Pressure-induced phase transition and structural changes under deviatoric stress of stishovite to CaCl2-like structure

Geochimicaet CosmochimicaActa, Vol. 60, No. 19, pp. 3657-3663, 19% Copyright 8 1996 Elsevier Science Ltd

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Pressure-induced phase transition and structural changes under deviatoric stress of stishovite to CaC12-like structure L. S. DUBROVINSKYand A. B. BELONOSHKO Theoretical Geochemistry Program, Institute of Earth Sciences, Uppsala University, S-752 36, Uppsala, Sweden (Received June 19, 1995; accepted in revised form June 11, 1996)

Abstract-Our recently developed interaction potential (Belonoshko and Dubrovinsky, 1995) was slightly corrected to reproduce experimental Raman frequencies at 1 bar and 300 K. Molecular and lattice dynamics study with the corrected potential shows that stishovite transforms to CaCl* structure under a hydrostatic pressure of slightly above 80 GPa and temperature 300 K. Any stress along a or b axis of stishovite leads to transformation of stishovite into CaCIZ structure. According to our calculations, under pressure above 40 GPa, a deviatoric stress of about 1.5-2.5 GPa is sufficient for the structural transformation to be observable experimentally by measuring Raman spectra. The Raman spectra simulated bv molecular and lattice dvnamics assuming small deviatoric stress are in good agreement with experi&ental observations by Kiigma et al. ( 1995’ 1. INTRODUCTION

potential models with pairwise atomic interactions reproduce Raman frequencies in close agreement with experiment (e.g., Parker and Price, 1989; Ghose et al., 1994). Recently, we developed an IP to describe interatomic interactions in silica (Belonoshko and Dubrovinsky, 1995). The functional form of this IP was suggested by Hofer and Ferreira ( 1966) and later tested for ionic-covalent compounds (Urusov, 1978; Urusov and Dubrovinsky, 1989, UNSOV et al., 1994). This P allows one to calculate structural and thermodynamic properties and phase transitions in better agreement with experiment than by using any other published IP (Belonoshko and Dubrovinsky, 1995). However, in order to achieve as close a reproduction of experimental Raman-active frequencies as possible, we corrected the IP slightly by including the upper (966.2 cm-‘, Bzs) and lower (231.6 cm-‘, B1,) experimental Raman-active frequencies at 300 K and 1 bar from Kingma et al. ( 1995) in the fitting procedure (Belonoshko and Dubrovinsky, 1995). For every given vector of the reciprocal lattice qwithin the first Brillouin zone, a set of frequencies wi is obtained by solving the corresponding determinantal equation

It is well recognized

that deviatoric stress can affect compression measurements (Meng et al., 1993; Weidner et al., 1992) and it is important to account for nonhydrostatic stress for accurate pressure calibration and determination of equation of state. However, it is not widely recognized that even a comparably small deviatoric stress is able to cause observable structural transition. The stress tensor in a diamond anvil cell (due to particulars of experimental setup) can be considered approximately as symmetric with three nonzero components, one perpendicular ( c3) to pressure transmitting diamond surfaces and two coplanar ones ( o1 ) . In this notation, pressure (P) is equal to ( cr3 + 201)/3 and the deviatoric stress is equal to o3 - ol. Hydrostatic (or rather quasihydrostatic) conditions can be created by the use of metal gasket and fluid pressure transmitting medium. Nonetheless, even at almost hydrostatic conditions, that is in the practical absence of a pressure gradient, stress might still remain (even if (03 + 2a,)/3 is constant, u3 - (T, might be quite large). Such a stress could have a profound effect on structural transformations. In this paper, we calculate the value of pressure for the phase transition of silica from stishovite to CaCIZ-type structure under hydrostatic conditions and examine the possibility of stress-induced stishovite-CaCl&e structural transition using molecular and lattice dynamics (MD and LD) with a recently developed interatomic potential (IP) for silica (Belonoshko and Dubrovinsky, 1995). The IP was slightly corrected to achieve more precise reproduction of experimentally measured Raman frequencies at ambient conditions. We found that the stishovite-CaClz-type transition occurs at 83 GPa and 300 K. According to our modeling, structural changes become observable at reasonable values of deviatoric stress (1.5-2.5 GPa) at pressures around 40 GPa. The calculated Raman spectra for such stressed conditions are in good agreement with experimental sented by Kingma et al. ( 1995 ) .

ID(q”) - I:(@(

= 0.

