Physics of the Earth and Planetary Interiors 217 (2013) 36–47
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Physics of the Earth and Planetary Interiors
journal homepage: www.elsevier.com/locate/pepi
Phase equilibria of (Mg,Fe)2SiO4 at the Earth’s upper mantle conditions
from first-principles studies
Yonggang G. Yu a,⇑, Victor L. Vinograd a, Björn Winkler a, Renata M. Wentzcovitch b
a
b
Institute of Geosciences, University of Frankfurt, Altenhöferallee 1, 60438 Frankfurt a.M., Germany
Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, MN 55455, USA
a r t i c l e
i n f o
Article history:
Received 4 September 2012
Received in revised form 2 January 2013
Accepted 11 January 2013
Available online 8 February 2013
Edited by Kei Hirose
Keywords:
Fe2SiO4
Fayalite
Wadsleyite
Ringwoodite (c-spinel)
Phase equilibrium
Thermodynamic properties
410-km and 520-km discontinuity
LDA + U
a b s t r a c t
Phase equilibria of a, b, and c (Mg,Fe)2SiO4 are important to understanding the mineralogy of the Earth’s
upper mantle. Using the first principles approach, we studied thermodynamic properties and phase stability fields of Fe2SiO4. We show that the correct phase transition sequence in Fe2SiO4 (a ? c) can be
obtained with the DFT + self-consistent Hubbard U method, while standard DFT methods (LSDA and rGGA) as well as the DFT + constant U method fail the task. The vibrational virtual crystal approximation
was used to derive the phonon density of state of the Fe2SiO4 polymorphs. High-pressure thermodynamic
properties of Fe2SiO4 are then derived with the aid of the quasi-harmonic approximation. They are in very
good agreement with experiments. The phase diagram of the (Mg,Fe)2SiO4 system is calculated under the
assumption of ideal mixing within a, b, and c solid solutions. The model permits the investigation of the
temperature and pressure effects on the phase boundaries. The widths of the divariant a–b and b-c loops
are barely sensitive to temperature between 1473 and 1873 K. This study shows the promise of applying
the DFT + self-consistent Hubbard U method to study phase equilibria of iron-bearing Earth minerals.
Ó 2013 Elsevier B.V. All rights reserved.
1. Introduction
Phase equilibria of (Mg,Fe)2SiO4 polymorphs are important for
understanding the seismic density and velocity profiles in the
Earth’s upper mantle and transition zone (e.g., Agee, 1998). The
pioneering experimental study of Ringwood (Ringwood, 1975) followed by successive investigations (Suito, 1977; Yagi et al., 1979;
Akaogi et al., 1984; Price et al., 1987; Katsura and Ito, 1989; Akaogi
et al., 1989; Morishima et al., 1994; Suzuki et al., 2000; Inoue et al.,
2006) provided persuasive arguments suggesting that the sharp
density increase at 410-km depth (5–10 km thick) reflects the
olivine (a) to wadsleyite (b) transition, while the much broader
(10–50 km thick) and more complex 520-km discontinuity (Wiggins and Helmberger, 1973; Shearer, 1990; Revenaugh and Jordan,
1991; Ryberg et al., 1997; Jones et al., 1992; Gossler and Kind,
1996; Nolet et al., 1994; Gu et al., 1998; Deuss and Woodhouse,
2001) may be caused by the wadsleyite (b) to ringwoodite (c)
transformation. Later studies suggested that the exsolution of calcium perovskite from garnet may also contribute to the velocity
jump at 520 km depth (Irifune and Ringwood, 1987; Gasparik,
1990; Weidner and Wang, 2000; Saikia et al., 2008).
⇑ Corresponding author.
E-mail address: [email protected] (Y.G. Yu).
0031-9201/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved.
http://dx.doi.org/10.1016/j.pepi.2013.01.004
Although the phase relations in the Mg;Fe2 SiO4 system are well
understood from experiments (Katsura and Ito, 1989; Akimoto,
1987; Frost, 2003; Katsura et al., 2004), its petrological modeling
still poses difficulties. Due to the instability of the b phase at high
Fe content the thermodynamic properties of b-Fe2SiO4 have been
uncertain. The mixing properties of the a- and c-type solid solutions are not well constrained either. Only a limited composition
range was covered by calorimetry measurements (Akaogi et al.,
1989). The lack of a reliable and self-consistent high-pressure thermodynamic data set for the ferromagnesium silicates ((Mg,Fe)2SiO4) has been a persistent challenge for phase equilibrium
assessment studies (Navrotsky and Akaogi, 1984; Bina and Wood,
1987; Akaogi et al., 1989; Fei and Saxena, 1986; Stixrude and Bukowinski, 1993; Frost, 2003; Stixrude and Lithgow-Bertelloni, 2011)
as many thermodynamic parameters have to be simultaneously fitted to reproduce few experimental constraints. The experimental
database has recently been substantially improved due to the element partitioning studies between the three phases (Frost, 2003).
On the other hand, the first principles based quasiharmonic calculations (e.g., Wentzcovitch et al., 2010b) have emerged as a reliable
tool in studying mineral properties at high pressures and temperatures. It has been shown that phase transitions in magnesium silicates can be reliably predicted (Tsuchiya et al., 2004; Yu et al.,
2007, 2008). Recently significant progress has been made in treating the strongly correlated behavior of the 3d electrons in iron-
Y.G. Yu et al. / Physics of the Earth and Planetary Interiors 217 (2013) 36–47
bearing silicates with the aid of the density functional theory + self-consistent Hubbard U (DFT + U sc ) method (Hsu et al.,
2011; Yu et al., 2012; Hsu et al., 2012).
In the past few decades a rich literature has been accumulated
on physical properties of the Mg2SiO4 polymorphs, including
their equations of state, compressibility, elasticity, thermal
expansivity, heat capacity, vibrational properties, etc. These data
have been reviewed and summarized in recent computational
studies (Wentzcovitch et al., 2010b; Yu and Wentzcovitch,
2006; Wu and Wentzcovitch, 2007; Li et al., 2007; Kiefer et al.,
2001). In comparison, much less knowledge has been gained on
the properties of Fe2SiO4 polymorphs. In this study the thermodynamic properties of the Fe2SiO4 polymorphs are systematically
investigated at high pressures. We demonstrate the importance
of using the self-consistent Hubbard U parameters in predicting
the phase transition sequence in Fe2SiO4, which forms the
basis to understanding phase equilibrium in the (Mg,Fe)2SiO4
system.
