Indian Journal of Pure & Applied Physics
Vol. 48, June 2010, pp. 403-409
Thermoelastic properties of NaCl crystal using
Holzapfel AP2 equation of state
P K Singh
Department of Physics, Institute of Basic Sciences, Khandari, Agra 282 002
E-mail: [email protected]
Received 25 September 2009, revised 5 January 2010; accepted 8 March 2010
Pressure P, bulk modulus K and its first and second pressure derivatives K′ and K′′, and the Grüneisen parameter γ and
its volume derivative q for NaCl crystal down to a compression, V/V0 = 0.65 have been studied. These properties have been
calculated along different isotherms at selected temperatures in the range 300-1050 K using Holzapfel AP2 equation of state.
The results obtained have been found to be in good agreement with the data reported in the literature. A reciprocal K-primed
equation, quadratic in P/K, has been found to satisfy the relationship between P, K and K′ along with different isotherms,
and is also consistent with the values of K′′ at different compressions and temperatures. Volume dependence of the
Grüneisen parameter has been discussed in terms of the existing recent formulations.
Keywords: Isothermal pressure-volume relationship, NaCl crystal, Grüneisen parameters, Pressure derivatives,
Bulk modulus, Equation of state
1 Introduction
NaCl is an important material widely used as a
pressure guage in high-pressure diffraction
experiments1,2. NaCl has a stable structure (B1) up to
a pressure of about 30 GPa. This pressure is
significantly higher than the value of bulk modulus
(K0=24GPa) for NaCl at ambient pressure3. Also, the
melting temperature of NaCl (Tm=1074K) is three to
four times larger than its Debye temperature3 θD. Thus
we have a wide range of pressures and temperatures
for NaCl in a given phase. It is therefore desirable to
investigate high derivative thermoelastic properties
and equation of state (EOS) for this material at high
pressures and high temperatures. The high derivative
properties include the pressure derivatives of bulk
modulus and volume derivatives of the Grüneisen
parameter4. These properties can be determined with
the help of an equation of state (EOS) having a
fundamental basis.
Serious efforts have been made to develop an
isothermal equation of state for NaCl5-8. In the model
of Decker5, the Grüneisen parameter γ is assumed to
be proportional to volume V, whereas Roy and
Roy8 have used the assumption that the product of
thermal expansivity α and isothermal bulk
modulus KT remains constant. These two assumptions
are approximately equivalent in view of the
relationship3:
γ=
αK TV
CV
…(1)
where CV is the specific heat at constant volume.
The results obtained by Roy and Roy8 are very
close to the Decker’s calculated isotherms. It has been
found by Brown7 that the largest contribution to
systematic uncertainty at the highest pressures arises
from the lack of knowledge of the volume
dependence of γ , and apparent errors in the Decker
scale are as large as about 3% for the 300 K isotherm.
The experimental value of pressure P=20GPa
determined by Li et al.9 using the ultrasonic and X-ray
techniques simultaneously is about 8% and 12%
higher than those inferred from the Brown7 and the
Decker5 pressure scales, respectively.
An alternative method, which does not require the
knowledge of volume dependence of γ, has been used
recently by Chauhan and Singh10. They have
mentioned the well known shortcomings11,12 of the
Birch equation, and emphasized the suitability of the
Vinet EOS13 based on the Rydberg potential
function14. However, the Vinet-Rydberg EOS is not
consistent with the asymptotic behaviour of solids at
extreme compression15,16. The Holzapfel EOS in the
AP2 form (Adapted polynomial of second order) has
been found to be more appropriate for studying the
high pressure behaviour of solids17,18. In the present
INDIAN J PURE & APPL PHYS, VOL 48, JUNE 2010
404
study, this EOS is used for determining isothermal
pressure-volume relationships and high derivative
thermoelastic properties of NaCl up to a pressure of
30 GPa at selected temperatures in the range
300-1050 K.
2 Theory
The Holzapfel AP2 EOS can be written as17,18:
P = 3K 0 x −5 (1 − x)[1 + c2 x(1 − x)] exp[c0 (1 − x)]
…(2)
where x = (V/V0)1/3, and
 3K 
c0 = − ln 0 
 PFG 0 
PFG0
Z 
= aFG  
 V0 
…(3)
5/ 3
x  dP 
 dP 
K = −V 
=− 

