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
ANALYSIS OF ISOBARIC EXPANSION DATA BASED
ON SOFT-SPHERE EQUATION OF STATE FOR LIQUID METALS
Pavel R. Levashov, Vladimir E. Fortov, Konstantin V. Khishchenko,
and Igor' V. Lomonosov
Institute for High Energy Densities, Russian Academy of Sciences,
Izhorskaya 13/19, Moscow 127412, Russia
Abstract. An analysis of isobaric expansion data for the liquid phase of refractory metals is presented. Simple equations of state based on soft-sphere model are used to generalize available experimental data on vapor pressure as well as isobaric and adiabatic expansion of liquid metals. In the
cases of discrepancy of isobaric expansion data obtained by different experimental teams several
equation-of-state variants are developed. It is shown for tungsten, tantalum, and vanadium that
higher value of(dV/8T)P leads to lower values of critical density and pressure.
INTRODUCTION
pendence of energy on temperature and volume to
take into account the influence of electrons and developed EOSs for thirteen elements [4]. New softsphere EOSs for eleven metals have been presented
in Ref. [5] based on recent data on isobaric and
adiabatic expansion and another procedure for determination of free parameters in the model [4]. The
EOSs [5] give the correct values of evaporation
temperature under the atmospheric pressure in contrast to Ref. [4]. In present work the procedure [5]
was used for EOS construction.
The range of applicability of resulting EOSs lies
in the following ranges: P < 50 kbar, VmQ < V, and
TmQ<T<20kK, where VmQ and TmQ are liquid specific volume and temperature at melting point under
the normal pressure. The EOSs give less accuracy
in a region of hot dense plasma because of rough
account of electrons influence.
The region of liquid on phase diagram of metals
is complicated for both theoretical and experimental
investigations [1]. Isobaric expansion (IEX) method
allows one to study thermophysical properties of
metals at relatively high pressures (up to 10 kbar)
and temperatures (up to 10 kK) along the liquidvapor coexistence curve [2]. Unfortunately there are
discrepancies in density and enthalpy measurements
for refractory metals (tungsten, tantalum, vanadium)
between results of different authors. The choice of
more reliable data is a complex problem because of
lack of experimental information for liquid metals.
In such a situation it is reasonable to use a simple
semiempirical equation of state (EOS) to generalize
available thermodynamic data.
SOFT-SPHERE EOS MODEL
RESULTS OF CALCULATION
The EOS model we used has been obtained as a
result of Monte-Carlo simulation of the system of
particles interacting with the soft-sphere potential
0(r) = s(cr/r)n [3], Young has corrected the de-
There are four sets of isobaric expansion measurements on tungsten: in water at P = I bar [6] and
1 kbar [7] and in the inert gas atmosphere at P = 2
71
[8] and 3 kbar [9]. The density values [7-9] correspond to each other in the limits of experimental
accuracy. The sound velocity in liquid metal has
been also measured [9]. Adiabatic expansion of
shock-compressed samples of tungsten with initial
porosity m = 2.\6 has been investigated in release
waves at pressures down to 0.3 kbar [10]. The critical point parameters have been estimated with the
help of the corresponding-states law [11], different
EOS models [12—14] as well as from an experimental data treatment [15-17].
iou
0.1
i/r, 10 K
0.2
0.3
FIGURE 2. Vapor pressure of tungsten. EOS calculations: solid
line — VI, dashed line — V2; handbooks data: 1 — [19], 2 —
[18]; CP estimations notation is the same as in Fig. 1.
103
102
0.1
10'
1
A g/cm3
10
1
10°
12
16
ID'1
20
r, io3 K
ID'2
FIGURE 1. Phase diagram for tungsten. P — isobars; R —
liquid-vapor equilibrium curve with critical point (CP); <p> —
half-sum of liquid and vapor densities; EOS calculations: a —
VI, b — V2; EEX data: 1 — [9], 2 — [6], 3 — [8], 4 — [7]; CP
evaluations: 5 — [12], 6 — [11], 7 — [15], 8 — [16], 9 — [13],
10 and 11 — this work, VI and V2 respectively.
a). Sound velocity vs. density in liquid tungsten at P = 3 kbar.
EOS calculations: solid line — VI, dashed line — V2; circles —
experimental data [9].
6
10
£/,km/s
FIGURE 3. Adiabatic expansion of tungsten. U— mass velocity; S\ and 5*2 — release isentropes of shock-compressed samples
with initial porosity m = 2.16 originated from Hugoniot states at
pressures PI = 1.16 Mbar and PI = 1.52 Mbar; EOS calculations:
a — VI, b — V2, c — multi-phase EOS [13]; 1 — experimental
data [10].
a). Pressure vs. density phase diagram for tungsten. EOS calculations: solid lines — VI, dashed lines — V2; R - two phase
region.
