CP620, Shock Compression of Condensed Matter - 2001
edited by M. D. Furnish, N. N. Thadhani, and Y. Horie
© 2002 American Institute of Physics 0-7354-0068-7/02/$ 19.00
CALCULATED HUGONIOT CURVES OF POROUS METAL:
COPPER, NICKEL, AND MOLYBDENUM
Y. Wang1'2, R. Ahuja1, B. Johansson1'3
1
Condensed Matter Theory Group, Department of Physics, Uppsala University,
Box 530, S-751 21, Uppsala, Sweden
2
Institute of Applied Physics and Computational Mathematics,
P.O. Box 8009, Beijing, 100088 P.R. China
3
Applied Materials Physics, Department of Materials Science and Engineering,
Royal Institute of Technology, S-100 44, Stockholm, Sweden
Abstract. Calculated Hugoniots curves of porous metal Cu, Ni, and Mo are reported using the newly
developed classical mean-field potential approach where both the cold and thermal parts of the
Helmholtz free-energy are derived entirely from the 0-K total energies and electronic density of states
calculated with the full-potential linearized augmented plane wave method within the generalized
gradient approximation. Our approach permits efficient computation and invokes no empirical
parameters. The calculation reproduces experimental data both at low and high porosities.
within which ground-state properties of many
physical systems can be calculated. However, the
ab initio thermodynamic study still remains a great
challenge to us.
2) A variety of semi-empirical models (13-15)
has been proposed to describe the shock-wave
behavior of porous materials during the past
decades. Among these, the Mie Gruneisen equation
of state EOS has extensively been adopted.
However, the success of the Mie Gruneisen EOS
depends on how to determine the Gruneisen
coefficient y which is usually taken as a function of
volume only. In contrast, the MFP approach does
not invoke the Gruneisen approximation, therefore,
the difficulties in determining the Gruneisen
coefficient have been circumvented.
3) To our best knowledge, there has been
lacking a first-principles calculation for the
Hugoniot of the porous materials. The present work
is unique in this regard.
INTRODUCTION
In order to proceed the hydrodynamic
calculations for solving problems in geophysics,
astrophysics, particles accelerator, fission and
fusion reactor, and etc, one needs an accurate
knowledge on the equation-of-state (EOS) and
related thermodynamic properties for a condensed
matter at high pressure and temperature. Due to
these
practical
applications,
considerable
experimental efforts (1-6) have been devoted to the
shock compression of materials since the shockwave data have been widely used as the reference
for extrapolating to other high pressure and high
temperature thermodynamic states.
As a continuation of the work in demonstrating
the newly developed first-principles mean-field
potential (MFP) approach (7-12), this work shows
how to calculate the Hugoniot of porous materials.
Some motivations behind the present work are:
1) Over the past decades, the density-functional
theory has successfully provided a framework
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as the input, and more dense points in the lattice
constant step of 0.005 a.u. are derived by cubic
spline interpolation for the convenience of onedimensional numerical enumeration. Out of the
lattice constant region of the LAPW calculations,
the extrapolation using the Lennard-Jones/Morse
function are invoked.
COMPUTATIONAL METHOD
Mean-field Potential Approach
The MFP approach (7, 8) is rather simple. Let us
consider a system with a given averaged atomic
volume V and temperature T. If we do not consider
the magnetic contribution, the Helmholtz freeenergy F( V, T) per ion can be written as
Computational Consideration on the
Hugoniot State of Porous materials
(1)
Hugoniot states, which are derived by the
conventional
shock-wave
technique,
are
characterized by using measurements of shock
velocity (D) and particle velocity (u) with V//V# =
(D-u)/D and PH = doDu where PH is the pressure
and d0 is the initial density. Through the RankineHugoniot relations, these data define a compression
curve [volume (V#) versus pressure (P//)] as a
function of known Hugoniot energy (£//).
vf<y,T) =
Ed(V,T) =
(2)
where V0 and EQ refer to the atomic volume and
energy at ambient conditions respectively.
