Computer simulation of the MgSiO3polymorphs | SpringerLink

Phys Chem Minerals (1992) 18:365-372

PHYSlCSCHEMISTRY [ NMINIRAIS 9 Springer-Verlag 1992

Computer Simulation of the MgSiO3 Polymorphs Masanori Matsui 1 and Geoffrey D. Price 2

1 Department of Earth and Planetary Sciences, Faculty of Science, Kyushu University, Hakozaki, Fukuoka, 812 Japan 2 Department of Geological Sciences, University College London, Gower Street, London WCIE 6BT, UK ReceivedAugust 15, 1991 / Accepted September 11, 1991 Abstract. Six polymorphs of MgSiO3 have been studied

using molecular dynamic (MD) simulation techniques, based on the empirical potential (MAMOK), which is composed of terms to describe pairwise additive Coulomb, van der Waals attraction, and repulsive interactions. Crystal structures, bulk moduli, volume thermal expansivities, and enthalpies were simulated for the known MgSiO3 polymorphs; orthoenstatite, clinoenstatite, protoenstatite, garnet, ilmenite, and perovskite. The simulated values compare very well with the available experimental data, and the results are quite satisfactory in view of the diversity of the crystal structures of the six polymorphs, the wide range of simulated properties, and the simplicity of the M A M O K potential. MD simulation was further successfully used to study the possibile existence of a post-protoenstatite phase at high temperature, and a C2/c phase at high pressure, both phases being suggested or inferred previously from experimental works.

Introduction

The polymorphs of MgSiO3 have been extensively studied at high temperatures and high pressures, because of their importance in determining the chemical and physical properties of the Earth's mantle. There are six known polymorphs of MgSiO3: orthoenstatite (ortho), clinoenstatite (clino), protoenstatite (proto), garnet, ilmenite, and perovskite. Both ortho and clino occur naturally; ortho is abundant in igneous and metamorphic rocks, while clino is often found in meteorites. When heated above 1273 K at ambient pressure, ortho transforms into proto (Smyth 1974a; Murakami et al. 1982), which is reported to be stable up to the incongruent melting at 1830 K (Kushiro 1972). Proto is unstable at room temperature. On cooling single crystals of proto, they transform into a mixture of ortho and clino (Smyth 1974 a; Murakami et al. 1982), however, this transformation behaviour contrasts with that observed when using powdered crystals, where the proto can be metastably

retained even at room temperature (Lee and Heuer 1987; Schrader et al. 1990). With increasing pressure at low temperature, ortho successively transforms to clino, to/%spinel (Mg2SiO4) plus stishovite (SiO2), to y-spinel (Mg2SiO4) plus stishovite, to ilmenite, and to perovskite (Ringwood 1975; Akaogi and Akimoto 1977; Ito and Navrotsky 1985). A different high-pressure phase transformation, however, occurs at high temperatures, where ortho successively transforms into clino, garnet, ilmenite, and perovskite (Kato and Kumazawa 1985; Sawamoto 1987). The three high-pressure phases, garnet, ilmenite and perovskite, are all quenchable to ambient conditions, and their crystal structures have been reported using single-crystal Xray analysis by Angel et al. (1989); Horiuchi et al. (1982, 1987), respectively. In previous papers (Price et al. 1987; Matsui 1988), we showed that by using lattice dynamic or MD techniques, it is possible quantitatively to simulate the structural and thermodynamical properties of minerals. The reliability of the calculated quantities is determined by the selection and generation of interatomic potential functions used for simulation (Price et al. 1989). Recently we have derived an empirical potential model, MAMOK, applicable to the MgSiO3 system, and have shown that the M A M O K potential can be successfully used to reproduce a wide range of structural and physical properties of MgSiO3 perovskite (Matsui 1988). The purpose of the work presented here is to reproduce or to predict the crystal structures, bulk moduli, thermal expansivities, and enthalpies of the six MgSiOa polymorphs, by using MD simulations with the MAMOK potential. The MD method has further been used to investigate a post-proto phase, which Murakami et al. (1982) suggested the possibility of existence of at temperatures between 1673 K and the incongruent melting temperature of 1830 K. The possible existence of another high-pressure phase with the C2/c pyroxene-type structure, which was proposed by Pacalo and Gasparik (1990) by analogy with MgGeO3, has also been investigated using our MD analysis.

