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**Research article**| 24 Mar 2022

# Ab initio thermal expansion and thermoelastic properties of ringwoodite (*γ*-Mg_{2}SiO_{4}) at mantle transition zone conditions

*γ*-Mg

_{2}SiO

_{4}) at mantle transition zone conditions

Donato Belmonte Mattia La Fortezza and Francesca Menescardi

^{1,2},

^{1},

^{1}

**Donato Belmonte et al.**Donato Belmonte Mattia La Fortezza and Francesca Menescardi

^{1,2},

^{1},

^{1}

^{1}DISTAV, University of Genoa, Genoa, 16132, Italy^{2}Italian National Antarctic Museum (MNA, Section of Genoa), University of Genoa, Genoa, 16132, Italy

^{1}DISTAV, University of Genoa, Genoa, 16132, Italy^{2}Italian National Antarctic Museum (MNA, Section of Genoa), University of Genoa, Genoa, 16132, Italy

**Correspondence**: Donato Belmonte (donato.belmonte@unige.it)

**Correspondence**: Donato Belmonte (donato.belmonte@unige.it)

Received: 06 Oct 2021 – Revised: 18 Feb 2022 – Accepted: 25 Feb 2022 – Published: 24 Mar 2022

Thermal convection in the Earth's mantle is driven by
lateral variations in temperature and density, which are substantially
controlled by the local volume thermal expansion of the constituent mineral
phases. Ringwoodite is a major component of the lower mantle transition
zone, but its thermal expansivity and thermoelastic properties are still
affected by large uncertainties. Ambient thermal expansion coefficient
(${\mathit{\alpha}}_{{V}_{\mathrm{0}}}$), for instance, can vary as much as 100 % according to different experimental investigations available from the literature. In this work, we perform ab initio density functional theory calculations of
vibrational properties of spinel-structured Mg_{2}SiO_{4} ringwoodite in order to provide reliable thermophysical data up to mantle transition zone conditions. Temperature- and pressure-dependent thermal expansivity has been obtained by phonon dispersion calculations in the framework of
quasi-harmonic approximation (QHA) up to 25 GPa and 2000 K. Theoretical
analysis of vibrational spectra reveals that accurate prediction of IR and
silent modes, along with their relative mode Grüneisen parameters, is
crucial to define thermal expansivity. A six-parameter analytical function
is able to reproduce ab initio values fairly well in the whole investigated
*P*–*T* range, i.e., ${\mathit{\alpha}}_{V}(P,T)=(\mathrm{1.6033}\times {\mathrm{10}}^{-\mathrm{5}}+\mathrm{8.839}\times {\mathrm{10}}^{-\mathrm{9}}T+\mathrm{11.586}\times {\mathrm{10}}^{-\mathrm{3}}{T}^{-\mathrm{1}}-\mathrm{6.055}{T}^{-\mathrm{2}}+\mathrm{804.31}{T}^{-\mathrm{3}})$ $\times \mathrm{exp}(-\mathrm{2.52}\times {\mathrm{10}}^{-\mathrm{2}}P)$, with temperature in kelvin and pressure in gigapascal. Ab initio static and isothermal bulk moduli have been derived
for ringwoodite along with their *P*, *T* and cross derivatives, i.e., *K*_{0} = 184.3 GPa, *K*_{T,300 K} = 176.6 GPa, ${K}_{\mathrm{0}}^{\prime}$ = 4.13, ${K}_{T,\mathrm{300}\phantom{\rule{0.125em}{0ex}}\mathrm{K}}^{\prime}$ = 4.16, ${\left(\frac{\partial {K}_{T}}{\partial T}\right)}_{P}$ = −0.0233 GPa K^{−1} and ${\left(\frac{{\partial}^{\mathrm{2}}{K}_{T}}{\partial P\partial T}\right)}_{\mathrm{0}}=\mathrm{1.0}\times {\mathrm{10}}^{-\mathrm{4}}$ K^{−1}. Computed thermal expansivity and thermoelastic properties support
the evidence that QHA performs remarkably well for Mg_{2}SiO_{4}
ringwoodite up to mantle transition zone temperatures. Since volume thermal
expansion of ringwoodite is strongly pressure-dependent and its pressure
dependence becomes more marked with the increasing temperature,
internally consistent assessments and empirical extrapolation of
thermoelastic data to deep mantle conditions should be taken with care to
avoid inaccurate or spurious predictions in phase equilibrium and mantle
convection numerical modeling.

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*γ*-Mg

_{2}SiO

_{4}) at mantle transition zone conditions, Eur. J. Mineral., 34, 167–182, https://doi.org/10.5194/ejm-34-167-2022, 2022.

Despite the development of internally consistent databases for planetary materials, thermodynamic properties of deep mantle minerals are still poorly constrained or even completely lacking at high pressure and temperature conditions (HP–HT). Relevant geophysical properties like volume thermal expansion and isothermal bulk modulus can be derived from experimental measurements at ambient pressure in the low- to medium-temperature range, but they are usually less defined at higher temperatures (and pressures) due to technical problems or conditions of the samples, especially when melting or decomposition may occur (Thieblot et al., 1998; Fiquet et al., 1999). For these reasons large extrapolation of thermophysical data is often invoked at HP–HT without any warranty of physical soundness (Helffrich, 1999). Inaccurate thermodynamic properties could in turn strongly affect the prediction of phase equilibrium and stability relations of mantle minerals at deep Earth conditions.

