CP620, Shock Compression of Condensed Matter - 2001
edited by M. D. Furnish, N. N. Thadhani, and Y. Horie
2002 American Institute of Physics 0-7354-0068-7
THEORETICAL EQUATION OF STATE FOR WATER AT HIGH
PRESSURES
Hermenzo D. Jones
Research & Technology Department
Naval Surface Warfare Center, Indian Head Division
Indian Head, MD 20640-5035
Abstract. An equation of state (EOS) for liquid water is constructed for high pressures. A perturbation
technique is used to calculate the thermodynamic properties of water. The intermolecular interaction is
described by a spherically symmetric, exponential-6 potential and an angular dependent, multipolar
contribution. Isothermal compression and the shock Hugoniot for water are well characterized by the theory.
divided, as suggested by Kang,4 into a reference
contribution, which is repulsive in character and a
perturbation which is attractive. With the prescription
of Verlet and Weis,5 the radial distribution function
for the reference system is akin to one for hard
spheres with a diameter which is a function of
temperature and density. The Helmholtz free energy
arising from the spherically, symmetric part of the
intermolecular potential can then be written
analytically, aside from several simple numerical
integrals.
The multipole corrections are derived from the
theory of Gubbins, Gray and Machado6 (GGM). The
same reference system formulated for the isotropic
part of the intermolecular potential is employed in the
calculation of the angular perturbation. Values of the
dipole moment and the quadrupole tensor elements
are based on the charge distribution given in Ref. 6.
and the imposition of the traceless condition for the
quadrupole tensor. These corrections are so large that
the Fade approximation, due to Stell, Rasaiah, and
others7"9 is used to represent the perturbation series in
order to obtain convergence.
Within the above frame work and assuming that the
internal molecular modes of water are unchanged
from the gaseous state, several calculations are
INTRODUCTION
Ree1'2 has demonstrated that multipolar effects are
important for water in the shock wave regime. In his
earlier work a central pair potential obtained by
averaging intermolecular potentials based on ab initio
quantum mechanical calculations was used. A
temperature dependent intermolecular potential was
employed to account for the angular dependence of
the molecular interactions in the later endeavor. The
lower pressure limit of these calculations was about
2.0 GPa. Multipolar effects are treated in a direct
manner in this analysis to describe water in the
pressure regime from 0.5 GPa to 100 GPa.
In this work a complete EOS for liquid water
based on intermolecular interactions is constructed
from liquid-state perturbation theory.
The
intermolecular potential is written as the sum of a
spherically symmetric, exponential-6 (exp-6)
interaction and angular dependent, multipole
corrections.
The isotropic contribution is treated with the
formalism of Weeks, Chandler and Anderson3
(WCA) where it is assumed that the repulsive forces
are dominant in dense fluids. The (exp-6) potential is
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performed. A room temperature isotherm and the
entropy and enthalpy on the 0.5 GPa isobar are
calculated and compared to experimental data.10"11 A
comparison of the theoretical shock Hugoniot and
experimental results12"15 is also presented.
free energy associated with (p(r) are given in an earlier
work.16
The multipole interactions are taken as
(4)
THEORY
The perturbation theory to be discussed here is
based on the assumption that the repulsive
intermolecular forces provide the dominant
characteristics of the material. This technique,
originated by WCA, should be quite applicable to the
high pressure domain. The interaction between two
molecules is taken as
where
1;nM'r)=—-——\———-——
^ 47i(-l) 2 / 4n(2M)!
J
-——\I
l 2
21+1 l (2/ + l)!(2/+l)! j
(1)
(5)
where
In Eq.(4) O denotes the orientation; C(l1l2l;m1m2m),
Ylm(O) and D^Q) are the Clebsch-Gordon
coefficient, spherical harmonic and generalized
spherical harmonic, respectively, as defined by
Rose.17 The Qln's are the spherical tensor components
of the multipole moments.18 In this work only dipole
and quadrupole components are considered.
The lowest-order contributions to the free energy
are evaluated by the techniques of GGM utilizing the
underlying hard sphere system of the isotropic
reference system which was defined earlier. These
results are combined in a Fade approximation7"9 so
that the total free energy per particle is written as
(2)
and the f
's are the position vector of the particles.
In the above, <D(r) contains the electronic repulsion
and
van
der
Waals
attraction, while
5(f f )
consists of the multipolar interactions.
Formulation of the EOS proceeds with the
construction of a reference system from the repulsive
part of O(r) and calculation of its free energy and
radial distribution function. Free energies from the
remaining molecular interactions are then computed
via perturbation theory.
In this work the spherically, symmetric potential
is given by
<D(r)=e[(6/a)exp[a(l -r
l -6/a)
(6)
Eq.(6) is a fundamental EOS from which all of the
thermodynamics of the systems can be obtained.
Here, f int contains the contribution from the
intramolecular vibrations and rotations, and f 0 arises
from the spherically symmetric reference potential.
The superscripts refer to the order of the perturbative
corrections.
(3)
In Eq.(3), 8 is the well depth, a is the steepness
parameter and rm is the position of the minimum.
Additional details on the reference system and the
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RESULTS AND DISCUSSION
For the numerical computations values of the exp-6
potential parameters are taken as a = 12.5, rm =
3.68xlO- 10 m and e/k = 99.0K. The dipole moment
and the quadrupole tensor elements are based on the
charge distribution given in Ref. 6 and the imposition
of the traceless condition for the quadrupole tensor.