(1)

where D(q) is the dynamic matrix (Born and Huang, 1954) and I is the identity matrix. Raman active frequencies were calculated from Eqn. 1 at q = 0 (Parker and Price, 1989). Table 1 shows that the experimental data can be reproduced within 10 cm-‘. MD simulations with the present IP (Table 2) show that thermo-elastic properties of silica polymorphs are almost the same as described previously (Belonoshko and Dubrovinsky, 1995). This is self-evident from the Fig. 1. In addition, elastic moduli were calculated using the earlier and current versions of the potentials. Table 3 shows the magnitude of changes which modification of IP has produced. We use a combination of MD and LD methods in the same way as described by Matsui and Tsuneyuki ( 1992). We calculated energy and volumetric properties of stishovite at constant (hydrostatic) pressures up to 100 GPa using the method of Parrinello and Rahman ( 1981). Then, at volumes corresponding to those pressures, keeping the volume and size of the computational cell in c direction constant, we calculated components of pressure tensor and energies for different a/b ratios (a, b, and c-dimensions of elementary cell; a = b + c in stishovite) . The computational cell consisted of 4 x 4 x 6 elementary cells translated in a, b, and c directions, respectively. We tested the precision of simulations using a computational cell consisting of 8 x 8 x 12 elementary cells. The test showed that a computational cell consisting of 4 X 4 X 6 unit cells is sufficient for obtaining reliable results. During each simulation, the system was equilibrated for 4000 timesteps. Next 4000 timesteps were used to calculate averages. Equilibration was monitored by observing fluctuations of intermediate averages. Coulombic part of interaction energy was calculated using the Ewald

spectra pre-

2. METHOD MD and LD are well established techniques of studying condensed

matter (e.g., Parker and Price, 1988). It has been also shown that 3657

L. S. Dubrovinsky and A. B. Belonoshko

3658

Table 1. Frequencies (Vi) and isothermal pnse~re shifts (rhrJdp)r of s~mo cal~~lotad IK and Kaman active bands at 1 bar and 300 K.

bKingma et al.( 1995) ’ Williams et al.( 1993) * HofIneister et al.( 1990) technique. Another series of MD simulations was done keeping volume constant and changing c/a compared to that in stishovite. Using the MD data obtained thus, we calculated elastic moduli (Barron and Klein, 1965). We also calculated elastic moduli using the second derivatives of Hehnholtz free energy with respect to bulk strains at given P, T conditions (Barron and Klein, 1965; Wall et al., 1993). according to c,j =

1

d2F

+6P,

i=j=

1,2,3

i = 1, j = 4, 5, 6 i = 2, j = 4, 5, 6 i = 3, j = 4, 5, 6

1, if

I

i=

3. RESULTS

Generally speaking, nonhydrostatic stress invariably results in structural deformations. The transformation from stishovite to CaC& structural type is simply a change of symmetry of the SiO,-octahedra from rhombic (mm2) in stishovite to monoclinic (2/m) in CaClz. This change of symmetry of octahedra produces the change of symmetry of the structure as a whole. This transition between structural types of tetragonal stishovite (space group P4/mnm) and rhombic CaClz (Pnnm) may be continuous if the unit cell

l,j=2,3

i=2,j=3

I

-$if

i=j=4,5,6

where F is Hehnholtz free energy, ci and cj are bulk strains, and cY are elastic moduli (in Voigt notation). Helmholtz free energy at given P, T was simulated using the following equation (Born and Huang, 1954; Parker and Price, 1989):

Table 2. Parameters of intemetion potential (the analytical form and paramtem of dispersive term are as in J3elonoshlco and Dubrovinsky (1995)).

Si 0

(3)

(2)

i = 4, j = 4, 5, 6

6=<

-exp(-z))),

where +,,,, is pair potential of interatomic interaction, k is Boltzmann’s constant, R is Planck’s constant, M is total number of phonon frequencies, and w, is frequency (for the interaction potential and method of its derivation, see Urusov and Dubrovinsky, 1989; Umsov et al., 1994; Belonoshko and Dubrovinsky, 1995). We calculated frequencies on a three-dimensional mesh of sixtyfour points within the first Brillouin zone. Parker and Price ( 1989) showed that for temperatures above 50 K, thermodynamic properties converge rapidly with the size of the mesh and for a number of materials there is only small difference in magnitude of the thermodynamic properties for a mesh containing more than eight distinct points.

