Thermodynamic Data PDF
Thermodynamic Data PDF
Thermodynamic Data PDF
Volume 10
Editor-in-Chief
Surendra K. Saxena
Editorial Board
Thermodynamic Data
Systematics and Estimation
With Contributions by
F. Baccarin A.B. Belonoshko M. Blander S.L. Chaplot N. Choudhury
S. Usha-Devi G. Fiquet D.G. Fraser S. Ghose P. Gillet A. Della Giusta
CM. Gramaccioli G. Grimvall A.F. Guillermet K. Heinzinger E. Ito
A.G. Kalinichev Y.H. Kim M.H. Manghnani L.C Ming A. Dal Negro
D.R. Neuville G. Ottonello T. Pilati K.R. Rao K. Refson P. Richet CR. Stover
V.S. Urusov J.-A. Xu
Springer-Verlag
New York Berlin Heidelberg London Paris
Tokyo Hong Kong Barcelona Budapest
Surendra K. Saxena
Planetary Geochemistry Program
Institute of Geology
Uppsala University
Box 555
S-75122 Uppsala
Sweden
987654 32 1
Preface
Walsh and Jessica Downey) for their invaluable support and the production
department for the care with which it handled the book.
Preface v
Contributors xi
Appendix 335
Index 359
Contributors
Introduction
port, spectroscopic, etc.) of the systems under study. In the case of simple
liquids-say, liquid noble gases-where the interactions between particles are
well known and where many-body non additive interactions do not play an
important role, the results of computer simulations have a high degree of relia-
bility and can be used as an "experimental" check against analytical theories (see,
e.g., Hansen and McDonald, 1976). In the case of complicated systems-say,
aqueous solutions in a wide range of temperatures, densities, and concentrations
that cannot yet be treated on a molecular level analytically-the computer
simulations can play the role of theory. They can predict the properties of fluids
that cannot be or not directly be measured, and they can explain macroscop-
ically measured properties on a molecular level.
Since the first MC and MD simulations of pure liquid water (Barker and
Watts, 1969; Rahman and Stillinger, 1971), great progress has been made in the
simulation studies of aqueous systems. One of the earliest significant results was
the ruling out of "iceberg" formation in liquid water. Computer simulations-in
spite of quite different intermolecular potentials employed-have unequivocally
shown that liquid water consists of a macroscopically connected, random net-
work of hydrogen bonds continuously undergoing topological reformations
(Geiger et al., 1979b). The anomalous properties of water arise from the competi-
tion between nearly tetrahedrally coordinated local patterns characterized by
strong hydrogen bonds and more compact arrangements characterized by more
strained and broken bonds (Stillinger, 1980).
Apart from numerous simulations of pure liquid water under ambient con-
ditions, much work has been done to study the thermodynamic, structural,
transport, and spectroscopic properties of dilute aqueous solutions of ions
(Mezei and Beveridge, 1981; Impey et aI., 1983; Bounds, 1985; Mills et al., 1986;
Guardia and Padro, 1990a) and hydrophobic solutes (Geiger et aI., 1979a;
Pangali et al., 1979; Okazaki et aI., 1981; Rapaport and Scheraga, 1982; Swope
and Andersen, 1984; Jorgensen et aI., 1985). These studies have been reviewed
already extensively (see, e.g., Wood, 1979; Beveridge et aI., 1983; Levesque et aI.,
1984). Futhermore, in recent years the dielectric properties of several water
models employed in computer simulations have been checked (Neumann, 1986;
Alper and Levy, 1989; Ruff and Distler, 1990). Guissani et ai. (1988) have simu-
lated the ionic equilibrium of water and found the MD results in very good
agreement with the experimental pH value.
The effects of temperature and pressure on the properties of water and aque-
ous solutions were also the subject of computer simulations. However, in most
studies either high pressures (Stillinger and Rahman, 1974b; Impey et aI., 1981;
Jancso et aI., 1984, 1985; Palinkas et aI., 1984; Rami Reddy and Berkowitz, 1987;
Madura et aI., 1988) or high temperatures (Stillinger and Rahman, 1972, 1974a;
Szasz and Heinzinger, 1983a; Jorgensen and Madura, 1985; De Pablo and
Prausnitz, 1989) were applied to the system. The term "high pressure" here
indicates pressures up to about 20 kbar, which is for pure water equivalent to
densities up to about 1.35 g/cm 3 , whereas "high temperature" usually means the
range from normal temperature to only about 500 K.
Computer Simulations of Aqueous Fluids 3
Except for LiCl solutions (Tamura et al., 1988), the solute concentrations
studied by computer simulations never exceeded 2.2 molal (Heinzinger and
Palinkas, 1985; Heinzinger, 1990), and thus, the effects of a solute concentration
on simulated properties have been scarcely studied so far.
On the other hand, in recent years the physical chemistry of aqueous solu-
tions at high temperatures and pressures has become a subject of great scientific
interest (Franck, 1970, 1981, 1987). While under normal conditions both ionic
and nonionic solutes mostly have a rather low solubility in water, such solutions
can become homogeneous in the whole concentration range at supercritical
temperatures. The importance of understanding the behavior of hot, concen-
trated aqueous systems, known in geochemistry as hydrothermal solutions, on
the molecular level has been stressed many times in the literature (Valyashko,
1977, 1990; Hamann, 1981; Todheide, 1982; Holloway, 1987).
The near-critical region of the phase diagram, where the properties of water
as well as those of aqueous solutions undergo drastic changes in a very narrow
range of temperatures and pressures, seems to be of primary importance from
various points of view. Because of the high compressibility of water under such
conditions, the solutions have very large excess volumes (Franck, 1970, 1987;
Zakirov and Kalinichev, 1980). This gives rise to significant changes in the
activity coefficients of the components (Shmulovich et al., 1980a,b, 1982a), which
have to be taken into account in any calculations of mineral-fluid equilibria (see,
e.g., Ferry and Baumgartner, 1987). Recent measurements (Tivey et al., 1990)
show that temperatures of seafloor hydrothermal vents can reach near-critical
values of 350 to 400°C, which is of great importance for their complex chemistry
(Von Damm, 1990). The possible significance of submarine near-critical springs
for the origin of life has also been discussed recently (Corliss, 1990).
Except for purely geological interest, there is an increasing demand for data
and models for aqueous solutions at high temperatures and pressures from
various groups of engineering and environmental* scientists (T6dheide, 1982;
Levelt Sengers, 1990). However, up to now fundamental work has lagged far
behind that devoted to solutions in ordinary liquid water.
Although computer simulation techniques have already proved to be one of
the most powerful tools to study usual aqueous solutions (Heinzinger and
Palinkas, 1985; Heinzinger, 1990) and permit transition from the "McMillan-
Mayer" to "Born-Oppenheimer" level of solution models (Friedman, 1981),
these methods still have not been applied systematically over a sufficiently wide
range of thermodynamic conditions required in geochemistry. Even the thermo-
dynamic and structural properties of the solvent itself-pure water-have not
yet been studied extensively by these methods at high temperatures and pres-
sures, although several important contributions should be mentioned in this
context.
The first MC simulation of supercritical steam (Beshinske and Lietzke, 1969)
*Information about the use of supercritical water reactors to destroy hazardous wastes has already
appeared in the literature [Environ. Sci. Technol.24, 1277 (1990)].
4 A.G. Kalinichev and K. Heinzinger
was published almost simultaneously with the first simulations of liquid water
(Barker and Watts, 1969). Only pressures and internal energies for very dilute
water vapor with a maximum density as low as 0.038 g/cm 3 were calculated in
this early paper. O'Shea and Tremaine (1980) have studied supercritical water at
higher densities. Kataoka (1987, 1989) has made the most extensive simulation
study of water over a very wide range of the phase diagram. However, in both
cases structural results were not reported. On the other hand, in the MD
simulations of Mountain (1989), only the radial distribution functions of ex-
panded water at high temperatures were analyzed. De Pablo et al. (1989, 1990)
have calculated from MC simulations the densities of water along the liquid-
vapor coexistence curve from 25°C to the critical point, while some properties
of water directly at the critical point were simulated in the work of Evans et al.
(1988). In this way, the thermodynamic properties and structure of dense super-
critical water have also been studied recently (Kalinichev, 1985b; 1986; 1991b;
Kalinichev and Heinzinger, 1992).
Unlike silicate melts (Kubicki and Lasaga, 1991), computer simulation meth-
ods have not yet been employed for the investigation of complex hydrothermal
solutions. The aim of this chapter is to introduce this powerful technique into
studies of aqueous systems of geochemical interest. Taking into account the
current progress in computer technology, we feel that these methods could in a
short time contribute a lot to the fundamental understanding of correlations
between the thermodynamic, structural, spectroscopic, and transport properties
of such systems on a molecular level.
The simulation methodology will be described briefly first, followed by a
discussion of the intermolecular potentials used in high-temperature and high-
pressure simulations. Then recent results on the thermodynamics of supercritical
water are presented. In the final two sections a review of the effects of tempera-
ture, pressure, and solute ions on the structure and dynamics of water and
aqueous fluids will be given.
Two methods of computer simulation have been developed: Monte Carlo (MC)
and molecular dynamics (MD). In both cases, the simulations are performed on
a small number of particles (atoms, ions, or molecules) 10 :s; N :s; 104 confined
in a (usually) cubic box. The interparticle interactions are described by pair
potentials, and it is generally assumed that the total potential energy of the
system can be developed as a sum of these pair potentials. Large numbers of
particle configurations are generated in both methods, and from this micro-
scopic information macroscopic properties (pressure, temperature, internal en-
ergy, heat capacity, etc.) can be calculated with the help of statistical mechanics.
It is not the aim of this chapter to discuss the theoretical fundamentals of
statistical physics and computer simulations, since many excellent sources are
Computer Simulations of Aqueous Fluids 5
currently available (see, e.g., McQuarrie, 1976; Hansen and McDonald, 1976;
Binder and Stauffer, 1984; Allen and Tildesley, 1987; Catlow et aI., 1990). How-
ever, some basic concepts and relationships should be mentioned here for
completeness.
where kB is the Boltzmann constant, mi and Vi are the masses and the velocities of
the particles in the box, respectively, and angular brackets denote the time
average of the system.
The pressure can be derived from the virial theorem
p = NkBT _ (~)/ ~
V 3V
r ..
\f1' , ,
F.) (2)
where V is the volume of the box and (r i ' FJ means the dot product of the
position and the force vectors of particle i.
The constant volume heat capacity Cv can be obtained from the temperature
fluctuation by
(3)
* The equivalence of ensemble and time averages, the so-called ergodic hypothesis, constitutes the
basis of statistical mechanics (e.g., McQuarrie, 1976).
Computer Simulations of Aqueous Fluids 7
ingly. Instead of the energy difference AU in Eq. (4), one should use the instant
enthalpy difference
AH = AU + PAV - kB T In(1 + AV/V)N, (5)
where P is the pressure (which is kept constant in this case) and V the volume of
the system.
In this ensemble, besides the trivial averages for configurational (i.e., due to
intermolecular interactions) enthalpy,
H = <U) + P<V) (6)
and molar volume,
(7)
such thermodynamic properties as the isobaric heat capacity Cp , isothermal
compressibility k, and thermal expansivity !1 can be easily calculated from the
corresponding fluctuation relations
(8)
One of the most obvious difficulties arises in both simulation methods from the
small system size, a consequence of the limited availability of computer time.
So-called periodic boundary conditions are therefore used in order to minimize
8 A.G. Kalinichev and K. Heinzinger
surface effects and to simulate more closely the properties of a bulk macroscopic
system. This means that the basic cubic box is assumed to be surrounded by
identical boxes in all directions infinitely. If a particle leaves the box through one
side, it enters simultaneously through the opposite side because of the identity of
the boxes. In this way, the problem of surfaces is circumvented at the expense of
the introduction of periodicity.
Whether the properties of a small infinitely periodic system and the macro-
scopic system which the model is designed to represent are the same depends on
the range of the intermolecular potential and property under investigation. For
short-range interactions either spherical or minimum image cutoff criteria are
commonly used (Allen and Tildesley, 1987). The latter means that each molecule
interacts only with the closest image of every other molecule in the basic box or
in its periodic replica. However, any realistic potential for water (not to mention
electrolyte solutions) contains long-range Coulomb interactions, which should
be properly taken into account. The treatment of these interactions is still under
discussion, and several methods are commonly used (see, e.g., Allen and
Tildesley, 1987), of which the Ewald summation is usually considered the most
satisfactory.
All existing schemes for handling long-range interactions are rather computer
time-consuming. Therefore, it makes sense to consider carefully the question of
whether they have to be included for the system under investigation before
extensive simulations are started. In the case of ionic interactions, where un-
screened Coulomb potential is used, there is no choice but to implement the
Ewald method for accurate calculations of thermodynamic and structural pro-
perties (Heinzinger and Palinkas, 1985; Heinzinger, 1990; Kubicki and Lasaga,
1991). On the other hand, as numerous studies with several intermolecular
potentials truncated at various distances and with different system sizes from 64
to 512 molecules have shown (Pangali et aI., 1980; Andrea et aI., 1983; Jorgensen
and Madura, 1985), thermodynamic properties and atom-atom distribution
functions of liquid water are very insensitive to the range of the potential beyond
~ 6 A. It means that even N = 64 already gives a good approximation to bulk
water as far as thermodynamic and structural properties are concerned. This is,
however, not the case for the dynamical properties of the system, orientational
correlations, and connected to them, dielectric properties, where long-range
interactions must be carefully accounted for (Andrea et aI., 1983; Neumann,
1986; Anderson et aI., 1987; Alper and Levy, 1989; Ruff and Diestler, 1990).
Contrary to the results for water under normal conditions, the analyses of
MC simulations for supercritical water have shown a slight, but noticable,
system-size dependence of thermodynamic and structural properties at densities
significantly lower than that of normal liquid water (Kalinichev, 1992). This
finding is in accordance with recent results on simple model fluids (Schaink and
Hoheisel, 1990), where the governing role of long-range attractive interactions
has been established for the correct description of the structural and dynamical
behavior of fluids.
The thermodynamic values calculated from classical simulations have to be
Computer Simulations of Aqueous Fluids 9
corrected for quantum effects. There are several methods to compute the quan-
tum corrections to the thermodynamic properties of liquid water (Owicki and
Scheraga, 1977; Beveridge et aI., 1983; Berens et aI., 1983). However, in the
high-temperature case, the relative importance of these corrections decreases
significantly, and they will be neglected therefore in this chapter.
o 50 100
n / 10 3 cant.
10
o 0.1 0.2 11 ps
Decisive for reliability of the properties calculated from the simulations are the
potential functions employed. The intermolecular interactions between water
molecules are far more complicated than those between particles of simple
liquids. The most importrant difference is the ability to form hydrogen bonds,
which makes water an associated liquid. An additional difficulty in the descrip-
tion of water interactions is the presence of substantial nonadditive three- and
higher-body terms, studied in detail by several authors (Hankins et aI., 1970;
Gellatly et ai., 1983; Clementi, 1985), which may raise doubts on the applicability
of the pair-additivity approximation ordinarily used in computer simulations.
On the other hand, the analysis of experimental shockwave data of water
has shown (Ree, 1982) that at high temperature (;;;:: 1300 K) and high pressure
(;;;:: 80 kbar), the intermolecular interactions of water become simpler. In this
case, it is possible to use a spherically symmetric model potential for the calcula-
tions of the water properties, either from computer simulations (Belonoshko and
Saxena, 1991) or from thermodynamic perturbation theory in a way similar to
that applied to nonpolar fluids (Shmulovich et aI., 1982b; Kalinichev, 1991a).
However, such simplifications exclude the possibility of understanding the com-
plex phenomena in aqueous fluids on a true molecular level, which is, actually,
the strongest advantage and the main objective of computer simulations.
The pair potential functions for the description of intermolecular interactions
used in computer simulations of aqueous systems can be grouped into two broad
classes as far as their origin is concerned: empirical and quantum mechanical
potentials. In the first case, all parameters of a model are adjusted to fit experi-
mental data for water from different sources, and thus necessarily incorporate
12 A.G. Kalinichev and K. Heinzinger
-q,
\
\
,0.8 A
.
\
<IlTd(~~ / lA H
+q
o
/0.8A
/ +q H b
I
I
I
-q
Q
Fig. 3. Schematic reperesentation of a water molecule in the (a) ST2 and (b) TIP4P and
MCYmodels.
Computer Simulations of Aqueous Fluids 13
LJ
Vij (r) = 4Eij [(r )12 - (r )6J '
(Jij (Jij
(12)
where i and j refer either to ions or water molecules, and a Coulomb term,
different for water-water, ion-water, and ion-ion interactions, given by
(13a)
(13b)
(13e)
The switching function Sww(r), in the Coulomb term of the water pair poten-
tial has been introduced by Rahman and Stillinger in order to reduce unrealistic
Coulomb forces between very close water molecules. d and r denote distances
between point charges and LJ centers, respectively, q being the charge in the ST2
model. The sign of the Coulomb term is correct if rx and f3 are chosen as odd for
positive and even for negative charges.
The LJ parameters for the cations are taken from the isoelectronic noble gases
(Hogervorst, 1973). If we compare e.g., Pauling radii, it is obvious that halide
ions have a larger ionic radius than the isoelectronic alkali ions. In order to
describe all interactions consistently, new LJ parameters had to be determined
for the halide ions on the basis of the Pauling radii (Palinkas et aI., 1977). With a
knowledge of the parameters for cation-cation and anion-anion interactions,
the parameters for cation-water and anion-water interactions have been deter-
mined by applying Kong's combination rules (Kong, 1973). The results of this
procedure are given in Table 1.
Evans (1986) has proposed recently a modified version of the ST2 potential
that includes atom-atom LJ terms centered both on the oxygen and hydrogen
atoms, thus eliminating the need to use the switching function in Eq. (13a). This
model has been employed in MD simulations of water at temperatutes up to
1273 K and at constant densities of 1.0 and 0.47 g/cm 3 (Evans et aI., 1988) and
14 A.G. Kalinichev and K. Heinzinger
604.6 111889 {
Uoo(r) = -r- + --;:s.sr - 2 2
1.045 exp[ -4(r - 3.4) J - exp[ -1.5(r - 4.5) J},
(14a)
16 A.G. Kalinichev and K. Heinzinger
{ 16.74 }
(14b)
- 1 + exp[5.493(r - 2.2)] ,
re and O(e are the corresponding equilibrium values (re = 0.9572 A,O(e = 104.52°).
The finally adopted parameter set is given in Table 2.
In the simulations of the electrolyte solutions with the BJH model for water,
the ion-oxygen and ion-hydrogen pair potentials (as well as the ion-ion ones
that are not very important in the case of the dilute solution) are derived from
ab initio calculations. They are presented in Table 3.
From the family of quantum mechanical water potentials, the MCY model
(Matsuoka et aI., 1976) should be mentioned in the context of high-pressure
and high-temperature simulations. This model has the same geometry as the
TIP4P potential [Fig. 3(b)], but a much more complicated functional form with
parameters derived from ab initio calculations. A flexible version of this model
(MCYL) has also been proposed (Lie and Clementi, 1986). The MCY model was
used by Impey et ai. (1981) in their MD studies of the structure of water at
elevated temperatures and high density, and by O'Shea and Tremaine (1980) in
the MC simulations of the thermodynamic properties ofliquid and supercritical
water. It is well known, however, that this potential reproduces poorly the
pressure at a given density (or the density at a given pressure) (Beveridge et aI.,
1983; Owicki and She raga, 1977). Even the addition of quantum mechanical
three- and four-body terms to the potential, though extremely demanding in
terms of computer time, did not improve the situation significantly (Clementi,
1985). A slightly different ab initio CC potential (Carravetta and Clementi, 1984)
has been used by Kataoka (1987, 1989) in extensive MD simulations of the
thermodynamic and transport properties of fluid water in a wide range of
conditions. A qualitative reproduction of the anomalous behavior of these pro-
perties has been achieved.
Thus, despite the great importance of quantum mechanical potentials from
a purely theoretical point of view, simple effective two-body potential functions
for water seem to be preferable for the extensive simulations of complex aqueous
systems of geochemical interest in the near future.
The thermodynamics of water in a wide range of temperatures (630 ::; T::; 780
K) and pressures (0.3 ::; P ::; 30 kbar) corresponding to densities 0.166 ::; (! ::;
1.284 g/cm 3 have recently been studied by both computer simulation methods
(Kalinichev, 1991 b; Kalinichev and Heinzinger, 1992). The rigid TIP4P potential
(Jorgensen et aI., 1983) has been used in connection with N PT-ensemble MC
simulations, whereas conventional NVE-ensemble MD simulations have been
performed with the flexible BJH water model (Bopp et aI., 1983).
In the MC simulations, two different system sizes were used (N = 64 and 216)
to assure the reliability of the minimum image cutoff principle applied here to
calculate the potential energy of the system. No long-range corrections were
made, as it was shown (Andrea et aI., 1983) that this method leads to only small
errors in the resulting thermodynamic and structural properties of liquid water,
even with N = 64.
The systems studied in the MD simulations consisted of 200 water molecules
in a cubic box with the side length adjusted to give the required density. The time
step of the integration was 1.5 x 10- 16 s and the simulations extended over
10,000 to 15,000 time steps. Ewald summation in tabulated form was used for the
-00
Table 4. Thermodynamic results of Monte Carlo simulations with the TIP4P potential.
T P n Vm Heonf Cp K cx·1OOO
Run (K) (kbar) N (106 conf.) (cm 3 /mol) (kJ/mol) (J/(mol· K) (I/Mbar) (I/K)
MC1 773 30.0 216 1.10 14.03 ± 0.02 12.35 ± 0.10 61.1 ± 2.2 6.7 ± 0.5 0.25 ± 0.03
(13.52t (15.17) (61.4) (8.0) (0.12)
MC2 773 10.0 64 0.37 18.20 ± 0.05 -8.91 ± 0.18 69.0 ± 5.2 32.7 ± 3.5 0.66 ± 0.08
(17.51) ( -9.55) (63.6) (23.1) (0.41)
MC3 773 1.0 64 1.60 43.13 ± 0.67 -11.53 ± 0.22 97.6 ± 4.6 780 ± 70 3.40 ± 0.3
(34.11) ( -14.70) (100.1) (543) (3.28)
MC4 773 0.5 64 1.10 83.34 ± 1.70 -6.07 ± 0.22 91.3 ± 5.3 2520 ± 340 4.20 ± 0.4
(70.11) ( -7.35) (130.4) (3263) (7.06)
MC5 673 10.0 64 1.10 17.12 ± 0.04 -13.14 ± 0.13 67.9 ± 2.3 19.7 ± 1.3 0.50 ± 0.04
(16.82) ( -12.98) (63.9) (19.8) (0.39)
64 1.10
>
MC6 673 1.0 31.23 ± 0.53 - 18.39 ± 0.29 99.0 ± 7.3 420 ± 50 3.10 ± 0.4 Q
(26.01) ( -21.22) (88.5) (211) (2.14) i7':
MC7 673 0.5 64 1.60 47.20 ± 0.99 -14.60 ± 0.25 126.3 ± 7.7 1680 ± 160 6.20 ± 0.6 e:..
5·
(31.17) ( -19.72) (122.3) (713) (4.89) r;.
::r
MC8 673 0.3 216 1.60 89.70 ± 1.50 -9.04 + 120.2 ± 10.8 4500 ± 600 7.25 ± 1.0 C1>
- 0.18 <
(50.31) ( -14.74) (451.8) (11712) (36.75) I»
::I
Po
• Values calculated from the HGK equation of state (Haar et aI., 1984) are given in parentheses for comparison as "experimental." ~
:I::
!J!.
::I
N
5·
(JQ
....C1>
Computer Simulations of Aqueous Fluids 19
Run
MD1 MD2 MD3 MD4 MD5
coulombic part of the interactions, and the "shifted-force" method (see, e.g.,
Allen and Tildesley, 1987) was used for the other parts of the BJH potential.
With this procedure, the total energy change AEIE during the simulation was
smaller than 5.10- 5 in all cases and the average temperature remained constant
without rescaling, which is very important for the reliability of the dynamical
properties calculated from velocity autocorrelation functions.
The calculated values of the thermodynamic properties for all thermody-
namic states studied are reported in Table 4 and 5. Thermodynamic properties
calculated via the HGK equation of state for water (Haar et aI., 1984), approved
by the International Association for the Properties of Steam (lAPS) as an
international standard up to 1273 K and 10 kbar, are also given there for
comparison. This equation reproduces virtually all available thermodynamic
measurements within the limits of their experimental accuracy and is used in the
present study as a reliable source of self-consistent "experimental" data.
The nonconfigurational contributions to the internal energy, enthalpy, and
heat capacities Cv and Cp were assumed to be identical with those for real gas
in its standard state (Woolley, 1980). The experimental configurational proper-
ties were estimated from the equation of state by
(16a)
and
Hconf = H(T,P) - Hig(T) + RT, (16b)
with the ideal-gas reference state at the given temperature. No other corrections
were added to any property reported.
20 A.G. Kalinichev and K. Heinzinger
Macroscopic Properties
p- V- T Relationships
A comparison of simulated and experimental molar volumes and pressures is
presented in Tables 4 and 5, and in Fig. 4 the positions of state points on the
thermodynamic surface of water, where the simulations have been performed,
are shown. The good agreement of calculated and experimental volumes for runs
MC1, MC2, and MC5 cannot be considered surprising, because the TIP4P
potential has already been carefully tested at liquidlike and higher densities up
to 373 K (Jorgensen and Madura, 1985) and up to 10 kbar at 298 K (Madura et
aI., 1988). In very recent MD simulations (Brodholt and Wood, 1990), the TIP4P
potential was successfully used even at a density almost twice as high as the
normal one and at a temperature of about 2000 K.
At lower pressures the volumes obtained in MC simulations become system-
atically higher than the experimental values. The reason for the agreement with
experiment in the first case, and for the discrepancy in the second, most probably
lies in the "effective" nature of the TIP4P potential, which was specially de-
veloped to reproduce correctly thermodynamic and structural properties of
liquid water under normal conditions and implicitly includes many-body effects
of intermolecular interactions. The relative influence of these effects should
obviously be density-dependent, which is not readily taken into account by the
present potential model and may lead to increasing disagreement with experi-
mental values as the density decreases. The same conclusion can be drawn from
the results of De Pablo and Prausnitz (1989) who made Gibbs-ensemble MC
DENSITY, 9 I em 3
1.5
1.0
10 P, kbar
Fig. 4. Simulated (symbols) and HGK (Haar et aI., 1984) (curves) isotherms of water
density. Full lines, 673 and 773 K; dashed lines, 723 and 823 K. Squares: MC simulations
at 673 K (filled) and 773 K (open). Circles: MD simulations at ~ 673 K (filled) and ~ 773
K (open). Triangle: MD simulation at 630 K. Arrows show the discrepancy in density
(MC) and pressure (MD) between simulations and experimental data.
Computer Simulations of Aqueous Fluids 21
30
20
10
o
-10
-20
0.1 10 P, kbar
Fig. 5. Simulated (squares) and HGK (Haar et aI., 1984) (curves) isotherms of configura-
tional enthalpy (Kalinichev, 1991b). Filled symbols, 673 K; open symbols, 773 K. Full
lines, 673 and 773 K; dashed lines, 723 and 823 K.
Computer Simulations of Aqueous Fluids 23
250 C p , J / ( K·mol )
200
150
100
50
Thus, the two water models used repoduce with reasonable accuracy both
temperature and density dependencies of enthalpy and internal energy over a
very wide range of thermodynamic parameters.
Heat Capacity
The agreement with experiment of the Cp values calculated from the MC simula-
tions [Eq. (8)] is again excellent at liquidlike densities and becomes significantly
worse only at the supercritical maxima of the heat capacity for both isotherms
(Table 4 and Fig. 6). As in the case of densities and enthalpies, the experimental
723 and 823 K isotherms (dashed lines in Fig. 6) suggest that the thermodynamic
properties of TIP4P water correspond to those of real water at a temperature
about 50° higher. The absence of the sharp maximum of Cp in the simulated 673
K isotherm clearly indicates that the critical point for this water model lies at
least 50° lower than the real one.
The values of Cv for the BJH potential (Table 5) are also in reasonable
agreement with experiment, although this property does not change much under
the conditions studied, and the error bounds for the calculated values are diffi-
cult to estimate.
10
K, 1/kbar
0.1 10 P, kbar
15
(\ a.·1000 I K-1
1\
\
10 \
\
\
0.1 10 P, kbar
and Figs. 7 and 8). Once again, the simulated isotherms at 673 and 773 K are
much closer to the experimental 723 and 823 K isotherms, respectively (dashed
lines in Figs. 7 and 8).
In accordance with earlier simulations with the TIPS2 water model
(Jorgensen, 1982a), the simulated thermal expansion coefficient at 30 kbar has
been reported to be approximately twice as large as the one calculated from the
HGK equation of state (Kalinichev, 1986). It was suggested that the HGK
equation may be incorrect in the region above ~ 20 kbar, because none of the
existent shock wave data at high pressures and temperatures were taken into
Computer Simulations of Aqueous Fluids 25
account during its parametrization. This assumption has been confirmed re-
cently by the work of Saul and Wagner (1989), who have shown that the HGK
equation can even lead to negative values ofthe thermal expansion coefficient at
higher pressures. Therefore, the values of IX given in Table 4 as "experimental"
may be underestimated at 30 kbar.
Microthermodynamic Properties
mol % a mol % b
3 3
2 2
o . OLL~~~~~~~~
-120 -80 -40 o -120 - 80 - 40 o
Eb, kJ/mol Eb, kJ/mol
temperature (Madura et at, 1988), the maxima of the distributions shift to lower
energies and their widths increase with increasing pressure.
It is interesting to note that under high-temperature conditions a certain
number of molecules even have a positive bonding energy. As the fraction of
such molecules obviously cannot increase significantly with decreasing pressure,
asymmetry of the distribution results at lower densities. At the very low density
of the run MD4 [dotted line in Fig. 9(b)], the distribution becomes even
bimodal, indicating that a water molecule can be found in two energetically
different environments with bonding energies distributed around ~ - 20 and
~ -5 kJ/moI. While the latter part of the distribution is obviously due to
non bonded, free water molecules, the former one might be considered an indica-
tion of the presence of weakly hydrogen-bonded molecules in significant
amounts even under these high-temperature, low-density conditions.
The bonding energy distribution for the water molecules in normal liquid
water is shown as a dotted line in Fig. 9(a). At 773 K and 10 kbar, the density
of supercritical water is virtually the same (~1 g/cm 3 ) as that of liquid water-at
298 K and 1 bar. Therefore, the difference between these two distributions
[curves 1 and 3 in Fig. 9[a]) can be considered an effect of temperature alone.
At the high temperature, the maximum is shifted by about 40 kJ/mol to higher
(less negative) energies and the distribution becomes significantly wider. The
comparison of both distributions clearly shows that a water molecule experi-
ences completely different energetic environments in these two thermodynamic
states, despite the fact that densities (and hence average intermolecular dis-
tances) are virtually the same in both cases.
(beyond the scale in Fig. 10), because relatively more pairs of molecules are
found at large intermolecular distances, having a near-zero interaction energy.
A comparison of the distributions at 773 K and 10 kbar and at 298 K and
1 bar [the second curve from the top and the dotted curve in Fig. 10(a), respec-
tively] indicates once again the opportunity to see the pure effect of a significant
temperature increase on the shape of the distribution along an isochore, as
densities at both thermodynamic states are virtually equal. The width and height
of the main maximum remain unchanged. However, in the attractive branch of
the distribution (negative energies) a significant amount of molecular pairs are
redistributed from the "hydrogen-bonding" range of energies (~- 25- -15
kJ/mol) to the "non bonding" range (~- 15- - 6 kJ/mol). In the repulsive
(positive) branch, the probability for a molecular pair to have a rather high
interaction energy between 10 and 20 kJ/mol noticeably increases, confirming
that repulsive interactions become a more important contribution to the ther-
modynamics of water at high temperatures.
10 <U), kJ I mol a b
5
O~~~~~~~~~/~~~~~~~
-5
-10
/
-15 /
/
3 4 5 6 r,A 3 4 5 6 r,A
Fig. 11. Average potential energy of a water molecule for normal- (left) and high- (right)
density water as a function of the 0-0 distance: total (solid lines), Coulombic (dashed
lines), and non-Coulombic (dotted lines) components (Jancso, et aI., 1984). Dash-dotted
lines in (a) and (b) represent high-temperature simulations MD3 and MD1, respectively
(see Table 5).
V
0 0
-5 -5
I ,
-10 -10 I ,
I ,
I,
-15 -15 ~I
2 3 4 5 6 7 r,A 2 3 4 5 6 7 r,A
Fig. 12. Distance dependence of the average potential energy between two water mole-
cules for supercritical water. (a) 773 K and 30, 10, 1, and 0.5 kbar (from the top to the
bottom). (b) Full line, 773 K and 10 kbar; dashed line, 298 K and 1 bar. The water density
is virtually the same in both thermodynamic states. Dotted line, non-Coulombic contri-
bution·to (U(r» (Kalinichev, 1991b).
The first properties derived from a computer simulation as far as the structure
of an aqueous system is concerned are the various radial distribution functions
(RDF), gi/r). These functions give the probability of finding a pair of atoms i
and j a distance r apart, relative to the probability expected for a completely
random distribution at the same density. The description of the structure in
terms of radial distribution functions is an essential constituent of any statistical-
mechanical treatment of a fluid (e.g., McQuarrie, 1976; Hansen and McDonald,
1976; Allen and Tildesley, 1987).
30 A.G. Kalinichev and K. Heinzinger
."
.v
.1'
J ~~••,
0 0
1 2 3 4 5 r, A
2 90H
,..
:\
1\
2 3 4 5 r, A
2 3 4 5 r, A
t
tion numbers, calculated as
where Qo is the number density of particles j, are also shown in Fig. 13 for 0-0
distributions as dotted and dash-dotted lines corresponding to the high- and
low-temperature conditions.
The effect of temperature on the water structure at a constant density can be
clearly seen from these curves. The characteristic second maximum of the
oxygen-oxygen RDF at ~4.5 A reflecting the tetrahedral ordering of water
molecules due to hydrogen bonding is completely absent under supercritical
conditions. Moreover, a pronounced minimum of the distribution appears in its
place. The comparison of 900(r) and noo(r) at low and high temperatures shows
that a significant redistribution of first and second neighbors from the
"hydrogen-bonding" regions of ~ 2.6 to 2.9 Aand ~ 3.8 to 5.0 Ato the inter-
mediate "nonbonding" region of ~ 3.1 to 3.8 Atakes place. Tanaka and Ohmine
(1987) have shown recently that the water molecular pairs at intermediate dis-
tances (~3.1- 3.4 A) are primarily responsible for repulsive interactions in liquid
water. Therefore, additional molecular density at these distances gives rise not
only to the increase of weakly bonded molecular pairs, but also to the increase
of repulsive interactions as well (see Fig. 10 and the discussion of pair interaction
energy distributions in the previous section).
The sharp first intermolecular peak of 90H(r) in normal liquid water becomes
significantly lower and much less pronounced under supercritical conditions,
although its presence alone clearly indicates that hydrogen bonding persists up
to 773 K. This peak disappears, with only a shoulder remaining, at a slightly
higher temperature (Mountain, 1989).
In Fig. 14 oxygen-oxygen, oxygen-hydrogen, and hydrogen-hydrogen RDFs
for the thermodynamic points studied in the supercritical MC simulations
(Kalinichev, 1991b, see also Table 4) are presented. The supercritical MD simu-
lations with the BJH potential (Table 5) resulted in RDFs very similar to those
obtained from the MC simulations with the TIP4P potential; therefore, they are
not presented here. Available experimental data on the X-ray diffraction of water
under supercritical conditions at 1 kbar pressure (Gorbaty and Demianets, 1983;
1985) are shown as dashed lines in Figs. 14 (a) and (d) for comparison. Though
the agreement is sufficiently satisfactory as far as the heights of the first maxi-
mum and the general shape of the curves are concerned, the disagreement in the
positions of the main maximum and in the different steepnesses of the repulsive
branches of the curves is rather disappointing. The comparison of simulated
RDFs at normal conditions with the X-ray data of Gorbaty and Demianets
shows a similar discrepancy in the steepness at small distances. A recent determi-
nation of the liquid water structure from neutron diffraction measurements
(Soper and Phillips, 1986) also gives a less steep repulsion branch of the RDF,
as compared with computer simulations even using different water models (e.g.,
Ruff and Diestler, 1990).
The general distance, temperature, and density dependence of 900(r) [Figs. 14
(a) and (d)] under supercritical conditions closely resemble that of a simple
liquid (e.g., argon), with the second maxima at distances approximately twice as
large as the first ones. Little can be said about the behavior of 9HH(r) [Figs. 14
32 A.G. Kalinichev and K. Heinzinger
90-0 a 90-0 d
5 5
0.5 kbar 0.3 kbar
4 4
2 10 kbar 2 1 kbar
30 kbar 10 kbar
0 0
0 2 4 6 8 r, A 0 2 4 6 8 r, A
5 5
90-H b 90-H e
4 0.5 kbar 4 0.3 kbar
2 10 kbar 2 1 kbar
30 kbar 10 kbar
0 0
0 2 4 6 8 r,A 0 2 4 6 8 r,A
5 5
9 H-H C 9 H-H f
4 0.5 kbar 4 0.3 kbar
2 10 kbar 2 1 kbar
30 kbar 10 kbar
0 0
0 2 4 6 8 r, A 0 2 4 6 8 r, A
Fig. 14. Simulated atom-atom radial distribution functions (full lines) for supercritical
water at 773 K (a, b, c) and 673 K (d, e, f) (Kalinichev, 1991b). Dashed lines represent
X-ray diffraction measurements (Gorbaty and Demianets, 1983; 1985; and private com-
munication) under the same conditions. Each curve is shifted by 1 relative to the previous
one.
Computer Simulations of Aqueous Fluids 33
(c) and (f)], except a general notion that these functions show almost no H-H
correlations under supercritical conditions. The shape of gOH(r) [Figs. 14 (b) and
(e)] remains virtually the same along the isotherms of 673 and 773 K over a very
wide range of pressures from 0.3 to 30 kbar. The small first peak at :::::: 1.8 A
indicates that hydrogen bonding still persists under these conditions.
If one adopts the simple geometrical convention that a hydrogen bond exists
if the O-H pair is separated by less than 2.4 A[the position of the first minimum
of gOH(r) in normal liquid water; see Fig. 13], the average number of hydrogen
bonds NB experienced by an oxygen atom in a molecule can be easily estimated.
Using this definition, Mountain (1989) has found that the number of hydrogen
bonds per molecule scales as a single function of temperature in the density
range 1.0 ~ (} ~ 0.6 g/cm 3 and the temperature range 270 :s;; T:s;; 1130 K. The
scaling does not work for densities less than ::::::0.45 g/cm 3 and breaks down for
densities greater than:::::: 1.0 g/cm 3 when NB saturates at a value of 2, while the
density can increase with growing pressure.
The pressure increase has a strong effect on the hydrogen bond structure of
water even at moderate temperatures. The projection densities for the eight
nearest-neighbor water molecules around a central one and separately for the
first four and the second four onto the xy-plane of a coordinate system as defined
in the insertion are shown in Fig. 15, calculated from MD simulations of pure
BJH water at 1 bar and 22 kbar (Palinkas et aI., 1984). Eight is about the number
of nearest neighbors in the high-pressure (HP) case. It can be clearly observed
that there is a decrease in the preference for the occupation of tetrahedral
positions in the HP water. The neighbors five to eight in HP water are seen to
be closer to the central molecule than the second neighbors in the normal-
pressure (NP) water, but they do not show any preference for occupying tetra-
hedral positions. Accordingly, it seems reasonable to conclude that the first
coordination sphere of HP water consists of neighbors of two different types.
Similar conclusions have been drawn from the results of other MD and MC
simulations, in which an interpenetration of the subsections of the random
hydrogen bond network has been proposed (Stillinger and Rahman, 1974b;
Impey et aI., 1981; Madura et aI., 1988).
A long standing-question was concerned with the effect of pressure on the
hydration sphere of ions. Conflicting conclusions-from a complete breakdown
to a strong enhancement-were drawn from various experimental investiga-
tions (Heinzinger, 1986). In order to clarify this point, two MD simulations of a
2.2 molal NaCl solution have been performed with the BJH model of water and
the Na+ -water and'Cl- -water pair potentials as given in Table 3. In each case,
the basic periodic cube contained 200 water molecules, 8 anions, and 8 cations.
The densities of the simulations were 1.0792 and 1.3067 g/cm 3 that are at 298 K
equivalent to pressures of 1 bar and 10 kbar, respectively. The simulations
extended over aboJ.lt S ps. The average temperatures of the simulations were 299
and 303 K for the normal-pressure and high-pressure runs, respectively (Jancso
et aI., 1985).
34 A.G. Kalinichev and K. Heinzinger
NP 1-8 HI" 1 - 8
Fig. 15. Densities of the projections of the oxygen atom positions of the eight, and
separately for the first and second four, nearest neighbor-water molecules around a
central one onto the xy-plane of a coordinate system as defined in the insertion, calcu-
lated from MD simulations of pure BJH water at 1 bar (NP, left) and 22 kbar (HP, right)
(Palinkas et ai., 1984).
10.--------,,------,20
gNao(r) / nNao(r) 5 9ClO (r) I
I
nCIO(r) 10
I I
I 15 I
I 4
I
I
5 I 10 3
5
2
5
~-L_il-~~~~_L--L~O ~-L~L-~-~~-~~~O
9 NaH (r) nNaH(r) 5 9 C1H (r) n C1H (r) 10
16
4
6 12
3
8 5
2
2 4
OL--L-~~-L--L-~-L~O L--L~L--L_L__L_~~~O
o 234 5 6 7 8 o 2345678
riA riA
Fig. 16. Ion-oxygen and ion-hydrogen radial distribution functions and running coor-
dination numbers for high-pressure (dashed) and normal-pressure (full) 2.2 molal NaCl
solutions (Jancs6 et al., 1985).
increase in the coordination number (the density is 21% higher), as well as the
slight decrease in peak height, seem to indicate a certain loss of structure of the
Na-hydration sphere with increasing pressure. The chloride hydration seems to
undergo greater changes with a pressure increase, suggesting that the hydration
layer of this ion is even less developed under pressure than it is normally, though
the preference for forming linear hydrogen bonds between Cl- and water is not
to be affected significantly (Fig. 17). Unlike Na+, the hydration number of Cl-
increases by almost two.
The orientation of the water molecules in the hydration shell of an ion can be
characterized by the cosine of the angle between the dipole moment direction of
the water molecule and the vector pointing from the oxygen toward the center
of the ion. In Fig. 17 distributions of (cos e) are shown for the hydration shells
of sodium and chloride ions in the NP and HP solutions. The increase in
pressure broadens the (cos e) distributions. This means that with increasing
pressure, a decrease in the preference for the trigonal orientation in the case of
Na- and for linear hydrogen bond formation in the case ofCI- can be observed.
A second computer simulation in which the effect of pressure on the hydra-
tion shells of ions has been investigated is reported by Bounds (1985). He per-
formed MD simulations where a Ca 2 + and Cl- were surrounded by 64 TIP4P
water molecules at room temperature and also pressures of 1 bar and about 10
kbar. As far as the Ca 2+ is concerned, the results are very similar to those found
for Na+: no change in the position of the firstmaximum (r M1 ) in gCaO(r), a small
decrease (hardly outside the limits of uncertainty) for r M1 in gCaH(r) and in the
average value of e, as well as a slightly larger hydration number (+ 0.3) in
the high-pressure case.
The effect of elevated temperature and pressure on the ion-oxygen RDFs has
W
0\
Table 6. Comparison of characteristic values of the radial distribution functions (intermolecular part) for normal-
pressure (1 bar; denoted by NP) and high-pressure (10 kbar; HP) 2.2 molal NaCI solutions (Jansco et aI., 1985), as
well as for pure water at normal pressure (0.9718 g/cm 3 ; 336 K) and high pressure (1.346 g/cm 3 ; 350 K) (Jansco
et aI., 1984). R k, rMk , and rm~ give the distances where gij(r) = 1 has i!s kth maximum and minimum, respectively.
The distances are given in A with an uncertainty of at least ±0.02 A. nij(r m1 ) is the running coordination number at
the first minimum of gij(r).
j Rl r Ml gikMl) R2 rml gij(rmd r M2 gii r M2) nij(rm1 )
P(eos e)
0.3
0.2
CI
0.1
-1 o
cos e
Fig. 17. Distribution of (cos 8) in the first hydration shells of the ions from MD simula-
tions of2.2 molal NaCI solutions at normal pressure (solid line) and high pressure (dashed
line). 8 is defined in the insertion. The distributions are normalized and given in arbitrary
units (Heinzinger, 1986).
16 .-----------------------------~
n 9 L10 (r)
II
II
14 II
II 40
II
12 II /
II /
II /
10 II / 30
/
/
/
8 /
/
/
/
20
6 I
/
I
/
4 I
/, 10
10
OL---~--~~~~--~----i-_o~
1 2 3 4 5 6 riA
Fig. 18. Ion-oxygen radial distribution functions and running coordination numbers
for a 0.55 molal LiI solution at 508 K (full) and 308 K (dashed) with the same density of
1.05 g/cm 3 . At the high temperature, the pressure is about 3 kbar (Szasz and Heinzinger,
1983a).
38 A.G. Kalinichev and K. Heinzinger
been investigated for a 0.55 molal LiI solution by MD simulations at 308 and
508 K and constant density, where the higher temperature corresponds to a
pressure of about 3 kbar (Szasz and Heinzinger, 1983a). It can be seen from Fig.
18 that the increase in temperature (and pressure) reduces significantly the height
of the first peak in gLiO(r) and broadens it (full line). At the same time, the gap
between the first and second hydration shell begins to get filled up. In this way,
the number of water molecules (six) in the first hydration shell of Li+ remains
constant, as can be seen from nLiO(r = 3 A). The second hydration shell of Li+
almost disappears at high temperature and pressure. Accordingly, there is a
significant difference between the two curves for nLiO(r) at r = 4.5 A that slowly
disappears with increasing distance. In the case of the iodide ion, the first
hydration shell is, even at normal temperature and pressure, not well pro-
nounced (lower part of Fig. 18). Consequently, not much change can be ob-
served. Only a slight smearing out of the first peak results from the temperature
(and pressure) increase.
Se1f-Diffusion Coefficients
The self-diffusion coefficients can be determined from a MD simulation either
by the mean square displacement
. \ [r(t) - r(0)]2)
D= 11m (19)
t-+ 00 6t
or, as done here, through the velocity autocorrelation functions (V ACF) with
the help of the Green-Kubo relation
D = lim -31
t-+ 00
it
0
(v(O)· v(t') >dt'. (20)
I I I I
o 0.2 0.4 0.6 0.8 t/ ps
Fig. 19. Normalized velocity autocorrelation functions for water molecules at high tem-
peratures and pressures. MDt (full), MD3 (dashed), MD5 (dotted), MD2 (dash-dotted),
and MD4 (long-dashed). See Table 5 for the thermodynamic conditions ofthe MD runs.
where N denotes the number of particles, NT the number of time averages, and
vj(t) the velocity of particle j at time t.
The velocity autocorrelation functions for water molecules at different tem-
peratures and densities calculated from MD simulations are shown in Fig. 19.
The effect of temperature on the VACFs is in close agreement with the results
of Stillinger and Rahman (1972) who simulated water at a temperature of about
600 K and a density of 1 g/cm 3 with an earlier version of the ST2 potential,
known as BNS. The density dependence of these functions is very similar to that
for water at normal temperatures (Jancso et aI., 1984), in agreement with the
similarity in the pressure-induced changes of the structural properties as men-
tioned above. It can be seen from Fig. 19 that the VACFs decay faster at the
higher density.
The corresponding self-diffusion coefficients of water, calculated from these
simulations, are given in Table 7. In the case of the lowest density studied (run
MD4), only a rough estimate of D can be made because of the very slow decay of
the velocity autocorrelation function under such conditions (the uppermost
curve in Fig. 19). The calculated values of D for the BJH water model employed
in the simulations are in good agreement with the simulations of Stillinger and
Rahman (1972) for the BNS potential and those of Brodholt and Wood (1990)
for the TIP4.P potential, as well as with the results of Kataoka (1989) who has
also calculated the viscosities and thermal conductivities of water in a wide range
of thermodynamic conditions from the MD simulations with the ab initio
Carravetta-Clementi water potential.