The calculated phase diagram for tungsten is
shown in Fig. 1. Two variants of EOS have been
constructed (VI and V2). The first variant VI is in
agreement with the measurements of density versus
temperature [6] and the second one V2 is in agreement with data [9]. In both cases the results of EOS
calculations correspond well to the experimental
information on enthalpy, evaporation temperature
under the normal pressure T^ = 5953 K [18], sound
velocity and vapor pressure (see Fig. 1 and 2). The
points of slope change on the dependences of pressure upon expansion velocity in adiabatic release
experiments correspond to the intersection of isentropes with the boundary of evaporation region
(Fig. 3). The initial states of the release isentropes
on shock Hugoniot lie beyond the region of appli72
ble 1 shows that the critical density and pressure
from the variant V2 are significantly lower in comparison with other estimations [2, 11-14]. Therefore
the experimental data [21] are probably less reliable
than that from Ref. [8].
cability of the soft-sphere EOS. Therefore we have
used the multi-phase EOS for tungsten [13] to determine the initial state parameters on two release
isentropes at pressure level P= 100 kbar. The
analysis of Fig. 1-3 shows that the variant V2 is
less reliable. It can be seen from the low value of
critical density and pressure in comparison with
other estimations (see also Table 1) as well as from
the very low pressure of the entrance points of isentropes into a two-phase region in comparison
with the experimental data [10]. The critical point
parameters for the variant VI of EOS lie very close
to the estimation from Ref. [13] and occupy an intermediate position between the other estimations.
There are few data on thermodynamic properties
of liquid tantalum and vanadium under static conditions. We have used density data [20] for the melting point under the normal pressure.
Tantalum has been investigated by isobaric expansion method at P = 2 [8] and 3 kbar [21]. The
sound velocity in liquid tantalum has been also
studied by the same technique [22]. The correction
of density and specific heat data from Ref. [21] has
been made with the help of redesigned optical pyrometer calibrating apparatus [23]. This has led to
the decrease of specific heat values by ~ 14 %. Note
that the two sets of density-measurement results [8,
21] don't correspond to each other in the limits of
experimental accuracy.
The calculated phase diagram for tantalum is
shown in Fig. 4. As well as in the case of tungsten,
two variants of EOS have been constructed (VI and
V2). The first variant VI is in agreement with the
results of density measurements [8] and the second
one V2 is based on the data [21]. In both cases the
results of EOS calculations agree well with the experimental information on enthalpy, sound velocity,
evaporation temperature value T^= 5623 K [18],
and vapor pressure. The analysis of Fig. 4 and Ta-
12
r, 103K
16
20
FIGURE 4. Phase diagram for tantalum. P — isobars; R —
coexistence curve with critical point (CP); <p> — half-sum of
liquid and vapor densities; EOS calculations: a — VI, b — V2;
EEX data: 1 — [8], 2 — [21], 3 — [23]; 4 — density at melting
point under the normal pressure [20]; CP evaluations: 5 — [12],
6 — [11], 7 — [2], 8 — [13], 9 — [14], 10 and 11 — this work,
VI and V2 respectively.
a). Sound velocity vs. density in liquid tantalum. EOS calculations: solid line— VI, dashed line— V2; down triangles —
experimental data [22].
The same analysis for vanadium shows that the
experimental information from Ref. [7] is more
preferable than the data [24] (see Table 1, VI for
vanadium is based on data [7], V2 — [24]).
TABLE 1. Estimations of critical point parameters for metals
Substance
W
Ta
V
rc»kK
15.37
13.59
14.46
13.40
9.49
8.06
Pcn kbar
10.5
3.1
8.6
2.8
7.8
2.6
pcn g/cm3
4.36
2.15
3.72
1.86
1.47
0.84
73
Zcr = Pcr/pcrRTcr
0.347
0.236
0.346
0.244
0.342
0.236
Rem.
VI
V2
VI
V2
VI
V2
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Furnish et al, AIP Conference Proceedings 505, New
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(2001).
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DISCUSSION AND CONCLUSION
It can be seen that for all three refractory metals
isobaric expansion experimental data with higher
value of (dVldT)P are in a worse agreement with
known critical point estimations and adiabatic release experiments. This fact can be explained by the
formation of dense opaque vapor layer around the
wire. If this is the case the temperature of surrounding vapor is less than that of the core because
of inner energy conversion into kinetic energy. The
density of wire is not uniform along the radius of
wire; therefore the measured density value is less
than that of core parameter. These reasons can result in the slope increase of density vs. temperature
experimental dependence. The higher values of
(dVldT)p give the higher values of heat capacity at
constant pressure CP, for example, up to IQ-12R for
molybdenum [13], where R is the specific gas constant.
It is necessary to note that the presented analysis
of isobaric expansion data doesn't take into account
the effects taking place during wire explosion process. Therefore a complete analysis needs further
investigations.
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