Ringwoodite is a major constituent phase of the lowermost mantle transition zone (MTZ), between ∼ 520 and 660 km depths, where its relative amount is estimated to be between ∼ 40 % vol and 60 % vol according to different compositional models of the Earth's mantle (Ringwood, 1975; Bass and Anderson, 1984). It is now generally accepted that spinel and post-spinel phase transformations of olivine play a key role in determining global mantle discontinuities in the middle and bottom part of the transition zone (Bina and Helffrich, 1994; Sinogeikin et al., 2003; Ishii et al., 2019), although the ultimate origin, sharpness and seismic signature of the 520 and 660 km discontinuities are still a matter of debate (e.g., Deuss and Woodhouse, 2001; Deuss et al., 2006; Saikia et al., 2008). This basically reflects the still unconstrained abundance of olivine versus non-olivine mineral phases (majoritic garnet above all) in this part of the mantle and the fact that their stability field, density and thermoelastic properties are roughly compatible with seismic impedance contrasts and velocity jumps observed by global seismology (Shearer, 1996; Deuss et al., 2006; Schmerr and Garnero, 2007). In any case, the thermodynamic behavior of ringwoodite has a significant impact on physicochemical processes of the mantle transition zone (Akaogi et al., 2007; Kojitani et al., 2016). The natural occurrence of ringwoodite with normal spinel structure in many shocked chondritic meteorites (e.g., Chen et al., 2004) and as a mineralogical inclusion in ultradeep diamonds from Juína, Brazil (Pearson et al., 2014), further supports this evidence. Structural inversion may have a potential effect (Kiefer et al., 1999; Panero, 2008; Bindi et al., 2018), but thermodynamic considerations suggest that Mg-endmember ringwoodite is unlikely to be stable in the inverse spinel phase in the Earth's transition zone.

In this work we perform ab initio density functional theory calculations of
vibrational properties and volume thermal expansion of spinel-structured
Mg_{2}SiO_{4} ringwoodite in the framework of quasi-harmonic
approximation (QHA) in order to provide reliable thermophysical data up to
mantle transition zone conditions. This study is part of a broader project
aimed at setting up a comprehensive high-pressure thermodynamic database for
solid-state and solid-liquid phase equilibrium calculations in geochemically
relevant multi-component systems by “ab initio assisted” computational
thermodynamics (Belmonte et al., 2017a, b). A detailed survey of the
available experimental and theoretical data on Raman and IR spectra,
isothermal bulk modulus, volume thermal expansion, and thermal expansivity of
Mg_{2}SiO_{4} ringwoodite is carried out to show how relevant current
uncertainties on the physicochemical behavior of this important building
block of planetary interiors might be. By giving a theoretical constraint to
the extrapolation of thermodynamic data at MTZ depths, ab initio
calculations performed in this work allow us to infer some relevant insights into
the role of ringwoodite in mantle dynamics, as briefly discussed in the
implications section.

Ab initio calculations in this work have been performed by using the LCAO (linear combination of atomic orbitals) approach with an all-electron Gaussian-type basis set as implemented in the CRYSTAL code (Dovesi et al., 2014). The hybrid B3LYP density functional, which contains 20 % of exact Hartree–Fock exchange mixed with generalized gradient approximation (GGA) exchange–correlation (Becke, 1993; Lee et al., 1988), has been employed due to its high performance on vibrational, elastic and thermodynamic properties of a large variety of insulating crystalline phases (e.g., Ottonello et al., 2010; Belmonte et al., 2013; Erba et al., 2014; Belmonte et al., 2016). The basis set for Mg, Si and O atoms and computational parameters are the same as those used in our previous investigations on dense magnesium silicates and oxides and can be found elsewhere (see De La Pierre and Belmonte, 2016 and Belmonte, 2017, for full details).

Phonon dispersion calculations at **q** points other than Γ
have been performed on large isotropic or anisotropic supercells with the
direct method (see Parlinski et al., 1997; Evarestov and Losev, 2009). In
order to check numerical convergence on calculated thermal expansion and
thermoelastic properties, a total of 18 **q** points in the first
Brillouin zone have been sampled by using 2 × 2 × 2,
3 × 1 × 1, 1 × 3 × 1, 1 × 1 × 3, 3 × 2 × 1 and 3 × 1 × 2
supercells and exploiting the cubic symmetry of the ringwoodite structure
(space group $Fd\stackrel{\mathrm{\u203e}}{\mathrm{3}}m)$. Once phonon modes have been computed on the
fully relaxed equilibrium structure, their volume dependence has been
defined in the framework of quasi-harmonic approximation (QHA) (e.g.,
Wallace, 1972). Mode Grüneisen parameters of phonon modes (*γ*_{i}) are thus obtained as follows:

where *ν*_{i} is the wavenumber of the *i*th vibrational mode of the
crystal lattice sampled at discrete **q** points in the first Brillouin
zone, and *V* is the volume of the unit cell. Thus, also the effect of phonon
dispersion on mode Grüneisen parameters has been defined by
the supercell approach in this work. Least-square fitting of vibrational frequencies
computed at five different volume conditions (i.e., $V/{V}_{\mathrm{0}}\cong \mathrm{0.89}$,
0.92, 0.96, 0.98 and 1.00) gives the values of *γ*_{i} for all modes.
Linear or second-order polynomial fitting accurately describes
the volume dependence of all the vibrational frequencies in the investigated
compression range. The calculated equation of state (EOS) for Mg_{2}SiO_{4}
ringwoodite allows us to convert volume dependence of vibrational frequencies
into pressure dependence for a direct comparison with experimental
measurements (see Sect. 3).

## 3.1 Vibrational properties at ambient and high-pressure conditions

The experimental characterization of the full vibrational spectra of deep
mantle minerals at ambient and non-ambient conditions is still challenging.
One of the reasons for the overwhelming success of theoretical investigations
based on density functional theory (DFT) combined with the quasi-harmonic
approximation (QHA) is that the full vibrational density of state (vDOS) of
crystalline solids can be directly provided by phonon dispersion
calculations. Physically consistent thermodynamic and thermoelastic
properties can then be predicted for a broad range of *P*–*T* conditions by
statistical mechanics (Wallace, 1972).

Even though there are several IR and Raman spectroscopic studies on ringwoodite, most of them have been performed to characterize natural samples of shocked meteorites (e.g., Guyot et al., 1986; Chen et al., 2004; Feng et al., 2011; Acosta-Maeda et al., 2013) or mineral inclusions in diamonds (Pearson et al., 2014), while only a few experimental investigations focused on the Mg end-member (Akaogi et al., 1984; McMillan and Akaogi, 1987; Chopelas et al., 1994).

