April 3, 2013 16:42 High Pressure Research wads˙thermo˙r

High Pressure ResearchVol. 00, No. 00, Month 2010, 1–11

ORIGINAL ARTICLE

Thermodynamic properties of anhydrous and hydrous wadsleyite,

β−Mg2SiO4

Sandro Jahna ∗, Roman Rahnera, Edgar Dachsb, Maria Mroskoa and Monika

Koch-Mullera,

aGFZ German Research Centre for Geosciences, Telegrafenberg, 14473 Potsdam,

Germany; bFachbereich Materialforschung und Physik, Universitat Salzburg,

Hellbrunnerstrasse 34, 5020 Salzburg, Austria

(April 2013)

Wadsleyite, β-Mg2SiO4, is an important high pressure mineral that may act as a sink ofwater in the Earth’s transition zone. In this study, we first determine the heat capacityand entropy of anhydrous wadsleyite at ambient pressure both experimentally and by first-principles simulations. The measured standard entropy S0 at 298 K is 86.7(11) J/(mol K).Then, extended simulations are performed to explore the effect of pressure and hydrationon these thermodynamic properties. The stability of several structural models of hydrogenincorporation is investigated and the lowest energy structures are used to derive the vibrationalentropy and the heat capacity of hydrous wadsleyite with a nominal content of 1.6 wt% and 3.3wt% H2O at ambient pressure and at 15 GPa. Considering an estimation of the configurationalentropy, S0 of hydrous wadsleyite with 1.6 wt% and 3.3 wt% H2O is about 3.5 and 5 J/(molK) higher than that of anhydrous wadsleyite at ambient pressure.

Keywords: thermodynamics; heat capacity; entropy; density functional theory; silicate

1. Introduction

The (Mg,Fe)2SiO4 polymorphs olivine (α), wadsleyite (β) and ringwoodite (γ)are considered important constituents of the Earth’s mantle. Three of the discon-tinuities in the wave velocities observed by seismologists in the deep Earth arethought to be related to phase transformations in which these polymorphs are in-volved. At about 410 km depth, which corresponds to a pressure of about 13 GPa,olivine transforms into the denser wadsleyite. At 520 km depth (∼18 GPa) wad-sleyite transforms into ringwoodite and at depths of about 660 km (∼24 GPa)ringwoodite decomposes into ferro-periclase, (Mg,Fe)O, and (Mg,Fe)-silicate per-ovskite, (Mg,Fe)SiO3. Recently, with higher seismic resolution it has been realizedthat the discontinuities vary on a global scale in depth [1, 2]. For instance, van derMeijde et al. [1] report depth differences of about 100 km for the 520-km disconti-nuity measured by seismology in Europe and Africa. Temperature variations alonecannot explain the observed seismic features and it is now widely discussed thatvariable hydrogen incorporation into the high pressure polymorphs of (Mg,Fe)2SiO4

may be responsible for the variation in the depth of the discontinuities.The (Mg,Fe)2SiO4 polymorphs belong to the group of nominally anhydrous min-

erals, which may incorporate hydrogen via point defects as hydroxyl groups. In this

∗Corresponding author. Email: jahn@gfz-potsdam.de

ISSN: 0895-7959 print/ISSN 1477-2299 onlinec© 2010 Taylor & FrancisDOI: 10.1080/08957959.YYYY.xxxxxxhttp://www.informaworld.com

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2 S. Jahn et al.

