The representative XRD patterns for calcite, dolomite and magnesite at elevated temperatures are shown in Figs. 2a-2c, and the unit-cell parameters are refined by the "unit cell" software (Holland and Redfern, 1997), and are listed in Table S2. The unit-cell parameters at ambient condition are a=4.998 2(1) Å, c=17.073 2(7) Å and V=369.38(2) Å3 for calcite; a=4.808 7(7) Å, c=16.027 8(5) Å and V=320.97(1) Å3 for dolomite; while a=4.645 3(1) Å, c=15.087 1(6) Å and V=281.94(2) Å3 for magnesite, which are consistent with the previous studies (Fahad and Saeed, 2018; Titschack et al., 2011; Falini et al., 1998; Effenberger et al., 1981). The a and c axes, as well as V, decrease in the order of calcite > dolomite > magnesite, with an increase of Mg percentage (Mg/(Mg+Ca)) in the compositions.
Figure 2. Selected powder XRD patterns at elevated temperature for (a) calcite, (b) dolomite and (c) magnesite, with the reflection peaks indexed.
Figure 3. Variations of the unit-cell parameters (a, c and V) with temperature for (a) calcite, (b) dolomite and (c) magnesite, which have been normalized to the ones at room temperature (a0, c0, and V0). Uncertainties of the unit-cell parameters are smaller than the sizes of the symbols.
The averaged thermal expansion coefficients for the axes (αa, αc) and volume (αV) are calculated in Table 1, and the volumetric thermal expansion coefficients are also fit for a linear function of temperature (α=a1×T+a0). Several high-temperature XRD experiments have been conducted on calcite, dolomite and magnesite (Wang et al., 2018; Merlini et al., 2016; Antao et al., 2009; Fei, 1995; Reeder and Markgraf, 1986; Markgraf and Reeder, 1985), however, the thermal expansion coefficients reported in these studies vary as a function of the temperatures range fit in each study. Therefore, to enable the direct comparison of these literature values with the present study, we recalculated the average volumetric thermal expansion coefficients from these earlier studies using the temperature range of this study (296-796 K), as shown in Fig. 4. The volumetric thermal expansion coefficients for calcite and dolomite measured in this study were consistent with previous measurements: 2.1×10-5-2.8×10-5 K-1 for calcite (Merlini et al., 2016; Antao et al., 2009; Fei, 1995; Markgraf and Reeder, 1985), 3.3×10-5-3.8×10-5 K-1 for dolomite (Merlini et al., 2016; Fei, 1995; Reeder and Markgraf, 1986) and 3.5(1)×10-5 K-1 for magnesite (Litasov et al., 2008).
Axial (10-5·K-1) Volumetric αa αc αV a1 (10-8·K-2) a0 (10-5·K-1) Calcite 0.09(3) 2.52(12) 2.71(9) 3.4(8) 1.2(5) Dolomite 0.64(3) 2.03(13) 3.30(14) 5.4(12) 0.4(6) Magnesite 0.67(3) 2.17(7) 3.51(10) 3.5(11) 1.6(6)
Table 1. The mean thermal expansion coefficients (10-5·K-1) for the axes and volume, the volumetric thermal expansion coefficients are also expressed in the form of α=a1×T+a0
Figure 4. Comparison among the mean thermal expansion coefficients for calcite, dolomite and magnesite of different studies.