366

Molecular Dynamics Simulations The M D simulations were carried o u t using the combination o f the constant-temperature (Nos6 1984) and the constant-pressure (Parrinello and R a h m a n 1981) M D methods, in a similar m a n n e r to o u r other recent investigations (Matsui 1988, 1989). Periodic b o u n d a r y conditions were imposed with the M D basic cells c o m p o s e d o f 6 unit cells (a x 2b x 3e, containing 480 atoms) for ortho, 12 unit cells (2a x 2b x 3e, 480 atoms) for b o t h clino and proto, 8 unit cells (2al x 2a2 x 2e, 1280 atoms) for garnet, 12 unit cells [3a1 x 2 (a~ + 2 a 2 ) • c, 360 atoms] for ilmenite, and 27 unit ceils (3a x 3b x 3c, 540 atoms) for perovskite. The equations o f m o t i o n were solved numerically with a time increment o f 1.0 fs. Equilibrium

Table 1. Potential parameters (MAMOK) used for simulation. The pairwise interaction between atoms i and j separated by the distance rij is of the form: V(rij) = qiqjri~ 1_ CiCjri~6 + f(Bi + Bj) x exp[(Ai + Aj - rij)/(Bi + Bj)]. f=4.184 kJ A -1 tool -1. Units: q in [e[, A and B in A, and C in A 3kJ1/2 mol- 1/a

Mg Si O

q

A

B

C

+1.565 + 2.329 - 1.298

1.0133 0.8436 1.7600

0.052 0.040 0.150

0 0 54.0

Table 2. Observed and simulated structural and physical properties of the six MgSiO3 polymorphs (P=0 GPa)

structures a n d energies were calculated f r o m time-averages taken over a sufficiently long time interval (5000 steps = 5 ps, typically), while the bulk moduli and thermal expansivities were evaluated using a numerical linear interpolation. The M D technique, based on a classical mechanics description o f the system, is valid only at the high temperature limit. Q u a n t u m corrections to the M D simulated properties were m a d e based on the W i g n e r - K i r k w o o d expansion o f the free energy in terms o f Planck constant, using the technique described previously (Matsui 1989). The M A M O K potential (Matsui 1988), used for the simulations, is listed in Table 1. Starting configurations to initiallize the M D simulations were taken f r o m the observed structures reported by Sasaki et al. (1982); Ohashi and Finger (1976); Murakami et al. (1982); Angel et al. (1989); Horiuchi et al. (1982, 1987) for the ortho, clino, proto, garnet, ilmenite, and perovskite p o l y m o r p h s , respectively. For c o m p u t a tional convenience in the simulation o f garnet, we assume the M g and Si ions are ordered on the two octahedral sites in the space g r o u p I41/a, as suggested f r o m an infrared, R a m a n and N M R stydy on MgSiO3 garnet by M c M i l l a n et al. (1989); however it is to be noted that the M g and Si ions have been reported to be partially [20 (6)%] disordered over the two octahedral sites, using a single crystal X - r a y analysis (Angel et al. 1989).

Phase

ortho

clino

proto

garnet

Space Group

Pbca

P21/c

Pbcn

I41/a

9.306 9.378 8.892 8.820 5.349 5.458 90.0 90.0 33.32 33.98

11.501 11.516 11.501 11.516 11.480 11.523 90.0 90.0 28.58 28.76

itmenite R3

perovskite Pbnm

4.728 4.740 4.728 4.740 13.56 13.33 120.0 120.0 26.35 26.03

4.775 4.772 4.929 4.925 6.897 6.942 90.0 90.0 24.44 24.56

Cell lengths, cell angle, and molar volume a

a/A b c fl or 7/~ V/(cm 3 mol- 1)

Obs Calc Obs Calc Obs Calc Obs Calc Obs Calc

18.227 18.146 8.819 8.727 5.179 5.262 90.0 90.0 31.33 31.36

9.605 9.600 8.813 8.672 5.166 5.244 108.5 108.6 31.22 31.14

Bulk modulus (Ko) b, volume thermal expansivity (~) ~, and enthalpy (H)

Ko/GPa ~/(10- 5 K- 1) H/(kJ tool -1)