study we concentrate on the Mg endmember of wadsleyite, β−Mg2SiO4, which cancontain a nominal H2O content of as high as 3.3 % by weight [3]. Anhydrouswadsleyite crystallizes in the orthorhombic system (space group Imma) and thestructure of hydrous wadsleyite is monoclinic in symmetry (space group I2/m) [4].Wadsleyite belongs to the group of spinelloids and consists of a slightly distortedcubic-closest packing of oxygen atoms. The Mg atoms are located at three distinctoctahedral sites M1, M2, and M3 and Si occupies the only tetrahedral site. Fur-thermore, there are four different oxygen sites with O1 being the only oxygen thatis not bonded to Si and therefore has the highest probability of being hydrated.Several models for H incorporation into the wadsleyite structure exist in the lit-erature, e.g. [5–8]. Most studies agree that the charge-balancing mechanism is thecreation of a Mg vacancy at the M3 site and the protonation of two undersaturatedoxygen atoms at the O1 site [5]. However, the exact H positions and OH-dipoleorientations are still discussed. Deon et al. [8] investigated synthetic wadsleyiteby polarized infrared (IR) and in-situ high-pressure IR spectroscopy, single-crystalX-ray refinement, electron microprobe and transmission electron microscopy. Theyproposed that hydrogen is mainly incorporated along the O1-H...O4 direction withan O-H...O angle of 164◦ and along the O3-H-O4 edge (see their Fig. 6). This as-signment is consistent with the observed pleochroic behavior of the OH bands andthe strong shift of the OH bands with increasing pressure to lower wavenumbersin their Fourier transform IR (FTIR) spectra. A recent neutron diffraction studyby Sano-Furukawa et al. [9] confirms the O1-H...O4 assignment but suggests alsothe existence of O1-H...O3 structures. The O1-H...O4 incorporation mechanismwas also found to provide the lowest energy configuration in electronic structurecalculations using density-functional theory (DFT) [7].

Experimental studies confirm a relationship between the variation in H2O contentof wadsleyite and the variation in its transformation pressure [10–13]. Wadsleyitecoexisting with olivine or ringwoodite has always a higher H2O content than theother polymorphs and, at least in experiments in the presence of water, the stabilityfield of wadsleyite increases to lower and higher pressure at the expense of olivineand ringwoodite, respectively [12, 13]. Mrosko et al. [14] studied the high pressurebehavior of orthorhombic wadsleyite, containing different amounts of H2O (nearlyanhydrous up to 1.2 wt%), in a diamond anvil cell via FTIR spectroscopy upto 24 GPa. Pressure-induced changes in the spectra were observed between 10.3and 11.4 GPa for the anhydrous sample and between 7.7 and 8.9 GPa for thehydrous sample and interpreted in both cases as structural changes of wadsleyitefrom orthorhombic to monoclinic symmetry. While wadsleyite with a high amountof H2O (up to 3.3 wt%) can have monoclinic symmetry at ambient conditions,Mrosko et al. [14] concluded that wadsleyite with lower H2O content (1.2 wt%)could display a similar but pressure-driven modification at 8.4 GPa and anhydrouswadsleyite transforms at higher pressures, at around 10 GPa, into the monoclinicphase.

Accurate thermodynamic data for the Mg2SiO4 polymorphs in the anhydrousand hydrous system are needed to evaluate the experimental results, e.g. the shiftof the phase boundaries as a function of H2O content. There are several studies onthe thermodynamic properties of anhydrous Mg2SiO4 polymorphs using differentmethods, including calorimetry, e.g. [15, 16], Raman spectroscopy [17] and molec-ular modeling [18–20]. However, there are only few estimates of entropies and heatcapacities of hydrous Mg2SiO4 polymorphs, which were derived from experimen-tally determined phase equilibria and have a rather large uncertainty [21, 22]. Inthis study we derive the entropy and heat capacity of anhydrous wadsleyite fromcalorimetric measurements and DFT calculations. As our synthesis experiments in

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Thermodynamics of wadsleyite 3

the system Mg2SiO4–H2O never resulted in a single-phase run product the reli-able calorimetric measurements of hydrous wadsleyite could not be performed. Wetherefore investigate the effect of hydrogen incorporation on the thermodynamicproperties using the computational approach only.

2. Methods

For the calorimetric measurements we synthesized about 30 mg of anhydrous wad-sleyite. To obtain sufficient sample material we performed four different high-pressure experiments in a multi-anvil apparatus. The purity and H2O content ofeach synthesis was determined by powder X-ray diffraction and FTIR spectroscopy,respectively. Synthesis conditions and characterization of the run products are de-scribed elsewhere [14]. Only pure, anhydrous wadsleyite powders were mixed to-gether for the calorimetric measurements.

We performed direct measurements of heat capacities at the University ofSalzburg using the heat capacity option of the Physical Properties MeasurementSystem PPMS (Quantum Design) and Differential Scanning Calorimetry (DSC) forlow and high temperatures, respectively. For the PPMS measurements 13.6(2) mgpowdered sample were sealed in a flat Al pan with lid and placed on a 4×4 mm widesample platform of the PPMS calorimeter. The measurements were performed onheating from 5 to 303 K in 40 - 50 steps. To improve statistics, every temperaturestep was measured three times. The conversion of heat capacities into entropieswas done by numerical integration using the Mathematica-functions Interpolationand NIntegrate. A detailed description of the setup, measuring procedure and datatreatment can be found in [23].