Our high-temperature XRD results support the general trend that the thermal expansivities (αV) increase in the order of calcite < dolomite < magnesite, at ambient pressure. The same trend of calcite < dolomite < magnesite was also observed for the isothermal bulk moduli (KT0) (Merlini et al., 2016; Redfern and Angel, 1999; Fiquet et al., 1994), as well as for the elastic bulk (KS0) and shear (G0) moduli (Yang et al., 2014; Lin, 2013; Chen et al., 2006). Thus, in the pressure range before any phase transitions at ambient temperature, the compressibilities for these three minerals show the order of calcite > dolomite > magnesite, which is opposite to the order of the thermal expansivity. In the crystal structures of calcite and magnesite (R3c) (Antao et al., 2009; Bromiley et al., 2007), both Ca2+ and Mg2+ cations are coordinated with six O2- anions, and the radii ratio of RCa2+: RO2- is equal to 0.707 at ambient condition, which is close to the upper limit (0.732) of the cation-to-anion radii ratio for the 6-fold coordination. Therefore the calcite structure is relatively unstable at high pressure conditions, and transforms to aragonite (9-fold coordination for Ca2+) at pressures as low as 2 GPa (above 550 K) (Suito et al., 2001); while the magnesite structure is stable up to more than 80 GPa (~1 700 km depth in the lower mantle) (Pickard and Needs, 2015; Boulard et al., 2011; Oganov et al., 2006; Isshiki et al., 2004). The more stable MgO6 octahedron of magnetite may contribute to its smaller compressibility with larger thermal expansion.
vi (∂vi/∂T)P γip (∂vi/∂P)T γiT ai (cm-1) (10-2·cm-1·K-1) (cm-1/GPa) (10-5·K-1) Calcite v4 711.2 -0.56(2) 0.29(2) 5.66a 0.58(2) 0.65(2) v2 873.4 -0.82(5) 0.35(2) 3.77a 0.31(1) -0.50(4) v3 1 397.9 -1.89(10) 0.50(3) 6.17b 0.32(1) 0.09(6) Dolomite v4 726.8 -0.91(5) 0.38(3) 1.31c 0.17(1) -0.69(9) v2 879.7 -1.20(5) 0.42(3) 0.27c 0.03(1) -1.27(10) v3 1 410.6 -2.35(7) 0.51(3) 3.36c 0.22(1) -0.94(10) Magnesite v4 756.2 -1.10(6) 0.41(3) 2.64(4)d 0.40(1) -0.05(7) v2 885.7 -1.25(7) 0.42(2) 1.16(7)d 0.15(1) -0.96(5) v3 1 422.0 -2.74(15) 0.55(3) 4.23(7)d 0.34(1) -0.73(10) a. Adams et al. (1980); b. Weir et al. (1959); c. Santillán and Williams (2004); d. Grzechnik et al. (1999).
Table 2. The IR internal modes Grüneisen parameters γiP, γiT and the anharmonic parameters ai of the calcite, dolomite and magnesite.
On the other hand, the c axis of these carbonate crystals shows significantly larger thermal expansivity (this study, Merlini et al., 2016; Anto et al., 2009; Zhang et al., 1997) and compressibility (Merlini et al., 2016; Redfern and Angel, 1999; Zhang and Reeder, 1999; Fiquet et al., 1994), as compared to the a axis. These phenomena could be explained by the internal structure of these phases in which the coplanar CO3 groups, serving as rigid bodies, stack perpendicular to the c axis (Antao et al., 2009, 2004; Bromiley et al., 2007). In this study, we observed a slightly positive thermal expansion coefficient for the a axis (αa) of calcite (+0.09×10-5 K-1), which is significantly smaller than that of dolomite and magnesite (0.64×10-5-0.67×10-5 K-1). However, previous studies have reported a negative coefficient of αa (-0.05×10-5- -0.01×10-5 K-1) for the calcite samples with high purity (Antao et al., 2009; Dove et al., 2005; Dove and Powell, 1989; Markgraf and Reeder, 1985), therefore, this discrepancy is likely due to some impurity (Mg/(Mg+Ca)=0.4%) in our natural calcite sample. Even a small amount of Mg or Fe content in the carbonate structures could have significant effect on the thermo-elastic properties, such as thermal expansion and compressibility (Merlini et al., 2016).
