First Principles Thermodynamics of Minerals at HP–HT Conditions: MgO as a Prototypical Material
Abstract
:1. Introduction
2. Materials and Methods
3. Results and Discussion
3.1. Thermal Expansion and Thermophysical Properties
3.2. Thermodynamic Properties
3.3. P–V–T Equation of State (EOS)
- (i)
- ab initio quasi-harmonic thermal pressure, in contrast to thermal expansivity, doesn’t display any deviation at P = 0 (i.e., V/V0 = 1.00) and high-temperature conditions (i.e., T > ΘD ≅ 945 K) (see also Section 3.1). This means that Pth is rather insensitive to QHA deviations at LP-HT conditions, at least for what concerns the accuracy of numerical predictions, as already noted in other theoretical works [61,105,106]. Thermal pressure is quite insensitive to compression, too, especially in the low- to intermediate-T range;
- (ii)
- ab initio quasi-harmonic calculations of thermal pressure reproduce quite well the values assessed from experiments at 1-bar pressure [57,102]. The slight differences with respect to the values of Isaak et al. [57] and Srivastava [102] at low temperature are due to the constraint Pth (300 K, 1 bar) = 0, assumed to be valid in those works; this assumption lacks a theoretical justification, because thermal vibrations obviously occur already at T = 0 K (zero-point motions), and their effects cannot be disregarded;
- (iii)
- Although the lower mantle consists primarily of bridgmanite, the thermal pressure of MgO seems to be representative of that of the lower mantle [103,104] owing to the peculiar thermoelastic and transport properties of periclase. Lower mantle thermodynamics and rheology are thus largely controlled by periclase and its Fe-bearing analogue (ferropericlase) [16,107,108,109,110].
3.4. Phase Equilibrium Calculation at HP–HT: Modelling High-Pressure Effects on Gibbs Free Energy
3.5. Phase Equilibrium Calculation at HP–HT: The Post-Spinel Phase Transformation
4. Conclusions
- At ambient pressure, ab initio thermophysical properties of MgO (such as α and KT) show a good agreement with experimental data up to relatively high temperatures (i.e., T ≈ 1800–2000 K), then deviations occur due to losing the power of the quasi-harmonic approximation at high-temperature (and low-pressure) conditions. Nevertheless, B3LYP results seem to show a less marked deviation in thermal expansivity with respect to other density functionals (e.g., LDA) under those conditions. Thermodynamic properties at low and up to relatively high temperatures are accurately defined by ab initio B3LYP phonon dispersion calculations. The level of accuracy with regard to current calorimetric results is around the experimental uncertainty (i.e., 1–2%) throughout the whole range of experimental measurement (i.e., T = 298.15–1000 K).
- A first principles Mie-Grüneisen EOS formalism (splitting pressure into static, zero-point and thermal contributions) gives reliable and physically-consistent P–V–T relations for MgO up to lower mantle conditions, provided that an accurate description of lattice vibrations and phonon density of state is available. Quasi-harmonic thermal pressures turn out to be rather insensitive to both volume compression and LP-HT deviations.
- P–V–T equations of state based on finite strain theory (such as HT-BM3-EOS) systematically predict anomalous thermodynamic functions (like negative thermal expansion or entropy increase with pressure) at HP–HT conditions. Nevertheless, spurious effects on the V(P,T) function become relevant for MgO at pressures higher than 60 GPa, but in most cases don’t have a huge impact on phase equilibrium calculations at deep mantle conditions as the ensuing Gibbs free energy differences tend to cancel out.
- Hybrid DFT and statistical mechanics calculations in the quasi-harmonic approximation accurately simulate phase reaction boundaries, such as the post-spinel phase transformation of ringwoodite (Mg2SiO4) to periclase (MgO) and bridgmanite (MgSiO3), at HP–HT conditions. Anharmonic effects, if any, are virtually irrelevant under those conditions. The hybrid B3LYP functional predicts accurate thermodynamic properties and, at the same time, reliable phase stability relations for magnesium oxide (and silicates) at HP–HT.