2. Methods
Our density functional (Kohn and Sham, 1965) calculations
were performed using the QUANTUM ESPRESSO package (Giannozzi
et al., 2009). Both the local density approximation (LDA, Perdew
and Zunger, 1981) and the generalized gradient approximation
(PBE-GGA, Perdew et al., 1996) functionals were used. The
DFT + Hubbard U method used here is based on a rotational invariant formulation (Cococcioni and de Gironcoli, 2005) of the standard LDA + U method (Anisimov et al., 1991). The self-consistent
U sc parameter is determined by the linear response method
(Cococcioni and de Gironcoli, 2005; Hsu et al., 2011). The U parameter derived from this method corresponds to the effective electronic interaction (U–J) as in Dudarev et al. (1998) (J is the
exchange interaction). The pseudopotentials (PP) have been extensively tested in the previous studies (Wentzcovitch et al., 2010b;
Umemoto et al., 2008). They include the Troullier and Martins
(1991) type O and Si PPs, the Mg PP generated by the method of
von Barth and Car, and an ultrasoft Fe PP (Vanderbilt, 1990). The
cutoff energy were 70 Ry for the wave function and 280 Ry
for the electron density. The k-point meshes used for a, b and
c Fe2SiO4 were 4 4 2, 4 4 4, and 4 4 4, respectively,
with a shift from the Brillouin–zone center (Monkhorst and Pack,
1976). Crystal structures were relaxed under hydrostatic pressure
using the variable cell shape molecular dynamics (Wentzcovitch,
1991; Wentzcovitch et al., 1993). Dynamical matrices of the three
Mg2SiO4 polymorphs were computed directly on the 2 2 2
q-point mesh (or finer meshes) using the density functional perturbation theory (Baroni et al., 2001). They were then interpolated to
denser q-point meshes (typically 8 8 8) to obtain the vibrational density of states (DOS). Quasiharmonic approximation
(QHA) was used to calculate the Gibbs free energy as a function
of pressure and temperature (P–T). To derive the vibrational DOS
of the iron end-members, we used the vibrational virtual crystal
approximation (VVCA) (e.g., Wentzcovitch et al., 2009; Wu et al.,
2009) in which the force constant matrices of Fe2SiO4 were approximated by those of Mg2SiO4, following which the dynamical matrice were formed and then diagonalized to obtain phonon
frequencies. This simple approximation is better than the Debye
model as it retains structural features in the vibrational DOS. The
effectiveness of this method is shown in Section 3.2. The magnetic
entropy of Fe2SiO4 arising from the disorder of magnetic moments
of ferrous iron in high spin state [2R lnð2S þ 1Þ = 26.763 J/mol/K per
formula unit] does not affect phase stability fields of the three
polymorphs (note that the degeneracy of t 2g orbitals is lifted due
to local distortions of FeO6 octahedra).
37
The thermodynamic data used here for Mg2SiO4 ringwoodite,
wadsleyite, and forsterite are adopted from the previous phase
boundary calculations for Mg2SiO4 (Yu et al., 2008), in which the
total energy convergence was carefully controlled to meet the
requirement for phase boundary calculations. An extensive discussion of the thermodynamic properties of Mg2SiO4 polymorphs can
be found in the previous publications (Yu and Wentzcovitch, 2006;
Wu and Wentzcovitch, 2007; Li et al., 2007). In this report the
emphasis is on Fe2SiO4 and the phase equilibrium in (Mg,Fe)2SiO4.
3. Results
3.1. Static equation of state of Fe2SiO4 fayalite and ringwoodite
Fig. 2a compares the static equation of states of fayalite calculated by four different DFT functionals: local spin-density approximation (LSDA), LDA + U sc , spin-polarized GGA (r-GGA), and
GGA + U sc , where U sc of the M1 and M2 sites are about 2.5 eV
(Table 1). In the pressure range of the transition zone (13–
23 GPa), we find that the variation of U sc is within the uncertainty
of the method (0.1 eV) to determine U. In specific, the pressure
derivative of the U sc parameter is about 1 meV/GPa, hence it is negligible to consider the pressure (or volume) dependence of U sc . We
find the antiferromagnetic (AF) state more stable than the ferromagnetic (FM) state. The magnetic moments of iron are parallel
within each edge-sharing octahedral chain running along the b axis
(Fig. 1a), but the moments of two adjacent chains are antiparallel.
This agrees with the previous studies (Cococcioni et al., 2003; Jiang
and Guo, 2004). The AF state is then adopted in all calculations.
LSDA underestimates the volume of fayalite. The r-GGA gives
correct equilibrium volume, but predicts a much smaller bulk
modulus (96.4 GPa) compared to the experimental value of 131–
136.3 GPa (Table 2). This shows that the r-GGA is inadequate for
studying compressibility. Both LDA + U sc and GGA + U sc substantially expand the unit-cell volume relative to the standard DFT calculation by 6.6% and 3.9%, respectively (Fig. 2). The bulk modulus is
improved (K S ¼ 150 GPa by LDA + U sc and 122 GPa by GGA + U sc ). It
might be conceivable that including on-site Coulomb repulsions
among 3d electrons (the U parameter as in the DFT + U sc methods)
shall expand the equilibrium volume, however, a rigorous mathematical derivation to reveal this effect still lacks at the current
stage. For the sake of consistency with our previous LDA study
on the thermodynamics of Mg2SiO4, here the thermodynamic
properties of Fe2SiO4 are calculated with the LDA + U sc .
The results on Fe2SiO4 ringwoodite (Fig. 2b) also support the
choice of the LDA + U sc . The bulk modulus and the unit-cell volume
(static values) as predicted by the LDA + U sc (207.5 GPa, 542.9 Å3)
are close to the experimental values (187–207 GPa, 559.3 Å3).
GGA + U sc predicts K S ¼ 171:1 GPa and V 0 ¼ 585:0 Å3 (Table 2).
We note that in our calculations for c-Fe2SiO4 an AF tetragonal
structure with c/a 0.98 (Fig. 1c) was adopted. In this structure
the orbital ordering occurs in alternating layers transverse to z axis,
such that the minority electrons in one layer occupy the dxz orbital
while those in the neighboring layers occupy the dyz orbital. This
type of orbital ordering is similar to that in KCuF3 (Liechtenstein
et al., 1995).