We will only consider the complete compacted
state of the porous material. In this case, one can
simply imagine that the EO in Eq. (2) for the porous
case is exactly the same as that of the nonporous
one. The only new thing for the porous case (18) is
that now the initial volume V0 = my«ormal, where m
is the initial porosity and y™rmai is ambient volume
of the nonporous material.
In the above equations Ec represents the static 0-K
total energy, F-wn the thermal free-energy of the
lattice ion, F€\ the free-energy due to the thermal
excitation of electrons, n(£, V) is the electronic
density of states (DOS), /is the Fermi distribution,
and eF is the electronic Fermi energy. It should be
mentioned that the essentially new work of the
MFP approach lies in the fact that mean-field
potential g(r,V), seen by the lattice ion, is simply
constructed in terms of the 0-K total energy. It is
shown that as a second order approximation of the
mean-field potential, the well-known Dugdale and
MacDonald expression of Grueneisen parameter is
explicitly deduced.
To calculate the 0-K total energy Ec in Eq. (1),
we employ the full-potential LAPW (16) method
within the GGA (17). In all the thermodynamic
calculations, we do not make any attempts to
analytically fit the LAPW calculated points since
the fitting might alter the original LAPW results.
The raw LAPW numerical points are directly taken
RESULTS AND DISCUSSIONS
The calculations are done at pressures up to 300
GPa. For Cu, we have considered the initial
porosity of m = 1.0, 1.13, 1.22, 1.41, 2.0, 3.0, 3.5,
and 4.0, for Ni, we have considered the initial
porosity of m = 1.0, 1.41, 1.72, 2.0, 2.7, 4.55, and
5.58, and for Mo, we have considered the initial
porosity of m = 1.0, 1.26, 1.82, 2.3, 3.5, 4.0, and
5.93. These initial porosities are from the existing
experimental data. Note that here m = 1.0 means
the nonporous material. The presently calculated
68
to treat the normal and anomalous compression
differently may now be that: (i) their method
needed the Griineisen parameter whose
dependences on the volume and temperature was
very difficult to determine and (ii) they neglected
the thermal electronic contribution.
EOS's are plotted in Figs. 1-3 for Cu, Ni, and Mo
respectively (by pressure against the reduced
density d/do, where do represents the ambient
density of the nonporous material).
300
250
-200
o
£150
£100
Cu
50
0.8
1.2
d/dO
1.4
1.6
FIGURE 1. The Hugoniot EOS for porous Cu. The solid lines,
from the right to the left, represent the presently calculated
results by initial porosity m - 1.0, 1.13, 1.22, 1.41, 2.0, 3.0,
3.5, and 4.0 respectively. The solid circles are shock-wave data
of the nonporous material by Mitchell and Nellis (1), the
plusses are from the LASL compilation (2), the crosses are
from Trunin el al (3) published in 1989, and the triangles are
the new data published by Trunin (4) in 1994.
FIGURE 2. The Hugoniot EOS for porous Ni. The solid lines,
from the right to the left, represent the presently calculated
results by initial porosity m =1.0, 1.41, 1.72, 2.0, 2.7, 4.55, and
5.58 respectively. The solid circles are shock-wave date of the
nonporous material from the LASL compilation (2), the
diamonds are from ATtshuler et al. (Ref. 5, nonporous
material), and the plusses are from Trunin et al. (3) published
in 1989.