(- 1

v acia4

’ 0, if

P=~,,+~~~(~+ln(l

4. e

P*

2.102 -1.051

0.35333 0.16557

c 1.28107 2.14450

0,

Cd1

20.047 5.611

1.2054 1.09%

kcahde

RoA ,

1.2274 0.6798

High-pressure phase transformation of stishovite

0.5

I

1.5

2

2.5

3

FIG. 1. Interaction energy as function of distances Si-0 (circles), O-O (inverted triangles), and Si-Si (triangles). The potential in this study (filled symbols) is compared with the one developed earlier (Belonoshko and Dubrovinsky, 1995) (open symbols).

of stishovite is stretched or compressed along one of the lattice axes a or b. Figure 2 shows MD calculated stress-strain relationship at constant values of volume and c for three different volumes, which under hydrostatic conditions correspond to pressures of 15.0,42.9, and 80.0 GPa. Likely the structure of the CaClz phase depends on the state of the applied stress. Table 3 shows values of elastic constants calculated by MD and LD. The elastic constants produced by the two methods are similar and at room pressure, they are close to the experimental data of Weidner et al. ( 1982). Figure 3

3659

shows the pressure dependence of elastic moduli. The value of cll-cl2 approaches zero at about 80 GPa. According to our calculations, therefore, stishovite transforms into CaC12 structure at P slightly above 80 GPa. In the MD simulation starting from stishovite structure (a/b = 1) at 100 GPa and 300 K, we observed that the stishovite structure transforms to CaC12-like structure with a ratio a/b = 1.053. Values of the elastic constants calculated by different methods are somewhat different. The difference is not critical, however, to make a judgment on value of pressure of the phase transition according to the two methods. Our calculated pressure dependencies of Raman frequencies under hydrostatic and nonhydrostatic conditions with methods described above are shown in Fig. 4. Among the eight theoretically possible Raman and IR active frequencies of stishovite, the pressure shifts have been experimentally measured only for six of them at 300 K (Williams et al., 1993; Kingma et al., 1995, and references therein). Table 1 shows the results of LD calculations of these frequencies and their pressure shifts at 300 K and 1 bar compared with corresponding experimental values. As one can see, isothermal pressure shifts of all the frequencies including all the Raman active bands are reproduced in good agreement with the experimental data. The reproduction of the frequencies of IR active .?Z,oscillations is somewhat worse. It may be related to the observations that the IP does not describe some kinds of movements in stishovite structure with sufficient accuracy (in particular, those having E, symmetry), as well as the fact that IR powder spectra of mtile-structured compounds are notorious for discrepancies between transmission and reflectance spectra of as much as several hundred wavenumbers for some bands, depending principally on particle size and geometry (Williams et al., 1993; Luxon and Summitt, 1969; Hofmeister et al., 1990). Experimental results by Kingma et al ( 1995) can be reproduced up to a pressure of about 40 GPa. Agreement with the experiment above 40 GPa can be achieved by assuming the presence of a nonhydrostatic deviatoric stress (and corresponding struc-

Table 3. Comparison of calculated and experimental elastic wnsmnts ( in GP

L. S. Dubrovinsky and A. B. Belonoshko

3660

causes structural changes observable experimentally by measuring Raman spectra. Of course, our consideration of the magnitude of stress presented above is very approximate and

(a) 1200

1

1.02

1.04

1.06

1.08

a/b

FIG. 2. Deviatoric stress-a/b-strain relationship as calculated using MD at three pressures. The value of stress necessary to produce

certain strain decreases with pressure.

tural change, Fig. 2) of about 1.5 GPa at 43 GPa and 2.5 GPa at 60 GPa. Calculated Raman frequencies are shown in Fig. 4. 4. DISCUSSION

The experimental data of Kingma et al. (1995) can be reasonably well reproduced by assuming that the structural change they observed was caused by a deviatoric stress of about 1.5 GPa (Fig. 4). However, the question remains whether or not our estimated deviatoric stress of the order of 1 GPa is realistic? Meng et al. ( 1993) measured deviator% stress in diamond anvil cell by compressing Au up to 32 GPa with Ne as the pressure-transmitting medium, essentially the same kind of experimental setup which was used by Kingma et al. ( 1995). Meng et al. ( 1993) observed (Fig. 5) that deviatoric stress is comparably small up to approximately 20 GPa and then increases rapidly. Simple extrapolation of their values of deviatoric stress to 50 GPa gives us a value of about 1.5 GPa. Note that under nonhydrostatic conditions stress may be as high as 15 GPa at a pressure of 100 GPa (Duffy et al., 1995). The probable explanation for the increasing stress might be that Ne stiffens much faster than stishovite (pressure derivative of bulk modulus of neon is significantly larger than that of stishovite; Finger et al., 1981; Hemley et al., 1989) and conditions of quasi-hydrostatic experiment become less hydrostatic with increasing pressure. Stress and nonhydrostaticity are not related directly; however, it is likely that at some (high) pressure, compressibility of Ne becomes about the same as that of stishovite. Therefore, the value of stress might be very high and should increase with pressure. Gold and stishovite are very different substances, of course. The value of stress in the Kingma et al. ( 1995) experiment might be higher or lower than 1.5 GPa. Nevertheless, taking into account that stishovite is highly anisotropic, we would say that 1.5 GPa is likely to be the lower limit of estimated stress. We showed (see above) that such a stress