dV
3


 dx 
…(6)
x  dK 
K′ = − 

3  dx 
…(7)
…(4)
with aFG=0.02337 GPa nm5. Z is the total number of
electrons in the volume V0. In the case of NaCl we
have Z=28 per molecule. This is to be multiplied by
the Avogadro number when V0 is given in the units of
cm3/mole. The constant c2 in Eq. (1) is related to K′0,
the value of K′=dK/dP at P = 0, as follows:
c2 =
and the material becomes more compressible at higher
temperatures.
In addition to P-V isotherms, the high derivative
properties along different isotherms can also be
calculated using values of input parameters
appropriately corresponding to each temperature. This
method has successfully been used by earlier
researchers10,19,21. The expressions for the bulk
modulus K and its pressure derivatives K′=dK/dP and
K′′=d2K/dP2 are obtained using the following
relationships:
3
( K 0′ − 3) − c0
2
…(5)
P-V relationships along different isotherms for NaCl
have been determined by using Eq. (2). The required
input data for V0, K0 and K′0 at different temperatures
have been taken from the literature3,10,19,20 (Table 1).
The volume ratio V/V0 in the table represents
V(T,P)/V(T,0) along different isotherms at selected
temperatures T. The amount of pressure required to
produce the same change in V/V0 decreases
continuously with the increase in temperature. This is
related to the fact that the bulk modulus becomes less
Table 1 — Values of input parameters for NaCl 3,10,19 used in
he Holzapfel AP2 EOS
T
(K)
V0=V(T,0)
(cm3/mol)
Ko
(GPa)
K'0
300
450
600
750
900
1050
27.015
27.523
28.101
28.757
29.517
30.419
24.0
21.6
19.0
16.5
14.1
11.7
5.35
5.51
5.73
6.03
6.40
6.85
PFG0
(GPa)
C0
1065.62
1033.05
997.87
960.19
919.39
874.39
2.70
2.77
2.87
2.97
3.08
3.22
KK ′′ =
x2
9K
 d 2K 
1
 2  − K ′ K ′ + 
 dx 
3



…(8)
where
dK
x  d 2 P  1  dP 
= −  2  − 

dx
3  dx  3  dx 
…(9)
and
d 2K
x  d 3P  2  d 2 P 

− 

=
−
3  dx 3  3  dx 2 
dx 2
…(10)
The pressure P as a function of x is given by Eq.
(2), the Holzapfel AP2 EOS. The high derivative
thermoelastic properties include the Grüneisen
parameter γ (Eq. 1) and its volume derivative q. The
modified free volume theory yields the following
expression4:
γ=
K′ 1
− −ε
2 6
…(11)
where
C2
0.83
0.99
1.22
1.58
2.02
2.57
ε=
f ( K − K ′P)
(3K − 2 fP )
…(12)
The parameter f takes different values for different
formulations of the Grüneisen parameter γ. Thus f=0
for Slater’s formula22, f=1 for the DugdaleMacDonald formula23, and f=2 for the Vashchenko-
SINGH: THERMOELASTIC PROPERTIES OF NaCl CRYSTAL
405
Table 2 — Pressure-volume relationships for NaCl along different isotherms at selected temperatures T
450 K
600 K
750 K
900 K
1050 K
300 K
V(T,P)
V(T,P)
P(GPa)
V(T,P) P(GPa) V(T,P) P(GPa) V(T,P) P(GPa) V(T,P)
P(GPa) V(T,P) P(GPa)
V(T,0) (cm3/mol)
(cm3/mol)
(cm3/mol)
(cm3/mol)
(cm3/mol)
(cm3/mol)
1
0.95
0.90
0.85
0.80
0.75
0.70
0.65
27.015
25.664
24.314
22.963
21.612
20.261
18.911
17.560
0
1.41
3.34
5.98
9.59
14.6
21.4
31.1
27.523
26.147
24.771
23.395
22.018
20.642
19.266
17.890
0
1.28
3.03
5.44
8.76
13.3
19.7
28.7
28.101
26.696
25.291
23.886
22.481
21.076
19.671
18.266
Zubarev formula24. Value of f for a given material can
also be determined by taking γ=γ0 at P=0. For NaCl,
γ=γ0=1.59 (Ref. 3) gives:
γ0 =
K 0′ 1 f
− −
2 6 3
…(13)
giving f = 2.98 for K′0 = 5.5. Equation (13) is obtained
from Eqs (11) and (12) at P=0, γ=γ0, and K′=K′0. The
second Grüneisen parameter q is obtained by
differentiating Eq. (11) as follows:
q=
V  dγ 
1  − KK ′′
dε 
+K

