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
PHASE DIAGRAMS AND THERMODYNAMIC PROPERTIES OF
METALS AT HIGH PRESSURES, HIGH TEMPERATURES
I. V. Lomonosov1, V. E. Fortov1, K.V.Khishchenko2, P.R.Levashov2
Institute of Problems of Chemical Physics RAS, Chernogolovka 142432, Moscow reg., RUSSIA
2
Institute for High Energy Density RASt hhorskaya str. 13/19, Moscow 127412, RUSSIA
Information on phase diagrams and thermodynamic properties of 30 simple and transition metals has
been analyzed with the use of multi-phase equation of state (EOS). The comparison has been done with
theoretical calculations of thermodynamic properties of solid, liquid, plasma, evaluations of the critical
point and with results of static and dynamic experiments. The major attention has been paid to revealing
of positions of high pressure melting and evaporating. We used direct temperature measurements which
allow one to obtaine the pressure-temperature phase diagram and indirect information on phase transitions resulting from shock-wave experiments. Conclusions are made for high-pressure melting and
evaporating, as well as for obtained parameters of the critical point.
INTRODUCTION
metals can be found somewhere [1,2]. The semiempirical EOS model is given by the free energy thermodynamic potential and accounts for the cold lattice contribution and thermal contribution of atoms
and electrons. The EOS provides for a correct description of phase boundaries - melting and evaporation, as well as effects of the first and primary
ionization. The thermal atoms contribution in the
model is different in solid and liquid state while the
thermal electrons' contribution is identical.
The phase diagrams of metals were calculated
with the use of the EOS, the comparison was done
for static data, principal, reflected and porous Hugoniots, release isentropes, isobaric expansion data,
density measurements in liquid metal, evaluations of
the critical point. It was demonstrated that the developed EOS describe available data with high accuracy and reliability [1,2].
One should note, that the traditional presentation
of the phase diagram deals with pressure-temperature (P - T) dependence [3,4] in a condensed state.
Our analysis includes available direct P -T data and
results of other indirect measurements on high pressure melting and evaporation.
Metals have been widely investigated at high
pressures, high temperatures. The available information includes theoretical calculations of thermodynamic properties of solid, liquid, plasma, evaluations of the critical point and numerous results of
static and dynamic experiments. These theoretical
and experimental data are separate partital derivatives (such as pressure, sound velocity, heat capacity, etc.) of thermodynamic potential known in specific regions of the whole phase diagram. We used a
multi-phase wide-range EOS [1] to calculate thermodynamic properties and phase diagrams for 30
simple and transition metals. The goals of this work
are analysis of phase diagrams of metals and comparison with available information. These results are
applied to understanding of positions of high pressure melting and evaporation.
EOS MODEL AND THERMODYNAMIC
PROPERTIES OF METALS
The detailed description of EOS model and results of calculations of phase diagrams for selected
Ill
Analogous direct P - T measurements of melting
have been done in dynamic experiments. Figure 2
illustrates the comparison with principal and porous
Hugoniot data for magnesium, melting temperatures
in re-shocked metal are given in Fig. 2,a.
Other indirect information on high pressure
melting can be obtained from different types of
shock-wave data. These are: precise data on principal and porous shock adiabats, sound velocity
measurements in shocked metal, information on
shock compression of cooled and heated samples.
MELTING
Direct measurements of melting at high pressure
have been fulfilled in static and dynamic experiments. Experiments done in traditional highpressure vessels typically are limited to 10 GPa
pressure [3,4], defining the initial slope of the
melting curve on P - 7-diagram. The information
obtained with the use of laser-heated diamond anvil
ceils (DAC) is available in megabar pressure range.
The comparison with DAC measurements in uranium is shown on Fig. 1.
r= 100(1 ooo /<•;_,„.-•
M
liquid
8
4
O
16
I2
ft
1
solid
10
0
20
40
60
80 100 120 140 160 180 200
14
12
16
18
20
Density, g/cm
Pressure, GPa
FIGURE 3. Phase diagram for molybdenum at high pressure.
Nomenclature is identical to Fig. 2.
a) Sound velocity in shocked molybdenum, points - [7J.
FIGURE 1. P• - ^-diagram for uranium at high pressure. M melting, H - shock adiabat, a,(3,y - solid phases, circles with
bold line - DAC data [5] (dashed lines - experimental error).
Temperature, 1000K
2.85 2.45 1.93
1.48
M
co
Q.
0
2.0
2.5
3.0
3.5
100- r=so(
10
Density, g/cm
12
14
16
18
20
22
24
26
Density, g/cm
FIGURE 2. Phase diagram for magnesium at high pressure. M melting, T - isotherms, m - porous Hugoniots, points - experimental data.
a) P - T-diagram for magnesium at high pressure. M - melting,
experimental data: 1 - [4], 2 - [6].
FIGURE 4. Phase diagram for bismuth at high pressure. Nomenclature is identical to Fig. 2, except: H^ and H^- shock
Hugoniots for cooled (T0=76 K) and liquid (T0=677 K) metal,
corresponding experimental data - stars asn closed curcles [8].
112
For example, melting effects are seen very well on
shock Hugoniots of porous molybdenum (porousities m=1.26,1.83) presented in Fig. 3. According to
[7], molybdenum melts in shock wave at pressure
390 GPa, see Fig. 3,a. The analysis of shock adiabats for cooled and heated liquid bismut [8] also
allows one to locate a position of high pressure
melting.
