Ab Initio Study of Elastic Properties of High-Pressure Polymorphs of

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Ab Initio Study of Elastic Properties of HighPressure Polymorphs of CO Phases II and V 2

Jae-Hyeon Parq, Sung Keun Lee, Sang-Mook Lee, and Jaejun Yu J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.6b07833 • Publication Date (Web): 13 Sep 2016 Downloaded from http://pubs.acs.org on September 27, 2016

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The Journal of Physical Chemistry

Ab Initio Study of Elastic Properties of High-Pressure Polymorphs of CO2 Phases II and V

Jae-Hyeon Parq, *,†,§,ǁ Sung Keun Lee,† Sang-Mook Lee,†,§ and Jaejun Yuǁ

† School of Earth and Environmental Sciences, Seoul National University, Seoul 08826, Korea § Quality of Life Technology Industrial Infrastructure Development Center, Seoul National University, Seoul 08826, Korea ǁ Department of Physics and Astronomy, Center for Theoretical Physics, Seoul National University, Seoul 08826, Korea

AUTHOR INFORMATION Corresponding Author * Phone: +82-2-878-7855. E-mail: [email protected]

ACS Paragon Plus Environment

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ABSTRACT

Elastic properties of prototypical CO2 polymorphs under compression are essential to understanding the nature of their pressure-induced structural changes. Despite the fundamental importance in physical chemistry and condensed matter physics and geophysical implications for the nature of fluids in the Earth and planetary interiors, the elastic properties of these polymorphs are not fully understood because of intrinsic uncertainty and difficulties in experimental estimation of elasticity. Theoretical calculations of elastic properties of high-pressure CO2 polymorphs allow us to reveal the previously unknown details of elasticity of the diverse polymorphs under extreme compression. As a step toward getting insights into the deep carbon cycle, we carried out density-functional-theory calculations and investigated the elastic constants, bulk modulus, shear modulus, Poisson ratios and acoustic wave velocities of CO2 polymorphs – II (tetragonal, P42/mnm), β-cristobalite-like V (VCR, tetragonal, I-42d) and tridymite-like V (VTD, orthorhombic, P212121) up to approximately 40 GPa. Particularly, the elastic properties and bulk moduli of all the three CO2 phases except the elastic constants of CO2 – II are the first calculation results, and the elastic constants and bulk modulus calculated for CO2 – II are improved. The change in elastic properties with varying pressure show distinct trends among CO2 – II, CO2 – VCR, and CO2 – VTD. Despite these differences, the bulk moduli for CO2 of phases I, II, VCR and VTD exhibit a gradual increase with increasing density without major discontinuity. On the basis of the calculated elastic properties of CO2 – II, CO2 – VCR, and CO2 – VTD and a comparison between these CO2 units and SiO2 materials, we suggest that these polymorphs may be classified into two groups: (1) a weakly connected group: CO2 – II, cristobalite, and tridymite and (2) a strongly connected group: CO2 – VCR, CO2 – VTD, and


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stishovite. This classification does not depend on crystal symmetry. For the bulk modulus of a CO2 solid is smaller than that of a SiO2 solid of the same density and the shear modulus of a CO2 solid is greater than that of a SiO2 solid of the same density. The elasticity of CO2 polymorphs shown here may hold some promise for investigating the elasticity of diverse solids consisting of oxide molecules under extreme pressure.

1. INTRODUCTION Carbon dioxide (CO2) is among the important archetypal low-Z oxides, which have fundamental implications for both physical chemistry and earth sciences. CO2 plays an essential role in the Earth’s shallow and deep carbon cycles as well as climate changes. Upon compression just like in the Earth’s interior and other planetary interiors in general, CO2 is subject to successive phase transitions from molecular fluid (usually at 1 atm) to dense solid polymorphs (at a high-pressure environment), similar to those observed in crystalline SiO2.1 Mechanistic studies of the phase transitions among the compressed CO2 phases may also provide important insights into the sequestration of atmospheric CO2 and the design of underground CO2 storage systems, which are being implemented for the reduction of CO2 to mitigate anthropogenic global climate changes. It has been suggested that crystalline CO2 can exist at the pressure range corresponding to the Earth’s lower mantle.2-4 Figure 1 illustrates a phase diagram of these CO2 polymorphs as suggested in Ref. 1. Although the phase stabilities among the diverse CO2 polymorphs remain to be explored, CO2 polymorphs are known to be largely divided into two groups: molecular solids at a low pressure range, such as phases I (dry ice), II,5 III,6 IV,7,8 and VII,9 and extended (polymeric) solids at a higher pressure range, such as V (VTD10 and VCR11),


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VI,12,13 VIII,14 amorphous,15 coesite-like,16 and ionic solid, 17 where the phase boundaries among the molecular phases of II, III and IV are yet to be confirmed. Contrary to CO2 polymorphs, the elastic constants and shear modulus of SiO2 polymorphs have been extensively studied and have provided an important understanding of the atomistic origin of elasticity,18,19 seismic properties,18-22 and anomalies, such as a negative Poisson ratio for crystobalite18,19,23 or seismic anisotropy.21 Because CO2 is isochemical to SiO2, it is expected that CO2 has a transition path similar to that of SiO2 upon compression. Many of the SiO2-like phases include the tridymite-like phase VTD (P212121)10 and the stishovite-like phase II (P42/mnm),5 amorphous (similar to silica glass),15 the coesite-like16 and β-cristobalite-like VCR (I-42d).11 Most SiO2-like CO2 phases have the same crystal symmetry corresponding to that of their SiO2 counterparts, respectively.1,24,25 Despite the similarity in their structures, the pressure-induced changes of the chemical bonding in CO2 polymorphs are significantly different from those of SiO2.26 Therefore, through the comparison with the well-known transition mechanisms and the elastic constants for SiO2 phases, the calculated elasticity for the CO2 phase may allow us to compare the universal densification mechanism for group (IV) oxides. The structural transitions between these CO2 phases are accompanied by the changes of their thermo-mechanical and transport properties.26,27 A knowledge of elasticity, in particular, and thus the effect of pressure on the directional changes of atomic arrangements within the lattice of CO2 polymorphs, is essential in understanding their diverse mechanical properties under compression. This understanding can also yield insight into the atomistic origin of seismicity in the interiors of planets with rich CO2 content.28 Furthermore, the degree of electron delocalization in the electronic structure of each CO2 polymorph is expected to vary from one phase to another.1,5,7,8,27,29-32 Because the degree of electron delocalization affects their elastic properties


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and their pressure dependences, an understanding of the elastic properties could consequently reveal the detailed nature of these phases from molecular solids to extended (covalently bonded) CO2 solids. The pressure-induced directional variations in lattice structure, symmetry and their atomic positions affect elastic constants (i.e., elements in the stiffness or compliance tensor), which in turn lead to the changes in mechanical moduli, such as bulk and shear modulus, acoustic wave velocity, and its anisotropy. For the detailed definition of elastic constants and moduli, see Section A in the Supporting Information. Despite their importance, direct experimental measurements of the elastic properties of CO2 phases at high pressure are challenging. The measurement of elastic constants and shear modulus for CO2 solids to date has been limited to phases I and III only up to 12 GPa.33,34 Since the information on elastic constants is difficult to be extracted from the experimental XRD data alone, such information at higher pressure range is currently not available.