The zone-center optic vibrational modes of ringwoodite with cubic spinel structure (space group $Fd\stackrel{\mathrm{\u203e}}{\mathrm{3}}m$, point group $m\stackrel{\mathrm{\u203e}}{\mathrm{3}}m)$ can be classified by symmetry analysis as follows:

where *A*_{1g}, *E*_{g} and *T*_{2g} modes (symmetric with respect to
inversion) are Raman active; *T*_{1u} modes (anti-symmetric with respect to
inversion) are IR active; and *A*_{2u}, *E*_{u}, *T*_{1g} and *T*_{2u} modes
are silent. Due to the high symmetry of the structure, *E*_{g} and *E*_{u}
modes are doubly degenerate, while *T*_{1g}, *T*_{1u}, *T*_{2g} and
*T*_{2u} modes are triply degenerate in the irreducible representations of
Eq. (2). Vibrational frequencies as computed in this work by ab initio B3LYP
calculations are compared to experimental results and other DFT calculations
(Piekarz et al., 2002; Yu and Wentzcovitch, 2006; Li et al., 2009; Hernández
et al., 2015) in Table S1 (see Supplement). Only transverse optic (TO) modes
have been computed, as LO–TO splitting is relevant just for the IR-active
*T*_{1u} modes (Piekarz et al., 2002), and its effect on thermodynamic
properties as determined by phonon dispersion calculations is negligible.

The experimental Raman spectrum of Mg_{2}SiO_{4} ringwoodite shows two
strong bands at 794–796 and 834–836 cm^{−1} (assigned to
*T*_{2g} and *A*_{1g} modes and corresponding to asymmetric and symmetric
stretching of ${\mathrm{SiO}}_{\mathrm{4}}^{\mathrm{4}-}$ tetrahedra, respectively), along with weaker peaks at 302, 370–372 and 600 cm^{−1} (McMillan and Akaogi, 1987; Chopelas
et al., 1994). Ab initio B3LYP calculations predict the position of the two
most intense Raman peaks within 3–5 cm^{−1} and with the correct symmetry
analysis. It is interesting to note that mode assignment in experimental
studies could be rather difficult. McMillan and Akaogi (1987) assigned the
Raman band at 794 cm^{−1} to the symmetric stretching vibration of Si–O
tetrahedra with *A*_{1g} symmetry in their unpolarized powder spectrum,
while Chopelas et al. (1994) correctly assigned it to asymmetric stretching
with *T*_{2g} symmetry based on single-crystal oriented spectra. This mode
assignment is confirmed also by other DFT studies (Piekarz et al., 2002; Yu
and Wentzcovitch, 2006; Li et al., 2009; Hernández et al., 2015). The weaker
Raman peaks are also accurately predicted, with a partial disagreement
(∼ 20 cm^{−1}) for the *T*_{2g} mode at 619 cm^{−1} (see
Table S1 in the Supplement). By comparing experimental and theoretical
investigations, the position of this peak is the most uncertain in the Raman
spectrum, located in the region between ∼ 570 and 620 cm^{−1}.

Four infrared bands are expected to be present in the IR spectrum of cubic
silicate spinels (White and DeAngelis, 1967; Preudhomme and Tarte, 1971),
although additional features could be present possibly due to some deviation
from the cubic symmetry, structural disorder or vibrational coupling between
octahedral and tetrahedral lattice modes (Jeanloz, 1980). Akaogi et al. (1984) observed only two major bands centered near 830 and 445 cm^{−1} for Mg_{2}SiO_{4} ringwoodite, along with minor features corresponding to
weak bands or shoulders at 920, 785, 510, 395 and 350 cm^{−1}. The
broadness of the major infrared bands found in experiments makes a
direct comparison with our ab initio calculations, which correctly predict
four IR-active *T*_{1u} modes with wavenumbers 341, 396, 545 and 802 cm^{−1} (Table S1) difficult. B3LYP results are quite different from those obtained by other GGA calculations (Piekarz et al., 2002; Hernández et al., 2015), except for the predicted IR modes in the low-frequency range, which may differ by less than 10 cm^{−1}. The same consideration roughly applies to computed frequencies for the silent modes.

In order to test the performance of ab initio B3LYP calculations, theoretical and experimental frequencies are compared by means of a global statistical index, which is defined as follows (see De La Pierre and Belmonte, 2016):

where *M* is the number of data considered in the statistics and $\left|\stackrel{\mathrm{\u203e}}{\mathrm{\Delta}}\right|$ is the mean absolute difference between calculated $\left({\mathit{\nu}}_{i}^{\mathrm{calc}}\right)$ and experimentally observed frequencies $\left({\mathit{\nu}}_{i}^{\mathrm{exp}}\right)$. Only the frequencies of the five Raman-active modes (i.e.,
*A*_{1g}, doubly degenerate *E*_{g} and triply degenerate *T*_{2g} modes)
have been included in the statistics, since two different experimental
studies give nearly coincident results (McMillan and Akaogi, 1987; Chopelas
et al., 1994), and a direct comparison between ab initio and experimental
data is not straightforward for the IR-active modes. In that case, *M*=12
and $\left|\stackrel{\mathrm{\u203e}}{\mathrm{\Delta}}\right|=\mathrm{7}$ cm^{−1} if frequencies computed
in this work at the B3LYP level of theory are compared with either the
experimental dataset by McMillan and Akaogi (1987) or that by Chopelas et
al. (1994) (Table S1). The agreement with observed Raman spectra is thus
excellent and markedly better than ab initio local density approximation (LDA) or GGA results, which give
$\left|\stackrel{\mathrm{\u203e}}{\mathrm{\Delta}}\right|=\mathrm{10}$–12 cm^{−1} and $\left|\stackrel{\mathrm{\u203e}}{\mathrm{\Delta}}\right|=\mathrm{24}$–25 cm^{−1}, respectively. The improved performance of
hybrid density functionals on vibrational properties of Mg silicates,
supported by several dedicated studies in the literature (Prencipe et al.,
2009; Demichelis et al., 2010; De La Pierre et al., 2011; De La Pierre and
Belmonte, 2016), is thus also confirmed in this work.