3.2. Thermodynamic properties of Fe2SiO4 polymorphs
Recently the thermal properties of Mg-silicates have been successfully described with the quasi-harmonic approximation
(QHA) (Wentzcovitch et al., 2010b,a; Wu and Wentzcovitch,
2011; Carrier et al., 2007). Here this approach is applied to Fe-silicates. In QHA the Helmholtz free energy is given by the following
equation:
38
Y.G. Yu et al. / Physics of the Earth and Planetary Interiors 217 (2013) 36–47
Fig. 1. (a) Fayalite, (b) wadsleyite, and (c) ringwoodite Fe2SiO4. Silicon atoms are
shown in blue, oxygen in red, and iron in green and purple. (a) the octahedral sites
are labeled as M1 and M2, and the dashed lines form octahedral chains. M1–O bond
lengths are 2.086 2, 2.094 2, 2.147 2 Å while M2–O lengths are 2.062, 2.083,
2.185 2, 2.228 2 Å. The M1–O–M2 angles (connecting the M1 and M2 sites) are
94.8 ° and 97.6 °. (b) wadsleyite (virtual end member, ferromagnetic). (c) a portion
of the spinel structure with isosurfaces of spin up 3d electron density. Fe atoms at
z = 14 (with dxz spin-up orbital) are in anti-ferromagnetic coupling with those at z = 12
(with five spin-up 3d orbitals, and one spin down dyz obital [notshown]). (For
interpretation of the references to color in this figure legend, the reader is referred
to the web version of this article.)
1X
hxj ðq; VÞ
2 q;j
X
þ kB T ln½1 expðhxj ðq; VÞ=kB TÞ;
FðV; TÞ ¼ U 0 ðVÞ þ
ð1Þ
q;j
where xj ðq; VÞ is the jth phonon mode at the q point in the first
Brillouin zone of the unit cell volume V.
In this work we do not include the energy of spin excitations
and the magnetic state is considered frozen. This contribution
might affect equations of state and phase boundaries somewhat.
However, such calculations are challenging especially for materials
with strongly correlated ions. Therefore in this study we will disregard this contribution. We derive the vibrational DOS of Fe2SiO4
polymorphs from the corresponding ones of Mg2SiO4 polymorphs
using the VVCA. Fig. 3 shows red shift of the center of mass of
the frequency domain, which manifests the effect of the atomic
mass change (Fe vs. Mg) on the vibrational spectrum.
The equation of state of fayalite derived from the QHA is compared with the experimental data and with the results on forsterite
in Figs. 4 and 5 and Table 3. At ambient conditions the LDA + U sc
method underestimates both the equilibrium volume (by 3%) and
the bulk modulus (by 5.9%). The discrepancy may be explained
by the different choices of the K 0 values in this work (4.0) and in
the experimental study (4.9) (Speziale et al., 2004). Above 4 GPa
the predicted bulk modulus is in good agreement with the experiment (Fig. 5).
The calculated temperature dependence of thermal expansivity
of fayalite is in excellent agreement with the experimental measurements by Suzuki et al. (1981) as shown in Fig. 5. This indicates
that the thermodynamic properties of Fe2SiO4 polymorphs can be
successfully predicted by combining the VVCA and QHA. The predicted expansivity of fayalite (2.58 10-5 K-1) is slightly smaller
than that of forsterite (2.66 10-5 K-1) (Table 3) at ambient conditions. At higher temperatures, especially above 500 K, the difference between the two values becomes more pronounced. The
predicted heat capacity at 300 K is in reasonable agreement with
the result by Robie et al. (1982). The effects of the lambda type
transition at 65 K (the paramagnetic to AF transition) and the short
range ordering of the magnetic moments near 16 K (the AF to the
canted AF transition) are beyond the scope of this study. The entropy of fayalite is larger than that of forsterite. This is consistent with
the differences between calculation and experiments in the C P
curves and in the vibrational DOS of fayalite and forsterite (Figs. 3
and 5, the smaller the center of mass in DOS, the larger the
entropy).
The agreement between calculation and experiments on the
thermodynamic properties of Fe2SiO4 ringwoodite is even better
than that for fayalite. The unit-cell volume is underestimated by
less than 2% (Fig. 6). The predicted value of the bulk modulus
(K S ¼ 199:8 GPa with K 0 ¼ 5:1) falls in the range of the experimental determinations, 187–207 GPa (Table 3). The discrepancies
among different experiments may have risen from the difficulty
to fit equations of state to the data collected in the limited compressional range (0–10 GPa) and from the difficulty in constraining
the K 0 value in experiments (Greenberg et al., 2011; Nestola et al.,
2011). The predicted temperature derivative of the bulk modulus
(dK S =dT) agrees well with the experimental data (0.022 GPa/K
(calculation) vs. 0.027 GPa/K (experiment)). The predicted heat
capacity curve (Fig. 7) at low temperatures follows the measurements by Yong et al. (2007). However, above 200 K the C P is
underestimated (Table 3). This is also true for fayalite (Fig. 5). It
shows that the VVCA is not fully satisfactory in reproducing the
vibrational DOS of Fe2SiO4 and that there is still room for refining
the model.
To assess the accuracy of the present model in predicting the
pressure dependence of vibrational modes, we computed the bulk
sound velocities of fayalite and ringwoodite at 973 and 1173 K and
compared them with the ultrasonic velocity measurements (Liu
et al., 2010). Fig. 8 shows that (1) V U of ringwoodite agrees well
with the experiment at high P–T; (2) in fayalite, although V U is
overestimated, the pressure dependence of V U at 973 K is well
reproduced. The deviation between the calculated and experimental values of dV U =dP for fayalite at 1173 K could be attributed
39
Y.G. Yu et al. / Physics of the Earth and Planetary Interiors 217 (2013) 36–47
(b) rw
(a) faya
350
600
Volume (°A3)
LSDA
LDA+Usc
σ-GGA
GGA+Usc
Zhang (1998)
Kudoh and Takeda (1986)
LSDA
LDA+Usc
σ-GGA
GGA+Usc
Nestola (2010)
Greenberg et al. (2011)
550
300
500
250
0
5
10
15
0
5
10
Pressure (GPa)
15
20
25
30
Pressure (GPa)
Fig. 2. Static (0 K) equation of state for Fe2SiO4 fayalite (a) and ringwoodite (b) calculated by four different functionals: LSDA, LDA + self-consistent U; r-GGA, and GGA + selfconsistent U. Experimental data at room temperature are shown as a reference (Zhang, 1998; Kudoh and Takeda, 1986; Nestola et al., 2010; Greenberg et al., 2011).