Indeed, the elastic constants and properties for phase I are well-

established,5,33-36 but the information for other high-pressure phases have not been fully confirmed, with the exception of the bulk modulus. For example, the bulk modulus of various CO2 phases, including I,5,33-36 II,5,30 III5,34 and V1,5,10,37, have been estimated from the equationof-state (EOS) of these phases derived from high-pressure X-ray diffraction (XRD) studies, whereas the bulk modulus and the pressure-derivative of bulk modulus have been derived from their pressure-volume relations fitted to the third-order Birch-Murnaghan (BM) EOS.5,30 Thus, it is cautioned that the rigorous experimental estimation of high-pressure bulk modulus can be somewhat challenging due partly to the solubility of CO2 in the pressure medium,1 nonhydrostatic effects,30,38 and a possible ambiguity in fitting procedure.30


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Theoretical calculations can provide a unique and complimentary opportunity in the study of the elastic and mechanical properties of CO2 polymorphs. First-principles calculations have been successfully used to predict the equilibrium structures of the CO2 phases and their relative stability at high pressure,3,4,13,31,39-42 elucidating the atomistic origin of their spectroscopic measurements.8,11,29-32,37,39,43 Furthermore, several other ab initio calculations based on density functional theory (DFT) have predicted the bulk modulus values for diverse CO2 polymorphs (phases I, II, III, IV, V, VI, and VII),8,31,32,37,39, which demonstrate the reliability and consistency of calculations in agreement with experimental bulk modulus measurements.29,32-36 In the previous ab initio DFT studies of CO2 polymorphs, however, the bulk modulus was estimated from the calculated EOS without looking into the details of the pressure-induced changes of elastic constants.8,31,32,37,39 Instead of such an indirect estimation of the bulk modulus, there is an alternative way to calculate elastic constants directly from the ab initio electronic structure calculations with a set of directional strains applied to the lattice. This direct method of calculations can provide more detailed information on elastic and mechanical properties, such as bulk modulus, shear modulus, Young’s modulus, Poisson ratio, and acoustic wave velocities. To the best of our knowledge, there is only one theoretical calculation result for the elastic constants of the CO2 phase II.30 Here, we present the results of ab initio DFT calculations with direct estimations of the elastic constants and the mechanical properties (bulk and shear moduli) of three CO2 polymorphs, phases II (P42/mnm), VCR (I-42d), and VTD (P212121). We selected these three phases because they have clearly identified geometries and counterparts in the known SiO2 phase diagram. Although it would be better to treat as many CO2 polymorphs as possible, the other CO2 polymorphs do not satisfy the following two conditions: the phases I, III, and VII do not have SiO2 counterparts, and geometric structure of the others has not been clearly identified.


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We also report the pressure-induced changes in the acoustic wave velocity of each polymorph for the first time from the calculations of the elastic constants and the shear and bulk moduli. In addition, we report the Poisson ratios as well as the wave velocity anisotropy by calculating the orientation dependence of the acoustic wave velocities for CO2 polymorphs. We finally discuss the effect of density on the elastic properties of CO2 and SiO2 polymorphs.

2. COMPUTATIONAL DETAILS We performed ab initio DFT calculations using the Vienna ab initio simulation package (VASP)44 with the projected augmented plane-wave (PAW)45,46 approach. We employed two different schemes of exchange-correlation energy functionals: the local density approximation (LDA) and the generalized gradient approximation (GGA) of Perdew-Burke-Ernzerhof (PBE).47 We present all the calculated results for comparison and crosscheck the calculated values, which vary according to the employed exchange-correlation schemes. In addition, we used the PBE with the long-range dispersion correction by Grimme (PBE-D2)48 to verify the results for possible molecular crystal, e.g., the phase II, in which the role of van der Waals interactions can be significant. The plane-wave basis set was truncated at a maximum energy of 1000 eV. The kpoints used for the Brillouin zone integration were generated using the Monkhorst-Pack method with grid sizes of 6 × 6 × 4 for phase VCR (I-42d) and 6 × 6 × 6 for phase II (P42/mnm) and phase VTD (P212121). The optimization of geometry was performed with the force criterion of 10-4 eV/Å by changing the lattice constants and relaxing atomic positions within the unit cell. Hydrostatic pressure was obtained from the diagonal components of the stress tensor with a threshold value of 5×10-5 GPa.


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We utilize the generalized Hooke’s law σi = Cijej in the Voigt notation, where σi denotes the stress tensor, Cij denotes the elastic constant tensor, and ej denotes the strain tensor. We also use the Voigt and the Reuss modulus schemes to estimate the elastic constants Cij, the bulk modulus K, and the shear modulus G (for the detailed description of the mathematical expressions, see Sections A-D in the Supporting Information49,50). Since the Voigt and Reuss schemes are known to set the upper and lower bound of bulk and shear moduli, respectively, for isotropic crystalline solids,49 we used both Voigt and Reuss schemes to estimate the bulk and shear moduli of each polymorph from the calculated elastic constant tensors. To determine the elastic constant tensor Cij for each crystal, we calculated total energies of the crystal for a set of strain vectors (e1, e2, e3, e4, e5, e6), which are chosen appropriately according to its symmetry. The stress tensor Cij of a tetragonal lattice has only six independent components, i.e., (C11, C12, C33, C44, C66), and that of an orthorhombic crystal has nine, i.e., (C11, C12, C13, C22, C23, C33, C44, C55, C66), following the crystal symmetry and the Onsager’s reciprocal relations Cij = Cji. For example, only four types of strain vectors, (e, 0, 0, 0, 0, 0), (0, 0, e, 0, 0, 0), (0, 0, 0, e, 0, 0), and (0, 0, 0, 0, 0, e), are necessary for the determination of Cij in a tetragonal lattice. When we take a sampling of strain vectors, e.g., e = 0, ±0.002, ±0.004, and ±0.008, corresponding to 0%, 0.2%, 0.4% and 0.8% of strains, respectively, total energy calculations for a tetragonal phase takes at least 4 × 7 = 28 distinct strain vectors for each crystal lattice, whereas those of an orthorhombic phase takes at least 6 × 7 = 42 distinct vectors. Therefore, we can obtain the elastic constant tensors through the linear square fit to the calculated total energies for the configurations with the strains of 0.2%, 0.4% and 0.8% to each crystal lattice constant. The elastic compliance tensors, which correspond to inverse elastic constant tensors, can then be easily obtained by computing the inverse matrices.