According to the mode distribution and selection rules (Falini et al., 1998; Böttcher et al., 1997, 1992; White, 1974), there are 30 vibrational modes total for calcite and magnesite (space group R3c), as well as for dolomite (space group R3). These vibrational modes include 3 acoustics modes and 27 optical modes (Raman-active, IR-active, and inactive). Among these optical modes, there are 15 lattice vibrational modes (external modes) and 12 internal modes for the CO3 groups, while the internal modes for magnesite and calcite can be divided into four groups: the v1 (symmetric stretching) modes contain 1A1g (Raman active)+1A1u (inactive); the v2 (out-of-plane bending) modes contain 1A2g(inactive)+1A2u (IR active); while the v3 (asymmetric stretching) and v4 (in-plane bending) modes are constituted of 2Eg (Raman active)+2Eu (IR active).
The representative mid-FTIR spectra of these three carbonates obtained at elevated temperatures from this study are shown in Figs. 5a-5c, which are in good agreement with the previous IR measurements (Lane and Christensen, 1997; Böttcher et al., 1992; Andersen et al., 1991; White, 1974). Three internal modes of CO3 groups were observed in this FTIR measurement: v4 (711-756 cm-1), v2 (873-885 cm-1) and v3 (1 397-1 422 cm-1), while in this Raman spectra, the internal vibrations of v4 (710-740 cm-1), v1 (1 085-1 097 cm-1) and v3 (1 431-1 446 cm-1) were detected and shown in Fig. 1. The FTIR and Raman spectra show that both the external (< 400 cm-1) and the internal Raman modes (> 700 cm-1) shift to higher frequency when comparing magnesite versus calcite. The structure refinements by single-crystal XRD (Markgraf and Reeder, 1985) show that the C-O covalent bond length (dC-O) for calcite (1.290 1(5) Å) is slightly longer than that for magnesite (1.289(1) Å), with the libration correction applied.
Figure 5. Typical FTIR spectra for (a) calcite, (b) dolomite and (c) magnesite at high temperatures.
The variations of the frequencies for the internal modes at elevated temperature are plotted in Fig. 6 with linear regression, and the fitted slopes ((∂vi/∂T)P, cm-1·K-1) are listed in Table 2. The internal vibration modes shift to lower frequency at elevated temperature, and for each carbonate, the magnitude for the temperature-dependence increases as the frequency increases: v4 (in-plane bending) < v2 (out-of-plane bending) < v3 (asymmetric stretching), which is consistent with the observations fo aragonite-group carbonates (aragonite, strontianite, cerussite and witherite) by high-temperature Raman and FTIR spectra (Wang et al., 2019).
Two contributions should be considered for the frequency shift of a given mode (vi): (1) a pure volume contribution due to compression or thermal expansion (quasi-harmonicity); (2) a volume independent contribution due to the intrinsic anharmonicity (Fujimori et al., 2002; Gillet et al., 1989). Thus, the isobaric (γiP) and isothermal (γiT) Grüneisen parameters, as well as the anharmonic parameters (ai) need to be calculated to separate these two contributions, as shown in Eqs. 2-4 below
where νi is the wavenumber of a given vibrational mode, ρ is the density, while α and KT are the thermal expansion coefficient and isothermal bulk modulus, respectively.
With the adoption of the thermal expansion coefficients, we calculated the IR-active mode Grüneisen parameters (γiP) for calcite, dolomite and magnesite, which are listed in Table 2. In addition, high-temperature Raman spectra were also measured for these three minerals (Gillet et al., 1993; Biellmann and Gillet, 1992; Liu and Mernagh, 1990). Here, we selected detailed datasets from Gillet et al. (1993) and Liu and Mernagh (1990) to calculate the isobar Grüneisen parameters for the Raman-active modes (please refer to Table S3), and the γiP parameters from both FTIR (this study, solid symbols) and Raman (Gillet et al., 1993; Liu and Mernagh, 1990, open symbols) measurements, as shown in Fig. 7a.