The heat capacity in the temperature range from 280 to 764 K was measuredusing a Perkin Elmer Diamond DSC at the University of Salzburg. The measure-ments were carried out under a flow of dried nitrogen gas with the calorimeterblock kept at 243(1) K. Thereby the differential heat flow between the powdersample (11.77(2) mg filled in an aluminum pan) and a reference substance withknown temperature-dependent heat capacity behavior (here: corundum) was de-termined in steps of 100 K. The reference heat capacity values of the corundumsingle-crystal were taken from the National Bureau of Standards Certificate [24]. Ablank run with an empty Al-pan is performed before the actual measurement andtherein-derived heat capacities are subtracted from the reference and the sampleruns, respectively, using Mathematica. The DSC measurements on the wadsleyitesample were repeated five times and then averaged to give a mean CP value andits uncertainty.

Electronic structure calculations in the framework of DFT were performed usingthe planewave code ABINIT [25]. The exchange-correlation functional was treatedeither in the local density approximation (LDA) [26] or in the generalized gradientapproximation according to Perdew et al. (PBE) [27]. A planewave cutoff of 1000eV was chosen in combination with optimized norm-conserving pseudopotentials[28]. We checked that calculations using a larger cutoff (1870 eV) yielded essentiallyidentical vibrational frequencies and thermodynamic properties. For Brillouin zonesampling, a 3x2x2 Monkhorst-Pack k-point set was used. Full geometry optimiza-tion was performed for anhydrous and hydrous wadsleyite at pressures of zero and15 GPa. The simulation cells contained 56 atoms in the orthorhombic cell of anhy-drous wadsleyite. For hydrous wadsleyite, either a single or two Mg atoms of theM3 site were replaced by two or four H atoms, which corresponds to nominal H2Ocontents of 1.6 wt% and 3.3 wt%, respectively. The corresponding simulation cellsthen contain 57 and 58 atoms. Assuming the M3 vacancy substitution mechanism,

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4 S. Jahn et al.

calculations starting with different positions of the H atoms were performed.The phonon frequencies at the Γ point of the Brillouin zone and the Born effective

charges were computed using linear response density functional perturbation theory[29]. For comparison of the vibrational properties of the DFT-optimized structuralmodel to experimental spectra, the low-frequency dielectric permittivity tensor,ǫij(ω), was derived from the response functions according to [30] with a dampingfactor of 2 cm−1 in the same way as in a previous study on hydrous silicates [31].The imaginary part of ǫij(ω), Im[ǫij(ω)], provides a first approximation to theIR spectrum. Heat capacities at constant volume (CV ) and vibrational entropies(Svib) within the framework of the harmonic approximation were derived throughthe construction of the phonon density of states from the phonon eigenstates andits integration according to [32]. The effect of explicit account of phonon dispersionon computed heat capacities and vibrational entropies was tested for anhydrouswadsleyite at 15 GPa. Using seven wavevectors q other than Γ we obtained CV andSvib values that differ by less than 0.3 J/mol K from the Γ point only calculation.

3. Results

3.1. Anhydrous wadsleyite

Measured and computed heat capacities and entropies of anhydrous wadsleyiteare shown in Fig. 1 and compared to literature data. Sample coupling values inthe PPMS measurement (a measure of the quality of the thermal contact betweensample and sample platform, e.g. [33]) are > 99% for T > 50 K and between98% and 99% for T < 50 K, indicating robust CP determinations. The PPMSdata of this study agree well with those of [16] for T > 100 K. The latter tendto be slightly larger by at most 0.8%. At T < 100 K, the CP data of [16] becomeincreasingly smaller compared to those of the present study by up to 50% at lowestT . DSC and PPMS measured heat capacities at 298 K deviate by only ∼0.5%,i.e. both methods agree within their 1σ-uncertainties of about 0.7% and 1.0%,respectively. The PPMS-measured heat capacity of anhydrous wadsleyite at 298 Kis CP = 116.0(11) J/(mol K) and the derived standard entropy is calculated to S0

= 86.7(11) J/(mol K) (Table 1).