Mg2SiO4
350
VDoS (arbitrary units)
Fe2SiO4
fayalite
Zhang (1998)
Kudoh and Takeda (1986)
Downs et al. (1996)
forsterite
°
wadsleyite
Volume (A3)
ringwoodite
300
olivine
0
200
400
600
800
1000
Frequency (cm-1)
Fig. 3. Vibrational density of states of Mg2SiO4 (in solid black lines) calculated from
direct phonon calculations, and those of Fe2SiO4 (in dashed red lines) derived from
Mg2SiO4 using a vibrational virtual crystal model. (For interpretation of the
references to color in this figure legend, the reader is referred to the web version of
this article.)
either to inaccuracy of the VVCA model or to the experimental
uncertainties.
Fe2SiO4 wadsleyite does not exist in nature and has not yet been
synthesized (Katsura and Ito, 1989; Finger et al., 1993). In our calculation, the FM configuration is found to be energetically more favorable than AF configurations. The calculated equation of state of
b-Fe2SiO4 in the FM configuration is consistent with single crystal
X-ray diffraction data of Fe-free and Fe-bearing (X Fe ¼ 0:25) wadsleyite (Fig. 9), if the Vegard’s law is used to extrapolate the unit-cell
volume from the intermediate to the end-member composition. Like
Fe2SiO4 fayalite and ringwoodite, b-Fe2SiO4 has a larger bulk modulus (5.1% larger) than the magnesium end member. But the difference in K S between b-Fe2SiO4 and b-Mg2SiO4 (8.5 GPa at ambient
conditions) decreases with pressure (Fig. 10). This is opposite to
the trend observed for ringwoodite (Fig. 7). Our results on b-Fe2SiO4
can thus serve as a reference, which allows for a comparison with
experiments at low iron content through interpolation.
3.3. Phase transitions in Fe2SiO4
Predicting phase relations in Fe2SiO4 from first-principles methods has been difficult due to the localized character of 3d electrons
250
0
5
10
15
Pressure (GPa)
Fig. 4. Equation of state of Fe2SiO4 fayalite calculated by LDA + U sc compared with
experimental results and with the results of forsterite (Mg2SiO4). The experimental
data are from Zhang (1998), Kudoh and Takeda (1986), Downs et al. (1996).
of iron. In Fig. 11, we compare the relative enthalpies of a, b, and c
Fe2SiO4 calculated by four different approaches. The r-GGA calculation gives a reasonable a ? c transition pressure (2 GPa), but
predicts wrong phase relations (Fig. 11a), because the b phase becomes more stable than a and c above 5.6 GPa. This contradicts
the experimentally observed phase transition sequence from a to
c at 2.75 GPa in Fe2SiO4 (the room temperature value (Yagi et al.,
1987)). When the GGA + constant U (4 eV) is used (Fig. 11b), c-spinel becomes the most stable phase above 38.4 GPa but still there
exists a finite stability field for the b phase (between 19.5 and
38.4 GPa). Only when the Hubbard U parameter is treated self-consistently (Table 1), the phase relations become qualitatively correct
(Fig. 11c and d). The GGA + U sc calculation (Fig. 11c) shows that a
transforms to c at 23.6 GPa, while b-Fe2SiO4 is metastable over the
whole pressure range. Along with the stable phase transition, two
metastable transitions are predicted—the b ? c at 16.5 GPa and
the a ? b at 26.5 GPa. The discrepancy in the a–c transition pressure (23.6 GPa from GGA + U sc versus 2.75 GPa from experiment) is
most likely systematic due to the inaccuracy of DFT functionals. It
40
Y.G. Yu et al. / Physics of the Earth and Planetary Interiors 217 (2013) 36–47
Faya 300 K
Table 2
Comparison of the static (0 K) equation of state of Fe2SiO4 fayalite and ringwoodite
calculated by four different functionals: LSDA, LDA + U sc ; r-GGA, and GGA + U sc .
Source of experimental data is shown in Table 3.
Forst 1000 K
Faya 1000 K
Forst 300 K
5
Faya: Suzuki et al. (1981)
Zha et al. (1996)
Speziale et al. (2004)
4
faya
3
-5
-1
KS (GPa)
α (10 K )
200
2
rw
V (Å3)
K (GPa)
K0
V (Å3)
K
(GPa)
K0
275.36
293.64
308.16
320.16
306.9–308.5
190.0
149.4
96.4
121.8
131–136.3
1.4
4.0
5.7
4.4
4.0–
4.9
524.48
542.88
566.08
585.04
559.30:2
173.2
207.5
189.1
171.1
187–
207
4.0
5.1
4.3
4.8
4.0–
5.6
150
1
0
0
5
10
15
0
500
Pressure (GPa)
150
15
300
-1
-1
S (J mol K )
-1
CP (J mol K-1)
1000
Temperature (K)
LSDA
LDA + U sc
r-GGA
GGA + U sc
Exp
100
50
200
100
Robie et al. (1982)
0
0
0
500
1000
1500
0
500
Temperature (K)
1000
15
Temperature (K)
Fig. 5. Thermodynamic properties of Fe2SiO4 fayalite from LDA + U sc calculations
compared with experiments and with the calculations of Mg2SiO4 forsterite.
Experimental data are from Zha et al. (1996), Speziale et al. (2004), Robie et al.
(1982), Suzuki et al. (1981). Detailed comparisons with experiments on Mg2SiO4
forsterite are referred to Li et al. (2007).