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All Cij of the phases were calculated from zero pressure to 44 GPa. The zero-pressure value of Cij represents the stress tensor at its local energy minimum for each phase. One can find the acoustic wave velocities from the Christoffel equation or from the Voigt-Reuss-Hill moduli as an average of the Voigt and the Reuss moduli derived from Cij (see the mathematical expression in Section E of the Supporting Information).49 The compressional P-wave velocity (Vp) and shear Swave velocity (Vs) averaged over all propagation directions is given by:

Vp =

( 3K + 4G )

3ρ

and VS = G ρ

(1)

where the bulk modulus K and shear modulus G are Voigt-Reuss-Hill averages, and ρ is the density. Finally the Poisson ratio of ν [= (3K – 2G)/(6K + 2G)] can also be derived from the Voigt-Reuss-Hill moduli. We note that the stability fields of phase II (P42/mnm), phase VCR (I-42d), and phase VTD (P212121) depend on the path taken during the synthetic process.1,29 Phase VCR can be directly synthesized from phase II, while phase VTD cannot be reached directly from phase II or phase VCR. Despite the path-dependent nature of the boundary of phase II, the lower limit of the stable pressure range of phase II can be relatively well-defined. The metastable CO2 – VCR and CO2 – VTD can be found below 10 GPa, although the phase VCR and the phase VTD are known to be stable in the ranges of approximately 30 to 40 GPa and above 40 GPa, respectively.1,10,37 These experimentally observable pressure ranges are taken into account when estimating the linear extrapolation of the bulk modulus at zero pressure K0 and its pressure derivative (dK/dP)0.

3. RESULTS AND DISCUSSION


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3.1. Equilibrium Lattice Constants and Geometries. Figure 2 shows the pressure-induced changes of the lattice constant for the three phases together with the equilibrium geometry of each phase, as obtained from the LDA calculations at each lattice constant and the corresponding hydrostatic pressure. The results are consistent with a previous LDA calculation result.11 The PBE calculations (see Figure S1 in the Supporting Information) also agree well with a previous study.29 We performed PBE-D2 calculation for CO2 – II (P42/mnm) and confirmed that our PBE-D2 lattice constants lie between LDA and PBE constants (see Figure S2 in the Supporting Information). Almost no difference between PBE and PBE-D2 lattice constants can be found in the stable pressure region of CO2 – II from 15 to 40 GPa, though some distinct difference remains significant at zero pressure. The bond lengths and angles exhibit a similar trend: a close agreement between PBE and PBE-D2 in the same stable pressure region but a sizable discrepancy at zero pressure (Tables 1, 2 & 3). Note that the lattice constants calculated by using the LDA exchange-correlation potential are systematically smaller than those calculated within the PBE or the PBE-D2. It is generally expected that the experimental values often lie between the LDA and PBE ones, which is a typical case of the DFT bulk calculations where LDA overestimates and PBE underestimates its binding energies. However, the calculated lattices constants deviate from the experimental results. We note that these are mainly due to measurement uncertainty (See below, section 3.2 for detailed discussion). The calculated lattice constants for CO2 – II (P42/mnm) and CO2 – VCR (I-42d) look satisfying this criterion if numerical and measurement errors are considered (see Tables 1 & 4 and Figure S3 in the Supporting Information).1,11,37 All lattice constants for both phases decrease as the pressure increases. CO2 – II (P42/mnm) exhibits a nonlinear feature, particularly at the lower pressure range, which is attributed to the


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molecular nature of its bonding characteristics. On the contrary, however, the impact of the D2 correction on CO2 – II, which is almost negligible in the pressure range of its stability field, indicates that CO2 – II in that pressure range from 15 to 40 GPa is not a purely molecular crystal.1,29 The calculated internal coordinates, bond lengths and angles for CO2 – II (P42/mnm) (Tables 1, 2 and 3) are generally consistent with experimental values,1,30 but it is noted that there exists a moderate degree of discrepancy among the results from earlier geometry measurements for the CO2 – II.5 The lattice constants of CO2 – VCR (I-42d) and CO2 – VTD (P212121) decrease linearly with pressure. The calculated internal geometric parameters for CO2 – VCR (I-42d) (Tables 4, 5 and 6) are in agreement with experimental values.1,37 In contrast, the topology of CO2 – VTD (P212121) is somewhat different from that of previous experiments (Table 7).1,29 The origin of the discrepancy for topology of CO2 – VTD (P212121) is not clear but seems to be related to the extra degrees of freedom in atomic positions within the unit cell. Note that our calculations were performed at 0 K although the two CO2 polymorphs – II (P42/mnm) and VCR (I-42d) – are stable in the temperature interval far above 0 K. CO2 – II (P42/mnm) and CO2 – VCR (I-42d) have higher crystal symmetry and thus even at 0 K the crystal structures may not deviate from those of high-temperature ones. In contrast, because of much lower crystal symmetry (and thus a much larger degree of geometric freedom) of CO2 – VTD (P212121), it is expected that the discrepancy between the current DFT calculation results at 0 K and experimental measurement for CO2 – VTD (P212121) at 1700–1800 K1 may prevail by huge temperature difference.

3.2. Elastic Properties.


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3.2.1. CO2 – II (P42/mnm). The calculated elastic constants for CO2 – II are shown in Figure 3a. For the pressure range of up to 40 GPa, the elastic constant C33 remains the largest and C44 is the smallest. Each component of Cij increases steadily and almost linearly with increasing pressure. PBE-D2 elastic constants were found to be located between LDA and PBE ones but closer to the PBE values (see Figure S4 in the Supporting Information). As mentioned in the Introduction section, there is only one theoretical calculation result for the elastic constants of the CO2 phase II.30 The comparison of the calculated and experimental results based on the previous study30 is presented in Figures S5 and S6 in the Supporting Information, where we present the calculated bulk modulus from the reported elastic constants of Ref. 30, and shows that our elastic constants are more consistent with their experiment than their elastic constants. In Figure 3b, the results of the bulk modulus for CO2 – II from LDA and PBE calculations are compared with the known experimental results. The calculated values increase from 82±10 GPa at 15 GPa to 179±14 GPa at 40 GPa, where the LDA and PBE values are set to the upper and the lower bounds, respectively. PBE-D2 bulk modulus values were found to be within these bounds but closer to the PBE values (see Figure S7 in the Supporting Information). The LDA result of the calculated bulk modulus at a relatively low-pressure range is in a reasonable agreement with a recent XRD study of Datchi and coworkers.30 Previous theoretical calculations31,32 derived from the P-V equation of state (the third-order BM-EOS) are also consistent with our results at the lower pressure range (0 to about 10 GPa). On the other hand, our results are somewhat different from the previous theoretical results at high pressure near 40 GPa. The bulk moduli from these earlier theoretical studies were determined by fitting the EOS below its stability field (e.g. pressure down to 10 GPa,31 or down to 0.2019 GPa32 while the experimental lowest pressure limit of CO2 – II is 15 GPa30). The linear extrapolation of the bulk modulus K0 and its