Figure 7. (a) Isobaric (γiP) and (b) isothermal (γiT) modes Grüneisen parameters of calcite, dolomite and magnesite, and the solid and open symbols represent the datasets from FTIR and Raman spectra, respectively (This study for high-T FTIR; Gillet et al. (1993) and Liu and Mernagh (1990) for high-T and high-P Raman; while Santillán and Williams (2004), Grzechnik et al. (1999), Adams and Williams (1980) and Weir et al. (1959) for high-P FTIR). The vertical error bars stand for the estimated uncertainties of the Grüneisen parameters, if larger than the sizes of the symbols.
High-pressure FTIR (Santillán and Williams, 2004; Grzechnik et al., 1999; Adams and Williams, 1980; Weir et al., 1959) and Raman (Gillet et al., 1993; Biellmann and Gillet, 1992; Liu and Mernagh, 1990) spectra of calcite, dolomite and magnesite have been well studied. Using the reported isothermal bulk moduli of KT0=73.4(2) GPa for calcite (Redfern and Angel, 1999), KT0=94(1) for dolomite (Merlini et al., 2016) for dolomite, and KT0=115(1) GPa for magnesite (Fiquet and Reynard, 1999), we calculated the isothermal Grüneisen parameters (γiT) for the IR-active (Table 2) and Raman-active (Table S3) internal modes, which are also plotted in Fig. 7b. Due to the uncertainties and discrepancies among different experiments, we could not draw any clear trends for these internal vibrations. But generally, the γiP and γiT mode Grüneisen parameters for the internal modes of CO3 group are in the range of 0-0.5, which are significantly smaller than those for the external modes (lattice modes) (Wang et al., 2019; Gillet et al., 1993).
Based on the derived Grüneisen parameters γiP and γiT, we evaluated the intrinsic anharmonic parameters (ai) for these IR-active (Table 2) and Raman-active (Table S3) modes (Fig. 8). The ai values are in the range of -0.96- +0.65×10-5 K-1 for the v4 (in-plane bending) modes; -1.27- -0.50×10-5 K-1 for the v2 (out-of-plane bending) modes; -0.35-+0.22×10-5 K-1 for the v1 (symmetric stretching) modes; while -0.94- +0.09×10-5 K-1 for the v3 (asymmetric stretching) modes. Similarly, for the aragonite-group carbonates (aragonite, strontianite, cerussite and witherite), the ai parameters for the internal modes are also in the range of -1.5-1.5×10-5 K-1 (Wang et al., 2019). The ai parameters of the vibrational modes are summarized in Table 3 for the phases of calcite, magnesite, and dolomite, and the ai parameters for the external modes in significant larger magnitudes (Matas et al., 2000; Gillet et al., 1993), as compared with those for the internal modes.
In this discussion, we provide robust evaluations of the anharmonic contribution to the thermodynamic properties (internal energy, heat capacities and entropy) of the calcite, dolomite and magnesite at high temperatures, on the basis of the intrinsic anharmonic parameters ai for the vibrational modes as summarized in Table 3. In the harmonic approximation, the internal energy (U), isochoric (CV) and isobaric (CP) heat capacities, as well as entropy (S) can be simulated as a function of temperature