The volume (or pressure) dependence of vibrational frequencies in the
framework of QHA is defined by the mode Grüneisen parameters (*γ*_{i}), which allow in turn the determination of fundamental thermodynamic
properties such as thermal expansion and *P*–*V*–*T* equation of state (e.g.,
Belmonte, 2017). Ab initio mode Grüneisen parameters of Raman-active
modes calculated in this work show an excellent agreement with the few
spectroscopic data available from the literature in the pressure range from
0 to 20 GPa (Chopelas et al., 1994; Chopelas, 2000). As far as we know,
there are no experimental data for IR spectra of Mg_{2}SiO_{4}
ringwoodite at high-pressure conditions. A comparison between theoretical
and experimental results is shown in Fig. 1 and Table S2. The calculated
*γ*_{i} values for the two most intense Raman bands in the 790–796
and 830–836 cm^{−1} frequency range are 1.4 and 1.1, respectively,
compared to observed values of 1.3 and 0.9 (Chopelas et al., 1994). The
predicted average mode Grüneisen parameter is 〈*γ*〉 = 1.19, very close to the experimental value of 1.11 derived
from high-pressure Raman spectra. By comparing different DFT simulations, although Raman frequencies change in terms of wavenumbers (see
Table S2), their mode Grüneisen parameters turn out to be quite similar
to each other. However, this is not the case for the IR-active and silent
modes. The pressure dependences of some of the *T*_{1u} modes (for instance
that in the 330–350 cm^{−1} region) and some of the silent modes as well
(e.g., the low-frequency *T*_{2u} mode) are sensibly different if B3LYP
results are compared to other GGA calculations (Piekarz et al., 2002;
Hernández et al., 2015). This is one of the reasons why marked differences
are also observed in the computed values of volume thermal expansion
coefficients (see Sect. 3.3 below). It is interesting to note, finally,
that the width of the phonon band gap predicted in this work at Γ
point is about 170 cm^{−1} at *P* = 0, thus sensibly narrower than that obtained by previous ab initio LDA and GGA investigations (Yu and
Wentzcovitch, 2006; Piekarz et al., 2002; Hernández et al., 2015). The
splitting between upper and lower phonon bands increases up to 260 cm^{−1}
at *P* = 27 GPa. This likely occurs because the highest-frequency *A*_{1g} and *T*_{2g} vibrational modes, which are mostly related to stretching of
Si–O stiff bonds in tetrahedral sites, display a steep pressure gradient (see
Fig. 1 and Table S2 in the Supplement). Therefore the high-frequency modes
are affected by pressure more than the medium- and low-frequency modes, as
observed also by Yu and Wentzcovitch (2006).

## 3.2 Thermoelastic properties

Thermoelastic properties of insulating and fixed-composition crystalline
phases can be defined solely by the statistical mechanics analysis of
vibrational modes of the crystal lattice (Born and Huang, 1954). The key
entity is represented by the *α**K*_{T} product, which can be derived
from the Helmholtz free energy *F*(*V*,*T*) by applying the thermodynamic
identity:

where *α* and *K*_{T} are the volume thermal expansion coefficient
(usually referred to as thermal expansivity) and the isothermal bulk
modulus, respectively. The statistical mechanics expression is the following
(see Belmonte, 2017):

where *R* is the universal gas constant, *Z* is the number of unit formula in
the unit cell, *V* is the molar volume, and *γ*_{i}(**q**,*V*) and
*X*_{i}(**q**,*V*) are the mode Grüneisen parameter and the
adimensional frequency of the *i*th vibrational mode, respectively, which
both depend on the **q** point sampling and the volume of the crystal.
The adimensional frequency can be simply obtained by converting the
vibrational frequency wavenumbers *ν*_{i}(**q**,*V*) in
angular frequencies *ω*_{i}(**q**,*V*) via the speed
of light in vacuum (*c*), i.e., ${\mathit{\omega}}_{i}(\mathbf{q},V)=\left(\mathrm{2}\mathit{\pi}c\right)\times {\mathit{\nu}}_{i}(\mathbf{q},V)$ and then applying

where ℏ and *k* are the Planck and Boltzmann constants, respectively.
By following a fully analytical approach, once the values of *α**K*_{T}
have been computed by Eq. (5), thermal expansivity *α*_{V} at given
*P*–*T* conditions can be defined if the values of the isothermal bulk
modulus *K*_{T} are also known.

^{a} Static values. ^{b} Values at *T* = 0 K including zero-point correction (ZPC). ^{c} Results given by linear fitting of published data. ^{d} Fixed value.

First-principle Mie–Grüneisen equation of state (FPMG-EOS) allow us to
define thermoelastic parameters (i.e., *K*_{T} and ${K}_{T}^{\prime}$) by fitting ab
initio *P*–*V*–*T* (or *F*–*V*–*T*) data obtained from phonon dispersion calculations (see Belmonte, 2017, for details). The ab initio B3LYP results of *K*_{T},
${K}_{T}^{\prime}={\left(\frac{\partial {K}_{T}}{\partial P}\right)}_{T}$, ${\left(\frac{\partial {K}_{T}}{\partial T}\right)}_{P}$ and $\left(\frac{{\partial}^{\mathrm{2}}{K}_{T}}{\partial P\partial T}\right)$ as computed in
this work for Mg_{2}SiO_{4} ringwoodite are shown in Fig. 2 and Table 1. Static values have been obtained by fitting energy–volume data (*E*–*V*) with
a third-order Birch–Murnaghan EOS.

