their parameters in order to find a better fit to experiments (e.g. Keane, 1953; Davis and Gordon, 1967: Freud and Ingalls, 1989 ;

Kumari and Dass, 1990). The other approach is to find a mathematical function which gives the best fit to the experiments (e.g. Mao, 1970; Huang and Chow, 1974; Roy and Roy, 2003; Saxena,

2004).

Roy and Roy (2005) give a good review and evaluate the fittings of the currently used EoSs.

Their proposed (Roy and Roy, 1999) three parameter empirical EoS is

 ln()1+ ap  V = V0 1−  , (32)  b + cp  where

1 1  ' '2 ' ''  a= 3 B +1 + 25B +18B − 32B B − 7 2  ()(0 0 0 0 0 ) 8B0  

1 1  ' '2 ' ''  b = 3()(B +1 + 25B +18B − 32B B − 7 )2 8  0 0 0 0 0 

1 1 1  ' '2 ' ''  ' 1  ' '2 ' ''  c = 3()(B +1 + 25B +18B − 32B B − 7 )2 ()B +1 − 3()(B +1 + 25B +18B − 32B B − 7 )2 16  0 0 0 0 0  0 8  0 0 0 0 0 

They used shock compression data of different metals (Nellis 1988) and the calculated EoS of halite (Decker, 1971) to evaluate the proposed equation up to ultra high .

- 12 - The empirical nature of these equations usually leads to a lack of generality and careful inspection reveals that a particular equation is typically gives excellent fitting only for special substances or a specially selected pressure and/or temperature range.

Many of the parameters in the EoS are inter-related, which adds to the complexity of calculations. The optimum values of each of the interrelated parameters have to be determined by confidence ellipses (Angel, 2000, Mattern et al., 2005). The thermodynamic description of solids is complicated, time consuming, labor intensive, and expensive.

3. Fundamental components of the volume in solid phase

Contrarily to gasses Avogadro’s principle does not apply to solids. in solid phase

occupies an initial volume []Vo at zero pressure and temperature.

m Vo = nVo , (33)

m In Eq. (33) n is the number of moles and Vo is the molar volume of the substance at zero pressure and temperature. The pressure modifies this initial volume by inducing elastic while the temperature by causing thermal deformation. Using equations (1) and (2) the actual volume at given pressure and temperature can be calculated (Cemic, 2005) by allowing one of the variables to change while the other one held constant

T p 1 α dT − dp ∫ Vp=0 ∫p=0 B V = V e T=0 or V = V e T=0 (34) []T p=0 0 []p T=0 0 and then

T p 1 α dT − dp ∫ Vp ∫p=0 B V = V e T=0 or V = V e T (35) []T p []p T=0 []p T []T p=0

- 13 - These two steps might be combined into one and the volume at a given p, and T can be calculated:

T p 1 T p 1 dT dp dT dp αVp=0 − αVp − ∫T=0 ∫p=0 BT ∫T=0 ∫p=0 BT=0 (36) Vp,T = V0e = V0e

The total volume change related to the temperature will be called thermal volume [Vth ]while the total volume change resulted from elastic deformation will be called elastic volume[Vel ].

The thermal volume at zero pressure is

 T   ∫αVdT  th T=0 [VT ]p=0 = Vo e −1 , (37)     while the elastic volume at zero temperature is

p 1  − dp   ∫ BT  el p=0 [Vp ]T=0 = Vo e −1 . (38)    

The thermal volume at pressure p is

p p T 1 1   − dp − dp αVdT ∫ BT ∫ BT  ∫  th th p=0 p=0 T=0 [VT ]p = [VT ]p=0 e = Voe e −1 , (39)     and the elastic volume at temperature T is

T T p  1  αVdT αVdT − dp ∫ ∫  ∫ BT  el el T=0 T=0 p=0 [Vp ]T = [Vp ]T=0 e = Voe e −1 . (40)    

The actual volume is the sum of the volume components:

el th el th []VT p = Vo + []Vp + []VT or [Vp ] = Vo + [Vp ] + [VT ] (41) T=0 p T T p=0

Since

- 14 - V = V []T p []p T (42) from Eq (41) follows that

th th el el [VT ]p −[VT ]p=0 = [Vp ]T −[Vp ]T=0 . (43)