Obs Calc Obs Calc Cale

108 84 2.5 3.8 --7187.7

92 2.5 4.1 --7187.7

112 88 4.0 6.0 --7186.5

154 137 2.2 2.2 --7151.0

212 247 224 250 2.4 3.2 2.8 2.9 --7157.0 --7150.0

Structural parameters are at 1353 K for proto, and at 300 K for the others. Observed values are from ortho : Sasaki et al. (1982); clino : Ohashi and Finger (1976); proto : Murakami et al. (1982); garnet: Angel et al. (1989); ilmenite: Horiuchi et al. (1982); perovskite: Horiuchi et al. (1987) b Ko at 300 K for the six polymorphs. Observed values are from ortho: Weidner et al. (1978); proto: Vaughan and Bass (1983) for (Mgl.6, Lio.2, 8Co.2)8i206, garnet: Akaogi et al. (1987); ilmenite: Weidner and Ito (1985); perovskite: Yeganeh-Haeri et al. (1989) o e between 1353 and 1633 K for proto, and between 300 and 500 K for the others. Observed values are from ortho: Skinner (1966); clino: Smyth (1974) for Mg0.31Fe0.6vCao.olsSiOa; proto: Murakami et al. (1984); garnet: Skinner (1966) for pyrope; ilmenite: Ashida et al. (1988); perovskite: Knittle et al. (1986), Fei et al. (1990)

367 Results and Discussion

Physical Properties

General

The bulk moduli of the MgSiO3 polymorphs have been measured accurately using Brillouin scattering experiments and/or hydrostatic compression experiments using X-ray data for ortho (Weidner et al. 1978), [Mgx.6, Lio.2, Sco.2)Si206]-proto (Vaughan and Bass 1983), garnet (Akaogi et al. 1987), ilmenite (Weidner and Ito 1985), and perovskite (Yeganeh-Haeri et al. 1989), while no measured value is available for clino. As can be seen in Table 2, our calculated bulk moduli at 300 K and 0 GPa agree well with experiment, with the errors having 22%, 21%, 11%, 6%, 1% for ortho, proto, garnet, ilmenite, and perovskite, respectively. The agreements with experiment for ortho and proto are not so good as those for ilmenite and perovskite. This might be expected by the fact that the M A M O K potential was originally developed for ilmenite and perovskite (Matsui 1988), in which each Si ion is coordinated by six O ions at distances around 1.80 A (see Table 3), while each Si ion in both ortho and proto is surrounded by four O ions with much shorter Si-O distances of around 1.63 A (see Table 3). Analogous overestimation, about 20%, might be expected in the simulated bulk modulus of clino (listed in Table 2), since the clino phase includes similar tetrahedral silicate chains as the ortho and proto phases. The degree of agreement between the observed and calculated bulk moduli for garnet, which contains both four- and six-coordinated Si ions, is between that for pyroxenes and for perovskite (or ilmenite). The potential model would be improved further by including covalent-bond energy terms, such as O-Si-O and Si-O-Si bond angle bendings, for the phases containing four-coordinated Si ions, as attempted in our previous studies of diopside (Matsui and Busing 1984) and orthosilocate (Price et al. 1987). Volume thermal expansion has been reported using high-temperature X-ray analyses, from room to high temperatures for ortho (Skinner 1966), Mgo.31Feo.67 Cao.o~sSiO3 clino (Smyth 1974b), pyrope (Mg3A12SiaO12 garnet; Skinner 1966), ilmenite (Ashida et al. 1988), and perovskite (Knittle etal. 1986; Fei etal. 1990), and from 1353 to 1633 K for proto (Murakami et al. 1982, 1984). In Table 2 we list the simulated and measured volume thermal expansivities between 1353 and 1633 K for proto, and between 300 and 500 K for the other phases. The simulated volume thermal expansivities for garnet, ilmenite, and perovskite are in close agreement with experimental values. However, the computed thermal expansivities for ortho, clino, and proto are all too big by 50 to 60%. These discrepancies may be partly due to the neglect of bond-angle bending terms for the tetrahedral silicate chains in ortho, clino and proto, as described above. Using calorimetric methods, the transformation enthalpies of ortho to garnet (Akaogi et al. 1991), to ilmenite (Ashida et al. 1988), and to perovskite (Ito et al. 1990) have been reported to be 31 (3) kJ mol -a at 983 K, 59 (4) kJ mo1-1 at 298 K, and 110 (4) kJ mol -~ at 298 K, respectively. The transformation enthalpy of ortho to garnet at 983 K combined with the heat capacity data