580
Fe2SiO4 rw
Greenberg et al. (2011)
Nestola et al. (2010)
Meng et al. (1994)
Mg2SiO4 rw
°
Volume (A3)
540
500
Since the calculated static phase transition pressures (P tr ) are
sensitive to DFT exchange–correlation functionals and the reason
why these functionals affect Ptr in such ways is still elusive, we
combined P tr from r-GGA with the relative enthalpy by the
GGA+U sc calculation. Namely the enthalpies from GGA + U sc calculations were corrected to reproduce the Patr!c from r-GGA, then by
Legendre transformation they were converted to the energy versus
volume relations which were then used within the QHA together
with the vibrational DOS to compute the Gibbs free energy as a
function of P–T. Thus we obtained a set of enthalpy versus pressure
curves for Fe2SiO4 (Fig. 12b). These functions still retain the relative energetics of a–b–c from the GGA + U sc calculations
(Fig. 11c). Based on this set of enthalpy data, we obtained the a–
c phase boundary for Fe2SiO4 (Fig. 13b). The predicted transition
pressure at 1173 K is 4.2 GPa, consistent with the experimental
determination, between 4.2–4.8 GPa (Yagi et al., 1987). The predicted Clapeyron slope of this transition at 1000 K of 2.1 MPa/K
is in close agreement with the estimate of 2.5 MPa/K by Akimoto
(1987).
By including the thermodynamic properties of Mg2SiO4 polymorphs from the previous calculations (Yu et al., 2008), we obtained the complete set of data necessary for describing the
phase transitions at both end-member compositions (Fig. 12a
and 13a). This dataset forms the basis for understanding the phase
equilibria in the (Mg,Fe)2SiO4 system.
3.4. Phase equilibria in (Mg,Fe)2SiO4 system
460
0
5
10
15
20
25
30
Pressure (GPa)
Fig. 6. Equation of state of Fe2SiO4 ringwoodite calculated by LDA + U sc compared
with experiments and with the results of Mg2SiO4-spinel. Experimental data are
from Greenberg et al. (2011), Nestola et al. (2010), Meng et al. (1994).
Table 1
The calculated self-consistent Hubbard U parameter (U sc ) in units of eV for Fe2+ in
Fe2SiO4 fayalite, wadsleyite, and ringwoodite (in high spin state)
faya
M1
2.53
M2
wads
M1
M2
M3
rw
M
2.52
2.82
2.91
2.83
2.53
can possibly be attributed to the lack of an explicit and self-consistent treatment of the J parameter (the exchange interaction) in this
DFT + U method. It is also possible that spin excitations might shift
these transition pressures. This should be investigated in the future. When LDA + U sc is used (Fig. 11d), the a ? c transition pressure is found at 18.5 GPa, which is better than the GGA + U sc result,
but is still too high compared with the experiment.
The free energy of mixing in a, b, and c (Mg,Fe)2SiO4 solid solutions is described here within the ideal mixing model (IMM). Within the IMM, the boundaries of a binary phase loop (phases a and b
with two end members A=Mg2SiO4 and B=Fe2SiO4) have the following analytic solution (see, e.g., Cemič, 2005):
xbB ¼
1 CA
CB CA
a
xB ¼ C B xbB
;
with
!
GbA GaA
;
C ¼ exp
RT
!
GbB GaB
C B ¼ exp
;
RT
A
where GaA and GaB are, respectively, the Gibbs free energies of the
end-members A and B in phase a. Here these functions are calculated using the GGA and GGA+U sc methods.
Fig. 14 shows the calculated binary phase loops of (Mg,Fe)2SiO4
at 1473 and 1873 K. At the Mg-rich composition, the two narrow
loops represent the divariant a–b and b–c fields (15—21.2 GPa).
For the most relevant composition in the upper mantle, (Mg0.9,-
41
Y.G. Yu et al. / Physics of the Earth and Planetary Interiors 217 (2013) 36–47
Table 3
Comparison of the experimental thermodynamics data and our calculations on Mg2SiO4 (by LDA) and Fe2SiO4 (by LDA + U sc ) at room conditions
Mg2SiO4
fo (cal)
fo (exp)
wa (cal)
wa (exp)
ri (cal)
ri (exp)
Fe2SiO4
fa (cal)
fa (exp)
wa (cal)
rw (cal)
rw (exp)
rw (exp)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
V (Å3)
K S (GPa)
dK S
dP
a (105 K 1)
C P (J1 mol1 K1)
c
290.3
289.2–291.91,2,3
541.34
535.3–539.34,5,6
527.5
526.7(3)9
127.4
1252,127.73
165.7
160(3)3,172(3)5
184.7
183(3)9
4.3
4.22
4.4
4.34, 6.3(7)5
4.26
4.2(3)9
2.66
2.773
2.21
2.067
1.97
1.9210
119.5
119.33
118.1
114.148
116.9
113.010
1.25
1.233
1.28
1.267
1.2
1.25 10
297.24
306.9–308.511,12,13,14
560.48
548.72
559.3 0:2 17,18
144.3
131–136.311,12,13
174.2
199.8
197–2071819,20,21
18717,20122
4.0
4.0–4.911,12,13
4.1
5.1
4.0–4.818,19,20,21
5.5–5.617,22
2.58
2.5715
2.35
2.15
126.1
131.916
126.4
126.4
131.123
1.32
13:9
1.36
1.40
-14.3
-22.2
27.0
(MPa K1)
Guyot et al. (1996).
Downs et al. (1996).
Gillet et al. (1991).
Hazen et al. (1990).
Hazen et al. (2000).
Horiuchi and Sawamoto (2000).
Suzuki et al. (1980).
Ashida et al. (1987).
Meng et al. (1994).
Chopelas et al. (1994).
Speziale et al. (2004).
Zhang (1998).
Andrault et al. (1995).
Kudoh and Takeda (1986).
Suzuki et al. (1981).
Robie et al. (1982).
Nestola et al. (2010).
(Greenberg et al. (2011)).
Liu et al. (2008).
Armentrout and Kavner (2011).
Hazen (1993).
Rigden and Jackson (1991).
Yong et al. (2007).