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pressure derivative (dK/dP)0 must exclude results at pressure less than 15 GPa where the nonlinear behavior in bulk modulus may appear. Otherwise, this would overestimate the moduli at higher pressure. Thus, it may not be appropriate to compare the previous theoretical results with ours. Further the pioneering experimental result by Yoo and coworkers5 matches well with our calculation results at higher pressure range near 40 GPa. We note that there exists a discrepancy between our calculations and the experimental results, which seems to reflect the intrinsic uncertainty in bulk modulus estimation via high-pressure XRD experiment. The discrepancy between those experimental results is primarily due to the intrinsic complexity in high-pressure experiments involving non-hydrostatic pressure conditions within gasket,30,38 potential interaction between CO2 fluids and pressure medium, and uncertainty in the estimation of the modulus from the fitting of the EOS.30 As listed in Table 8, our bulk modulus K0 is estimated from Voigt-Reuss-Hill values to be 31.7 GPa, 14.2 GPa and 25.9 GPa (slightly larger values than the bulk modulus calculated at zero pressure) within LDA, PBE and PBE-D2, respectively, and its pressure derivative (dK/dP)0 is 4.12, 3.89 and 3.67 within LDA, PBE and PBE-D2, respectively. The calculated shear modulus is also presented in Figure 3c. The trend in the calculated elastic properties of CO2 – II under increasing pressure is due to the pressure-induced structural changes. The relatively small mechanical moduli at low-pressure range are attributed to weak intermolecular interactions. With increasing pressure, intermolecular orbital overlap increases, leading to an increase in the bulk and the shear moduli. The LDA shear modulus increases from 24±12 GPa at 15 GPa to 49±27 GPa at 40 GPa, whereas a finite and consistent difference lies inbetween the LDA and PBE results. PBE-D2 shear modulus values were found to be close to the PBE values (see Figure S8 in the Supporting Information). Such similarity between PBE and


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PBE-D2 calculations is also found in the calculation of acoustic wave velocities and Poison ratio of CO2 – II (see Figures S9 & S10 in the Supporting Information). The D2 correction appears to have little effect on the elastic properties, like the geometry of CO2 – II, which may be another indication that supports the view that CO2 – II is not purely molecular.1,29 As for the CO2 polymorph studied here (stable > 15 GPa), the effect of dispersion term on the structure and elasticity is not significant (as shown in Tables 2 & 3 and Figures 2, 3, S1-S8). However, the effect could be prominent for the other lower pressure phases (e.g., phases I and VII), and thus, the effect of the dispersion correction for those phases needs to be performed. 3.2.2 CO2 – VCR (I-42d). Figure 4 shows the elastic constants, bulk modulus, and shear modulus of CO2 – VCR. With increasing pressure, the elastic constants increase linearly in the pressure range from 0 to 39 GPa. Whereas the pressure gradient of Cij in CO2 – II (P42/mnm) seems to be correlated with the magnitude of Cij at 1 atm, that is, the larger Cij at 1 atm, the larger the pressure gradient of Cij, the pressure gradient of Cij in CO2 – VCR does not exhibit such a trend. Another difference is found in elastic constants at zero pressure: most elastic constants of CO2 – VCR except C12 are larger than 100 GPa at zero pressure. In contrast, C12 at 1 atm has a negative value, indicating that the transition from the tetragonal to orthorhombic structure can be prohibited by contraction in the [010] direction when uniaxially compressed along the [100] direction. Bulk and shear moduli for CO2 – VCR (Figures 4 b and c) within its stability field, which is known to lie from approximately 30 to 40 GPa, increase from 264±24 (at 30 GPa) to 297±28 GPa (at 39 GPa) and from 163±29 (at 30 GPa) to 173±35 GPa (at 39 GPa), respectively. The Voigt bulk and shear moduli tend to have larger values at zero pressure and exhibit greater pressure dependency than those of CO2 – II (P42/mnm). Figure 4b also shows the bulk modulus


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from previous theoretical calculations32,39 and experimental measurements.37 Taking into consideration the experimental or theoretical uncertainty, these results are generally consistent with the current results. Large discrepancy look distinct for the measurement of one group which may have suffered from non-hydrostatic effects30,38 and ambiguity of EOS fitting.30 Previous calculations could also be affected by inaccuracy of EOS fitting (i.e., sensitive to numerical errors). We estimated the K0 and (dK/dP)0 of CO2 – VCR with varying pressures at 10, 30, 39 and 41 GPa. From Voigt-Reuss-Hill bulk modulus values, the estimated K0 within LDA and PBE is 158.8 GPa and 143.3 GPa, respectively, and (dK/dP)0 within LDA and PBE is 3.73 and 3.59, respectively (Table 8). The elastic properties of CO2 – VCR reflect its polymeric structure, where all bonds have a covalent character, which leads to large mechanical moduli and elastic constants. 3.2.3 CO2 – VTD (P212121). Figure 5 shows the elastic constants and the mechanical moduli of CO2 – VTD. The values are significantly different from those of CO2 – II (P42/mnm) but show similarity to those for CO2 – VCR (I-42d). It appears that the elastic constants increase linearly with pressure from 0 to 44 GPa, and, as in CO2 – VCR, the pressure gradient of Cij in CO2 – VTD is much larger than that of CO2 – II. The similarity of trends in elastic properties between CO2 – VTD and CO2 – VCR can be attributed to their polymeric character. Although the C11, C22 and C33 values are much larger than the other values, as shown in Figure 5; the C55 value decreases and becomes negative beyond approximately 24 GPa for LDA and 26 GPa for PBE. Our bulk modulus within LDA coincides with a previous DFT calculation within the LDA39 and is close to the experimental value (which is suspected of having non-hydrostatic effects30,38 and ambiguity of fitting30) above 40 GPa.1 From Voigt-Reuss-Hill bulk modulus values, the K0 and (dK/dP)0 of CO2 – VTD can be calculated from the four pressure points of 10,