It is important to note that no symmetry constraints on the six basic cell parameters (three cell lengths and three cell angles) or on the atomic positions in the basic cell are imposed in our MD simulations. Therefore, the space group symmetry and translational symmetry in the basic cell are determined solely by the nature of the potentials, and hence the simulation can be used to examine the stability of crystal lattices. No significant deviation from the observed crystal symmetry was found for any of the polymorphs studied. We note that the calculated properties of perovskite listed below are slightly different from those given in our earlier publication (Matsui 1988), because the earlier values were obtained from raw MD results without quantum correction, while the present results include a quantum correction.

Structures The simulated cell parameters and interatomic distances for all the phases exept proto, at 300 K and 0 GPa, are given in Tables 2 and 3, together with the observed values for comparison. The crystal structure of proto has only been reported at temperatures of 1353, 1533 and 1633 K, using high-temperature single-crystal X-ray analysis (Murakami et al. 1982, 1984), so for proto we list the details of the structure simulated at 1353 K in Tables 2 and 3 for comparison with experiment. The agreement between the observed and calculated structures are good for all six polymorphs. The calculated cell parameters are correct to within 1.6%, 1.6%, 2.0%, 0.4%, 1.7%, and 0.7% of those observed for ortho, clino, proto, garnet, ilmenite, and perovskite, respectively. The error in molar volume is within 1% for each polymorph, except that for proto, in which the calculated molar volume is 2% too large when compared with the observed value. The simulated molar volumes at 300 K and 0 GPa increase in the order, perovskite, ilmenite, garnet, clino, ortho, and the same is true for the observed values. The nearest-neighbour Mg-O and Si-O distances are also reproduced quite accurately, as shown in Table 3. The calculated mean Si-O distances agree with experimental data to within 0.01 A for each polymorph, except for some in garnet, in which the errors are 0.04 A for Si3-O and 0.02 A for Si4-O. The errors in the mean Mg-O distances are within 0.01 A for both ortho and clino, 0.04 A for proto, 0.03 A for garnet, 0.02 A for ilmenite, and 0.02 A for perovskite. The maximum discrepancy in the individual Si-O distances is less than 0.04 A for each polymorph, and the mean deviations in the individual Mg-O distances are 0.05, 0.04, 0.07, 0.06, 0.01, and 0.02 A for ortho, clino, proto, garnet, ilmenite, and perovskite, respectively.

368 Table 3. Comparison a between observed and simulated Si-O and Mg-O distances (A) in the six polymorphs (P = 0 GPa) Obs

Catc

Obs

Calc

Obs

Calc

SiA-01A SiA-02A SiA-03A SiA-03A (SiA-40)

1.61 1.59 1.67 1.65 1.63

1.63 1.60 1.64 1.62 1.62

SiB-03B (SiB-40)

1.68 1.64

1.64 1.63

~Mgl-60)

2.08

2.08

Mgl-01A Mgl-01A Mgl-01B Mgl-01B Mgl-02A Mgl-02B

2,15 2.03 2.17 2.07 2.01 2.05

2.19 2.04 2.19 2.08 1.97 2.03

Mg2-01A Mg2-01B Mg2-02A Mg2-02B Mg2-03A Mg2-03B (Mg2-60)

2.09 2.06 2.03 1.99 2.29 2.45 2.15

2.02 1.99 1.99 1.95 2.35 2.60 2.15

SiB-01B SiB-02B SiB-03B

1.62 1.59 1.68

1.64 1.61 1.64

SiA-01A SiA-02A SiA-03A SiA-03A ~SiA-40)

1.61 1.59 1.67 1.65 1.63

1.63 1.60 1.64 1.62 1.62

SiB-03B (SiB-40)

1.68 1.64

1.64 1.63

~Mgl-60)

2.08

2.08

Mgl-01A Mgl-01A Mgl-01B Mgl-01 B Mgl-02A Mgl-02B

2.14 2.03 2.18 2.07 2.01 2.04

2.18 2.04 2.18 2.09 1.97 2.02

Mg2-01A Mg2-01B Mg2-02A Mg2-02B Mg2-03A Mg2-03B (Mg2-60)