The non-zero multipole moments of interest are
^i 2 =1.86xlO- I 8 (esu)
Q xx =2.63xlO- 26 (esu)
Q yy =-2.50xlO- 26 (esu)
Q zz =-0.13xlO- 26 (esu).
200
400
600
800
1000 1200 1400 1600 1800 2000
T(K)
With
these
parameters
established,
the
thermodynamic behavior of water is now investigated.
In Figs. 1 and 2 , the theoretical entropy and
relative enthalpy along the 0.5 GPa isobar, are seen
to closely follow the experimental results.11 In this
pressure regime the multipolar energy is nearly
seventy per cent of the ideal gas contribution at a
temperature of 800 K and continues to make it's
presence felt as the temperature reaches 1400 K.
60
200
400
600
FIGURE 2. Relative enthalpy vs temperature for liquid water for
a pressure of 0.5 GPa. The experimental data points from Chase,
Curnutt, Prophet, McDonald and Syverud11 are represented by the
inverted triangles.
800 1000 1200 1400 1600 1800 2000
T(K)
0.80
0.84
0.88
0.92
0.96
1.00
V/VO
FIGURE 1. Entropy vs temperature for liquid water for a pressure
of 0.5 GPa. The experimental data points from Chase, Curnutt,
Prophet, McDonald and Syverud1' are represented by the inverted
triangles.
FIGURE 3. Pressure vs compression for liquid water for a
temperature of 298.15 K. The experimental points from Adams 10
are represented by the inverted triangles.
105
Figure 3 demonstrates that the room temperature
isotherm of Adams10 compares favorably with the
theory. At this temperature the multipolar energy is
dominant. The compressibility associated with the
angular contribution is comparable to that from the
exp-6 potential.
As a final application of this technique, the shock
Hugoniot of liquid water is considered. The initial
conditions for the material are taken as T0 = 293 K
and PO = 998.2 kg/m3. Inspection of the shock
Hugoniot in the pressure-volume plane in Fig. 4
shows that there is good agreement between the
theory and experimental data12"15 from 0.1 GPa to 90
GPa. For P* 10 GPa and T « 1000 K, the angular
effects are still substantial. The magnitude of the
multipolar energy is 80% of that from the ideal gas
contribution, and ZMP * -0.25 ZEXP, where ZMP is
the angular compressibility, and ZEXP is the
compressibility from the exp-6 potential. For higher
pressures and temperatures the multipolar effects
decline rapidly. This is in agreement with Ree's
study1
where he notes that the long-range
electrostatic interactions vanish for pressures above
25 GPa.
WALSH & RICE
MITCHELL&NELLIS
LYZENGA, ETAL
LYSNE
THEORY
FIGURE 4. Pressure vs compression for the shock Hugoniot
for liquid water. The data points, represented by the open
triangles, inverted triangles, squares and circles and are given
in Refs. 12-15, respectively.
3. Weeks,J.D., Chandler, D., and Anderson, H.C., J.
Chem. Phys. 54, 5237 (1971).
4. Kang, H.S., Lee, C.S., Ree, T., and Ree, F.H., J.
Chem. Phys. 85, 414(1985).
5. Verlet, L., and Weis, J., Phys. Rev. A5, 939 (1972).
6. Gubbins, K.E., Gray, C.G., and Machado, J.R.S.,
Molec. Phys. 42, 817(1981).
7. Stell, G., Rasaiah, J.C., and. Narang, H., Molec. Phys.
23,393 (1972).
8. Stell, G., Rasaiah, J.C., and. Narang, H., Molec. Phys.
27, 1393 (1974).
9. Rushbrooke, G.S., Stell, G. and Hoye, J.S., Molec.
Phys. 26, 1199 (1973).
10. Adams, J., J. Am. Chem. Soc. 53, 3769 (1931).
11. Chase, M.W.,. CurnuttJ.L, Prophet, H., McDonald,
R.A., and Syverud, A.M., J. Phys. Chem. Ref. Data 4, 1
(1975).
12. Lysne, P.C., J. Geo. Res. 75, 4375 (1970).
13. Walsh, J.M., and Rice, M.H., J. Chem. Phys. 26, 815
(1957).
14. Mitchell, A.C. andNellis, W.J., J. Chem. Phys. 76,
6273 (1982).
15. G.A. Lyzenga, TJ. Ahrens, WJ. Nellis and A.C.
Mitchell, J. Chem. Phys. 76, 6282 (1982).
16. Jones, H. and Gray, M.V., J. Appl. Phys. 53, 6604
(1982).
17. Rose, M.E., Elementary Theory of Angular
Momentum, J. Wiley and Sons, Inc., New York (1957).
18. Armstrong, R.L., Blumenfeld, S.M., and Gray, C.G.,
Can. J. Phys. 46, 1331(1968).
SUMMARY
Thermodynamic properties for high pressures have
been calculated with liquid-state perturbation theory
for fluids with angular dependent intermolecular
interactions. Theoretical predictions for the
temperature dependence of the entropy and enthalpy
on the 0.5 Gpa isobar for water are in good agreement
with experimental results. Isothermal compression at room
temperature is well described by the theory. The calculated
shock Hugoniot for water is also in close agreement with
experimental results.
ACKNOWLEDGMENT
This work was supported by the Independent
Research Program of the Naval Surface Weapons
Center, Indian Head Division.
REFERENCES
1. Ree, F.H., J. Chem. Phys. 76, 6287 (1982)
2. Ree, F.H., J. Chem. Phys. 84, 5845 (1986).
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