The VACFs for the sodium ions, chloride ions, and solvent water at normal
40 A.G. Kalinichev and K. Heinzinger
1.0
1.0
1.0 H-----'-.<~----------:::_,-;-~___:::ii'_==_=::_::_:=-~""'4
pressure (1 bar) and high pressure (10 kbar) calculated from MD simulations of
2.2 molal NaCl solutions at normal temperature are shown in Fig. 20 (Jancso et
aI., 1985). The pressure dependence found for solvent water is very similar to the
one for pure water. In all three cases, the velocity autocorrelation function
decays faster at the higher pressure. The pressure dependence ofthe self-diffusion
coefficients of the ions and of solvent water has been calculated from the two
simulations of the 2.2 molal NaCI solution with the help of the VACFs according
to Eq. (20). The results are compared with the available experimental data in
Table 8. Measurements of the pressure dependence of the self-diffusion coeffi-
cients of the ions in aqueous solutions are not known.
The self-diffusion coefficient of the solvent water has been corrected for the
different temperatures of the NP- and HP-MD simulations by using the temper-
ature dependence of D of pure water (Weingartner, 1982). The comparison of the
self-diffusion coefficients obtained from the MD simulations with the experimen-
tal results shows that the simulation leads to smaller values in all cases. By
considering the uncertainties in the experimental results for DNa + and DC1 -
as indicated by the values reported from different laboratories, the differences
between simulation and experiments seem to be small enough to justify the
conclusion that the changes in the self-diffusion coefficients for water and the
ions with increasing pressure, as calculated from the simulations, are reliable, at
least qualitatively.
Table 8 shows that at normal temperature the 21% density increase is accom-
panied by about a 20% decrease in the self-diffusion coefficient of solvent water
Dw' (Separate calculations lead to similar changes for all three water subsystems:
bulk water, hydration water ofNa+ and of Cl-). In pure water at 350 K, a 38%
density increase yielded about a 35% decrease in the self-diffusion coefficient
(Jancso et aI., 1984), which suggests that at high pressures the increased steric
42 A.G. Kalinichev and K. Heinzinger
Hindered Translations
The pronounced effects of temperature and pressure (density) are also reflected
in the spectral densities of the hindered translational motions for the water
molecules that have been calculated by Fourier transformation
h-
f(v) = V 2nc
foo0 (v(O)'v(t)
(V(0)2) cos(2ncvt)dt (22)
from the normalized velocity autocorrelation functions. Such spectra for nor-
mal- and high-density simulations at low temperature (Jancs6 et aI., 1984) are
shown in Fig. 21 as the solid and dashed lines, respectively. Similar spectra from
high-temperature simulations (Kalinichev and Heinzinger, 1992) are shown in
Fig. 21 as dotted and dash-dotted lines, respectively, for the normal- and high-
density cases. In normal liquid water, the distinct peak at ~ 50 cm -1 is usually
assigned to the hydrogen bond 0-0-0 flexing motion and the broad peak
around ~ 200 cm -1 to 0-0 stretching motions (see, e.g., Sceats and Rice, 1980).
The first peak strongly decreases with isothermally rising density, and both of
the peaks have completely disappeared at high temperature, indicating a signifi-
cant breakdown of the hydrogen bond structure of water due to increased
density and temperature. The same temperature dependence has been observed
already by Stillinger and Rahman (1972, 1974a) in MD simulations of water at
elevated temperatures using BNS and ST2 potentials. Both the low- and high-
Computer Simulations of Aqueous Fluids 43
f(v)
0.1
temperature simulations show in the region between 300 and 400 cm -1 at high
density about twice the intensity than at normal density.
It is interesting to note that the effects of pressure and temperature on the
spectral densities in the frequency region corresponding to the hydrogen bond
bending and stretching motions are very similar to that of Li+ as obtained from
an MD simulation of a 2.2 molal LiI solution with the ST2 water model (Szasz
and Heinzinger, 1983b). The changes with pressure in the spectral densities of
Na+ and Cl- calculated by Fourier transformation [Eq. (22)] of the corre-
sponding velocity autocorrelation functions (Fig. 20) are rather small. They do
not give any hint for understanding the different pressure dependencies of the
self-diffusion coefficients of the two ions discussed above (Jancso et aI., 1985).
to the O-H bonds in the plane of the molecule (VI and v2 ). and 3) perpendicular
to the plane of the molecule (PI and P2).
By using capital letters to denote the projections of the hydrogen velocities
onto the corresponding unit vectors. the following quantities are defined:
Ql = U1 + U2• Rx = VI - V2 •
Q2 = VI + V2• Ry = PI + P2 • (23)
Q3 = U1 - U2• Rz = PI - P2 •
where Ql. Q2. and Q3 describe approximately the three normal mode vibrations
1.0 z
~
r:\~
r.
kY
0.5
r-\'
\... \
\ ..\ \
\"-.\ \.
1". • '\,.
x
\ .. \ .
\ ... \
,..
"-
---------- ----
1.0 l-+-r-~-_===""-~~~~=-;-::-=-=-=--. . . . ..
,.,""...---1
~
~\
1.\
I~
0.5 h'\
.. \
,.\ \ y
\ ..•\ \
\ ..
\ .. \
"-
--
I -\1'.;.:"'·r-;;;>~:;;.;·"'-:;..:-
1.0. , ... .p ' . · ""-~;;,;;;;;...;;--======.....,
... . .~.-
. / ..
\:~(..'
0.5
Fig. 22. Velocity autocorrelation functions for the three librations of water molecules at
high temperatures and different densities. MDl (full). MD3 (dashed), MD5 (dotted), MD2
(dash-dotted), and MD4 (long-dashed). See Table 5 for the thermodynamic conditions of
the MD runs. The librations are calculated separately for the three components in a
molecule-fixed coordinate system as defined in the insertion.
Computer Simulations of Aqueous Fluids 45
1.0
0.5
°1 a b
0.0
~nA.
I~V v
-0.5
-1.0
1.0
0.5
°2
0.0
A1\ A
~"
-0.5
-1.0
1.0 03
0.5
0.0
Inn n n
~Vvv
-0.5
-1.0
0 0.1 0.2 II ps o 0.1 0.2 II ps
Fig. 23. Velocity autocorrelation functions for the three intramolecular vibrations of
water molecules at high temperatures and highest (a) and lowest (b) densities studied
(runs MDl and MD4, respectively). QI' Q2' and Q3 denote the symmetric stretching,
bending, and asymmetric stretching modes, respectively.
A-Y
f (v) z Fig. 24. Spectral densities (in arbi-
trary units) of molecular librations
in high-temperature water at differ-
ent densities. See Fig. 22 for the
notation.
\
\
f (v 2 )
(\
I\
/-1t ~
I- 1
I!/
z
\
- ---
1000 1500 2000 v2 ' cm· 1
f (v3 )
2500
1.86 and 1.97 D for the dipole moment, respectively (Jancso et at, 1984). Both the
increases of temperature and of density lead to an increase in the intramolecular
0- H distance and a decrease in the H -0- H angle. These changes in geome-
try, together with the partial charges on the 0 and H atoms, lead to the
temperature and density dependence of the dipole moment for the water mole-
cules. From an empirical relationship for the rate of decrease of the O-H
stretching frequency with the intramolecular O-H distance (~20.000 cm-1j,.\.)
48 A.G. Kalinichev and K. Heinzinger
(La Placa et aI., 1973), a redshift of about 55 cm- 1 relative to liquid water at
normal temperature is expected for the vibrational frequencies of the run MD3
at a temperature of 680 K, but at a density close to that of normal water.
However, a blueshift of the same magnitude is estimated, based on the symmetric
stretching frequency for normal liquid BJH water (VI = 3475 cm- l ) (Bopp, 1987)
in reasonable agreement with the experimental data of Lindner (1970), from
which a blueshift of about 120 cm- l might be estimated. These results seem to
indicate that the simple empirical relationship is not valid any more when both
temperature and pressure (density) can influence independently the average
intramolecular geometry.
The temperature and density dependence of the position of the maximum in
the spectral densities of the symmetric stretching frequency VI (Fig. 25 and Table
9) is in a good agreement with available experimental data, whereas all simulated
spectra are approximately twice as wide at half-height as the corresponding
experimental ones. The very weak temperature and density dependence of the
bending frequency V2 is in agreement with the Raman measurements of Ratcliffe
and Irish (1982) who found that this frequency (1638 em-I) does not change
within experimental error up to 573 K. No experimental data exist for the
asymmetric stretching frequencies V3 for H 2 0 under the thermodynamic condi-
tions of the MD simulations reported. However, the temperature and density
Computer Simulations of Aqueous Fluids 49
Conclusions
In this chapter, the concept and application of computer simulation methods are
presented with particular attention to aqueous systems at high temperatures and
pressures. The statistical mechanical formulas for thermodynamic, structural,
and kinetic properties are presented, including the treatment of time correlation
functions for the analysis of the spectroscopic properties of water. The current
status of available intermolecular potentials for aqueous electrolyte solutions is
briefly summarized. Particular emphasis is placed on the properties of super-
critical water in a wide range of densities.
It is obvious that computer simulation studies of water and aqueous solutions
at high temperatures and pressures are just at their earliest stage as far as
geochemical relevance is concerned. However, the results presented here seem to
demonstrate that simulation methods can contribute significantly to a better
understanding of the properties of hydrothermal and metamorphic fluids.
There is undoubtedly strong need for the improvement of the potentials
employed in the simulations, since all empirical intermolecular potentials used
so far in computer simulations have been parameterized to reproduce correctly
a set of liquid water properties under normal conditions. This leads to some
quantitative disagreements between the simulated and measured properties of
aqueous systems at high temperatures and pressures. The reason of these dis-
agreements lies, most probably, in the "effective" nature of the potentials used,
which takes into account the many-body effects of intermolecular interactions
only implicitly. Many-body effects can be accounted for by the polarization
models of water, which can also describe the dissociation of water molecules
(Stillinger and David, 1978). The effect of dissociation, which can be neglected
under the thermodynamic conditions presently studied (Holzapfel, 1969; Tanger
and Pitzer, 1989), has obviously to be taken into account at higher temperatures
that are of geochemical interest (Helgeson, 1981; Sverjensky, 1987). So far,
however, only one simulation of this kind has been performed (Demetros and
David, 1982).
The direct successive inclusion of three- and higher-body terms from ab initio
calculations (Clementi, 1985), though very impressive, can hardly be considered
50 A.G. Kalinichev and K. Heinzinger
Acknowledgments
This study was carried out during the research stay of one of the authors
(A.K.) at the Max-Planck-Institut fUr Chemie (Otto-Hahn-Institut) within the
scope of the Alexander von Humboldt Fellowship Program. Financial support
by the Alexander von Humboldt-Foundation and Deutsche Forschungsgemein-
schaft is gratefully acknowledged.
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Chapter 2
Estimating Thermodynamic Properties
by Molecular Dynamics Simulations:
The Properties of Fluids at High
Pressures and Temperatures
D.G. Fraser and K. Refson
Introduction
Computer simulations of geochemical reactions are beginning to enable experi-
mentalists to explore processes which are difficult or impossible to reproduce in
the laboratory because of either their kinetics or the physical conditions in-
volved. In some cases, it has even been claimed that the calculated data are more
reliable than available experimental results (Lie and Clementi, 1986), and in the
case of H 2 0, such a claim was vindicated by subsequent better experimental
data (Soper and Phillips, 1986). The widespread availability of mini-super-com-
puters has made such calculations tractable in most laboratories, and since their
first applications to geochemistry in the mid-1970s (Woodcock et aI., 1976;
Matsui et aI., 1981; Angell et aI., 1982), very rapid progress has been made.
Recent reviews of applications to silicate minerals (Parker and Price, 1990) and
diffusion in silicate melts (Kubicki and Lasaga, 1990) have appeared and give
good summaries of these areas. It is predictable that such calculations will
become increasingly important to experimentalists over the next few years as a
prelude to designing new experiments or to help pinpoint crucial experiments in
difficult areas so that a good understanding of simulation techniques and their
basic assumptions will be essential.
The use of thermodynamics and the application of experimental data ob-
tained from studies of simple systems to geological problems require both accu-
rate standard state data for minerals, melts, and volatile fluids and a detailed
knowledge of the effects of mixing. Standard state data are available for many
simple systems of geological interest although the range of pressures and temper-
atures of measurement is often limited. In addition, the wide range of possible
mineral compositions in natural rocks makes it necessary to have robust
theoretical mixing models for interpolation and extrapolation. Slow reaction
kinetics also make some experiments impossible to carry out in reasonable times
so that alternative means of investigation are required.
Estimating Thermodynamic Properties by Molecular Dynamics Simulations 61
Gp T
,
= Gpo , T + f -'
p
po
RT
P dP.
(3)
For real gases, the ideal gas equation rapidly becomes invalid at pressures
above even a moderate to high vacuum as the fluid density increases. This was
interpreted by Van der Waals (1873) as being the result of the existence of
long-range attractive forces in the fluid opposed by short-range repulsive forces
acting to define the contribution to the fluid volume of the volumes of molecules
themselves and is expressed by the Van der Waals equation of state
The concept that long-range attractive forces and short-range repulsive forces
determine the P- V- T properties of fluids is still central to attempts to simulate
the behaviour of fluids at high pressures and temperatures.
The Van der Waals equation can be used only over relatively limited ranges of
pressure, and many attempts have been made to describe the behaviour of real
fluids over more extended pressure ranges. The virial equation, for example, has
the form
62 D.G. Fraser and K. Refson
PV _ 1 B(T) qT)
RT- +V+y+ .. ·, (5)
where again V is the molar volume of the fluid and B(T) and qT) are empirical
coefficients which are properties of the fluid in question. The second virial
coefficient carries important information about the interactions of pairs of mole-
cules that is important in determining the pair potential U(r) needed for the
calculations described below. However, the virial equation does not converge at
high pressures and it depends on empirical knowledge of the potentials.
Other empirical approaches to describing the properties of high-pressure
fluids include the law of corresponding states and the Redlich-Kwong and
modified Redlich-Kwong equations.
In the former, the equation of state is expressed in terms of so-called "re-
duced" variables PR , VR , and TR , in which the P, V, and T are related to the
respective values at the critical point: Pc, Vc, and Tc
T
TR = - · (6)
Tc
Using these reduced variables, we see that the properties of many volatile
fluids are similar so that the properties of fluids which have not been investigated
at high pressures may be estimated from experimental data for other fluids for
which data already exist using correlations based on the reduced variables (e.g.,
Wood and Fraser, 1976).
The Redlich-Kwong equation (Redlich and Kwong, 1949) can be considered
an extension to the Van der Waals equation. It also has two empirical variables
a and b
Holloway (1977, 1981) and Kerrick and Jacobs (1981) have both used further
modified Redlich-Kwong (MRK) equations in which Temperature, or Tem-
perature and Pressure dependent terms are introduced into the coefficients a and
b to improve the ability of the equations to describe the behaviour of H 2 0, CO2 ,
and H 2 0-C02 mixtures at high temperatures and pressures, and this approach
has been followed by several subsequent authors (Bottinga and Richet, 1981;
Halbach and Chatterjee, 1982). A review of equations of state of fluids is given
by Saxena and Fei (1987).
the molecular level (e.g., Allen and Tildesley, 1987). Such methods have been
extensively investigated for H 20 at low pressures (Jorgensen, 1982; Jorgensen et
aI., 1983; Jorgensen and Madura, 1985; Lie and Clementi, 1986) and the inter-
molecular interactions under those conditions are quite well known. More
recently, calculations based on the work of these authors have been performed
to estimate the P- V- T properties of H 20 to 1 Mbar and 2000 K (Fraser and
Refson, 1990; Belonoshko and Saxena, 1991) and some calculations over a
limited pressure range were reported by Brodholt and Wood (1990). Similar
studies of the thermodynamic properties of N2 by molecular dynamics simula-
tion have been successful in predicting structural phase transformations in solid
N2 up to 100 GPa (1 Mbar) (Belak et aI., 1990; Nose and Klein, 1983).
Statistical mechanics allows bulk thermodynamic properties to be calculated
from a knowledge of the motions and energy states of the constituent particles
at the atomic or molecular level. The fundamental assumption is the ergodic
hypothesis that a time-averaged property of a given system is equivalent to the
instantaneous average of the property calculated for many such systems. Two
main methods are used. Monte Carlo (MC) methods examine the behaviour of
an ensemble of particles moved at random. Molecular dynamics (MO) uses the
Newton-Euler equations of motion to calculate the motion of each particle in
the system.
In MO, models of the long- and short-range intermolecular forces in a system
are used to calculate the accelerations and velocities of individual molecules
using the equations of simple mechanics. By knowing the exact positions and
velocities of molecules, the pressure and temperature, for example, may be
calculated for a given fluid density or volume, thus yielding an equation of state
for the fluid. In addition, because the dynamical trajectories of individual parti-
cles are calculated, dynamic properties of the fluid such as diffusion rates and
viscosity may be calculated.
The potential energy of an assemblage of molecules may be expanded in terms
of single-body, two-body, three-body, etc. interactions. Thus,
(8)
where the first term represents the effects of external fields and interactions with
the container walls and is normally zero, the second term all the pairwise
interactions, the third term all three-body interactions, etc., and the terms ri or
ri , rj , etc. give the coordinates of individual atoms, pairs, and so on. Interactions
for terms higher than three are very small and are usually ignored in such
calculations. In addition, in order to economize on computer time, it is usual to
incorporate the three-body interactions into the second term by using an effec-
tive pair potential vtff (ri' rJ Thus, the PE is given by
(9)
The description of the effective pairwise potential defines the potential energy
well for the system. This potential is usually designed to include London or
dispersion forces and Coulomb interactions in the case of charged or polar
64 D.G. Fraser and K. Refson
systems. One common expression for these interactions is the 12-6 Lennard-
Jones potential which describes the potential energy well in the form
(10)
He 0.08655 0.2602
Ne 0.3492 0.2755
Ar 1.177 0.3350
Kr 1.661 0.3581
Xe 2.336 0.3790
N2 0.8663 0.3632
O2 1.050 0.3382
CO2 2.039 0.3762
CH 4 1.341 0.3721
H 2O 0.64869 0.3154
Estimating Thermodynamic Properties by Molecular Dynamics Simulations 65
by Ross and Ree (1980) and Belonoshko and Saxena (1991). The Lennard-Jones
formulation, however, is extensively used in the literature and we have used this
in our simulations, together with a description of the geometry and mass and
charge distribution of the H 2 0 molecule based on structural observations. The
coefficients used in these simulations thus have the merit of relating to the
physical properties of real molecules and should provide the best basis for
extrapolation beyond the bounds of existing experimental data. This will be
particularly important if fluid mixtures are to be considered successfully.
We have used a 12-6 Lennard-Jones model to investigate the behaviour of
H 2 0 at high pressures and temperatures. A correct description of the inter-
molecular potentials according to (9) is essential if the calculated thermody-
namic quantities of the bulk fluid are to agree with experimental data. Much
research has already been done on simulating the properties of H 2 0 at low
temperatures and pressures because of its importance as a solvent and in biologi-
cal systems (Lie and Clementi, 1986, Jorgensen, 1981, 1982; Jorgensen and
Madura, 1985; Jorgensen et aI., 1983). In addition to a small number of ab initio
models of intermolecular potentials for water based on quantum chemistry, of
which one of the best known is the MCY potential of Matsuoka et ai. (1976),
there are dozens of semiempirical potentials. A comparison of results obtained
using six such models (SPC, TIP3P, BF, TIPS2, and TIP4P) is given by
Jorgensen et ai. (1983) who show that most achieve good agreement with experi-
mental data and that TIPS2 and TIP4P give particularly good agreement with
experimental data for the 0-0 radial distribution function obtained by neu-
tron diffraction (Thiessen and Narten, 1982). Simulations of the high-pressure
properties of water using MCY water have been reported by Impey et ai. (1981)
and similar calculations using TIP4P have been used to calculate accurate
densities of H 2 0 at up to 10 kbar at 298 K (Reddy and Berkowitz, 1987;
Madura, et aI., 1988). Recently, we have investigated the use of this potential at
much higher pressures and temperatures and have reported calculated P- V- T
properties of water up to 300 kbar and 1500 K elsewhere (Fraser and Refson,
1990).
Simulation Conditions
In the TIP4P model, as in MCY, the water molecule is considered to be rigid
with the geometry and point mass and charge distribution shown in Fig. 2. The
0.15 A
H H
The TIP4P model of the water
Fig. 2. q q
molecule. (+0.52) (+0.52)
66 D.G. Fraser and K. Refson
interatomic angle H-O-H is to4· 52° and the O-H bond length 0.9572 A,
which are the values measured for a single water molecule in the gas phase
(Benedict et aI., 1956). Although the molecules can rotate freely in the MD cell,
subject to the constraints of the potential energy equation, the intramolecular
geometry is fixed and pressure-independent. In our simulations, a molecular
dynamics cell containing 256 H 2 0 molecules was used as a microcanonical
ensemble of fixed energy, volume, and number of particles (N, V, E). The cell
satisfied periodic boundary conditions so that molecules leaving the right-hand
side of the cell reentered at the left-hand side and so on in other dimensions.
Within the box all molecules were subjected to forces as a result of the potential
described above. The equations of motion were integrated using a modified
Beeman algorithm (Beeman, 1976, Refson and Pawley, 1987) with a time step of
0.2 femto-seconds (1 fs = to- 15 s). The size of time steps was chosen so as to
keep the total energy constant and give sufficiently accurate measurements of
the paths of the molecules as they move with respect to each other's potential
energy wells. The long-range Coulombic term was calculated using the Ewald
sum technique (Berthault, 1952). After initial trials it was found sufficient for
convergence to carry out the calculations to 9 Ain real space and 2 A-I in
reciprocal space, beyond which the calculations produced minimal further
changes in the potential energy of the system. The parameters used in conduc-
ting the calculation are summarized in Table 2. The positions, velocities, and
orientations of all 256 molecules were recorded every 50 time steps (10 fs) so that
the dynamics ofthe system could be observed for each temperature and pressure.
The initial configuration was a regular array of randomly orientated H 2 0
molecule positions, slanted obliquely across the MD cell.
Results
0
~
0
1000K rho=1.75 0
o
0 0 1000K rho=1.75 e;-
<'? 0 n
0
E..
>- I>'
e> >-
.....
Q)
0
0 e> 0
c 0 Q) 0 CI
LU 0 c co '<
co LU ::s
I>'
E 0 ~ 0 3
Q) 0 0
(5 .... I- 0 n·
<D oo
c..
... \FJ.
0
.... . a·
0
M.
E..
........
0 .. ..... .. -..-.- ... ..- ~~
C)'
.:. -............................... -. ........... ........... ...........- ...........
L,-
~.
o
1000 2000 4000 5000
::s
oo
0 1000 2000 3000 4000 5000 o 3000
Fig. 3. Evolution of calculated potential, rotational, kinetic, and total energies for TIP4P water at 1000 K and
0\
density = 1.75 over 5000 time steps. Velocity Scaling was carried out for the first 2500 time steps. -..J
68 D.G. Fraser and K. Refson
steps to give the desired value of temperature. Changes in the calculated values
were no longer significant after around 1500 steps and the scaling was turned otT
after 2500 steps, thus fixing the total energy as shown in Fig. 3. In carrying out
runs to determine thermodynamic quantities, measurements were made in this
way for 5000 time steps for each change in density, and final data were obtained
by accumulating and averaging the values over the last 1000 time steps. Longer
Estimating Thermodynamic Properties by Molecular Dynamics Simulations 69
2.25
0
0
0
0
C\I
0
0
0
10
~ 2.0
(5
~>- 0
~ 0
CIl
0 1.75
w ~
c:
til
0
I-
0
. . . .: : : : : : : : : g~~g 1.0
0
0
10
Fig. 4. Variations in energy ofTIP4P water with temperature for different densities.
runs of up to 20,000 time steps for sample runs at density 2.0 and T = 1500 K
produced identical results. The complete configuration was stored at the end of
each run and used as a starting configuration for subsequent simulations for the
same density at different temperatures. The order in which calculations at the
different temperatures were made was randomized. In addition, some points
were calculated both up- and down- temperature so as to simulate experimental
"reversals" as a check on equilibrium.
Calculations were carried out over a range of temperatures for fluid densities
from 0.25 to 2.25, and tl].e P- V- T resu\ts are given in Table 3. A plot of the total
energy of the system for these densities as a function of temperature is shown in
Fig. 4. For a given temperature, the total energy varies with density and, with
increasing fluid density, first decreases before increasing rapidly beyond a den-
sity of around 1.0. These data reflect the shape of the potential energy well.
Discussion
In order to assess how these calculations using the TIP4P model compare with
experimental P- V- T measurements, data at 1000 K from Table 3 are compared
with available experimental measurements in Fig. 5. There are few experimental
measurements of the high-pressure and temperature properties of H 2 0. Hydro-
static measurements have been made by Burnham et al. (1969) up to 1173 K and
70 D.G. Fraser and K. Refson
ca
'0
::i:
a "!
0,
0
...J
~
~
<Xl
ci
Fig. 5. The P- V properties ofTIP4P water along the 1000 K isotherm. Experimental data
are from Rice and Walsh (1957), Mitchell and Nellis (1982), Lysenga et al. (1982), and
Burnham et al. (1969).
8.9 kbar and were extrapolated to 1273 K and 10.0 kbar. At higher pressures,
several shock-wave measurements of fluid densities were made without direct
measurements of temperature (Rice and Walsh, 1957, Mitchell and Nellis, 1982).
In addition, a very limited number of shock-wave data are also available for
which temperatures were simultaneously determined (Lysenga et aI., 1982).
These lie in the P- T range 48.9 GPa (489 kbar) to 80.0 GPa (800 kbar) and 3280
to 5270 K and are included in Fig. 5 for comparison.
The results of our MD simulations using TIP4P agree surprisingly well with
the available experimental data, despite the rigidity of the model molecule. At
low pressures the calculated P- V- T data lie close to the isotherms of Burnham
et ai. (1969). The Rice and Walsh (1957) data appear to agree with both our
calculations and the data of Mitchell and Nellis (1982) up to about 50 kbar.
Beyond 50 kbar, our calculated P- V data lie close to the data of Mitchell and
Nellis (1982), but the Rice and Walsh data indicate higher densities. The temper-
atures measured in the experiments of Lysenga et ai. (1982) are much higher, but
their observed densities in the range 500 to 800 kbar are similar to those calcu-
lated by us using the simple TIP4P potential and the temperature dependence
is quite small. Thus, it seems that this simple approach is capable of yielding
P- V- T data which agree with experiment over a wide range of conditions.
Estimating Thermodynamic Properties by Molecular Dynamics Simulations 71
0
N
~ III
.s
CI)
E
:l
~ C!
800 K + 800 K
0 1000 K
It)
0 /'::,. 1125 K
0 1250 K
0 X 1500 K
0
0
Fig. 6. The P- V- T surface of TIP4P water calculated using the TIP4P water model.
The entire P- V- T surface calculated for H 20 using the above model is shown
in Fig. 6. The data were fitted first to quadratics on each isotherm. Inspection of
the residuals indicated significant cubic terms and these were incorporated
before the full P- V- T fit was attempted. Fitting in this way to the simulated data
gives the following equation of state for water:
1
V= -0.39675 + 0.810g lO P - 0.013875(loglOP)2 + 0.023657(loglOP )3
679.202 217.76(loglOP)
+ T T (11)
Curves calculated using this equation are shown together with the data in Fig.
6. The specific volume of water at 1000 K, 10 kbar using this equation is 1.29
cm 3 . g-l, which can be compared with the value of 1.253 cm 3 • g-l tabulated by
Burnham et al. (1969), and we expect the agreement to be rather better at higher
pressures as deviations related to the flexibility of the H 20 molecule are domi-
nated by the short-range "hard-sphere" repulsions. Comparisons of these data
with the results of the simulations of Belonoshko and Saxena (1991) and with
the equation of Halbach and Chatterjee (1982) based on a modified Redlich-
Kwong model fitted to the data of Burnham et al. and Rice and Walsh (1957)
are shown in Fig. 7.
72 D.G. Fraser and K. Refson
~
~
()
()
Lyzenga et al. 1982
~
.S;
<ll Mitchell and Nellis 1982
E
::J
(5 Calculated· (Present Work)
> '<I:
iii ~
C>
.......
~
co
0
Fig. 7. Comparison of the equation of state [Eq. (12)] of H 2 0 calculated from the
present work with the equation of Belonoshko and Saxena (1991), based on a single-site
model and with the extended Redlich-Kwong equation of Halbach and Chatterjee
(1982).
Structural Information
0-0 for densities ranging from 0.25 to 2.5 g' cm -3 at 1000 K. The plot of gOH(r)
shows the sharp peak at 0.9572 Acorresponding to the fixed O-H distance of
the TIP4P H 2 0 molecule. It also shows a general increase in ordering and a shift
of peak positions with increasing density to closer distances of approach. The
inflection and decrease in the number of pairs at large distances is a computa-
tional artifact caused by the compression of the 256 water molecules into vol-
umes less than the cell volume at high pressure so that the RDF (radial distribu-
tion function) falls off. Similar features can be observed in gHH(r). However, it
should be noted that at the highest pressures, a significant number of H-H
approaches can be seen at distances smaller than the intramolecular fixed H - H
distance of TIP4P.
The 0-0 RDF, goo(r), measures the distributions of 0-0 distances be-
ween molecules. It shows greatly increased order for relative densities above
1.75. Reddy and Berkowitz (1987) found that for calculations using TIP4P at 298
K, increasing the pressure up to 6 kbar caused a decrease in intensity of the
0-0 peak at around 2.75 Aand a disappearance of the peak at 4.33 A, which
they interpreted in terms of a disappearance of the tetrahedral low-pressure
structure of water with increasing pressure. At the much higher pressures and
temperatures studied by us, we confirm the movement of the peak positions of
goo(r) to smaller values with increasing pressure. However, several marked
new effects are apparent. The small second peak of Reddy and Berkowitz (1987)
is not present at the high pressures and temperatures of these runs and indicates
that the low-temperature and -pressure tetrahedral framework of H 2 0 has
disappeared. The degree of order increases strongly however with growing pres-
sure. The small broad peak in goo(r) at around 6 Aat low pressure moves rapidly
to smaller distances and clearly splits into two at densities of 2.0 and above. In
addition, the first radial distribution peak moves to smaller distances, becomes
much sharper, and rapidly increases in intensity at density 2.0. These features
strongly suggest crystallization in the system. Although crystallization kinetics
may be slow, a number of "reversals" carried out by repeating calculations both
up- and down-temperature from configurations previously stored at the end of
runs made for other conditions give good agreement and indicate that equilib-
rium was attained in the runs. Furthermore, as shown in Fig. 3, the translational
and rotational energies had clearly converged so that at each point on the curve,
we believe equilibrium was achieved. To minimize systematic errors, the data
were collected for each density in a randomized order of temperatures.
In order to investigate the apparent phase transition in more detail, we
recorded the atomic coordinates in the MD simulation every 50 time steps and
followed the trajectories of the H 2 0 molecules as a film sequence. At low
pressures, the TIP4P water molecules are seen to move randomly in the disor-
dered fluid. At high pressures, however, while the H-atoms are disordered, the
O-atoms define a lattice structure. At low temperatures and high pressures
above about 20 kbar (2GPa), ice exists as the proton-ordered BCC (body-
centred cubic) ice VIII structure (e.g., Demontis et at, 1988). On heating, this
undergoes a phase transition to form the cubic orientation ally disordered phase
ice VII (Wong and Whalley, 1976). Comparison of gHH(r) and goo(r) in Fig. 8
O-H
1000 K
'"
.5
0.5
. 5
1.25
10
Dist in Angstroms) a
0-0
1000 K
'"
10
Dist in Angstroms) b
Fig. 8. Pairwise radial distribution functions gOH{r), gHH{r), and goo{r) at 1000 K. The
curves for different densities are each offset by 0.5 units on the y-axis. Note that peak
positions move to smaller distances with increasing pressure and the second nearest-
neighbor peak around 4.5 Ain goo{r) becomes resolved above a density of2.0. Also shown
(continued)
H-H
1000 K
0.25
0.5
0.75
1.25
10
Dist in Angstroms) c
0-0
0
oj
0
oj
'" .'.
C!
'"ci
o
ci
10
Dist in Angstroms) d
for comparison is the pairwise radial distribution function goo{r) at 1000 K for den-
sity = 2.25 and the sharp peaks of goo{r) calculated for a crystalline Bee structure like
that of ice VII. The fall-off in the rdf at about go increasing with density is an artifact of
the computational procedure and has no physical meaning.
76 D.G. Fraser and K. Refson
shows these effects clearly, with gHH(r) showing very little order indeed in con-
trast to the high degree of order exhibited by goo(r). We have calculated goo(r)
for a body-centred cubic lattice like ice VII and this is also shown for comparison
in Fig. 8. The agreement between the peak positions suggests that TIP4P water
at 1000 K, 500 kbar has an ice VII structure. Since no dissociation is possible in
this model, ionic contributions could not be investigated.
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Chapter 3
Equations of State of Fluids at High
Temperature and Pressure (Water,
Carbon Dioxide, Methane, Carbon
Monoxide, Oxygen, and Hydrogen)
A.B. Belonoshko and S.K. Saxena
Introduction
The problem of calculating properties of fluids at high temperature (T) and
pressure (P) remains one of the main problems of physical chemistry. More than
100 years have passed since the contemporary approach of studying a fluid state
was devised (van der Waals, 1881). The comprehensive state of understanding of
the fluid state was described in a review of Barker and Henderson (1976).
There are many reasons for the attention this problem has received. First, an
understanding of the nature of the fluid state is of basic scientific interest.
Second, the experimental (static) measurements of the P- V- T properties of fluids
are restricted to comparatively low values of T and P (about 1000 K and 10,000
bars). Investigators in many branches of science including physical geochemistry
need to know the properties of fluids at a much higher pressure and temperature
than mentioned above. The list of applications of the equation of state (EOS) at
extremal T and P is quite extensive.
In addition to the static measurements of P- V- T properties, the shock-wave
experiment is a very important source of data. The precision of the latter is much
lower as compared to the former. Moreover, temperature in a shock-wave
experiment is not actually measured. The two exceptions known to the authors
are the experiments of Lysenga et al. (1982) and Radousky et al. (1990). To
acquire simultaneously P, V, and T, one has to apply some form of EOS.
Therefore, it is understandable that the shock-wave P- V- T data (but only P- V)
are not purely experimental, but have a model-dependent character.
There is a very wide..group of EOS with polynomial formulation. The coeffi-
cients of these polynomials are usually calculated with the least-squares method
over a range of experimental (static) data that exist. The extrapolation with
these EOSs leads to large errors, because at high-temperature and -pressure
conditions, the polynomial term with the highest degree dominates and calcu-
lated values of pressure or volume depend practically only on the value of
80 A.B. Belonoshko and S.K. Saxena
the coefficient of this term. Occasionally, it may not even make any physical
sense. It is for this reason that the use of such EOSs is recommended mostly for
interpolation and some limited extrapolation (see Saxena and Fei, 1987a). Some
polynomial EOSs (Saul and Wagner, 1989; Hill, 1990) have been formulated on
the basis of shock-wave data (e.g., Walsh and Rice, 1957). However, as men-
tioned above, these data are model-dependent and therefore the EOSs are valid
only as models to describe initial P- V experimental shock-wave data (e.g., Rice
and Walsh, 1957). All other EOSs offluids, which claim to be valid for extrapola-
tion, could be divided into two large groups. Each of these two groups can be
further classified as has seen done by Belonoshko and Saxena (1991b), but here
we want to emphasize only one of their main distinctions.
The first group of EOSs consists of equations with parameters calculated
directly from experimental data. The ideas of molecular size and interaction
potentials are used to obtain a proper formulation of such EOSs. This approach
has had a long history starting with the paper of van der Waals (1881) (Hol-
loway, 1977; Bottinga and Richet, 1981; Delany and Helgeson, 1978; Ferry and
Baumgartner, 1987; Fuller, 1976; Grevel, ·1990; Halbach and Chatterjee, 1982;
Kerrick and Jacobs, 1981; Mills et aI., 1977; Mitchell and Nellis, 1982; Redlich
and Kwong, 1949; Rice and Walsh, 1957; Rimbach and Chatterjee, 1987; Saxena
and Fei, 1987a; Saxena and Fei, 1987b; Shmonov and Shmulovich, 1974; Tait,
1889; Tziklis et aI., 1975). The list of references is not exhaustive and several
important studies for low P- V- T data are not mentioned here.
To the second group belong the equations formulated to a considerable
degree on the basis of statistical physics using the methods of computer simula-
tion: molecular dynamics (MD) and Monte Carlo (MC). The main concept in
this approach is one of molecular interaction. There exist numerous techniques
to calculate the P- V- T properties if the interaction potential (IP) is known (Allen
and Tildesley, 1987; Barker and Henderson, 1976; Brown, 1987; Boublik, 1977;
Carnahan and Starling, 1969, 1972; Fiorese, 1980; Johnson and Shaw, 1985;
Kalinichev and Heinzinger, 1991; Kataoka, 1987; Luckas and Lucas, 1989;
Nellis et aI., 1983; Ree, 1982; Ross, 1987; Ross and Ree, 1980; Ross et aI., 1983;
Saager and Fischer, 1990;Saager et aI., 1990; Shmulovich et aI., 1982; van
Waveren et aI., 1986; Weeks et aI.,1971; Wentorf et al., 1950). The precision of
the calculated data with such techniques is about the same as the precision in
the estimation of the IP. Two of the most precise methods are MD and Me. The
very first attempt to calculate properties of liquid argon with the MD approach
resulted in a highly accurate reproduction of the existing experimental P- V- T
data (Verlet, 1967, 1968). Since that time, the methods of computer "experi-
ments" have been applied to solve a wide variety of problems in fields as different
as molecular biology and aerodynamics.
We consider that the EOS-based MD or MC methods of simulation are
better for extrapolation at high temperature and pressure than the EOSs of
the first group. The EOSs of the first group are based on experimental data in the
range of low temperature and pressure. These regressed coefficients could be
wrong at a higher temperature and pressure. Therefore, the authors of these
Equations of State of Fluids at High Temperature and Pressure 81
EOSs must restrict the range of applicability of their equations which in most
cases must be adhered to drastically. The EOSs of the second group are based
on the fundamental properties of the matter energy of intermolecular interac-
tion. If we can calculate IP from experimental data with a theoretically based
method with high precision, we can apply MD and calculate P- V- T within the
precision of the IP calculation. The nature of IP is the same under any tempera-
ture and pressure, as long as the species remain unchanged. It is evident for
monatomic (simple) fluids and should be discussed in more detail for molecular
fluids.
In this chapter, we review our recent calculations following the method
of MD simulation. We calculated IP in the range of the experimental static
P- V- T data. Using such IPs, we then simulated P- V- T data up to 4000 K and 1
Mbar with the MD technique for six species: water, carbon dioxide, methane,
carbon monoxide, oxygen, and hydrogen. The simulated data were then fitted
with a viriallike (Tait, 1889) equation for use in phase equilibrium computations
for chemical systems at high to ultra-high pressures and high temperatures.
The aim of this section is not to present a complete review of the method of MD.
It has been done many times before (e.g., Allen and Tidesley, 1987; Boublik, 1977;
Kubicki and Lasaga, 1990). We present only the basic idea of the approach. For
the sake of simplicity and brevity, many important details are omitted in the
following description. We can imagine fluid (as well as solid, glass, melt, plasma,
and so on) as a set of N particles (atoms, ions, molecules). If the initial coordi-
nates and velocities are known, we can solve Newtonian equations of motion
(2)
where uij is the potential energy of interaction between particles i and j, or IP.
Rene Descartes (1644) in "Principia Philosophiae" claimed that if he could
know the coordinates and velocities of all bodies in the universe, he could
calculate the future of the universe at any given time. This is exactly the idea of
MD. Unfortunately, Rene Descartes had no computers to realize his dream.
Furthemore, for some reasons it was a "little bit" difficult to know the coordi-
nates and velocities of all bodies. As soon as computers were available, the MD
approach became the method of choice. Although we still do not know the initial
82 A.B. Belonoshko and S.K. Saxena
P = NkT _ ~
V 3 V i=l
Nf f dUij
drij
j>i
r ..
'J'
(4)
where Vi is the velocity of the ith particle, m the mass of the particle, and k the
Boltzmann constant.
Strictly speaking for the exact prediction of the properties of matter, it is neces-
sary to solve the many-body Schrodinger equation describing the motion of all
nuclei and electrons. However, the following assumptions help us to simplify
the calculations:
1. The Born-Oppenheimer (1927) approximation that the potential energy of a
system of N atoms depends only on coordinates of nuclei.
2. The molecules are rigid.
3. The interactions are pair-additive.
These assumptions reduce the problem to that of the calculation of the
potential energy of N molecules as follows:
(5)
Hence, the calculation of configuration energy and the forces in Eq. (1)
requires the knowledge of only the pair interactions between molecules, which
depend on their positions and coordination rand n. However, it should be
pointed out that the assumptions 2) and 3) can (and sometimes do) lead to
remarkable errors in calculated values. Due to quantum effects, assumption 1)
becomes untenable for light atoms like H or He at low temperatures. The
Equations of State of Fluids at High Temperature and Pressure 83
necessity of accounting for the orientation of the molecules does not allow us to
use simple analytical techniques to calculate thermodynamic properties. How-
ever, it is evident that under high enough temperatures any fluid can be imagined
as an ideal gas. Of course, this temperature should be sufficiently high for a dense
fluid to behave as an ideal gas. In any case, there must exist a low temperature
at which we can describe the fluid in terms of a simple liquid; in other words,
interaction in such fluids need not depend on the orientation of molecules above
a certain "critical" temperature. This is true for monatomic and spherically
symmetric molecules. It is enough for a fluid to be orientationally disordered to
become a simple liquid. The question is, what is the value of such a critical
temperature? The answer to this question one can only obtain posteriori, that
is, by trying to calculate the fluid properties by disregarding the orientational
effect and comparing the calculated data with the experimental.
Water among the fluids of a C-O-H composition is the most structured
species due to hydrogen bonds. If we could show that under some not very
high temperature it is possible to disregard the orientational interaction, then
we could also ensure that such a temperature would be lower for other species.
Gorbaty and Demjanetz (1983) have demonstrated experimentally that the wa-
ter structure is very similar to that of a simple liquid under high temperature.
The temperature and pressure of their experiments were up to 1000 bar and 773
K. Stillinger and Rahman's (1974) MD simulation of water showed that as the
density of water increased from 1 to 1.346 g/cm 3 , the average coordination
number increased from 5.8 to 11.8. The latter value is typical for a simple fluid
as a condensed phase. Later it was confirmed by Kalinichev (1986,1991, and his
chapter in this volume) that the high-temperature structure of water is indeed
similar to the structure of a simple liquid. Undoubtedly, the contribution of
hydrogen bond energy to the energy of interaction must be taken into account
even for the simple liquid structure. However, as soon as high temperature leads
to orientational disorder, this contribution can be effectively taken into account
with the calculation of spherically symmetric IP (Belonoshko, 1989; Belonoshko
and Shmulovich, 1986, 1987; Belonoshko and Saxena, 1991a).
An interaction between molecules of simple liquids depends only on the
distance between them, which is the distance between their centers of mass.
There are two types of IP most widely used: the Lennard-Jones (LJ) potential
(6)
(7)
and Nellis et al. (1983) to model shock-wave data. On the other hand, LJ IP
allows us to reproduce experimental P- V- T data with good precision in the
range of moderate densities (VIVo about 0.9-1.1) (Verlet, 1967; Barker and
Henderson, 1976; Allen and Tildesley, 1987). It is natural to assume that IP such
as LJ IP with equivalent r about (J would allow us to reproduce the experimental
P- V- T in the range of moderate densities and the IP of the theoretically correct
form such as exp-6 IP would be useful over a wide range of P- V- T.
The parameters of r*, c in Eq. (7) and (J, c in (6) have identical sense. The
parameter c represents the depth of the energetic well, i.e., the minimum of the
potential, and parameters r* and (J are responsible for the location of this well.
Ifr* = 21/6 (J, Eqs. (6) and (7) will result in the potential minimum. It means that
if CLl = cexp -6' then LJ and exp-6 will be very similar, especially at (J < r < r*
exactly in the range of moderate densities. Hence, we can try to calculate param-
eters (J and c of LJ IP and afterwards from MD simulation to calculate parame-
ter r:t.. It would be ideal to calculate all three parameters of exp-6 IP from a MD
simulation. However, due to purely technical restrictions we are forced to adopt
this solution.
The most well developed analytical theory of liquid state is perturbation
theory (Weeks et aI., 1971). This theory allows one to connect the parameters
of the IP and properties of a fluid. Shmulovich et al. (1982) obtained a rather
simple analytical expression for the pressure of LJ fluid. This expression was
parametrized to coincide with MD simulation results for pressure under given
volume, temperature, and parameters (J and c. As one could expect, the pressure
at high density in this method became too high due to the extreme stiffness of
LJ IP. Nevertheless, the EOSs for CO 2 , CH 4 , and Ar up to 10 Kbar and under
supercritical temperatures reproduced experimental values of pressure rather
well. Their success prompted us to use this method of calculation of r* as 21/6(J
and c for each of the gases and to calculate afterwards with these parameters the
parameter r:t. from MD simulations. The calculation was done by minimizing
(8)
E/k(K) a(A) rx
Table 3. Potential minima location divided 21/6(a) and depth of well e/k calculated
earlier.
The P- V- T data for six species in the C-O-H system were simulated with
the MD approach using potential (7) with the parameters in Table 2. The
technical details of simulation were described previously (Belonoshko and
Saxena, 1991a, 1991b). The goal was to get the set of P-V-T points more or less
uniformly distributed in the temperature and pressure space. For that purpose,
pressure under specified T and V has been calculated. The range of pressures,
volumes, and temperatures is presented in Table 4.
The statistical errors of calculated T and P are of the same magnitude as
those calculated with the commonly adopted methods (Allen and Tildesley,
1987; Hill, 1962). It should be taken into account that the lower the pressure, the
larger the error of pressure calculation.
°
As discussed recently by Belonoshko and Saxena (1991), the SW data are
quite consistent with our simulated data on H 2 0, CO2 , CH 4 , 2 , and H 2 .
Note that this consistency exists only in a broad sense. In comparing the simu-
lated and the SW data, it is important to note that the estimated temperature on
the Hugoniots has a large uncertainty. There are several experimental (SW) data
for which the temperatures (as evaluated by Nellis et al., 1981; Ross and Ree,
1980; Ross, 1987) were above the upper limit of our calculations (approximately
4000 K). Our results on CO do not fit the SW data above about 230 Kbar. Nellis
et al. (1981) found that the products of the SW experiments above 230 Kbar are
solid carbon, O 2 CO, and CO 2 , Of course, even a small amount of a solid phase
precipitated from a fluid would decrease the total pressure drastically. The
comparison of our results with the results of other models and experimental
P- V- T data (Babb et al., 1968; Bottinga and Richet, 1981; Grevel, 1990; Halbach
and Chatterjee, 1982; Shmonov and Shmulovich, 1974; Shmulovich and
Shmonov, 1978; Tziklis and Koulikova, 1965; Tziklis et al.. 1975) is discussed
below.
The MD-simulated P- V- T data together with experimental P- V- T data in the
range above 5 Kbar as referred to before are fitted with the following polynomial
in Tand V:
abc
P = v+ V 2 + Vm' (9)
where
Equation (9) without the third term is very similar to the equation of Tait
(1889). Sysoyev (1980) showed with the theory of perturbation of liquid that the
Tait equation appears theoreticaly reliable. Vasserman and Rabinovitch (1968)
also applied an EOS of that kind to describe the properties of liquid air and its
components. Spiridonov and Kvasov (1986) reviewed an EOS for dense fluids
and showed that an EOS similar to the EOS (9) is the most reliable and
effective for a description of the properties of the dense fluid phase.
The coefficients and some other data on the fits are shown in Table 5.