 = 
γ  dV T γ  2
dP 
…(14)
where
K
[ fKK ′′P + εK (3K ′ − 2 f )]
dε
=−
dP
(3K − 2 fP)
…(15)
Eqations (11-15) have been used for determining
γ and q at different compressions and temperatures.
3 Results and Discussion
The results for P, K, K′and KK′′obtained with the
help of Eqs (2-10) for NaCl along different isotherms
are given in Tables 2-5. There is good agreement
between the results obtained in the present study and
the pressure, volume, bulk modulus data for NaCl
reported in the literature1,3,6,10. For studying high
derivative properties, it has been found that the results
given in Tables 2-4 for P, K and K′ plotted in Fig. 1 as
1/K′ versus P/K satisfy the following relationship
proposed by Shanker et al.25:
1
P
P
= A + B  + C  
K′
K
K
2
…(16)
0
1.13
2.69
4.86
7.87
12.1
17.9
26.2
28.757
27.319
25.881
24.443
23.006
21.568
20.130
18.692
0
0.99
2.38
4.32
7.04
10.9
16.2
23.9
29.517
28.041
26.565
25.089
23.614
22.138
20.662
19.186
0
0.85
2.07
3.79
6.22
9.67
14.6
21.6
30.419
28.898
27.377
25.856
24.335
22.814
21.293
19.772
0
0.71
1.75
3.25
5.38
8.45
12.8
19.2
Table 3 — Values of bulk modulus K(GPa) for NaCl at different
compressions and temperatures
V(T,P)
V(T,0)
300K
1
0.95
0.90
0.85
0.80
0.75
0.70
0.65
24.0
31.2
40.4
52.2
67.4
87.1
113
148
Bulk modulus K(GPa)
450 K
600 K
750 K 900 K 1050 K
21.6
28.3
36.9
47.9
62.1
80.6
105
138
19.0
25.2
33.0
43.2
56.4
73.8
96.8
128
16.5
22.2
29.4
38.9
51.2
67.5
89.3
119
14.1
19.2
22.9
34.6
46.1
61.3
81.7
109
11.7
16.3
22.3
30.2
40.7
54.7
73.6
99.4
Table 4 — Values of pressure derivative of bulk modulus (K') for
NaCl at different compressions and temperatures
V(T,P)
V(T,0)
300 K
1
0.95
0.90
0.85
0.80
0.75
0.70
0.65
5.35
4.95
4.61
4.33
4.09
3.89
3.70
3.54
Pressure derivative of bulk modulus K'
450 K 600 K 750 K 900 K 1050 K
5.51
5.07
4.72
4.42
4.17
3.95
3.76
3.59
5.73
5.25
4.86
4.54
4.27
4.04
3.84
3.66
6.03
5.49
5.05
4.70
4.41
4.16
3.94
3.75
6.40
5.77
5.28
4.88
4.56
4.29
4.05
3.84
6.86
6.11
5.54
5.10
4.74
4.44
4.18
3.95
Table 5 — Values of second pressure derivatives of bulk modulus
for NaCl at different compressions and temperatures
V(T,P)
V(T,0)
1
0.95
0.90
0.85
0.80
0.75
0.70
0.65
Second pressure derivatives of bulk modulus KK"
300 K 450 K 600 K 750 K 900 K 1050 K
−9.10
−6.97
−5.47
−4.37
−3.55
−2.92
−2.42
−2.03
−9.79
−7.43
−5.79
−4.60
−3.72
−3.05
−2.53
−2.11
−10.84 −12.49 −14.62 −17.58
−8.13 −9.17 −10.49 −12.24
−6.27 −6.97 −7.83 −8.94
−4.94 −5.43 −6.03 −6.77
−3.97 −4.33 −4.75 −5.26
−3.24 −3.50 −3.81 −4.18
−2.67 −2.87 −3.10 −3.38
−2.23 −2.38 −2.56 −2.76
406
INDIAN J PURE & APPL PHYS, VOL 48, JUNE 2010
Fig. 1 — Plots between 1/K′ and P/K for NaCl along different isotherms
where A = 1/ K′0, B = −K0K′′0/K′02, and C=( K′∝/
K′02)[ K0 K′′0+ K′0(K′0− K′∝)]. The parameters A, B,
and C are found to be temperature-dependent
(Table 6). The validity of Eq. (16) has been discussed
recently by Shanker et al.25,26 and Kushwah and
Bharadwaj27. The second pressure derivative of bulk
modulus is obtained from Eq. (16) as follows:
P 
P 