The compressed part of the thermal volume is the same as the expanded part of the elastic volume. Since the volume difference in Eq. (43) both temperature and pressure dependent I will

th−el call this volume difference to thermo-elastic volume [∆Vp ]T

th−el th th el el [Vp ]T = [VT ]p −[VT ]p=0 = [Vp ]T −[Vp ]T=0 . (44)

The thermoelastic volume can be calculated as:

T p    1  αVdT − dp  ∫   ∫ BT  th−el T=0 p=0 [Vp ]T = Vo e −1 e −1 . (45)        

It can be concluded that the actual volume comprises from four distinct volume parts, initial

th el volume, thermal volume at zero pressure [VT ] , elastic volume at zero temperature [Vp ] , p=0 T=0

th−el and thermo-elastic volume []Vp (Fig. 2). T

th el th−el V = Vo + []VT + [Vp ] + []Vp . (46) p=0 T=0 T

These fundamental volume components are related to the thermo-physical variables as:

Vo = f ()n , (47)

th [VT ]p=0 = f ()n,T , (48)

el [Vp ]T=0 = f ()n,p , (49)

- 15 - and

th−el [Vp ]T = f ()n,T,p . (50)

4. New description for the pressure-temperature-volume relationship

Recent study (Garai, 2007) suggested that the mechanical equivalency of or the first law of thermodynamics is correct only if the energy from the mechanical is conserved by the same physical process as the heat. This condition is not satisfied in solid phase; therefore, heat and work is not interchangeable and they must be treated separately. The lack of

‘communication’ between the heat and work results that thermoelastic volume should not exist

Eq. (50). Thus the volume should comprise only from the initial, thermal, and elastic .

Deducting the thermal-elastic volume from the actual volume gives

T p  1  αVdT − dp  ∫ ∫ BT  th−el T=0 p=0 V −[Vp ]T = Vo e + e −1 . (51)    

th−el The elimination of the thermoelastic volume requires the transformation of the V −[Vp ]T volume to the actual volume V:

T p  1  αVdT − dp  ∫ ∫ BT  th−el T=0 p=0 V −[Vp ]T = Vo e + e −1 ⇒ V . (52)    

This transformation can be achieved by redefining the volume coefficient of thermal expansion and the bulk modulus

α ⇒ α B ⇒ B . Vp o T o (53)

- 16 - Superscript o is used for the new parameters which are defined as:

1 ∂V th α o ≡ (54) Vp=0 ∂T and

∂p B ≡ −V . (55) o T=0 ∂V el

Using the new definition of the volume coefficient of thermal expansion the thermal volume is

 T   ∫ αodT  th T=0 Vo = Vo e −1 (56)     while the elastic volume can be calculated as:

p 1  − dp   ∫ Bo  el p=0 Vo = Vo e −1. (57)    

Subscript o is used to indicate that these fundamental volume parts were determined by using Bo

and α o . The actual volume is the sum of the initial, thermal and elastic volumes (Fig. 1)

el th V = Vo + Vo + Vo . (58)

Substituting the fundamental volume components gives the actual volume

p T  1  − dp αodT  ∫ Bo ∫  p=0 T=0 V = Vo e + e −1 . (59)    

Eq. (59) is identical with the required expression given in Eq. (52) except the conventional volume coefficient of expansion and bulk modulus has been replaced with the newly defined ones. Thus the transformation of the volume is completed by the introduction of the new definitions [Eq. (54) and (55)].

- 17 -

4.1 The affect of pressure and temperature

Equation (59) assumes a constant value for volume coefficient of thermal expansion and the bulk modulus. In order to take into consideration the pressure and temperature affect on these parameters linear factors are introduced. The bulk modulus is approximated as

Bo ()p,T = ap + bT + Bo . (60)

where a and b are linear factors relating to the pressure and temperature respectively. Equation

(59) can be rewritten then as:

p T  1  − dp αodT  ∫ ap+Bo ∫  p=0 T=0 V = Vo e + e −1 (Universal 4) (61)     and

p T  1  − dp αodT  ∫ ap+bT+Bo ∫  p=0 T=0 V = Vo e + e −1 (Universal 5). (62)    

The numbers after Universal refers to the number of parameters in the equation.