The average error in pressure is between 3.27 to 5.53%, which corresponds to an
average error of 1 to 2% in volume. In the range of low temperature and high
pressure, i.e., in the range of high compressibility (Z around 100 and higher), the
maximum error is about 15 to 20%. Therefore, the error in volume in these
points is of the same magnitude, i.e., to 2%. The quality of the fit of the calculated
data to the experimental data and a comparison with the high P- T data from
other equations are shown in Fig. 1. There are two SW experiments with
simultaneously measured temperature. The comparison of calculated and mea-
sured volumes for H 2 0 and CH 4 under experimental T and P is shown in
Table 6. The molar volumes and fugacities are given in the appendix. The
fugacities of the fluids at T and P > 5 Kbar may be accomplished by using the
following equation for the experimentally determined fugacities, as reviewed by
·L-__~r-------------------------~8;:em~'
culated from the existing equa-
tions and the experimental data
as referred in the text and com- ------
------ -------------
pared to the data from Eq. (9)
together with MD-simulated
CII
points (stars): (a) H 2 0, (b) CO2 ,
°
(c) CH4 , (d) CO, (e) 2 , (0 H 2 •
'"'15 1ddddc -
::I
III
III
CII
MD points
- - - - Halbach and Chatterjee, 1882
- - - EOS (this work)
,",10
~
1\
II
c
c
c
c
10 c
'a:I"'
~ 111
CII
=''=.!"' II
CII
~'"'
****It - )0) points
00000 - ShDlODOY aDd ShmulOYlch, 11178
- - - lOS (thU work)
cccee - BotUqa aDd Blchet, 11181
,_ 1_
b, Temperature, K
0* 0 0 0 0 r:Jc 0 0 15 em'
0 0 20 em'
25 em'
30 em'
35 em'
40 em'
***** - MD points
Grevel, 1990
00000 - EOS (this work)
10DO 1200 "DO 1100
c Temperature, K
10600
. . . . . - 573.15 K
••••• - 473.15 K Babb et al., 1988
HOO
- - - EOS (thil work)
'as"
.0
8500
Q)
'"
=:l 71100
III
III
Q)
~ '" 8500
5500
4600
28 30 32 38 38
12000
'" 10000
as
.0
--
......
.....
EOS (thia work)
373.15 K
- 473.15 K Tziklia and
- 573.15 K Koulikova. 1965
rJmaI- 673.15 K
Q)
; 8000
III
III
Q)
'"
~
1000
~OO+---~-.--~--.---r--,--~--,---~-'
U U U ~ H "
e Mole volume, em3
8000
-- - EOS (thil work)
••••• - 298.15 K
7500 ..... - 323.15 K Tziklil et aI., 1975
..... - 373.15 K
!I!I!I!I!I - 423.15 K
~7000 !I
.0
ajlliOO
'=:l"
:8000
Q)
'"
~ IiIiOO
46OO+-----~----._----~----r_~--r---~
18 18 20
f Mole volume. ems
Fig. 1. (Continued.)
Equations of State of Fluids at High Temperature and Pressure 91
between shock densities of 1.6 and 1.9 g/cm 3 at a temperature of 1300 K or more.
However, direct structural measurements of shock-wave compressed water with
Raman spectrometry did not show the presence of such ions in water (Holmes
et aI., 1985). Radousky et al. (1990) also indicated the significant stability of the
shock-wave compressed methane up to very high pressures and temperatures.
Thus, the data on the dissociation and stability of high-TP fluids are controver-
sial. We must wait for experimental determinations.
The fluids in the temperature and pressure range investigated may undergo
solid-fluid transformation (Grace and Kennedy, 1967; Stishov, 1974). One can-
not apply the theory ofliquid state in this case, as was done for high compressed
fluid (see Brown, 1987 and Ree, 1982). The advantage of our approach is that the
MD technique allows us to calculate properties both in the liquid and solid state.
Conclusions
With the MD simulation, we have calculated the properties of six fluids of
C-O-H composition (water, carbon dioxide, methane, carbon monoxide, oxy-
gen, hydrogen), the most commonly occurring species in natural fluids. The
theory of perturbation of liquid and direct MD calculation has been used
for the calculation of the potential of intermolecular interaction. The IP adopted
is of the oc-exp-6 type. The data obtained with a MD simulation up to 4000 K
and 1 Mbar, together with the low-temperature and -pressure data, were fitted
to obtain an EOS for six species. The range of applicability of the EOS obtained
IS
5000 bar < P < 1 Mbar,
T.pec < T < 4000 K,
where T.pec is 400 K for all species except water. T.pec for water equals 700 K. A
comparison with the available experimental data shows that our EOSs repro-
duce these data with a maximum error of about 5 to 6% on volume and fugacity.
It makes these EOSs useful for geochemical calculations in the specified range
of T and P, i.e., under the Earth's mantle conditions.
Acknowledgments
This research was financially supported by the Swedish Natural Science Re-
search Council (NFR).
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Equations of State of Fluids at High Temperature and Pressure 97
Introduction
anharmonicity and the simple way it can be taken into account when high-
pressure and high-temperature spectroscopic measurements are available.
In fact, the property calculated from spectroscopic data is the isochoric heat
capacity (Cv ), which is related to Cp by
Cv = Cp - TVIX 2 K T , (1)
where IX is the thermal expansion coefficient and K T the isothermal bulk mod-
ulus. However, thermal expansion coefficients and bulk moduli are generally
poorly known at high temperatures. As pointed out by Richet et al. (1982), Fei
and Saxena (1987), and Saxena (1989), Cp measurements and Cv calculations
enable one to place constraints on these parameters through Eq. (1). We con-
clude this review with a check of the consistency between high-temperature
thermal and volume properties. The temperature dependence of the bulk mod-
ulus, the pressure dependence of the thermal expansion coefficient, and a few
relationships that have been proposed recently to correlate IX, K T, and Cp will be
included in the discussion.
Calorimetry
Measurements
Heat capacities are either measured directly or determined from relative en-
thalpy measurements through
(2)
In any case, one actually measures the change in enthalpy resulting from a given
temperature increase. Basically, the difference between relative enthalpy mea-
surements by drop calorimetry and direct Cp measurements by adiabatic or
differential scanning calorimetry is that in the latter case, only the change in
temperature is small enough that the differential of Eq. (2) can be approximated
by a finite difference. With most techniques, it is necessary to enclose the sample
in a container whose heat capacity or enthalpy must be measured first in order
to separate its contribution to the measured property from that of the sample
itself. With adiabatic and drop calorimetry, the tight control of the thermal
conditions enables one to perform these measurements only once, whereas they
have to be made repeatedly with DSC.
Another important point is that heat capacity measurements are needed from
the (generally high) temperature of geochemical interest down to very low tem-
peratures. Of course, the reason is that third-law entropies are determined with
S= fT CPdT. (3)
o T '
100 P. Riehet, P. Gillet, and G. Fiquet
Adiabatic Calorimetry
The most accurate method for Cp determinations is adiabatic calorimetry (e.g.,
Robie and Hemingway, 1972). With this technique, the sample is enclosed in a
small calorimeter that is electrically heated from the inside and enclosed in a
shield separating the calorimeter itself from an outer zone that is constantly
maintained at the same temperature as the calorimeter. No heat is exchanged
between the calorimeter and the exterior in this way; hence, the term adiabatic
is applied to the shield and the technique. Typically, the electrical energy sup-
plied is that needed to increase the temperature of the calorimeter by a few
degrees. This has long been done manually, but measurements are no longer
tedious, with automated equipment by which a product is run smoothly from
liquid helium to room temperature within a week.
The major difficulty stems from the decrease of Cp at lower temperature,
which necessitates samples of 10 to 20 g or more in order to measure significant
amounts of heat at the lowest temperatures. This need of big samples is the main
constraint on adiabatic calorimetry. As for the temperature limitation of the
technique, it results from the increasing difficulty of maintaining adiabatic con-
ditions when radiative heat transfer becomes important at high temperatures.
The upper limit of adiabatic calorimetry is thus about 1000 K, even though only
a few laboratories succeed in making measurements well above room tempera-
ture (i.e., Gmnvold, 1967).
The accuracy of the technique depends on the temperature range of the
measurements. The relative contribution of the calorimeter itself to the mea-
sured heat increases with decreasing temperatures and it can represent 90% of
the total measured heat at 10 K. The uncertainties can be greater than 10%
below 10 K, but this is not much of a problem because Cp data in this tempera-
ture range usually make a negligible contribution to the third-law entropy. In
contrast, errors can be as small as 0.2% from about 50 K to room temperature
(e.g., Krupka et al., 1985). Standard entropies can thus be known with the same
uncertainty, as listed in the thermodynamic tables of Robie et al. (1979).
2000r-----------------------------~
• Gronvold et al.
Il Hemingway
1600 ··
-~
o
1200 .·
--..,
E
c.
800
o
400 ·...
jl/\ ....
I l - . · - O --c.. -re,cooO • • ~~..-[]_. ----c-.--.
o
830 840 850 860
T (K)
Fig. 1. Heat capacity of quartz at the r:t.-P transition. Adiabatic data of Grenvold et al.
(1989) and DSC measurements of Hemingway (1987). The difference in the transition
temperature is likely due to the fact that the r:t.-P transition temperature varies slightly
from one sample to another.
102 P. Richet, P. Gillet, and G. Fiquet
Drop Calorimetry
Above 1000 K, it becomes more advantageous to make one measurement only at
high temperatures, namely, that of the temperature of the sample, and to per-
form the calorimetric measurement near room temperature where heat exchange
with the exterior can be controlled tightly. With this technique, which has long
been used for geochemical purposed (e.g., White, 1919; Kelley et aI., 1953; Kelley,
1960), a sample is heated to a given temperature T and then rapidly dropped into
a calorimeter at a low temperature To, where the relative enthalpy HT - HTo
is measured. Of course, some heat is lost during the drop by conduction and
radiation. Experience shows that the heat loss can be accounted for by the
measurements on the empty crucible because this loss is the same for the empty
and the loaded crucible.
Various kinds of calorimeters can be used to measure relative enthalpies.
With copper-block or water calorimeters, the enthalpy of the sample is deter-
mined from the temperature rise of the calorimeter once thermal equilibrium is
reached with the initially hot sample. The changing temperature difference
between the calorimeter and the exterior makes it difficult to control heat leaks
to the exterior. These problems are avoided with isothermal calorimeters in
which a univariant phase change is monitored, such as the melting of ice at 273
K, or of diphenyl-ether at 300 K. Drop calorimetry with an ice calorimeter
constitutes, in fact, the primary high-temperature calorimeteric apparatus since
the NBS Cp data for sapphire, which are generally used for thermal calibrations,
have been obtained in this way up to 2200 K (Ditmars and Douglas, 1971;
Ditmars et aI., 1982).
In Fig. 2 we represent schematically the equipment we have recently set up to
Fig.2. (a) and (b). Schematic representation of drop-calorimetry set-up for measurements
between 400 and 1100 K. The ice calorimeter consists essentially of a vessel maintained
at 273.15 K and filled with water and mercury. The water is partly crystallized prior to
the measurements. Some ice melts when heat is supplied. Since the volume of the calorim-
eter is constant, mercury then enters the vessel to compensate for the negative volume of
melting of ice. This quantity of mercury is proportional to the amount of heat, and it is
simply given by the weight change of a beaker from which mercury flows into the
calorimeter. Key to letters shown: A, alumina-sheathed Pt-PtRh 10% thermocouples,
with the hot junctions shown by the two dots within the crucible; B, to potentiometric
equipment for temperature measurements; C, water cooler; D, alumina tubes; E, alumina
shield; F, silver heat pipe; G, Kanthal-wound furnace; H, alumina rod holder for the
crucible; J, Pt-Rh 15% crucible; K, hook for suspending the crucible; L, hydrolic jack; M,
gates; N, to vacuum pump and argon gas cylinder; 0, calibrated glass capillary for
measuring the continuous slight heat supply from the exterior; P, beaker of mercury; Q,
calorimeter well; R, mercury-tempering glass coil; S, evacuated external vessel; T, internal
vessel; V, water; V, ice mantle; W, copper block fitting the shape of the crucible for
speeding up heat exchange with the calorimeter; X, mercury; Y, ice bath.
Macroscopic and Microscopic Approaches 103
complement between 400 and 1100 K the measurements previously made in our
laboratory between 800 and 1800 K as described by Richet et al. (1982). The ice
calorimeter is similar to that made at NBS (see Ditmars and Douglas, 1971) and
has already been described in detail (DenieJou et aI., 1971; Richet et aI., 1982).
With an ice calorimeter, the errors on enthalpies are small enough that the
uncertainties of the reported results originate mostly from the measurement of
the temperature itself or from temperature gradients within the crucible. Tem-
k.------~
I~----C
1__- - - -- 0
G A
,
0 H
C K
M F
N
0
0
J
Q
R
5
...
I
U
V
W 0
~
IS
X
Y
~ 2 em
10cm
a b
104 P. Riehet, P. Gillet, and G. Fiquet
perature gradients are reduced to a fraction of a degree over a few cm with the
simple silver heat pipe shown in Fig. 2. In addition, the measurement of tempera-
ture with two termocouples within the crucible itself, with the same technique as
described in Richet et al. (1982), allows not only precise observations, but also
rapid detection of thermal stability. Runs can thus be kept as short as possible,
which makes measurements possible on metastable phases that do not trans-
form within less than 1 h.
The example of sapphire will be used to illustrate the capabilities of drop
calorimetry at high and intermediate temperatures. Results for oc-A1 2 0 3 ob-
tained with the apparatus of Fig. 2 are listed in Table 1 and plotted in Fig. 3(a)
along with those of Ditmars et al. (1982) and Richet et al. (1982). If we fit the
results of Table 2 and Richet et aI. (1982) in the form
130r---------------------------~
120 c-Ol'"
-
c,...........
arrr'
~ .,-r:r
yrSf
--
110
'0
E
-:I /
. .... •
100 I~
E •
CJ /
•
90 :l c
•
Richet et al (1982)
This Work
NBS
80
a 400 800 1200 1600 2000
T (K)
125
115
-
~
0
105 c
.I'
/C
-
E r#
-.. c
-:I
95 c
c "
C.
0 9
g "
I
85 B c Gronvold
B • NBS
• This Work
75
b 300 500 700 900
T (K)
Fig. 3. Properties of ex-A1 2 0 3 (sapphire). (a) Comparison of mean heat capacities, em =
(HT - H273.1S)/(T - 273.15), obtained in this work and by Richet et al. (1982) with the
NBS-fitted values (Ditmars and Douglas, 1973; Ditmars et aI., 1982). (b) Heat capacity as
given by the adiabatic measurements of Grmmvold (1967) and the drop-calorimetry
results of NBS (Ditmars et aI., 1982) and our laboratory (Table 1 and Richet et aI., 1982).
the sample attains the same final state at the reference temperature To, regardless
of the initial temperature T from which a given phase is dropped. For kinetic
reasons, it may thus be needed to set To higher than T in instances like or-
der-disorder reactions. The problem with such transposed drop-calorimetry
experiments is that the heat exchanged cannot be measured as accurately at 1000
K or more as near room temperature. Errors are about 1% for the enthalpies,
which is generally acceptable for the enthalpy itself, but they preclude reliable Cp
106 P. Richet, P. Gillet, and G. Fiquet
Eq. (6) Eq. (8) Eq. (9) Eq. (10) Eq. (11)
Representation of measurements medio. excel. good excel. excel.
Low-temperature extrapolationa good medio. bad bad bad
High-temperature extrapolation of
DSC measurements bad bad medio. medio. medio.
Drop calorimetry data up to
1800 K medio. bad good good excel.
a Extrapolation to lower temperatures for phases that are stable at high temperatures only.
determinations since derivative properties are about one order of magnitude less
accurate than the original data.
Cp Equations
Theoretical treatments of the thermodynamics of anharmonic oscillators do not
lead to analytical expressions for the heat capacity. The Cp-temperature relation-
ships that are used to obtain the representations of the experimental data needed
for thermodynamic calculations are thus purely empirical. In addition, none of
the rather simple equations used as flexible enough to represent the O'-shape of
Cp VS. T curves below room temperature. The well-known consequence is that
standard entropies cannot be determined from low-temperature extrapolations
of data measured above room temperature. They must be obtained from adia-
batic measurements performed up to room temperature that are usually
smoothed numerically.
Above 298 K, in contrast, the temperature dependence of Cp is usually slight
enough that comparably simple equations allow one to account for the experi-
mental data. Can these equations be extrapolated safely to determine the en-
tropy and other thermodynamic functions of minerals at mantle temperatures?
This would be especially useful when only DSC data are available, and thus we
will discuss briefly some of the equations that are either used extensively or have
been proposed recently for high-temperature extrapolation purposes.
As long as computing facilities have been limited, the expression proposed by
Maier and Kelley (1932),
c
Cp =a+bT+ T2 , (6)
has remained the most popular. The drawbacks of this expression for accounting
for measurements from room to high temperature led Chipman and Fontana
(1935) to early recommend instead
d
Cp = a + bT + T O•s ' (7)
Macroscopic and Microscopic Approaches 107
The main advantage of this equation is its flexibility, which indeed enables one
to reproduce well the experimental data from room temperature to at least 2000
K. This equation has thus been used in the well-known thermodynamic tables
of Robie et al. (1979). Its main limitation is that it frequently gives clearly
incorrect heat capacities when extrapolated to high temperatures because the bT
and eT2 terms become predominant.
Guessing systematics in Cp variations, Holland (1981) suggested to uses dum-
my Cp data to constrain Eq. (8) at high temperatures, but the validity of the
assumptions made is not readily tested since they rest on data that are already
extrapolated. Looking for an equation that would provide instead a correct
built-in high-temperature behavior, Berman and Brown (1985) and Berman
(1988) proposed a development in reciprocal temperature
ko.s k2 k3
Cp = ko + TO.s + T2 + T3· (9)
Basically, the same form has been selected by Fei and Saxena (1987) who
switched from a ko.sITo. s to a kilT term and also included another term linear
in temperature, as justified by the presence of the TVrt. 2 KT product in Eq. (1)
kl k2 k3
Cp = ko + bT + T + T2 + T 3· (10)
kl k2 k3
Cp = ko + kinIn T + T + T2 + T3· (11)
The relative merits of these equations have been discussed by Richet and
Fiquet (1991) whose conclusions are summarized in Table 2. With Eqs. (9), (10),
and especially (11), one can achieve both accuracy of representation of the
experimental data and reliability of high-temperature Cp extrapolations. A
somewhat obvious result, however, is that a prerequisite for reliable high-tem-
perature extrapolations is the availability of data at temperatures as high as
possible. The properties of a number of important minerals, including oxides, are
thus known satisfactorily at 2000 K or more when the measurements extend
beyond 1500 K. However, a number of mantle minerals can be studied at best
by DSC only over narrow temperature intervals. In such a case, no empirical
equation guarantees correct Cp extrapolations up to 2000 K.
108 P. Richet, P. Gillet, and G. Fiquet
250,....---------------,
200
entropy for oxide and silicate minerals. Following Fyfe et al. (1958), Holland
(1989) then modified this procedure to also take into account the slight depen-
dence of the entropy on molar volume through a simple formula based on the
Einstein and Debye theories of solids
S = kV + Ln;S;. (12)
In this equation, the parameter k is 1 J/K cm 3 , n; is the number of moles of oxide
i, and S; are the empirically determined coefficients.
In contrast to simpler additive schemes, the models of Robinson and Haas
and Holland are very useful in estimating standard entropies. With Holland's
model, for instance, the standard entropy of jadeite is predicted to be 134.8 J/mol
K. This result is in good agreement with the experimental measurement of
133.5 ± 1.2 J/mol K, whereas the sum of the entropies of quartz and nepheline
is clearly discrepant at 165.8 J/mol K. Prediction of the entropy for polymorphic
modifications provides a more stringent test since the influence of chemical
effects are separated from those of chemical factors. Silicon is four-fold coordi-
nated in both coesite and quartz. The entropy difference between these forms
cannot be predicted with the model of Robinson and Haas (1983), whereas the
model of Holland (1989) gives 38.1 and 40.1 J/mol K for the standard entropies
of coesite and quartz, respectively. This method yields entropies in the right
order since the experimental values are 39.4 ± 0.4 and 40.4 ± 0.2 J/mol K, respec-
tively (Holm et al., 1967; Richet et al., 1982). On the other hand, the polymorphs
of NaAISi04 , nepheline and carnegieite, do not seem to show major coordina-
tion differences. According to Holland's model, the higher molar volume of
carnegieite should translate into an entropy 2 J/mol K higher than for nepheline.
This is at variance with the calorimetric standard entropies 124.3 ± 1.3 and
118.7 ± 0.2 J/mol K for nepheline and carnegieite, respectively (Kelley et al.,
1953; Richet et al., 1990).
Additional calorimetric and structural data will probably allow further refine-
ment of these models. The main limitation of such empirical approaches for
mantle minerals is, in fact, the very lack of data that can be used to obtain and
check the validity of their empirical coefficients. Low-temperature calorimetric
data for six-fold coordinated Si are limited to the observations of Holm et al.
(1967) for stishovite. No mineral with twelve-fold coordinated Ca, Mg, or Fe has
been investigated and the trend shown by the coefficients of Holland (1989) for
the various coordination states of these oxides is erratic. This shortcoming thus
prevents prediction of the entropy of a number of high-pressure minerals, such
as those with perovskite structures.
Cristob.
gonal GeO z, cristobalite and
quartz (SiOz), carnegieite and ne-
pheline (NaAlSi04 ), and pseudo-
Q' 75
wollastonite and wollastonite
o (CaSi0 3 ). Solid and open sym-
---
E bols for the high- and low-
.- - .-..: .- - - .
"'") Carneg. ~ _.
• -_. PsWoli. temperature forms, respectively .
Co
()
65 Data from Richet (1990) and
.-~.=--=-.-----. Woli.
t=="~ Richet et al. (1982, 1990, 1991).
60 c __ c __ c--c--c--c--c
Nephel.
55~~~~~~~~~~~~~~~
T (K)
additive models increase with temperature for spinel, diopside, and forsterite.
For forsterite, an excess Cp of about 6% is observed at 2000 K with respect to
the constituting oxides. These deviations are all positive and they likely result
from the nonadditivity of the anharmonic contribution to Cp that, as described
in the next section, is smaller for the individual oxides than for the other
minerals. The puzzling aspect of these deviations, however, is that they are
apparently unrelated to the kind of coordination polyhedra present in the
minerals.
To look in another way at the relationship between the structure and the
high-temperature heat capacity, we plotted recent calorimetric data for pairs of
polymorphs in Fig. 5. The main feature is the existence of a general trend,
whereby the high-temperature form has higher Cp above 1000 K. The Cp con-
trast is small between wollastonite and pseudo wollastonite, which is consistent
with the structural similarities between these CaSi03 polymorphs. The minor Cp
difference between cristobalite and quartz is also expected in view of the tetra-
hedral coordination of Si in both Si02 forms. Along the same lines, the greater
difference between the Ge0 2 modifications could be related to a major coordina-
tion difference for Ge between the four- and six-fold coordination states of the
hexagonal and tetragonal forms, respectively. The problem, however, is that the
Cp difference does not conform to the expected trend according to which denser
phases have a higher entropy at high temperatures (Navrotsky, 1980). On the
other hand, the Cp difference reaches more than 10% between nepheline and
carnegieite, which are structurally analogous to tridymite and cristobalite, re-
spectively. In summary, Si02 and CaSi03 polymorphs conform to the trends
assumed from structural arguments, whereas Ge0 2 and NaAISi04 show un-
expected differences. Hence, it does not seem possible to predict currently the
magnitude of Cp differences at high temperatures from simple crystallochemieal
arguments.
Macroscopic and Microscopic Approaches 111
General Remarks
The specific variations of the heat capacity and entropy of minerals discussed in
the previous sections show that the empirical estimation of these properties can
be fraught with difficulties. Calculation of the heat capacity from statistical
methods thus appears to provide an alternative for mantle minerals or other
phases for which calorimetric data are lacking. In addition to their practical
interest, these calculations also constitute the best way for understanding the
specificity of the variations of these properties. Starting with the well-known
harmonic modeling, we will then focus on anharmonic factors that are likely at
the root of the complex high-temperature variations reviewed in the previous
section.
The property evaluated most readily from spectroscopic data is the isochoric
heat capacity Cv, from which Cp is obtained through Eq. (I). Strictly speaking,
defects and impurities also contribute to the heat capacity, but their contribution
is small enough that they can be safely neglected for oxides and silicates. In
addition, important crystal-field and magnetic contributions can exist for com-
pounds like iron-bearing minerals, which must be evaluated specifically (see
Wood, 1981). Leaving aside these factors, one assumes in the harmonic approxi-
mation that the lattice energy of the crystal Vo is constant at constant volume.
Then, Cv is determined only from the derivative of the vibrational internal energy
(Vvib )
(13)
a
1100 1100
1000 1000
-
900 900
-...
E
0
(1)
.c
E
:J
c:
(1)
>
m
~
--
;>
C)
-15
-0.5
B
-1.4
5
-1
tion of each mode to the heat capacity is that of an Einstein oscillator. For the
ith mode, one has
(hvdkTf exp(hvdkT)
(14)
Cvi = [exp(hvdkT) - lY ,
where hand k are the Planck and Boltzmann constants, respectively. Heat
Macroscopic and Microscopic Approaches 113
The high-temperature limit of Eq. (15) is k for one oscillator. For one mole of a
crystal with N atoms in the formula unit, the high-temperature limit of Cv is thus
3N R, which is the so-called limit of Dulong and Petit already mentioned in the
previous section.
Densities of states are most rigorously determined from lattice dynamical
modeling of inelastic neutron scattering measurements (e.g., Ghose, 1988). Un-
fortunately, this very time-consuming work with experiments requiring about 50
g of sample is restricted to a few minerals (i.e., Barron et aI., 1976; Rao et aI.,
1988). The density of states has also been calculated for minerals rather simple
structurally from interatomic potentials derived from crystallographic and elas-
tic data (e.g., Price et aI., 1987). The good results obtained from simpler modeling
suggests, however, that these detailed calculations of g(v) are not necessarily
needed in heat capacity calculations. As shown in Fig. 6(a), if dispersion is the
basic feature of acoustic modes, dispersion effects are generally slight for the
optic modes. We will thus restrict ourselves to these simpler models, whereby the
variations of the mode frequencies with the wave vector over the first Brillouin
zone are neglected for the optic modes.
Kieffer Model
In the widely used model of Kieffer (1979), for instance, the three acoustic modes
are treated in a Debye-like fashion with sinusoidal dependences of the frequen-
cies on the wave vectors. And the optic modes are counted either as discrete
modes or continua, delimited by low- and high-frequency cutoffs, over which a
uniform distribution of models is assumed (Fig. 6). The resulting simplification
to the actual density states is shown in Fig. 6(b) for quartz. With proper selection
of the number of modes and boundaries of the optic continua, one usually
reproduces the low-temperature calorimetric measurements when the relevant
acoustic and spectroscopic data are known. Extensive reviews and numerous
applications of these calculations have been published (Kieffer, 1979, 1980, 1985;
Ross et aI., 1986; Hofmeister, 1987; Hofmeister et aI., 1987; Chopelas, 1990;
Hofmeister and Chopelas, 1991) and thus we will just present a few new or recent
examples to illustrate some of the possibilities offered by this model.
Below 50 K, only the acoustic modes and the optic modes with wave numbers
lower than about 100 cm -1 contribute significantly to Cv ' At these temperatures,
Cv is thus very sensitive on the lower end of the density of states, i.e., on the
114 P. Richet, P. Gillet, and G. Fiquet
-- ;:-
Ol
86
2
3650
120
0
Wave number
1000
-
~
0
800
.,
-
.....E 600
a. 400
0 • Model values
-- Experimental
200
1400
1200
S2' 1000
'0
.,.E
-
800
en 600
• Model values
400
- Experimental·
200
O+-~---r--r-~--~~--~~--~~
--
;>
0)
206 modes 48 modes 4 modes
500
-~
"0
400
,.€
-C-
O 200
300
• Model values
Experimental
100
0
0 200 400 600 800
T (K)
800
Q' 600
0
,--
-
E
400
en • Model values
200 - Experimental
O~~~-.--~--.---~-'r-~--,
--
;>
0)
103 modes
500
~ -
-.-,
'0 450
E
C-
0 400 o Cp Clinozo AI
• Cp Clinozo Fe
350
250 350 450 550 650 750
T (K)
-
700
~
-.-,
'0 600
E
500
Ul
0 S Clinozo AI
400
• S Clinozo Fe
300
250 350 450 550 650 750
T (K)
Fig. 8. (Continued.)
118 P. Richet, P. Gillet, and G. Fiquet
for the elino-ortho transition, with the ortho form the high-pressure polymorph.
It has been suggested instead that elinozo"isite is the high-pressure and low-
temperature form (see Newton, 1987), but the difference of only 0.5 J/mol K
between the calorimetric (Perkins et aI., 1980) and the calculated standard
entropy of orthozo"isite lends independent support to the conelusion drawn from
vibrational modeling.
60
~ GeOz Tltra
'0
8 40
~ STiSHOVITE
•• - Co
-C ' p
~ 100 100
'0
8
~ SO 50
rn
calculated standard entropy ofstishovite is 30.5 J/mol K, which confirms the low
calorimetric result of 27.8 J/mol K of Holm et al. (1967).
Anharmonic Modeling
Vibrational Anharmonicity
A harmonic solid would not undergo thermal expansion. Hence, the TVa 2 Kr
term of Eg. (1) represents an anharmonic contribution that is usually found to
be significant above room temperature in Cp calculations. As illustrated by the
examples of the previous section, satisfactory results can be obtained in this way
up to about 1000 K when the harmonic approximation is used for Cv ' At higher
temperatures, however, one observes that the difference between Cv and the limit
of Dulong and Petit depends specifically on the mineral considered. For peri-
clase and lime, Cv actually tends toward this limit, whereas it exceeds 3N R by
5% at 2000 K for forsterite (Fig. 10). As will be described below, this excess Cv
is most simply ascribed to the intrinsic anharmonicity of the vibrational modes.
210
Forsterite
190
170
150
130
400 800 1200 1600 2000
60
MgO
-..
-5
~ 55
-
0
50
~
'-"
Q..
U 45
40
400 800 1200 1600 2000
55
45
T (K)
Fig. ZO. Heat capacities of forsterite, periclase, and lime: Experimental Cp and Cv as
calculated by Gillet et al. (1991a) and Fiquet (1980) with Eq. (1) from the experimental
Cp , K T , and rJ. data of Gillet et al. (1991a), Richet and Fiquet (1991), Isaak et al. (1989a,b),
Chang and Graham (1977), and Smith and Leider (1968). The horizontal lines are the
Dulong and Petit harmonic limits. Error bars for Cv included as resulting from those on
KT and rJ..
Macroscopic and Microscopic Approaches 121
(16)
Anharmonic Calculations
That the a i parameters are not zero shows that vibrational modes are not strictly
harmonic, but it is necessary to assess the influence of anharmonicity on thermo-
dynamic properties. This can be done simply in a quasiharmonic way, by taking
into account (oln vJoT)v when differentiating the partition function to derive the
internal energy of an anharmonic oscillator. With this approximation, one finds
that
(18a)
122 P. Richet, P. Gillet, and G. Fiquet
a -...
Q.
m
I
(!)
2,0
..,.
I
-
6
! i II F
T""
-
a..
l-
1,0
I i n internal modes
--c: I HI
f't)
;>
IH t I
-
lattice modes
f't) 0,0
0 200 400 600 800 1000
b
-...
I
10
f H f
~ 8
II)
ff
-
6
T""
- c..
6
4
I fn
f internal modes
t-
f't)
~
;>
2 lattice modes
fn Hf f i
-
c:
f't) 0
0 200 400 600 800 1000
Wave number (cm- 1j
c
0
! p.
-... -I
f I I! ft t
Ii jllil
I
-2
~
internal modes
II)
-3
-
6
T""
-4
lattice modes
m -5
-6
0 200 400 600 800 1000
Wave number (cm-1)
Fig. 11. Relative frequency shifts with pressure (a) and temperature (b) of the Raman
modes as a function of wave number and anharmonic parameters (c) of forsterite.
Macroscopic and Microscopic Approaches 123
where Uih is the harmonic energy. Summing over all the modes with respect to
temperature, then one obtains the anharmonic ev • The calculation is somewhat
tedious (see Gillet et aI., 1991a), but it gives eventually a very simple result,
namely,
(18b)
where e~i is the harmonic ev, as given by Eq. (13), of the ith continuum or
discrete mode.
Of course, all the ai parameters cannot be determined since only part of the
vibrational modes are optically active. When using a Kieffer model, the simplest
way to evaluate the anharmonic contribution consists of averaging the a/s over
the same continua as used for calculating the harmonic part of ev • The anhar-
monic parameters of the acoustic modes could be determined from acoustic
measurements as a function of pressure and temperature. For minerals with a
great number of optical modes, however, the contribution of the acoustic modes
to the heat capacity is small and their anharmonicity can be neglected.
The results plotted in Fig. 12 for forsterite summarize the main features of
these calculations. As already noted, the difference between ep and ev is negligi-
ble below room temperature, above which the different density of states that can
be set up from available spectroscopic measurements leads to the same heat
capacities. The TVIX 2 KT and intrinsic anharmonicity terms begin to be signifi-
cant at 300 and 1300 K, respectively. Without the latter contribution, the differ-
ence between the experimental and calculated ep data would reach 5% at 2000
K. If forsterite is representative of mantle minerals in this respect, then the
conclusion is that reliable predictions of heat capacities under mantle conditions
will require not only good thermal expansion data, but also comprehensive
spectroscopic information in order to evaluate the anharmonic contribution to
ev•
Griineisen Parameters
Spectroscopic measurements can also be used to calculate the Griineisen param-
eter Y that is needed for relating P, T, and V along adiabatic paths. Macro-
scopically, this parameter is
(19)
~( Vi) evi
Ym = KT L. oln oP TLeVi· (20)
A comparison of Griineisen parameters evaluated with Eqs. (19) and (20) shows
that the latter values are generally too small with respect to the former, and that
the discrepancy seems to increase with temperature (Hemley et aI., 1989; Gillet
et aI., 1991a). As an example, one obtains 1.28 at 300 K and 1.06 at 2000 K for
a
MODEL I
.....
-1.5 ~
~ -2.5 -1
b
(J/mol/K) ANHAAMON I C MODELS
o
WAVENUMBER (em-1) 175 ~
1
OGPJimH ~_
MODEL II
'"
-2 160
HARI-IONIC MODELS
~ -1
52
145
105 644 825 975
WAVENUMBER (em-1)
65 r--1~-] 115 I
300
I
500
I
700
I
900
I
1 iOO 1300
I I
1500
I
1700
I
1900
i
;ti
and anharmonic modeling, compared to the Dulong and Petit limit. The various densities of states shown in (a) yield almost ~
the same heat capacities, as shown by the slight differences between the curves drawn in (b).
Macroscopic and Microscopic Approaches 125
forsterite with Eq. (19), whereas Eq. (20) gives 1.22 and 0.83 at the same tempera-
tures (Gillet et at, 1991a).
To determine whether anharmonicity could be responsible for this discrep-
ancy, Gillet et at (1991a) carried out an anharmonic calculation of Ym and
obtained in the limit of high temperatures
For forsterite, one finds actually that anharmonicity contributes little to Ym and
another reason for the discripancy has thus to be found. In fact, the lack of
high-temperature data for (Oln v;/8Ph makes it necessary to use the room-
temperature slope of the frequency shifts with pressure for evaluating Eqs. (20)
and (21). Contrary to what is often assumed, these slopes could increase with
increasing temperature. This is suggested by the simple argument depicted in
Fig. 13, which rests on the assumption that frequencies are mainly determined
by the volume. If true, such increases with a temperature of (Oln v;/8Ph could
-
the 1-bar frequencies are also
850
measured values. At this tem-
0r;-
-
perature, the molar volume of
E 42
forsterite is 45.22 cm 3 and is () 840
back to 43.67 cm 3 , its 298-K, 10.
0 10 20 30 40 50
Q)
lobar value, at a pressure of .!l
42 kbar according to the elastic E
data of Isaak et al. (1989a) used :J
l:
with a Birch-Murnaghan equa-
Q) 240
tion of state. Hence, frequencies >
«S
(Ov) =0.l2 cm· 1/kbar
should be similar at 298 K and
3: oP 298
1 bar, and at 1300 K and 42
kbar if they depend primarily
on volume. With this assump- 230 (Ov) =0.26 cm· 1/kbar
tion and a linear pressure de-
pendence of the frequencies, one
oP 1300
finds that (evleP) generally in-
42
creases with temperature in a
specific way, as illustrated by 220
0 10 20 30 40 50
the two Raman-active modes
considered in this figure. P (kbar)
126 P. Richet, P. Gillet, and G. Fiquet
-,...
~
I
5
-
It)
I
,...
0
3
~
2
300 500 700 900 1100 1300 1500 1700 1900
T (K)
Fig. 14. Thermal expansion offorsterite. Curve 1 from Suzuki et al. (1984); curve 2 from
Kajiyoshi (1986); curve 3 from Fei and Saxena (1987); thick curve from Cp - Cv inversion
from Gillet et al. (1991a).
Macroscopic and Microscopic Approaches 127
inverted values of K ro and (dKr/dT) are also in excellent agreement with the
experimental results oflsaak et al. (1989a). Such results thus suggest that reliable
estimates of the high-temperature thermal expansion and bulk modulus at 1 bar
can be obtained through anharmonic modeling from the room-temperature
properties and the experimental Cp and calculated C. data.
Acknowledgments
We gratefully thank A. Le Cleac'h, F. Guyot and O. Vidal for their work related
to the topics discussed in this paper and Y. Bottinga, M. Madon, and B. Reynard
for helpful comments. This work was supported by the DBT program, "Fluides
et cinetiques," and grant CNRS-INSU-DBT 241.
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Cosmochim. Acta 44, 61-84.
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Mineral. 16,83-97.
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Chapter 5
Thermodynamics of Silicate Melts:
Configurational Properties
P. Richet and D.R. Neuville
Introduction
The ease with which a liquid adjusts to the shape of its container is a well-known
consequence of the hallmark of the molten state, atomic mobility. Atomic mobil-
ity is the very reason why liquids flow, even though another salient feature
evident through daily experience is that the viscosity increases when the temper-
ature decreases. In fact, if crystallization does not occur, the viscosity eventually
becomes so high that flow can no longer take place during the timescale of an
experiment. The resulting material is a glass, i.e., a solid with the frozen-in
disordered atomic arrangement of a liquid. Glasses have been produced for
millennia, but the kinetic nature of the liquid-glass transition and its influence
on the properties of glasses have long remained elusive. We will not specifically
address these aspects, however, because they have already been extensively
discussed in the geochemical literature from a relaxational (Dingwell and Webb,
1990) or thermochemical standpoint (Richet and Bottinga, 1983, 1986). In this
review, we will focus on features of liquids that are directly related to atomic
mobility, namely, the existence of those contributions to physical properties of
liquids that have been termed configurational (Simon, 1931; Bernal, 1936).
Configurational properties are, in effect, at the root of the main differences
between liquids and solids. To illustrate the importance of configurational as-
pects, consider the second-order thermodynamic properties of diopside and
CaMgSi 2 0 6 glass and liquid (Table 1), which are representative of silicates in
this respect. Above room temperature, the glass and crystalline phases have
similar heat capacities (Cp ), thermal expansion coefficients (IX), and compressi-
bilities (p = 1jKT' where KT is the isothermal bulk modulus). In contrast, the
heat capacity of the liquid is about 30% higher than those of the solid forms and
the IX and KT differences are greater still. These differences can have far-reaching
consequences when integrated as a function of temperature or pressure, as
illustrated by two simple examples. Moderate extrapolation of the Cp data for
Thermodynamics of Silicate Melts: Configurational Properties 133
·Cp from Richel and Fiquet (1991); IX from Finger and Ohashi (1976), to ± 10- 5 K -1 ; and KT from
Levien and Prewitt (1981).
bCp from Richet et al: (1986); IX and KT from Soga et al. (1979).
cCp from Richet and Bottinga (1984b); IX from Bottinga et al. (1982), and KT from Rivers and
Carmichael (1987).
700~~~~~~~~~~~~
Diop ide
600
Tg
~ 500 'f'--
;,,;' - Crystal
'0 400 -/;/
..§ 300 / /
""")
'-' Glass /"
"•
00 200 ,,/ T0
I
100 / scon".'_ __
O~
/ ww~~~~~~~~
o 500 1000 1500 2000
T (K)
Fig. 1. Entropy of diopside and CaMgSi 2 0 6 glass and liquid, and configurational en-
tropy of the amorphous phases. The extrapolated entropy of the supercooled liquid,
indicated by the dot-dashed curve, is equal to the entropy of the crystal at To . (Note that
the configurational entropy of the liquid, whose extrapolation is shown as a dotted line,
would vanish at a temperature higher than To because the vibrational entropy of the
amorphous phase is higher than that of the crystal.) Data from Krupka et al. (1985),
Richet and Bottinga (1984b), Richet and Fiquet (1991), and Richet et al. (1986).
-e
80 \ pressure at 1670 K as calculated
,.-..
\. from the Birch-Murnaghan equa-
0 75 \. Liquid
--e
\. tions of state reported by Bottinga
<')
70 (1986).
~
'-' 65
;;>
60
55
0 40 80 120 160 200
P (kbar)
tant the role they play in phase equilibria involving a liquid phase can be.
Thermodynamic modeling is a necessary step for interpolating or extrapolating
the limited amount of measurements that can be made in view of the consider-
able ranges of temperature, pressure, and chemical composition of natural mag-
mas (e.g., Bottinga and Weill, 1970, 1972; Carmichael et aI., 1977). For this
purpose, one thus needs to understand the complex way configurational proper-
ties depend on these factors. Even though direct structural information is be-
coming available through techniques like nuclear magnetic resonance (NMR)
spectroscopy (e.g., Stebbins and Farnan, 1989; Farnan and Stebbins, 1990),
thermodynamic data are still much more abundant than high-temperature
structural information for melts. An essentially phenomenological approach will
thus be followed in this review after an introductory, qualitative discussion of
configurational changes and their influence on thermodynamic properties.
The configurational heat capacity will be used first to illustrate the specific
dependences of configurational properties on temperature and composition.
This property is one of the authors' favorite because it has great importance in
thermodynamic calculations, and it will also exert strong constraints on the
quantitative modeling of the temperature-dependent changes in the relative
abundances of the various structural entities present in a melt. In addition, the
configurational heat capacity determines the temperature dependence of the
configurational entropy that has a major effect on the relaxational properties of
melts, of which viscosity is the most important. Quantitative modeling of vis cos-
ity will thus be also reviewed briefly within the framework of the configurational
entropy theory, which requires understanding the composition and temperature
dependencies of configurational heat capacity. In turn, this theory allows deter-
minations of configurational entropies from viscosity measurements, and these
data will be valuable for improving or setting up the thermodynamic models of
silicate melts that are needed to predict solid-liquid, gas-liquid, or liquid-liquid
phase equilibria. Finally, we will conclude this review, admittedly biased by the
authors' own interests, with a cursory examination of configurational effects on
volume, particularly with respect to high-pressure fusion.
Thermodynamics of Silicate Melts: Configurational Properties 135
Configurational Properties
Configurational Changes
aeon! .
r----------~
'- 'b
: a VI
I
i
I
I
:
I
DISTANCE
or the lack of this configurational contribution. Hence, the glass transition can
be viewed as the onset of the exploration by matter of positions characterized
by higher values of interatomic potentials (e.g., Goldstein, 1969). This spreading
of configurations over states of higher and higher potential energy is the main
feature of atomic mobility, no matter how complex this dynamical process may
be at a microscopic level.
Turning now to the volume changes of an amorphous substance, one notes
that a general feature of interatomic potentials is their anharmonic nature, i.e.,
the forces exerted on the vibrating atoms are not strictly proportional to the
displacements from their equilibrium positions. Increasing vibrational ampli-
tudes thus results in increases of interatomic distances (Fig. 3). As solids, liquids
have also such an anharmonic vibrational expansion, but the configurations of
higher energy that begin to be explored above the glass transition are generally
associated with increases in interatomic distances. This is the reason why the
thermal expansion coefficient usually increases markedly at the glass transition.
(Note, however, that a positive configurational thermal expansion coefficient is
not a thermodynamic requirement. In a few instances, a densification of the
structure is observed with increasing temperatures, as illustrated by the well-
known negative ex of water below 4°C.)
Consider finally the compressibility. In minerals, the existence of long-range
order restricts the compression mechanisms to changes in bond angles or dis-
tances if the crystalline structure is to be preserved. As a result, the compressi-
bility of a given phase is somewhat limited. Important compaction takes place
through phase transitions characterized generally by increases in coordination
numbers. Within a given structure, the effects of pressure can be viewed as
inducing only variations in the interatomic potentials characterized by shorter
equilibrium distances and steeper slopes around the minima. Because the shape
of these potentials determines the vibrational energy levels, this compression will
be termed vibrational as long as no significant changes in short-range order take
place in the crystal.
In a liquid, this vibrational compaction also exists, but it is considerably
enhanced by mechanisms like a progressive switch of cations toward higher
oxygen coordination polyhedra, which provides a continuous path for markedly
increasing the density (e.g., Waff, 1975; Stolper and Ahrens, 1987). The signature
of the availability of configurational states of higher density is the common
four- or five-fold increases in compressibility observed on melting or at the glass
transition (see Table 1). This configurational part of the compressibility can be
positive only (e.g., Davies and Jones, 1953). On the other hand, glass transition
temperatures vary with pressure (e.g., Rosenhauer et aI., 1979), which would
make it useful to describe the glass transition (for a given frequency, see below) as
a curve in the P- T plane. However, the effects of pressure and temperature on
the properties of glasses are actually of a different nature, for glasses quenched at
room temperature from pressures ofa few tens ofkbar undergo permanent com-
paction. A high pressure can thus induce irreversible configurational changes at
temperatures at which the substance is said to be a glass. Hence, for given fre-
Thermodynamics of Silicate Melts: Configurational Properties 137
The strong rate dependence of the glass transition shows simply that the kinetics
of configurational changes increases tremendously with temperature. In a quali-
tative way, the time for structural relaxation is a measure of the time required to
switch from an initial to a final configuration in order to regain internal equilib-
rium after a change in temperature or, more generally, in any other state vari-
able. Hence, the glass transition range is usually defined as the temperature
interval where time-dependent properties are observed because of the similitude
between relaxation times and measurement timescales. For usual calorimetry or
volume measurements, experimental timescales are about 1 to 15 min., and this
period is similar to the relaxation times of liquids with viscosities the order of
10 13 poises. The temperature at which this viscosity is attained is thus opera-
tionally used as the definition of the glass transition temperature, below which
materials cooled at rates the order of 10°/min show solidlike behavior.
Practically speaking, the distinction between vibrational and configurational
properties makes sense only if there is some way to determine their relative
importance. For this purpose, one must note that atomic vibrations have a
period the order of 10- 12 to 10- 14 s in solids and in liquids as well. As long as
relaxation times are long with respect to 10- 12 to 10- 14 s, vibrations in a liquid
can thus be considered as taking place instantaneously in a fixed structural
environment. Whenever this assumption is met, physical properties can be sepa-
rated into independent vibrational and configurational contributions. This is
actually the case when the viscosity is higher than about 1 poise (0.1 Pa s), as
generally found for silicate melts at temperatures of geochemical interest.
In a few instances, configurational and vibrational properties can be deter-
mined from measurements made on different timescales. Consider, for example,
the measurement of the adiabatic compressibility by an ultrasonic method,
whereby the speed of sound (c) in the material is most conveniently measured
at MHz frequencies (Fig. 4). The passage of the compression wave will result
in configurational changes only if the temperature is high enough that these
changes can take place faster than the 10- 6 s period of the perturbation. Experi-
138 P. Richet and D.R. Neuville
4000
'<ii'
E
'-'
3500
3000
1300
Fig. 4. Speed of sound at various frequencies for amorphous Na 2 Si 2 0 s . The Co and Coo
lines, referring to sound velocities at zero and infinite frequencies, represent the equilib-
rium and vibrational values, respectively. At these very short timescales, note that liquid-
like behavior begins to be observed when the viscosity is as low as 106 poises and that a
fully relaxed property is measured at viscosities lower than 10 3 poises. The adiabatic
compressibility is Ps = -1/V(iJV/iJP)s = l/pc 2 , where p is the density. Data from Nikonov
et al. (1982).
mentally, this happens only when the viscosity is lower than about 10 3 poises.
