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
EQUATION OF STATE AND PHASE DIAGRAM OF IRON
V.V. Dremov, A.L. Kutepov, A.V.Petrovtsev, A.T. Sapozhnikov
Russian Federation Nuclear Centre-Institute of Technical Physics, P.O. Box 245, Snezhinsk,
Chelyabinsk region 456770, Russia.
Abstract. On the basis of latest experimental data and ab-initio calculations the multy-phase equation
of state for iron taking into account a, e, y and liquid phases has been constructed and the phase
diagram has been calculated. Free energy of solid phases is written as a sum of four terms
corresponding to static lattice potential, lattice phonon free energy, magnetic term (for a phase only)
and free energy due to thermal exitations of electrons. The last term instead of traditionally used
square law was determined using Fermi-Dirac statistics and the one-particle spectrum of a crystal
calculated for T=OK. Lattice contribution is described by high-temperature expansion of Debay model
and static potential is an approximation of ab-initio calculations and experimental data on static
compression.To evaluate the equation of state for a liquid phase we used Grover model. Obtained
results are in good agreement with thermophysical, phase boundary and shock wave measurements.
Up-to-date ideas about layout of iron phase diagram at high pressures and temperatures have been
discussed.
data by different authors and on results of ab-initio
calculations performed in our group.
INTRODUCTION
Thermodynamic properties of iron are of interest
by reasons of iron application as an important
structural material. References to geophysical theory
of iron inner core of the Earth also initiate interest to
high-presure and high-temperature properties of
iron. Numerical modelling of dynamic processes,
polymorphic transitions and melting in shock waves
requires multi-phase equation of state.
But despite of extensive experimental and
theoretical treatment there is no consensus among
different groups of scientists about iron phase
diagram. Two of the vexed questions are melting
line position at high pressures (>100GPa) and
existence of dhcp phase stability field below the
melting line.
In this paper we present equation of state for iron
including a, s, y and liquid phases. When fitting
parameters of the EOSs we based on experimental
EQUATION OF STATE
Helmholtz free energy for solid phases is written
as a sum of four terms: static lattice potential,
quasiharmonic phonon free energy, the free energy
arising due to thermal excitation of electrons and the
last term is the magnetic contribution to the free
energy that is used for a- phase only due to its
ferromagnetism. The static lattice potential was
taken in the form of Vinet-Ferrante-Ross-Smith [1].
Phonon free energy is described by high-temperature expansion of Debay model. Magnetic free
energy has been derived from
Andrews? s
approximation of the experimental data on heat
capacity [2]. The form of the EOS is similar to that
used by Boettger and Wallace [3] for a and s iron.
At high temperatures (round 103K and above), the
87
contribution to the free energy from the thermal
excitation of electrons essentially differs (see, for
example, [4]) from the traditionally used square law
that leads to inadequate description of Hugoniot for
wave amplitudes > 100 GPa and puts the problem of
deriving a new EOS which would be more accurate
in the region of high temperatures. In this paper the
contribution from the thermal excitation of electrons
was determined using Fermi-Dirac statistics and the
one-particle spectrum of a crystal calculated at
T=OK. In the zone structure calculations we used the
total-potential
linear augmented plane waves
(LAPW) method. The standard LAPW basis was
completed with local functions strictly bounded by
the MT-spheres with zero values and normal
derivatives on the boundaries of the MT-spheres.
The local functions were presented as the linear
combinations of solutions of the radial equation
with the spherically symmetric component of a
potential and also the first and the second
derivatives of these solutions with respect to energy.
They appeared important for Fe in context of the
relaxation of the linear approximation.
To construct EOS for liquid iron the Grover
model of a liquid [5] has been used. This model
uses solid EOS to construct the EOS for
corresponding liquid. In our evaluation the EOS of
y-phase parameters of which has been slightly
changed to fit a set of experimental data has been
taken as the basic solid EOS.
Gamma
Alpha
7.4
E7'2
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I 68
Liqud
6.6
6.4
6.2
500
250
200
I 150
50
9
10
13
Density (g/ccm)
FIGURE 2 Iron Hugoniots. Solid lines - calculation. Dots-[10],
stars-[l l],dimonds-[12], squares-! 13], circles -[14], plus -[15].
PHASE DIAGRAM
7.8
7.6
Comparison of calculated and experimental data
on thermal expansion of iron at normal pressure is
shown on Fig.l. One can see that calculated thermal
expansion for liquid overtakes the experimental one
as temperature increases. We found that in this
region density of iron is strongly ruled by electronic
term. These calculation are well consisted with those
reported in [6].
Comparison of calculated and experimental
Hugoniots of a and s iron is presented on Fig.2.
Comparison of calculated Hugoniot of preheated yFe (centered at 1573K) with experimental data [9] is
shown on Fig.4.
1000
1500
2000
Temperature (K)
FIGURE 1 Thermal expansion of iron. Open circles
experimental data [7-8]. Solid line - calculation.
Figures 3-5 depict layout of the iron phase
diagram in different ranges of pressure and
temperature.
At low pressures and temperatures (see Fig.3)
where oc-e, ot-y transitions take place there is no
much discrepancy in experimental data
and
equilibrium phase boundaries are well defined. As
for the phase diagram at high pressures and
temperatures there are two main groups of
experimental data referring to experiments on laserheated dimond anvil cells (LHDAC) [16,20,24-25]
and to shock temperature measurements [21,23].
The shock sound velocity experiments [22,26] also
should be added to experimental data on phase tran-
sition of iron at high pressures.