Investigating highly symmetrical atomic arrangements linear correlation between the volume coefficient of thermal expansion and the thermal was detected (Garai, 2006).

Based on this correlation the integral

T α dT (63) ∫ o T=0

is approximated by an area of trapezoid (Fig. 3) The integral below the Debye temperature [Tθ ]

- 18 - is then

T≤Tθ 2 α oT  α o ()TdT ≈   (64) ∫ 2T T=0  θ  T≤Tθ while at temperatures higher than the Debye temperature is

T T>T α ()T.dT ≈ []α (T − T ) θ (65) ∫ o o θ Tθ Tθ

Combining the two parts Eqs. (64) and (65) gives the general formula

T α T 2  T  α ()T dT ≈ I ()T o + []1− I ()T α T − θ  , (66) ∫ o A A o T=0 2Tθ  2  where

1 if T ≤ Tθ IA ()T =  . (67) 0 if T > Tθ

Substituting Eq. (66) into Eq. (61) gives

p  1 2  − dp αoT  Tθ   ∫ ap+Bo IA ()T +[]1−IA ()T αo  T−   p=0 2Tθ  2  V = Vo e + e −1 (Debye 4). (68)    

Assuming linear pressure dependence for the Debye temperature requires the introduction of an additional multiplier c

Tθ ()p = cp + Tθ (p = 0 ). (69)

Equation (68) can be written then as:

p  1 2  − dp αoT  cp+Tθ   ∫ ap+Bo IA ()T +[]1−IA ()T αo  T−   p=0 2(cp+Tθ )  2  V = Vo e + e −1 (Debye + pressure 5). (70)    

- 19 - The validity of Eqs. (61), (62), (68), and (70) will be tested to experiments and I will call these equations to Universal 4 parameter, 5 parameter, Debye, and Debye plus pressure respectively.

Equation (61) has an analytical solution for the pressure

 V αT  ln − e +1  Vo  p = Bo . (71)  V αT  − a ln − e +1 −1  Vo 

and for the temperature

−p  V b  ln − e ap +Bo +1   Vo (72) T =   . α o

Equation (62) has analytical solution only for the pressure

 V αT  ln − e +1  Vo  p = ()Bo + cT . (73)  V αT  − a ln − e +1 −1  Vo 

Iteration was used to calculate the temperature from Eq. (62).

5. Testing the EoS to experiments of perovskite

Perovskite, the most abundant of the mantle, has been extensively investigated at high pressures and temperatures. The availability of a wide range of pressure and temperature experiments makes this mineral ideal for thermodynamic studies. Experiments up to 25-30 GPa pressure usually use multi-anvil apparatus while at higher pressures diamond anvil cells (DAC) are used. The experimental results of multi anvil press (Funamori, et al., 1996; Wang et al.,

- 20 - 1994; Morishima et al., 1994; Utsumi et al., 1995) and diamond anvil (Fiquet et al., 1998; 2000;

Saxena et al., 1999) are used in this study (Fig. 4).

5.1 Fitting criteria

The fitting accuracy of empirical EoSs with the same number of parameters is evaluated by correlation coefficients and/or root-mean-square deviations (RMSDs). The number of wiggles of the data deviation curve allows detecting the possible standard error in the fitting equation (Roy and Roy, 2005). The fit quality of models using different numbers of parameters can not be evaluated by their correlation coefficients only (Lindsey, 2004; Burnham and Anderson, 2002;

2004). The test devised assessing the right level of complexity is the Akaike Information

Criteria AIC (Akaike, 1973; 1974). Assuming normally distributed errors, the criterion is calculated as:

 RSS  AIC= 2k + n ln  , (74)  n  where n is the number of observations, RSS is the residual sum of squares, and k is the number of parameters. The preferred model is the one which has the lowest AIC value.

5.2 Fitting parameters

The calculated fitting parameters, correlation coefficient, RMSD, and AIC are given in Table

1 for Eqs. (61), (62), (68), and (70). The best fit is achieved by the universal 5 parameter equation [Eq. (62)]. The approximations used for the volume coefficient of thermal expansion in Eqs. (68) and (70) did not increase the fitting and better fit was achieved by assuming constant

- 21 - value for the volume coefficient of thermal expansion. Based on visual inspection the residuals seem to be random (Fig. 5). The RMSD or uncertainty of the universal 5 parameter equation is

0.05 cm3, 0.8 GPa, and 123 K for the volume, pressure, and temperature respectively. These values are in the range of the uncertainties of the experiments, since the laser heated DAC data have an order of magnitude uncertainty in the temperature measurement (Shim and Duffy, 2000).