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20, 30, and 44 GPa. The estimated K0 within LDA and PBE is 152.7 GPa and 138.7 GPa, respectively, and (dK/dP)0 within LDA and PBE are 3.94 and 3.72, respectively (Table 8). The negative value of C55 above 25 GPa for LDA is not compatible with the Born stability condition, i.e.,

C11+C22-2C12 > 0,

C22+C33-2C23

> 0,

Cii > 0

for all

i,

and

C11+C22+C33+2C12+2C13+2C23 > 0 for orthorhombic lattice.51 We suggest that this negative C55 value must be an artifact since DFT calculations for tridymite-like structure28,39,52,53 were inaccurate or the entropic contribution may be crucial for the stabilization of VTD phase above 1300 K. A future DFT molecular dynamics study at finite temperature may provide better constraints on the pressure dependence of C55. The negative C55 leads to the negative Reuss shear modulus (Figure 5c). However, according to our calculation, the Voigt moduli of CO2 – VTD (Figure 5 b & c) increase with increasing pressure with positive pressure dependence. Hence, in this study, the Voigt moduli for CO2 – VTD were employed instead of the Voigt-Reuss-Hill moduli and were used to calculate the acoustic wave velocity.

3.3. Comparison among CO2 – II (P42/mnm), CO2 – VCR (I-42d) and CO2 – VTD (P212121). Distinctive trends in the elastic constants and their pressure dependence and pressure-induced changes in mechanical moduli for CO2 – II, CO2 – VCR, or CO2 – VTD are presented in the current study. These trends confirm that the nature of bonding and connectivity among CO2 units are clearly manifested in the calculated elastic properties. The elastic properties of solids depend on their crystal structure and lattice symmetry. Based on the distances among CO2 clusters, CO2 – II can be classified as a molecular solid, which differs from the CO2 – VCR or CO2 – VTD


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phases of dense and covalently bonded polymeric crystal.1,30,32 The pressure-induced changes in the seismic wave velocity may also elucidate the nature of bonding in the CO2 phases. Figure 6 shows the acoustic wave velocities for these three phases with varying pressure. In particular, the figure exhibits a distinct trend among CO2 – I (Pa-3) (calculated from the available experimental bulk and shear modulus34), CO2 – II, CO2 – V (VCR and VTD). The average P-wave velocities for CO2 – VCR (from 11.3 km/s at 30 GPa to 11.6 km/s at 39 GPa) or for CO2 – VTD (from 11.5 km/s at 40 GPa to 11.8 km/s at 44 GPa) look much larger than that for CO2 – II (from 6.5 km/s at 15 GPa to 8.7 km/s at 39 GPa). Large difference is also found in comparison of the average S-wave velocities for CO2 – VCR (6.5 km/s at 30 GPa to 6.6 km/s at 39 GPa) or for CO2 – VTD (from 6.4 km/s at 40 GPa to 6.5 km/s at 44 GPa) with that for CO2 – II (from 2.9 km/s at 15 GPa to 3.8 km/s at 39 GPa). Although the pressure-induced increases in seismic wave velocities for CO2 – VTD and CO2 – VCR are not significant, the pressure-induced increases in the average P-wave velocity for CO2 – II are more drastic: the average slopes of the P-wave velocity (pressure-gradient of seismic wave velocity) curves are 0.092 km/s·GPa for CO2 – II, 0.033 km/s·GPa for CO2 – VCR, and 0.075 km/s/GPa for CO2 – VTD. The average slopes of the S-wave velocity for CO2 – II are also larger than those for CO2 – VCR (0.011 km/s·GPa), and CO2 – VTD (0.025 km/s·GPa). These changes reflect pressure-induced changes in density and elastic moduli (see the results below). Figure 7 shows the Poisson ratios of CO2 – I (Pa-3), CO2 – II, CO2 – VCR and CO2 – VTD within the experimentally observable pressure range of each polymorph. The calculated Poisson ratio of CO2 – II ranges from 0.37 at 15 GPa and 0.38 at 39 GPa on average. While the experimental Poisson ratio of CO2 – I slightly increases from 0.34 at 1.5 GPa to 0.37 at 9.8 GPa and approaches that of CO2 – II, which is almost constant in the pressure range of 15–39 GPa.


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Poisson ratios of CO2 – VCR and CO2 – VTD are quite smaller and have more variation than that of CO2 – II. With increasing pressure, the average values of the calculated Poisson ratio of CO2 – VCR increases from 0.20 at 10 GPa to 0.26 at 39 GPa and that of CO2 – VTD increases from 0.22 at 20 GPa to 0.28 at 44 GPa. The bulk modulus may provide useful insights into the strength of the intermolecular interaction.5,30,32 The calculated bulk modulus values shown in Figures 3b, 4b, and 5b and in Table 8 tend to support the molecular nature of CO2 – II; the K0 of CO2 – II is closer to that of CO2 – I (3 – 10 GPa5,33-36) than that of CO2 – VCR or VTD (140 – 158 GPa), and the bulk modulus in the stable pressure range of CO2 – II is closer to that of CO2 – I (3 – 62 GPa34) than those of CO2 – VCR and VTD (240 – 326 GPa). X-ray Raman spectroscopy (oxygen K-edge feature) may provide additional insights into the distinction.57-59 In case of O2, for example, a previous O Kedge XRS study of compressed O2 has demonstrated that electron delocalization increases with increasing pressure.60 Figure 8a illustrates the relationship between the bulk moduli and the densities of the CO2 polymorphs. The bulk moduli for CO2 – I, CO2 – II, CO2 – VCR and CO2 – VTD increase gradually with increasing density without major discontinuity. Bulk moduli of CO2 solids may be expressed by one function depending only on density, but this needs to be confirmed by experiments. Figure 8b shows the effect of density on the shear modulus of the phases studied here. Again, the shear modulus generally increases with the density of the phases. However, the figure also shows a distinctive discontinuity upon the phase transition from CO2 – II to CO2 – V. The trend in shear modulus can in turn be used to classify the phase in the distinctive mechanical properties (i.e., molecular solids vs. extended solids).


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Figure 9 shows the effect of density on the P- and S-wave velocity of the CO2 phases (and the comparison between SiO2 and CO2 phases is described below). The P- and S-wave velocities increase with increasing density. Pronounced S-wave velocity discontinuity is also observed upon the phase transition from CO2 – II to CO2 – V, as expected from the large difference in their shear modulus values. Because the P-wave velocity is largely dependent on bulk modulus that does not change significantly upon phase transition among CO2 phases, the large discontinuity in the P-wave velocity has not been observed upon phase transition (see section 3.4 for further discussion). Figures 10-11 show the orientation dependence of the seismic wave velocities for CO2 - II and CO2 - VCR; that for CO2 - VTD has not been calculated due to the negative value of C55. Both results within LDA and PBE show similar trends: the seismic velocity anisotropy (difference in seismic wave velocity) in CO2 - II is most prominent in the [110] direction, whereas the anisotropy is small in the [100] and [010] directions. The seismic wave velocity anisotropy for CO2-VCR is strongest in both the [011] and [101] directions, whereas the anisotropy is smallest in the [110] direction. The results show that the seismic anisotropy for CO2 - II and CO2 - VCR show a distinctive trend. The current results confirm that the distinct changes in seismic wave velocity reflect the difference in the atomic structure and the connectivity of the CO2 units in the phases.