0 200 400 600 800Temperature (K)

0

50

100

150

200

Hea

t cap

acity

(J/

mol

K)

Akaogi et al. (2007)Ashida et al. (1987)this study (PPMS)this study (DSC)

(a)

0 200 400 600 800Temperature (K)

0

50

100

150

200

250

Ent

ropy

(J/

mol

K)

(b)

Figure 1. Heat capacities (a) and entropies (b) of anhydrous wadsleyite at ambient pressure. Black circlesand red squares refer to PPMS and DSC measurements, blue lines to LDA and maroon lines to PBEcalculations. Solid and dashed lines in (a) show computed CV and CP within harmonic approximation asdescribed in the text. Results from previous measurements [16, 35] are shown for comparison. Experimentaland DFT data of this study are also provided in the electronic supplement of this article.

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Thermodynamics of wadsleyite 5

The DFT-optimized cell parameters of anhydrous wadsleyite at ambient condi-tions and at 15 GPa are given in Table 1. The computed a and c lattice parametersat ambient conditions show the usual systematic behavior of the two functionalsused, i.e. LDA gives about 1% lower and PBE about 1% larger lattice parametersthan the experiment [8]. The b parameter is slightly smaller than the experimentalvalue for both functionals. Using the fully optimized structure, vibrational frequen-cies and thermodynamic properties are computed. The harmonic heat capacity atconstant volume, CV , and the respective entropy at ambient pressure are shownas full lines in Fig. 1. The computed CV and Svib at 298 K for both pressures arelisted in Table 1. In all cases, LDA predictions of CV and Svib are smaller thanthose of PBE. Increasing pressure results in a decrease in both heat capacity andentropy.

Table 1. Lattice parameters, heat capacities and entropies of anhydrous and hydrous wadsleyite at pressures of 0 GPa (am-

bient) and 15 GPa, T =298 K, from DFT calculations compared to experimental data of this study and results from the

literature. The standard entropy S0 of anhydrous wadsleyite is determined by the vibrational entropy (Svib) only. Configura-

tional contributions (Sconfig) to the entropy of hydrous wadsleyite are estimated in the Discussion.

a (A) b (A) c (A) β (deg) CV (J/mol K) Svib (J/mol K)

0 GPa, anhydrousDFT LDA (this study) 5.632(1) 11.290(1) 8.144(1) 115.5(5) 86.4(5)DFT PBE (this study) 5.715(1) 11.429(1) 8.310(1) 118.2(5) 91.7(5)Experiment (this study) 116.0(11) (CP ) 86.7(11) (S0)Experiment 5.694a 11.459a 8.253a 116.25(19)b (CP ) 86.4(4)b (S0)DFT LDA 118.1c 88.67c

B3LYP 5.734d 11.521d 8.332d 85.662d

15 GPa, anhydrousDFT LDA (this study) 5.508(1) 11.049(1) 7.913(1) 108.7(5) 76.8(5)DFT PBE (this study) 5.578(1) 11.160(1) 8.042(1) 111.0(5) 80.2(5)Experiment (extrapolated) 5.565e 11.170e 7.982e

0 GPa, 1.6 wt% H2ODFT LDA (this study) 5.616(1) 11.353(1) 8.108(1) 91.2(1) 114.8(5) 87.3(5)DFT PBE (this study) 5.700(1) 11.515(1) 8.283(1) 91.3(1) 117.6(5) 93.1(5)Experiment 5.673(2)e 11.551(3)e 8.251(1)e 90.06(2)e

Experiment 5.683(2)f 11.514(2)f 8.247(2)f 90.02f

DFT PBE 5.720g 11.575g 8.278g 91.14g

15 GPa, 1.6 wt% H2ODFT LDA (this study) 5.488(1) 11.085(1) 7.875(1) 91.2(1) 107.9(5) 77.5(5)DFT PBE (this study) 5.557(1) 11.209(1) 8.006(1) 91.2(1) 110.3(5) 81.0(5)Experiment (extrapolated) 5.546e 11.224e 7.976e