KK ′′ = − K ′ 2  B + 2C 1 − K ′ 
K 
K 

…(17)
Values of A, B and C given in Table 6 when
substituted in Eqs (16) and (17) yield good agreement
with the values of K′ and K K′′ given in Tables 4 and
5 determined in the present study using the Holzapfel
AP2 EOS. Thus Eqs (16) and (17) are compatible
with the results based on the Holzapfel AP2 EOS.
The Grüneisen parameter γ and its volume
derivative q along different isotherms have been
determined with the help of Eqs (11) and (14) using
the values of P, K, K′ and K K′′ given in Tables 2-5.
First ε and K(dε/dP) are determined using Eqs (12)
and (15) at different compressions and temperatures.
The results are given in Tables 7 and 8. Variations in
the values of γ as well as q are quite significant
(Figs 2 and 3). Both γ and q increase with the increase
in temperature, and decrease with the increase in
pressure. The results for γ and q obtained in the
Table 6 — Values of the parameters for NaCl appearing
in Eq. (16)
T (K)
A
B
C
300
450
600
750
900
1050
0.187
0.181
0.175
0.166
0.156
0.146
0.318
0.323
0.330
0.343
0.357
0.375
0.618
0.625
0.631
0.634
0.638
0.637
Table 7 — Values of the parameter ε for NaCl
based on Eq. (12)
ε
V(T,P)
V(T,0)
300 K
450 K
600 K
750 K
900 K
1050 K
1
0.95
0.90
0.85
0.80
0.75
0.70
0.65
0.920
0.779
0.672
0.587
0.520
0.466
0.422
0.386
0.920
0.774
0.664
0.579
0.512
0.458
0.414
0.379
0.920
0.767
0.654
0.567
0.500
0.447
0.404
0.370
0.920
0.757
0.640
0.553
0.486
0.434
0.393
0.359
0.920
0.746
0.625
0.537
0.471
0.420
0.381
0.349
0.920
0.733
0.607
0.519
0.454
0.405
0.368
0.338
present study are based on the free volume
formulation derived from the fundamental
relationship between thermal pressure and thermal
energy3 using the pressure derivatives of bulk
modulus determined from the Holzapfel AP2 EOS.
This EOS has been found compatible with the results
SINGH: THERMOELASTIC PROPERTIES OF NaCl CRYSTAL
based on the ab initio molecular dynamics17,18. Values
of γ as well as q (Figs 2 and 3) increase substantially
with the increase in temperature. This finding is
consistent with the critical analysis of equations of
state for NaCl presented by Dorogokupets28. He
predicted that the strongest temperature dependence is
characteristic of γ at constant pressure, and this is
arising from the intrinsic anharmoncity of the solid.
Along different isotherms γ decreases with the
increasing pressure, i.e. decreasing volume. Earlier
researchers have used the following relationship3 :
γ V 
= 
γ 0  V0 
q
…(18)
Table 8 — Values of K(dε/dP) for NaCl based on Eq. (15)
V(T,P)
V(T,0)
300 K
450K
1
0.95
0.90
0.85
0.80
0.75
0.70
0.65
−3.23
−2.32
−1.71
−1.27
−0.96
−0.73
−0.56
−0.43
−3.37
−2.29
−1.73
−1.28
−0.96
−0.72
−0.55
−0.42
K(dε/dP)
600 K 750 K
−3.58
−2.48
−1.77
−1.29
−0.95
0.71
−0.54
−0.41
−3.86
−2.60
−1.81
−1.29
−0.94
−0.69
−0.52
−0.39
900 K
1050 K
−4.20
−2.72
−1.84
−1.29
−0.92
−0.67
−0.49
−0.37
−4.62
−2.87
−1.87
−1.28
−0.89
−0.64
−0.47
−0.34
407
Eq. (18) is valid only when q is assumed to remain
constant. However, it has been found that q does not
remain constant, and changes significantly along
different isotherms with the change in pressure or
volume (Fig. 3). It may be more appropriate to write29
q V 
= 
q0  V0 
λ
…(19)
where λ is the Jeanloz parameter. To examine the
validity of Eq. (19), the plots between ln(q/q0) and
ln(V/V0) are shown in Fig. 4. The nature of plots
reveals that λ in Eq. (19) does not remain constant
with the increase in compression. The slope of the
plots in Fig. 4 decreases with the increase in pressure
along different isotherms leading to the conclusion
that λ decreases with the increase in pressure. The
simple relationships such as Eqs (18) and (19) are
therefore inadequate for representing the volume
dependences of γ and q. A model for the volume
dependence of γ consistent with the generalized freevolume theory has been developed by Burakovsky
and Preston30 as follows :
n
1/ 3
V 
V 
γ = γ ∞ + a1   + a2  
 V0 
 V0 
Fig. 2 — Volume dependence of γ for NaCl along different isotherms
…(20)
408
INDIAN J PURE & APPL PHYS, VOL 48, JUNE 2010
Fig. 3 — Volume dependence of q for NaCl along different isotherms
Fig. 4 — Plots between ln(q/q0) and ln(V/V0) for NaCl along different isotherms
SINGH: THERMOELASTIC PROPERTIES OF NaCl CRYSTAL
where γ∝ is the value of γ at extreme compression
V→0. Burakovsky and Preston29 have taken γ∝=1/2,
based on the Thomas-Fermi theory. Here a1, a2 and n
are positive constants for a given material, and n is
taken greater than 1 in Eq. (20) for γ (Refs 30, 31).
Thus the two volume-dependent terms in Eq. (20)
represent concave up and concave down variations
respectively. Eq. (20) on differentiation with respect
to V yields32 :
V
q=
γ
n
1/ 3
1 V 
a2  V  
 dγ 