Fe-rw 300 K
Mg-rw 300 K
Fe-rw 1000 K
280
Mg-rw 1000 K
6.8
4
3
Liu et al. (2010)
973 K
6.6
1173 K
2
6.4
200
1
160
0
0
5
10
15
0
200
400
Pressure (GPa)
600
800 1000 1200 1400
Temperature (K)
VΦ (km/s)
240
α (10-5 K-1)
KS (GPa)
Liu et al. (2010)
Rigden and Jackson (1991)
rw
6.2
6
faya
5.8
300
5.6
-1
-1
S (J mol K )
CP (J mol K-1)
150
-1
dK S
dT
100
50
200
5.4
0
100
4
6
8
10
Pressure (GPa)
Yong et al. (2007)
0
2
0
0
200
400
600
800 1000 1200 1400
Temperature (K)
0
200
400
600
800 1000 1200 1400
Temperature (K)
Fig. 7. Thermodynamic properties of Fe2SiO4 ringwoodite from LDA + U sc calculations and those of Mg2SiO4 ringwoodite calculated by LDA in comparison with
experiments (Liu et al., 2008; Rigden and Jackson, 1991; Yong et al., 2007). Detailed
comparisons with experiments on Mg2SiO4 ringwoodite are referred to Yu and
Wentzcovitch (2006).
Fe0.1)2SiO4, we find that with increasing pressure, the transformations of a into b and b into c occur through the divariant fields of
Fig. 8. Pressure dependence of bulk sound velocity (V U ) of Fe2SiO4 fayalite and
ringwoodite compared with experiments (Liu et al., 2010). Solid and dashed lines
are from LDA + U sc calculations at 973 and 1173 K, respectively.
a + b, and b + c. The assemblage of a + b is stable from 15.6 to
16.0 GPa (at 1473 K). This is consistent with the width of the
410-km discontinuity of <10 km from seismic observations. The
b + c field for (Mg0.9,Fe0.1)2SiO4 extends from 18.9 to 19.8 GPa (at
1473 K), indicating the width of the b ? c transition to be less than
30-km (1 GPa). The predicted phase relations are in a very good
42
Y.G. Yu et al. / Physics of the Earth and Planetary Interiors 217 (2013) 36–47
400
600
ΔH (meV / Fe2SiO4)
°
Volume (A3)
400
(a) σ-GGA
Fe-wads
Fe25 Hazen et al. (2000)
Fe00 Hazen et al. (2000)
Mg-wads
(b) GGA + U (4 eV)
200
200
β
0
γ
0
α
α
β
-200
550
-400
-10
-400
-5
0
5
10
400
0
10
20
30
ΔH (meV / Fe2SiO4)
500
0
5
10
15
Pressure (GPa)
Fig. 9. Equation of state of Fe2SiO4 wadsleyite (virtual end member) from this
calculation in comparison with experiments ((Mg1x ,Fex )2SiO4 with x ¼ 0 and 25%
from Hazen et al., 2000) and with the calculations on the Mg2SiO4 counterpart.
250
200
200
0
0
α
γ
α
β
γ
-200
β
-200
-400
-400
0
10
20
Pressure (GPa)
30
40
0
10
20
30
40
Pressure (GPa)
3
-1
200
2.5
-5
α (10 K )
KS (GPa)
3.5
2
ciated with the transitions in the two end-members. The eutectoid
point is also shifted from X Fe ¼ 0:27 and P triple ¼ 15:0 GPa at 1473 K
to X Fe ¼ 0:31 and P triple ¼ 15:6 GPa at 1873 K. For (Mg0.9,Fe0.1)2SiO4,
however, the widths of the a–b and b–c loops are barely affected.
1.5
1
0.5
150
0
0
5
10
15
0
Pressure (GPa)
500
1000
15
Temperature (K)
S (J mol-1 K-1)
150
-1
(d) LDA + Usc
Fig. 11. Static enthalpies of the three Fe2SiO4 polymorphs, in which the a phase is
taken as the reference. Four different functionals were used: (a) r-GGA, (b) GGA + U
with U ¼ 4 eV, (c) GGA + U sc , and (d) LDA + U sc .
4
Fe-wads 300 K
Mg-wads 300 K
Fe-wads 1000 K
Mg-wads 1000 K
40
400
(c) GGA + Usc
-1
CP (J mol K )
γ
-200
100
50
4. Discussion
300
4.1. The effects of different approximations used in the study
200
The approximations used in this study need further clarifications. For Mg2SiO4 the LDA functional gives superior results for
the equation of state when the quantum zero point motion and finite temperature effects are included in the calculation based on
QHA (Wentzcovitch et al., 2010b). In strongly correlated materials,
such as Fe2SiO4, LDA + U sc predicts the equation of state in a
slightly better agreement with experiments than GGA + U sc . This
is seen, for example, in Fig. 2. Since GGA results give, in general,
softer bulk modulus and elastic tensors, pressure corrections had
to be applied in previous GGA calculations of Fe2SiO4 (Stackhouse
et al., 2010). Only limited agreement with experiments were found
in the previous calculations of equation of state parameters for Fe2SiO4 (Derzsi et al., 2011) and Mg2SiO4 (Piekarz et al., 2002) using
GGA functionals. On the other hand, the PBE-GGA outperforms
the LDA in predicting transition pressures, because the former usually overestimates the transition pressure by less than 4 GPa, while
the latter tends to dramatically underestimate it by about 6–13
GPa in comparison with experiments (Wentzcovitch et al.,
2010b). The reason is that the atomization energy is better modeled by the PBE-GGA than by LDA, which can be seen from
Fig. 15 (Ernzerhof and Scuseria, 1999). Our study further shows
that choosing the DFT + U sc method is crucial for reproducing correct enthalpy-pressure relations for a, b and c Fe2SiO4. Both the rGGA and GGA + U (4 eV) failed the task. On the other hand, none of
these functionals (r-GGA, GGA + U (4 eV), LDA + U sc , and
GGA + U sc ) were successful in predicting the phase transition pressures of Fe2SiO4 quantitatively (Fig. 11). It is, therefore, important
to explore more advanced theoretical methods, such as the exact
100
0
0
0
500
1000
Temperature (K)
1500
0
500
1000
15
Temperature (K)
Fig. 10. Thermodynamic properties of Fe2SiO4 wadsleyite (virtual end-member)
calculated by LDA + U sc and those of Mg2SiO4 wadsleyite calculated by LDA.