- The Clapeyron slope of post-spinel phase transformation is correctly predicted by ab initio B3LYP calculations, both in sign and magnitude. The calculated value in this work (i.e., dP/dT = −2.53 MPa/K) is in good agreement with the majority of experimental investigations, especially with those based on calorimetric measurements. Although the Clapeyron slope of the phase boundary is almost insensitive to the exchange-correlation term, its P–T location strongly depends upon the choice of the density functional: B3LYP results are in qualitative agreement with previous GGA calculations, while LDA drastically underestimates transformation pressures.
- The calculated density change across the post-spinel phase boundary at P–T conditions compatible with the 660-km seismic discontinuity in the Earth’s mantle is different from that inferred by geophysical models. This suggests either an incomplete dissociation reaction of ringwoodite to periclase + bridgmanite or the occurrence of multiple mineral phase changes in mantle aggregates at those depths.
Acknowledgments
Conflicts of Interest
Appendix A
T (K) | KT (GPa) | α × 105 (K−1) | VT 1 (cm3/mol) | CV (J/mol·K) | CP (J/mol·K) | S (J/mol·K) | γth 2 |
---|---|---|---|---|---|---|---|
0 | 161.5 | 0.00 | 11.406 | 0.000 | 0.000 | 0.000 | - |
100 | 161.4 | 0.61 | 11.412 | 7.532 | 7.539 | 2.290 | 1.496 |
298.15 | 157.7 | 2.97 | 11.444 | 36.142 | 36.616 | 26.251 | 1.483 |
300 | 157.7 | 2.98 | 11.444 | 36.272 | 36.753 | 26.478 | 1.483 |
400 | 155.2 | 3.42 | 11.481 | 41.339 | 42.174 | 37.870 | 1.475 |
500 | 152.5 | 3.70 | 11.522 | 44.069 | 45.269 | 47.640 | 1.474 |
600 | 149.8 | 3.89 | 11.566 | 45.674 | 47.246 | 56.080 | 1.475 |
700 | 147.0 | 4.04 | 11.613 | 46.687 | 48.641 | 63.474 | 1.478 |
800 | 144.2 | 4.18 | 11.660 | 47.364 | 49.711 | 70.042 | 1.483 |
900 | 141.3 | 4.30 | 11.710 | 47.837 | 50.592 | 75.947 | 1.488 |
1000 | 138.4 | 4.42 | 11.760 | 48.180 | 51.361 | 81.314 | 1.493 |
1100 | 135.5 | 4.54 | 11.812 | 48.437 | 52.061 | 86.236 | 1.499 |
1200 | 132.5 | 4.66 | 11.866 | 48.633 | 52.724 | 90.787 | 1.505 |
1300 | 129.5 | 4.78 | 11.922 | 48.787 | 53.370 | 95.026 | 1.512 |
1400 | 126.5 | 4.90 | 11.979 | 48.910 | 54.010 | 98.999 | 1.519 |
1500 | 123.4 | 5.03 | 12.039 | 49.009 | 54.657 | 102.744 | 1.526 |
1600 | 120.4 | 5.17 | 12.100 | 49.090 | 55.320 | 106.293 | 1.534 |
1700 | 117.3 | 5.31 | 12.164 | 49.158 | 56.005 | 109.672 | 1.542 |
1800 | 114.1 | 5.47 | 12.231 | 49.215 | 56.719 | 112.902 | 1.550 |
1900 | 111.0 | 5.63 | 12.300 | 49.263 | 57.471 | 116.003 | 1.559 |
2000 | 107.8 | 5.80 | 12.372 | 49.304 | 58.265 | 118.988 | 1.568 |