0 GPa, 3.3 wt% H2ODFT PBE (this study) 5.665(1) 11.600(1) 8.291(1) 90.0(1) 117.2(5) 94.0(5)DFT PBE 5.694g 11.654g 8.272g 90.0g

15 GPa, 3.3 wt% H2ODFT PBE (this study) 5.523(1) 11.272(1) 7.982(1) 90.0(1) 109.6(5) 81.3(5)

a Experimental lattice parameters [8]. b Experimental data [16]. c DFT LDA calculations [19]. d B3LYP electronicstructure calculations [20]. e Experimental data [36], which cover a pressure range up to 7.3 GPa (anhydrous) and 9.6GPa (hydrous with 1.66 wt% H2O); the data at 15 GPa are linear extrapolations of the measured lattice parameters. f

Experimental lattice parameters of deuterated wadsleyite [9] (samples had about 2 wt% D2O in the starting material).g DFT calculations, models 1 (1VM3, 1.6 wt% H2O) and model 2 (2VM3, 3.3 wt% H2O) of ref. [7].

3.2. Hydrous wadsleyite

For DFT modeling of hydrous wadsleyite it was assumed that hydrogen is incor-porated close to Mg vacancies at the M3 site. Full geometry optimization wasperformed with ABINIT starting from several different initial positions for two hy-drogen atoms replacing one Mg atom in the orthorhombic unit cell. In agreementwith earlier studies [7], the lowest energy structures for both 1.6 and 3.3 wt% H2Oare obtained when the two or four hydrogen atoms bond to oxygen at O1 sites and

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6 S. Jahn et al.

the OH dipoles are oriented almost parallel to O1-H...O4 (see Fig. 2a). All otherstructural models had energies of at least 0.4 eV/cell higher than the low energystructures. This result is independent on the functional or pressure. In addition tothe lowest energy configuration with hydration on O1-H...O4 edges only, structuraloptimization of the cells with 1.6 wt% H2O also resulted in configurations with oneH on an O1-H...O4 edge and the other H on O3-H...O3, O3-H...O4 or O4-H...O3.Combinations of O1-H...O4 and O1-H...O3 were not observed. The optimization ofa configuration where both H were positioned on O3-H-O4 edges yielded an energyof about 1.7 eV larger than the lowest energy structure with two O1-H...O4.

To allow comparison to measured IR spectra of hydrous wadsleyite [8], we com-puted the low-frequency dielectric permittivity tensor. Its imaginary part as a firstapproximation to the IR spectrum is shown in Fig. 2b for the PBE calculations at0 GPa and 15 GPa. In the frequency range of OH-stretching vibrations a singlepeak is observed, which consists of two vibrational contributions of very similarfrequencies: 3256/3258 cm−1 at 0 GPa and 3079/3080 cm−1 at 15 GPa. Usingthe LDA functional, the frequencies are almost 10% lower (2994/2997 cm−1 at 0GPa). The low energy configurations of wadsleyite with 3.3 wt% H2O show verysimilar spectra with 4 OH bands between 3322 and 3329 cm−1 at 0 GPa and be-tween 3164 and 3173 cm−1 at 15 GPa (PBE functional, thin lines in Fig. 2b). Thehigher energy structures with 1.6 wt% H2O, one H on O1-H...O4 and the otheron O3-H...O3, O3-H...O4 or O4-H...O3 show a splitting of the OH frequencies, e.g.3209/3407 cm−1 for the lowest energy O3-H...O4 structure and 2926/3400 cm−1

for the lowest energy O3-H...O3 structure.For the lowest energy configurations, the vibrational contribution to the entropy

Svib at ambient conditions increases by 0.9 (LDA) to 1.4 J/(mol K) (PBE) forwadsleyite with 1.6 wt% H2O and by 2.3 J/(mol K) (PBE) for wadsleyite with3.3 wt% H2O compared to the anhydrous phase (see Table 1). At 15 GPa androom temperature, a smaller increase is observed: 0.7 to 0.8 J/(mol K) for 1.6 wt%H2O and 1.1 J/(mol K) for 3.3 wt% H2O. The respective CV of the hydrous modelsare somewhat lower than the anhydrous ones at 298 K (Table 1). The higher energystructures with 1.6 wt% H2O yield Svib between 93.4 and 93.9 J/(mol K) (PBE)for the O3-H...O3 configurations and those O3-H...O4 or O4-H...O3 configurations,which connect to a different O4 than the second OH group (O1-H...O4). In the caseswhere the two OH groups connect to the same O4 (O1-H...O4 and O3-H...O4 orO4-H...O3), the vibrational entropy is somewhat larger (94.8 to 95.0 J/(mol K)).