3.4. Comparison to SiO2. The crystal structure of CO2 – II is the same as (or isosymmetric to) that of stishovite (P42/mnm). Other than the crystal symmetry, however, the characteristics of CO2 – II and stishovite are very much different: stishovite is much denser than CO2 – II. Stishovite also has a c/a ratio far from one, which is related to its Si-O-Si angles of approximately 80°,61 whereas the C-O-C angles of CO2 – II range from 110° to 120°. This leads


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to noticeable differences in the bulk and shear moduli and their pressure dependences (see Figure 8). CO2 – VCR was given its name from its structural similarity to β-cristobalite. Both CO2 – VCR and β-cristobalite have the same I-42d symmetry,24,25 but CO2 – VCR is denser and harder than βcristobalite. Consequently, CO2 – VCR is less compressible than cristobalite (Figure 8a). Furthermore, cristobalite has a negative Poisson ratio related to its auxetic behavior (expanding when compressed)18,23 which CO2 – VCR does not have. CO2 – VTD has an identical crystal structure to that of SiO2 tryidymite. It may not be practical to compare the elastic properties of CO2 – VTD and those of tridymite, partly because of the inaccuracy of calculations for tridymitelike materials and the lack of experimental data for tridymite. Nevertheless, taking into account the differences in their density and stability fields, one can expect that the mechanical moduli of tridymite are much smaller than those of CO2 – VTD. Although CO2 – VTD (P212121) has a structure distorted from the OC phase (C2221) tridymite, CO2 – VTD’s elastic properties are not affected by the distortion.39 Although the elastic moduli for the CO2 and SiO2 phases (up to stishovite) generally increase with density, the bulk modulus of a CO2 solid is smaller than that of a SiO2 solid of the same density (particularly for coesite) and the shear modulus of a CO2 solid is greater than that of a SiO2 solid of the same density (Figure 8). All of these comparisons reveal tendencies of the elastic properties of CO2 and SiO2 solids. On the basis of the density and elastic properties of CO2 – II, CO2 – VCR and CO2 – VTD and comparison to the properties of SiO2 materials, these polymorphs may be classified into two groups: (1) a weakly connected group: CO2 – II, cristobalite, tridymite and (2) a strongly connected group: CO2 – VCR, CO2 – VTD and stishovite. Note that this classification does not depend on crystal symmetry. The shear modulus in Figure


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8b substantiates this classification because a similar pattern is found in the acoustic wave velocities with respect to density (Figure 9). The shear modulus values of CO2 phases and SiO2 phases at identical density are largely different while bulk modulus values of those phases are rather similar. Therefore, the calculated P-wave velocities (strongly dependent on bulk modulus) of CO2 and SiO2 phases at the corresponding density are rather similar, whereas the S-wave velocities (dependent on shear modulus) for these phases are distinct. The results imply that the bulk modulus and P-wave velocity is less sensitive to variation of chemical composition if the density of the phases are identical. In Figures 8a & 9a, the P-wave velocity (or bulk modulus) differences of the same density between CO2 and SiO2 in the strongly connected group, which can be estimated by extrapolation, look a little bit larger than those in the weakly connected group. This might be related to the theory that the transition mechanism of CO2 (from the weakly connected group to the strongly connected group or vice versa) is different from that of SiO2.26 Whereas we provide novel relationship between seismic wave velocity and density in the current study, the chemical origin of the observed trends needs further discussion.

4. CONCLUSIONS We report herein the calculated elastic properties of CO2 – II, CO2 – VCR and CO2 – VTD with varying pressure up to approximately 40 GPa using ab initio DFT calculations within the LDA and the PBE exchange-correlation functionals. To our knowledge, the elastic properties except elastic constants of CO2 – II and bulk moduli of all the three CO2 phases are the first calculation results, and elastic constants and bulk modulus of CO2 – II is improved. The calculation of elastic properties of high pressure CO2 polymorphs is important to experimentalists because their


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measurement is difficult and has been inaccurately or hardly performed. The equilibrium lattice constants and geometries are consistent with the experimental results except for those of CO2 – VTD. In CO2 – II, the calculated elastic constants increase systematically with increasing pressure. The estimated bulk moduli of these phases gradually increase with increasing pressure and are largely consistent with earlier experimental as well as with previous theoretical calculations. The calculated elastic constants and bulk modulus of CO2 – VTD differ significantly from those of CO2 – II but are similar to those of CO2 – VCR. The estimated shear moduli of these phases also increase with increasing pressure. The distinctive pressure-dependence of the elastic constants, mechanical moduli, acoustic wave velocities, and Poisson ratios among CO2 – II and CO2 – VCR or CO2 – VTD reflects the differences in the bonding nature in each of the polymorphs [i.e., molecular solids (CO2 – II) vs. polymeric crystals (CO2 – VCR or CO2 – VTD )] suggested by previous studies. Despite these phase differences, the bulk moduli for CO2 of phases I, II, VCR and VTD increase gradually with increasing density without major discontinuity, which looks like suggesting universal CO2 bulk modulus formula depending only on density. Although the elastic moduli for the CO2 and SiO2 phases (up to stishovite) generally increase with density, the bulk modulus of a CO2 solid is smaller than that of a SiO2 solid of the same density and the shear modulus of a CO2 solid is greater than that of a SiO2 solid of the same density. On the basis of the calculated elastic properties of CO2 – II, CO2 – VCR and CO2 – VTD and their comparison with SiO2 materials, these polymorphs may be classified into two groups: (1) a weakly connected group: CO2 – II, cristobalite, and tridymite and (2) a strongly connected group: CO2 – VCR, CO2 – VTD and stishovite. Note that this classification does not depend on crystal symmetry. Our report of elasticity of CO2 phases may hold some promise for studying the elasticity of diverse solids consisting of oxide molecules under extreme compression.