At higher viscosities, only vibrations relax completely and a rapidly decreasing
fraction at most of the configurational part of the compressibility can be mea-
sured. Hence, the equilibrium compressibility of the melt can be determined
solely at low viscosities and there exists a temperature interval, of the order of a
few hundred degrees, where the sound velocity depends on the frequency.
It is generally difficult to measure in such a way the frequency dependence of
the heat capacity or the thermal expansion coefficient, especially at the high
temperatures at which silicates are molten. The simplest assumption that can be
made is that there is no discontinuity in vibrational properties at the glass
transition. A priori, this assumption can be justified by data like those discussed
above for CaMgSi 2 0 6 (Table 1), which suggest that second-order thermody-
namic properties are mainly determined by the solid or liquid nature and not by
the crystalline or amorphous state of a phase. (We will emphasize that this is true
above room temperature only. As mentioned in the previous section, vibrational
properties depend sensitively on the structure below 200 K). If the vibrational
properties of a glass do not differ considerably from those of the isochemical
crystal stable at the same pressure, then glasses with different configurational
states should exhibit still smaller differences. In other words, the configurational
state of a liquid should exert little influence on its vibrational properties, which
can be obtained from high-temperature extrapolation of the glass properties.
The most extensive data for second-order properties deal with heat capacity.
Thermodynamics of Silicate Melts: Configurational Properties 139
Heat Capacity
For silicates, the glass transition takes place when the heat capacity of the glass
(Cpg ) is within a few percent of 3R/g atom, with R = gas constant [Fig. 5(a)].
This is the Dulong and Petit harmonic limit for the isochoric heat capacity from
which Cp differs little for silicate glasses (Haggerty et aI., 1968; Richet and
Bottinga, 1986). Under usual cooling conditions, glass transition temperatures
of silicates are generally lower than 1200 K. A significant anharmonic contribu-
tion to Cpg is not expected below 1200 to 1500 K, and the available data
discussed below do not show any progressive increase of the heat capacity of the
liquid (Cpl ) beginning above these temperatures. For sodium-silicate liquids, for
instance, the heat capacity is constant to within ± 1% from 800 to 1800 K.
Hence, anharmonicity is unlikely a major feature and the vibrational Cp of
silicate melts should not vary significantly with temperature above I'y. This leads
immediately to a simple equation
c; - Cpg(I'y). (1)
The validity of this equation, which is unlikely to be absolute (Goldstein,
1976), has been justified more rigorously elsewhere for silicates (Richet et aI.,
1986). This expression provides a simple means of determining calorimetrically
the configurational heat capacity, and it also shows that most of the temperature
dependence, if any, of Cpl can be ascribed to temperature-dependent configura-
tional changes in the liquid. As already noted, the vibrational Cp is a linear
function of composition. Deviations of Cpl from additive variations are thus
attributable to the composition dependence of the configurational heat capacity.
Practically speaking, the only apparent difficulty in using Eq. (1) is that the
glass transition of most silicates of geochemical interest is higher than the
temperature range at which direct Cp measurements can be made accurately.
140 P. Richet and D.R. Neuville
Mg72 I 13 C> 0 o· 0 ~
~ .• e
___ 30 ceo 0 ~ . •• .
~ e- ~ o-o-o-o-o-o -0-<>
-
aAISi04 0- T
•• • C.76/ 11
E 28 ••
o
eo:
Co ~-------~---------
J:l
U 24 fl'
fl'
{J"O'
e-e-
22~~~~~~~~~~-L~~~~
80
~
~ 75
e
--
~ 70
E
U
65
Fig. 5. Calorimetric effects of the glass transition for the calcium and magnesium alumi-
nosilicates of Table 2: (a) true heat capacity, obtained by differentiation of the fitted
enthalpy data of Table 2 shown in (b) in the form of mean heat capacities. The heat
capacities of NaAISi04 and NaAISi 3 0 s (Richet et aI., 1990; Richet and Bottinga, 1984a)
are included for comparison in (a), where the Dulong and Petit limit of 3R/g atom is
shown as the horizontal dashed line.
Ca76/11 b Mg72/13 c
BT.1 807.5 33.689 BU.15 881.0 39.189
BT.14 898.5 40.232 BU.17 929.8 42.854
BT.4 993.1 47.254 BU.19 976.1 46.441
BT.10 1060.8 52J89 BU13 992.7 47.748
BT.9 1129.2 57.542 BUI 1074.7 53.880
BTJ 1158J 59.755 BU18 1124.1 57.674
BT.8 1165.8 60.495 BU6 1173.7 61.837
BT.7 1260.8 68.730 BUll 1201.8 64.594
BT.5 1345.9 76.420 BU20 1351.6 78.853
BT.12 1414.2 82.594 BU9 1395.2 82.827
BT.13 1669.1 105.91 BUlO 1770.0 119.07
BT.6 1728.8 111.61 BU.21 1820.1 123.87
"Drop-calorimetry measurements made on synthetic samples prepared from oxide (or carbonate)
mixes with the ice calorimeter and high-temperature equipment as described by Richet and Bottinga
(1984a, b).
bCa76jll: 76.5,11.75, and 11.75 mol %Si02 , A1 2 0 3 , and CaO, respectively. For the liquid phase,
HT - H273 = -46,766 + 91.537 T (Jjmol). For the glass phase, HT - H273 = - 24,830 + 64.842 T
+ 5.943 10- 3 T2 + 18.233 10 5jT (Jjmol), with a glass transition temperature of 1199 K.
'Mg72jI3: 72.5, 13.75, and 13.75 mol %Si02 , A1 2 0 3 , and MgO, respectively. For the liquid phase,
HT - H273 = -45,932 + 89.232 T + 2.24 10- 3 T2 (Jjmol). For the glass phase, HT - H273 =
-25,532 + 65.890 T + 5.87910- 3 T2 + 19.381 10 5jT (Jjmol), with a glass transition temperature
of 1153 K.
--...e
~
'-'
15 (b) temperature-dependent values
for MO = K 20. For reasons of
c
o
consistency, all the vibrational
IJ Co 10 heat capacities have been assumed
U to be 3R/g atom because of the
lack of experimental Cp data for
5 ~~~I~~I~~~~~~~I~~
some of these glasses at the glass
40 50 60 70 80 90 100
transition. Data from Stebbins
a moI% Si0 2
et al. (1984) for BaSi 20 s and
Li 2Si0 3 , and Richet et al. (1984)
25 ~~~lnn~I~~~~~nn~~~ and Richet and Bottinga (1985) for
1800 K
the other compositions.
,.-..
-e
~ 20
~
o
15
1200 K
...c
'-'
o
IJ Co 10
U
5
40 50 60 70 80 90 100
b moI% Si02
40
o
\ NTS2
-e
\
\
o
-...--=
o 30
~
25
o 20
(J
Co
U 15
- --:=----.. ~-:...- - - - - - - --
NS2
10 L.....J....J....1-'--"--'-....I.....J-'--.L.....L....I.....JL.......I.....L.....L.....I-L.......I.....L.....L.....I-LJ
10
and Bottinga (1984a, b), and " =-
Table 2. U 5
o
40 50 60 70 80 90 100
mol % Si02
This increase of c~onf with decreasing Si02 content is preserved for alumino-
silicate melts when Al is exchanged for Si along MAI0 2-Si0 2 joins (Fig. 8), on
which lie a number of important mineral compositions. Again, however, the
temperature and composition variations depend specifically on the cation M.
The configurational heat capacities of Na-bearing melts are the lowest, but they
vary most with temperature. In addition, they do not follow a linear variation
with composition down to pure Si02. The composition of the two products
investigated in this study was thus chosen to complement previous observations
and determine whether calcium and magnesium aluminosilicates behave in the
same way. In both cases, the temperature dependence of c~onf is significant only
for AI-rich, Si-poor liquids. But c~onf varies linearly with composition from pure
Si02 to cordierite composition (Mg2AI4Sis018) at least, whereas calcium com-
positions show deviations from linearity that seem, however, less important than
for sodium aluminosilicates.
Even though the structural literature for silicate glasses is extensive, it has
limited predictive power to account for these c~onf data in view of the specificity
of interactions in melts that are clearly nonexistent in the glassy phases. Note in
this respect that the glass transition of silicates is likely triggered by the onset of
oxygen mobility, as suggested by Richet and Bottinga (1983) from an examina-
tion of oxygen diffusivity data on both sides of the glass transition (Yinnon and
Cooper, 1980) and observed in NMR experiments by Farnan and Stebbins
(1990). In addition, the temperature-independent, additive functions of composi-
tion found for the configurational heat capacity of the binary systems of Fig. 6(a)
suggest, as pointed out by Riehet and Bottinga (1985), that c~onf is mainly
determined by the basic structural units which exist from pure Si02 to metasili-
cate compositions at least. The temperature-induced changes would thus affect
primarily the short-range order, a conclusion consistent with the thermody-
namic interpretation of the viscosity data reviewed in the fifth section.
Specifically, changes in silicon coordination from four to five, or dispropor-
tionation reactions of the so-called Q species of the form 2Q3 = Q2 + Q4 are
associated with enthalpies of about 30 kJ Imol (Stebbins, 1988). Over the 1000 K
144 P. Richet and n.R. Neuville
interval that can be generally investigated above the glass transition, the c;onf
data indicate changes in configurational enthalpy of silicate melts ranging from
15 to 25 kJ/g atom, or from 45 to 75 kJ/mol on an oxide formula basis. A
comparison of such figures with the above enthalpies of reaction shows at once
the importance of practically unquenchable effects such as coordination or
Q-species distribution changes. Further work is thus badly needed to relate
quantitatively the structural information to the thermodynamic data.
Finally, we will also point out briefly correlations between the variations of
the heat capacity and of some other physical properties of aluminosilicate melts.
From the high-temperature viscosity data, for instance, Bottinga and Weill
(1972) assumed that tetrahedrally coordinated aluminum is more strongly asso-
ciated with alkali than with alkaline-earth cations, in the order K, Na, Ca, and
Mg. On the other hand, molecular orbital calculations have indicated a decrease
in the same order ofthe strength of AI-O bonds (Navrostsky et aI., 1985). Finally,
enthalpies of mixing also become progressively more negative in the same order
along the same joins (Navrotsky et aI., 1982; Roy and Navrotsky, 1984). This is
in agreement with increasing liquid immiscibility, which is usually viewed as
resulting from the competition of cations for oxygen bonding. Not surprinsingly,
these data thus also point to a central role for oxygen atoms in temperature-
induced configurational changes.
Entropy
Structural relaxation can take place in glasses well below the glass transition
range (e.g., Johari, 1976). As shown by available data, however, the calorimetric
consequences of these changes are not significant for silicates. Without serious
° °
errors, one can assume that the configurational entropy of silicate glasses re-
mains constant from the glass transition down to K. In other words, the
residual entropy of a glass at K is the configurational entropy of the liquid
frozen in at the glass transition
(2)
crystal from °
law is valid for a crystalline form of the material, Cp measurements for this
K to the melting point If yield the absolute entropy at If.
Determination of the entropy of fusion, usually from measurements of the
enthalpy of fusion, then gives the absolute entropy of the liquid at If. Finally,
°
measurements of the heat capacity from If back to K for the supercooled
liquid and the glass give SiO). In summary, one obtains
Thermodynamics of Silicate Melts: Configurational Properties 145
sconf(T) =
9
I°Tt C
~
T
+ dT + t1S'f + fT. ---.!!!:dT
C
~
T
+ fO
~
C
~dT
T'
(3)
where Cpc is the heat capacity of the crystal and t1Sf its entropy of melting. As
illustrated in Fig. 1, all the data needed to evaluate Eq. (3) play an important
role. This is the reason why low-temperature Cp data are badly needed for both
the crystal and glass. Below room temperature, the heat capacity contrast be-
tween the glass and crystal indeed results in important variations temperature
the entropy difference between the crystal and the glass for a number of composi-
tions. These trends are shown in Fig. 9 for a few materials, demonstrating the
predominantly low-temperature origin of the differences in vibrational entropy
between amorphous and crystalline phases.
In fact, glasses are nonequilibrium susbstances because of the kinetic nature
of the glass transition, and discussion of their thermodynamic properties is not
as straightforward as implicitly assumed in this paper. As discussed in previous
reviews, knowledge of the pressure and temperature is insufficient to specify the
state of a glass, and the fictive temperature introduced by Tool and Eichlin
(1931) represents the simplest one-parameter way to specify the thermal history
of a glass (e.g., Riehet and Bottinga, 1983). For a glass formed not too slowly by
continuous cooling of a liquid (i.e., at rates of 10 K/min or more), the fictive
temperature is simply the temperature at which the glass transition took place.
Then, one shows that glasses with the same composition, but different thermal
histories, have different entropies. As already discussed, available data do not
indicate a significant dependence of the heat capacity on the thermal history of
Table 3. Comparisons between residual entropies, Sg(O), obtained from viscosity and
calorimetry measurements, and between residual entropies and entropies of (Ca,Mg) or
(Si,Al) disordering (Sd)'·
silicate glasses. From Eq. (3), the entropy difference between glasses with differ-
ent fictive temperatures Tl and Tz is thus
The residual entropies have also been listed in Table 3 on a g atom basis to take
into account the widely different numbers of atoms in the formula units. At this
point, it is useful to separate somewhat arbitrarily the configurational entropy
into two parts. In an instance like NaAISi 3 0 s , there is an obvious contribution
that results from possible (Si,AI) disordering. We will call chemical that part of
the entropy which originates in the mixing of different elements on structurally
equivalent sites. For a glass like SiOz, the distributions ofSi-O-Si and O-Si-O
angles and of interatomic distances essentially represent configurational en-
tropy. More generally, these distributions for the various elements of an amor-
phous phase determine, in principle, the various coordination states, the way
bridging and nonbridging oxygens are bonded to Si (Q species distribution), etc.,
and intermediate-range order as well. We will term topological this contribution
to the entropy that does not result from element mixing.
In the glassy state, the relative importance of these contributions appears to
depend sensitively on composition, as suggested in Table 3 by comparisons of
residual entropies with entropies calculated for ideal Ca,Mg and Si,AI disor-
dering. On a g atom basis, pure SiOz probably gives a lower bound to the
topological entropy of a glass, namely, about 1.7 Jig atom K. The lower value
for NaAISi04 glass would thus suggest very limited Si,AI disordering, whereas
such an aluminum avoidance would not hold for NaAISi 3 0 s whose residual
entropy is high enough to suggest complete Si,AI disorder. Indeed, these calo-
rimetric interpretations are consistent with conclusions drawn independently
from NMR or Raman spectroscopic observations (Murdoch et al., 1985; Matson
et al., 1986).
As pointed out in the next section, however, residual entropies are important
only in the neighborhood of the glass transition. At high temperatures, they
represent only a small fraction of sconf, which as an integrated form of c;onf, must
be interpreted in the same way as the configurational heat capacity. Indeed, part
of the specificity of the c;onf data described in the third section for aluminosilicates
also likely results from disordering effects since the configurational heat capacity
Thermodynamics of Silicate Melts: Configurational Properties 147
also has chemical and topological parts. For instance, progressive (Si,AI) disor-
dering in the liquid state would be consistent with the higher c~onf of NaAISi04
with regard to that of NaAISi 3 Os [Fig. 5(a)]. In this respect, it is probably not
fortuitous that the data for aluminosilicates suggest a coupling between the
temperature dependence of c~onf and its nonlinear variations with composition.
On the other hand, the distributions of bridging and non bridging oxygens
among the Q species have been investigated extensively in the glassy state (e.g.,
Maekawa et aI., 1991). Unfortunately, their variations with temperature are not
known. More generally, attempts to translate quantitatively into thermody-
namic terms the available high-temperature structural information are still very
few (e.g., Stebbins, 1988). If this subject is well beyond the scope of our review,
we will nonetheless emphasize once more the considerable differences between
glasses and melts, and the resulting need for a thermodynamic evaluation of the
importance of structural factors through their contribution to configurational
properties.
T.
dT. (5)
Since c;onf is necessarily positive, sconf can only increase with temperature.
Less obvious features are the changes in the relative importance of the topologi-
cal and chemical contributions to the configurational entropy. Especially when
5
,-...
~
Fig. 9. Low-temperature entropy e 4
....Q
-
difference between the glass and
crystal forms of pyrope (Mg 3 -
AI 2 Si 3 0 12 ), albite (NaAISi 3 O s ),
=
ell 3
~
wollastonite (CaSi0 3 ), anorthite '-'
70~~~~~~~~~~~~~~~~~
--
60
-e
~ 50
40
5
... 30 tr-+-_ _--
=
8
00.
20 o 0 0 0 0
(Ca,Mg) smix
10
mixing can be assumed ideal, there is no evidence for significant variations of the
latter contribution, which becomes smaller and smaller with respect to the
former when the temperature increases. This is shown in Fig. 10 for configura-
tional entropies of liquids along the join CaSi03 -MgSi03 , and this fact has
important consequences regarding the relaxation properties of melts that will be
discussed in the next section.
However, these calorimetric determinations of configurational entropies have
two serious shortcomings. First, the use of Eq. (2) is restricted to liquids having
the same composition as a crystal whose entropy of melting can be measured,
i.e., in general, a congruently melting compound. This prevents systematic de-
terminations for intermediate compositions of a solution. Second, as evident
in Fig. 1, a calorimetrically determined residual entropy represents a small
difference between great numbers. Hence, such determinations can be beset by
important errors, especially when some of the data needed to evaluate Eq. (2) are
not so well known, as are the entropies of fusion of pyrope (Mg 3 AI 2 Si 3 0 12 )
and enstatite (MgSi0 3 ), for example. The possibility of determining more accu-
rately the configurational entropy from viscosity data is thus interesting, espe-
cially since any liquid can be investigated, regardless of the complexity of its
composition.
,I t' •
°
~","Py.t. .:;Di
AQ Ab~
13 ,I ..-!.. I : ~
I I
12
, r ;
,
~ ..
11 J.' ~ /
,~
,I Ab:'
i i /~d
• '
10 f Jd ,i l
I /
9
Q) Si0 2 /.'
I/)
'0 8
a. //
C'""7 I '
OJ
.2 @
i
e
6
5
II + NS z
o
3 4 5 6 7 8 9 10 11 12 13 14 15
10yr
Fig, 11. Viscosity-temperature relationships for a few silicate melts. Abbreviations: Ab
(NaAISi 3 0 s ); An (CaAlzSizOs); Di (CaMgSi z0 6 ); Jd (NaAISi z0 6 ); KS n (K 2 0.nSiO z);
NS n (NazO.nSiO z); Ne (NaAISi04 ); Or (KAlSi 3 0 s ); Py (Mg 3 Al zSi 3 0 12 ). Experimental
data as reviewed in Richet (1984) and Neuville and Richet (1991).
forming liquids is that it can span more than 13 orders of magnitude, as shown
in Fig. 11 for a few compositions. The viscosity of these melts may differ by many
orders of magnitude and the deviations of log 1'/ from linear variations of the
reciprocal temperature, i.e., from Arrhenius laws, also depend specifically on
composition. Regardless of these important features, however, a common trait
is that the slopes of the viscosity curves in Fig. 11 increase when the temperature
150 P. Richet and D.R. Neuville
decreases. In many cases, this increase is strong enough that the viscosity would
seem to become infinite at a given temperature.
Now, consider a hypothetical liquid with zero configurational entropy. There
would be no way to switch any structural entity from one place to another, and
the viscosity would thus be infinite. If only two configurations were available for
the whole liquid, then mass transfer would require a simultaneous displacement
of all entities. The probability of such a cooperative event would be extremely
small, but not zero, and the viscosity would be extremely high, but no longer
infinite. When the configurational entropy increases, the cooperative rearrange-
ments of the structure required for mass transfer can take place independently
in smaller and smaller regions of the liquid. Correlatively, the viscosity thus
decreases when the configurational entropy increases. These are the basic ideas
of the theory of relaxation processes of Adam and Gibbs (1965), according to
which the probability for a cooperative rearrangement is
I - A Be
og 1'/ - e + Tsconf' (7)
--....
~ 12
cQ.
l02SK
-ceJl9
' -'
bution of the mixing entropy to the total sconf in Eq. (7), which can be effective
at lower temperatures only, i.e., at high viscosities.
Entropy Modeling
In Fig. 13, the entropies obtained from the viscosity data of Fig. 12 for molten
pyroxenes clearly show the maximum due to mixing. As expected from (Si,AI)
mixing, analogous effects on the entropy are also observed along MAI02-Si02
joins, but not along other binary joins of the CaO-MgO-AI203-Si02 system
(Neuville, unpublished results). In fact, for both molten pyroxenes and garnets,
entropies of mixing are consistent with the ideal mixing of Ca and Mg (Neuville
and Richet, 1991). That (Ca,Mg) mixing does not seem to be affected by the
presence of aluminum has the noteworthy consequence that the problem of
determining entropies of mixing in the quaternary system CaO-MgO-Al2 0 3-
Si02 reduces to the same problem for the two limiting alumino silicate ternaries.
On the other hand, a preliminary analysis of the viscosity of mixed alkali silicate
liquids suggested that Na and K are mixing pairwise (Richet, 1984), and not
individually as alkaline-earth cations. This would be consistent with a clustering
of the two nonbridging oxygen entities Na-O formed when Na 20 is introduced
and reacts with a bridging oxygen. In alumino silicate liquids, alkali elements are
assumed to maintain electrical neutrality around tetrahedrally coordinated AP+
ions, and viscosity data could thus enable one to determine whether the alkali
ions mix individually or not.
These examples show not only how entropy models can be set up from
viscosity data, but also that models obtained for simple systems should be
applicable to systems of more immediate geochemical utility like the joins inves-
tigated by Hummel and Arndt (1985) or Tauber and Arndt (1987). The applica-
tion of these models to phase-equilibria calculations might appear doubtful,
however, since only configurational entropies would be taken into account. In
Volume
Thermal Expansion
Most of this review has been devoted to the thermal aspects of configurational
properties. The reason lies not only in their importance, but also in the wealth
of quantitative information that has been recently gathered and which contrasts
with the scarce data available for volume properties. However, the configura-
tional contribution to the thermal expansion coefficient of silicate liquids can
produce important effects, as shown in Fig. 15 for Na 2 0-Si02 liquids. In the
glassy state, the molar volume increases with the Si02 content, whereas the
converse holds true above 1200 K in the liquid state. Even though volume is a
property more prone to straightforward theoretical evaluation than entropy, the
way configurational properties result in such crossovers is still poorly known.
Part of our ignorance originates in the lack or in the uncertainties of thermal
expansion data. As noted by Bottinga et al. (1983), one finds discrepancies of
more than 40% in the reported thermal expansion coefficients for the most
extensively studied melts, namely, binary Si02 -Na 2 0 liquids. Indeed, there is
no direct way of directly measuring IX and this property is usually determined
from the differentiation of high-temperature volume measurements having un-
154 P. Richet and D.R. Neuville
28 ,-
Fig. IS. Volume of sodium silicate
NazO·SiOz Liquids . ~.-t:;'- glasses and liquids as a function of
,; ......
'-.~ .;::::- temperature at 1 bar. Numbers are
:::;- .27
-
,//
,.,. molar Na 2 0 contents. Experimen·
e
Q ~
r:~:::;:;':?~~:
24 I I
K cm 3 jmol 105 K- 1
Na 2 0
20.3 760 25.49 27.21 1.3 7.1 4.5 (0.8) 5.5
24.0 756 25.15 27.28 1.3 8.8 6.0 (0.8) 6.8
31.1 743 24.71 27.47 1.7 11.4 9.1 (0.9) 6.7
33.8 732 24.55 27.55 1.8 12.2 9.9 (0.9) 6.8
37.2 720 24.42 27.61 2.0 12.9 10.9 (1.4) 6.5
K 20
17.3 781 28.05 30.08 1.0 7.8 5.1 (Ll) 7.8
23.2 764 28.53 31.29 1.3 10.2 8.7 (0.7) 7.8
27.8 733 28.82 32.19 1.5 11.8 10.7 (0.5) 7.9
31.8 723 29.09 32.92 1.6 13.0 12.0 (0.5) 8.1
a VTgand VI673 are the molar volumes at the glass transition temperature T, and at 1673 K,
respectively. For liquids, IX'HT and lX 'm are the thermal expansion coefficients at 1673 K and mean
values between T, and 1673 K obtained from volume measurements, respectively (Bockris et aI.,
1956; Shermer, 1956).
2000
1800
-- - - ---- --.
" ,,
,-...
~ Spinel
,
\
'-" 1600 \
\
Eo-; \
\
1400
1200
0 50 100 150 200
P (kbar)
Fig. 16. Phase relationships between fayalite, Fe 2 Si04 spinel, and Fe 2 Si04 liquid (exper-
imental data from Akimoto et aI., 1967). The dashed line within the spinel (and post-
spinel) stability fields is the metastable extension of the fayalite melting curve as calcu-
lated by Richard and Richet (1990).
4000~------~------~------~------~
Liquid
3000
Stishovite
2000
1000
O~~--~~------~------~------J
o 100 200 300 400
P (Kbar)
Fig. 17. Schematic melting relationships of Si02 polymorphs. The dotted line represents
an assumed constant glass transition temperature, below which amorphization takes
place. (Adapted from Hemley et a!., 1988, with thermodynamic calculations reported by
Richet, 1988.)
Acknowledgments
We thank Y. Bottinga, F. Guyot, and A.M. Lejeune for helpful comments. This
research was supported by grant CNRS-INSU-DBT 293.
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Chapter 6
Crystal Chemical and Energetic
Characterization of Solid Solution
v.s. Urusov
Introduction
The golden age of classic crystal chemistry (1920s-30s) yielded many well-
known empirical rules and generalizations concerning the formation of solid
solutions or isomorphous mixtures (mixed crystals). Among them are Vegard's
rule of additive dependence oflattice spacings on composition, Goldschmidt and
Hume-Roseri's rules of maximal 15% difference of ionic or atomic radii for the
existence of wide miscibility, Goldschmidt and Fersman's rules of substitution
polarity (in relation to sizes and charges of the ions replacing each other), and
the criteria of proximity of polarizabilities or electronegativities of substituents,
etc. In the sections that follow we will return to an analysis of these rules from a
more sophisticated and modem point of view.
On the other hand, at the same time the development of the thermodynamics
of solutions made it clear that the configurational entropy of a random mixture
(gaseous, liquid, or solid) could be considered a common cause of miscibility at
elevated and high temperatures. If interatomic (intermolecular) forces are attrac-
tive or slightly repulsive (as in gases), there is no obstacle for a mixture to be
stable at all temperatures or even to form ordered (intermediate) compounds
during cooling. Only when repulsive interatomic forces dominate does a solu-
tion tend to decompose into phases that are close in their composition to pure
components. Such a decomposition is most typical for solid solutions. Therefore,
solid solution formation is very often accompanied by the occurrence of repul-
sive forces, as expressed in a qualitative manner by the classical crystal chemical
rules of isomorphism mentioned above.
The description of these forces in a quantitative way is believed to be the main
purpose of contemporary crystal chemical and energetic analysis of solid solu-
tions. This survey is devoted to the development and achievements of that
theory. The key ideas concern an approach with which one is able to present
the energetics of substitution solid solutions by simple and rather accurate
Crystal Chemical and Energetic Characterization of Solid Solution 163
analytical forms and with which one can explain many observations and some-
times predict the stability conditions of a solid solution.
Already in the early 1920s Grimm and Herzfeld (1923) had begun the analysis
of energetic effects of formation of binary ionic solid solutions and tried to solve
the following equation:
(1)
where 8.Hm is mixing enthalpy, V(x) is lattice energy of a solid solution, Vl and
V2 are lattice energies of pure components, Xl and X2 are their molar fractions.
They used the Born-Lande model to express V and to obtain the analytical
form of 8.Hm • They concluded that theoretical estimates of 8.Hm for some alkali
halide solid solutions had a positive sign and fitted a repulsion parameter in the
Born-Lande equation to obtain the correct 8.Hm values. This was a very useful
result because 8.Hm is usually a very small portion of the lattice energy V, not
more than 0.5%. This quantity is less than the errors in V values, estimated to
be about 1 to 2%. Therefore, there is a cancellation of the main errors in Eq. (1).
Later some investigators (Tobolsky, 1942; Wallace, 1949; Fineman, 1950; for
references, see Urusov, 1977) developed this approach and found that the mixing
enthalpy 8.Hm had to be expressed as a function of the square of the size
difference parameter (j = 8.R/R, where 8.R = R2 - R l , Rl and R2 are inter-
atomic distances in pure components, and R is the average interatomic distance
in the solid solution. However, 8.Hm , calculated by these authors for several
systems of alkali halide solid solutions, did not agree very well with calorimetric
measurements.
More successful attempts to solve the same problem were performed by
Wasastjerna (1949) and his Finnish successors (Hovi, 1950; Hietala, 1963a, b).
They used Vegard's rule R = xlR l + x 2R 2 in the form
(2)
and the expansion oflattice energy into power series with the small-size parame-
ter (j < 1. The restriction of this series up to quadratic terms again gives 8.Hm as
a function of (j2. Wasastjerna accounted for the local displacements of a common
ion C in the solid solution (AX1' Bx2 )C with an NaCl-type structure if this ion is
surrounded by ions of various kinds, A and B. Then he divided the total distance
2R between A and B in the following proportion:
(3)
164 v.s. Urusov
The Wasastjema-Hovi model was analyzed in detail by Urusov (1977). Here
we concentrate somewhat more on Hietala's (1963a) theory that is rather differ-
ent from the previous treatment. He used Taylor's series expansion for energy
U(R) - U(Ro) = U'(Ro)AR + (t)U"(R o)(AR)2 + ...
(4)
where AR = R - Ro; considering that the first derivative of the lattice energy
U'(Ro) = 0 at equilibrium state (R = R o, T= 0). Hietala also considered the
local atomic displacement when the surrounding of a common ion is asymmet-
ric. He presented the change of the interatomic distances (A - C) and (B - C)
in the form
RI = R - u, R2 = R + u,
where u is the displacement of ion C from the center of the linear bonds A-C-B
or B-C-A. Solving the system of the two equations
9
ARm = (;dX (VI) (AR)2
1 X2 PI R; . (7)
For alkali halides with an NaCI-type structure, VIP is nearly constant and
equal to 630 kJ/mol with error bars of a few percent. In other words, the
properties of both end-members are very close and VdPI ~ V2/P2. Thus, one
finally has
AR)2
ARm = 1420X 1 X2 ( R; = X I X 2 Q, (8)
9
ARm = 1.25(4")X (VI) (AR)2
1 X2 PI R; ~ (T)XIX2
11 (VI)
PI (AR)2
R; . (9)
Hietala's model takes into account the non parabolic character of the mixing
Crystal Chemical and Energetic Characterization of Solid Solution 165
enthalpy by adding the third-order term in the expansion (4). In this case, one
can rewrite Eq. (8) in the following form:
AHm = x l x2 Q[1 + B(XI - X2)]' (10)
where B = 0.6(A. -1 )(AR). The second term in square brackets in Eq. (to) makes
the AHm parabola skew. Due to the asymmetry of AHm (10), its values are
relatively higher at Xl > X 2 , i.e., when molar fractions of the component 1 with
a shorter interatomic distance Rl ( < R 2 ) are more than 0.5. This conclusion is at
least in qualitative agreement with available experimental data.
There exist many extensive calculations of the properties of ionic solid solu-
tions in a nonanalytical way. Durham and Hawkins (1951) and Lister and
Meyers (1958) performed detailed analysis of the local atomic displacements and
mixing enthalpies for those alkali halide systems that were investigated calo-
rimetrically (Lister and Meyers, 1958).
Many other authors followed Mott-Littleton's theory (Mott and Littleton,
1938) to calculate the local distortions and energetic effects of solution for
isolated impurity ion in alkali halides (Brauer, 1953; Fumi and Tosi, 1966;
Hardy, 1962; Fukai, 1963; Dick and Das, 1962; Tosi and Doyama, 1966;
McDonald, 1966; Douglas, 1966; Kristofel, 1968; see references in Urusov, 1977).
The theory was expanded to the full composition range of alkali halide solid
solutions by Fancher and Barsch (1969). Their calculations as well as the calcula-
tions using Wasastjerna-Hovi's and Hietala's theories are in excellent agree-
ment with the experimental measurements of mixing enthalpies (Lister and
Meyers, 1958) for continuous solid solutions in systems KBr-KI, NaI-KI,
NaBr-NaI, NaBr-KBr, NaCI-KCI, NaCI-NaBr.
In conclusion, we note that Mott-Littleton's theory was used repeatedly for
the prediction of local distortions around isolated bivalent impurity and corre-
sponding solution energies in alkali halides (Bowman, 1973; Stoneham, 1975;
Urusov et ai., 1980; Bandyopadhyay and Deb, 1988). There are few experimental
data to be compared with these theoretical estimates. They come mainly from
studies of transport properties, in particular, electrical-conductivity measure-
ments. One can find at least semiquantative confirmation of the theoretical
predictions.
where E(x) is the atomization energy of a solid solution, E1 and E2 are the
atomization energies of pure components. It was shown by Urusov (1969a) that
when both components are similarly ionic, flHm preserves the general form of
the ionic model [Eqs. (7) and (9)] and the ionicity degree h is not involved at the
first approximation in the final expression. Moreover, in order to include an
explicit dependence of interaction parameter Q on the composition it was
proposed that the size parameter be used in the form of c5 = flR/R = flR/
(x1R1 + x 2R2) instead of c5 1 = flR/Rl in Eqs. (7) and (9). Then we finally have
for any isovalent solid mixture of isostructural components an expression very
similar to Eq. (7) (Urusov, 1968a, 1974, 1975, 1977):
flH m = (9V)
413 2
X 1 x 2 c5· (12)
The size parameter c5 in the last equation varies from c5 2 = flR/R2 toc5 1 =
flR/Rl within the whole range of compositions, from Xl = 0 and X2 = 1, and
vice versa. Hence, simple linear interpolation offers the following useful
approximation:
flH m = (9V)
413 2
X 1 X 2 (x 2 c5 1 + x 1 c522 ) = X1 X2(X2Ql + X1 Q2), (13)
which expresses energetic asymmetry in an explicit form. One can compare Eq.
(13) with the well-known representation of the excess free energy of mixing in a
sub regular model (Saxena, 1973)
(14)
where the Margules parameters W1 and W2 are accordingly in close relation to
Ql = (9V/4j3)c5f and Q2 = (9V/4j3)c5r
Let us define an asymmetry parameter rJ
rJ = ~: = (~:y. (15)
From Hietala's Eq. (10), the asymmetry parameter may be simply presented by
the formula
rJH ~ 1 + 1.2(A -1 )(flR). (16)
Table 1 shows a comparison between rJ and rJH from Eqs. (15) and (16) and the
V
If = pE, (17)
if we accept that on the average, for halides /; = 0.8, IlX = 3.0, for oxides /; = 0.6,
IlX = 2.0, for chalcogenides /; = 0.3, IlX = 1.0, and for carbides /; = 0.2, IlX = 0.5.
Finally we have the so-called "crystal chemical form" of the formula for the
IlHm calculations
(19)
By using Eg. (19), it is possible to predict the values of the mixing enthalpy
IlHm or interaction parameter Q = IlHm/(xl X2). The calculated Q are given in
Table 2 and compared with available experimental estimates (Urusov, 1977;
Davies and Navrotsky, 1983). The table includes both rock salt and cesium
chloride structures and examples of cubic perovskite and silicate spinel struc-
tures. If one takes into acco).lnt large uncertainties of the empirical data, espe-
cially for oxides, then one may observe fair agreement between these two
columns of values.
168 V.S. Urusov
Table 2. Predicted and experimental enthalpy interaction parameter (kJ) for some
alkali halide and oxide binary solid solutions.
System Q,exp. Q,calc. System Q,exp. Q,calc.
CsCl-CsBr 4.40 3.65 CoO-MgO 0 0.62
CsBr-CsI 8.78 8.04 CoO-NiO 0 0.31
TlCl-TlBr 3.31 2.93 FeO-MgO 5.02 2.49
KI-RbI 2.41 2.17 CoO-MnO 6.01 7.88
KCl-RbCl 3.37 2.97 MgO-MnO 17.08 12.87
KBr-KCl 4.03 3.38 MnO-NiO 9.99 17.74
NaBr-NaCl 5.27 5.07 CaO-SrO 28.80 30.38
KBr-KI 7.44 7.07 SrO-BaO 37.45 33.44
NaBr-KBr 12.27 15.40 SrTi03 -BaTi03 9.53 8.36
NaI-KI 10.61 11.92 y-Mg 2 Si04 -y-Fe 2 Si04 7.8 6.5
NaBr-NaI 7.98 9.76
KCl-NaCl 18.48 19.04
It is obvious that for all cubic crystals, the size parameter b may be expressed
by means of the relative difference of the lattice parameter a
b = (a 2 - ad _ _A.a
(20)
a (x 1 a1 +x 2 a2 ) a
(21)
That is not the case for crystals of lower symmetry. Then the latter definition
of by (21) becomes useful for a rough calculation within a framework of the
pseudo isotropic approach. For example, the exact value of the size parameter
for spinel-type y-Mg 2Si04 - y-Fe2Si04 cubic solid solutions b = A.a/a = 0.021.
Using this b value, m = 3, v = 6, ZA = 2, ZB = 4 (SiOt-), one obtains from Eq.
(19) Q = 6.51 kJ. That does not stand in contradiction with the experimental
estimate: Q = 7.8 kJ (Akaogi et aI., 1989). Practically speaking, the same value
of the "volume" parameter by = 0.020 could be calculated from Eq. (21) for
orthorhombic olivine-type IX-Mg 2Si04 - IX-Fe 2Si04 solid solutions. However,
an experimental estimate of the interaction parameters in this system seems to
be about a factor of 2 larger: Ql = 16.8, Q2 = 8.4 kJ (Wood and Kleppa, 1981).
A much better estimation of the size parameter for olivine solid solutions would
come from the relative difference of M-Si distances: ~1-Si = 0.018, ~2-Si =
0.029. Using these values, one obtains from Eq. (19) Q = 5.0 - 13.0 kJ, which is
in satisfactory agreement with the experimental asymmetry of mixing enthalpy.
No constraints exist to apply Eq. (19) to sufficiently covalent systems,
for instance, semiconductor binary alloys with tetrahedral zincblende and a
Crystal Chemical and Energetic Characterization of Solid Solution 169
al., 1987; Suh and Talwar, 1989). The interaction parameters Q, calculated from
Eqs. (19) and (22) and obtained by the methods of Fedders and Muller (1984)
and Suh and Talwar (1989), are compared in Table 3 with some experimental
data. As may be seen in Table 3, all these estimates correlate satisfactorily.
Ifproperties of pure components are remarkably different, i.e., VdP1 =f. V2/P2'
the approximation of Eq. (7) is not completely valid. Then one can use the
following correction to Eq. (7) (Hietala, 1963a):
where s = (R 2P1)/(R 1P2) and Qd is the so-called deformation energy. For alkali
halide solid solutions, s ~ 1 within deviations only of a few percent. Such is not
the case for solid solutions like NaCl-AgCl and NaBr-AgBr. Although the
end-members in both these systems have very close lattice parameters (15 = 0.017
and 0.035), their elastic properties are significantly different. Thus, the coeffi-
cients of compressibility Pare equal to 2.26 and 2.48 Mbar- 1 for AgCl and AgBr
and about two times less than 4.26 and 5.08 for NaCI and NaBr. Hence, s values
are 0.54 for (Ag,Na)CI and 0.51 for (Ag,Na)Br systems and the correction factors
2s/(1 + s) in Eq. (23) will be 0.70 and 0.67, respectively.
Table 4 shows a comparison of theoretical estimations of AHm by means of
Eq. (23) and experimental data (Kleppa and Meschel, 1965). Very sharp discrep-
ancy between these two sets of AHm values is evident. It means that other
forces in addition to short-range repulsion have to be involved to explain this
contradiction.
It is not questionable that bonding characters are rather different in alkali
and silver halides. In other words, the previous assumption of proximity of
covalency of both components is not valid for Na-Ag substitutions and the
approximation /1 ~ /2 is not fulfilled.
Now suppose the bonding character J;(x) of a solid solution obeys the simple
additivity rule (Urusov, 1968b):
(24)
Here /1 and /2 are the fractional bonding characters of pure components. Using
(24), one obtains the following expression for mixing enthalpy:
(25)
where Qd is mainly due to elastic forces and Qc is a chemical term. Qd coincides
with the deformation energy in Eq. (23); Qc describes a change of long-range
electrostatic energy and could be expressed as follows:
(26)
where A is the Madelung constant, characteristic for a given structure type, and
z+ and z- are the formal charges of the cation and anion.
()
...
~
a
()
::r
i""
8-
tTl
Table 4. Comparison between theoretical and experimental mixing enthalpies ARm (J/mol) in the systems NaCl-AgCl and NaBr-AgBr. ~
~::to
NaCl-AgCl NaBr-AgBr n
()
Eq. (25) Eq. (25) ::r
x(NaCI) Eq. (23) I'l.f = 0.113 Exp. x (NaBr) Eq. (23) Aj= 0.101 Exp.
0.85 41 1450 1465 ± 400 0.87 145 1090 1130±3oo
i
~.
0.75 58 2130 2430 ± 330 0.77 230 1720 1845 ± 375 ::to
o
0.60 76 2730 2640 ± 200 0.70 270 2050 2175 ± 375 ::I
0.50 82 2850 2720 ± 130 0.65 300 2225 2510 ± 420 o-.
en
0.35 73 2595 2430 ± 200 0.50 330 2460 2135 ± 210
~
0.15 41 1455 1420 ± 80 0.45 325 2430 2135 ± 170 P-
en
0.27 260 1950 1880 ± 125 o
i:
::to
o
::I
...-
-.I
...-
172 v.s. Urusov
Equation (25) provides an excellent fit with experimental measurements of
AHm for NaCI-AgCI and NaBr-AgBr systems at Ai; = 0.113 ± 0.002 and
0.101 ± 0.005 (see Table 4). One may compare the above-mentioned Ai; with
estimates of the differences of ionicity degrees for these two pairs of the end-
members by the well-known Pauling (Pauling, 1960) scale of electronegativities
(Ai; = 0.067 and 0.068) and Phillips' (Phillips, 1973) spectroscopic definition of
ionicity (Ai; = 0.079 and 0.084).
The discussion up to now has dealt only with the case of isostructural compo-
nents of a solid mixture. If the structure types of the end-members are different,
theoretical analysis leads to a new form (Urusov, 1969b)
(27)
where AHtr is the enthalpy of polymorphous transition of end-member 2 to a
structure type of end-member 1. The component 2 must undergo the phase
transition in order to be soluble in component 1. If the phase transition under
consideration is accompanied by a coordination number change, then AHtr may
be identified with the site preference energy. Thus, the octahedral and tetra-
hedral site preference energies of some divalent cations were calculated from
experimentally determined terminal solubilites in binary oxide and chalcogenide
systems with a rock salt or nickel arsenide structures of the end-member 1 and
wurtzite or sphalerite structures of the end-member 2 (Urusov, 1969b; Davies
and Navrotsky, 1983).
problem. Wasastjerna's assumption, Eq. (3) (with the second-order terms ne-
glected), leads to the following individual bond lengths A-C and B-C in the
asymmetric linear chain A-C-B for the NaCl structure type of solid solution:
(28)
Practically speaking, the same result comes from the minimization of solid
solution energy by Hietala (1963a):
2 (1 - 2x 2),
- = R - u = Rl - (AR)
Rl
- =R
R2 2 (1
+ u = R2 + (AR) - 2xd·
Here it was taken into account that the average interatomic distance R obeys
Vegard's rule (R = Rl + x 2AR = R2 - xlAR) and local displacement u is only
very slightly dependent on the composition, u ~ AR/2.
At a random distribution of the substitutents, the probability of the bond
configuration A-C-A occurring is proportional to xi, the probability of the
bond configuration B-C-B-x~, and the probability of the mixed arrangement
A-C-B-2xlX2' It makes possible the calculation of the average distances of
both different types, full numbers of which are proportional to Xl and X2
accordingly
(29)
x 2R B- C = X~(R2 - xlAR) + X1 X2 [ R2 + (A2R)(1- 2Xd]
It means that a maximal increase of the shorter bond A-C as well as maximal
decrease of the larger bond B-C in dilute solutions (X2 -+ l,x l -+ 1) consists of
1/2 of the value of the difference in interatomic distances AR. In other words,
relaxation parameter A is equal to 1/2 for the NaCl structure type of solid
solutions. It is also clear from Eqs. (29) that the average shortest distance
(R = xlR A - C + x 2RB-c = xlR l + x 2R 2) obeys Vegard's rule at least in the first
approximation. This simple consideration is in very good agreement with nu-
merous calculations of lattice distortions around isovalent impurity in an ionic
174 V.S. Urusov
crystal of the NaCI structure, using the Mott-Littleton theory (see "Energetics
of Ionic Solid Solutions"). A typical result of such calculations is the fact that the
displacement of the nearest neighbors around the impurity ion is nearly half the
difference between the interatomic distances: i5R ::::; m~R.
A more detailed picture of the atomic local displacements is obtainable by
DLS modeling, which consists of the least-square optimization of individual
bond lengths, i.e., their fitting some standard distances (which represent the
distances in end-members in the case of solid solutions). The application of DLS
modeling to isovalent solid solutions of different structures made it possible for
Dollase (1980) to introduce the notion of "site compliance." The latter means the
actual fraction of increasing (or decreasing) the bond length relative to the
difference of the bond lengths in end-members in the limit of infinite dilution (a
very small amount of impurity atoms). Dollase concluded (Dollase, 1980) that
the compliance parameter Cs is inversely proportional to the coordination num-
ber of the nearest neighbors of the impurity ion because they suffer the largest
displacements (Fig. 1). Thus, less close-packed structures (ZnS, and Re0 3 types)
are characterized by larger changes in bond lengths, whereas close-packed ones
(NaCl and CsCI types) exhibit relatively smaller changes in bond lengths. It
should be noted that the predicted change of bond lengths in the NaCl-structure
solid solutions is about 50%, this result being close to the above-mentioned
theoretical estimates of the structure relaxation by entirely independent
approaches.
It is very difficult to obtain experimental data on the local structure of a solid
solution by traditional diffraction methods that carry information on average
atomic coordinates. Nevertheless, early X-ray diffraction studies of solid solu-
tions (Wasastjerna, 1945; Iveronova, 1954) detected a decrease in reflection
intensities as compared to the case of pure crystals. This fact cannot be explained
only by thermal vibrations and requires the assumption of noticeable static
displacements of ions in a nonmixed sublattice from their regular positions. An
analysis of such effects showed that local displacements are on the order of the
difference of interatomic distances of end-members; no strict correlation, how-
ever, was observed between them.
About 10 years ago it became possible to determine individual bond lengths
in mixed crystals by the EXAFS method (extended X-Ray absorption fine struc-
Cs
09
07 K4
05 Fig. 1. Correlation between site compliance pa-
rameter Cs and first coordination number. Points
0.3 and dotted line, DLS method, solid line, calcu-
0 2 1/ 6 8 lated, Eq. (36).
Crystal Chemical and Energetic Characterization of Solid Solution 175
ture). EXAFS spectroscopy is a very useful tool for characterizing lattice distor-
tions around an impurity atom as it probes one particular element at a time and
provides information about the number, distance (with accuracy of 0.005 A), and
chemical identity of the nearest neighbors. In 1979 and 80 this method allowed
considerable local displacements of atoms around impurities to be detected in
metal alloys and then in mixed halogenides and chalcogenides. For instance, in
(K,Rb)Br and Rb(Br,l) solid solutions, maximum change in individual distances
amounts to about 40 to 50% (Boyce and Mikkelsen, 1985). Relatively large
changes in the shortest distances, about 40%, were also observed for Sr-substi-
tuted fluorite (Ca,Sr)F2 (Vernon and Stearns, 1984).
On the other hand, maximum changes in the bond lengths of mixed tetra-
hedral semiconductors with a ZnS structure are much less, about half that
for essentially ionic solid solutions of the NaCl structure. For instance, in
a (Ga,ln)As system the maximum change in the nearest-neighbor distances
Ga-As and In-As corresponding to the dilute solutions amounts only to 20 to
25% of the difference A.R between the bond lengths in the end-members (Mik-
kelsen and Boyce, 1983).