4. Discussion

4.1. Anhydrous wadsleyite

The entropy and heat capacity values of anhydrous wadsleyite derived fromour calorimetric measurements are in good agreement with previous measure-ments by Akaogi et al. [16] (see Table 1). Both calorimetric data sets are con-sistent with entropies derived from vibrational spectra using the Kieffer model(S0 =87.4 J/(mol K) [37] and 85.1 J/(mol K) [17]). However, the standard en-tropies derived from thermodynamic measurements and vibrational spectroscopyare substantially lower than those derived from phase equilibria and used in in-ternally consistent thermodynamic models (S0 = 95 J/(mol K) [21, 22, 38] and91.32 J/(mol K) [39]). In the latter case, S0 appears as a free fit parameter thatis not well constrained. We suggest to use a more realistic value of S0 of about86-87 J/(mol K) in future thermodynamic models.

Electronic structure calculations of the present study but also from previous

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Thermodynamics of wadsleyite 7

(a)

0 250 500 750 1000

wavenumber (cm-1

)

0

5

10

Im[ε

(ω)]

(a.

u.)

3000 32500

1

2

3

(b)15 GPa

0 GPa

Figure 2. (a) DFT-optimized lowest energy structure of hydrogen incorporation into wadsleyite with1.6 wt% H2O. Hydrogen atoms (white balls) are located close to Mg vacancies at the M3 site (at thecenter of the octahedral wire frame). They are solely bonded to oxygen atoms at O1 sites (red balls)and directed towards oxygens at O4 sites. Yellow octahedra and blue tetrahedra represent MgO6 andSiO4 units. (b) Isotropic average of the imaginary part of the low-frequency dielectric permittivity tensor,Im[ǫ(ω)], of the structure shown in (a) from DFT-PBE calulations at 0 GPa (blue lines) and 15 GPa (reddashed lines). The thin lines in the high frequency range refer to the low energy configuration with 3.3wt% H2O.

investigations of anhydrous wadsleyite [19, 20] yield heat capacities and entropiesthat are broadly consistent with the experimental data (see Table 1). In orderto link our computed harmonic heat capacities at constant volume, CV to themeasured CP , we added the anharmonic correction term using the relation

CP = CV + TV α2KT (1)

As the correction is small at least at low temperatures we refrained from a fullquasiharmonic derivation of the thermal expansion coefficient α and the isothermalbulk modulus KT (as performed in [19, 20]) and used instead experimental valuesof α and KT from [16] and references therein. The resulting CP , shown in Fig. 1 asdashed lines, of the LDA calculations are very similar to the experimental valuesup to at least 600 K whereas the PBE values are somewhat larger over the wholetemperature range. Increasing pressure results in a decrease of both CV and Svib

(Table 1), which is consistent with DFT calculations by Wu and Wentzcovitch [19].The heat capacities above ambient T can be represented by a Berman and Brown-

type CP polynomial [34]. To constrain CP beyond the experimental temperaturerange we jointly fit the DSC-measured CP and the computed (DFT PBE) CP

between 1000 K and 2000 K and obtain

CP (T ) = 207(5) − 846(190)T−0.5− 6.6(17) × 106T−2 + 8.4(34) × 108T−3 (2)

where CP and T are given in units of J/(mol K) and Kelvin. Compared to the CP

polynomial proposed by [35], equation 2 yields CP s that are generally larger byseveral percent. A maximum deviation of about 4% is observed at 700 K, whereasit is 1–3% in the temperature range between 1000 K and 2000 K. Note that theharmonic approximation becomes increasingly inappropriate towards high temper-atures and therefore the high temperature data of this fit have to be used with care.However, if the DFT results are not used as a constraint for the high temperaturebehavior the exptrapolation beyond the experimental range becomes even less reli-able. As an independent test, we computed the high temperature CP using classical

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8 S. Jahn et al.

molecular dynamics simulations with a polarizable ionic potential that has beenshown to be accurate for Mg2SiO4 phases in a wide range of pressures and tem-peratures [40]. These simulations yield CP of 184(2) J/(mol K) at 2000 K, whichis consistent with the prediction of our fitted polynomial (Eq. 2) of 187 J/(mol K).