Pressure, GPa

W 300

Q 200 8 c;i G & 0’

100

J

0 0

30

60

90

Pressure, GPa FIG. 3. Pressure dependence of elastic constants of stishovite calculated by MD and LD at temperature 300 K (a). Values of c, ,-cIz (which is indicative of stishovite stability) calculated by the two methods are shown separately (b). The calculated elastic constants are in reasonable agreement with the experimental data by Weidner et al. (1982).

High-pressure phase transformation of stishovite

3661

As ----.--oa____B___

Pressure (GPa) FIG. 4. Calculated (lines) pressure dependencies of Raman frequencies of stishovite compared to experimental (spheres) data (Kingma et al., 1995). Assignment of modes is as given by Kingma et al. ( 1995). Ramanfrequencies at pressures above 40 GPa were calculated assuming deformed stishovite (CaC&) structure under stress. The value of stress varied from 1.5-2.5 GPa at pressures from 40-60 GPa (for corresponding value of strain, see Fig. 2). The appearance of higher frequency branch of B,, + AS mode was calculated assuming the same stress causing c/a deformation of stishovite structure. Pressure dependence of B,, + A, without stress is shown by dashed line.

we do not pretend to provide exact values of deviatoric stress. Another, quite plausible explanation for the appearance of deviatoric stress is that diamond surfaces start to touch the sample at some (high) pressure. If this is so, then the

observation that any medium used which is more compressible than Ne (hydrogen or helium) lowers the value of pressure at which structural changes occur (H.-K. Mao, pers. commun.) finds perfect explanation. Because the more compressible medium would lead to earlier touch of diamonds (or grains of sample start to touching each other), deviatoric stress will appear earlier and this, in turn, will lead to the distortion of stishovite structure. For example, J. Akella (pers. commun.) reported that they can not reach pressures higher tban 450 kbar with Ar as a pressurizing medium because diamonds start to touch the sample. This is quite

close to the value reported by Kingma et al. ( 1995) as the pressure of the transition to CaClAke structure. The stishovite-CaC& type transition was predicted for the first time by Hemley et al. (1985) on the basis of modified electron-gas approximation. The pressure of transition was estimated to be at least 150 GPa. Using an ab initio IP (Tsuneyuki et al., 1989) and Matsui and Tsuneyki (1992) calculated about the same transition pressure. The pressure calculated by Cohen ( 1992) using LAPW method is equal to 45 GPa and very close to the recent experimental value. The value of the transition pressure calculated by MD using PIB ++ is equal to 75 GPa (Kingma et al., 1995) which is close to our prediction. Deviatoric stress can be significantly decreased by heating up an experimental cell (Meng et al., 1993) by a few hundred degrees. It means that, according to this study, if the results

L. S. Dubrovinsky and A. B. Belonoshko

3662

REFERENCES Barron T. H. and Klein M. L. (1965) Second-order elastic constants of a solid under stress. Proc. Phys. Sot. 85, 523-532. Belonoshko A. B. and Dubrovinsky L. S. ( 1995) Molecular dynamics of stishovite melting. Geochim. Cosmochim. Acta 59, 18831889. Born M. and Huang K. ( 1954) Dynamical Theory of Crystal Luttices. Clarendon Press. Cohen R. E. ( 1991) Bonding and elasticity of stishovite SiO, at high pressure: Linearized augmented plane wave calculations. Amer. Mineral. 20

30

Pressure, GPa FIG. 5. Dependence of deviatoric stress in diamond anvil cell containing Au and Ne on pressure (symbols-measured by Meng et al. (1993), solid curve-fit to experimental data). The experimental data were fitted with the function 0.00623 X P + 0.000459 X PZ (P in GPa), which is regular polynomial of second degree (coefficient a0 set to zero for obvious reason). The expression is not intended to have any physical meaning and is used for rather short extrapolation to estimate possible value of deviatoric stress at 50 GPa.