Therefore, the position of melting region occures defined at moderate pressures from compendium's data on phase diagrams [3,4] and at high
pressures on the base of direct and indirect measurements.
sure - expansion velocity (P - U) dependence also
include indirect information on evaporation. It is
seen as change ofP-U slope, see Fig. 6.
m=2.17
10
CO
QL
O
to
CO
10°
10'2
2
EVAPORATION
4
6
Expansion velocity, km/s
Direct P - T measurements of evaporating region encounter serious difficulties due to high parameters of critical points for metals (of the order of
1 GPa and 10000 K). These data have been obtained for lead in experiments on isentropic expansion of shocked sample [9], see Fig. 5.
FIGURE 6. P - ^/-diagram for porous tungsten, m - porous
shock adiabat, S - equilibrium (solid lines) and metastable
(dashed lines) release isentropes, arrows indicate evaporation,
points with bars - experimental data [10].
15-
I
10-
'55
4
8
12
16
20
Temperature, 1000 K
10°
FIGURE 7. Phase diagram for tungsten in the region of lower
density. M and R - boundaries of melting and evaporation, CP critical point, P - isobars, S - release isentropes, open points isobaric expansion data, closed points - evaluations of the critical point, dashed line - evaporation region from EOS [11].
Pressure, GPa
FIGURE 5. P - 7-diagram for lead in the evaporation region. R
- evaporation with critical point CP, PJJ - initial shock pressure
on release isentropes, open points - experiment [9], closed
points - evaluations of the critical point.
The EOS calculations show, that both isentropes S j
and S2 [10] are subcritical entering into evaporation
region from liquid phase. The analysis of evaporation region with the critical point is more complicated for indirect data. It requires to account for
other available information like isobaric expansion
(IEX) data [12] and evaluations of the critical point.
Calculations with the use of this EOS and another
One should note that experimental isentropes come
into liquid-gas region (R on Fig. 5) from liquid
(PH= 100-220 GPa) and gas (PH=270 and 370 GPa)
phases, allowing a direct determination of evaporation curve R.
More traditional data on release expansion of
shocked solid and porous samples in form of pres113
EOS for liquid metals [11] demonstrated, that the
correct description of evaporation pressure on release isentropes [10], IEX and shock-wave data in
liquid state corresponds to a quite defined position
of evaporation region, see Fig. 7,
REFERENCES
1. Fortov, V.E., and Lomonosov, I.V., J, Pure and Appl,
Chem, 69, 893-904(1997).
2. Fortov, V.E., Khishchenko, K.V., Levashov, P.R.,
and Lomonosov, I.V,, NucL Instrum. Methods Phys,
Res, A415, 604-608 (1998),
3. Young, D, A,, Phase Diagrams of the Elements, Berkeley; Univ. of California Press, 1991,
4. Tonkov, E, Yu,, Phase Diagrams of Elements at High
Pressure, Moscow: Nauka, 1977 [in Russian].
5. Yoo, C.-S,, Cynn, H., and Soderlind, P., Phys, Rev,
857,10359-10362(1998).
6. Urtiew, P.A., and Grover, R., J. Appl Phvs, 48, 11221126(1977).
7. Hixson, R.S., Boness, D.A,, and Shaner, J.W., Phys.
Rev, Lett, 62, 637-640 (1989),
8. Trunin, R.F., et al,, Teplofiz. Visokh, Temper, 33(2),
222-226 (1995) [in Russian].
9. Ternovoi, V.Ya., Fortov, V.E., Kvitov, S.V., and
Nikolaev, D.N., in Shock Compression of Condensed
Matter-1995, eds. S.C. Schmidt, W.C.Tao. AIP
Press: New York, 1996, p.81-84.
10. Zhernokletov, M.V,, et al., in: Shock Compression of
Condensed
Matter-1999,
eds.
M.D.Furnish,
L.C.Chhabildas, R.S.Hixson, AIP Press: New York,
2000, p. 193-196.
11. Levashov, P.R., Fortov, V.E., Khishchenko, K.V..
Lomonosov, I.V., in: Shock Compression of Condensed Matter-1999, eds. M, D. Furnish, L, C,
Chhabildas, R.S.Hixson, AIP Press: New York, 2000,
p,89-92,
12. Gathers, G. R., Rep. Progr, Phys. 49, 341-396
(1986).
CONCLUSION
Wide-range multi-phase EOS have been constructed for 30 simple and transition metals. The
analysis of thermodynamic properties and phase
diagrams of metals at high pressure, high temperature demonstrated high accuracy and reliability of
developed EOS, A detailed consideration of direct
P - T arid indirect data allows one to make a conclusions on positions of high pressure melting and
evaporation regions. Summary of parameters of
critical points is given in Table 1,
Table 1. Critical points of metals
Be
Mg
Na
Zr
Hf
V
Nb
Ta
Cr
Mo
W
Fe
Co
Ni
Zn
Cd
Ag
Au
Re
Ir
Pt
Sn
Bi
U
Pc>
kbar
2,87
2.46
0.47
9.88
11.74
9.19
11.06
9.93
9.91
7,59
11.80
11.31
5.55
10,42
3.28
0.87
10,64
6,14
15.91
13.40
6.21
2.39
2,25
7.70
T
ic,
K
8877
3957
2473
14860
15810
9915
19180
13530
7797
10180
15750
8787
9157
7547
3079
2510
7053
8515
18710
16220
11430
8175
4869
9637
sc,
PC.
3
g/CM
0,398
0.553
0,240
1.634
3.610
1.631
1.701
4,263
2.660
3.690
4,854
2.183
1.890
2.092
2.381
2.283
3.279
6,061
6.024
6.061
5.236
1,592
3,937
4.505
J/g/K
13,18
3.789
3.281
1.693
0.885
2.718
2.023
0.923
2.332
1.520
0.837
2.496
2.458
2.518
1.468
0.840
1.118
0.624
0.824
0.780
0.807
1.123
0.529
0.727
114