4.2. Hydrous wadsleyite

Our DFT calculations suggest that the hydration of wadsleyite leads to a smallincrease of Svib. For H2O contents of 1.6 wt% and 3.3 wt%, the vibrational en-tropies at standard conditions are 0.9 to 1.4 J/(mol K) and 2.3 J/(mol K) largerthan those of anhydrous wadsleyite (Table 1). This assumes that the lowest energyconfigurations of H incorporation dominate at ambient temperature. This smallincrease in Svib is in contrast with the S0 optimized for the internally consistentthermodynamic data set by Komabayashi et al. [21, 22]. Their S0 values for anhy-drous and hydrous wadsleyite are 95 and 84 J/(mol K), respectively. Hence, theypredict a much larger difference and a decrease in S0 with increasing H2O content,which is not plausible. As already stated above, using a more realistic S0 value foranhydrous wadsleyite in their thermodynamic data set would resolve part of thediscrepancy. To assess the merit of our DFT predictions we discuss in the followingthe structural model and the possible contribution of configurational entropy.

The lowest energy structural models proposed here (Fig. 2a) are consistent withearlier DFT simulations [7] and also agree with recent experimental observations[8, 9] with regard to O1 being the major hydration site and the O1-H vectorlying predominantly along a M3 edge pointing towards O4. The lowest energyconfiguration of the M3 vacancy hydrogen incorporation mechanism studied hereis obtained when both hydrogens are placed on the two long O1...O4 edges ofan M3 octahedron. Other configurations with one hydrogen on an O1...O4 edgeand the other e.g. on an O3...O3 or an O3...O4 edge (as suggested in [8]) yieldsignificantly larger energies (at least 0.4 eV per simulation cell). They are thereforeunlikely to occur in a relatively ordered structural environment that is imposed bya periodic simulation cell containing only 57 atoms. For wadsleyite with 1.6 wt%H2O, hydration leads to a monoclinic distortion of the unit cell (see Table 1)of about 1.2 degrees, which is larger than that observed experimentally. This isplausible if it is considered that the defect structure in the simulations is fullyordered. A random distribution or partial clustering of M3 vacancies is expectedto reduce the monoclinic distortion. The low energy configuration considered forthe case of 3.3 wt% H2O even results in an orthorhombic cell again (Table 1). Thetotal energy of wadsleyite with 1.6 wt% H2O is very similar to that of a phasecomposed of dry wadsleyite and wadsleyite with 3.3 wt% H2O. This suggests arelatively random distribution of M3 vacancies under the condition that they arecoupled to two vacant O1 hydration sites.

The computed vibrational spectra (Fig. 2b) show a relatively well defined sin-gle peak in the OH stretching region above 3000 cm−1, which is composed oftwo (1.6 wt%) or four (3.3 wt%) very similar eigenfrequencies. Coupling betweenadjacent hydrated M3 vacancies leads to a shift to higher wavenumbers in thelatter case compared to the former. The vibrational frequencies of isolated or clus-tered hydrated M3 vacancies may be related to the two major OH bands in theIR spectra of hydrated wadsleyite. The pressure evolution of the OH bands ν2

and ν3 reported by Deon et al. [8] shows frequency shifts of -8.8 cm−1/GPa and-7.5 cm−1/GPa in the pressure range from 0 to 16 GPa. In comparison, the DFT-PBE calculations yield frequency shifts with a rate of -11.8 cm−1/GPa for thelower and -10.3 cm−1/GPa for the higher frequency bands. At ambient pressures,

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Thermodynamics of wadsleyite 9

the absolute computed frequencies are about 2% lower than the experimental ones,e.g. 3256-3258 cm−1 (1.6 wt% calculation) vs. 3326 cm−1 (ν2) or 3322-3329 cm−1

(3.3 wt% calculation) vs. 3383 cm−1 (ν3). The experimental spectra appear to bemore complex [6, 8] and consist of a number of distinct modes that are not coveredby the present calculations. Deon et al. [8] find another band with similar pleochroicand high pressure behavior but appearing at higher wavenumbers. Furthermore,four bands between 3580 and 3680 cm−1 with a negligible pressure dependencewere reported. This suggests that other hydrogen incorporation mechanisms mustbe present. According to [7], Mg vacancies on the M2 site are energetically unfavor-able. Future studies are needed to investigate the possibility of other incorporationmechanisms or disorder and their effect on the vibrational spectra and eventuallyon the thermodynamic properties of hydrous wadsleyite.