Static values calculated in this work for bulk modulus (i.e., *K*_{0} = 184.3 GPa) are in remarkable agreement with experimental results, with the
observed isothermal values being around 182–185 GPa at ambient conditions (Hazen,
1993; Meng et al., 1994; Katsura et al., 2004, 2010; see also Table 1).
On the other hand, literature values of ${K}_{\mathrm{0}}^{\prime}$ seem to be more
scattered, ranging from 4.2 up to almost 5.0. Our calculated values for the
bulk modulus pressure derivative (i.e., ${K}_{\mathrm{0}}^{\prime}$ = 4.13 at the static
level, ${K}_{T,\mathrm{300}}^{\prime}$ = 4.16 at *T* = 300 K and ambient pressure) are much
closer to the experimental values of Meng et al. (1994) and to some
vibrationally constrained thermodynamic assessments as well (e.g., Jacobs et
al., 2017) (Table 1). A thorough comparison between different ab initio DFT
simulations shows that B3LYP results are intermediate between those obtained
by LDA and GGA, respectively, as expected from previous theoretical studies
on the elasticity and EOS of several high-pressure minerals with hybrid
density functionals (e.g., Erba et al., 2014; Prencipe et al., 2014; Ulian
and Valdrè, 2018). In particular, LDA calculations give an upper bound
to the calculated static or room-temperature bulk moduli, yielding values in
the range 184–208 GPa (Kiefer et al., 1997, 1999; Yu and Wentzcovitch,
2006; Panero, 2008; Núñez-Valdez et al., 2011), while a lower bound
of 175–178 GPa is provided by GGA-based DFT or MD simulations (Piekarz et al.,
2002; Li et al., 2006, 2009; Panero, 2008; Hernández et al., 2015). It is
interesting to note that vibrational effects on EOS parameters are relevant
as *K*_{T} and ${K}_{T}^{\prime}$ at *T* = 300 K are rather different from static
values. The former is lowered by ∼ 8 GPa (i.e., about 4 %), and
the latter increases from 4.13 to 4.16 (Table 1). The effect of zero-point
motions alone decreases the bulk modulus by ∼ 2.5 % over the
entire temperature range (see Fig. 2). Those outcomes on ringwoodite are
thus consistent with that observed for other high-pressure Mg silicates,
like for instance bridgmanite (Oganov et al., 2001), suggesting that
vibrational effects on thermoelastic properties cannot be disregarded in DFT
simulations even at ambient *P*–*T* conditions (see Belmonte, 2017).

The temperature dependence of the isothermal bulk modulus of
Mg_{2}SiO_{4} ringwoodite at *P* = 0 GPa is shown in Fig. 2. Ab initio B3LYP results compare favorably with literature values, especially in the medium- to high-temperature range. The value inferred for ${\left(\frac{\partial {K}_{T}}{\partial T}\right)}_{P}$ by linear fitting of ab
initio data is −0.0233 GPa K^{−1}, which is not far from the available
experimental results (Meng et al., 1993, 1994; Katsura et al., 2010). The
only exception is represented by the markedly negative value given by
Katsura et al. (2004) (see Table 1). Interestingly, the computed ${\left(\frac{\partial {K}_{T}}{\partial T}\right)}_{P}$ is in excellent agreement
with that obtained by some thermodynamic assessments based on the Helmholtz
free energy, hence constrained by vibrational theory (Dorogokupets et al.,
2015; Jacobs et al., 2017). Furthermore, the isothermal bulk modulus values
computed by ab initio B3LYP at HT conditions are nearly coincident with
those predicted by Frost (2003) at *T* = 1673 K and by the ab initio MD
simulation of Li et al. (2006) at *T* = 2000 K (see Fig. 2). For instance, Li et al. (2006) obtain K_{T} = 138 ± 6 GPa at *T* = 2000 K, compared to our value of 137.6 GPa at the same temperature. This remarkable
agreement supports the evidence that QHA performs well for Mg_{2}SiO_{4}
ringwoodite in the whole *P*–*T* range of the mantle transition zone.
Conversely, other “internally consistent” thermodynamic assessments give
*K*_{T} values for ringwoodite which are overestimated (Holland and Powell,
2011) or possibly affected by physical unsoundness (Fabrichnaya et al.,
2004).

The mixed *P*–*T* derivative of the isothermal bulk modulus has also been
constrained in this work. Ab initio B3LYP calculations give ${\left(\frac{{\partial}^{\mathrm{2}}{K}_{T}}{\partial P\partial T}\right)}_{\mathrm{0}}\cong \mathrm{1.0}\times {\mathrm{10}}^{-\mathrm{4}}$ K^{−1} at room temperature, which is virtually
identical to the assessed value by Dorogokupets et al. (2015) and in
qualitative agreement with the experimental results of Meng et al. (1994)
within their quite large uncertainties (see Table 1). First-principle MD
simulations of Li et al. (2006) provide a higher ${K}_{T}^{\prime}$ of 5.2 ± 0.3 at *T* = 2000 K (against ${K}_{T,\mathrm{2000}}^{\prime}$ = 4.33 predicted in this
work), but also in this case the agreement seems to be reasonable in view of
the huge uncertainties which currently affect the definition of
thermoelastic properties of deep mantle minerals at simultaneous high-pressure and high-temperature conditions.

## 3.3 Thermal expansion and thermal expansivity

It is well known that hybrid functionals like B3LYP, which are based on the
generalized gradient approximation for the DFT exchange–correlation term,
overestimate molar volumes of magnesium silicates by ∼ 1 %–2 %,
essentially due to an underbinding trend predicted for Mg–O and Si–O bond
lengths (Kohanoff, 2006; Ottonello et al., 2009; Belmonte et al., 2014). The
optimized unit cell volume of ringwoodite at the athermal limit (i.e., *T* = 0 K, *P* = 0 GPa) is *V*_{0} = 530.9 Å^{3}, that is 1 % higher
than experimental values obtained by X-ray diffraction at ambient conditions
(Sasaki et al., 1982; Katsura et al., 2004). Nevertheless, zero-pressure
relative volume thermal expansion (i.e., $V/{V}_{\mathrm{0}}$) and thermal expansivity
(*α*_{V}) are accurately reproduced in the framework of QHA up to a
limit, which, as a rule of thumb, lies within the Debye temperature and two-thirds of the melting point of the substance. This limit is purely empirical in nature and may change according to the physicochemical properties of the solid phase. In other words, deviations from the observed trend may occur due to the onset of anharmonic effects at temperature conditions not so far from
the melting point. In such a case, the crystalline substance can be referred
to as a “quasi-harmonic” solid. Ab initio B3LYP results of volume thermal
expansion and thermal expansivity are shown in Figs. 3 and 4,
respectively.