 = a1n  +   
3  V0  
 dV  T γ   V0 

…(21)
and
n
1/ 3

a2  V  
2 V 
a1n   +   
9  V0  

 V0 
V  dq 
 − q …(22)
λ= 
 =
q  dV  T
  V  n a  V 1/ 3 
a1n  + 2   
3  V0  
  V0 

It is thus evident from Eqs (21) and (22) that q and
λ both depend on volume. Eqs (18) and (19), which
are based on the assumptions that q and λ are
constants, are not adequate. The results obtained in
the present study (Figs 2-4) support this finding.
Finally, it should be mentioned that Stacey and
Davis33 have found two thermodynamic constraints,
viz. K′∝ must be greater than 5/3 and γ∝ greater than
2/3. The Stacey reciprocal K-primed EOS, which
satisfies these constraints, can be obtained from Eq.
(16) when the quadratic term is neglected i.e. C(P/K)2.
The results obtained by Kushwah and Bharadwaj34 for
NaCl using the Stacey EOS are in good agreement
with the results derived from the Holzapfel AP2 EOS
used in the present study. The Holzapfel AP2 EOS
(Eq. 2) can be generalized by taking x −3K′∞ in place of
x−5. It is found that values of K′∝ close to 5/3 produce
agreement between the Holzapfel EOS and the Stacey
EOS for the compressions down to V/V0 = 0.60 in
case of NaCl.
409
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