It was also observed that the distributions of the Ga-As and In-As distances
in the solid solution are nearly the same as in the end-members, which is in
accordance with the fact that the nearest environment of both cations is uniform
and consists only of arsenic atoms. As distinct from this, there occurs a bimodal
distance distribution around As that corresponds to the mixed cation (Ga,In)
environment of this atom.
The distances between the second nearest neighbors (cation-cation and
anion-anion) considerably differ in their character. EXAFS data indicate the
existence of two difference As-As distances in the solid solution: Shorter dis-
tances correspond to the As-Ga-As configuration and longer distances to the
As-In-As configuration. The weighted average of these two distances corre-
sponds to Vegard's rule. It is clear that the anion packing in the solid solution
is strongly distorted, as compared to the regular cubic (closest) packing of anions
in the end-members.
A different picture is observed in the case of the distances of the cation-cation
second nearest neighbors. All the interatomic distances Ga-Ga, In-In, and
In-Ga (Ga-In) vary to obey Vegard's rule within deviations on the order of 0.05
A, which follows from the virtual crystal model. This means that the atoms in
the mixed (cation) sublattice occupy nearly regular positions, and distortions of
the ideal packing are relatively slight.
Some other semiconductor alloys were investigated recently by EXAFS:
Ga(As,P) (Sasaki et aI., 1986), (Ga,In)P (Mikkelsen and Boyce, 1983), etc. All the
studies allows us to conclude that real maximum changes of interatomic dis-
tances are only about 20 to 30% in such systems, i.e., relaxation parameter A is
approximately 0.7 to 0.8. The latter is sufficiently larger than A for ionic NaCl-
structure solid solutions (",0.5), being in accordance with the site compliance
prediction by DLS modeling (Dollase, 1980), Fig. 1.
Finally, let us note that there are extensive calculations of lattice distortions
176 v.s. Urusov
associated with substitutional defects in semiconductors, using semiempirical
pseudopotential (Srivastava and Weaire, 1987), tight-binding (Talwar et aI.,
1987), and valence force field (Martins and Zunger, 1984) methods. They lend
support to existing EXAFS experimental data: Predicted relaxation parameter
A is in range of 0.6 to 0.8 for most semiconductors. This is also close to the
theoretical prediction by DLS modeling (Fig. 1).
As follows from the experimental and theoretical results considered above, the
largest displacements in a solid solution structure (Axl' Bx2 )C are experienced
by atoms C in that sublattice where no mixing takes place. As for atoms A and
B, they form the nearly nondistorted packing. Other consequences of this as-
sumption are as follows.
If the environment of the common atom C is uniform, i.e., if it consists either
of atoms A or of atoms B, then all the distances A-C or B-C are the same and
equal to the mean R(x); see Eq. (2). If the environment of the atom C is mixed,
i.e., consists of some atoms A and B, then bond chains A-C-B appear in the
structure and the atom C is displaced from its ideal position on the middle of
the bond toward the smaller atom. Let, for example, the atom A be larger than
B. In this case, the atom C is displaced from the center of the A-C-B chain
toward B and the distance A-C becomes equal to R + u, where U is a certain
displacement of the atom C.
A change in the distance C-B in the A-C-B chain is dependent on the bond
angle LA -C-B and can be represented in a first approximation as
U1 = UCOsoc, (30)
where oc = LA-C-B. In the NaCI structure (octahedral arrangement), the
A-C-B chain is linear, oc = 180°, and, consequently, U 1 = - U (Fig. 2). In the ZnS
structure (tetrahedral coordination), oc = 109°28' and, therefore, U 1 = (-1/3)u
U ul!.
oR=u +3(-r +Jlf)
P =8(x,"32 +:12%/)
(Fig. 3). In the CsCI structure with a cubic environment of the central atom, there
exist three types of chains with the angles 180°, 109°28', and 70°32' (Fig. 4). From
Eq. (30) it follows that in this case, U 1 = - U, u~ = (± 1/3)u.
Now let us try to estimate primary displacements using the simple model of
radial forces. Denote by e(R) the energy of a certain pair of bound atoms
separated by a distance R from each other. The energy change in the formation
of the solid solution may be represented as
AE = Nv{xi[e1(R) - e1(R 1 )] + xHe 2(R) - e2(R 2)]
+ x 1 x 2[e 1 (R 1 ) - e1 (R 1 )] + x 2x 1 [e 2(R 2) - G2(R 2)]} (31)
Here N is Avogadro's number, v is the coordination number, e1 (R 1 ) and e2 (R 2 )
are the energies of pairwise interactions in the end-members, e1 (R) and G2(R) are
the energies of two bond types (A-C and B-C) at a mean distance R in symmet-
ric bond chains, and e1 (Rd and e2 (R 2 ) are the energies of these bond types in
nonsymmetric bond chains. The distances Rl and R2 depend on displacements
of C atoms in accordance with the bond angles as shown by the previous Eq.
(30). For example, for the NaCl structure R2 = R + u, Rl = R - u.
Expand the energy AE into a Taylor series. Leaving the first- and second-
order terms and taking into account that e'(R) = 0 (at T = 0 K) in the equilib-
rium state, we have e(R) - e(Ro) = (t)e"(Ro)(R - RO)2. If we assume that the
properties of the components are close, i.e., e'{(R 1 ) ~ e;(R 2) = e"(R), we obtain
from (31)
178 v.s. Urusov
dE = (t)Nve"(R) [xi(R - Rl)2 + x~(R - R2)2 + Xl x 2(R - Rl - U)2
+ XlX2(R - R2 + U)2]. (32)
Using Vegard's rule, we can rewrite expression (32) as follows:
dE = (t)Xl X2Nve"(R) [2Xl x 2(dRf + (x2dR - U)2 + (- Xl dR + U)2]. (33)
Minimizing dE as a function of the displacement u, we arrive at the condition
ddE = (XlX2)
du -2- Nve"(R)( - 2X2dR + 2u - 2Xl dR + 2u) = 0,
(35)
3
Cs =-, (36)
v1
which is compared in Fig. 1 with the DLS results.
One can easily see from the previous discussion that all primary displace-
ments in a solid solution structure completely compensate for one another and
cannot be the reason for deviations from Vegard's rule [Eq. (2)]. Nevertheless,
the deviations from this additivity rule may be observed very often through
accurate X-ray measurements. Usually, at least in simple binary systems, they
could be described by a second-degree parabola
(37)
where d is positive in a majority of systems. Let us construct a possible geometric
model of the deviations from Vegard's rule.
(43)
It was found in the preceding section that u = AR/2 for the NaCl structural type.
Accordingly,
(44)
aR B3 = 4XlX2(~). (45)
and, subject to u = (~)~R for the ZnS (B3) structure, we observe that
9 (AR)2
aR B3 = (4)X l X 2 --Y' (46)
with substitutions in one, two, three and four vertices of the cube. For B2 the
ratio of the number of secondary displacements to that of primary displacements
is I = 3.25, in accordance with the fact that the second (V2 = 6), third (V3 = 12),
and fourth (V4 = 8) neighbors of substitutional defects take part in these dis-
placements. In this case, summation (42) yields
UR B2 = 8x l X 2 (~). (47)
uR = Vl X l X 2 (~). (49)
whereas Eqs. (44), (46), and (48) can be written in the form
Table 5. Measured and calculated deviations (A· 104 ) from Vegard's rule.
NaCI-KCl NaCl-NaBr KCl-KBr RbI-RbBr
parameters a and c from linearity for intermediate compositions does not exceed
0.012 to 0.015 A.
It is possible that the accuracy of predictions by the geometric model worsens
with decreasing packing density. This model is based on the assumption that a
mixed sublattice forms a regular packing, i.e., the coordination polyhedrons
around the central atom (in a nonmixed sublattice) are undistorted. In fact, this
is not the case, as can be seen from the EXAFS data for the InAs-GaAs system
(Mikkelsen and Boyce, 1983). Thus, the In-In distances are systematically 0.08
A larger than the Ga-Ga distances for the same compositions of the solid
solution, whereas the Ga-In and In-Ga distances are of intermediate values
close to the additive ones. Therefore, solid solution relaxation occurs in both
sublattices, although to different degrees, and both primary and secondary
changes in bond lengths with respect to those in the end-members could be less
than in the geometric model.
In general, due to positive deviations from the additivity of bond lengths and
lattice parameters, Retger's rule of molar volume additivity
(51)
provides normally better approximation than Vegard's rule for most available
measurements (Slagle and McKinstry, 1966; Ahtee, 1969; Urusov, 1977).
In addition, as it became clear many years ago (Zen, 1956), Retger's rule
indeed corresponds to the slight positive departures from Vegard's rule. At the
same time, the validity of the former means that the mixing volume A Vm appears
to be close to zero in most simple cases.
Taking into account that (as/aVl)T = (Xl/Pl and (as/avzh = (Xz/Pz, where (Xl
and (Xz are the volume thermal expansion coefficients and f31 and f3z are the
coefficients of isothermal compressibility and assuming approximately (Xdf31 ~
(Xz/Pz, we notice that the third term in Eq. (53) is close to zero, while the fourth
one is the nonconfigurational entropy of mixing. Remember the well-known
Maxwell relationship
(
a2S) a2p 8 (54)
aV2 T = [(aV}y(aT)v] = PV'
where 8 = (a In p/aT)v. If we assume now that 81 /Pl Vl ~ 82 /P2 V2 = 8/PV, then
finally we obtain
(55)
184 V.s. Urusov
By comparing Eq. (55) with Eq. (7), and keeping in mind that (ARjR)2 ~
!(AVjVf, one can easily see the following relation:
1
or t = 20. (56)
The average value of 0 for the alkali halides is about 1.8 (± 0.3)· 10- 4 K -1. Hence,
t = 2800 ± 500 K, this being coincident with the above-mentioned empirical
estimation, Eq. (52).
Finally, one can write the useful parabolic expression:
ASvib = X i X 2 Q. (57)
t
It means that, in fact, there exists a correlation between the vibrational entropy
and the size parameter (j, which was first found out empirically for alkali halide
solid solutions by Ahtee and Inkinen (1970). The common value of t of about
2800 K could be used not only for alkali halide or metallic solid solutions,
however the universality of this parameter is now open to question (Urusov and
Kravchuk, 1976, 1983).
Immiscibility occurs when positive mixing enthalpy AHm outweighs the con-
figurational ASc and nonconfigurational contributions in entropy terms:
T(dSc + dSvib ). At higher temperatures, the latter may be larger than dHm
and the mutual solubility of components is complete, at low temperatures
AHm> TASm, and a result is a miscibility gap. Under these conditions, when the
composition dependence of AHm and ASvib can be represented by a second-
degree parabola, Eqs. (8) and (57), the critical solution temperature T.,r can be
calculated from the following equation (Ahtee and Inkinen, 1970; Urusov, 1977):
T = Q (58)
cr (2kN + Qjt)'
where k is Boltzmann's constant. Inasmuch as the interaction parameter Q is
(j2-dependent, it is evident that T.,r is also a function of the size parameter (j. The
calculated T.,r are compared with the experimental estimates in Table 6 for some
simple halide Na-K systems.
First, one can see the obvious decrease of T.,r as (j values decrease. Second, one
T:
can easily note that the theoretical calculations of r without involving the
correction of ASvib strongly overestimate the critical temperatures, while inclu-
sion of the effect of ASvib by means of Eq. (58) draws hearer the calculated and
the experimental T.,r.
In fact, the miscribility gaps are asymmetric because of the energetic asym-
metry, i.e., the dependence of Q on composition. The greater the asymmetry
Crystal Chemical and Energetic Characterization of Solid Solution 185
parameter 1] [see Eq. (15)], the greater the deviations of T.,r/Tmax and Xcr from
their values, predicted by the simple regular model (T.,r/Tmax = 1, Xcr = 0.5). Here
T.,r/Tmax is the reduced critical solution temperature, Tmax being the value of T.,,,
corresponding to the maximal value of Ql (or maximal size parameter <5 1 ) for a
system under consideration. Table 7 shows predicted critical temperatures and
compositions of mixing for various asymmetry parameters.
The asymmetric solvi as a function of the size parameters were calculated
for various binary systems (oxides, chalcogenides, etc.) by some investigators
(Urusov, 1974, 1975, 1977; Davies and Navrotsky, 1983). The predicted miscibility
gaps are in very reasonable agreement with the available experimental data, in
particular, for AI 2 0 3 -Fe 2 0 3 , Ti0 2 -Sn02 , CaS-MgS, HgS-HgTe, etc.
Later some useful approximate correlations were established between proper-
ties of simple binary melts and solid solutions (Urusov and Kravchuk, 1976;
Kravchuk and Urusov, 1978). For instance, the mixing enthalpy of melts very
simply anticorrelates with that of solid solutions
AH~ ~ -O.2AH~. (59)
Here s denotes solid and I the liquid phases. Moreover, the universal correlation
between mixing enthalpy and excess (vibrational?) entropy, discussed in the
preceding section, is also valid for ionic melts
I AH~
ASvib = - - , (60)
t
where t has the same empirical value, being equal to about 2800 K.
186 V.s. Urusov
It follows from the above that a close relationship exists between all energetic
parameters governing the distribution of the isovalent impurity during crystalli-
zation from melt. Taking into account previously established correlations, one
can write an expression for the partition coefficient Ki of the i-component
between solid and liquid phases in the following form:
Here AHml and Tml are the melting enthalpy and melting temperatures, and ARt
is the partial mixing enthalpy of the i-component in the solid phase.
For the simplest case of infinite dilution (i.e., without regard for the Ki
dependence on composition), Eq. (61) is written as
AHml)(1
In Ki = ( kN
1)
T - Tml -
(1.2Q)(1
kN
1) .
T - 2800 (62)
main factors affect enthalpy of mixing and therefore the boundaries and stability
field of a solid solution.
The first and, possibly, the most important is the size difference, i.e., the value
of the size parameter (j. Let us write (j in the conventional form
size mismatch.
One can use the electronegativity difference IlX as a good measure of the
ionicity degree difference 4/;. In the above examples of Na-Ag substitutions,
IlX is equal to 1.0 and Ill; is equal to 0.10 to 0.11. Thus, one can easily explain
the absence or limitation of isomorphous substitutions between K + and Ag+
(llr = 18%, IlX = 1.1), Ca2+ and Cd2+ (Ilr = S%, IlX = 0.6) or Hg2+ (llr = 8%,
IlX = 0.9), Mg2+ and Zn2+ (Ilr = 12%, IlX = 0.4), although there are extensive
or even complete substitutions between Na+ and K+ (Ilr = 36%,IlX = 0.1), K+
and Rb+ (Ilr = 12%,IlX = 0.0), Ca2+ and Sr2+ (Ilr = 16%,llx = 0.1), Sr2+ and
Ba2+ (llr = 1S%, IlX = 0.1), Zn2+ and Cd2+ (llr = 19%, IlX = 0.1) with similar
or larger Ilr, but smaller IlX. In other words, a IlX value of about 0.4 represents
an upper limit, so that larger values of IlX keep solid solubility from being
anywhere efficient.
At last, the fourth main factor is structure difference, as seen from Eq. (27). It
concerns nonisostructural systems, which usually show only limited solubility.
It should be borne in mind that the enthalpy IlHtr of the "forced" structure
transition of "guest" end-member 2, on dissolving in the "host" end-member 1
with another structure, is positive. The higher the positive IlHtr> the more limited
the solubility of the guest end-member in the host one. Let us consider only one
example: The solubility of alabandite MnS (the NaCl structure) in sphalerite
ZnS reaches 40 mol % at 6OOD C, although the size mismatch is significant
(llr = 0.08 A), but the solubility of CoS (the NiAs structure) in ZnS is remarkably
less, about 30 mol % at 8S0 C, although the sizes of Zn2+ and Co2+ are
D
known that there are unstable red modifications of MnS with sphalerite and
wurtzite structures, but any analogous modifications of CoS do not exist.
The effect of temperature on solid solubility is unambiguous: due to the
mixing entropy contribution, the higher the temperature the more extensive the
solubility. Only a few examples of a slight temperature influence on solid solu-
tion boundaries are known. One of the most important is FeS solubility in
sphalerite ZnS, because this fact forms the basis for well-known sphalerite
geo- and cosmo barometer.
As for the pressure influence on solid solubility, the situation is simple only
when components have different structures. Then the pressure increase expands
the stability field of a denser structure, usually a structure with larger coordina-
tion numbers. And the opposite is true: the close-packed component is pushed
out of the loose matrix. It is the case for Fe in sphalerite; the terminal solubility
of FeS in ZnS at 500°C falls from about 50 mol % at 1 bar to nearly 10% at 2
GPa due to a volume gain of about 6 cm 3 /mol FeS.
When components are isostructural, the prediction of pressure influence is
somewhat complicated. At first glance, the problem reduces to the determination
of the sign of deviations from Retger's rule, Eq. (51). If the deviations are positive
(positive mixing volume AVm ), the pressure increase corresponds to a solubility
decrease, and vice versa. This is true, but in the author's opinion, another factor
is most important. It is the dependence of the interaction parameter Q on
pressure. It was demonstrated (Fancher and Barsch, 1971) by detailed calcula-
tions for alkali halide solid solutions that they have to decompose under high
pressure in good agreement with experimental evidence.
Indeed, as follows from the previous discussion, Q is proportional to VIP,
which is growing when pressure builds up. The calculations of l1P values at
various pressures and corresponding changes of Q were first undertaken by the
author (Urusov, 1977) and later refined (Kirkinsky and Fursenko, 1980) by
accounting for the different compressibility of end-members. The definite result
of these calculations is a strong increase of interaction parameter Q, as well as
the critical solution temperature 7;,r at a pressure rise. This conclusion is consis-
tent with most available experimental data. For instance, the predicted increase
of 7;,r for the NaCl-KCl miscibility gap is equal to about 200°C at 20 kbar,
being very close to the experimental estimates (Bhardway and Roy, 1971). The
following mineral systems also reveal the effect of solid solubility depression
under high pressure: NaAISi 3 0 8 -KAISi 3 0 8 , Mg 2 Si04 -CaMgSi04 , Mg 2 Si 2 0 6 -
CaMgSi 2 0 6 (Urusov, 1977).
It is probable that one has to take into account the above-mentioned effect if
one intends to predict the mixing properties of a solid solution at high pressures.
For instance, let us compare the size parameters 6 and the values of VIP (kJ) for
the series of MgSi0 3 -FeSi0 3 solid solutions of various structures: pyroxene
(6 = 0.016; VIP = 3350), garnet (0.010; 4400), ilmenite (0.006; 5530), perovskite
(0.013; 6040). One can see that, despite decreasing 6 in the direction from low- to
high-pressure phases, the product (VIP)6 2 and, hence, the interaction parameter
Q are likely to be largest for the densest polymorphs of the perovskite structure.
190 V.S. Urusov
It means that, as a rule, high pressure could suppress the stability field of a solid
solution. However, this assumption needs further refinement.
Concluding Remarks
The author well understands that this chapter severely suffers from incom-
pleteness; in particular, this contribution deals only with an isovalent type of
solid solution. There is no discussion of the crystal chemistry and energetics of
heterovalent solutions, which, in fact, represent many varieties. Our knowledge
of this field is limited, but some useful semiquantative formulations have been
established by Urusov (1977 and elsewhere). The complications concerning the
multicomponent and multisite solid solutions are also amenable to similar
theoretical treatment. The simplest case of isovalent and isostructural solid
solution was chosen as the primary theme in this chapter to demonstrate the
fundamentals of such an approach. The author believes that in the near future
there will be a rapid development of the crystal energetic theory of solid
solutions.
Acknowledgments
The author thanks I.P. Deineko for great help in the preparation of this manu-
script, V.P. Volkov, A.S. Marfunin, and N.R. Khisina took part in numerous
useful discussions.
References
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Akaogi, M., Ito, E., and Navrotsky, A. (1989). Olivine-modified spinel-spinel transitions
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Anderson, D.L. and Anderson, O.L. (1970). The bulk modulus-volume relationship for
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Bandyopadhyay, S. and Deb, S.K. (1988). Divalent defects in alkali halides. Ind. J. Phys.
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Barrett, W.T. and Wallace, W.E. (1954). Studies of NaCl-KCl solid solutions. I. Heats
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Bhardway, M.e. and Roy, R. (1971). Effect of high pressure on crystal solubility in the
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Born, M. (1923). Atomtheorie des festen Zustandes. Berlin.
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Fancher, D.L. and Barsch, G.R. (1969). Lattice theory of alkali halide solid solutions. I.
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Fancher, D.L. and Barsch, G.R. (1971). Lattice theory of alkali halide solid solutions. III.
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192 V.S. Urusov
Introduction
1c
1
1------- b -------i
Fig. 1. The crystal structure of C2/c pyroxene projected onto the (100) plane. The
geometry of the new M2' site as determined by Oal Negro et al. (1982) has been added to
the diagram.
196 G. Ottonello, A. Della Giusta, A. Dal Negro, and F. Baccarin
from the original drawing of Cameron and Papike (1980) by the identification of
a nonequivalent M2' site, made by Dal Negro et aI. (1982); the atom nomencla-
ture is after Burnham et al. (1967). Although the Ml polyhedron is quite regular,
a notable variation is shown by the MI-02 bond distances (2.021-2.051 A),
which reflect the occupancy by ions smaller than Mg and Fe2+, i.e., Ti 4 +, Fe3+,
AI3+, and Cr3+. The occupancy by the latter ions produces a strong angular
distortion of the polyhedron. The crystal-chemical characteristics of the M2
polyhedron largely depend on the Ca occupancy, but in a nonlinear and rather
complex fashion (Dal Negro et aI., 1982); moreover, the reconaissance of an M2'
site, nonequivalent to M2, by Fourier difference synthesis over a large number
of natural samples adds further complexity to the M2 geometry (Dal Negro et
aI., 1982).
The detailed crystallographic studies existing On a large number of natural
and synthetic samples (Clark et aI., 1969; Bruno et aI., 1982; Carbonin et aI.,
1991; and references therein) made it possible to treat all nonequivalent inter-
ionic distances in the asymmetric unit of C2/c pyroxenes in the same manner as
proposed by Ottonello et aI. (1989) and Della Giusta et aI. (1990) for Pbnm
silicate olivines. This leads to simple proportionality expressions involving
atomic proportions on sites
(1)
Table 3. DLS optimization of a chemically complex C2jc pyroxene after four iteration
cycles.
Table 3. (Continued.)
M2 (Ca O. 762 , Na O.096 ' Fe O.024 ' Mg O. ll1 , Mn O. OO7 )
Ml (Mg O. 781 , Fe O. 053 ' Al o . 1l6 , Cr O. 037 ' Ti o .013 )
Composition T (Si1. 917 ' Al o .o83 )
Atom 1 Atom 2 Deale Dobs Do-Dc Weight
A B C f3
calc 9.709 8.874 5.263 106.34
obs 9.711(1) 8.874(2) 5.263(1) 106.33(1)
the structure until the discrepancy between the predicted distances and the
simulated ones is minimum. This method was first developed by Meier and
Villiger (1969); we adopt here an updated version, DLS-76 by Baerlocher et al.
(1977). The minimized function is expressed by:
(3)
where Uj is the weight assigned to distance j and Dref j the refined distance. The
summation occurs over all nonequivalent distances in the asymmetric unit. The
weight factor is proportional to the Pauling's strength of the bond: the higher
the bond strength, the higher the weighting factor. By a trial-and-error proce-
dure, we found that appropriate weighting factors are the following: Uj = 0.166
for Na-O and all 0-0 distances; Uj = 0.250 for Ca-O; Uj = 0.333 for Mg-O,
Fe2+ -0, Mn-O; Uj = 0.5 for AI-O, Fe3+ -0, Cr-O; Uj=0.666 for Ti-O and
Uj = 1.0 for Si-O.
In Table 3 we see the results of the DLS procedure on a chemically complex
C2/c pyroxene. In this case, convergency was achieved in four steps. Although
the refined distances do not differ by more than 0.0001 A from the calculated
ones, the refinement modifies cell edges on the third decimal place. The resulting
atom positional parameters, which are the first essential step in building the
interionic potential model, are strictly representative of C2/c geometry.
200 G. Ottonello, A. Della Giusta, A. Dal Negro, and F. Baccarin
(5)
(6)
(7)
where, Z;, Zj are the formal charges of the i, j ions; b;j' dd;j' dqij are the repulsive
and dispersive (dipole-dipole and dipole-quadrupole) coefficients between the
i, j ions. Cij, q, ct d , and ct q are structure coefficients that depend on atomic
I'
Mg 2Si 20 6
u(o.O)
I(g)
1-'
Hf(o.o)
12I~ 1 21 •
E.,.
(8)
2 E,.M, + 2 E,.,; + 3 E•. o"
2 Mg(s) + 2 Si(s) + 3 02(g) 2 Mg(g) + 2 Si(g) + 60(g)
where I Mg is the ionization energy for the metal Mg to its formal valency state
in the crystal (2 + ), Eao is the electron affinity for oxygen, Es are the sublimation
energies of metals, and Ed02 is the dissociation energy of molecular oxygen.
There is a main difficulty in applying the Hess additivity rule to the energy
balance (8), namely, that most of the quantities involved are not specifically
available at zero-point conditions. Hence, the Born-Haber-Fayans cycle is
commonly referred to with 298.15 K, P = 1 bar conditions, is neglected, and the
effect of P change the thermal expansion of gaseous (ideal) ions from 0 to
Energy Model for C2/c Pyroxenes 201
_ 27
d%-- lXilXj (lXiEi IXjEj)
2· -+- , (11)
8 e (Ei + Ej ) ni nj
where
lXi' IXj = free ion polarizabilities for i, j ions
Ei , Ej = mean excitation energy
ni' nj = effective electrons
(12)
(13)
with
n;, nj = "outer" electrons for i,j ions
e= electronic charge
m = rest mass of electron
h = Planck's constant
(see Ottonello, 1987, and Ottonello et aI., 1989, 1990, for an extended discussion
of the energy terms involved and the resulting interionic coefficients). The bij
202 G. OttonelIo, A. Della Giusta, A. Dal Negro, and F. Baccarin
bij = bpijexp (~ ,
r. + r.) (14)
where
= "repulsive" radii for i,j ions
r i , rj
In the Born-Mayer approach, b has a constant value in the family of salts and
p is variable from salt to salt, whereas in the Huggins-Mayer treatment, b is
variable from salt to salt and p is constant (Tosi, 1964).
Adopting the generalized Huggins-Mayer approach b can be regarded as an
arbitrary constant since the single ion repulsive contributions
(16)
become characteristic parameters of ions (Tosi and Fumi, 1964). Stemming from
Eq. (16), the evaluation of bij repulsive coefficients can be readily generalized for
chemically complex mixtures as shown by Ottonello (1987). The generalization
implies that the b factor of common ions be linearly related to the molar
proportions of end-member components in the mixture, i.e., for a binary
(AxBI-x)(AyBl-y)Si206 clinopyroxene
radii of ions to be coincident with ionic radii, this will reduce the number of
variables to four (bMg2Si206; bCaMgSi206; bCaFeSi206; bFezSi20J Obviously, as we know
the enthalpy of the four end-members at 298.15 K, 1 bar and we have four b
factors to evaluate, the system can be solved for whatever hardness factor is in
the family of salts. However, we wish for an energy model that predicts interac-
tion energies in chemically complex mixtures. Therefore, we are forced to a
nonlinear minimization procedure where the input data are all the existing
energy data in chemically complex cpx and the solving set of factors is the
minimum number of parameters necessary to achieve a sufficiently precise energy
simulation.
In the "mixture" notation, i.e., "solid phases containing more than one substance
when all substances are treated in the same way" (see IUPAC, 1979), the stan-
dard state is identical for all components in the mixture and is normally that one
of "pure component at P, T of interest." The molar Gibbs free energy of a
mixture is moreover constituted by three terms
where
Xi = molar fraction of component i
p? = standard-state chemical potential of component i
Gmix id = ideal molar Gibbs free energy of mixing
(23)
XM,s terms in (23) are atomic fractions of the various metals on sites where
mixing takes place (i.e., MI, M2, T for C2/c pyroxenes). The excess molar Gibbs
free energy of mixing is usually expressed in terms of macroscopic interaction
parameters, i.e., for a binary A-B mixture
(24)
The Margules parameter li'(p, T) in a regular mixture varies with P, T conditions
and it can easily be shown by partial derivation over the two intensive variables
204 G. Ottonello, A. Della Giusta, A. Dal Negro, and F. Baccarin
As in our notation,
IIi!
1"',
= G~ = Hi! I '(P,T)
- TSi! '(P,T)
(26)
if we consider simply the enthalpy of a given binary mixture at P, T conditions,
we will have
-
Hi(p,n = H- 0p " Tr + fTTr Cp/dT + fPP r [-
V; - T aT dP.
(av;)] (28)
+f
T
H,'o,n = Hi!'O,Tr) ·dT.
Cp,. (29)
Tr
We must then introduce the following considerations: (1) Based on (25), the
interaction parameter WH represents the virtual macroscopic interaction of the
substance at zero-point conditions (actually, in most cases, WH is no more than
a fitting parameter relating the measured excess energy terms at various P, T
conditions to composition).
(2) The internal disorder (long-range plus short-range terms) for all compo-
nents of interest here does not vary with temperature, thus, their Cp functions
(30) do not carry any configurational information.
(3) Short-range ordering is implicitly excluded by interionic potential models,
which are based on fixed atomic positional parameters, valid in the whole
crystal.
With the above provisos, H~ix,Pr, Tr is the enthalpy of the mixture at P" T,. and
at the same state of configurational disorder observed at Pr , T.
Energy Model for C2/c Pyroxenes 205
From Eq. (31) or its analogous forms, we can reduce the experimental infor-
mation on the high P, T enthalpy of the various binary cpx mixtures existing in
the literature to data that which can be reconverted to lattice energy through the
thermochemical cycle (9). Once the repulsive energy is parameterized with a
nonlinear minimization of the obtained U values, the reversed operation will
lead to Bmix , p. T estimates in the whole compositional range of interest.
We wish to stress here that there is no empiricism involved in the reduction
of experimental data. Both the internal energy U and enthalpy H are, in fact,
state functions and their relations are commonly described by Legendre trans-
forms involving the state parameters. The most familiar of the Legendre trans-
forms relating enthalpy to internal energy is
dH = dU + PdV + VdP. (32)
Relation (32) is, however, not particularly interesting to us as it relates U and H
through volume and pressure variations. For our purposes, it is better to express
both U and H as functions of the intensive variables P, T
(34)
(35)
(36)
(37)
(38)
with
oc = isobaric thermal expansion coefficient
P= isothermal compressibility coefficient.
Let us now consider the partial derivatives (35) and (37). Cp is mainly vibrational
and, if we consider substances that do not change their configurational disorder
with T, then Cp is simply and solely vibrational. The PVoc contribution in (35)
reflects the anharmonicity of the vibrational terms, so (aUjaT)p is intrinsically
vibrational and cannot in any case be evaluated by a static potential model (this
206 G. Ottonello, A. Della Giusta, A. Dal Negro, and F. Baccarin
If the mixture is ideal, then by definition, the enthalpy change involved in (39)
and (40) is also zero; hence,
1 -0 1-0 2-0
:3 H Fe2Si206. T r. P r = :3 H Mg 2Si 20 6• T ro P r + :3 H CaFeSi20 6• T ro P r
(41)
Table 4. Energy values for C2/c pyroxene. EBH = Born-Haber-Fayans thermochemical cycle. All values are in kJ/mol.
Asterisks mark adopted values. tTl
::s
0
....
(JQ
Component EBH HLp. u Ee EDD EDQ ER '<
Table 5. Entropy at 1',. = 298.15 K, Pr = 1 bar and coefficients of the Cp function for C2/c pyroxene end-members. Limits of validity of the
Cp functions are also given. Cp = A + BT + CT- 2 + ET- 3 + FT- 1j2 + GT- 1 • Data are in J/mol x K.
Compound stP r A x lO-2 B X lO3 C X lO-6 E X lO-8 F G X 10- 4 1iim
Mg 2Si 20 6 134.85 2.889 3.764 -2.70 9.224 0 -3.876 1830"
G,b,e
Fe 2Si 20 6 188.84 3.0514 6.524 -2.088 9.224 0 -3.876 -
CaMgSi 20 6 143.093 2.2121 32.80 -6.586 0 0 0 1664G,d
CaFeSi 20 6 170.289 2.2933 34.18 -6.28 0 0 0 - G,. P
CaAl 2Si06 156.0 2.324 21.845 -7.2788 0 0 0 16Q04 0
G,d
NaAISi 20 6 134.72 2.6566 13.86 3.294 -4.930 0 0 - 0-
-
NaFeSi 20 6 153.55 1.9313 80.793 -3.9581 0 0 0 95Qf ::s
g.
2.1932 41.882 -3.2133 0 0 0 950-105Qf .0
2.lO33 45.564 -3.2133 0 0 0 1050-14Q()f ?>
CaMnSi 20 6 173.31 2.2667 27.93 -7.lO65 0 -104.59 0 16649 0g.
NaCrSi 20 6 149.64 2.4649 18.249 3.5215 -4.93 492.317 0 iii"
CaTiAl 20 6 175.80 2.5088 -2.839 -7.2633 0 -5.616 0 844i C'l
2.3655 24.884 -8.2651 0 -5.616 0 844-18ooi 2'
m
.P'
-
• Saxena (1990).
?>
bSaxena and Chatterjee (1986). 0
'Entropy value calculated to be consistent with energy and interactions in the cpx quadrilateral (see text). a
d Robie et al. (1978).
z
C1>
(JQ
e Saxena et al. (1986).
...$)
fHelgeson et aI. (1978). ::s
'"
Q.
• Calculated with the exchange reaction (42).
~
hCalculated with the exchange reaction (43). t:=
iCalculated with the exchange reaction (44). g'"
'So"
Energy Model for C2/c Pyroxenes 209
a Present work.
bSaxena and Eriksson (1983).
'Saxena (1990).
dSkinner (1966).
e Akimoto (1972).
models in predicting the volume are due to the large dependence of ~ V on the
interaction parameters adopted to describe mixing within each class of sub-
stances. Estimates of ~ V for reaction (45) range thus from 0 (Holland et al., 1979)
to 0.355 (Lindsley et al., 1981), 0.6188 (Nickel and Brey, 1984), 0.9754 (Davidson
and Lindsley, 1989), and 2.0 cm 3 /mol (Saxena, 1981).
The structural predictions of our model for the four limiting binary joins
in the quadrilateral are compared in Figs. 2 a through d with experimental
evidence and theoretical estimates of the C2/c molar volumes. In comparing the
results, one should keep in mind that the differences in molar volumes bear the
maximum bias as three distinct distances and one angle are involved in the
calculation. Discrepancies in the observed and calculated interionic distances are
generally lower by a factor of 10 (see, e.g., Table 3). In Fig. 2(a} we see that there
is good agreement between model predictions and experimental observations
on the join MgzSiz06-CaMgSiz06 in the range 0.6 < X(CaMgSi z0 6 } < 1.0.
Notably, the model reproduces fairly well the convex upward inflection observed
by Clark et al. (1962). For X(CaMgSi z0 6} < 0.6, the model predicts molar vol-
umes substantially higher than those observed by Newton et al. (1979). However,
the two extreme compositions of Newton et al. (1979) [i.e., X(CaMgSi z0 6} = 0.2
and 0.33, respectively] should be discarded as their diffraction patterns exhibit
some complexities not in line with a simple C2/c symmetry (see Newton et al.,
1979). Moreover, the model estimate for pure C2/c Mg zSi z0 6 falls within the
volumetric range obtained by various authors from energy considerations.
For the join CaMgSi z0 6-CaFeSi z0 6, the model predictions are very close to
the experimental observations of Turnock et al. (1973), with the exception of the
small inflection observed by these authors for the single composition
X(CaFeSi z0 6) = 0.6, which is not reproduced at all [see Fig 2(b)]. The model
volumes along the join FezSiz06-CaFeSiz06 are slightly larger than the values
obtained by Lindsley et al. (1969) and in closer agreement with the results of
Davidson and Lindsley (1989) and Syono et al. (1971) for C2/c Fe zSi z0 6 [Fig.
2(c)]. Along the join MgzSiz06-FezSiz06' there is no experimental information
and the model predicts a slight positive excess volume of mixing, which has, as
we will see later, some significant consequences on the intrinsic stability of C2/c
Ca-poor phases at high P. Due to the impressive experimental work of Lindsley
and his coworkers, the miscibility of C2/c and Pbca substances along the joins
MgzSiz06-CaMgSiz06 and FezSiz06-CaFeSiz06 is well known at various P,
Tconditions (Lindsley, 1981; Lindsley et al., 1981; Davidson and Lindsley, 1989;
and the references therein). The mixing properties within the C2/c class, based
on Lindsley's data, are usually reconverted to a macroscopic subregular
Margules-type formulation (see Table 7). For the join MgzSiz06-FezSiz06'
there is general consensus about an essentially ideal mixing behavior within the
C2/c structural class (Sack, 1980; Saxena, 1983; Saxena et al., 1986) as far as
enthalpic and entropic terms are concerned. One can now parameterize the
lattice energy of C2/c pyroxenes in the cpx quadrilateral stemming from the
energies calculated for the three joins and the four end-members of interest
involved in the calculations.
212 G. Ottonello, A. Della Giusta, A. Dal Negro, and F. Baccarin
66
65
64
63~--~--~----~---L----L---~--~----~---L--~
X CaMgSi206
- model prediction -+- Newton et al.(1979) ........ Clark et al.(1962)
a 0 Nickel and BreY(19S4) x D.L.(19S9) ¢ Saxena(19S1)
68.0
67.5
67.0
66.5
66.0~--~--~----~--~--~----~--~--~----~--~
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
X CaFeSi206
b - model prediction -+- Turnock et al.(1973) * D.L. (1989)
Fig. 2. Structure model predictions (a-d) for the four limiting C2/c binary joins in the
pyroxene quadrilateral, compared with experimental values and theoretical estimates.
D.L. (1989) = Davidson and Lindsley (1989).
Energy Model for C2/c Pyroxenes 213
68,0
67,0
66,0
65,0~--~--~--~----~--~--~--~----~--~--~
0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 O,g 1,0
X CaFeSi206
~ model prediction -+- Lindsley et al,(1969)
C * Syono et al.(1971) o D,L,(19a9)
66
65
64
63~--~--~--~~--~--~--~----~---L--~--~
0,0 0,1 0,2 0,3 0.4 0,5 0,6 0,7 0,8 O,g 1.0
X Fe2Si206
~ model prediction + D,L,(19a9)
d * Saxena(19a1) o Nickel and Brey(19a4)
Fig. 2. (Continued.)
N
......
.l>-
Table 7. Macroscopic subregular interaction parameters for C2/c pyroxenes. Data are in J/mol (WH ), J/mol x K (Ws), J/mol x bar (Wy ).
I = Mg zSi 2 0 6 , 2 = CaMgSi z0 6 , 3 = CaFeSi z0 6 , 4 = Fe zSi z0 6 , 5 = CaAl zSi06 , 6 = NaAISi z0 6 •
WH 12 WH 21 WH 13 WH 31 WH 14 WH 41 WH 23 WH 32
52,971 29,085 -960.5 1190.1
23,663 32,467 50,774 27,841 -1558.6 2518.6 13,984 17,958
25,484 31,216
93,300 -20,000 0 0 12,000 12,000
WH 24 WH 42 WH 34 WH 43 WH 25 WH 52 WH 26 WH 62
P
Q
....
22,590 46,604 25,462 31,693 28,243 44.063 0
::s
b ~
17,327 49,128 20,099 16,109 25,397 29,175
20,697 16,941
.0
24,000 ~
15,000
f tl
41,214 ~
;-
Ws 12 Ws 21 as 13 Ws 31 as 14 as 41 Ws 23 as 32 Cl
a·
15.887 3.538 0.041 -0.081 '"
F
0.016 -0.007 38.907 24.609 24.389 24.651 -0.001 0.004 ~
d
0.0 0.0 tl
45.0 -28.0 0.0 0.0 0.0 0.0 e:..
z
<1l
(J<l
Ws 24 Ws42 Ws 34 as 43 Ws25 Ws 52 as 26 Ws 62 ...,
.0
-0.003 -0.008 I>'
3.940 14.996 -15.036 -3.879 ::s
0-
23.620 41.017 -0.034 0.033 9.046 19.027 'Tl
0.0 0.0 I:C
I>'
0.0 0.0 (")
(")
f -12.259 30.920 I>'
...,
S·
m
::I
CD
....
(Jq
'<
0
==
Q..
Table 7. (Continued.) ~
0'
....
Wy 12 Wy 21 Wy 13 Wy 31 Wy 14 Wy 41 Wy 23 Wy 32 n
N
N
VI
-
216 G. Ottonello, A. Della Giusta, A. Dal Negro, and F. Baccarin
o~----~------~-----U------~----~------~----~
0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
hardness factor (A)
-0- cpx-quadrllateral -t-- OaMgS1206-0aA12SI06
....... OaMgS1206-NaAIS1206
Fig. 3. Fit of the repulsive energy model in the various investigated systems at different
conditions. Results are relative to model (2), i.e., U = Ee + ER + ED.
The parameterization of the lattice energy was performed with the method
previously outlined with a nonlinear minimization procedure (James and Roos,
1977) involving 72 distinct compositions in the P, T range of 1 to 20000 bar and
298.15 to 1700 K, respectively. Calculations were done with different values of
the hardness parameter for the class of substances (the Huggins-Mayer ap-
proach). As shown in Fig. 3, the best results were achieved with a hardness factor
of 0.5 A. The variables involved in the calculations are the repulsive radii of
cations on sites M1 and M2 (five values as Ca is assumed to be stable on the M2
site at all P, T, X conditions) and the four repulsive factors b of the end-members.
The best fit of the interionic potential calculations is obtained if we consider all
energy constituents (Le., U = Ee + ER + ED + ECFS ; see Table 8). Disregarding
the crystal field stabilization energy terms has, however, a very limited effect on
the model as the mean error rises only from 268 to 311 J/moi. The following
arguments indeed suggest that we disregard the ECFS contribution to the lattice
energy of C2/c pyroxenes:
1. No precise CFS estimates are available for the C2/c structure (case 3 in Table
8 was evaluated adopting the canonical CFS term of Dunitz and Orgel, 1957,
Le., 49.7 kJ/mol for Fe2+ in both Ml and M2 sites.
2. As the Ml and M2 sites are differently distorted, one would, however, expect
slightly different ECFS for M 1 and M2 sites; moreover, the different degeneracy
states for 3d electrons in the M1 and M2 sites would result in electronic
entropy contributions not readily quantifiable in chemically complex mix-
tures (see Ottonello et ai., 1990).
Table 8. Parameters of the repulsive energy obtained by nonlinear minimization of binary-join energetics. p, r are in (A); b factors are in
llmol x 10 19 • g
cia
(a) Optimization on the joins Mg2Si206-CaMgSi206, Fe2Si206-CaFeSi206' Mg2Si206-Fe2Si206 '<
p = 0.25 p = 0.50 ~
o
U = Ee + ER + ED U = Ee + ER U = Ee + ER + ED U = Ee + ER + ED + E CFS ~
b Mg 2 Si 2 0 6 3.23953 3.07047 3.39063 3.39038
...0'
(')
b Fe 2 Si 2 0 6 3.117605 2.87426 3.02727 3.08709 ~
b CaMgSi 20 6 3.39273 3.11362 3.34256 3.34270 '"0
'<
b CaFeSi 20 6 3.23925 2.92203 3.12258 3.14985 ...o
r Mg, M1 0.76363 0.50023 0.54599 0.54735 ><
g
r Mg,M2 0.78276 0.70205 0.62340 0.62187 ~
r Fe2+,M1 0.87501 0.71456 0.81996 0.81964
r Fe2+, M2 0.78288 0.74346 0.84855 0.83934
rCa, M2 0.91566 0.81855 0.85120 0.85005
LX 2 0.0550 0.0285 0.0020 0.0015
Mean error 1.660 1.195 0.311 0.268
(kllmol)
Table 8. (Continued.)
p
(d) Extension of (a) to minor components· (p = 0.50)
U = Ec + ER + ED
Q
o
-
b Mg 2 Si 2 0 6 = 3.39063 b Fe 2 Si 2 0 6 = 3.02727 b CaMgSi 2 0 6 = 3.34256 ~
j5"
b CaFeSi 2 0 6 = 3.12258 b CaMnSi 2 0 6 = 2.56038 b CaAI 2 Si 2 0 6 = 2.41396
b CaTiAI 2 0 6 = 2.18952 b NaAISi 2 0 6 = 2.66178 b NaFeSi 2 0 6 = 3.89261 ~
b NaCrSi 2 0 6 = 1.63984 ~
r Fe2+, Ml = 0.81996 r Fe2+, M2 = 0.84855 r Mg, M 1 = 0.54599 p;
r Mg, M2 = 0.62340 rCa, M2 = 0.85120 rNa, M2 = 0.0 9.
r AI, Ml = 0.83292 r AI, T = 0.05071 r Mn, Ml = 1.12930 ~
=
r Fe3+, Ml = 0.01587 r Cr3+, Ml = 1.34900 r Ti 4 +, Ml = 0.05143 !"
~
·Concentration limits of validity: Mg 2 Si 2 0 6 = 1.0, Fe2Si206 = 1.0, CaMgSi 20 6 = 1.0, CaFeSi 20 6 = 1.0, NaAlSi 20 6 = 0.1, CaMnSi 20 6 = 0.1, tl
CaAl 2Si06 = 0.1, CaTiAl 20 6 = 0.05. NaFeSi 20 6 = 0.1, NaCrSi 2 0 6 = 0.1.
e:..
f8-
:-r1
t:I::I
~.
::s
Energy Model for C2/c Pyroxenes 219
As Ca is always stable on the M2 site, this configurational term does not coincide
with the maximum disorder within the quadrilateral, but only along the edge
Mg2Si206-Fe2Si206'
GT p in Table (9) is the Gibbs free energy of the substance at T = 1773 K,
P = 20 Kbar. GT,p values were obtained by the model (and analogous values
calculated at different T, P conditions) through the relation (22) for the calcula-
tion of the Gibbs free energy of mixing terms. These terms were then reconverted
to the usual subregular macroscopic formulation and are shown in Table 7.
The macroscopic interaction factors W were calculated with two distinct ideal
mixing contributions, i.e.,
(49)
0.000 1.000 -33,321.4 -38,878.6 6127.5 -437.7 -132.6 -3201.8 143.1 66.234 -2843.8 549.9 137.7 -3681.1
0.200 1.000 -33,324.1 -38,867.0 6129.2 -448.5 -137.9 -3163.9 148.5 66.404 -2804.3 557.1 138.1 -36~4.1
0.400 1.000 -33,327.5 -38,855.9 6130.4 -458.9 -143.0 -3126.6 152.7 66.572 -2765.6 563.1 138.4 -3625.7
0.600 1.000 -33,330.8 -38,844.5 6130.7 -469.1 -147.9 -3089.3 156.3 66.761 -2726.8 568.6 138.8 -3596.2
0.800 1.000 -33,334.4 -38,832.9 6130.2 -479.1 -152.6 -3052.3 159.6 66.931 -2688.3 573.6 139.2 -3566.3
1.000 1.000 -33,338.8 -38,821.9 6129.1 -488.8 -157.2 -3016.1 162.5 67.108 -2650.5 578.3 139.5 -3536.5
p
1.200 1.000 -33,343.6 -38,811.0 6127.2 -498.2 -161.6 -2980.3 165.0 67.284 -2613.3 582.7 139.9 -3506.6 Q
1.400 1.000 -33,348.1 -38,799.4 6124.7 -507.5 -165.9 -2944.1 167.2 67.454 -2575.7 586.7 140.2 -3475.8 o
-a.