2100 | 104.6 | 5.98 | 12.446 | 49.340 | 59.109 | 121.872 | 1.577 |
2200 | 101.3 | 6.17 | 12.524 | 49.371 | 60.011 | 124.665 | 1.587 |
2300 | 98.1 | 6.38 | 12.605 | 49.398 | 60.978 | 127.376 | 1.597 |
2400 | 94.8 | 6.61 | 12.688 | 49.421 | 62.020 | 130.013 | 1.607 |
2500 | 91.4 | 6.85 | 12.775 | 49.442 | 63.147 | 132.583 | 1.618 |
2600 | 88.1 | 7.12 | 12.865 | 49.461 | 64.374 | 135.093 | 1.630 |
2700 | 84.7 | 7.40 | 12.959 | 49.477 | 65.712 | 137.546 | 1.642 |
2800 | 81.2 | 7.72 | 13.056 | 49.492 | 67.180 | 139.947 | 1.654 |
2900 | 77.8 | 8.06 | 13.157 | 49.505 | 68.800 | 142.300 | 1.667 |
3000 | 74.3 | 8.45 | 13.262 | 49.517 | 70.594 | 144.608 | 1.680 |
T (K) | Uvib (kJ/mol) | Hvib (kJ/mol) | Svib (J/mol·K) | Fvib (kJ/mol) | Gvib (kJ/mol) |
---|---|---|---|---|---|
0 | 14.555 1 | 14.556 | 0.000 | 14.555 | 14.556 |
100 | 14.735 | 14.736 | 2.290 | 14.506 | 14.507 |
298.15 | 19.581 | 19.582 | 26.080 | 11.805 | 11.806 |
300 | 19.648 | 19.649 | 26.304 | 11.757 | 11.758 |
400 | 23.556 | 23.557 | 37.513 | 8.551 | 8.552 |
500 | 27.839 | 27.840 | 47.059 | 4.309 | 4.310 |
600 | 32.333 | 32.334 | 55.248 | −0.816 | −0.815 |
700 | 36.954 | 36.955 | 62.370 | −6.705 | −6.704 |
800 | 41.659 | 41.660 | 68.651 | −13.262 | −13.261 |
900 | 46.421 | 46.422 | 74.259 | −20.412 | −20.411 |
1000 | 51.222 | 51.223 | 79.318 | −28.096 | −28.095 |
1100 | 56.054 | 56.055 | 83.922 | −36.261 | −36.259 |
1200 | 60.908 | 60.909 | 88.146 | −44.867 | −44.866 |
1300 | 65.779 | 65.780 | 92.045 | −53.879 | −53.878 |
1400 | 70.664 | 70.665 | 95.665 | −63.267 | −63.266 |
1500 | 75.560 | 75.561 | 99.043 | −73.004 | −73.003 |
1600 | 80.465 | 80.466 | 102.208 | −83.069 | −83.067 |
1700 | 85.378 | 85.379 | 105.187 | −93.439 | −93.438 |
1800 | 90.296 | 90.297 | 107.998 | −104.101 | −104.099 |
1900 | 95.220 | 95.221 | 110.660 | −115.035 | −115.033 |
2000 | 100.149 | 100.150 | 113.188 | −126.228 | −126.226 |
2100 | 105.081 | 105.082 | 115.595 | −137.668 | −137.667 |
2200 | 110.017 | 110.018 | 117.891 | −149.343 | −149.341 |
2300 | 114.955 | 114.956 | 120.086 | −161.243 | −161.241 |
2400 | 119.896 | 119.897 | 122.189 | −173.357 | −173.356 |
2500 | 124.839 | 124.840 | 124.207 | −185.678 | −185.677 |
2600 | 129.784 | 129.785 | 126.146 | −198.196 | −198.195 |
2700 | 134.731 | 134.732 | 128.013 | −210.905 | −210.903 |
2800 | 139.680 | 139.681 | 129.813 | −223.796 | −223.795 |
2900 | 144.630 | 144.631 | 131.550 | −236.865 | −236.863 |
3000 | 149.581 | 149.582 | 133.228 | −250.104 | −250.103 |
Appendix B
Thermophysical Properties | MgO (pc) 1 | MgSiO3 (bgm) 2 | γ-Mg2SiO4 (rng) 3 |
---|---|---|---|
K0 (GPa) | 167.01 | 249.2 | 196.4 |