The values of *α*_{V} computed at zero pressure and discrete
temperatures by the statistical thermodynamic approach described in Sect. 3.2 are fitted by a suitable polynomial function in the temperature range
from *T*_{r} = 298.15 K to *T* = 2000 K, as follows:

Fitted numerical coefficients of the polynomial function at *P* = 0 are
*α*_{0} = 1.6033 × 10^{−5} K^{−1}, *α*_{1} = 8.839 × 10^{−9} K^{−2}, *α*_{2} = 11.586 × 10^{−3}, *α*_{3} = −6.055 K and *α*_{4} = 804.31 K^{2} (see Table 2). Relative volume thermal expansion can
thus be obtained by analytical integration of *α*_{V}, i.e.,

A huge uncertainty exists on the ambient value of volume thermal expansion
coefficient of Mg_{2}SiO_{4} ringwoodite (Nestola, 2016), with
experimental values ranging from ${\mathit{\alpha}}_{{V}_{\mathrm{0}}}$ = 1.51 × 10^{−5} K^{−1} (Ming et al., 1992) to ${\mathit{\alpha}}_{{V}_{\mathrm{0}}}$ = 3.07 × 10^{−5} K^{−1} (Inoue et al., 2004). ${\mathit{\alpha}}_{{V}_{\mathrm{0}}}$
can thus vary as much as 100 % according to different experimental
investigations. The computed value in this work, i.e., ${\mathit{\alpha}}_{{V}_{\mathrm{0}}}$ = 1.98 × 10^{−5} K^{−1}, is not far from the experimental value
${\mathit{\alpha}}_{{V}_{\mathrm{0}}}$ = (1.86 ± 0.09) × 10^{−5} K^{−1}
obtained by Suzuki et al. (1979). Both values are quite different from those
determined by other experimental works (Meng et al., 1994; Katsura et al.,
2004). Again, ambient thermal expansion coefficients calculated by lattice
dynamics modeling or other vibrationally constrained assessments are in
remarkable agreement with our result (i.e., ${\mathit{\alpha}}_{{V}_{\mathrm{0}}}$ = 1.84 ± 0.10 × 10^{−5} K^{−1} according to Chopelas, 2000;
${\mathit{\alpha}}_{{V}_{\mathrm{0}}}$ = 1.89 × 10^{−5} K^{−1}, Dorogokupets et
al., 2015; ${\mathit{\alpha}}_{{V}_{\mathrm{0}}}$ = 1.99 × 10^{−5} K^{−1}, Jacobs
et al., 2017). This result can be interpreted as a clear sign of
thermodynamic soundness of thermal expansion data.

The calculated trend of volume thermal expansion up to high-temperature
conditions provides some insights into the thermodynamic behavior of
ringwoodite at deep mantle conditions (Figs. 3 and 4). Ab initio thermal
expansivity as calculated in this work is relatively different from the HT
extrapolation of experimental data (see Fig. 4a). In particular, B3LYP
gives *α*_{V} values which are higher than those determined by Suzuki
et al. (1979) at *T* > 1000 K but much lower than the observed
trend by Katsura et al. (2004). However, the optimized trend of Katsura et
al. (2010) is in excellent agreement with our calculations in the whole
temperature range, except for the ambient value (i.e., ${\mathit{\alpha}}_{{V}_{\mathrm{0}}}$ = 2.45 ± 0.05 × 10^{−5} K^{−1}), which seems to be
strongly overestimated with respect to our result and to that observed by
Suzuki et al. (1979) as well. This leads in turn to a different trend in the
relative volume thermal expansion (Fig. 3). The evidence that different
DFT and QHA simulations give quite similar values of ${\mathit{\alpha}}_{{V}_{\mathrm{0}}}$ (see
Piekarz et al., 2002; Yu and Wentzcovitch, 2006; Hernández et al., 2015)
strengthens the ability of ab initio calculations to identify and select
accurate experimental results for thermal expansivity. For instance, the
*α*_{V} values determined by Ming et al. (1992) are likely affected by
severe underestimation (Fig. 4a). On the other hand, the temperature
dependence of thermal expansivity may change among different theoretical
simulations. Ab initio B3LYP gives ${\left(\frac{\partial {\mathit{\alpha}}_{V}}{\partial T}\right)}_{{P}_{\mathrm{0}}}$ similar to those calculated by Chopelas (2000) and Dorogokupets et al. (2015), but sensibly different from that obtained by other ab initio DFT–GGA calculations (Piekarz et al., 2002) (see Fig. 4b). Interestingly, *α*_{V} does not display any inflection
point in the entire T range investigated in this work, thus confirming that
(i) QHA performs exceptionally well for Mg_{2}SiO_{4} ringwoodite up to
deep mantle temperatures and (ii) ringwoodite can be considered a
“quasi-harmonic” solid in the Earth's mantle transition zone. Further
supporting evidence is that the thermal expansivity predicted at *T* = 2000 K and room pressure by the first-principle molecular dynamics of Li et al. (2006) are almost identical to that predicted by this work, with the former being
equal to 3.85 × 10^{−5} K^{−1} and the latter around
3.81 × 10^{−5} K^{−1} (Fig. 4b). This confirms the
performance of ab initio B3LYP and QHA calculations since MD simulations, which
are able to catch anharmonic effects, give nearly coincident results on
thermal expansivity in the high-temperature range. Therefore, intrinsic
anharmonic effects do seem to be negligible for Mg_{2}SiO_{4}
ringwoodite even at HT (and low-pressure) conditions.