Detailed comparisons with experiments on Mg2SiO4 wadsleyite are referred to Wu
and Wentzcovitch (2007).
correspondence with the a–b and b–c loops determined by Katsura
and Ito (1989), except that the a–b and b–c transitions in Mg2SiO4
are overestimated by 2 GPa in GGA, a common trend found also
for phase transitions in MgSiO3 polymorphs (majorite, ilmenite,
and perovskite (Yu et al., 2011)). At high iron content (x > 0.5),
our calculation correctly locates the a–c phase loop, showing that
no phase stability field exists for the a–b or the b–c loop. The difference in the a–c phase loops determined from our calculations
and from the experiment (Akimoto, 1987) mainly occurs due to
the discrepancy in the transition pressures of the two end-member
components. The a ? c transition pressure in Fe2SiO4 is underestimated by 1 GPa, while the transition pressures in Mg2SiO4 (a ? b
and b ? c) are overestimated by 2 GPa (Fig. 13).
When temperature is raised to 1873 K, the binary phase loops
are shifted to higher pressures (by 1 GPa in Mg2SiO4 and by
0.7 GPa in Fe2SiO4), because of the positive Clapeyron slopes asso-
43
Y.G. Yu et al. / Physics of the Earth and Planetary Interiors 217 (2013) 36–47
ΔH (meV per formula)
300
300
(a) Mg2SiO4 [GGA]
200
β
100
0
(b) Fe2SiO4 [GGA+Usc]
w pressure correction
200
γ
100
0
α
-100
-100
-200
-200
-300
α
β
γ
-300
0
5
10
15
20
Pressure (GPa)
25
-5
0
5
10
15
20
Pressure (GPa)
Fig. 12. Relative static enthalpies of the end members used in our phase equilibrium study. In (a) the Mg2SiO4 components were obtained from the PBE-GGA calculations (the
same as in the previous calculations (Yu et al., 2008)); in (b) the Fe2SiO4 components were based on the GGA + U sc calculations as in Fig. 11c, but a constant pressure shift is
applied (for details see Section 3.3).
exchange in DFT (Engel and Schmid, 2009) and the quantum Monte
Carlo method (e.g. Driver et al., 2010). Recently, hybrid functionals
were used to study electronic and infrared vibrational spectrum of
fayalite (Noël et al., 2012).
In our study, the vibrational DOS of the Fe2SiO4 polymorphs
were obtained using the VVCA based on the dynamical matrices
of Mg2SiO4. This approach avoids frozen phonon calculations for
Fe2SiO4 using the DFT + U method, which is highly computationally
intensive (e.g. Metsue and Tsuchiya, 2011). On the other hand, the
VVCA compromises the accuracy in phonon frequency calculations,
because of neglecting the changes in macroscopic dielectric constants and Born effective charges due to iron substitution for magnesium. However, the calculated thermodynamic properties agree
well with the experiments (Table 3), suggesting that atomic mass
plays a major role in modifying vibrational DOS.
4.2. Electronic structures of Fe2SiO4
Fig. 13. Phase boundaries of (a) Mg2SiO4 and (b) Fe2SiO4 compared with experiments. The experimental data for Mg2SiO4 are from Katsura and Ito (1989), : a
phase, : b phase, and M : c phase. The experimental data for Fe2SiO4 are from Yagi
et al. (1987)), : a phase and M : c phase. The calculated Clapeyron slopes (at
1500 K) are 2.5 MPa/K and 3.5 MPa/K, respectively, for a–b and b–c transitions in
Mg2SiO4; that for the a–c transition in Fe2SiO4 is 2.1 MPa/K.
The magnetic susceptibility study (Santoro et al., 1966) and
heat capacity measurements (Robie et al., 1982) on fayalite
revealed two magnetic transitions under cooling at 65 K and
16–23 K, respectively. The first one is the paramagnetic to AF transition (Néel temperature), while the second one is attributed to the
linear AF to partially canted AF transition. In fayalite oxygen forms
a distorted hexagonal close-packed sublattice, while Si and Fe are
arranged locally in a way that each SiO4 tetrahedron is surrounded
by three FeO6 octahedra by sharing faces, two of which are of M1type and one of M2-type. This configuration creates edge-sharing
M1 and M2 sites, leading to zigzag M1–M2–M1–M2–. . .chains
extending along the b-axis. Our study shows that the magnetic moments of iron are parallel within each chain and antiparallel between two adjacent chains (Fig. 1a), consistent with earlier
works (Cococcioni et al., 2003). This magnetic configuration is explained by the super-exchange interaction (Anderson, 1950) between two iron atoms via a bridging oxygen atom: the
ferromagnetic coupling is more favorable when the \M1–O–M2
angle is close to 90° (the case of edge-sharing octahedra Fig. 1a),
whereas the AF alignment is more preferable when the angle is
close to 180° (the case of corner-sharing octahedra). Indeed the
average connecting angle for the edge-sharing octahedra within
the chain is 96° and the angle for the corner-sharing octahedra
belonging to two different chains is 117°. Mössbauer spectroscopy
and neutron diffraction measurements show that magnetic moments on M1 sites are collinear and those on M2 sites are canted
(Fuess et al., 1988). This correlates with the structural difference
of the M1 and M2 sites. The M2 site is more distorted than M1
44
Y.G. Yu et al. / Physics of the Earth and Planetary Interiors 217 (2013) 36–47
(a) 1473 K
(b) 1873 K
20
20
β
Pressure (GPa)
β
β+γ
α+β
15
γ
γ
15
α+γ
10
α
10
α
5
DFT
Akimoto (1987)
0
Mg2SiO4
0.2
0.4
DFT
Katsura & Ito (1989)
Katsura et al. (2004)
5
Akaogi et al. (1989)
0.6
0.8
XFe
1
0
0.2
0.4
0.6
XFe
Fe2SiO4 Mg2SiO4
0.8
1
Fe2SiO4
4
90
80
(a) faya
Fe-3d ↑
Fe-3d ↓
O-2p
LDA
PBE-GGA
0
70
60
40
30
20
10
0
Fig. 15. An illustration of the errors in atomization energy for selected molecules
obtained by LDA and PBE-GGA functionals in comparison with experiments. Data
source: Table 1 in Ernzerhof and Scuseria (1999).
in terms of the variances of the Fe–O bond lengths and of the
\O–Fe–O angles (for details see the caption of Fig. 1a).