2.09 2.05 2.03 1.99 2.28 2.41 2.14

2.03 1.99 2.00 1.95 2.32 2.50 2.13

SiB-01B SiB-02B SiB-03B

1.62 1.59 1.68

1.64 1.60 1.65

1.61 1.60 1.65 1.65 1.63

1.63 1.61 1.62 1.63 1.62

Mgl-01 ( x 2) Mgl-01 ( • 2) Mgl-02 ( x 2) (Mgl-60)

2.08 2.26 2.03 2.12

2.09 2.40 1.99 2.16

Mg2-01 ( x 2) Mg2-02 ( x 2) Mg2-03 ( x 2) (Mg2-60)

2.08 2.05 2.44 2.19

2.00 2.00 2.56 2.19

Sil-02 ( x 2) Sil-03 ( x 2) Sil-06 ( x 2) ~Sil-60)

1.79 1.84 1.79 1.81

1.78 1.84 1.78 1.80

Si4-04 (Si4-40)

1.64 1.64

1.62 1.62

2.12 2.33 2.27 2.43 2.15 2.25 2.13 2.58 2.28

2.07 2.44 2.17 2.58 2.17 2.21 2.07 2.79 2.31

2.31 2.39 2.18 2.24 2.28

2.34 2.47 2.20 2.17 2.30

1.62 1.62 1.63

Mgl-01 Mgl-02 Mgl-03 Mgl-03 Mgl-04 Mgl-04 Mgl-05 Mgl-06 (Mgl-80)

Mg2-01 ( x 2) Mg2-02 ( x 2) Mg2-05 ( • 2) Mg2-06 ( • 2) (Mg2-80)

Si2-05 ( x 4)

1.62

1.62

Si3-06 ( x 4)

1.65

1.61

Mg3-01 ( • 2) Mg3-04 ( x 2) Mg3-05 ( x 2) (Mg3-60)

2.00 1.98 2.01 2.00

2.03 1.99 2.05 2.03

Si4-01 Si4-02 Si4-03

1.62 1.66 1.64 1.77 1.83

1.74 1.86

Mg-0 ( x 2) Mg-0 ( • 2)

1.99 2.16

1.98 2.15

(Si-60) (Mg-60)

1.80 2.08

1.80 2.06

1.80 1.80 1.78 1.79

1.81 1.78 1.79 1.79

Mg-01 Mg-01 Mg-01 Mg-01 Mg-02 ( • 2) Mg-02 ( x 2)

2.01 2.10 2.85 2.96 2.05 2.28

2.00 2.12 2.85 2.93 2.04 2.29

Mg-02 ( x 2) Mg-02 ( x 2) (Mg-60) (Mg-80) (Mg-120)

2.43 3.12 2.13 2.20 2.47

2.46 3.08 2.13 2.22 2.47

Orthoenstatite

Clinoenstatite

Protoenstatite

Si-01 Si-02 Si-03 Si-03 (Si-40) Garnet

Ilmenite

Si-0 ( x 3) Si-0 ( x 3) Perovskite

Si-01 ( x 2) Si-02 ( x 2) Si-02 ( x 2) (Si-60)

" Values are at 1353 K for proto, and 300 K for the others See foot-note a in Table 2 for the references of the observed values

o n o r t h o ( K r u p k a et al. 1985) a n d o n g a r n e t (Yusa et al. m a n u s c r i p t in p r e p a r a t i o n ) yields the t r a n s f o r m a t i o n enthalpy o f o r t h o to g a r n e t at 298 K o f 32 (3) kJ m o l - 1 . T h u s the enthalpies are m e a s u r e d to increase in the order: o r t h o < g a r n e t < i l m e n i t e < perovskite (1), at r o o m t e m p e r a t u r e a n d zero pressure.

The m e a s u r e d e n t h a l p y of f o r m a t i o n of clino at 298 K is - 3 5 . 5 kJ tool -1 (Robie et al. 1978), which is w i t h i n the reported range o f the m e a s u r e d e n t h a l p y o f f o r m a t i o n o f o r t h o at 298 K, which s p a n f r o m - 3 6 . 8 (0.7) kJ mo1-1 ( C h a r l u et al. 1975)0 - 3 6 . 0 (0.9) kJ t o o l - 1 ( C h a t i l l o n - C o l i n e t et al. 1983), to - 3 3 . 9 (1.8) kJ

369

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