Traditionally, the experimental data are
partitioned into the normal compressibility and the
anomalous compressibility depending on m. For
the normal compressibility, as the pressure
increasing the volume decreases as usual. The most
intrigued case is the anomalous compressibility
where, instead of decreasing with increasing
pressure, the volume increases, and then, when the
pressure is high enough the compressibility become
normal
By the MFP approach by which the Griineisen
parameter is no longer invoked, the so-called
normal and anomalous behaviors have been
reproduced simply in the same framework as the
previous work (8). The interesting point is that the
method of setting the initial parameters E0 and V0
in the present work is exactly the same as that for
the normal compressibility of the previous
empirical work by Dijken and De Hosson (15). The
reason why the work of Dijken and De Hosson had
SUMMARY
In Summary, the Hugoniot equation-of-states of
porous metals Cu, Ni, and Mo have been calculated
using the MFP approach which does not invoke
any adjustable parameters and Griineisen
parameter. This work might be the first ab initio
calculation on the shock-wave compression of
porous materials and the so-called anomalous
behaviors on compressibility are reproduced.
ACKNOWLEDGEMENTS
This work is supported by Chinese National
PAN-DENG Project (Grant No.95-YU-41),
Swedish Foundation for Strategic Research (SSF),
69
Swedish Natural Science Research Council (NFR),
and Goran Gustavsson Foundation.
0.6
0.8
1
1.2
1.4
3.
Trunin, R. F., Simakov, G. V., Sutulov, Yu. N.,
Medvedev, A. B., Rogozkin, B. D., and Fyodorov,
Yu. E., Zh. Eksp. Teor. Fiz. 96, 1024 (1989) [Sov.
Phys. JETP 69, 580 (1989)].
4. Trunin, R. F., Phys. Usp. 37, 1123 (1994).
5. ATtshuler, L. V., Bakanova, A. A., and Trunin, R.
F., Zh. Eksp. Teor. Fit. 42, 91 (1962) [Sov. Phys.
JETP 15, 65 (1962)].
6. Hixson, R. S. and Fritz, J. N., J. Appl. Phys. 71,
1721 (1992).
7. Wang, Y., Phys. Rev. B 61, Rl 1863 (2000).
8. Wang, Y., Chen, D., and Zhang, X., Phys. Rev. Lett.
84, 3220(2000).
9. Wang, Y., and Sun, Y., J. Phys.: Condens. Matter
12,L311 (2000).
10. Wang, Y., and Li, L., Phys. Rev. B 62, 196 (2000).
11. Wang, Y., and Sun, Y., Chinese Phys. Lett.
(Accepted).
12. Li, L., and Wang, Y., Phys. Rev. B 63, 245108
(2000)..
1.6
13. Boshoff-Mostert, L., and Vilioen, H. J., J.
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14. Wu, Q., and Jing, F., /. Appl. Phys. 80, 4343
(1996).
15. Dijken, D. K., and De Hosson, J. T. M., J.
Appl. Phys. 75, 809 (1994).
FIGURE 3. The Hugoniot EOS for porous Mo. The solid lines,
from the right to the left, represent the presently calculated
results by initial porosity m =1.0, 1.26, 1.82, 2.3, 3.5, 4.0, and
5.93 respectively. The solid circles are shock-wave date of the
nonporous material from the LASL compilation (2), the
diamonds from Hixson and Fritz (Ref. 6, nonporous material),
and the plusses are from Trunin el al (3) published in 1989.
16. Blaha, P., Schwarz, K., and Luitz, J., WIEN97, A
full potential linearized augmented plane wave
package for calculating crystal properties
(Karlheinz Schwarz, Techn. Universitat Wien,
Austria, ISBN 3-9501031-0-4, 1999).
17. Perdew, J. P., Burke, S., and Ernzerhof, M., Phys.
Rev. Lett. 77, 3865 (1996).
18. Zeldovich, Yu. B., and Raizen, Yu. P., Physics of
Shock Waves and Non-stationary Hydro dynamic
Phenomena (New York, Plenum Press, 1967).
REFERENCES
1.
2.
Mitchell, A. C, and Nellis, W. J., /. Appl Phys. 52,
3363(1981).
Los Alamos Shock Hugoniot Data, edited by S. P.
Marsh (University of California Press, Berkeley,
1980).
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