The comparison between ab initio calculations and internally consistent
thermodynamic assessments deserves special attention. Figure 4c shows that
thermal expansivity assessed via mathematical procedures able to reproduce
selected phase equilibria of Mg_{2}SiO_{4} ringwoodite (e.g., Fabrichnaya
et al., 2004; Holland and Powell, 2011) turns out to be inaccurate at mantle
temperatures. For instance, Holland and Powell (2011) used a thermal
pressure Tait EOS, along with some assumptions on thermoelastic properties,
to predict thermal expansion of mantle minerals at HP–HT conditions. *α*_{V} values obtained for Mg_{2}SiO_{4} ringwoodite are underestimated
by 10 %–15 % in the high-temperature range (see Fig. 4c). The same also applies to the relative volume thermal expansion trend depicted in Fig. 3. This is perhaps even more evident if thermal expansion data by Fabrichnaya et al. (2004) are considered in Fig. 4c, which basically means that
internally consistent thermodynamic datasets currently developed for deep
mantle minerals are not necessarily also physically consistent. In this
respect, it does not seem fortuitous that thermodynamic assessments based on
Helmholtz free energy and/or constrained by vibrational theory (e.g.,
Dorogokupets et al., 2015; Jacobs et al., 2017) provide thermal expansion
data very close to the ab initio results obtained in this work (Figs. 3
and 4c).

The coupling between thermal expansivity and isothermal bulk modulus
provides a further clue on the physical soundness of thermoelastic
properties and the robustness of their extrapolation to deep mantle
temperatures. This is not a trivial task from an experimental point of view,
since *α**K*_{T} values are difficult to be constrained at those
conditions. Kojitani et al. (2016) tried to infer *K*_{T} of
Mg_{2}SiO_{4} ringwoodite by fitting experimental *P*–*V*–*T* data of Katsura et al. (2004) with a high-temperature third-order Birch–Murnaghan EOS and by assuming *α*_{V} of Suzuki et al. (1979) as fixed in the fitting. They obtain ${\left(\frac{\partial {K}_{T}}{\partial T}\right)}_{P}=-\mathrm{0.019}\left(\mathrm{3}\right)$ GPa K^{−1}, which is much less negative than any other theoretical or
experimental determinations (see Table 1). This is most likely due to the
fact that the two experimental datasets cannot be reconciled by the
assessment. Ottonello et al. (2009) obtain a value of ${\left(\frac{\partial {K}_{T}}{\partial T}\right)}_{P}=-\mathrm{0.0104}$ GPa K^{−1}, which is even less negative and leads to an overestimation of *K*_{T} values of ringwoodite at HT. In that case, an overestimation may
occur because the assessment of thermoelastic properties is not based on the
full vDOS but rather on a semiclassical thermodynamic approximation
employing the logarithmic volume derivative of the Grüneisen parameter
(the so-called *q*^{ht} parameter), which is difficult to constrain by
lattice dynamics calculations (Anderson, 1995). The average value of the *α**K*_{T} product calculated for mantle phases over an appropriate
temperature range can give useful insights into the accuracy of thermoelastic
data (Stixrude and Lithgow-Bertelloni, 2005). If we consider the average
value of *α**K*_{T} of Mg_{2}SiO_{4} ringwoodite over a *T* range from
300 to 1000 K, namely where the temperature gradient of *α*_{V} is
steeper, we obtain 45.6 bar K^{−1} according to our calculations.
Vibrationally constrained thermodynamic assessments (e.g., Stixrude and
Lithgow-Bertelloni, 2005) give values around 44 bar K^{−1}, in excellent
agreement with the ab initio value determined by Eq. (4), while other
internally consistent thermodynamic databases could give clearly inaccurate
results (e.g., ${\stackrel{\mathrm{\u203e}}{\mathit{\alpha}{K}_{T}}}_{\text{300\u20131000\hspace{0.17em}K}}$ = 37.3 bar K^{−1} according
to Fabrichnaya et al., 2004). As for experimental data, there is a marked
difference between the dataset of Katsura et al. (2004) and that of Katsura
et al. (2010), with the latter being much closer to ab initio results than the
former (i.e., ${\stackrel{\mathrm{\u203e}}{\mathit{\alpha}{K}_{T}}}_{\text{300\u20131000\hspace{0.17em}K}}$ = 52.7 and 48.0 bar K^{−1}, respectively). This confirms the predictive power of state-of-the-art
ab initio calculations in constraining the extrapolation of experimental
data at deep mantle conditions.

Thermal expansivity is a key parameter for mantle dynamics. Buoyancy forces
that control thermal convection in the Earth's mantle are driven by lateral
variations in temperature and density, which are in turn determined by the
local volume thermal expansion of the constituent mineral phases
(Christensen and Yuen, 1985). Thermal expansivity is commonly assumed as
either constant or simply pressure-dependent by most numerical
simulations of mantle convection, while simultaneous *P*–*T* dependence is only rarely taken into account (Schmeling et al., 2003; Tosi et al., 2013).

In this work, *α*_{V}(*P*,*T*) and *K*_{T}(*P*,*T*) functions of
Mg_{2}SiO_{4} ringwoodite are computed in a broad range of *P*–*T*
conditions compatible with the stability of this phase in the Earth's mantle
transition zone (i.e., *P* = 0–25 GPa, *T* = 298.15–2000 K). The
following six-parameter analytical function is able to reproduce ab initio
values in the whole investigated *P*–*T* range with a maximum deviation of a few percent:

and therefore it has been adopted in this work. The coefficient *α*_{5} in Eq. (9) has been fitted starting from the polynomial function
at *P* = 0 (see Eq. 7), and then applying an exponential pressure dependence
(Table 2). A similar parametrization was also implemented by Tosi et al. (2013) in their numerical modeling of mantle convection due to its
relatively simple form.