1.600 1.000 -33,353.6 -38,788.6 6121.4 -516.5 -170.0 -2909.1 169.0 67.629 -2539.1 590.3 140.6 -3445.3
~o
1.800 1.000 -33,359.1 -38,777.4 6117.5 -525.3 -173.9 -2873.9 170.3 67.806 -2502.5 593.4 141.0 -3413.7
2.000 1.000 -33,364.6 -38,766.0 6113.0 -533.8 -177.7 -2838.8 170.3 67.976 -2465.9 595.3 141.3 -3380.0
?>
0.000 0.800 -33,379.4 -38,939.7 6122.3 -430.8 -131.1 -3175.0 145.5 65.885 -2816.5 552.8 137.0 -3659.7 ~
ji;
0.332 0.800 -33,385.5 -38,922.6 6128.8 -451.0 -140.7 -3099.8 155.5 66.191 -2738.4 566.4 137.6 -3605.0
0.668 0.800 -33,393.6 -38,905.6 6132.2 -470.2 -149.7 -3026.8 162.8 66.509 -2662.3 577.3 138.2 -3547.8 9-
s::
1.000 0.800 -33,401.3 -38,888.6 6134.3 -488.8 -158.3 -2953.2 168.8 66.822 -2585.8 587.0 138.7 -3487.8 !4
.P'
1.332 0.800 -33,410.7 -38,872.3 6134.5 -506.6 -166.3 -2881.4 173.6 67.129 -2511.0 595.5 139.3 -3427.5 ?>
1.668 0.800 -33,421.8 -38,855.8 6131.6 -523.8 -173.8 -2811.3 177.2 67.441 -2437.9 602.7 139.9 -3366.6
2.000 0.800 -33,432.5 -38,839.1 6128.0 -540.4 -181.0 -2740.7 178.2 67.749 -2364.3 607.3 140.4 -3300.6
oe:..
0.000 0.600 -33,439.0 -39,003.8 6118.0 -423.6 -129.5 -3149.7 145.2 65.506 -2790.8 553.0 136.3 -3635.1
0.286 0.600 -33,445.8 -38,991.8 6127.6 -443.0 -138.6 -3075.4 .155.4 65.796 -2713.5 566.9 136.8 -3581.9
0.570 0.600 -33,453.1 -38,980.0 6136.1 -461.9 -147.3 -3001.4 163.1 66.085 -2636.5 578.1 137.2 -3524.4
0.856 0.600 -33,461.1 -38,968.1 6142.8 -480.2 -155.6 -2928.2 169.5 66.368 -2560.3 588.2 137.7 -3465.6
f8-
1.144 0.600 -33,470.8 -38,956.9 6147.7 -498.1 -163.5 -2856.7 174.9 66.651 -2485.8 597.3 138.1 -3406.7 ~
1.430 0.600 -33,480.9 -38,946.2 6151.9 -515.4 -171.1 -2785.6 179.4 66.935 -2411.7 605.4 138.6 -3346.5 txl
1.714 0.600 -33,490.9 -38,935.3 6155.1 -532.4 -178.4 -2714.3 182.6 67.212 -2337.5 612.2 139.0 -3284.1
2.000 0.600 -33,501.7 -38,924.3 6157.0 -548.9 -185.4 -2643.9 183.3 67.476 -2264.0 616.5 139.4 -3217.9
I·
Table 9. (Continued.) trl
~
0.000 0.400 -33,500.6 -39,071.0 6113.9 -415.9 -127.6 -3126.6 143.5 65.138 -2767.2 551.8 135.6 -3610.1 ~
0.250 0.400 -33,507.9 -39,063.6 6127.0 -434.9 -136.4 -3052.7 154.0 65.405 - 2690.3 565.9 136.0 -3557.7 ~
0.500 0.400 -33,515.8 -39,056.7 6139.5 -453.6 -144.9 -2979.4 161.9 65.661 -2614.0 577.4 136.3 -3501.6 o
0.750 0.400 -33,524.0 -39,050.0 6151.2 -472.0 -153.2 -2906.3 168.6 65.930 -2537.9 587.8 136.7 -3443.5 ft
1.000 0.400 -33,533.0 -39,044.0 6162.2 -490.0 -161.2 -2834.1 174.5 66.186 -2462.8 597.3 137.7 -3384.9 0'
...,
1.250 0.400 -33,541.6 - 39,037.4 6172.6 -507.8 -169.0 -2761.4 179.5 66.443 -2387.1 606.0 137.4 -3324.2 (")
1.500 0.400 -33,551.6 -39,032.2 6182.5 -525.3 -176.7 -2690.3 183.6 66.685 -2313.0 613.7 137.7 -3263.5 ~
1.750 0.400 -33,561.3 -39,026.5 6191.8 -542.6 -184.1 -2618.7 186.6 66.942 - 2238.5 620.3 138.0 -3200.4 :1
2.000 0.400 -33,572.2 -39,021.8 6200.7 -559.7 -191.4 -2548.4 187.0 67.184 -2165.1 624.4 138.4 -3133.9 ~
0.000 0.200 -33,564.6 -39,141.5 6109.5 -407.4 -125.3 -3105.9 140.3 64.740 -2746.0 549.1 134.8 -3584.9
0.222 0.200 -33,572.7 - 39,138.9 6126.5 -426.3 -134.0 -3032.7 151.0 65.002 -2669.8 563.4 135.1 -3533.7 ~
0.444 0.200 -33,580.8 -39,136.5 6143.2 -445.1 -142.5 -2959.6 159.1 65.238 - 2593.8 575.2 135.4 -3478.3
0.668 0.200 -33,589.4 -39,134.4 6159.5 -463.6 -150.8 -2886.9 166.2 65.488 -2518.1 585.9 135.7 -3421.2
0.890 0.200 -33,598.4 -39,133.4 6176.2 -482.0 -159.1 -2814.7 172.4 65.711 -2442.9 595.7 136.0 -3363.2
1.110 0.200 -33,606.4 -39,132.1 6193.2 -500.3 -167.2 -2741.5 177.8 65.948 - 2366.8 604.8 136.2 -3302.9
1.332 0.200 -33,615.3 -39,131.3 6209.8 -518.6 -175.3 -2669.2 182.5 66.185 -2291.4 613.1 136.5 -3242.0
1.556 0.200 -33,624.6 -39,130.7 6226.2 -536.8 -183.3 -2597.3 186.3 66.401 -2216.5 620.6 136.7 -3180.2
1.778 0.200 -33,634.1 -39,130.9 6243.1 -555.0 -191.3 -2525.5 189.1 66.618 -2141.7 627.0 137.0 -3116.4
2.000 0.200 -33,643.9 -39,131.6 6260.2 -573.2 -199.4 -2454.1 189.3 66.834 -2067.4 630.8 137.2 -3048.7
0.000 0.000 - 33,631.6 -39,215.7 6104.6 -398.0 -122.5 -3088.0 134.4 64.353 -2727.7 543.7 134.0 -3557.8
0.200 0.000 -33,640.2 - 39,217.4 6125.3 -417.0 -131.1 -3015.4 145.3 64.584 -2652.1 558.2 134.3 -3507.7
0.400 0.000 -33,648.5 -39,219.2 6146.3 -435.9 -139.7 -2942.5 153.6 64.802 - 2576.2 570.2 134.5 -3452.8
0.800 0.000 -33,665.9 - 39,225.3 6189.8 -473.6 -156.8 -2797.4 167.4 65.238 -2425.2 591.2 134.9 -3338.6
1.000 0.000 -33,674.6 -39,229.1 6212.5 -492.6 -165.4 -2724.9 173.2 65.449 -2349.7 600.6 135.1 -3279.6
1.200 0.000 -33,683.3 - 39,233.4 6235.7 -511.7 -174.0 -2652.4 178.3 65.654 -2274.2 609.4 135.3 -3219.4
1.400 0.000 -33,692.1 -39,238.2 6259.7 -530.9 -182.7 -2579.9 182.7 65.866 -2198.7 617.4 135.5 -3158.0
1.600 0.000 -33,700.6 - 39,243.2 6284.6 -550.4 -191.6 -2507.2 186.3 66.070 -2123.0 624.7 135.7 -3094.9
1.800 0.000 -33,709.5 -39,249.0 6310.0 -570.0 -200.6 -2434.9 188.8 66.262 -2047.7 630.8 135.8 -3030.4
tv
2.000 0.000 -33,717.8 - 39,254.6 6336.6 -590.0 -209.7 -2361.9 188.8 66.439 -1971.7 634.5 136.0 -2960.8 tv
-
222 G. Ottonello, A. Della Giusta, A. Dal Negro, and F. Baccarin
6000.00
4000.00
2000 .00
0.00 12 0 0
-2000.00
4000.00
2000 .00
0.00
-2000.00
- 4000.00
1 5 00
Fig. 4. Molar Gibbs free energy of mixing terms predicted by interionic potential calcula-
tions for the binary joins (a) Mg2Si206- CaMgSi206' (b) Fe2Si206-CaFeSi206' (c)
Mg2Si206-Fe2Si206' compared with experimental evidences (a, b) and theoretical esti-
mates (c). Values are relative to different T (K) conditions and P = 1 bar.
Energy Model for C2/c Pyroxenes 223
0.00
-5000.00
-10000.00
-15000.00
-20000.00
join Mg2Si206-Fe2Si206
Gmixing O!mole)
o ideol mixing (2 sites)
* structure-energy model
- 2 5000.00 -t-rTTTTT"T-rrrrrTTTTTTTrlrTT"TTTTTTTT"T-r-r-r-rrTTTTT"T""T"T"1'-'-'-ro
C 0.00 0 .20 0.40 0.60 0.80 10C
Fig. 4. (Continued.)
We see in Table 7 that the adoption of forms (48) or (49) does not imply any
substantial modification of the WH or Wv terms, but results in different Ws.
Moreover, we report that attempts to fit Eq. (49) to the binary joins Mg 2Si 20 6-
CaMgSi 20 6 and Fe2Si206-CaFeSi206 gave unsatisfactory results.
The Gibbs free energy of mixing terms obtained by the model on three
binary joins of the cpx quadrilateral are compared in Fig. 4 with experimentally
derived and/or estimated values. The agreement between the predicted and
observed values for the joins Fe2Si206-CaFeSi206 and Mg2Si206- CaMgSi206
is rather impressive [see Figs. 4(a) and (b)]. The molar Gibbs free energy of the
mixing terms depicted by the model for the join Mg2Si206- Fe2Si206 at P = 1
bar conditions is essentially that of an ideal mixture with negligible asymmetry
[Fig. 4(c)]. For the remaining three binary joins in the pyroxene quadrilateral,
there is no direct experimental information, only estimates (Saxena et aI., 1986).
The highest bias between model and estimated values is observed for the join
Mg2Si206- CaFeSi206 at low T (see Table 7). In Fig. 5 we see stereographic
representations of the Gibbs free energy of mixing terms within the quadrilateral
at various P, T conditions based on the structure-energy calculations. As we see,
the Gibbs free energy of a mixing surface is not regularly upward-concave,
as would be expected from a Kohler-type combination of binary interaction
~
""'\
""'\
m
....
~ ~ <b I
() I\) .......
() ~
~ ~ ~
~~ ......,~
\.... '()
I
.... I
~~ ~ 18
~ ~
~ ....~ I
~~ I
~#> ~
~ t\
....~
I
~I~~
q§b
c:z:," structure-energy model
T=500 'C P=1 bor T=1000°C P=1 bor
Fig. 5. Molar Gibbs free energy of mixing terms depicted by the model within the pyroxene quadrilateral at various T, P conditions (a-c). All data
are relative to the C2/c structure. In (d) the results of the Kohler model at T = lOOO°C, P = 1 bar, obtained by adopting the WH , Ws , Wy binary
interaction terms of Table 7 and disregarding ternary and quaternary terms, are represented for comparative purposes. The compositional coordinates
are those of Table 9.
Energy Model for C2/c Pyroxenes 225
c...
10
..0
/I
(L
U
o
/I
%, I-
e
I
(])
""0
0' o
~ E
0' c...
(])
CD
""0
o '-
E 10
...0
>.~
Ol
'- lSI
~ ~
(]) /I
I(L
(])
'- £.J
.3()lSI
lSI
:J ill
96'tl-6- tB'6i:CL- S9'LtLt:J- es'.98tl!e- '- ..-
.....> II
(Slow//) X}W D (J) I-
226 G. Ottonello, A. Della Giusta, A. Dal Negro, and F. Baccarin
0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00
1.00 1.00
0.80 0.80
0.60 0.60
0.40 0.40
0.20 0.20
0.00 0.00
0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00
Fig. 6. Topographic representation of the molar Gibbs free energy of mixing terms
depicted by the model at T = l000°C , P = 1 bar. Compositional coordinates as in Fig.
5. The heavy line is the miscibility gap limb observed at the same T, P conditions by
Davidson and Lindsley (1989).
0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00
1.00 1.00
0.80 0.80
0.60 0.60
0.40 0.40
0.20 0.20
Fig. 7. Effect of pressure on the molar Gibbs free energy of mixing of C2/c pyroxenes
in the quadrilateral. Dashed lines are relative to T = 1500 C, P = 1 bar conditions; D
solid lines are relative to T = 1500 C, P = 40 Kbar. The dashed area encompasses
D
The joins CaMgSi z0 6 -CaAl zSi06 and CaMgSi z0 6 -NaAISi z0 6 have been care-
fully investigated by various authors due to the importance of Tschermak's and
jadeiitic molecules in natural C2/c pyroxenes. Figure 8 shows calculated vol-
umes along the join CaAl zSi06-CaMgSi z0 6 compared with various experimen-
tal results. According to the structural simulation, the excess volume of mixing
is virtually negligible, whereas the experimental data seem to indicate a certain
negative deviation from ideal in the CaMgSi z0 6 -rich region. The discrepancies
between the various experimental data at parity of composition are essentially
identical in magnitude to the discrepancy between model and experiment, and
no definite conclusion can be reached. In Fig. 9 we see that the agreement
between structural predictions and experiments is quite satisfactory for the join
NaAISi z0 6 -CaMgSi z0 6 . This fact is remarkable due to the volumetric change
involved in the binary. The mixing properties of CaMgSi z0 6 -CaAl zSi06 based
on the calorimetric data of Newton et al. (1977) and the analysis of Gasparik and
Lindsley (1980) of the phase equilibrium data were converted by Ganguly and
Saxena (1987) to a subregular Margules formulation (Table 7). Adopting their
formulation and the end-member parameters listed in Tables 4 through 6, we
calculated 18 distinct compositions to constrain the interionic potential calcula-
tions. The repulsive energy was parametrized at various p factors. The best
results were obtained with p = 0.25. As we see in Fig. 3, the value of p is tightly
constrained and the mean error is higher than the corresponding error for the
228 G. Ottonello, A. Della Giusta, A. Dal Negro, and F. Baccarin
v (em3/mole)
67r-------------------------------------------.
66
65
64
63~--~--~----~--~--~--~L---~---L--~--~
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
X CaMgSi206
model prediction -t- Newton et al.(1tl77) ""*- Doroshev et al.(1987)
o Wood et al.(1978) x Clark et al.(1962)
Fig. 8. Structure model predictions for the join CaAI 2 Si06-CaMgSi 2 0 6 , compared with
various experimental sources.
v (em3 / mole)
67r----------------------------------------------.
66
65
64
63
62
61
60~--~--~----~---L----~--~--~----~---L--~
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
X CaMgSi206
~ model prediction + Wood et al. (1980) ""*- Wood et al. (1980)
Fig. 9. Structure model predictions for the join NaAISi 2 0 6-CaMgSi 2 0 6 , compared
with the results of Wood et al. (1980). Crosses indicate two natural samples.
Energy Model for C2jc Pyroxenes 229
8000.00
4000.00
-4000.00
-8000.00
join CaMgSi206-CaAI2Si06
Gmixing U/mole)
o Ganguly and Saxena (1987)
* structure-energy model
- 12000.00 -rrTTTTTT""TT"rTTTTTTTTTr1rTT'rrTTTTTTT....-nrTT"TTTTTT"TT1rTTT1
0.00 0.20 0.40 0.60 0.80 1.00
Fig. 10. Molar Gibbs free energy of mixing terms predicted by the model for the join
CaMgSi z0 6-CaAl zSi06 at various T (K) conditions (P = 1 bar), compared with the
subregular Margules formulation of Ganguly and Saxena (1987).
cpx quadrilateral at the best-fit condition. The energies obtained via interionic
potential calculations were then converted to a subregular macroscopic model,
as was previously done for the cpx quadrilateral (Table 7).
The calculated Gibbs free energy of mixing terms is shown in Fig. 10, where
it is compared with the subregular formulation of Ganguly and Saxena (1987).
As we see, there are marked discrepancies at low T in the CaAl 2 Si06 -rich region,
while agreement is better at a high temperature.
The mixing properties in the binary CaMgSi 2 0 6- NaAISi 2 0 6 have been in-
vestigated by various authors (Ganguly, 1973; Wood et aI., 1980; Holland, 1983;
Gasparik, 1985; Cohen, 1986). Gasparik (1985) based on his own phase-equilib-
rium study and on the data of Holland (1983) proposed a Redlich-Kister
formulation for macroscopic interactions as follows:
AO = 12,600 - 9.45 T (J/mol), (51a)
Al = 12,600 - 7.60 T, (SIb)
A2 = -21,400 + 16.2 T, (SIc)
Gmixexc = X(1 - X)[AO + Al(2x - 1) + A2(2x - 1)2], (52)
with X = X(NaAISi 2 0 6 ) .
Cohen (1986), based on the calorimetric data of Wood et al. (1980) and phase-
230 G. Ottonello, A. Della Giusta, A. Dal Negro, and F. Baccarin
10000.00
5000.00
0.00
-5000.00
-10000.00
join CoM9~i206-NoAISi206
Gmixing \j/mole)
o Gosponk (1985)
* structure-energy model
-15000 .00 ~~~~~nn. .~"no~no~nonnTTnn"nnTn
0.00 0.20 0.40 0.60 0.80 1.00
a
8000.00
4000.00
....
••••••••_ ~o ~ •••:
~~oo--/-
join CoM9Si206-NoAISi206
Gmixing U/mole)
o CQhen (1986)
* structure-energy model
-1 6000.00 +rTTTTTl:-rrrTTTTTTTl:-rrrTTTTTTTTTTlrTTrTTTTTTTl--rrrTTn
LOC
b
Fig. 11. Molar Gibbs free energy of mixing terms predicted by the model for the join
CaMgSi 2 0 6 - NaAlSi 2 0 6 at various T (K) and P = 1 bar, compared with the (a) Van
Laar formulations of Gasparik (1985) and (b) Cohen (1986).
Energy Model for C2/c Pyroxenes 231
The ultimate goal of the model is the prediction of energy of the C2/c pyroxenes
in a chemical system representative of the actual chemical complexity of the
natural phases. The model must then include ferric (NaFeSi 2 0 6 ), chromian
(NaCrSi 2 0 6 ), manganoan (CaMnSi 2 0 6 ), and titanian (CaTiAI 2 0 6 ) compo-
nents, that normally occur as "minor" components in natural C2/c pyroxenes.
Only for two of the above listed components (i.e., acmite NaFeSi 2 0 6 and
johannsenite CaMnSi 2 0 6 ) are jj~ p enthalpy data available (and we still have
some doubts about the value assig~ed to acmite; see Table 4), whereas the jj~ p r
r>
between repulsive parameters, ionic radii, and other atomistic properties of the
substituting ions. It has, in fact, been evident since 1931 that the "short-range"
interactions derived through quantum-mechanical calculations for closed-shell
atoms can be expressed as exponential functions of the distance, i.e.,
-Yijrij )
~REP = b exp ( -----a;;- , (54)
where ao is the Bohr's radius and Yij is a short-range interaction constant that
was shown by Zener (1931) to be related to the ionization potentials of the
substituting ions (Il' 12 ) through
Moreover, as shown by Della Giusta et al. (1990), the radius of the ion in the
crystal (ri ) is related to its ionization potential (Ia through screen constants
(C1,C2) as follows:
ri = [(C3.~~aW4)}4/4. (57)
Bokreta and Ottonello (1987) have shown that in garnets the b factors of end-
member components are linearly related to ionic radii and free ion polarizabi-
lities on sites X and Y
(58)
As for Eq. (57), relation (58) can be converted to a simple linear dependency on
rio Indeed we see in Fig. 12 that such a simple relationship does exist for C2/c
pyroxenes and is analytically represented by
b = 3.0044 + 0.5686riM, - 0.6326riM2 + O.l589riT • (59)
Relation (59) that is based on repulsive radii coincident with the ionic radii of
Shannon (1976) (and not on the optimized radii of Table 8) leads to a I:X 2 of
0.0158 on eight values. The maximum error is observed for NaFeSi 2 0 6 (210
kJ/mol). The mean error is 76 kJ/mol. Because of these large approximations, the
values listed in Table 4 for NaCrSi 2 0 6 and CaTiAl 2 0 6 should be considered
preliminary at present.
Besides the immediate utilization of relation (59) for predictive purposes, we
emphasize that the system of Eqs. (54-59) indicates that the short-range interac-
tions can be reconverted to a discrete set of parameters, each one singularly
dependent on the interionic distance, has great heuristic validity, and should be
further investigated to assess the energetics of trace components.
Energy Model for C2/c Pyroxenes 233
b calculated
3,0.--------------------------------------------.
+
2,9
2,8
2,7 +
2,6~----------~----------~----------~----------~
Let us assume that the predicted Ht,Pr values of NaCrSi 2 0 6 and CaTiAl 2 0 6
are sufficiently accurate. We want now a set of internally consistent p, b, r factors
capable of reproducing the energetics of C2/c pyroxenes in the Na-Mg-Ca-
Mn-Fe-AI-Cr-Ti-Si-O space. We have already seen that the cpx quadrilateral
is optimized at p = 0.5, while the binaries CaMgSi 2 0 6 -NaAISi 2 0 6 and
CaMgSi 2 0 6-CaTiAI2 0 6 need a hardness factor of 0.25. One would be tempted
to follow the suggestions of Gilbert (1968) and assign to each atom its own
hardness factor (see, e.g., Miyamoto et ai., 1982). In this way, the number of
variables would become sufficiently large to fit whatever energetic term is in
the whole compositional space (and even more I). We prefer, however, to re-
tain the Huggins-Mayer formulation that proved so satisfactory for the cpx
quadrilateral.
For this purpose, it is sufficient to limit the compositional space for minor
components within the molar proportions observed in the natural phases. The
results of the extended calculation and new compositional limits are shown in
Table 8(d) and in Table 9 are listed energy values obtained within the pyroxene
quadrilateral (all relative to the C2/c structural form). In evaluating the extended
model, we assumed that acmite and ureyite behave similarly to jadeite in their
mixing properties with the diopsidic components (according to Popp and
Gilbert, 1972; acmite and jadeite mix ideally). We assumed, moreover, that
234 G. Ottonello, A. Della Giusta, A. Dal Negro, and F. Baccarin
Summary
Acknowledgments
This work was supported by MURST project "Crystal chemistry and Thermo-
dynamics of Minerals."
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sur la connaissance affinee des structures cristallines. C.N.R.S. M em. 69, 1-206.
Wagman, D.D., Evans, W.H., Parker, V.B., Halow, I., Bailey, S.M., and Schumm, R.H.
(1981). Selected values of chemical thermodynamic properties. Nbs Tech. Note, 270-8,
pp.I-134.
Wood, B.1., Holland, T.1B., Newton, R.C. and Kleppa, O.J. (1980). Thermochemistry of
jadeite-diopside pyroxenes. Geochim. Cosmochim. Acta 44, 1363-1371.
Zener, C. (1931). Interchange oftranslational, rotational and vibrational energy in molec-
ular collisions. Phys. Rev. 37, 556-569.
Chapter 8
Practical Problems in Calculating
Thermodynamic Functions for
Crystalline Substances from Empirical
Force Fields
C.M. Gramaccioli and T. Pilati
Introduction
The correlation between the structure and thermodynamic properties of min-
erals has long been considered fundamental. Besides the first-law functions,
it is possible to calculate entropy and free energy for a pure ideal crystal if
the vibrational properties are taken into account. These calculations imply
the precise evaluation of spectroscopic data (Raman, IR, phonon dispersion
curves) and allow good estimates of elastic properties and atomic displacement
parameters.
Whereas the improvement in computer design strongly favors development
in this field, there are still several difficulties with respect to general application
of these calculations. There is, however, evidence that the difficulties may be
overcome, provided adequate simplification in computing techniques is consid-
ered. In this chapter, we show that close relationships can be observed between
a Raphson-Newton process of energy minimization and the calculation of
vibration frequencies; based on these techniques, it is possible to check the
nature of a minimum and its physical significance. Thus, we can come much
closer to a prediction of the entropy and free energy of a crystalline substance.
Crystallographic information (including optical data) has long been consid-
ered a collection of the "essential" physical properties of a certain mineral. This
happens especially because of the well-known traditional connections between
crystallography and mineralogy: For a great majority of minerals, there are
accurate measurements of unit-cell parameters and X-ray powder diffraction
data, together with the values of the refractive indices and the orientation of the
indicatrix. For the most important species, these properties (including atomic
coordinates, displacement parameters, and other information obtained from
careful crystal-structure determination and refinement) have been investigated
in a substantial number of specimens by different authors.
On the other hand, in general, the amount of crystallographic information
240 CM. Gramaccioli and T. Pilati
available in the literature can hardly be compared with the relative scarcity of
reliable measurements of thermodynamic properties. The latter are, however,
continuously gaining in attention from mineralogists; for petrologists they are
even more important for immediate use than the crystal-structure data. At the
same time, spectroscopic information is also increasing in importance; for these
reasons, in the last several years less unilateral knowledge of the physical-
chemical properties of minerals has been developing.
Since all these data of a different nature for a certain mineral must be inter-
connected, the derivation of mutually consistent thermodynamic, spectroscopic,
or elastic data from the crystal structure on a routine basis from a few parame-
ters of general use would be a fundamental achievement; another very important
(and closely related) problem is deriving the thermodynamically stable crystal
structure under certain conditions, and possibly reproducing all the details,
including thermal parameters, within the range of the experimental error. All
this would emphasize the importance of crystallography in earth sciences, in a
new role.
For some of these problems, or at least for some aspects of the situation,
well-known solutions have already been available for a long time. For instance,
there are satisfactory models (and computer programs) for deriving the stable
crystal structure, starting from atomic charges and empirical atom-atom inter-
action parameters (see, e.g., Busing, 1981; Catlow and Mackrodt, 1982; Matsui
and Matsumoto, 1982; Busing and Matsui, 1984; Matsui and Busing, 1984a;
Parker et aI., 1983; Price and Parker, 1984; Price et aI., 1985; Burnham, 1990);
most of these models and programs are based on an evaluation of the first
derivatives of the crystal energy E with respect to atomic coordinates and
unit-cell parameters. There are also good models for predicting the heat of
formation for entire groups of compounds with surprising accuracy if their
structure is known and similar results can be obtained for elastic constants (see,
e.g., Vieillard, 1982; Catti, 1982, 1989; Matsui and Busing, 1984a, b; Ottonello,
1986; Au and Weidner, 1986). The Debye theory with adequate modifications is
a "classic" example for deriving the specific heat or other temperature-dependent
properties of many crystalline substances. In several cases, these calculations
have also been successful in eliminating substantial experimental errors, or in
providing a useful interpolation of experimental data (see, e.g., Kieffer, 1985).
However, in general, these procedures are not fully consistent with each other,
and even when they are theoretically, it is often impossible to formulate a
reasonable assumption about the value of corresponding parameters for differ-
ent minerals; similarly, in general, it is difficult to be confident about the success-
ful use of the same parameters in calculating different properties of a particular
mineral. For instance, the Debye model implies at least one empirical parameter
Practical Problems in Calculating Thermodynamic Functions 241
to be used for a certain substance only, and this parameter should be found only
a posteriori on a "best-fit" basis to the experimental data unique to this sub-
stance. Similarly, wide differences in the atomic charge and VFF parameters
(such as bond-stretching constants) or in repulsive short-range terms are en-
countered, depending not only on the particular mineral, but also the author,
model, and/or particular properties that are the main thrust of the work.
In deriving consistent models, the same force field should be employed in
different kinds of calculations, and the greater number of applications could
provide a substantially wider basis for deducing the necessary empirical con-
stants on best-fit grounds. For instance, the energy minimum criterion could be
improved by verifying whether its "walls" are consistent with a real minimum of
the energy function, and with the experimental values for the vibrational fre-
quencies of the crystal. All this implies abandoning the Debye theory in favor of
more complex models, mainly based on Born-von Karman's lattice-dynamical
procedure (see, e.g., Born and Huang, 1954; Cochran, 1973; Willis and Pryor,
1975; Ghose, 1985). This procedure is based on the harmonic approximation and
invariance of the equations of motion for appropriate translations within the
crystal. The calculations can be brought to the following eigenvalue-eigenvector
form:
(1)
where w( = 21tv) are the angular frequencies, and D(q) is the dynamical matrix
for a certain value of the wave vector q. For q = 0, most of the vibrational
frequencies (v's) correspond to IR- or Raman-active vibrational modes, and a
check with experimental results is relatively easy. For nonzero values of q, the
experimental phonon dispersion curves can be used instead for comparison.
However, very few experimental data of this kind have been obtained so far (e.g.,
by inelastic neutron scattering).
The elements of D(q) are given by
(2)
where
(3)
(see, e.g., Willis and Pryor, 1975, Eq. 3.lOb). Here E is the potential energy, x& is
a coordinate of the kth atom in the unit cell; Xjk' is a coordinate of the k'th atom
in the crystal, related to x~, by a lattice translation r' (l = 0 when r' = 0), ~r' is
the distance between the two atoms involved, mk and mk , are the masses of the
atoms k and k', respectively. The summation 1:, is extended (in principle) to all
the translated units in the crystal.
By introducing mass-weighted coordinates
(4)
242 C.M. Gramaccioli and T. Pilati
(6)
Since for a crystal, the contributions to ~I are nonzero only at definite points in
the lattice, the elements of D(q) are the Fourier transforms of the cp;~k' (0, I)/V,
where V is the volume of the crystal
Let us consider the nature of cpW (0, I) or cp;~k' (0, I): They are the second
derivatives of the crystal energy with respect to the atomic coordinates or
mass-weighted atomic coordinates, respectively. These second derivatives can be
obtained like the corresponding first derivatives, starting from an adequate
model, and the same empirical parameters could be used in both static and
dynamical calculations. Because they can be derived from more general princi-
ples on an atom-atom basis, these parameters are no longer of one substance
only and should be capable of transfer from one mineral to another. A notable
result of this kind for minerals has been achieved by Price et al. (1987a, b) for the
natural polymorphs of Mg 2 Si04 (forsterite, wadsleyite, ringwoodite) and by
Lam et al. (1990) for forsterite.
The use of consistent models for both the vibrational and static properties of
crystals would imply definite advantages also in the range of the application and
in the quality of the results. For instance, in agreement with the temperature-
dependent character of thermodynamic functions, the vibrational contribution
Ev to the total energy is the following:
where Vi is the frequency of the ith mode; g(Vi) is a normalized density of states
function [~ig(vJAvi = nf' where nf is the number of degrees of freedom consid-
ered in the calculations for the formula unit], and Eo is the zero-point energy:
(9)
Practical Problems in Calculating Thermodynamic Functions 243
where hand k are the Planck and Boltzmann constants, respectively. Therefore,
the calculated value for energy becomes temperature-dependent, even in the
harmonic approximation, and the zero-point term can also be evaluated. Con-
trary to widespread belief, this term is not quite negligible: For instance, accord-
ing to our calculations, it amounts for forsterite to 14.1 kcal mol-t, to be
compared with a total vibrational energy of 18.2 kcal mol- 1 at room tempera-
ture. Therefore, the packing energy minimum without any additional vibrational
contribution does not correspond to the structure at 0 K, nor to any physically
significant counterpart.
For entropy the vibrational contribution is the following:
S= T
EVib - [(hVi)J
kT Avi·
3R~ig(vJln 1 - exp (10)
CV = 3R~ig(Vi) (k;
hV.)2 exp (hV.) k; - 1J-2 Avi·
k; [exp (hV.) (11)
(12)
where ep,i is the mass-adjusted polarization vector for the atom p [which is part
of an eigenvector g of the dynamical matrix D(q)], n the total number of unit
Ei
cells in the crystal, and the average energy of the ith mode
This evaluation of the U's may be important not only for the "academic" exercise
of reproducing the details of crystal structure and discussing their accuracy, but
also for a general check of the behavior of the vibrational model through the
entire Brillouin zone, and not at a certain point only, as for Raman- and infrared-
active frequencies. Here, a further motive for using crystallographic information
in connection with other properties is evident: For a detailed discussion of this
point, see, e.g., Willis and Pryor (1975) or Gramaccioli (1987).
Since most minerals are not pure compounds, one might wonder whether this
effort of calculating properties is indeed useful for wide application, e.g., the
244 C.M. Gramaccioli and T. Pilati
quently suggested [Choudhury et al. (1989)]. All these procedures are substan-
tially more difficult to deal with, not only because of their complexity, but also
because they require the introduction of a number of additional empirical con-
stants, many of which are hardly significant. However, although the rigid-ion
model does not interpret the dielectric properties of the crystal, it can neverthe-
less give good to excellent results in interpreting the vibrational frequencies of
important minerals (see, e.g., Elcombe, 1967; Iishi, 1978b, c; Iishi et aI., 1979). As
an example, a comparison of the experimental Raman-active vibration fre-
quencies for forsterite with the results of various calculations is given in Table 1:
In spite of its greater complexity, the shell model (of Price et aI.) does not yield
decidedly better agreement with experimental data than the rigid-ion model
Table 1. Examples of vibrational frequencies (cm -1) for forsterite at the f -point (for the
complete set, see Pilati et aI., 199Oc). In the first column, the range of experimental
measurements obtained by different authors is reported; the following columns, in
sequence, report the calculated values ofIishi (1978b), Pilati et al. (199Oc), Price et al.
(1987a), and Rao et al. (1988).
used by the other authors. Also the values of thermodynamic functions afforded
by the rigid-ion model are quite satisfactory; an example for forsterite is given in
Table 2.
In building up dynamical matrices to be diagonalized, Coulombic interaction
between charged atoms is another difficult problem to be accounted for ade-
quately, even with the simplest possible models like the rigid-ion one. This
happens because of the very slow convergence of atom-atom interaction energy
on increasing the maximum interaction distance; for this reason, summations are
generally carried out on the reciprocal lattice, following a well-known procedure
devised by Ewald (1921) in connection with the evaluation of Madelung con-
stants. This procedure was extended to the evaluation of Fourier-transformed
second derivatives for dynamical matrices in a correct way for the first time by
Kellermann (1940). Although the papers just cited are "classic" and well estab-
lished, the application to the general case involves considerable complexity, both
in understanding the principle and in applying it to practice.
Another basic difficulty is the present virtual lack of empirical atom-atom
energy fuctions that can be applied in general. For static energy calculations, the
best-fit values for the atomic (ionic) radius and charge are often sufficient to
obtain a reasonable value of packing energy and to have the actual crystal
structure reproduced satisfactorily; the situation is, however, much more delicate
when the same functions are tested on vibrational grounds.
unit cell and to the unit-cell parameters. As we have seen [see Eqs. (2,3,5-7)],
vibrational calculations (in the harmonic approximation) are instead essentially
based on second derivatives of energy with respect to atomic positions. The
evaluation of the whole set of second derivatives surely is a much longer proce-
dure than evaluating first derivatives; however, this brings definite advantages,
even if calculations are performed only for a static model. Especially if the
potential is obtained from the best fit of non vibrational data, there is, in fact, no
reason why the second derivatives for a certain minimum should be compatible
with the actual vibration frequencies. The situation is not rarely still worse, the
second derivatives' matrix being nonpositive definite and leading to imaginary
frequencies or, in other words, to the instability of the calculated structure; a
similar argument can be extended to any dynamical matrix, in general. In the
practice of energy minimization, a process using first derivatives only, such as,
for instance, the steepest descent, might be definitely simpler to consider and use
than a method using also second derivatives, like the Raphson-Newton proce-
dure. However, if mass-weighted atomic coordinates are used, the eigenvalues
and eigenvectors of the second derivatives' matrix 0(0) at convergence should
correspond to the vibrational frequencies of the crystal for q = 0, i.e., the results
could be immediately compared with other easily accessible data, such as
Raman- and infrared-spectra. A possible scheme could be the following:
O(O)Ax' =- F. (14)
Rigid-Body Models
In all these procedures, the size of the matrices could be substantially reduced if
some parts of the structure are considered "rigid bodies." This can be done most
easily for molecular crystals (see, e.g., Gramaccioli, 1987), but even for ionic
crystals such as olivine (forsterite) good results can be achieved (see, e.g., Ghose
et aI., 1987; Rao et aI., 1988, and the corresponding column in Table 1). Figure
1 reports some of the lower phonon dispersion curves obtained by the authors
using this approximation (and the rigid-ion model): The agreement with experi-
mental measurements is striking.
6. 6. 62 63
.0 <0
3e 38
32 32
---.
2. 28
...i
0
II i
... '6
'2
~ '2
~ 8
",
Q/b· Q/b e
38 38
32 32
28
2<
20
'8
., '2
o 0.1 0.2 0.3 0.. 0.5 0.. 0.3 0.2 0.1 0 o 0.1 0.2 03 O. 0.5 0.. 0.3 0.2 O. I 0
Q/c· Q/c·
Fig. 1. Comparison of experimental and theoretical phonon dispersion curves for for-
sterite (after Ghose et aI., 1987, Pergamon Press pic.); continuous lines are from calcu-
lations, full circles are experimental values. The calculated curves are not fitted to
experimental data, but were predictions before the experiments were conducted.
Practical Problems in Calculating Thermodynamic Functions 249
where X;k and XJk are mass-weighted coordinates of the atoms k and k'. For a
nonlinear isolated molecule or group, there are six eigenvalues of Ds equal to
zero, corresponding to rigid translations or rotations of the group (or to their
linear combinations).
For a rigid-body model, Ms consists of six columns only (five for a linear
group), and the operation described in Eq. (15) reduces the order of D(q) to 6Z,
where Z is the number of these groups in the primitive unit cell. In view of the
degeneracy corresponding to the zero eigenvalue, a diagonalization of Ds or
similar matrices by any routine would not necessarily afford eigenvectors corre-
sponding to a pure translational or rotational motion, or if this happens, these
translations and rotations are not necessarily relative to the reference axes. For
the sake of a better interpretation of the motion, it is advisable to replace these
vectors by the following:
m1= m2 = m3= m4 = ms= m6=
1X1 0 0 0 xd321 Y 1 f331
0 1X1 0 -Zlf311 0 Xl f331
0 0 a1 Y1 f311 -X 1f321 0
, (17)
1X2 0 0 0 Z2 f322 - Y2f332
0 1X2 0 -z2f312 0 X2f332
0 0 1X2 Y2f312 -X 2f322 0
where ak = (mklm)1/2, f3ik = (mkII;)1/2 (m being the mass and Ii the ith principal
moment of inertia of the group). This normal coordinate basis has the advantage
that internal movements of the group corresponding to the lowest frequencies
(which define the paths of easiest deformation) can be easily included, just by
adding further columns to the M's.
By using this procedure, the reduction of the order of the matrix can be quite
considerable for complex organic molecular crystals, but also for minerals the
250 C.M. Gramaccioli and T. Pilati
advantage is evident. For instance, for a-sulfur, the order of the dynamical
matrix is 192 (96 for q = 0), and it can be reduced to 80(40) by taking into
account the two lowest-frequency internal modes (both twofold degenerate),
practically without affecting the results.
The use of rigid or partially rigid models does not necessarily imply neglect
of the contribution of the higher internal frequencies; these contributions (to the
values of thermodynamic functions or the Us) can be simply added to the total,
by assuming them to be constant throughout the entire Brillouin zone, with no
mixing with the lattice modes (Gramaccioli and Filippini, 1983). This approxi-
mation is reasonable on both theoretical and experimental grounds.
An example of the success of this procedure in interpreting the experimental
data for a-sulphur is given in Tables 3 (thermodynamic functions) and 4 (atomic
displacement parameters). The tensor W = <qiq) indicates the importance and
coupling of the external and lowest-frequency internal modes and is reported as
an example in Table 5.
calc obs
a The calculated value for the heat of sublimation without considering the zero-point energy
is reported within parentheses; the worse agreement with the experimental result of the
corrected value derives from not having considered this effect in deriving the van der Waals
potential (Rinaldi and Pawley, 1975).
S1 114 53 12 13 5 2
119 56 11 6 4 7
S2 107 51 16 -16 2 2
108 49 17 -6 6 -8
S3 83 67 14 3 -4 7
85 57 16 14 -3 2
S4 67 84 11 0 5 -2
74 87 11 -12 6 0
'"t)
Pl
~
n'
a
...'"t)
o
~
(1)
8
'"
Er
Table 5. The tensor W for IX-sulfur (atomic mass units x A2; referred to the principal axes of inertia of the molecule, after Gramaccioli and n
Filippini, 1983). For q7 - qs and q9 - qlO, which are the internal mode frequencies considered here, the calculated frequency for the free a
(")
~
molecule is reported in the first column.
e-
5'
ql(B) q2(A) q3(A) q4(B) qs(A) q6(A) q7(A) qs(B) q9(A) qlo(B) O<l
....,
::r
(1)
ql 6.420 0.000 0.000 -0.193 0.000 0.000 0.000 -0.119 0.000 0.035
q2 8.637 -0.743 0.000 1.207 -1.349 -0.445 0.000 0.079 0.000 Elo
p..
q3 5.486 0.000 -0.980 0.075 0.128 0.000 -0.044 0.000 '<
::l
q4 3.447 0.000 0.000 0.000 0.045 0.000 -0.054 ~
IV
Vl
252 C.M. Gramaccioli and T. Pilati
(b)
0.20 1.0
0.8
O.S
0.10
0·4 Ri9ld mol.cular ian mo~1.
RigId ,on mO~1 & K,.tf.r"s
modfl l
O.bye mO~1
o so 100
TEMPERATURE (K)
Fig. 2. Comparison of experimental specific heat data with calculated results (after Rao
et ai., 1988). (a) Low-temperature region showing good agreement of the continuous line
given by the rigid-body (molecular ion) model with the experimental data shown by the
filled circles. No fitting is attempted. (b) Comparison of experimental data (continuous
line) with results of various theoretical models as indicated in the figure. The discrepancy
between the results of the rigid-body model and those from the experiment is due to
omission of the internal modes' contribution.
Likewise for forsterite, the order of the complete dynamical matrix is 168 (84),
and it can be reduced to 48 (24), if the Si04 tetrahedron is considered rigid and
the two Mg atoms independent points. The values of the specific heat at various
temperatures calculated by Rao et al. (1988) are reported in Fig. 2: The disagree-
ment at higher temperatures can probably be substantially eliminated if the
contribution of the internal modes as specified above is introduced.
A similar procedure can be used (with evident advantage) in the Raphson-
Newton minimization. For instance, if a single independent group is present in
the crystal, for a single step we have [corresponding to Eq. (14)]:
(18)
Here, M is the eigenvector matrix (in this case, unique), and ~q is the column
matrix corresponding to the shifts in normal coordinates. These shifts can be
easily referred to mass-weighted (and Cartesian) coordinates, since
MAq = Ax'. (19)
Here, Ax' corresponds to shifts in mass-weighted coordinates, as is evident from
the above equations.
r----,r--~-----.---.--___,_--._-__,--,-
r =2
+ r=1
1. 30 (!) r=O
n
Fig. 3. Change in the estimated Beq values (A2) for the Cl atom in NaCl as a function of
the Brillouin-zone sampling (from Pilati et aI., 1990a). Each curve joins the corresponding
points belonging to the same sequence, but differing in the number n of sampling intervals
along each reciprocal axis.
254 C.M. Gramaccioli and T. Pilati
Mg(1}
0.45
Mg(2)
0.40
0(1)
g- 0.35
m
r=1
X t = 0.4
+ t =0.375
0.30 r =0 * t =0.5
Si
0.25
o
n
Fig. 4. Change of the Beq values (A 2) for various atoms in the forsterite structure at 298
K as a function of the Brillouin-zone sampling (from Pilati et aI., 1990a). The kind of
sequence is determined by the parameters rand t.
Practical Problems in Calculating Thermodynamic Functions 255
estimations of the atomic displacement parameters (B's) for alkali halides; for
thermodynamic functions, in general, a reasonable estimate can be obtained
even with a considerably smaller number of points [on the order of 20-30, or
even less (see, e.g., Price, 1987b, or Pilati et aI., 1990a)]. However, if a suitable
progression is chosen, even 100 to 200 points can be sufficient for determining
all properties within the corresponding experimental accuracy (Filippini et aI.,
1976; Kroon and Vos, 1978, 1979; Pilati et at, 1990a). Figures 3 and 4 show
convergence as a plot of the equivalent Debije-Waller factor Beq against the
number of grid spacing with several kinds of sequence for NaCl and forsterite.
In our computer programs, the above-mentioned simplifications have been
considered; moreover, satisfactory coding for the atom-atom interactions has
been developed, so that the correct contribution to dynamical matrices is ob-
tained in any case even for nonzero values of the wave vector q and in the
presence of interactions involving more than two atoms, such as bond-angle
bending and torsion. For Coulombic interactions, a new routine has been writ-
ten that can be easily understood on crystallographic terms; this routine per-
forms summations in the reciprocal lattice, following a modification of Bertaut's
method (Bertaut, 1952, 1978a, b, 1983, 1985, 1986; Pilati et at, 1990b). Besides
its simplicity in application, this routine has the advantage of considering the
contribution of the macroscopic field quite naturally, leading to the so-called
TO-LO splitting for infrared-active modes.
An Example: F orsterite
Since forsterite has so far been the best example for such calculations, the atomic
displacement parameters as calculated by Pilati et at (1990c) are here reported
in Table 6, together with the corresponding experimental values, obtained by
different authors from accurate X-ray diffraction measurements and subsequent
crystal-structure refinement. Similarly, a drawing ofthe thermal ellipsoids by the
ORTEPII program (Johnson, 1976) is reported in Fig. 5. These results show
remarkable agreement, with the exception of Hazel)'s (1976), in which, however,
considerable systematic errors are present. An interesting feature is the consider-
able importance of the zero-point contribution (see Table 7), which corresponds
to about one-half of the total mean-square displacement at room temperature.
All this confirms the validity of the rigid-ion model, at least for most purposes.
The reason why the rigid-ion model works, in spite of its approximation, can be
ascribed to the essential good agreement of the lowest calculated frequencies
with the experimental data (see Ghose et at, 1987); for the highest values (where
experimental phonon dispersion curves are not available), there might be consid-
erable differences. This is similar to what happens for alkali halides, and espe-
cially here the shell model should prove its advantage (see, e.g., Cochran, 1973,
Woods et aI., 1960, 1963, and Fig. 6). However, since in practice only the lowest
energy levels are occupied, at least for reasonable values of temperature, this
explains the good agreement for thermodynamic functions and atomic displace-
ment parameters.
256 C.M. Gramaccioli and T. Pilati
Table 6. (Continued.)
'L x
O(3)
obs. calc.
Fig. 5. An ORTEPII drawing (projection on the mirror plane {01O}) of the calculated
and observed thermal ellipsoids at 99.9% probability for the asymmetric unit offorsterite
at room temperature (from Pilati et aI., 1990c).
"
'.-
o 0
o 0
5
'1
Fig. 6. Phonon dispersion curves for sodium iodide around 100 K (after Woods et al.,
1963). The measured points are compared with calculations based on the rigid-ion
(dashed) or simple shell model (continuous lines). The broken vertical line indicates the
[110J zone boundary. The essential agreement of the two models for the lower branches
is evident.
Table 8. Frequencies (cm -1) at the r point (q = 0), corresponding to the packing energy "minimum" of Miyamoto et al. (1982) and evaluated R
e
using the same potential. JJ.
Au 394i 196i 166 207 396 534 603 712 918 1678 ~
Ag 265i 238 287 426 449 613 1001 1136 1437 1817 1874 8
195 310 331 429 493 578 683 784 922 1069 1641 1797 1862 o
Q.
Btu
'<
B 2u 336i 133 409 449 476 597 716 891 1646 ::l
I»
B3u 104 263 347 395 585 636 676 737 910 1105 1477 1808 1878 a
(=i.
BIg 489i 67i 292 563 595 1206 1691 >Tj
B 2g 133 272 378 451 524 623 1144 1214 1457 1828 1908 l::
IV
VI
IQ
260 CM. Gramaccioli and T. Pilati
according to Iishi, 1978c; see Table 1). Therefore, the lack of physical validity of
this solution is evident even when a smaller matrix is used.
References
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structure incommensurable. e.R. Acad. Sci. Ser. II 302,1137-1142.
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Chapter 9
Predictions of the Entropies of Molecules and
Condensed Matter
M. Blander and c.R. Stover*
Introduction
In this paper, we discuss a statistical mechanical theory for calculating the stan-
dard nonelectronic entropies (S~) and free energy functions [(G~ - H~98)/T] of
substances at high temperatures. These quantities are important and are often
the only unknown data necessary for determining free energies of compounds,
which is necessary for the calculation of chemical and phase equilibria. This lack
of data is particularly significant at high temperatures that are important in the
genesis of magmas and metamorphic rocks.