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Supporting Information Available: Mathematical expressions (the generalized Hooke’s law, the

Voigt notation, the Voigt and the Reuss modulus schemes, relationship between a strain tensor and the lattice constants, and acoustic wave velocity formula) and Figures S1-S12 (comparison of the calculated elastic constants between the current study and those of Ref. 30, lattice constants and elastic constants calculated within the PBE, and lattice constants and elastic properties calculated within the PBE-D2). ACKNOWLEDGMENT This work was supported by the Technology Innovation Program (10036459, Development of center to support QoLT industry and infrastructures) funded by the MOTIE/KEIT, the Korea Institute of Energy Technology Evaluation and Planning (KETEP) and the Ministry of Trade, Industry & Energy (MOTIE) of the Republic of Korea (No. 20168510030830), and the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST. No. 20090092790) to SML and (2014-053-046) to SKL. JY acknowledges a support by the NRF funded by the Ministry of Science, ICT & Future Planning (no. 2013R1A2A2A01067950). We thank Prof. C. S. Yoo for helpful discussion. We deeply appreciate constructive and careful comments from the 3 anonymous reviewers, which greatly improve the quality and clarity of the manuscript


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Implication for triclustered MgSiO3 melt in Earth's mantle, Proc. Natl. Acad. Sci. 2008, 105, 7925–7929. (59) Lee, S. K.; Eng, P.; Mao, H. K. Probing of pressure-induced bonding transitions in crystalline and amorphous earth materials: Insights from x-ray Raman scattering at high pressure. Rev. Mineral. Geochem. 2014, 78, 139–174. (60) Meng, Y.; Eng, P. J.; Tse, J. S.; Shaw, D. M.; Hu, M. Y.; Shu, J.; Gramsch, S. A.; Kao, C.; Hemley, R. J.; Mao, H. Inelastic x-ray scattering of dense solid oxygen: Evidence for intermolecular bonding. Proc. Natl. Acad. Sci. 2008, 105, 11640–11644. (61) Ross, N. L.; Shu, J.-F.; Hazen R. M. High-pressure crystal chemistry of stishovite. Am. Mineral. 1990, 75, 739–747.


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Table 1. Lattice Constants, Density and Internal Coordinates of the Atomic Positions in CO2 – II (P42/mnm) at 25.8 GPa LDA

PBE

PBE-D2

exp.30

a (Å)

3.483

3.565

3.554

3.516

b (Å)

3.483

3.565

3.554

3.516

c (Å)

3.939

4.093

4.083

4.104

ρ (g/cm3)

3.059

2.810

2.835

2.881

C

(0 0 0)

(0 0 0)

(0 0 0)

(0 0 0)

O

(0.766 0.766 0)

(0.770 0.770 0)

(0.769 0.769 0)

(0.770 0.770 0)

Table 2. C-O Bond Lengths of CO2 – II (P42/mnm) pressure

C-On. (Å)

(GPa)

LDA

PBE

PBED2

0

1.161

1.170

10

1.158

15

C-On.n. (Å) LDA

PBE

PBED2

1.169

2.796

3.326

3.016

1.165

1.164

2.522

2.668

2.651

1.156

1.163

1.162

2.460

2.581

2.568

25.8

1.153

1.160

1.159

1.14 30

2.365

2.458

2.450

2.45 30

28

1.153

1.159

1.158

1.366 5

2.350

2.439

2.431

2.360 5

39

1.151

1.156

1.156

1.187 1

2.285

2.360

2.354

2.368 1

44

1.150

1.155

1.154

2.260

2.331

2.325

exp.

exp.

On. = nearest oxygen On.n. = next nearest oxygen


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Table 3. On.n.-C-On.n. Bond Angles of CO2 – II (P42/mnm) pressure

0 GPa

10 GPa

15 GPa

25.8 GPa

28 GPa

39 GPa

44 GPa

LDA

103.1°

110.3°

111.4°

112.8°

112.9°

113.6°

113.8°

PBE

102.4°

109.9°

111.2°

112.7°

112.9°

113.7°

114.0°

PBE-D2

104.6°

111.4°

112.0°

112.9°

113.0°

113.6°

113.9°

113.6° 30

122.6° 5

117.1° 1

exp. On.n. = next nearest oxygen

Table 4. Lattice Constants, Density and Internal Coordinates of the Atomic Positions in CO2 – VCR (I-42d) at 39 GPa LDA

PBE

exp.1

a (Å)

3.533

3.611

3.594

b (Å)

3.533

3.611

3.594

c (Å)

5.903

5.928

5.917

ρ (g/cm3)

3.967

3.782

3.824

C

(0 0 0)

(0 0 0)

(0 0 0)

O

(0.202 0.250 0.125)

(0.198 0.250 0.125)

(0.213 0.250 0.125)


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Page 32 of 49

Table 5. C-O Bond Lengths of CO2 – VCR (I-42d) pressure

0 GPa

10 GPa

30 GPa

39 GPa

41 GPa

LDA

1.383 Å

1.373 Å

1.359 Å

1.354 Å

1.353 Å

PBE

1.403 Å

1.392 Å

1.375 Å

1.369 Å

1.368 Å

1.393 Å 1

1.38 Å 37

exp.

Table 6. O-C-O Bond Angles of CO2 – VCR (I-42d) Solids pressure

O-C-O

C-O-C

(GPa)

LDA

PBE

0

106.5°

106.5°

115.5°

115.6°

106.6°

106.6°

115.3°

115.5°

107.0°

106.8°

114.5°

114.9°

107.3°

107.0°

106.4° 1

114.0°

114.5°

115.9° 1

107.3°

107.1°

106.6° 37

113.8°

114.4°

115.4° 37

10

30

39

41

exp.

LDA

PBE

123.4°

124.6°

120.9°

121.9°

117.5°

118.3°

116.4°

117.1°

113.3° 1

116.2°

116.9°

113.2° 37


exp.

32

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Table 7. Lattice Constants, Density and Internal Coordinates of the Atomic Positions in CO2 – VTD (P212121) at 44 GPa LDA

PBE

exp.1

a (Å)

6.609

6.698

6.266

b (Å)

3.648

3.706

4.386

c (Å)

6.224

6.314

6.087

ρ (g/cm3)

3.896

3.730

3.495

C1

(0.417 0.493 0.179)

(0.417 0.493 0.180)

(0.444 0.704 0.164)

C2

(0.089 0.408 0.304)

(0.089 0.411 0.304)

(0.062 0.578 0.311)

O1

(0.596 0.215 0.491)

(0.596 0.211 0.492)

(0.491 0.127 0.525)

O2

(0.254 0.279 0.203)

(0.254 0.283 0.204)

(0.262 0.527 0.258)

O3

(0.430 0.742 0.346)

(0.431 0.743 0.345)

(0.429 0.963 0.232)

O4

(0.923 0.269 0.207)

(0.924 0.271 0.208)

(0.948 0.432 0.163)