K′0 | 3.95 | 4.2 | 4.322 |
(dK/dT)P (bar/K) | −308.0 | −280.0 | −104.108 |
α0 × 107 | 0.2608 | 0.0818 | 0.0 |
α 1 × 107 | −104.77954 | 198.2 | 323.0 |
α 2 × 103 | 42.9578 | 0.0 | −7.3084 |
α3 | −17.04307 | −0.474 | 1.3745 |
α 4 | 2122.59 | 0.0 | −150.406 |
a × 10−2 | 0.44202 | 1.1012 | 1.5346 |
b × 102 | 0.39852 | 0.95903 | 2.1405 |
c × 10−6 | −1.2908 | −3.8879 | −4.749 |
d × 105 | 0.12101 | 0.099651 | −0.308 |
e × 10−2 | 0.93607 | 1.9259 | 1.2708 |
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T (K) | KT (GPa) B3LYP | KT (GPa) LDA [58] | KT (GPa) LDA [59] | KT (GPa) GGA [61] | KT (GPa) Exp. [57] |
---|---|---|---|---|---|
T = 0 K | 167.01 1; 161.53 2 | 173 1; 164 2 | 168.8 2 | 181.24 1; 173.48 2 | - |
T = 300 K | 157.69 | 159 | 163.2 | 170.53 | 161.6 ± 0.6 |
T =1000 K | 138.38 | 139 | 143.0 | 152.59 | 141.4 |
T = 2000 K | 107.78 | 108 | 114.9 | 127.72 | 110.7 |
T = 3000 K | 74.26 | 74 | 86.7 | 106.11 | - |
T (K) | K′T (GPa) B3LYP | K′T (GPa) LDA [59] | K′T (GPa) GGA [61] | KT (GPa) Assessment [63] |
---|---|---|---|---|
T = 0 K | 3.95 1; 4.01 2 | - | 3.997 1; 4.014 2 | - |
T = 300 K | 4.04 | 4.11 | 4.036 | 3.99 |
T =1000 K | 4.22 | 4.43 | 4.130 | 4.15 |
T = 2000 K | 4.52 | 4.65 | 4.244 | 4.46 |
T = 3000 K | 4.95 | 4.93 | 4.331 | 4.98 |
T (K) | 2 × 2 × 2 ab initio B3LYP | 4 × 4 × 4 ab initio B3LYP | 5 × 5 × 5 ab initio B3LYP | NIST-JANAF [74] |
---|---|---|---|---|
298.15 | 34.4 | 36.5 | 36.6 | 37.1 |
700 | 45.9 | 48.5 | 48.6 | 48.7 |
1000 | 48.5 | 51.1 | 51.3 | 51.2 |
1800 | 53.8 | 56.5 | 56.7 | 54.9 |
V/V0 | T (K) | Pst (GPa) 1 | PZPC (GPa) 2 | Pth (GPa) 2 | P (GPa) 2 |
---|---|---|---|---|---|
1.00 | 300 | 0.00 | 1.77 | 0.67 | 2.44 |
0.95 | 300 | 9.48 | 1.86 | 0.58 | 11.92 |
0.90 | 300 | 21.67 | 2.00 | 0.51 | 24.17 |
0.85 | 300 | 37.42 | 2.20 | 0.45 | 40.07 |
0.80 | 300 | 57.93 | 2.50 | 0.41 | 60.84 |
0.75 | 300 | 84.87 | 2.96 | 0.40 | 88.23 |
0.70 | 300 | 120.63 | 3.69 | 0.40 | 124.72 |
0.65 | 300 | 168.74 | 4.84 | 0.42 | 174.01 |
1.00 | 500 | 0.00 | 1.77 | 1.72 | 3.49 |
0.95 | 500 | 9.48 | 1.86 | 1.57 | 12.90 |
0.90 | 500 | 21.67 | 2.00 | 1.43 | 25.10 |
0.85 | 500 | 37.42 | 2.20 | 1.34 | 40.97 |
0.80 | 500 | 57.93 | 2.50 | 1.30 | 61.73 |
0.75 | 500 | 84.87 | 2.96 | 1.32 | 89.15 |
0.70 | 500 | 120.63 | 3.69 | 1.40 | 125.72 |
0.65 | 500 | 168.74 | 4.84 | 1.56 | 175.14 |
1.00 | 1000 | 0.00 | 1.77 | 4.70 | 6.47 |
0.95 | 1000 | 9.48 | 1.86 | 4.40 | 15.74 |
0.90 | 1000 | 21.67 | 2.00 | 4.19 | 27.85 |
0.85 | 1000 | 37.42 | 2.20 | 4.07 | 43.70 |
0.80 | 1000 | 57.93 | 2.50 | 4.11 | 64.54 |
0.75 | 1000 | 84.87 | 2.96 | 4.33 | 92.16 |