Accurate enthalpies of formation of compounds are relatively easy to measure
calorimetrically. In order to calculate the free energies of compounds from
enthalpies of formation at different temperatures, one needs a knowledge of the
entropies as a function of temperature. On the other hand, with a knowledge of
the free energy functions of a compound as a function of temperature, one needs
only a single measurement of the enthalpy of formation of that compound to
determine the free energies over a range of temperatures. Thus, since there are
very many substances for which such enthalpies offormation are the only known
thermodynamic data, a major expansion of standard tables of free energies of
formation is possible if one can predict entropies or free energy functions for
these substances.
In prior papers (Frurip and Blander, 1980, Frurip et aI., 1982a, b; Blander and
Stover, 1985), we have shown that the high-temperature (700 K) entropies and
* Summer 1983 Student Research Participant from Swarthmore College, Swarthmore, PA 19081.
The submitted manuscript has been authored by a contractor of the U.S. Government under
contract No. W-31-109-ENG-38. Accordingly, the U.S. Government retains a nonexclusive, royaJty-
free license to publish or reproduce the published form of this contribution, or allow others to do
so, for U.S. Government purposes.
Predictions of the Entropies 265
free energy functions of all the nonhydrogenic vapor molecules in the JANAF
tables (Chase et al., 1986; Barin and Knacke, 1973; Barin et al., 1977) and of
molten alkali halides are in good agreement with the predictions of this theory.
This means that one can predict these quantities with only a knowledge of a size
parameter and of the masses of the atoms. For molecules, it means that one does
not need to know the vibrational frequencies, which are unknown and/or diffi-
cult to measure for many high-temperature molecules. In this paper, we will
review the theory and detail the proof of a modification that increases its
accuracy (Blander and Stover, 1985; Stover, 1983). We will then explore the
potential utility of the theory for solids by comparison with data on the entropies
of alkali halides and alkaline earth oxides and sulfides. These preliminary calcu-
lations suggest that our approach could be used to make predictions of entropies
and free energy functions of solids with applications to simple solids of impor-
tance in geology being possible at present. Ultimately, it should be possible to
extend this theory to the more complex solids found in many geologic systems.
The development and extension of the theory would enable one to greatly
expand the available data on free energies of formation of compounds at high
temperatures and improve the ability to calculate chemical and phase equilibria
in geologic systems.
The standard entropy of a molecular gas containing one or two different ele-
ments A and/or B, S7, is given by the expression
s7 = {I
R +
3(nA + nB )
2
I [ZQe
+n N
n (2nmikT)
i=A,B h2
3n 2
d]
+
To
aT
In Qe
+
Td
dT
In Z}
'
(1)
where Z is the configurational integral
Z = _1_
I I
nA·nB·
f···f v
e-pudr 1 .. ·dr nA dr 1 "·dr nB' (2)
in which f3 = l/kT, U is the total potential between the atoms of the molecule
AnABnB , and dr i are volume elements in real space that contain the center of an
atom. The integral (2) is integrated over all possible configurations of the mole-
cule in the volume V. In addition, Qe is the electronic partition function, mi the
atomic mass, h Planck's constant, k Boltzmann's constant, T temperature, and
N Avogadro's number. The molecule is thus considered to be a collection of the
atoms that interact with the total potential U, which is a function of the atom
coordinates. The total potential is assumed to be the sum of the pair potentials.
In the theory, one first performs a dimensional analysis of the configurational
integral for an ionic molecule AnABnB , where A is a cation and B an anion. The
266 M. Blander and c.R. Stover
(3)
When one considers two such ionic molecules, one with an interionic size
parameter d and a second (test salt) with a size parameter do, then the relative
values of the total potential are given by
where g is a scaling parameter dido. From this, one can show that the configura-
tional integrals for the two molecules are related through the expression
(4)
where Z is the integral for any salt and Zo the integral for the test salt. In the
original paper (Frurip and Blander, 1980), it was shown that at constant V and
T, the nonelectronic entropies for a molecule with the size parameter d are given
by the expression
where (Jo(T, V) is a universal constant for all molecules with the given number of
atoms (nA + nB ), and mA and mB are the atomic masses of the A and B atoms,
respectively. A very similar expression has been shown to be valid for the free
energy function (Go - H~98)/T. A correction to the coefficient of the Ind term
was deduced (Blander and Stover, 1985; Stover, 1983) that improved the accu-
racy of the predictions for vapor molecules from the theoretical expression (5).
Using the scaling property of Z with g for fixed do as given above and writing
out (oZ/oT)g and (oZ/ogh in terms of Zd o, we can easily show that (oZ/og)T =
l/g[3(nA + nB - I)Z + T(oZ/oT)g] at constant V. By using an expression for S
in terms of Z, it follows that
[ O(S/R)J
olng Y,T =
[
3(nA + nB - a ( ----aT
OlnZ)J
1) + aT T
2
'
where the expression on the right-hand side is the coefficient of the In d term in
the equation for SIR and 0/oT[T2(0 In Z/oT)] is a correction term to this
coefficient in Eq. (5). Defining %T[T 2(o In Z/oT)] as b, we can show that b =
«puf> - <PU>2 and is a positive number equal to [(Cy/R) - 3(nA + nB )/2],
where Cy is the constant volume heat capacity. Because of the significant im-
provement in the accuracy of the predictions of relative values of entropies (and
free energy functions), it is recommended that this correction term be included
Predictions of the Entropies 267
+ [Cv(T) + 3(nA + nB ) _
R 2
3J In d. (6)
Equation (6) has been derived for ionic molecules. It was shown that the differ-
ence between such ionic molecules and realistic molecules with the same number
of atoms is equal to the differences between the constant volume heat capacities
for these two types of molecules (Frurip and Blander, 1980). At high tempera-
tures, these differences are very small for nonhydrogenic vapor molecules, in-
dicating that Eq. (6) is valid for all such molecules at high temperatures.
Successful comparisons of Eqs. (5) or (6) with data on all the molecules, clusters,
and molecular ions in the JANAF tables and on all 20 molten alkali halides
(Blander and Stover, 1985) have been made using Eqs. (5) (Frurip and Blander,
1980; Frurip et al., 1982a, b) and (6). (Blander and Stover, 1985) In this approach,
the condensed phase is equivalent to a very large molecule with the order of
Avogadro's number of atoms. For example, plots of the entropies vs. the last two
terms in Eq. (6) were made. (Blander and Stover, 1985) Calculated values of the
intercepts and standard deviations are given in Table 1 for AB 4 , AB s , and AB 6 ,
as well as for the molten alkali halides. Part of the deviations from Eq. (6) are,
of course, due to the inaccuracies in the known data. In any case, the data are
consistent with the predictions of this simple statistical mechanical equation
Intercept SD
Moleculeb 1000 K 5000 K 1000 K 5000K
AB4 17.347 38.139 0.605 0.596
ABs 15.549 41.122 0.814 0.807
AB6 12.046 42.379 1.159 1.174
2.548' 0.768'
AB(liq.) -0.338 d 0.402 d
"Taken from Blander and Stover (1985).
b AB4 = TiBr4, CCI 4, CF4, TiCI 4, PbCI4, SiF4, SiI 4, SF4, ZrCI 4, ZrI 4 , ZrBr4 , ZrF4 , PbBr4, PbI 4,
PbF4, SiCI 4; ABs = ClFs, NbCl s, PCl s, TaC1 s, 1Fs, PFs, NbBrs; AB6 = MoCI 6, WC1 6, MoF6, SF6,
WF6, WBr6; AB(liq.) = a1120 alkali halides.
cd = Interatomic spacing of the vapor molecules in A.
d d 3 = V in cm 3 , where V is the molar volume at room temperature.
268 M. Blander and c.R. Stover
with relatively small standard deviations. In some prior cases, it was shown that
free energy functions were also in accord with a similar equation. In cases in
which dimensionless entropies or free energy functions deviated by more than 2
from these predictions, it was shown that the data were incorrect. Most signifi-
cant for geologists is the fact that the entropies of molten alkali halides are
consistent with Eq. (6). This presents the possibility that data for condensed
phases might be consistent with Eq. (6). Equation (6) is valid for gaseous mole-
cules that have different types of bonding and structures, with (10 being a function
only of the total number of atoms (nA + nB) in the molecule. As will be seen
below, this universality does not appear to hold for crystalline solids. However,
our preliminary examination of some simple solids indicates that the theory
could prove to be useful when (10 is restricted to a subclass of solids of a given
stoichiometry.
We performed calculations on simple 1 : 1 mostly cubic solids to illustrate the
potential of Eq. (6). We do not have good values of Cy/R for such a calculation
in which the coefficient of the lnd term in Eq. (6) is (Cy/R + 3). The average
values of Cp/R at 1200 K for the alkali halides and for alkaline earth oxides and
sulfides are about 8.39 and 6.69, respectively. Since Cp > Cy , we will assume an
effective value of 8 for (Cy / R + 3) in our calculations on 1: 1 solids. Since the
correlations we are making are not very sensitive to this quantity, a more precise
value is not warranted at present. In Fig. 1 we exhibit a plot of the dimensionless
24.---------------------------~
22 .. Temperature
0700 K
20 01200 K
18 ......... ~ .........:........ ~
a:
o~ 16
(J)
14 ...... .; ......... ; .........:........ .
··· ... ...
·· . . . .
· . .
....... ,, ..........,......... , .................. .
12
.. ..
8;-~~~-r~~-r-r~-r~-T~~
· . .
12 14 16 18 20 22 24 26
3/2 In mArne + 8 In d
Fig. 1. Comparisons of measured entropies of solid alkali halides at 700 and 1200 K with
the predictions ofthe theory. The theory is represented by the solid lines that have a unit
slope. All 20 alkali halides are represented at 700 K, and, at 1200 K, data were not
available for six alkali halides (RbF, RbCI, RbBr, RbI, CsBr, and CsI).
Predictions of the Entropies 269
entropies of alkali halides at 700 and 1200 K vs. 3/2 In (mAm B) + 81n d, where
mA and mB are the atomic weights of the cation and anion, respectively, and d is
an interionic distance between the cation and the anion that is taken to be
proportional to the cube root of the molar volume at room temperature. The
average values of the first two terms on the RHS of Eq. (6) are - 4.266 ± 0.592
at 700 K and 0.108 ± 0.622 at 1200 K. The points for LiCl, LiBr, and LiI in these
two plots lie about 1-1.4 dimensionless entropy units above the predicted
values plotted from these intercepts [i.e., the lines exhibited in Fig. 1 with a slope
of 1 and intercepts given by the first two terms in Eq. (6)]. Because of high
anion/cation radius ratios for these three salts, the cations will have considerably
more freedom of motion in their lattice sites, which might be expected to increase
the entropies of these salts. If these three salts are excluded from the averaging,
the values of the intercepts are -4.449 ± 0.388 at 700 K and -0.181 ± 0.237 at
1200 K. We thus see that, except for LiCl, LiBr, and LiI, the alkali halides are
consistent with the predictions of Eq. (6), with a relatively small standard devia-
tion and a maximum deviation of about 0.5 dimensionless units for NaF. With
all the lithium halides included, the standard deviations are larger and the
maximum deviation is about 1.4 dimensionless units for LiI.
Analogous series of compounds are formed by alkaline earth oxides (BeO,
MgO, CaO, SrO, BaO) and their corresponding sulfides. In Fig. 2 are plotted
values of SO/R Barin and Knacke (1973); Barin et al. (1977); M.W. Chase et al.
(1986); L.B. Pankratz (1982) for these two series vs. 3/2 In mAmE + 8/3 In V, where
V, the molar volume near room temperature, is taken to be proportional to d 3 •
The average values of the intercepts are for the oxides - 6.464 ± 0.735 at 700 K
and - 3.045 ± 0.768 at 1200 K and for the sulfides -6.690 ± 0.782 at 700 K and
- 3.128 ± 0.827 at 1200 K. The closeness of the values of these intercepts for the
oxides to those for the sulfides at the same temperature suggests that these two
populations can be combined to give common intercepts of -6.577 ± 0.725 at
700 K and of -3.086 ± 0.754 at 1200 K that were used to plot the lines in Fig.
2. It should be noted that the points for BeS in Fig. 2 deviate most from the
predictions of theory. BeS has a very high anion/cation radius ratio (analogous
to LiCl, LiBr, and LiI), which can lead to higher entropies than predicted in a
manner that is similar to the lithium halides. If BeS is omitted from the calcula-
tion, the average values of the intercepts at 700 and 1200K are -6.713 ± 0.621
and -3.243 ± 0.754, respectively.
The difference between the intercepts for the alkali halides and those for these
oxides and sulfides is 2.1 at 700 K and 3.2 at 1200 K. The reasons for these
differences are not clear. A clue to part of the difference can be deduced from the
relation mentioned earlier, which to first-order terms equates deviations from
the theoretical predictions to differences in the constant volume heat capacities.
For example, at 1200 K, the differences between the average value of Cp/R for
the 14 alkali halides that are included in the plot of Fig. 2 and the average value
of Cp/R for the alkaline earth oxides and sulfides is 1.70. This is substantial but
not as large as the difference in the intercepts. Of course, the differences in Cp are
not exactly equal to the differences in Cv but are not likely to be very different.
In any case, the predicted correlations appear to be valid for each of these two
270 M. Blander and C.R. Stover
20.---------------------------~
18 .................................. .
14
a:
o'i- 12
(f)
10
o Sulfides
6
44-~~~~~~~~~~~~-+~~
10 12 14 16 18 20 22 24
3/2 In mArne + 8 In d
Fig. 2. Comparisons of measured entropies of alkaline earth oxides and sulfides at 700
and 1200 K with the predictions ofthe theory. The theory is represented by the solid lines
that have a unit slope.
Discussion
A statistical mechanical theory that leads to a simple expression for the relative
values of nonelectronic entropies (and free energy functions) of substances seems
to apply to vapor molecules, simple liquids (alkali halides), and solids (alkali
halides and alkaline earth oxides and sulfides). For vapor molecules, the theory
is universal and applies to all non-hydrogenic molecules with the same total
number of atoms, regardless of differences in structure and/or bonding. For such
molecules, the theory greatly simplifies the calculation of entropies and free
energy functions. This should permit one to greatly expand tables of thermo-
dynamic data on standard free energies of formation of vapor compounds since
there are many compounds for which the enthalpies of formation are known
(mostly at 298 K) and entropies and free energy functions are unknown. From
this theory, one can deduce that the universality is, at least in part, related to the
Predictions of the Entropies 271
fact that the high-temperature heat capacities of vapor molecules with any given
total number of atoms differ little from each other.
For condensed matter, this universality is not likely to hold since the high-
temperature heat capacities of, e.g., 1: 1 and 1: 2 halides (e.g., KBr and CaBr 2 )
are significantly larger than those of the corresponding 1 : 1 and 1 : 2 oxides and
sulfides (e.g., CaO and Ti0 2 ). As a consequence, this theory can only be applied
to a class of materials such as, e.g., all halides or all oxides and sulfides of a given
stoichiometry. Further study is required to define the range of the classes of
solids (and liquids if enough data are available) to which such a theory applies.
For example, our preliminary examination of 1 : 2 solid oxides (e.g., Ti0 2 , Si02 ,
Th0 2 , etc.) indicated that the theory applies. In addition, further theoretical
studies of the differences in values of (To for different classes of solids of the same
stoichiometry are needed. Differences in structure will have to be taken into
account. For example, most of the compounds plotted in Figs. 1 and 2 are cubic,
some being body-centered, some face-centered. Neither beryllium oxide nor
beryllium sulfide are cubic. The aim is to clearly demonstrate that the theory is
a useful tool for making predictions. For geologists, this theory has the potential
to significantly expand tables of needed thermodynamic data on free energies of
formation of compounds and to improve capabilities for calculations of chemi-
cal and phase equilibria. This potential can be realized by extending the theory
to predict entropies and free energy functions for simple and complex solid
compounds.
Acknowledgment
This work was performed under the auspices of the u.s. Department of Energy,
Division of Materials Sciences, under Contract No. W-31-109-ENG-38.
References
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Springer-Verlag, New York.
Barin, I., Knacke, 0., and Kubaschewski, O. (1977). Supplement, Thermochemical Prop-
erties of Inorganic Substances. Springer-Verlag, New York.
Blander, M. and Stover, e. (1985). High Temp. Sci. 19, 231-241.
Chase, M.W. et al. (1986). JANAF Thermochemical Tables, 3rd ed., Parts I and II.
American Chemical Society and American Institute Physics Washington, D.e.
Frurip, DJ. and Blander, M. (1980). J. Chem. Phys. 73, 509.
Frurip, DJ., Blander, M., and Chatillon, e. (1982a). Statistical mechanical predictions of
entropies and free energy functions for small clusters of atoms, in Metal Bonding and
Interactions in High Temperature Systems with Emphasis on Alkali Metals, lL. Gole
and W.e. Stwalley, eds., ACS Symposium Series, Washington, D.e., Vol. 179.
Frurip, DJ., Chatillon, e., and Blander, M. (1982b). J. Phys. Chem. 86, 647.
Pankratz, L.B. (1982). Properties of elements and oxides. U.S. Bureau of Mines Bull. 672,
U.S. Govt. Printing Office, Washington, D.e.
Stover, C.R. (1983). Entropies for MX2-type molecules and a classical statistical mechani-
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Argonne Nat. Lab. Argonne, IL.
Chapter 10
Systematics of Bonding Properties and
Vibrational Entropy in Compounds
G. Grimvall and A. Fernandez Guillermet
Introduction
Svib(T) = t 'onax
Sosc(w, T)F(w)dw. (2)
r
ture expansion of SD is
1000
900
:.::
800
(/)
CD
700
600 0
SOO 1S00 2000 2500
T (K)
Fig. 1. The entropy Debye temperature es(T) for three phases of aluminum silicate,
AlzSiO s : andalusite, kyanite, and sillimanite. Experimental (solid line) and estimated
(dashed line) data from the JANAF thermochemical tables (1985). We also show the heat
capacity Debye temperature edT) for andalusite; dashed curve.
e
compare s of two crystalline phases of the same compound. Their vibrational
e
entropy difference !J.S is a constant at high T. When their s differ by only a small
amount !J.es , we have (with !J.S being an average per atom)
(5)
The entropy difference between the a-phase and the f)-phase of cryolite
(Na 3 AIF6) at 1000 K is given (JANAF, 1985) as !J.S' = (509.179 - 497.868)
J/(K·mol). Then, !J.es/es = -!J.S'/(30R) = -0.045.
There are two fundamentally different reasons for the temperature depen-
dence of es(T): one related to low temperatures and one to high temperatures. If
the actual system does not have a Debye spectrum, the solution es(T) will vary
with the temperature when T < es. For high temperatures and strictly harmonic
vibrations, es(T) asymptotically approaches a value that represents the loga-
rithmically averaged phonon frequency. This is seen if (2) is expanded at high
temperatures and then compared with (4). For a real solid, the solution es(T)
derived from experiments almost always decreases with T when T > es. That is
caused by anharmonic effects in the lattice vibrations, mainly indirectly through
the thermal expansion.
The general shape of es(T) in Fig. 1 is characteristic of most solids. In
particular, one does not expect any humps or irregularities for T > es. There-
fore, plots of es(T) can be used as a check on experimental data, see a discussion
of irregularities in the heat capacity of Ca (Grimvall and Rosen, 1982). Often,
8s(T) has a minimum at low temperatures (T/es '" 0.1) (Rosen and Grimvall,
1983). However, Be and Mg indicate that es(T) may decrease continuously with
increasing T from 0 K (Rosen and Grimvall, 1983). The precise behavior depends
Systematics of Bonding Properties and Vibrational Entropy 275
,----
I
1000
g
CD
T (K)
Fig. 2. Two Debye temperatures, 8elast(T) and 8s (T), for corundum (IX-AI 2 0 3 ). 8elast(T) is
from Goto et. al. (1989) and based on their measured elastic constants. 8s(T) is evaluated
by us from the entropy data of Ditmars and Douglas (1971) below 100 K and data given
in the JANAF thermochemical tables (1985) at higher temperatures.
on how the Debye model can reproduce the properties of the real phonon
spectrum. See also Fig. 2.
The maximum value of Os(T) in Fig. 1 gives a good account of the loga-
rithmically averaged phonon frequency wlog through the relation
(6)
The initial decrease of Os with T at high temperatures reflects how wlog decreases.
At very high temperatures, the anharmonic effects are complicated, but Os(T) is
still a very useful parameter in the modeling of the high-temperature thermody-
namic properties of solids.
For a quick calculation of (}s(T) from a given table of entropy data, one may
use Eq. (4) at high temperatures and an interpolation in the tables of the Debye
model entropy at intermediate and low temperatures (T < Os/2). See, e.g., tables
in the American Institute of Physics Handbook (1972).
Another chapter in these proceedings deals with the Komada-Westrum
modeling of realistic phonon spectra. In that approach, the thermal properties
are described by a characteristic temperature OKw(T) that normally varies very
little with T. However, their OKw(T) would also decrease with T if the model is
used to represent the high-temperature vibrational entropy, and it would fail
to represent a heat capacity per atom that is larger than 3kB .
The ordinary heat capacity Debye temperature referring to low temperatures,
where the Debye T3 law holds, can also be calculated from the elastic constants
cij. Thus, we define the Debye temperature 0elast. Since the elastic constants vary
with temperature, for the same reason that Os(T) varies with T when T > Os, we
get a temperature-dependent (}elast(T). Figure 2 compares (}e1ast(T) from measure-
276 G. Grimvall and A. Fernandez Guillermet
ments of cij(T) (Goto et aI., 1989) with Os(T) from entropy data (JANAF, 1985;
Ditmars and Douglas, 1971) for corundum (a-AI203). Because Os and Oel,s! give
different weights of the phonon spectrum, we do not expect their temperature
dependence to be identical, but they have the same value in the limit T --+0.
(7)
The effective mass Meff is the logarithmically averaged mass in the compound.
For a crystal AaBbCc' one has
(a + b + c)lnMeff = a1nMa + b1nMb + clnMc. (8)
As an example, for sodium chloride Meff = (MN.MCl)1/2. The effective force
constant ks is a very complicated average over all interatomic forces in the solid.
Here we only use the fact that ks measures the bonding strength. One should
note that ks refers to an average over all bonds in a solid and not only to those
that, e.g., are important for the hardness.
It is natural to assume that the interatomic forces are similar in chemically
related compounds. If ks is identical in two compounds, their Os (at high T)
would differ only by the effect introduced through the effective mass. As an
example, consider ZrTe s and HITes that have the same crystal structure. The
heat capacities have been measured and at room temperature they are not far
from the Dulong-Petit value 18R per formula unit (R is the gas constant).
Therefore, the standard entropies °S(298.15 K) should give Os values that repre-
sent the logarithmically averaged phonon frequencies. Shaviv et ai. (1989) ob-
tained °S(HITe s) - °S(ZrTe s ) = 32.99R - 31.96R = 1.03R. The effective mass
ratio is
Meff(HITes) = (MHf)1/6 ~ 1.118.
Meff(ZrTe s ) Mzr
Hence, by Eqs. (4) and (7), the entropy difference per mol of formula units at high
Systematics of Bonding Properties and Vibrational Entropy 277
temperatures and from the mass effect alone would be 18R In(1.118)1/2 : : : : 1.004R,
i.e., in a remarkable agreement with the measured entropy difference, 1.03R.
ZrTe s and HITes are examples of what is termed low-dimensional solids. Their
crystal structure is chainlike and in some respects they can be viewed as having
dimensions between 1 and 2. Therefore, the low-temperature heat capacity
deviates from the usual T3 law. At room temperature, however, one is in the
classical regime of lattice vibrations for these compounds, and the separation of
masses from force constants in Os holds in spite of the special crystal structure.
We remark here that Latimer's (1951) rule for the estimation of the vibra-
tional entropy of compounds is closely related to the separation of interatomic
forces and atomic masses. Briefly, this rule has had some success because it
correctly accounts for the mass effect in vibrational entropy (Grimvall and
Rosen, 1983; Grimvall1983).
Anderson and Nafe (1965) noted, for the bulk modulus B, that 10g(B) varies
almost linearly with 10g(Q) for many chemically related solids. Here, Q is the
volume per atom in the compound. We take a similar approach and investigate
trends in the effective force constants ks for compounds that have the same
crystal structure, i.e., alkali halides and some transition metal compounds. Fig-
ures 3 and 4 show ks vs. Q in logarithmic plots. There is a clear correlation
between ks and Q for the alkali halides, but it is not as good as for the bulk
modulus, and lithium halides and alkali hydrides fall on trend lines different
from those in Fig. 3 (Haglund and Grimvall, 1991). The force constants ks of the
NaCl-structure carbides, nitrides, and oxides formed with titanium and vana-
dium, respectively, follow two simple but different trends when plotted vs. Q
(Fig. 4).
100 ,,
.. NaF
,,
'-., KF
E
'-
z ,., NaCl
50 -
NaB~,. KCl
" • KBr
,
Fig. 3. A logarithmic plot of the average Nal.', • Kl
interatomic force constant ks, calculated "
from entropy data, vs. the volume per 30 I:i:.-----,L----'---'---:c':--'----"
10 20 50
atom Q for some alkali halides. Data from
Haglund and Grimvall (1991).
278 G. Grimvall and A. Fernandez Guillermet
,,
400 VC, /
age atomic volume Q. Data from
'"
",'TiN
E
350
,
I
/ '"
/
Fernandez Guillermet and Grimvall
(1989a).
"-
z VN,' '" '"
300
, 4
,,
I
Vl TiO
.:.::
250
VO~
200 '"::'8.5=-----:'9.0=------;;9.~5-----;;10~0-----'
y
z
15
i Ti~' and Grimvall (1989a).
o
~
SeC ' \ VN
TiO',,,
~ 1 , VO
'e
ne (e/aJ
We have noted in previous work that the quantity Es with the dimension of
energy
(9)
may show more regularity than ks alone. Let the effective interatomic forces in
a solid be described by a single central potential V(r) = Vocp(rja), where the
length parameter a but not the strength Vo varies between solids. Then these
solids would have atomic volumes Q that scale as a 3 but Es would be the same
(Grim vall and Rosen, 1983). That model gives a crude description of the alkali
halides (see Fig. 3, where the slope of the straight line corresponds to kS QO.83
being a constant), but fails for the transition metal compounds considered in Fig.
4. This difference is not surprising since the two groups of solids have very
different chemical bonding. However, when Es from Eq. (9) is plotted vs. the
average number of valence electrons per atom in the compound ne , very regular
behavior emerges for the transition metal compounds (Fig. 5). A discussion of
the carbides has been given elsewhere (Fernandez Guillermet and Grimvall,
1989a), and we only sketch the arguments. The density of states N(E) for the
Systematics of Bonding Properties and Vibrational Entropy 279
The phase diagram of an alloy system can be calculated if one knows the molar
Gibbs energy
G = H - TS (10)
of its various competing phases as a function of temperature, pressure, and
composition. In Eq. (10), H is the enthalpy and S the entropy. For a stoichio-
metric compound AaBb at constant atmospheric pressure Po, one can write the
Gibbs energy °GAaBb as
where
Acknowledgments
This work was supported in part by the Swedish Board for Technical Develop-
ment and by the Swedish Natural Science Research Council.
References
American Institute of Physics Handbook (1972). D.E. Gray, ed., McGraw-Hill, New York,
4-114.
Anderson, O.L. and Nafe, 1.E. (1965). The bulk modulus-volume relationship for oxide
compounds and related geophysical problems. J. Geophys. Res. 70, 3951-3963.
de Boer, F.R., Boom, R., Mattens, W.C.M., Miedema, A.R., and Niessen, A.K. (1989).
Cohesion in Metals. North Holland, Amsterdam.
Ditmars, D.A. and Douglas, T.B. (1971). Measurement of the relative enthalpy of pure
IX-AI 20 3 (NBS heat capacity and enthalpy standard reference material no. 720) from
273 K to 1173 K. J. Res. NBS. 75A, 401-420.
Fernandez Guillermet, A. (1991). Predictive approach to thermodynamic properties of
metastable Cr3 C carbide Int. J. Thermophys. 12,919-936.
Fernandez Guillermet, A. and Frisk, K. (1991). Thermodynamic properties ofNi nitrides
and phase stability in the Ni-N system. Int. J. Thermophys.12, 417-431.
Fernandez Guillermet, A. and Grimvall, G. (1989a). Cohesive properties and vibrational
entropy of 3d transition-metal compounds: MX (NaCl) compounds (X = C, N, 0, S),
complex carbides, and nitrides. Phys. Rev. B40, 10582-10593.
Fernandez Guillermet A. and Grimvall, G. (1989b). Homology of interatomic forces and
Debye temperatures in transition metals. Phys. Rev. B40, 1521-1527.
Fernandez Guillermet, A. and Grimvall, G. (1989c). Thermodynamic properties of tech-
netium. J. Less-Common Metals 147, 195-211.
Fernandez Guillermet, A. and Grimvall, G. (1990). Correlations for bonding properties
and vibrational entropy in 3d-transition metal compounds, with application to the
CALPHAD treatment of a metastable Cr-C phase. Z. Metallk. 81, 521-524.
Fernandez Guillermet, A. and Grimvall, G. (1991). Bonding properties and vibrational
entropy of transition metal MeB2 (AIB 2) diborides. J. Less-Common Metals 169,
257-281.
Fernandez Guillermet, A. and Grimvall, G. (1992). Cohesive properties and vibrational
entropy of 3d-transition metal carbides. 1. Phys. Chern. Solids 53, 105-125
Fernandez Guillermet, A. and Huang, W. (1991). Thermodynamic analysis of stable and
metastable carbides in the Mn-V-C system and predicted phase diagram. Int. J. Ther-
mophys. 12,1077-1102.
Fernandez Guillermet, A. and Jonsson, S. (1992). Predictive approach to thermody-
namical properties of Co nitrides and phase stability in the Co-N system. Z. Metallk.
83,21-31.
Goto, T., Anderson, O.L., Ohno, I., and Yamamoto, S. (1989). Elastic constants of
corundum up to 1825 K. J. Geophys. Res. 94(B6), 7588-7602.
Grimvall, G. (1983). Standard entropies of compounds: Theoretrical aspects of Latimer's
rule. Int. J. Thermophys. 4, 363-367.
Grimvall, G. and Rosen, 1. (1982). Heat capacity of fcc calcium. Int. J. Thermophys. 3,
251-257.
Grimvall, G. and Rosen, 1. (1983). Vibrational entropy of polyatomic solids: Metal
carbides, metal borides, and alkali halides. Int. J. Thermophys. 4, 139-147.
282 G. Grimvall and A. Fernandez Guillermet
Grimvall, G. and Thiessen, M. (1986). The strength of interatomic forces, in 2nd Interna-
tional Coriference on Science Hard Materials, Institute Physics Coriference Series, Vol.
75, Adam Hilger, Bristol, pp. 61-67.
Grimvall, G., Thiessen, M., and Fernandez Guillermet, A. (1987). Thermodynamic prop-
erties of tungsten. Phys. Rev. B36, 7816-7826.
Haglund, 1. and Grimvall, G. (1991). Lattice vibrations and bonding in alkali hydrides
and alkali halides. Unpublished.
Haglund, 1., Grimvall, G., Jarlborg, T., and Fernandez Guillermet, A. (1991). Band
structure and cohesive properties of NaCl-structure transition-metal carbides and
nitrides. Phys. Rev. B43, 14400-14408.
JANAF Thermochemical Tables, 3rd ed. (1985). M.W. Chase, C.A. Davies, 1.R. Downey,
Jr., DJ. Frurip, R.A. McDonald, and A.N. Syverud, eds., J. Phys. Chem. Ref. Data 14,
Supplement 1.
Latimer, W.M. (1951). Methods of estimating the entropies of solid compounds. J. Am.
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Rosen, 1. and Grimvall, G. (1983). Anharmonic lattice vibrations in simple metals. Phys.
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Shaviv, R., Westrum, Jr., E.F., Fjellvag, H., and Kjekshus, A. (1989). ZrTe s and HITes:
The heat capacity and derived thermophysical properties from 6 to 344 K. J. Solid
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Ceramic Materials. Materials Science Monographs, No. 49, Elsevier, Amsterdam.
Chapter 11
Phonon Density of States and
Thermod ynamic Properties of Minerals
Subrata Ghose, Narayani Choudhury, S.L. Chaplot, and K.R. Rao
Introduction
An important objective of the earth sciences is to develop the capability of
predicting the thermodynamic properties of minerals and their phase relations
under various pressure-temperature conditions in the earth, moon and the ter-
restrial planets. Experimentally, thermodynamic properties such as specific heat
can be measured by adiabatic and differential scanning calorimetry. However,
there are cases, in which the determination of the low temperature specific heat
by adiabatic calorimetry is not feasible due to the paucity of materials. Such is
the case for the high-pressure and high-temperature magnesium silicate phases:
Mg 2 Si04 (f3- and )I-spinel) and MgSi0 3 (ilmenite, perovskite and garnet) consid-
ered to be stable in the earth's mantle. These phases are synthesized in cubic-
anvil and split-sphere apparati in 10 to 20 mg quantities, that are barely ade-
quate for the specific heat measurement by differential scanning calorimetry
(DSC), usually in the range 300 to 900 K. Hence, the ability to correctly predict
the low temperature specific heat of these phases would be extremely useful.
Such a theoretical treatment should at the same time provide an understanding
of the thermodynamic properties of minerals at the atomistic level. This was the
goal aimed at the Mineralogical Society of America Short Course organized
by S.W. Kieffer and A. Navrotsky on "Macroscopic to microscopic: Atomic
environments to mineral thermodynamics" held at Washington College,
Chestertown, Maryland in spring, 1985. At least for minerals of medium struc-
tural complexity, this goal is in sight. It is now possible to theoretically explore
the entire spectrum of thermal vibrations of crystals by lattice dynamical meth-
ods. The vibrational average, known as the phonon density of states, g(w), is the
basis for the calculation of thermodynamic properties such as internal energy,
free energy, specific heat, and entropy. The accuracy of the theoretical calcula-
tion of g(w) can be tested against the generalized density of states G(E), which
can be measured by inelastic neutron scattering on large powder samples. Such
284 S. Ghose, N. Choudhury, S.L. Chaplot, and K.R. Rao
where Wj(q) gives the dispersion relation, i.e., the frequency dependence with
wave vector q for the jth phonon branch in the Brillouin zone. A is a normaliza-
Partition function z
= U(V) + Eyib
Specific heat Cv = (dEjdT)
Entropy S = (dF/dT)
Phonon Density of States and Thermodynamic Properties of Minerals 285
f g(w)dw = 1. (2)
Once g(w) is known, the free energy, internal energy, specific heat and entropy
can be calculated (Table 1). The specific heat at constant volume, Cv(T), e.g., is
given by:
hw )2
f( kBT exp(hw/kBT)
Cv(T) = kB [exp(hw/kBT) _ 1]2 g(w)dw, (3)
where, h is the Planch's constant, h/2n; kB the Boltzman constant, and T the
temperature.
Einstein Model
The simplest approximation of the phonon density of states is given by the
Einstein model, in which all the atoms are assumed to oscillate independently of
each other with the same mean frequency, WE' the Einstein frequency. In this
model, each atom is connected by an elastic spring to its equilibrium position,
but there are no interactions with the neighboring atoms. The mean energy of a
linear harmonic oscillator of frequency w in thermal equilibrium at temperature
T is given by the Planck's formula if we assume no zero-point energy
hw
exp(hw/kBT) - 1.
(4)
Hence, the internal energy E of the crystal of r atoms in the primitive unit cell
and N unit cells at temperature T is given by
E= 3rNhw
(5)
exp(hw/kBT) - 1
The density of states is given by
gE(W) = 3rNc5(w - WE)' (6)
In terms of the Einstein temperature lJE , the Einstein specific heat
)2 eVT
lJE
CE = 3rNkB ( T (e~/T _ 1)2' (7)
where lJE = hWE/kB. At high temperatures, the specific heat is correctly predicted,
where it approaches the Dulong-Petit limit, Cv -. 3rNkB' but it is underesti-
mated at low temperatures, giving an exponential decrease instead of T3 depen-
dence as experimentally observed. This deficiency is due to the fact that the
assumed single Einstein frequency lies somewhere in between the optic modes
and the acoustic modes, and at low temperatures mostly the low frequency
acoustic modes and not the high frequency optic modes are excited.
286 s. Ghose, N. Choudhury, S.L. Chap\ot, and K.R. Rao
Debye Model
where V is the volume of the crystal. In a crystal with r atoms in the primitive
cell and N unit cells, the total number offrequencies is 3rN, and hence
where the intergration is over all allowed frequencies. Since this integral would
diverge to infinity if it is extended to include infinitely high frequencies, Debye
assumed that it should be used only to a maximum or cut-off frequency W D
known as the Debye frequency. Hence, in the Debye model the allowed vibra-
tional modes exist in the frequency range 0 to W D with no modes of higher
frequency. Therefore for W > WD, gD(W) = 0, and for W < WD, gD(W) = 9rNw2/
w~. In terms of the Debye temperature ()D' defined as
where x = hw/kB T.
At temperatures much higher than the Debye temperature, CD = 3rNkB.
At low temperatures (T « (}D)' the Debye specific heat
12n:4
CD(T) = -5- rNkB
(T)3
(}D ' (13)
be used in the above equation, such that CD (Tj8D ) = Cv(T), is not a constant but
is a function of temperature. The T3 law that depends on a constant 8D is only
accurate over a small temperature range. 8D is still a useful parameter character-
istic of a given material and reflects the strength of the interatomic forces. For
example, for diamond with strong covalent forces, 8D is close to 2000 K, whereas
for the soft potassium metal it is 100 K.
To circumvent the deficiencies of both the Einstein and Debye models, a hybrid
model can be developed (Briiesch, 1982) in which the density of states gDE(W) is
given by
(14)
In this model the acoustic modes are described by a Debye model, whereas the
optic modes are described by an Einstein model. The density of states corre-
sponding to the three models and the general case are illustrated in Fig. 1.
In the hybrid model,
(15)
where WE is the mean optical frequency. The Debye density of states gD(W) is the
same as before, but the normalizing condition is now
f WD
o gD(w)dw = 3rN. (16)
The resulting specific heat CDE(T) gives the classical result 3rNkB at high temper-
atures. At low temperatures the Einstein term can be neglected, and
w ZB
0)
q w
w
I,IE(W)
wE
b) ,.'
"
~~,-;-
"" ,.
,,,,."
q,.
q w
wE
w ze
l,Io(W)
Wo
c)
- - - - _........._'------ q ~~--~-~_~~.w
qo wE
Fig. 1. Phonon dispersion and phonon density of states for a crystal with two atoms in
the unit cell (a) qualitative general behavior, (b) Einstein approximation, (c) Debye
approximation, and (d) hybrid Einstein-Debye model (after Briiesch, 1982).
from spectral data. Isolated high-frequency optic modes (e.g., Si-O and O-H
stretching modes) are represented as separate Einstein oscillators with frequen-
cies WEl, W E2 , ... etc., or by a second optic continuum. A phonon density of
states so constructed for forsterite is shown in Fig. 2. This model has been
successfully used to estimate the specific heats of a large number of rock-forming
silicates (Kieffer, 1985) and a number of the high-pressure magnesium silicate
Phonon Density of States and Thermodynamic Properties of Minerals 289
I
phases considered to be stable in the earth's mantle (Akaogi et at, 1984; Ashida
et at, 1988). In spite of its success, this model is empirical in nature and requires
extensive prior knowledge of elastic constants and frequencies of optic phonons
from IR and Raman data. In case inelastic neutron scattering results are not
available, the neglect of low-frequency optic phonons that are IR- and Raman-
inactive may result in an underestimation of the low-temperature specific heat.
On the other hand, at high temperatures where the Einstein oscillator model
would be valid, this model might give satisfactory results.
Lattice Dynamics
k
z
Z B
A
U
R
E
;\
H
Q
112g,
r .1 y
ky
D S
kx
Fig. 3. Brillouin zone for an orthorhombic lattice. The principal symmetry directions ~,
~, and A are shown.
where the suffices (x, /3, y denote the Cartesian coordinates. The second derivative
ifJaP U~,) is the negative of the force constant and gives the force in the (X-
direction on the ion Uk) when the ion (l'k') is displaced by a (small) unit displace-
ment in the /3-direction. For a perfect crystal with uniform translational symme-
try, no residual stresses remain. Hence,
11' )
~ ifJaP ( kk' = O. (23)
where,
We then have 3rN coupled equations of motion. The standard technique that is
used in solving these equations involves making a transformation to a set of
normal coordinates. The squares of the frequencies are the eigenvalues of the
dynamical matrix. The displacements of the atoms in one of these normal modes
labeled by (qJ correspond to a wavelike displacement of atoms and is given by
(28)
where P (;) are the normal coordinates and e(kl~) is the eigenvector of the
normal mode (qj), where j runs from 1 to 3N and is used to distinguish between
the 3N normal modes at q; wj(q)/2n is the frequency of the normal mode. The
frequencies of the normal modes (eigenvalues) and their eigenvectors are deter-
mined by diagonalizing the dynamical matrix through a solution of the secular
equation
(29)
Phonon Density of States and Thermodynamic Properties of Minerals 293
L e:(klqJea(klqj)/M
ka
k = bjr' (30)
Evib(Wi) = ( ni + ~) hWi'
where i represents phonons in state qj. Here, nj is the thermal occupancy number
of phonons given by eqn. (21). The total vibrational energy of a crystal is
given by a summation over all the phonon modes in the Brillouin zone.
The thermodynamic properties of a crystal, namely, the free energy, internal
energy, specific heat, and entropy, are obtained from the partition function Z,
defined as
(32)
The thermodynamic properties of the crystal obtained from the partition func-
tion can be expressed as averages over the phonon density of states g(w) and are
given in Table 1.
The external pressure P acting on the crystal is related to the volume deriva-
tive of the free energy F
_ of _ [OU(V)
P - - oV - - av + OFvibJ
oV '
oU(V) 1
= - ~:1-
uV
+-
Vo
L IjEvib(WJ
j
(33)
(34)
294 S. Ghose, N. Choudhury, S.L. Chapiot, and K.R. Rao
The negative volume derivative of the static lattice energy, U(V) can be ex-
pressed in terms of the bulk modulus B at zero pressure (Briiesch, 1982)
_ au(V) = _ B(V - Yo)
(36)
av Vo
where Vo is the crystal volume at 0 K.
The coefficient of volume thermal expansion IXv of the crystal, expressed as
the fractional change of volume with the temperature at constant pressure, is
given by
(37)
where Cvj(T) is the contribution from phonons in the state i to the specific heat at
constant volume. In the Griineisen approximation, all phonon modes in the
Brillouin zone are assumed to be characterized by an average Griineisen con-
r.
stant In this approximation
single crystals (Dorner, 1982; Sk6ld and Price, 1986). The coherent scattering
cross-section of the neutron dependent on energy E and scattering angle !l due
to the creation ( + ) and annihilation ( - ) of a single phonon of frequency Wj(q)
is given by
d2 (Jcoh (Q w)
d!ldE '
Here, the scattering vector, Q = k1 - ko, where k1 and ko are scattered and
incident neutron wave vectors and [niq) + ± t tJ
is the population factor, the
upper and lower - and + signs indicate phonon annihilation and phonon cre-
ation respectively, and t is a reciprocal lattice vector.
The delta functions indicate the conservation of energy and momentum
during the scattering process
Q = ± q + t, (41)
E1 - Eo = ±hwj(q). (42)
Fj(Q) = L bk
k
fo.)
Mk
e-Wk(Q)eiQ.rk. (43)
The sum extends over all atoms of the unit cell. Here Mk is the mass of the kth
atom, bk = neutron scattering length averaged over various isotopes, fk the
position vector, and e- Wk the Debye-Waller factor for the kth atom. The dy-
namical structure factor can be calculated from polarization vectors based on a
lattice dynamical model. Good agreement between the observed and calculated
dynamical structure factors along with the phonon dispersion and phonon
density of states provides a stringent test for the accuracy of the interatomic
potential used in the lattice dynamical calculation.
In the technique known as coherent inelastic neutron spectroscopy, scattered
neutron groups are observed whenever the energy and momentum conservation
laws are simultaneously satisfied at certain wand q. The wavelength or velocity
of the neutrons can be measured in two ways: (1) Bragg scattering from a single
crystal monochromator or analyzer using the Bragg equation, A = 2d hkl sin 0,
and (2) time of flight method, where the transit time of neutrons over a finite
flight path is electronically measured, from which the neutron velocity can be
determined. For phonon measurements, a single crystal triple axis spectrometer
based on the Bragg law is most widely used. The most commonly used method
is the constant Q method, where the momentum transfer is constant, as opposed
to the constant E method, where the energy transfer is constant. For further
details, see Ghose (1988) and the references therein.
296 s. Ghose, N. Choudhury, S.L. Chap\ot, and K.R. Rao
The generalized phonon density of states G(E)(E = hw), as determined by
inelastic neutron scattering from a large powder sample, is qitTerent from g(w), in
that G(E) is a sum of the partial components of the density of states due to the
various species of atoms, weighted by their scattering cross sections. G(E) also
needs correction for multi phonon scattering. Hence, a direct determination of
g(w) from G(E) is not possible. However, when g(w) is available from a lattice
dynamical calculation, G(E) can always be calculated and compared with the
experimentally measured values.
An incoherent approximation to the coherent scattering (Placzek and Van
Hove, 1955) is usually used to derive the scattering from a powder sample
where the sum over q is over all q satisfying the relation Q = t ± q, in which t
is the reciprocal lattice vector. The scattering then provides the one phonon
density of states weighted by the scattering length and the population factor.
Any inelastic neutron scattering spectrum contains a contribution from multi-
phonon scattering that has to be estimated and subtracted from the experimen-
tally observed spectrum G(E) to obtain the one-phonon density of states. Usu-
ally, multiphonon scattering contributes a continuous spectrum and etTectively
increases the background. The coherent neutron scattering measures the scatter-
ing function in terms of Q and E, which in the conventional harmonic phonon
expansion can be written as:
S(Q, E) = S(O) + S(1) + s(n), (45)
where S(O), S(1) and s(n) represent elastic, one-phonon, and multiphonon scatter-
ing, respectively. The multiphonon contribution to the total scattering is esti-
mated in the incoherent approximation using Sjolander's formalism (Rao et
aI., 1988), in which the total scattering is treated as a sum of the partial compo-
nents of the density of states from the various species of atoms, weighted by
the squares of their scattering lengths.
The time-of-flight method using a pulsed spallation neutron source is the
most commonly used technique, which is capable of the necessary energy resolu-
tion and covers the energy range needed for both external and internal vibra-
tions in solids. The low-resolution medium-energy chopper spectrometer
(LRMECS) at the Argonne National Laboratory is such an instrument using the
Intense Pulsed Neutron Source (IPNS) (Fig. 4). The chopper is phased to the
source and gives pulses of monochromatic neutrons at the sample. Scattered
neutrons are time-analyzed over a heavily shielded flight path of 2.5 m, whereas
the incident flight path is 7.5 m. The neutron chopper has a 5-J.ls burst time to
give around 6% resolution of the incident energy. The range of detector angles
from -10 to 120° gives a wide range ofsimultaneous Q vectors from 1 to 20 A-1
Phonon Density of States and Thermodynamic Properties of Minerals 297
Collimators:
IPNS Shield
Soller slil Beam defining
\
f l B t t - - - - - - Beam F4 -
Beam scraper
upstream -
Hydrogenous shielding
at 100 me V energy transfers. This instrument has been used to measure the
generalized density of states from forsterite and fayalite powder samples at
several temperatures.
ation of the usual Pbnm setting) with four formula units, i.e., 28 atoms in
the unit cell. The unit cell dimensions offorsterite are a = 10.190, b = 5.987, and
c = 4.753 A, and of fayalite: a = 10.471, b = 6.086 and c = 4.818 A. They are
isostructural with two types of [MO 6J octahedra forming serrated edge-sharing
chains running parallel to the b-axis, cross-linked by isolated [Si04 J tetrahedra
sharing edges with adjacent octahedra (Fig. 5). Since the unit cell contains 28
atoms, there are 3 x 28 = 84 independent phonon branches! To simplify the
task, a rigid-molecular-ion model was adopted by Rao et al. (1988) for the
calculation of the external modes, in which the "molecule", i.e., the [Si04 Jgroup,
was assumed to be rigid with only three translational and three rotational
degrees of freedom, whereas the two M atoms (M 1 and M2) each have three
degrees of translational freedom, thereby reducing the number of independent
phonon branches to be calculated to 48. The potential energy of the crystal was
assumed to contain two-body potentials containing a Coulombic and a repul-
sive term
e2 Z(Kk)Z(K'k') [ br ]
V(r) = 4neo r + aexp - R(Kk) + R(K'k') , (46)
where Zk and Rk are the "effective charge" and "effective radius" of the kth ion,
and K is the rigid unit, r the interatomic separation, e the electron charge and eo
is the vacuum permittivity. Constants a and b were set equal to 1822 eV and
12.364, respectively. Zk values were adjusted to maintain charge neutrality. The
effective charges and radii were treated as adjustable parameters, which were
optimized with respect to the known crystal structure and the elastic constants
(Rao et al., 1987, 1988; Ghose et al., 1987, 1991). The optimized values used for
the phonon dispersi'on calculations of forsterite are given below
Mg1 Mg2 Si 01 02 03
Charge 1.60 1.80 1.0 -1.2 -1.0 -1.10
Radius (A) 1.68 1.73 1.00 1.55 1.45 1.50
Similar values were used for the preliminary calculations on fayalite. The
phonon dispersion relations for the external modes of forsterite calculated along
the three principal symmetry directions are shown in Fig. 6 (Rao et al., 1988).