200
10
choosed a mid position for melting line of iron in
the region of e-L equilibrium (see Fig.5). Position
of triple point y-e-L calculated with our EOSs is
closer to that obtained by Boehler [20] but the slope
of melting line in the region of 8- phase is greater
(see Figs.4-5) than that in [20]. Calculated melting
line in the region of y-phase is well consited with
LHDAC data [16,20].
15
Pressure (GPa)
FIGURE 3 Low pressure-temperature phase diagram of iron.
Solid line - this work, black dots - data from [16], squares [17], triangles -[18], circles-[19].
The shock sound velocity measurements of
Brown and McQueen [22] has detected two
discontinuities
at 200GPa and 243 GPa
corresponding to solid-solid phase transition and
melting. Corresponding temperatures were not
measured but estimated from the shock energy (see
Fig.5). Shock temperature measurements [21,23]
are consistent with data [22] concerning pressure of
Hugoniot and melting curve intersection. But data
on temperature of the intersection are different in
all three papers. LHDAC data give the melting line
at pressure >100GPa lying much lower. Difference
in temperature between [20] and [21] is -2000K at
200GPa and -1500K between [20] and [23]. In
[27] it has been suggested that the discreapancy in
this measurements arises due to metastability and
superheating of shocked solid phase far above
equilibrium melting line. It should be noted that
more recent shock temperature measurements [23]
give lower temperature of melting than earliest
measurements [21] and vise versa more recent
LHDAC experiments [16] using double sided
configuration and very stable YLF laser give
higher temperatures for melting line than previous
LHDAC data [20]. So, there is a converging of
experimental data obtained by different methods to
some average position.
When fitting parameters of our EOSs we
Pressure (GPa)
FIGURE 4 Medium pressure-temperature phase diagram of
iron. Solid lines - this work, black dots - data from [20],
triangles - [21], diamonds-[16], hexagrams and dashed line are
experimental [9] and calculated Hugoniot of preheated y-Fe,
dotted line - melting line from experimental data [21].
Mentioned above discontinuity in shock sound
velocity measurements at ~200GPa refers according
to [22] to solid-solid phase transition and could be
linked with s(hcp) to P(dhcp) transition. Stability
field for the last was supposed in papers [20,24-25]
to be at T>1500K and P>40GPa. But in more recent
shock sound velocity experiments by Nguyen and
Holmes [26] only one but rather extended
discontinuity in the region 200-25 OGPa takes place
and corresponds to melting. From the other hand in
[16] it was reported that no new phases other than e
and y at pressures above 12GPa and temperatures
above HOOK were observed.
Our ab-initio calculations show that dhcp phase
takes an intermediate position (in the sence of its
enthalpy of formation at T=OK) between fee and hep
phases. So it is possible that at T>OK there is a
region of dhcp-phase stability somewhere between
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7. Touloukian Y.S.,Kirby R.K., Taylor R.E.,Desai P.D.,
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the APS Topical Conference on Shock Compression
of Condensed Matter, 1997, pp. 133-136.
10. McQueen R.G., Marsh S.P, Taylor J.W., Fritz J.N.,
Carter W.J., in High-Velocity Impact Phenomena,
1970
11.Kozlov E.A., High Pressure Research, 10, 541-582
(1992).
12. Zhernokletov M.V., Zubarev V.N., Trunin R.F.,
Fortov V.E. Experimental Data on Shock
Compressibility and Adiabatic Expansion of
Condensed Matters at High Energy Density.
Chernogolovka, 1996, p.385 (in Russian).
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4872-4887(1974).
14. Bancroft D., Peterson E.L.,Minshall S., J.Appl.Phys.,
27,291-298(1956).
15. Hixson R.S., Fritz J.N. "Shock Comperssion of Iron",
in Proceedings of the APS Conference on Shock
Compression of Condensed Matter, 1991, pp.69-70.
16. Shen G., Mao H., Hemley R.J., Duffy T.S., Rivers
M.L., Geophys.Res.Lett, 25, 373-376 (1998)
17. Johnson P.C., Stein B.A., Davis R.S., J.AppLPhys,
33, 557 (1962).
18. Kaufman L., Clougherty E.V., Weiss R.J., Acta
MetalL, 11, 323 (1963).
19. Bundy P.P., J.Appl.Phys., 36, 616 (1965).
20. Boehler R., Nature, 363, 534-536 (1993).
21. Williams Q., Jeanloz R., Bass J., Svendsen B., Ahrens
T.J., Science, 236, 181(1987).
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7485-7494(1986).
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26. Nguyen J., Holmes N.C., "Iron Sound Velocities in
Shock Wave Experiments", in Proceedings of the
APS Topical Conference on Shock Compression of
Condensed Matter, 1999, pp.81-84.
27. Bonnes D.A., "Metastability in Shocked Iron:
hep and fee phases. However, we did't take into
account dhcp phase as a part of our multy-phase
EOS by two reasons. First, there are no sufficient
experimental data on dhcp phase. Second, according
to our ab-initio calculations its thermodynamic
properties are very close to those of hep phase so
that in hydrodynamic calculations dhcp phase could
be ignored even if it is indeed exists.
150
200
250
300
Pressure (GPa)
FIGURE 5 High pressure-temperature phase diagram of iron.
Black dots - melting line data from [20], squares - Brown and
McQueen data [22], x -Hugoniot [23], circles-Hugoniot [21],
dotted and dashed lines -melting lines from experimental data
[23] and [21]. Solid lines are calculated ones: 1-melting line, 2-sHugoniot, 3- liquid Hugoniot.
ACKNOWLEDGEMENTS
The work has been performed under partial
support by the Contract 1000530009-35 between
RFNC-VNIITF and LANL.
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