In fayalite, the calculated U sc parameters for M1 and M2 sites
are 2.5 eV. This value is consistent with the U sc parameters for
high spin Fe2+ in MgSiO3-type perovskite (U = 2.9 eV (Hsu et al.,
2010)) and post-perovskite (U = 2.9 eV (Yu et al., 2012)). The projected electronic density of states on M1 site (Fig. 16a) shows that
the five spin-up 3d orbitals are fully occupied, and a band gap of
1.5 eV exists within the spin-down d band which splits the dxy orbital from the others. This agrees with the previous calculation (Jiang
and Guo, 2004). The previous inelastic neutron-scattering study by
Aronson et al. (2007) suggested the importance of spin–orbital
interactions on non-dispersing magnetic excitations at low temperatures, and implied the distinct contributions to heat capacity
from the M1 and M2 iron sites. They suggested that the M1 site
DOS (arbitrary unit)
50
Si2
Na22
CO
F2 2
H2O
O2
NO 4
N2H
N2 OH
H3CO
H2C
O
HC
CON
HC
CNH6
C2 4
C2H2
C2H
LiF
Li2 l
HC2
SH3
PH2
PH 4
SiH3
SiH2
SiH
HF
H2O
OH3
NH2
NH
NH4
CH3
CH2
CH
CH
BeH
LiH
theory - exp in atomization energy (kcal/mol)
Fig. 14. Binary phase loops of a-b-c (Mg,Fe)2SiO4 from DFT + U sc calculations (solid blue lines) compared to experiments, at (a) 1473 K and (b) 1873 K. The data points in (a)
are from Akimoto, 1987 — : a phase, M: a + c phase, : cphase; the dashed lines there serve to guide the eye. In (b) the binary phase loops at the magnesium-rich end are
from experiments by Katsura et al. (2004); Katsura and Ito (1989), and the dashed loop at the iron-rich end is from Akaogi et al. (1989). (For interpretation of the references to
color in this figure legend, the reader is referred to the web version of this article.)
-4
4
(b) wads
0
-4
4
(c) rw
0
-4
-10
-5
0
5
Energy (eV)
Fig. 16. The projected electronic density of states of Fe2SiO4 (a) fayalite, (b)
wadsleyite, and (c) ringwoodite calculated by LDA + U sc .
is responsible for the broad bump (near 20 K) and the M2 site for
the sharp lambda transition (65 K) in the experimental C P curve
(Robie et al., 1982). Unfortunately our study did not take into account the spin–orbital coupling or collinear magnetic moments.
Further calculation is needed to elucidate the magnetic transitions
and the associated heat capacity variation.
In the calculation, we assume that b-Fe2SiO4 is isostructural
with b-Mg2SiO4 (space group Imma). In this structure, since FeO6
Y.G. Yu et al. / Physics of the Earth and Planetary Interiors 217 (2013) 36–47
octahedra share edges, one expects the FM coupling among iron
atoms to be more preferable than other configurations according
to the rule of super-exchange interactions. Indeed the FM configuration easily converges in electronic structure calculations. A few
other magnetic configurations were tried but encountered convergence problems. The calculated Hubbard U parameters for M1, M2
and M3 sites are 2.82, 2.92 and 2.83 eV, respectively. This correlates with the experimental finding showing that the M1 and M3
sites are enriched in Fe2+ [b-(Mg0.75Fe0.25)2SiO4 (Hazen et al.,
2000)]. Note that the average U in wadsleyite (about 2.8 eV) is larger than that in olivine (2.5 eV). This may be associated with why
b-Fe2SiO4 is less stable than a-Fe2SiO4.
4.3. Phase equilibria calculations and their implications
In the previous calculation by Akaogi et al. (1989), enthalpy and
entropy of b-Fe2SiO4 were obtained by fitting the univariant line of
the experimental binary phase diagram of Katsura and Ito (1989),
while these parameters of a and c-Fe2SiO4 were adjusted to reproduce the a–c phase boundary of Yagi et al. (1987). The model of
Akaogi et al. (1989), therefore, is consistent with the experiments
by Katsura and Ito (1989) and Yagi et al. (1987). The equations of
state they have used for Mg2SiO4 and Fe2SiO4 were not well constrained from today’s point of view, and the determined values of
the excess mixing parameters (unit cell volume, enthalpy, and entropy) of the a, b, and c solid solutions were, by no means, unique.
Our QHA calculations based on LDA + U sc provide reliable equation
of state parameters and thermodynamic properties as a function of
P-T, however, difficulties in determining absolute values of transition pressures still remain due to uncertainties of DFT functionals.
Considering the uncertainties associated with the enthalpies of the
Fe2SiO4 polymorphs, we see no sense in varying the mixing parameters. Therefore the IMM was adopted in the present study.
The 520-km discontinuity is believed to be a broad seismic
reflector (with 10– 50 km width, 0.4–2 GPa) (Shearer, 2000),
although its fine structure features are still controversial among
seismic studies (Gu et al., 1998; Deuss and Woodhouse, 2001). In
the previous study (Yu et al., 2008), we found that the b to c transition in Mg2SiO4 in the context of the pyrolite composition model
is incapable to account for the 1.3–2.9% density discontinuity at
520-km depth (Lawrence and Shearer, 2006). The narrow b–c binary phase loop predicted from this study supports the view that the
pyroxene/garnet/Ca-pv system needs to be taken into account to
interpret the broad 520-km discontinuity profile (Saikia et al.,
2008).
5. Conclusions
The use of the DFT + U sc method made it possible to predict the
topologically correct sequence of phase transitions in Fe2SiO4 polymorphs. The use of Hubbard U parameter is necessary to retain the
localized character of d orbitals of iron and to predict the correct
insulator-type electronic band structure, which is missing in LSDA
or r-GGA calculations. With the aid of VVCA and QHA calculations,
the LDA + U sc results greatly improve the equation of state parameters relative to r-GGA results, especially for the bulk modulus.
Using the ideal solid solution model, the binary phase diagram of
(Mg,Fe)2SiO4 can be obtained in good correspondence with experimental data. The dependence of the phase relations on temperature is small. The widths of the divariant a–b and b–c loops are
barely sensitive to the temperature change within the interval of
1473–1873 K.
45
Acknowledgments
This research was supported by the Alexander von Humboldt
foundation. RMW acknowledges support also from NSF Grants
EAR-1019853 and EAR-0817202. Calculations were performed on
the CSC supercomputers at the University of Frankfurt. We thank
J.D. Gale for a helpful discussion. The phonon density of states
for Mg2SiO4 wadsleyite is based on the work by Zhongqing Wu.
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