One-phonon dynamical structure factors were also calculated using the rigid-
molecular-ion model, which were used as guides for assigning the observed
phonon peaks during the inelastic neutron-scattering experiments.
Phonon dispersion relations in single crystals of forsterite (3.4 x 1 x 0.6 cm)
and fayalite (cylinder of 6 x 0.6 cm length) (kindly provided by Prof. H. Takei,
University of Tokyo) were measured by inelastic neutron scattering at the
Brookhaven National Laboratory, using a triple-axis spectrometer and the High
Flux Beam Reactor (HFBR) (Ghose et al., 1988, 1991; Rao et al., 1988). Over 150
phonon measurements were made for forsterite and 70 for fayalite along the
three principal directions in the xy-, yz-, and xz-planes. Some of the selected
6., 6..
600 600 I
Fig. 6. The calculated phonon dispersion relation of the external modes in forsterite (Rao
et aI., 1988).
300 S. Ghose, N. Choudhury, S.L. Chaplot, and K.R. Rao
:E, (1.5.',0)
.. -.
1\'
til
I-
~ .: ~: :! "" ; .... :~<lr '5 00 6 ·e O
..
Z
::)
0 6. 1 ( 6, 4,0 ) 6. 4 i .(6.1 .6,0)
u
Z
0
a:
I-
::)
• .
UJ •
Z •
" :."
. 1\3
(0.4,~ 4) (0, 4
1\3
,~ . 2)
EN ERGY (meV)
Fig. 7. Selected acoustic and optic phonon measurements in a single crystal offorsterite.
The numbers in parentheses indicate the coordinates of the wave vector involved in
the neutron scattering process in units of reciprocal lattice vectors (Ghose et aI., 1987
Pergamon Press, pic.).
0·5 a
11- o 0 0.50"- 0
~, ~4 ~2 ~J
> 30
CII
--
E
>
I!)
20
10
IX
UJ
Z
UJ o 0 0.5 b* --- 0
Z
0 1\2 I\J
z
0
J: 30
Q.
20
10
INS Raman IR
104
144 142 144
184 183
192 192
200 201
258 260
315 318 313
325 324 323
with parameters C, D, n, and roo To simulate the O-Si-O bond bending poten-
tial, the 0-0 interaction within the same silicate group is of the form
e2 Z2(0)
v(ro-o) = -4- - - + saexp [ -2 bro-oJ
() -
W
-6- (48)
m;o ro-o R 0 ro-o
with parameters sand w. The parameters Z(O), R(O), a, and b were the same as
in the rigid-molecular-ion model. The other parameters were optimized to the
-1
followmg values: C = 1, D = 3.4eV, n = 10.5 A , ro = 1.61 A, s = 55, and
• 0 0
w = 1250 eV A6. The computer program DISPR (Chaplot, 1978) was used in the
numerical computations. The total one-phonon density of states, g(w) of for-
sterite derived from the rigid-ion model by sampling 27 wave vectors in the
irreducible Brillouin zone is shown in Fig. 9. The partial density of states in the
rigid-molecular-ion model from external modes only due to the translations of
the two Mg atoms and the translation and rotation of the silicate group are
shown in Fig. lOa, b.
Figures 11 and 12 show the comparison of the observed and calculated
neutron spectra, including the one phonon and multiphonon contributions and
intensity broadening due to instrumental resolution at 300 K for forsterite and at
300 and 17 K in fayalite respectively. The inelastic neutron-scattering measure-
ment of G(E) was made on synthetic powder samples ("" 50 g) in both cases
by the time-of-flight method using the low-resolution medium-energy chopper
spectrometer and the Intense Pulsed Neutron Source at the Argonne National
Laboratory. The agreement between the theoretical and experimental curves at
300 K for both forsterite and fayalite is very satisfactory. The peaks in the 20 to
80 me V region in the neutron spectra of fayalite are shifted to lower energies
compared to those in forsterite. However, the neutron spectrum of fayalite at 17
K in the 20 to 40 meV region shows a marked deviation from that at 300 K and
the calculated spectrum at 17 K based only on lattice contribution. Since fayalite
is paramagnetic at 300 K that undergoes a transition to the antiferromagnetic
state at 65 K (Fuess et aI., 1988), this deviation is most likely due to the spin-
wave (magnon) contribution. The magnetic transition causes a A-type anomaly
in the specific heat curve of fayalite at 65 K (Robie et al., 1982b). In both sets of
FREQUENCY. W (eml)
3
;02
(b)
VI
UJ
<
til
u.. 0.1
o
>
t-
Vi Fig. 9. Total one phonon density of states
Z
UJ calculated from the rigid-ion model of for-
o O~--~--~--~~~~~~
200 400 600 BOO 1000 1200 sterite. Note the energy gap in the 700 to
FREQUENCY W (eml) 850 cm- 1 region (Rao et a!., 1988).
Phonon Density of States and Thermodynamic Properties of Minerals 303
Mg I-T Mg n-T B
.'1
W 0.';5
~
<f) OL.....<"-----'----'----'----'-----"~
u..
o Ir-------------~
> 0.15r-
t:
<f)
Z 0.10
UJ
Cl
0.05
,.......,.-..,....--,--..--...,.----r--,..---,
10.0 r-----r..__.......
8.0
..
•
.....
1",
... ,. :.
....
, " A
~ - I
'v!o'-; f'~.
'\..,
6.0
;;:;- 4.0
.
!::
~ 20
>-
~
Q:
o~-~-~-~--~-~-~--~~~
I-
iD
It:
~
>- 100
Fig. 11. Generalized density of I- B
states, G(E) of forsterite from iii
z 8.0 Total with
UJ
a powder sample (a) inelastic lz 60 / , / ' " multi phonon
contribution
neutron scattering spectrum and /
\
,..,
4.0 I \
(b) theoretical spectrum from \
II "\
the rigid-ion model, including 20 One Phonon./' \ I \
only \ I \
multiphonon contribution and
resolution broadening (Rao et 20 40 GO 80 100 120 120 160 180
ai., 1988). E (meV)
304 s. Ghose, N. Choudhury, S.L. Chaplot, and K.R. Rao
~-
Fig. 12. Comparison of the inelastic neu-
tron scattering spectra of fayalite with the
17 K
calculated spectra based on the rigid-ion
300 K
model. (a) Experimental spectra at 300 and
17 K, including multiphonon contribu-
tions. (b) Calculated phonon spectra with
one-phonon and multi-phonon contribu-
tions at 300 K (full line) and at 17 K
~2~----------------------~
(dash-dot line). The dashed line gives the
iii (b) one phonon contribution to the phonon
z
ILl spectra (Price et aI., 1991).
o
20 60 160
ENERGY (m~V)
spectra, the peaks below 80 meV are mainly from translation of the Mg or Fe
atoms and external vibrations of the silicate groups, whereas above 80 me V the
peaks are due to the internal vibrations of the silicate group. The tail above
140 meV is mainly due to the multiphonon contributions in both systems. The
significant aspect about the one-phonon density of states g(w) in both cases is
the band gap between 700 cm- 1 (87 meV) and about 850 cm- 1 (105 me V),
principally separating the external from internal vibrations. This band gap has
significant implications for the thermodynamic properties of olivines.
Once g(w) is known, the lattice specific heat at constant volume Cv can be
calculated. Results of these calculations for forsterite using the rigid-molecular-
ion, rigid-ion, Kieffer, and Debye models are shown along with the calori-
metrically measured values at constant pressure Cp by Robie et al. (1982a) in
Fig. 13. The specific heat computed from the rigid-molecular-ion model shows
excellent agreement (within 1%) with experimental values in the low-tempera-
ture region (0-100 K). The discrepancy between the two increases with increa-
sing temperature because this model neglects the high-frequency internal modes,
which are excited only at high temperatures. Since all degrees offreedom are not
considered, the Dulong-Petit limit at high temperature is never reached. In the
rigid-ion model all degrees of freedom are taken into account and the agreement
is much more satisfactory. Beyond 400 K, the difference between the calculated
Cv and the experimental Cp is due to the neglect of anharmonic corrections.
Choudhury et al. (1989) estimated the bulk modulus B and the thermal expan-
sion (Xv from quasiharmonic lattice dynamical calculations and used these values
to estimate the anharmonic corrections according to the equation (39). At 300
K, the correction is on the order of 1% increasing to about 5% at 1200 K. With
these corrections, which are important at high temperatures, the agreement
between computed and experimental values of Cp(T) is within 1% at high tem-
peratures up to 1200 K. The estimated weighted average of mode Griineisen
Phonon Density of States and Thermodynamic Properties of Minerals 305
( b)
0.20 1.0
0.8
0.6
" .____ e-.--
~ .
0.10 I/ e E.llpeonmental
0.4 '/ Rigid molecular ion model
• Rigid Ion model & Kieffer's
model
Oeby. model
o 50 100
TEMPERATURE (K)
Fig. 13. Comparison of experimental specific heat, Cp of forsterite (Robie et aI., 1982a)
with theoretical specific heat, Cv (a) Theoretical results (0-100 K) based on rigid-molecular-
ion model (full line) and experimental data (filled circle). (b) Comparison of experimen-
tal data (full line) with theoretical models. The discrepancy between the results of the
rigid-molecular-ion model and experimental results above 100 K is due to the neglect of
internal vibrations that are excited at higher temperatures (Rao et aI., 1988).
~200~------------------------------=<
lC
-0
E
~
~
W
:I:
U
~ 100
U
W
0..
II)
r
parameters, of all modes in the entire frequency range is 1.14. The specific heat
calculated from the Debye model shows considerable deviation from the experi-
mental values. This is not surprising in view of the fact that the Debye density
of states is a poor approximation of the true density of states of forsterite.
Furthermore, considerable variation of the Debye temperature with tempera-
ture (Kieffer, 1985) indicates that the Debye model is inadequate for forsterite.
On the other hand, the Kieffer model predicts the specific heat quite well,
although agreement in the low-temperature region (0-100 K) is not as good as
that predicted by the rigid-molecular-ion model.
The computed specific heat Cv of fayalite is compared with the experimental
values (Robie et aI., 1982b) in Fig. 14. The measured specific heat shows a A-peak
306 S. Ghose, N. Choudhury, S.L. Chapiot, and K.R. Rao
(49)
where e is the electron charge, qi and qj are ionic charges of ions i and j, A ij , Bij'
and Cij are parameters of the short-range potential, and rij is the ionic separation.
Included was Si-O-Si bond bending term of the type
(50)
where kij is a spring constant, eijk the O-Si-O bond angle, and eo the tetrahedral
angle. The polarizability of the oxygen atom was described by a shell model. The
phonon dispersion relations along the r-X-R-r direction were calculated using
this potential. The one-phonon density of states constructed by sampling 8 and
27 grid points in the Brillouin zone (Price et aI., 1987b) is very similar to that
constructed by Rao et al. (1987, 1988). The calculated specific heat shows good
agreement with the calorimetric measurements between 80 to 300 K, but the
discrepancy in the 10 to 80 K range is quite large (10-20%).
Price et al. (1987) have also calculated the specific heats of f3 and l' (spinel)
phases of Mg 2 Si04 using the same potential, as for forsterite. The predicted bulk
moduli are up to 35% too stiff and the thermal expansion coefficients too low,
indicating that the interatomic potential needs further improvement. Neverthe-
less, the calculated specific heats show fair agreement with those measured by
differential scanning calorimetry in the 300 to 1000 K range by Watanabe (1982).
Because of the lack of experimental data in the low temperature region (0-300
K), it is not possible to evaluate the accuracy of the predicted low temperature
specific heat.
Phonon Density of States and Thermodynamic Properties of Minerals 307
boring modes) leads to a transition to the tetragonal structure and at about 1500
K to the cubic structure due to the unstable R 25 mode. MgSi0 3 perovskite
synthesized at 26 GPa and 1873 K shows twin domains with {1l0} and {1l2}
reflection planes, which may be associated with the cubic to tetragonal and
tetragonal to orthorhombic phase transitions, respectively, on cooling (Wang et
al., 1990). The equation of state of orthorhombic perovskite has been calculated
by Wolf and Bukowinski (1985), Hemley et al. (1987), and Cohen (1987). The
experimentally observed equation of state (volume dependence on pressure at
ambient temperature) (Knittle and Jeanloz, 1987) has been fitted very well by the
potential-induced breathing (PIB) model (Cohen, 1987). This model also has
been used successfully to calculate the zone-center optic phonons in the orthor-
hombic perovskite (Hemley et al., 1989). In this Gordon-Kim-type model, the
electron density of the crystal is approximated by a superposition of component
ion electron densities, which are calculated by using the local density approxi-
mation with self-interaction corrections. The electrostatic field of the crystal is
simulated by a Watson sphere in each atomic calculation. The radius of the
Watson sphere is chosen such that it gives an electrostatic potential within the
sphere equal to the Madelung potential at the crystallographic site.
Choudhury et al. (1988) have calculated the phonon dispersion relations
along the three principal symmetry directions (L, ~, A) of orthorhombic MgSi0 3
perovskite using the rigid-ion model in the quasi harmonic approximation. The
interatomic potential, consisting of a Coulomb interaction and a Born-Mayer-
type repulsive interaction, is similar to that used by Rao et al. (1988) for for-
sterite. The final parameters Z(k) and R(k) used are 2.2, 2.3, and -1.5 for charges
and 1.66, 1.20, and 1.70 Afor the radii of Mg, Si, and 0 respectively; these have
been revised from our initial report (Choudhury et al., 1988) as they yielded
elastic constants that were systematically lower than the experimental values
(Yeganeh-Haeri et al., 1989). The phonon dispersion relations are shown in Fig.
16 and the phonon density of states in Fig. 17. The phonon density of states has
peaks around 400, 600, and 900 cm -1 and no band gap, as in forsterite. The band
gap in the phonon density of states of forsterite is essentially due to the large
1 2 0 01:,. ,
800
~2
1:,
__ 400
'E 0
.!!. 0 0.5.· 0 0 0.5.· 0
i1200~~, ~2
~,
g 800
If 400
z
1120~0.0'5b'
A, 0 0~20'5b'
A, 0
Fig. 16. MgSi0 3 perovskite (orthorhombic),
800
phonon dispersion relation along the three
400
principal symmetry directions: a* (~), b* (~),
o and c* (A) directions. There are 60 phonon
o 0.5e· 0 0
PHONON WAVE VECTOR branches in each direction.
Phonon Density of States and Thermodynamic Properties of Minerals 309
III
W
!(
Iii
I&.
o
>-
I-
iii
zW
Q
':":.: 0.8
'",
::!.
S
:z:
o
~ 0.4
~
III
b = 7.903, and c = 5.557 A. There are four formula units, i.e., 32 atoms in the
unit cell. The crystal structure consists of chains of edge-sharing [AI06 ] octa-
hedra running parallel to the c-axis, cross-linked by dimers of edge-sharing
[AIOs] trigonal bipyramids and tetrahedral [Si04 ] groups (Fig. 19). A transfer-
able set of interatomic potential valid for all three Al 2 SiO s polymorphs (andalu-
site, sillimanite and kyanite) was used for the lattice dynamical calculations,
which was very similar in nature to the one used for the Mg 2 Si04 polymorphs by
Price et ai. (1987b). Because of the variable coordination of half of the Al atoms
(four, five, and six in sillimanite, andalusite, and kyanite, respectively), the poten-
tial used for andalusite is not expected to be very accurate. The calculated
phonon frequencies at the zone center show moderate agreement with experi-
mental IR and Raman measurements (see also Iishi et aI., 1979). The phonon
dispersion relation along c* was calculated and some of the low-frequency
phonon branches were measured by inelastic neutron scattering. Although the
qualitative agreement is good and the calculated acoustic branches match the
experimental measurements well, the optic branches show large discrepancies
between the calculated and measured ones. The calculated phonon density of
states is shown in Fig. 20, which also illustrates the phonon density of states
based on IR and Raman measurements by Kieffer (1985) and Salje and Werneke
(1982). Apparently, there are no band gaps in the phonon density of states.
Perhaps this is due to the fact that the internal vibration frequencies of the
[AIOs] group span the band gap between the ranges of external vibration
frequencies (translations of Al atoms, and translation and rotation ofthe [Si04 ]
group) and the internal vibration frequencies of the [Si04 ] group. If this argu-
Phonon Density of States and Thermodynamic Properties of Minerals 311
o 500 1000
FREQUENCY (1/CM)
i 160 ANDALUSITE
~
3 120
>-
~
III
Il.
rJ 80
iii
CD
l:
40
ment is correct, small band gaps are expected to occur in the phonon density of
states of sillimanite and kyanite. The calculated specific heat of andalusite (Fig.
21) shows good agreement with the experimental DSC measurements by Salje
and Werneke (1982) in the 150 to 500 K range. However, a comparison of the
calculated low-temperature specific heat (5-300 K) with the experimental mea-
surements by adiabatic calorimetry (Robie et aI., 1984) is necessary to evaluate
the accuracy of the potential model used. It will also be very interesting to
calculate the phonon density of states of sillimanite to see if the predicted band
gap exists and, if so, to explore its effect on thermodynamic properties. Winkler
et al. (1991) have also calculated the specific heat of sillimanite by lattice dy-
312 S. Ghose, N. Choudhury, S.L. Chaplot, and K.R. Rao
namical methods, but no details are given. The calculated values are in good
agreement with DSC measurements by Salje and Werneke (1982), but the agree-
ment in the low-temperature region (0-80 K) is poor. The reason for this
discrepancy again is most likely due to the approximate nature of the inter-
atomic potential used rather than the poor sampling of the Brillouin zone.
Conclusions
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Wolf, F.H. and Jeanloz, R. (1985). Lattice dynamics and structural distortions of CaSi03
and MgSi0 3 perovskites. Geophys. Res. Lett. 12,413-416.
Yeganeh-Haeri, A., Weidner, DJ., and Ito, E. (1989). Single-crystal elastic moduli of
magnesium metasilicate perovskite, in Perovskite: A Structure of Great Interest to
Geophysics and Materials Science, A. Navrotsky and DJ. Weidner, eds., pp. 13-25.
American Geophysical Union, Washington, D.C., pp. 13-25.
Chapter 12
Thermal Expansion Studies of
(Mg,Fe)2Si04 -Spinels Using
Synchrotron Radiation
L.c. Ming, M.H. Manghnani, Y.H. Kim,* S. Usha-Devi,t
J.-A. Xu, and E. Ito
Introduction
On the basis of high-pressure-temperature studies of the systems (Mg,FehSi04 ,
(Mg,Fe)Si03 , and (Mg,FehAI 2 Si 3 0 4 , it is now well recognized that seismic
discontinuity at 400-km depth is due to the olivine-spinel transformation in
(Mg,FehSi04 and that the (Mg,FehSi04 -spinel is one of the most abundant
mineral phases in the transition zone (e.g., see Ringwood, 1975; Weidner, 1985;
Weidner and Ito, 1987; Bina and Wood, 1987; Katsura and Ito, 1989). An
accurate knowledge of the density (p) as a function of pressure and temperature
of this phase is therefore of paramount importance in realistically modeling the
Earth's transition zone in terms of mineralogical assemblages. The density-
pressure relationship for (Mg,FehSi04 -spinels and their structural analogs has
been studied quite extensively using ultrasonic interferometry (e.g., see Mizutani
et aI., 1970; Chung, 1971; Wang and Simmons, 1972; Liebermann, 1975), Bril-
louin scattering (Weidner et al., 1984), and X-ray diffraction (Mizukami et aI.,
1975; Mao et aI., 1969; Wilburn and Bassett, 1976; Sato, 1977). However, data for
the density-temperature relationship for (Mg,FehSi04 -spinels are not so well
established, mainly due to one of three problems: 1) Data for the Mg 2 Si04 -spinel
are limited to a single experiment (Suzuki et aI., 1979); 2) the thermal expansion
measurements have been made in a narrow range of temperature, as in the case
of the Fe 2 Si04 -spinel where data have been obtained only between 8 and 398°C
(Mao et aI., 1969); and 3) anomalously high thermal expansion values have been
obtained in the case of single-crystal measurement of the Fe 2 Si04 -spinel at
temperatures above "'400°C (Takeuchi et aI., 1984; Yamanaka, 1986).
The first two problems are related to the small amounts of the high-pressure
phases that are generally available for the powder X-ray diffraction technique
employed. In addition, the high-pressure phase, which is metastable at ambient
conditions, converts back to the low-pressure phase on heating at 0.1 MPa.
Therefore, unless the data are acquired rapidly so as to outpace the conversion
316 L.c. Ming, M.H. Manghnani, Y.H. Kim, S. Usha-Devi, J.-A. Xu and E.lto
Experimental Methods
Samples
Instrumentation
The spinel sample was heated either in air or a vacuum (10- 4 -10- 5 torr) in an
improved diamond-anvil cell (DAC) equipped with a micro-resistive heater
(Ming et at, 1987) and X-rayed under in-situ high-temperature conditions using
energy-dispersive X-ray diffraction techniques. All the measurements were car-
ried out at Stanford Synchrotron Radiation Laboratory (SSRL). The experimen-
tal setup at SSRL and the temperature calibration procedures have been de-
scribed previously (Ming et aI., 1983, 1987).
The sample temperature was monitored within ± 3°C by a precalibrated
Pt-Pt 10% Rh thermocouple placed adjacent to the sample (Ming et aI., 1983,
1987). All the X-ray diffraction data were collected by an intrinsic Ge detector
with 2() ~ 13°. Runs conducted in a vacuum were carried out to 900°C and those
in air were limited to less than 650°C in order to prevent the oxidation of the
tungsten carbide and the graphitization of diamond anvils. With the storage ring
operating at 3.0 Ge V and 30 to 60 rnA, each spectrum was collected for 5 min in
the live-time mode.
Data Reduction
In the case of a cubic material, where (d/do)(hkl) = (a/a o ), the thermal expansion
can be evaluated by using either Eq. (1) or (2). For noncubic materials, either Eq.
(1) or (2) first provides the values for each d(hkl)' which can, in turn, be fitted by
a least-squares program to yield the lattice parameters and thus the thermal
expansion. It is evident that the same result will be obtained if the same diffrac-
318 L.c. Ming, M.H. Manghnani, Y.H. Kim, S. Usha-Devi, J -A. Xu and E. Ito
(112)
OLIVINE
~m
("0)
(122)
121"
(042)
gm
>-
J-
C/) - h
Z
W
I-
Z
--------~--~'---~d
-------~---- c
SPINEL
9.96 16.25 22.53 28.82 35.11 41.40
PHOTON ENERGY
Fig. 1. Energy dispersive X-ray diffraction spectrum of a (Mg O.4 ,Fe o.6 h Si04 -spinel in a
vacuum (10- 4 - ro- 5 torr) and at a temperature (in 0c) of (a) 330, (b) 408, (c) 490, (d) 570,
(e) 650; (f), (g), and (h) are at 740°C at 2-min intervals. The data also show the back
transition of spinel ~ olivine at high temperatures (Ming and Manghnani, 1988).
tion lines are used in both equations. Based on four to five diffraction peaks
observed at high temperatures, the values of a/ao and thus l1Vo in this study
have been determined to be ±O.1 to 0.15% and OJ to 0.45%, respectively.
Results in a Vacuum
Thermal expansion measurements on Mg 2Si04 -, (Mgo.s,Feo.2hSi04 -, and
(MgQ.4,Fe o.6 hSi04-spinels have been carried out to 900, 735, and 768°C, respec-
Thermal Expansion Studies of (Mg,FehSi04 -Spinels 319
tively. The results are listed in Table 1 and plotted in Figs. 2 to 4. These data can
be expressed in terms of second-order polynomials. For Mg 2Si04,
For (Mgo.s,Feo.2hSi04,
Mg 2 Si04
1339 21 1.000 15.1
1341 220 1.0045(12) 21.1
1342 286 1.0046(35) 22.1
1343 314 1.0058(15) 22.5
1344 396 1.0081(18) 23.3
1345 416 1.0069(12) 23.5
1346 450 1.0075(17) 23.8
1347 498 1.0101(22) 24.2
1348 584 1.0126(11) 24.8
1349 648 1.0137(52) 25.1
1350 673 1.0152(43) 25.2
1351 704 1.0182(12)"
1352 761 1.0202(13t
1353 795 1.0226(22)"
1354 810 1.0193(27)"
1355 900 1.0246(66)"
320 L.c. Ming, M.H. Manghnani, Y.H. Kim, S. Usha-Devi, J.-A. Xu and E. Ito
Table 1. (Continued.)
( Mgo.sFeo.2)2Si04
1246 21 1.()()()() 17.6
1247 154 1.0019(21) 22.3
1248 182 1.0018(18) 23.0
1249 229 1.0032(26) 23.8
1250 262 1.0065(46) 24.4
1251 285 1.0060(19) 24.7
1252 342 1.0065(12) 25.4
1254 374 1.0074(13) 25.8
1255 426 1.0118(25) 26.2
1256 478 1.0117(27) 26.7
1257 532 1.0113(43) 27.0
1258 564 1.0142(27) 27.3
1259 596 1.0155(20) 27.4
1260 614 1.0140(22) 27.6
1261 632 1.0185(21) 27.7
1262 650 1.0168(17) 27.8
1263 670 1.0208(20)"
1264 682 1.0208(20)"
1265 700 1.0224(09)"
1266 735 1.0205(45)"
1267 748 1.0241(17)"
( MgO.4Feo.6)2Si04
1302 21 1.0000 18.2
1303 128 1.0017(21) 21.7
1304 228 1.0040(13) 23.4
1305 292 1.0057(17) 24.2
1306 350 1.0060(18) 24.8
1307 407 1.0088(20) 25.2
1308 443 1.0117(14) 25.5
1309 471 1.0123(18) 25.7
1310 486 1.0106(32) 25.8
1311 503 1.0113(37) 25.8
1312 536 1.0124(36) 26.0
1313 571 1.0136(42) 26.2
1314 600 1.0138(15) 26.4
1315 644 1.0155(13) 26.6
1316 650 1.0149(22) 26.8
1317 685 1.0153(21) 26.8
1318 712 1.0155(23) 26.9
1319 724 1.0170(12) 26.9
1320 733 1.0169(15) 27.0
1321 753 1.0196(10) 27.0
1322 768 1.0196(05) 27.1
"Not included in the final refined analysis; see explanation in the text.
Thermal Expansion Studies of (Mg,Fe)2Si04-Spinels 321
1.03 r-;============;--------~
o Exp. Dala (in va c)
- Theory
o
1.02
o
>
......
>
1.01
1.00 4iJo'~---'-----'-------'------'------.J
o 200 400 600 800
Temperature, °C
1.03 r.=============;----------,
o Exp. Data (in vac)
- Theory
o
1.02
o
>
......
>
1.01
1.004o":=:::..----'------'-------'--------l
o 200 400 600 800
Temperature, °C
1.03 r,:::==========::::::;--------~
o Exp. Data (in vac)
- Theory
1.02
c
;>
......
;>
1.01
1.00 0Y!F'----2--'O'-0----4-'0-0----6--'-0-0- - - - - - 1
800
Temperature, °c
Fig. 4. Thermal expansion data of a (MgQ.4,Feo.6)2Si04 -spinel (in a vacuum). Results
obtained on the basis of Griineisen theory are shown in a solid line.
(7)
where p is the number of atoms in the molecular formula, R the gas constant,
and (}D the Debye temperature. Hill and Jackson (1990) suggested that the
thermal expansion of a material at a given temperature T with respect to the
reference temperature (room temperature TR ) can be written as
V(T) {2k + 1- J[1 - 4kE((}D' T)/QJ} (8)
{2k + 1 -
~--
Table 3. A comparison of the Debye temperature (BD ) and the Griineisen parameter (y)
of (Mg,Feb Si04-spinels between the best-fit solutions obtained in this study and those
calculated on the basis of the thermochemical data (Watanabe, 1987).
y y
On the basis of the heat capacity measurement (Cp ) for the temperature range
between 350 and 700 K for four (Mg,Fe)2Si04-spinels, Watanabe (1987) calcu-
lated their thermal gamma (Ylb) and the Debye temperature ((}D) using the pub-
324 L.e. Ming, M.H. Manghnani, Y.H. Kim, S. Usha-Devi, J.-A. Xu and E. Ito
lished values of y, Yo, and Ks. His results are also given in Table 4 for compari-
son. As can be seen, the best values of y an (}D obtained in this study are too high.
The most probable reason is that the present experimental data are not accurate
enough to permit a two-parameter fit. This situation can be greatly improved if
Watanabe's (}D is assumed and we allow only the Yth to be evaluated using Eq.
(8). Results given in Table 4 show fairly good agreement with those of Watanabe
(1987). Both data sets indicate that the value of Y increases slightly with an
increasing Fe content, in spite of the fact that the values obtained in this study
are systematically lower that those of Watanabe's. The slightly higher values of
Watanabe (1987) are probably a result of the higher (xo used for both MgSi04-
and Fe 2Si04-spinels in his calculation.
The thermal expansion of Mg 2 Si04 (in polycrystalline form) has been studied
in air to 700°C using an angular-dispersive X-ray diffraction technique (Suzuki
et aI., 1979). They performed two different fitting procedures for evaluating their
results. Their unconstrained solution I yields 1.43 and 1198 K for y and (}D'
respectively. By assuming the value of (}D to be 849 K based on ultrasonic
measurements (Liebermann, 1975), they obtained in their solution II a value of
1.27 for y. By adjusting the values of Vo and Ko to those used in this study, we
obtain values of y as 1.27 and 1.13 for their solutions I and II, respectively. Our
thermal expansion data yield a value of y = 1.06 at (}D = 849 K, which is in fairly
good agreement with the y = 1.13 obtained by Suzuki et al. in their solution II.
Table 5 summarizes the values of (}D and y obtained by different investigators
using various experimental methods and/or assumptions.
Once the y and (}D are known, the value of V(T)/VR as a function of tem-
perature can be calculated using Eq. (8). Plotting the calculated V(T)/VR for
Mg 2Si04, (Mgo.8,Feo.2)2Si04' and (Mgo.4,Feo.6hSi04 in Figs. 2, 3, and 4,
respectively, we found some anomalous high values of V(T)/VR for Mg 2Si04
and (Mgo.8,Feo.2hSi04' but not (Mgo.4,Feo.6hSi04' A similar study of the
Fe 2Si04-spinel also did not show such an anomaly at high temperatures (Man-
ghnani et aI., 1991). This phenomenon may be related to some unknown struc-
tural change to the Mg-rich spinels at high temperatures, as indicated in the case
of MgSi0 3 -perovskite (Wang et aI., 1991). It may also be attributed to the y ~ p
Thermal Expansion Studies of (Mg,FehSi04 -Spinels 325
transition prior to finally reverting to the olivine, as can be expected from the
well-established P - X phase diagram of (Mg,Fe)2Si04 -spinels (e.g., Akimoto,
1987; Katsura and Ito, 1989). The reason that we fail to distinguish the f3 phase
from the y phase in our X-ray diffraction spectrum is probably because the five
or seven observed diffraction peaks are the major ones common to both phases
(Ito et al., 1974; Moore and Smith, 1970). Although the f3 phase has more lines,
326 L.c. Ming, M.H. Manghnani, Y.H. Kim, S. Usha-Devi, loA. Xu and E.lto
they are not observed in this study. This may be due, in part, to the weak
intensity of those additional peaks of the p phase and due, in part, to the low
resolution power of the energy-dispersive X-ray diffraction method employed.
Further studies with an angular-dispersive X-ray diffraction method employing
a wide-angle position-sensitive detector would be valuable in elucidating the
cause of such behavior. In order to deduce accurately the value of y and hence (X
for the spinel phase, we repeated the calculation described earlier without includ-
ing the anomalous data points at high temperatures. The final recalculated
values are given at the last row of Table 4. The relationship between y and the
composition of (Mg,FehSi04-spinels can be expressed linearly as y = 1.09 +
0.16 XFe with a correlation coefficient of 0.946.
Using the refined values of y and eD' we calculated the thermal expansion
coefficient «(X) as a function of temperature for Mg 2Si04, (Mgo.8,Feo.2hSi04'
and (Mgo.4,Feo.6hSi04' Results are listed in the last column of Table 1 and are
also plotted together with those of Fe2Si04 (Manghnani et aI., 1991) in Fig. 5.
This result shows that the values of (X for (Mg,FehSi04-spinels increase slightly
with increasing Fe content at room temperature, but that such a relationship
becomes much less obvious at some high temperatures. Values of (xo of
(Mg,FehSi04-spinels obtained in this study and those from other studies are
plotted in Fig. 6. In general, there is good agreement among the various studies.
However, our results for Mg 2Si04 and Fe 2Si04 are slightly lower than previous
experimental results (Suzuki et at, 1979; Mao et aI., 1969), and the theoretical
value obtained for Mg 2Si04 is slightly lower than experimental values. On the
basis of results obtained from synchrotron studies and treated with Griineisen
theory (i.e., this study; Manghnani et aI., 1991), the compositional dependence of
35
30
t;t;bA tn
ff}ft1l/&o+d'*<r 0
t;
wbA Pl:~+ 0
t; Q-0 ~o 0
t; B- 0
t; ill 00
0
0
t;Fa lOO (Manghnanl et al., 1991)
20 t;
+ Fo 40 (this study)
+ o Foso (this study)
0 o Fo lOO (this study)
0
15
o 200 400 600 800 1000
Temperature, °C
30r-------------------------------------~
. 'CI)
25
QI
-=
iii'~ 20 ...---
.-I
~
-=
15
Mol. % of Mg
35
o This study
x Suzuki, 1979
'V Fei and Saxena, 1986
30 'V
";' 'V
'V 0
CI)
'V Xo [lJ
QI 0 0 0
W
-= 'V 0
0
'C
25 DO 0
<:.-I 'V
x
'V
DO
0
~ w
20
)(
0
15
0 200 400 600 800 \000
Temperature, °c
Fig. 7. A comparison of the IX of Mg 2Si04-spinels in this study and those previously
reported.
328 L.c. Ming, M.H. Manghnani, Y.H. Kim, S. Usha-Devi, I-A. Xu and E. Ito
Observations in Air
Table 6. (Continued.)
1.04
o O.2Fe
+ O.4Fe
1.03 o O.6Fe +
+ 0
Q 1.02
0 0
;;. 0
.......
;;. 1.01 0 00 00
0 0
000
++
~ o[J)J -l@ 0 ~ 0 0
1.00 +
0
0
0.99
0 200 400 600 800
Temperature, °C
sion decreases at temperatures between 300 and 400°C and then increases very
rapidly at temperatures above 400°C.
Figures 9 and 10 show the comparison ofthe thermal expansion data obtained
in air and a vacuum for (Mgo.s,Feo.2hSi04 and (Mgo.4,Feo.6hSi04' respec-
tively. It is found that in the case ofa Fe-poor spinel such as (Mgo.s,Feo.2hSi04'
the thermal expansion data obtained in air are similar to those obtained in a
vacuum; however, in the case of Fe-rich spinels such as (Mgo.6,Feo.4hSi04 and
(Mgo.4,Feo.6)2Si04' the results obtained in air are very much different from
those obtained in a vacuum. Similar anomalous behavior has also been recently
observed in a Fe2Si04-spinel (Manghnani et aI., 1991).
Anomalous thermal expansion has also been observed in other materials such
as BaTi03 (Shirane and Takeda, 1950), PbTi03 (Shirane and Hoshino, 1951),
and Zr02 (Adams et aI., 1985) and has been correlated to phase transition at high
temperature. This is b,ecause the high-temperature phase has a smaller volume
than that of the low-temperature phase. In view of the fact that the molar volume
of the olivine (IX) phase is '" 8 to 10% larger than that of the spinel (y) phase
(Ringwood and Major, 1970), it is difficult to reconcile our anomalous thermal
expansion data with the y --+ IX transition for the Fe-rich spinels.
In a recent study on the back transition of (Fe, Mgh Si04-spinels at high
temperatures, we found that the Fe-rich spinels are oxidized when heated in air
(Ming et aI., 1991). Figure 11 shows the energy spectrum of (Mgo.4,Feo.6hSi04
at various temperatures, indicating that the formation of the Fe 20 3 (hematite)
phase starts between 298 and 388°C and its amount increases with increasing
temperature. Because all the diffraction peaks of the spinel phase are present up
1.03 r;::::======::;----------~
I + ~n v~cuum I
0
ID aIr
o
o
rn 0
1.02
~ .flo
i> o
..... 0+0
i> o
1.01 +
rlr
-4ld- .jO
+ ++
+ DO
++ 0
1.004l...B----~-------'------'-------'
o. 200 400 600 ~OO
Temperature, °C
Fig. 9. The thermal expansion data of a (Mgo.s,Feo.2hSi04 -spinel obtained in air appear
to be the same as those obtained in a vacuum.
Thermal Expansion Studies of (Mg,Fe)2Si04-Spinels 331
1.03
o in vacuum I
I + in air
+
1.02 00
+ I][)
~ 00
Q 0
>
...... 1.01
0000 ~
0
> 0
+
++ ljl + ~ + +
+0+
1.00 + 0
+
+
0.99'--_ _ _--'-_ _ _ _"--_ _ _---'-_ _ _----.J
o 200 400 600 800
Temperature, °C
Fig. 10. The thermal expansion data of a (Mgo.4 ,Feo.6 hSi04-spinel obtained in air
appear to be quite different from those obtained in a vacuum.
! . .
0
T,·C
! !
23
23
>-
I-
410
enz 486
w ---",.J 594
I- 582
Z
--~_~1509
_ _-"'_~ 472
_ _-",~~ 430
388
298
262
- - - - - <; -: --<;'---:::---...-~
-
E
ft".,
""-.'"
0- --_
"-0
228
Spln!1 - - ~ :;; .:
to 596°C (see Fig. 11), it is a certainty that as the oxidation of the Fe-rich spinel
phase proceeds, the spinel structure remains intact. Most likely, the Fe atoms on
the surface of the spinel grains combine with oxygen to form Fe 2 0 3 , thus leaving
vacant sites behind. The Fe atoms next to the vacant sites then hop into these
sites and the oxidation process continues. It is clear that as this oxidation
proceeds at high temperatures, the spinel phase becomes a defect structure with
many vacant Fe sites inside the structure. Such a defect structure will result in a
decrease of the zero-pressure density and at the same time in an increase of the
thermal expansion coefficient. If the defect structure is formed fast enough at
some temperature Tn, it is anticipated that, as the temperature increases, the
thermal expansion will first increase slowly, then will have a sudden decrease
drop at Tn, and finally above Tn, will increase rapidly. This represents the most
probable scenario of the thermal expansivity of the Fe-rich spinels in air.
Acknowledgment
The authors thank the staff of the Stanford Synchrotron Radiation Laboratory
for providing the facilities for this research. The authors would also like to thank
S.K. Saxena, R. Jeanloz, and W.A. Bassett for reviewing this paper; X. Li and V.
Askarpour for helpful discussions; L.-J. Wang for some data analysis, and J.
Balogh for modifying the high P-T DAC used in our study. This research was
supported by the Earth Scienc~ Division, National Science Foundation (NSF
Grant EAR87-08482). School of Ocean and Earth Science and Technology
contribution number No. 2683.
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Appendix
Tables of Molar Volumes and Fugacities
°
of Fluids on C-O-H System (H 2 0, CO 2 ,
CH 4 , CO, 2 , H 2 ) up to 1 Mbar and
4000K
The values of calculated molar volume and fugacity have been calculated. Molar
volume was calculated as a solution ofEq. (9) in Chap. 3. The equation has been
solved with a Newton-Raphson technique.
The value of fugacity under the specified temperature and pressure was
calculated with the numerical integration of the VdP term as follows:
The coefficients for the calculation of RTlnf(5 Kbar, T) with expression (10)
are listed in Table 5. The computer program in Fortran for the calculation of
fluid properties (~G, fugacity, molar volume) is available on request.
P (GPa)
Table 1. (Continued.)
P (GPa)
P (GPa)
T(K) 0.5 0.6 0.7 0.8 0.9 1.0
373.15 34.101 32.589 31.417 30.463 29.664 28.978
473.15 35.828 34.005 32.623 31.519 30.605 29.829
573.15 37.732 35.546 33.925 32.651 31.610 30.733
673.15 39.794 37.206 35.321 33.858 32.676 31.691
773.15 41.983 38.969 36.799 35.135 33.801 32.698
873.15 44.257 40.811 38.347 36.471 34.978 33.751
973.15 46.574 42.706 39.948 37.857 36.200 34.845
1073.15 48.892 44.629 41.584 39.279 37.458 35.972
1173.15 51.182 46.555 43.238 40.725 38.742 37.126
1273.15 53.422 48.465 44.895 42.184 40.042 38.298
1500.00 58.261 52.675 48.600 45.482 43.009 40.991
2000.00 67.674 61.104 56.208 52.400 49.341 46.822
2500.00 75.640 68.393 62.924 58.623 55.136 52.243
3000.00 82.542 74.777 68.867 64.187 60.370 57.184
3500.00 88.653 80.463 74.193 69.203 65.115 61.690
4000.00 94.155 85.603 79.026 73.772 69.453 65.825
2.0 3.0 4.0 5.0 6.0 7.0
373.15 24.988 22.972 21.651 20.681 19.921 19.301
473.15 25.440 23.288 21.895 20.881 20.092 19.449
573.15 25.911 23.614 22.147 21.087 20.266 19.600
673.15 26.401 23.951 22.406 21.298 20.445 19.755
773.15 26.909 24.298 22.673 21.515 20.627 19.913
873.15 27.436 24.657 22.946 21.737 20.814 20.075
973.15 27.980 25.025 23.227 21.964 21.005 20.240
1073.15 28.541 25.404 23.514 22.197 21.201 20.408
1183.15 29.118 25.792 23.809 22.434 21.400 20.580
1273.15 29.709 26.190 24.110 22.677 21.604 20.755
1500.00 31.095 27.125 24.817 23.247 22.080 21.165
2000.00 34.295 29.315 26.480 24.585 23.200 22.127
2500.00 37.547 31.613 28.248 26.019 24.403 23.162
3000.00 40.723 33.940 30.075 27.517 25.669 24.256
3500.00 43.761 36.238 31.917 29.049 26.977 25.393
4000.00 46.641 38.473 33.742 30.588 28.304 26.558
338 Tables of Molar Volumes and Fugacities of Fluids on C-O-H System
Table 2. (Continued.)
P (GPa)
P(GPa)
Table 3. (Continued.)
P (GPa)
P (GPa)
T(K) 0.5 0.6 0.7 0.8 0.9 1.0
373.15 33.032 31.364 30.069 29.018 28.139 27.387
473.15 34.632 32.682 31.194 30.003 29.017 28.181
573.15 36.365 34.097 32.395 31.049 29.945 29.017
673.15 38.226 35.607 33.669 32.153 30.923 29.894
773.15 40.200 37.204 35.012 33.315 31.947 30.812
873.15 42.271 38.380 36.419 34.529 33.016 31.768
973.15 44.418 40.621 37.883 35.791 34.127 32.760
1073.15 46.620 42.415 39.394 37.096 35.275 33.785
1173.15 48.857 44.250 40.944 38.436 36.455 34.840
1273.15 51.113 46.112 42.524 39.806 37.663 35.920
1500.00 56.232 50.384 46.176 42.989 40.480 38.446
2000.00 67.187 59.701 54.266 50.125 46.856 44.203
2500.00 77.497 68.602 62.101 57.125 53.180 49.968
3000.00 87.209 71.037 69.580 63.852 59.300 55.585
3500.00 96.430 85.065 76.717 70.296 65.184 61.006
4000.00 105.255 92.749 83.558 76.481 70.843 66.231
2.0 3.0 4.0 5.0 6.0 7.0
373.15 23.075 20.946 19.573 18.575 17.800 17.172
473.15 23.496 21.240 19.801 18.754 17.961 17.313
573.15 23.930 21.542 20.035 18.956 18.125 17.456
673.15 24.380 21.852 20.275 19.152 18.292 17.601
773.15 24.843 22.170 20.519 19.352 18.462 17.749
873.15 25.320 22.495 20.769 19.556 18.634 17.900
973.15 25.811 22.829 21.024 19.763 18.810 18.052
1073.15 26.315 23.169 21.283 19.974 18.989 18.207
1173.15 26.832 23.517 21.548 20.189 19.170 18.365
1273.15 27.361 23.873 21.818 20.408 19.354 18.524
1500.00 28.603 24.704 22.447 20.917 19.783 18.895
2000.00 31.502 26.646 23.914 22.099 20.776 19.752
2500.00 34.542 28.705 25.472 23.355 21.829 20.660
3000.00 37.635 30.836 27.098 24.669 22.932 21.611
3500.00 40.720 33.001 28.766 26.025 24.075 22.599
4000.00 43.761 35.172 30.456 27.409 25.246 23.614
342 Tables of Molar Volumes and Fugacities of Fluids on C-O-H System
Table 4. (Continued.)
P (GPa)
Table 5. (Continued.)
P(GPa)
Table 6. (Continued.)
P (GPa)
P (GPa)
Table 7. (Continued.)
P (GPa)
P (GPa)
Table 8. (Continued.)
P (GPa)
Table 9. (Continued.)
P(GPa)
P(GPa)
P(GPa)
P (GPa)
M N
Madelung constant, 170,246 N 2 , 63, 64
magnetic contributions, 111 NaAISi04, 140, 145-147
364 Index
W molar volume, 22
Wasastjerna-Hovi model, 164, 165 neutron diffraction, 31
Wasastjerna's assumption, 173 phase diagram, 4,21
water, P-V-T,22,63,65,69-72,76
ab-initio, 16,21,40, 72, 76 quantum corrections, 9
BJH model, 17, 19,23,31,33,34,43, self-diffusion coefficients, 39
45,48 shock wave data, 11, 70
fJ-tridymite, 50 simple point charge model, 15
central face model, 15 specific volume, 71
compressibility, 3, 21, 24 supercritical, 4, 8, 17,25-29, 32, 39,48,
critical point, 4, 13, 15,21,23,40 49
density, 20-22, 67 thermal expansivity, 24, 25
dipole moment, 12 TIP4P model, 14, 17,20-23,29,65-73,
enthalpy, 22 76
equation of state, 71, 72, 76 X-ray diffraction, 31, 32
freezing point, 21 well depth, 66
heat capacity Cp, 23 wollastonite, 110, 147
heat capacity Cv, 23 wurtzite, 172, 189
HGK equation of state, 18-20,24
in quartz, 50
intermolecular potentials, 11, 15 X
internal energy, 22 xe,64
liquid-vapor coexistence, 4, 7, 15,21 x-ray measurement, 179
Lennard-Jones model, 65
MC simulation, 9, 10, 15, 17,20,22,23,
28,31,33 Z
MCY model, 17 zero-point conditions, 204
MD simulation, 9, 10, 13, 15, 17, 19,22, zero-point energy, 135, 242
27,31,33,34,45,48,68,70,73 zone-center optic phonons, 308
melting point, 21 zone-center phonons, 301