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Table 8. The Bulk Modulus Values at Zero Pressure K0 and Pressure Derivatives (dK/dP)0 for CO2 – II (P42/mnm), CO2 – VCR (I-42d), and CO2 – VTD (P212121) phase

K0 (GPa)

(dK/dP)0

method

reference

CO2 – II

131

2.1

exp. (XRD)

ref. 5

16.5

5.24

exp. (XRD)

ref. 30

7.46

6.29

calc. (DFT-D3)

ref. 32

4.37

6.66

calc. (DFT-PBE)

ref. 31

31.7

4.12

calc. (DFT-LDA)

this study

14.2

3.89

calc. (DFT-PBE)

this study

25.9

3.67

calc. (DFT-D2)

this study

136

3.7

exp. (XRD)

ref. 37

126.4

6.6

exp. (XRD)

ref. 1

149.1

3.6

calc. (DFT-LDA)

ref. 39

144

3.95

calc. (DFT-LDA)

ref. 37

135

3.82

calc. (DFT-PBE)

ref. 37

136.1

4.0

calc. (DFT-D3)

ref. 32

158.8

3.73

calc. (DFT-LDA)

this study

143.3

3.59

calc. (DFT-PBE)

this study

365

0.8

exp. (XRD)

ref. 5

326.7

1.0

exp. (XRD)

ref. 10

267.5

1.5

exp. (XRD)

ref. 1

151.5

3.9

calc. (DFT-LDA)

ref. 39

CO2 – VCR

CO2 – VTD


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152.7

3.94

calc. (DFT-LDA)

this study

138.7

3.72

calc. (DFT-PBE)

this study

Figure Captions: Figure 1. Phase diagram of CO2 polymorphs based on Ref. 1. CO2 – V is divided into two polymorphs: β-cristobalite-like V (VCR) and tridymite-like V (VTD). The letters a, c, and i indicate amorphous CO2, coesite-like CO2, and extended ionic CO2, respectively. Dashedlines are approximated lines due to the discrepancy among the experiments or the pathdependence of the phase boundaries. Figure 2. (Top) Pressure dependence of the lattice constants of CO2 – II (P42/mnm), CO2 – VCR (I-42d) and CO2 – VTD (P212121) calculated within the LDA. A previous calculation for CO2 – II within the LDA (Santoro: Ref. 11) is also shown. (Bottom) Equilibrium crystal structures of the phases as labeled. CO2 – II at 28 GPa, and CO2 – VCR and VTD at 30 GPa. Figure 3. Calculated pressure dependence of (a) elastic constants, (b) bulk modulus and (c) shear modulus of CO2 – II (P42/mnm). Only the result within the LDA is shown for elastic constants, but elastic constants calculated within the PBE and PBE-D2 (Figure S4 in the Supporting Information) have similar behavior. The bulk modulus values of other studies are also shown (Yoo: experimental values from Ref. 5; Datchi: experimental values from Ref. 30; Bonev: theoretical values from Ref. 31; Gohr: theoretical values from Ref. 32). Figure 4. Calculated pressure dependence of (a) elastic constants, (b) bulk modulus and (c) shear modulus of CO2 – VCR (I-42d). Only the result within the LDA is shown for elastic constants,


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Page 36 of 49

but elastic constants calculated within the PBE (Figure S11 in the Supporting Information) have similar behavior. The bulk modulus values of other studies are also shown (Datchi: experimental values from Ref. 37; Yoo: experimental values from Ref. 1; Dong: theoretical values from Ref. 39; and Gohr: theoretical values from Ref. 32). Figure 5. Calculated pressure dependence of (a) elastic constants, (b) bulk modulus and (c) shear modulus of CO2 – VTD (P212121). Only the result within the LDA is shown for elastic constants, but elastic constants calculated within the PBE (Figure S12 in the Supporting Information) have similar behavior. The bulk modulus values of other studies are also shown (Yoo: experimental values from Ref. 1; Dong: theoretical values from Ref. 39). Figure 6. Longitudinal wave (P wave) and shear wave (S wave) velocities of CO2 – I (Pa-3), CO2 – II (P42/mnm), CO2 – VCR (I-42d) and CO2 – VTD (P212121) in the definite stable pressure range of each phase. The values of CO2 – II, CO2 – VCR and CO2 – VTD are based on our calculations (the Voigt-Reuss-Hill averages except for VTD, which takes the Voigt scheme). The experimental values of CO2 – I were estimated from Ref. 34. Figure 7. Pressure variation of the Poisson ratio of CO2 – I (Pa-3), CO2 – II (P42/mnm), CO2 – VCR (I-42d) and CO2 – VTD (P212121). The values of CO2 – II, CO2 – VCR and CO2 – VTD are based on our calculations (the Voigt-Reuss-Hill averages except for VTD, which takes the Voigt scheme). The experimental values of CO2 – I was estimated from Ref. 34. Figure 8. (a) Bulk modulus and (b) shear modulus of CO2 – I (Pa-3), CO2 – II (P42/mnm), CO2 – VCR (I-42d) and CO2 – VTD (P212121) with respect to density (CO2 – I: Ref. 34). The mechanical moduli of related SiO2 polymorphs from literature are also shown (stishovite from Ref. 54, cristobalite from Refs. 18 and 23, tridymite from Ref. 55, and coesite from Ref.


36

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56). The values of CO2 – II, CO2 – VCR and CO2 – VTD are from our calculations (the VoigtReuss-Hill averages except for VTD, which takes the Voigt scheme). Figure 9. Calculated acoustic (a) longitudinal P wave and (b) transverse S wave velocities of CO2 – II (P42/mnm), CO2 – VCR (I-42d) and CO2 – VTD (P212121) with respect to density (estimated from the Voigt-Reuss-Hill averages except for VTD, which takes the Voigt scheme). The seismic wave velocities of related SiO2 polymorphs from literature are also shown (stishovite from Ref. 54, cristobalite from Refs. 18 and 23, tridymite from Ref. 55, and coesite from Ref. 56). Figure 10. Orientation dependence of the acoustic wave velocities of CO2 – II (P42/mnm) calculated (a) within the LDA and (b) within the PBE. Three curves are calculated velocities of one fast longitudinal (P) and two slow transverse (S) waves. Figure 11. Orientation dependence of the acoustic wave velocities of CO2 – VCR (I-42d) calculated (a) within the LDA and (b) within the PBE. Three curves are calculated velocities of one fast longitudinal (P) and two slow transverse (S) waves.


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Figure 1.


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Figure 2.


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Figure 3.


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Figure 4.


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Figure 5.


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Figure 6.


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Figure 7.


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Figure 8.


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Figure 9.


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Figure 10.


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Figure 11.


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TOC graphic


49

Ab Initio Study of Elastic Properties of High-Pressure Polymorphs of

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