0.70 | 1000 | 120.63 | 3.69 | 4.80 | 129.11 |
0.65 | 1000 | 168.74 | 4.84 | 5.57 | 179.16 |
1.00 | 2000 | 0.00 | 1.77 | 10.90 | 12.67 |
0.95 | 2000 | 9.48 | 1.86 | 10.37 | 21.71 |
0.90 | 2000 | 21.67 | 2.00 | 10.02 | 33.68 |
0.85 | 2000 | 37.42 | 2.20 | 9.92 | 49.55 |
0.80 | 2000 | 57.93 | 2.50 | 10.20 | 70.63 |
0.75 | 2000 | 84.87 | 2.96 | 10.96 | 98.78 |
0.70 | 2000 | 120.63 | 3.69 | 12.380 | 136.69 |
0.65 | 2000 | 168.74 | 4.84 | 14.70 | 188.29 |
1.00 | 3000 | 0.00 | 1.77 | 17.16 | 18.93 |
0.95 | 3000 | 9.48 | 1.86 | 16.40 | 27.74 |
0.90 | 3000 | 21.67 | 2.00 | 15.92 | 39.59 |
0.85 | 3000 | 37.42 | 2.20 | 15.87 | 55.49 |
0.80 | 3000 | 57.93 | 2.50 | 16.40 | 76.83 |
0.75 | 3000 | 84.87 | 2.96 | 17.73 | 105.55 |
0.70 | 3000 | 120.63 | 3.69 | 20.16 | 144.47 |
0.65 | 3000 | 168.74 | 4.84 | 24.11 | 197.70 |
Post-Spinel Reaction | Δ (kJ/mol) | Δ (J/mol·K) | Δ (cm3/mol) | dP/dT (MPa/K) |
---|---|---|---|---|
Ab initio (B3LYP, this work) | 90.69 | 3.25 | −3.52 | −2.53 |
Exp. [125] (calorimetry) | 96.8 ± 5.8 | 11.1 ± 3.7 | −3.79 ± 0.04 | −4.0 ± 2.0 1 |
Exp. [126] (calorimetry) | 86.1 ± 3.6 | 1.2 ± 2.3 | −3.84 | −3.0 ± 1.0 1 |
Exp. [131] (calorimetry) | 88.4 ± 2.5 | 1.4 | −3.80 | −2.6 ± 0.2 1 |
Exp. [134] (calorimetry) | 78.54 ± 2.28 | 2.1 ± 0.6 | −3.79 | −1.5 ± 0.6 1 |
Exp. [135] (spectroscopy) | 99.0 2 | 9.5 | −3.77 | −2.5 ± 0.4 |
Exp. [121] | - | - | - | −2.3 ± 0.4 1 |
Exp. [124,127] | - | - | - | −2.8 |
Exp. [129] | - | - | - | −1.2 ± 0.8 |
Exp. [130] | - | - | - | −1.3 ± 0.3 |
Exp. [132] | - | - | - | −2.0 |
Exp. [133] | - | - | - | −0.55 ± 0.15 |
Ab initio (LDA/GGA) [137] | - | - | - | −2.6/−2.9 |
Ab initio (GGA) [138,139] | ≈82 3 | 3.78 | −3.82 | −3.9 ± 1.3 |
Density | This Work (Ab Initio B3LYP) | PREM [123] | SF’99 [142] | Yu07 [137] |
---|---|---|---|---|
ρ (g/cm3) at depth < 660 km (rng) | 3.70 1 | 3.99 | - | 3.81 3 |
3.76 2 | ||||
ρ (g/cm3) at depth > 660 km (pc + bgm aggregate) | 3.98 1 | 4.38 | - | 4.13 3 |
4.04 2 | ||||
Δρ/ρ (%) (rng → pc + pv) | 7.1% 1 | 9.3% | 5 ± 1% | 7.9% 3 |
7.2% 2 |
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Belmonte, D. First Principles Thermodynamics of Minerals at HP–HT Conditions: MgO as a Prototypical Material. Minerals 2017, 7, 183. https://doi.org/10.3390/min7100183
Belmonte D. First Principles Thermodynamics of Minerals at HP–HT Conditions: MgO as a Prototypical Material. Minerals. 2017; 7(10):183. https://doi.org/10.3390/min7100183
Chicago/Turabian StyleBelmonte, Donato. 2017. "First Principles Thermodynamics of Minerals at HP–HT Conditions: MgO as a Prototypical Material" Minerals 7, no. 10: 183. https://doi.org/10.3390/min7100183