Modes Type Calcite Magnesite Dolomite Number of Oscillators Frequency (cm-1) ai (10-5·K-1) Frequency (cm-1) ai (10-5·K-1) Modes Type Number of Oscillators Frequency (cm-1) ai (10-5·K-1) A2u Acoustics 1 173a -15a 273a -2a Au Acoustics 1 203a -7.8a Eu Acoustics 2 98a -15a 139a -2a Eu Acoustics 2 104a -7.8a Eu Infrared 2 92c -15a 225c -2a Eu Infrared 2 150c -7.8a A2u Infrared 1 102c -15a 230c -2a Au Infrared 1 146c -7.8a Eg Raman 2 155.9b -16.9b 213b 0.3b Eg Raman 2 178b -8.4b A2g Inactive 1 190d -15a 311d -2a Ag Raman 1 235d -7.8a Eu Infrared 2 223c -15a 301c -2a Eu Infrared 2 255c -7.8a Eg Raman 2 276b -13b 329b -2.5b Eg Raman 2 300b -7.3b Eu Infrared 2 297c -15a 356c -2a Eu Infrared 2 345c -7.8a A1u Inactive 1 289d -15a 365d -2a Au Infrared 1 314c -7.8a A2u Infrared 1 303c -15a 362c -2a Au Infrared 1 361c -7.8a A2g Inactive 1 311d -15a 359d -2a Ag Raman 1 335b, c -3.7b Eg Raman 2 711b -0.3b 738b 0.44b Eg Raman 2 724b 0.2b Eu Infrared 2 711.2* 0.65* 756.2* -0.05* Eu Infrared 2 726.8* -0.69* A2u Infrared 1 873.4* -0.5* 885.7* -0.96* Au Infrared 1 879.7* -1.27* A2g Inactive 1 882d -0.5* 888d -0.96* Ag Raman 1 879b -1.27* A1g Raman 1 1 088.4b -0.1b 1 094b 0.21b Ag Raman 1 1 097b 0b A1u Inactive 1 1 088d -0.1b 1 099d 0.21b Au Infrared 1 1 096d 0b Eu Infrared 2 1 397.9* 0.09* 1 422* -0.73* Eu Infrared 2 1 410.6* -0.94* Eg Raman 2 1 432.4b -1.3b 1 444b -1.42b Eg Raman 2 1 439b -1.9b *. This study; superscript: a. Matas et al. (2000); b. Gillet et al. (1993); c. White (1974); d. Valenzano et al. (2007).
Table 3. The frequencies and intrinsic anharmonic parameters (ai) for the vibrational modes of calcite, magnesite and dolomite
where N is the total number of atoms inside a unit cell of a mineral phase (N=10 for these three carbonates), and vi is the fundamental frequency of the i-th vibrational mode. While Uvih and Cvih are the internal energy and isochoric heat capacity of the i-th vibration, in the quasi-harmonic case.
However, such harmonic approximation becomes inaccurate at elevated temperatures, as the anharmonic contribution becomes more severe (Dorogokupets and Oganov, 2004; Oganov and Dorogokupets, 2004). Therefore, at high temperatures the incorporation of an anharmonic correction is increasingly paramount. The anharmonic corrections to the internal energy and heat capacity CV are expressed below
Based on the equations above, we derived the thermodynamic properties U(T)-U(0K), CV, CP and S(T)-S(0K) for the phases of calcite, magnesite as well as dolomite. The results for the harmonic approximation are shown in Figs. 9a, 9c, 9e, 9g), while the anharmonic corrections (the differences calculated between the anharmonic and harmonic cases) are shown in Figs. 9b, 9d, 9f, 9h). The trends of the internal energies decrease in the order of calcite > dolomite > magnesite, since the frequencies of the vibrational modes systematically increase in the order of calcite < dolomite < magnesite. The internal energy is inversely correlating with the frequency vi of the i-th microscopic vibration (Eq. 5). For each carbonate phase in this study, the molar heat capacity, CV should approach the limit of 3nR at extremely high temperature (R is the molar gas constant, and n=5 for these carbonates).
Figure 9. The thermodynamic properties of internal energy U (a), (b); heat capacities CV (c), (d); and CP (e), (f) as well as entropy S (g), (h), as a function of temperature, for the phases of calcite, magnesite and dolomite. The models from harmonic approximation are shown on the left (a), (c), (e), (g), while the anharmonic corrections (differences between the anharmonic and harmonic cases) are plotted on the right (b), (d), (f), (h).