Figure 5 shows ab initio B3LYP thermal expansivity calculated at different
pressure conditions (i.e., *P* = 0, 5, 10, 21 and 25 GPa). *α*_{V}(*P*,*T*) is strongly pressure-dependent and does not show any inflection point in
the whole *P*–*T* range investigated in this work, thus supporting again the
quasi-harmonic nature of ringwoodite at mantle transition zone conditions.
The experimental values obtained by Katsura et al. (2004) at *P* = 21 GPa
are in qualitative agreement with our calculations up to about 1000 K and then
deviate at higher temperatures. Even though experimental uncertainties in
*α*_{V} values are not provided at *P* = 21 GPa, overestimation of
room-temperature *α*_{V} seems to also be kept at high pressures.

Even more interesting is the analysis of ab initio volume thermal expansion
calculated up to mantle transition zone pressures along different isotherms
(i.e., *T* = 500, 1000, 1500 and 2000 K) (Fig. 6). By the computed
trends in Fig. 6 it is quite evident that the pressure dependence of volume
thermal expansion of ringwoodite becomes more marked with increasing
temperature. This means that inaccurate extrapolations of low-pressure data
at mantle transition zone conditions could lead to large errors in predicted
molar volume and density values at HP–HT, thus affecting phase equilibrium
calculations and large-scale numerical modeling of mantle convection at
those conditions. Experimental values obtained by in situ X-ray diffraction
in a multianvil apparatus by Katsura et al. (2004), although sensibly
scattered, turn out to be in good agreement with ab initio results. The
combined *P*–*T* effects on thermal expansivity of ringwoodite are shown by a
contour plot in Fig. 7, which displays calculated *α*_{V}(*P*,*T*)
values within the stability field of the spinel-structured Mg_{2}SiO_{4}
polymorph at depths of the lower mantle transition zone (i.e.,
> 520 km) as inferred by global geophysical models (Dziewonski and Anderson,
1981; Brown and Shankland, 1981). Since contour lines change their slope
going from shallow to deep MTZ conditions, the increased pressure dependence
of thermal expansivity with temperature is highlighted even more clearly. Ab
initio DFT–QHA simulations thus provide a fundamental constraint to interpret
deep mantle processes under a mineral physics and thermodynamic perspective.

We showed that modern DFT–QHA calculations with the hybrid functional B3LYP
are able to accurately predict volume thermal expansion and thermoelastic
properties of Mg_{2}SiO_{4} ringwoodite up to mantle transition zone
conditions. Ab initio results of *α*_{V}(*P*,*T*), $V/{V}_{\mathrm{0}}(P,T)$ and *K*_{T}(*P*,*T*) give a physical constraint to the HP–HT
extrapolation of thermoelastic properties as determined by laboratory
experiments, thus enhancing our understanding of mantle processes (e.g.,
thermal convection) in planetary interiors. Theoretical analysis of lattice
dynamics allowed the interpretation of vibrational spectra at both ambient and
high-pressure conditions by suggesting that IR-active and silent modes,
along with their mode Grüneisen parameters, play a key role in
determining volume thermal expansion of ringwoodite, particularly at high
temperatures and pressures. Since literature data on those vibrational modes
are still scarce or controversial, the huge uncertainties that currently
affect thermal expansivity of this mantle mineral may be readily understood.
In this respect, vibrationally constrained thermoelastic properties as
computed by ab initio methods revealed that physical unsoundness could be
concealed under internally consistent thermodynamic datasets developed for
mantle phases. Ab initio thermal expansivity and isothermal bulk modulus
calculated in this work in a broad range of *P*–*T* conditions (i.e., *P* = 0–25 GPa, *T* = 0–2000 K) support the evidence of a quasi-harmonic behavior
of ringwoodite in the Earth's mantle transition zone. In fact,
high-temperature values of *α*_{V} and *K*_{T} predicted in this
work by DFT–QHA calculations turn out to be nearly coincident with the
results of the first-principle molecular dynamics simulation by Li et al. (2006), stressing that intrinsic anharmonic effects are virtually absent at MTZ conditions. Finally, since thermal expansivity of Mg_{2}SiO_{4}
ringwoodite is strongly pressure-dependent and its pressure dependence
increases with increasing temperature (see Figs. 5–7), extrapolation of
low-pressure thermoelastic data to deep mantle conditions should be taken
with care to avoid inaccurate or spurious numerical predictions.

All data derived from this research are presented in the main text, enclosed tables and figures, and Supplement.

The supplement related to this article is available online at: https://doi.org/10.5194/ejm-34-167-2022-supplement.

DB designed the research, performed the calculations and wrote the paper. DB, MLF and FM worked on data interpretation, discussed the results and commented on the paper.

The contact author has declared that neither they nor their co-authors have any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This article is part of the special issue “Probing the Earth: experiments and mineral physics at mantle depths”. It is not associated with a conference.

Financial support by the Italian MIUR PRIN 2017 (project number: 2017KY5ZX8) is warmly acknowledged. We also acknowledge the CINECA award under the ISCRA initiative (ISCRA C Project THOMMY, HP10CNF2NR), for the availability of high-performance-computing resources and support.

This research has been supported by the Ministero dell'Istruzione, dell'Università e della Ricerca (grant no. MIUR PRIN 2017 – project 2017KY5ZX8).

This paper was edited by Monika Koch-Müller and reviewed by two anonymous referees.

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- Abstract
- Introduction
- Computational method
- Results and discussion
- Implications: thermal expansivity at mantle transition zone conditions
- Conclusions
- Data availability
- Author contributions
- Competing interests
- Disclaimer
- Special issue statement
- Acknowledgements
- Financial support
- Review statement
- References
- Supplement

_{2}SiO

_{4}ringwoodite, a major mineral phase of the Earth's mantle transition zone. We tried to understand why current data on volume thermal expansion are still controversial by performing a detailed analysis of vibrational spectra. We proposed a reliable parametrization for thermal expansivity of ringwoodite in the transition zone which could be useful for numerical simulations of mantle convection.

- Abstract
- Introduction
- Computational method
- Results and discussion
- Implications: thermal expansivity at mantle transition zone conditions
- Conclusions
- Data availability
- Author contributions
- Competing interests
- Disclaimer
- Special issue statement
- Acknowledgements
- Financial support
- Review statement
- References
- Supplement