So far, the DFT-derived entropies only contained the vibrational contribution. Ifhydroxyl groups are distributed in the wadsleyite structure more or less randomly,configurational entropy Sconfig has to be added to Svib presented in Table 1 toderive S0. In the following we estimate Sconfig for hydrous wadsleyite that containseither 0.25 or 0.50 H atoms per formula unit, i.e. for Mg0.875H0.25SiO4 (1.6 wt%H2O) and for Mg0.75H0.5SiO4 (3.3 wt% H2O).

We employ the structural considerations underlying the thermodynamic modelof Frost and Dolejs [11], which provides a better description of the phase dia-gram of hydrous Mg2SiO4 than an earlier model by Wood [41]. The main differ-ence between the two models lies in the degree of ordering of hydroxyl groupsand Mg vacancies. Frost and Dolejs [11] assumed that two hydroxyl groups areassociated with one Mg vacancy, which is plausible due to local charge balancerequirements. Further, Mg vacancies are assumed to be distributed on one half ofthe M3 sites (here called M3b sites). The O1 site as the dominant hydration siteis confirmed by number of recent studies including the present one [5, 7, 9]. Thus,the structural model for hydrogen incorporation may be described by the formulaMgM3a

0.5 (Mg,V)M3b0.5 MgM1−M2Si(O,OH)O1

0.5OO2−O43.5 . Ideal mixing of Mg vacancies (V )

leads to

Sconfig = −R[

XV ln((XV )0.5) + (1 − XV ) ln((1 − XV )0.5)]

(3)

with XV the mole fraction of V on the M3b site, i.e. XV = 0.25 for 1.6 wt% H2Oand 0.25 OH per formula unit; XV = 0.5 for 3.3 wt% H2O and 0.5 OH per formulaunit. The power of 0.5 arises from the M3b site multiplicity. In this formulation it isassumed that the positions of the OH groups are fixed due to local charge balancerequirements (mainly on O1-H...O4 edges) and do not contribute to Sconfig. Theapplication of this model to wadsleyite containing 1.6 wt% H2O (XV = 0.25) and3.3 wt% H2O (XV = 0.5) yields Sconfig of 2.3 J/(mol K) and 2.9 J/(mol K),respectively.

The resulting S0 including configurational entropy would increase to 89.6 J/(molK) (DFT-LDA) and 95.4 J/(mol K) (DFT-PBE) for hydrous wadsleyite with 1.6wt% H2O and to 90.2 J/(mol K) (DFT-LDA) and 96.0 J/(mol K) (DFT-PBE) forhydrous wadsleyite with 3.3 wt% H2O at ambient pressure. Hence, the standardentropy of hydrous wadsleyite with 1.6 and 3.3 wt% H2O is predicted to be about3.5 J/(mol K) and 5 J/(mol K) larger than that of anhydrous wadsleyite. Notethat the possible presence of competing hydrogen incorporation mechanisms orsomewhat different mixing behavior still puts some uncertainties on these values.

In conclusion, S0 of anhydrous wadsleyite is now well constrained and shouldbe used in thermodynamic models. Our study supports the presumably dominanthydrogen incorporation mechanism via Mg vacancies at M3 sites and hydration of

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10 REFERENCES

O1 with the OH dipole pointing to O4 but other hydroxyl groups must be present aswell to explain experimentally observed spectral features. Using density-functionalperturbation theory and thermodynamic modeling we predict standard entropiesof hydrous wadsleyite, which still need to be validated by experiment.

5. Acknowledgments

We thank Artur Benisek for his support during the thermodynamic measurements,Matthias Gottschalk and two anonymous reviewers for helpful comments and dis-cussions. This work was supported by DFG projects KO1260/11-1 within priorityprogram SPP1236 and JA1469/4-1 within the Emmy-Noether program.

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