76, 733-142.

Cohen R. E. (1992) First principles predictions of elasticity and phase transitions in high pressure SiO, and geophysical implications. In High-Pressure Research: Application to Earth and Planetary Sciences (ed. Syono Y. and Manghnani M. H.), pp. 425432. Terra Sci. Pub]. Co. Dubrovinsky L. S., Belonoshko A. B., Dubrovinsky N. A., and Saxena S. K. ( 1996) New high-pressure silica phases obtained by computer simulation. In High Pressure Science and Technology (ed. W. Trezeciakowski). World Sci. Duffy T. S., Hemley R. J., and Mao H. K. (1995) Equation of state and shear strength at multimegabar pressures: magnesium oxide to 227 GPa. Phys. Rev. Lert. 74, 1371-1374. Finger L. W., Hazen R. M., Zou G., Mao H. K., and Bell P. M. ( 1981) Structure and compression of crystalline argon and neon at high pressure and room temperature. Appl. Phys. Lett. 39,892894.

by Kingma et al. ( 1995) were affected by stress, measured pressure of structural transition is likely to increase significantly with temperature. However, the ultimate determination of value of deviatoric stress can be done only by carrying out in situ X-ray measurements. 5. CONCLUSIONS

MD and LD simulations with a slightly modified version of the IP reported earlier (Belonoshko and Dubrovinsky, 1995) suggest that stishovite transforms to CaCl,-type structure at pressures above 800 kbar at 300 K. Calculated elastic moduli and Raman frequencies are in good agreement with the experimental data at room conditions. Experimental data by Kingma et al. ( 1995) are well reproduced up to 40 GPa in the simulations assuming hydrostatic conditions. Above pressure 40 GPa, our calculated and measured (Kingma et al., 1995) Raman spectra are consistent only if there was a deviatoric stress in the experiment. Our potential model gives accurate results for other silica phase transformation such as quartz-coesite and coesite-stishovite (Dubrovinsky et al., 1996). Therefore, it might be advisable to further examine the stishovite to CaC&-type structure with accurate structural refinements under a carefully controlled stress environment. Acknowledgments-Help and suggestions of S. K. Saxena and V. Swamy are gratefully acknowledged. Comments and discussion with M. Bukowinski were very helpful and improved the paper. Authors are thankful to K. Refson for providing computer program Moldy and to J. Akella, R. Boehler, M. Catti, D. Mao, A. Navrotsky, and D. Weidner for discussion. Comments of R. Cohen and R. Hemley were helpful. Insightful reviews by G. Ottonello and an anonymous reviewer are appreciated. Calculations were done using IBM SP2 in High-Performance Computing Center in Maui (Hawaii, USA). The research was supported by Swedish National Science Council (Naturvetenskapliga ForskningrAdet) Grant No. G-Gu 06901-301 and Royal Swedish Academy of Sciences. Editorial

handling:

J. Tossell

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Kingma K. J., Cohen R. E., Hemley R. J., and Mao H. K. (1995) Transformation of stishovite to a denser phase at lower-mantle pressures. Nature 374, 243-245. Kittel C. ( 1976) Inwoduction to Solid State Physics, 4th ed. Wiley. Luxon J. T. and Summitt R. (1969) Interpretation of the infrared absorption spectra of stannic oxide and titanium dioxide (rutile) powders. J. Chem. Phys. 50, 1366- 1370. Matsui Y. and Tsuneyuki S. (1992) Molecular dynamics satudy of rutile-CaCl,-type phase transition of SiOp. In High-Pressure Research: Application io Earth and Planetary Sciences (ed. Syono Y. and Manghnani M. H.), pp. 433-439. Terra Sci. Publ. Co. Meng Y., Weidner D. J., and Fei Y. (1993) Deviatoric stress in a quasi-hydrostatic diamond anvil1 cell: effect on the volume-based pressure calibration. Geophys. Res. Lett. 20, 1147- 1150. Parker S. C. and Price G. D. (1988) Computer modelling of the structure and thermodynamic properties of silicate minerals. In Computer Modelling of Fluids, Polymers and Solids; Proc. NATO Adv. Study Inst. on Computer Modelling of Fluids, Polymers, and Solids (ed. Catlow C. R. A. et al.), pp. 405-431. Kluwer.

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