The anharmonic contributions to the thermodynamic properties are positive (Eqs. 9, 10), because nearly all the ai parameters are in negative values. At the high temperature of 800 K, the anharmonic contributions are 11.5%, 6.5% and 1.8% to the internal energy of calcite, dolomite and magnesite, respectively; 16%, 8.4% and 2.1% to the heat capacities, individually; while 7.9%, 4.2% and 1.0% to the entropies, respectively. The anharmonic parameters from the external modes (in the frequency range below 400 cm-1) play more important role than those from the internal modes (700-1 450 cm-1), and the high-pressure and high-temperature Raman studies (Matas et al., 2000; Gillet et al., 1993) indicate that the absolute values of ai parameters for the external vibrations shows the order of calcite (-17×10-5- -15×10-5 K-1) > Dolomite (-9×10-5- -7×10-5 K-1) > magnesite (-3×10-5- -2×10-5 K-1).
On the other hand, the intrinsic anharmonicity is also important and responsible for some isotope phenomena such as pressure effects on isotope equilibrium fractionations between minerals. Polyakov (1998) and Gillet et al., (1996) constrained the reduced isotopic partition function ratios (β-factor) for oxygen isotope (18O/16O) in calcite with the anharmonic correction: the anharmonic correction to the value of 103·lnβ decreases from 0.60 at 300 K to 0.37 at 1 200 K, which is comparable to the typical values of the anharmonic correction in gas molecules. This calculation method has been well established and accepted for evaluating the equilibrium isotope fractionation β-factor of various isotopes, such as O, C, Mg and Ca, etc. Our calculation above reveals that the anharmonic contribution to the thermodynamic properties for calcite is 7-8 times of that for magnesite. Hence, the anharmonic correction to 103·lnβ should be of much smaller values for magnesite at elevated temperatures.
A rigorous relationship between pressure-dependence and temperature-dependence of the β-factor is established (Polyako, 1998; Polyakov and Kharlashina, 1994) as
The Grüneisen parameter, γ, with the anharmonic correction is expressed below
where the dimensionless frequency ui=h·vi/(kB·T), γi and ai are the isothermal Grüneisen parameter and intrinsic anharmonic parameter, respectively, while * denotes for the heavier isotopes. According to Eq. 12, the external modes at lower frequencies account for larger weights in determination of the anharmonic contribution to the γ parameter (Reynard and Caracas, 2009). For the case of calcite, the anharmonic correction to the pressure-dependence of β-factor (103·∂lnβ/∂P)T is less than 0.02 GPa-1, in the temperature range up to 1 200 K (Polyakov, 1998). Hence, such anharmonic correction for magnesite would be no more than 0.003 GPa-1. Even when extrapolated to 80 GPa, which is close to the phase transition boundary of magnesite, the anharmonic correction to 103·lnβ is about 0.25, which is also smaller than that for the typical value of the anharmonic correction in gas.
Carbonate cycle has drawn lots of attention from the whole society of Earth sciences, which has important implication to the composition and evolution history of the Earth. The calcite-group carbonates, including calcite, dolomite and magnesite, are the most abundant carbon-bearing minerals in the crust, which serves as carriers of carbon from the Earth's surface to the deep interior (Dasgupta and Hirschmann, 2010). Hence, evaluations of the thermodynamic properties (such as P-V-T Equations of State, elasticities, capacities, and equilibrium isotope fractionation) are essential for constraining the physical and chemical behaviors of the carbonate in the deep Earth. However, such thermodynamic properties at the P-T conditions in the mantle are not measurable at low-pressure conditions in the laboratory, while calculations provide us with good way for such simulation. In addition, the widely used Debye model, which only considered the vibrations of atoms inside the crystal structure in harmonic approximation, should be inaccurate especially in high-T conditions, and then the anharmonic correction is necessary and important for such thermodynamic modeling (Dorogokupets and Oganov, 2004; Polyakov, 1998; Gillet, 1996). This study provides robust and detailed evaluation of the anharmonic contribution to the thermodynamics of the calcite-group minerals at high-temperature condition