The uncertainty is significantly smaller if an electrical heater is used in the DAC. Having the same uncertainty from the fitting of the universal as from the experiments is a clear indicative that the proposed EoS correctly describes the P-V-T relationship of perovskite.

Starting from 300K the experiments were separated into 200 K wide temperature groups.

Using the three most widely used isothermal EoS, Birch-Murnaghan, Vinet, and Roy & Roy, equations (23), (30), and (32) respectively the fitting parameters were determined for each of these temperature range. Using averages determined from the overall fitting the RMSD, and

AIC was calculated for the universal 4 and 5 parameter equations in of the temperature range.

The fitting parameters were also determined for the conventional bulk modulus [ K T conv.] as:

−p ap+KT (75) V = V0T e (Conventional).

The calculated values of the five parameter universal EoS have equal or better fitting parameters than any of the investigated isothermal EoSs (Table 2). The four parameters universal EoS has better fitting parameters in seven temperature ranges than the conventional bulk modulus equation. The only exception is the highest temperature range (1700-1900K). The better fitting indicates that the proposed new definition of the bulk modulus describe the volume pressure relationship more accurately than the conventional one.

- 22 - 6. Conclusions

Assuming that adiabatic conditions do not exist in elastic solid phase universal P-V-T EoS has been derived by using the newly defined expressions of the volume coefficient of thermal expansion and the bulk modulus. Using the high pressure and temperature experiments of perovskite, the fitting parameters, correlation coefficients, RMSD, and AIC were calculated.

The calculated fitting parameters of the new universal EoS are superior to the Birch-Murnaghan,

Vinet, and Roy & Roy equations. Additional advantage of the proposed universal EoS is its simplicity and the fact that it allows the back and forward calculation of any of its quantities.

Acknowledgement

I would like to thank Alexandre Laugier for his encouragement and Mike Sukop for reading and commenting the manuscript. This research was supported by Florida International University

Dissertation Year Fellowship.

- 23 - References

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- 25 -

Figure 1. Thermo-physical relationships (a) solid phase conventional description (b) proposed new description (c) new description at constant volume. (The arrow ↔ represent a reversible while → represents an irreversible relationship or process.)

- 26 -

Figure 2. The fundamental volume components of the actual volume, in accordance to the conventional thermo-physical description of solids.

Figure 3. Approximation used for the volume coefficient of thermal expansion in Eqs. (68), and

(70).

- 27 -

Figure 4. Pressure-temperature range covered by the data.

- 28 -

Figure 5. The residuals plotted against (a) volume (b) pressure (c) temperature.

- 29 - 3 -5 -1 0-109 GPa [N=257] Ko Vo[cm ] αo [10 K ] a b c R RMSD AIC [GPa] V(p,T) (Univ. 4) 274.2 24.345 1.451 1.291 0.99934298 0.055 -1486.1 V(p,T) (Univ. 5) 280.2 24.232 2.362 1.405 -0.0277 0.99957336 0.044 -1595.0 p(V,T) ( Univ. 4) 276.3 24.370 1.303 1.277 0.99932559 1.038 27.3 p(V,T) ( Univ. 5) 282.9 24.210 2.499 1.406 -0.0319 0.99960000 0.797 -106.7 T(V,p) ( Univ. 4) 269.8 24.326 1.620 1.323 0.95180023 144.2 2563.1 T(V,p) ( Univ. 5) 123.1 2474.0 V(p,T)(Debye; 4) 273.8 24.483 1.774 1.305 0.9991373 0.063 -1416.1 V(p,T)(Debye+pres; 5) 273.5 24.433 3.857 1.467 51.6 0.9992888 0.057 -1463.7 Average ( Univ. 4 ) 273.4 24.347 1.458 1.297 Average ( Univ. 5) 281.5 24.221 2.431 1.405 -0.0298

Table 1 P-V-T fitting parameters and results.

- 31 -

- 33 -

Table 2 P-V fitting parameters and results

- 34 -