Volume 123, Issue 9 p. 7347-7360
Research Article
Free Access

Equation of State of Polycrystalline Stishovite Across the Tetragonal-Orthorhombic Phase Transition

Johannes Buchen

Corresponding Author

Johannes Buchen

Bayerisches Geoinstitut, Universität Bayreuth, Bayreuth, Germany

Correspondence to: J. Buchen,

[email protected]

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Hauke Marquardt

Hauke Marquardt

Bayerisches Geoinstitut, Universität Bayreuth, Bayreuth, Germany

Department of Earth Sciences, University of Oxford, Oxford, UK

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Kirsten Schulze

Kirsten Schulze

Bayerisches Geoinstitut, Universität Bayreuth, Bayreuth, Germany

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Sergio Speziale

Sergio Speziale

Deutsches GeoForschungsZentrum, Potsdam, Germany

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Tiziana Boffa Ballaran

Tiziana Boffa Ballaran

Bayerisches Geoinstitut, Universität Bayreuth, Bayreuth, Germany

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Norimasa Nishiyama

Norimasa Nishiyama

Deutsches Elektronen-Synchrotron (DESY), Hamburg, Germany

Now at Laboratory for Materials and Structures, Tokyo Institute of Technology, Yokohama, Japan

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Michael Hanfland

Michael Hanfland

European Synchrotron Radiation Facility, Grenoble, France

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First published: 16 July 2018
Citations: 18
This article was corrected on 3 DEC 2018. See the end of the full text for details.

Abstract

High-pressure silica polymorphs may contribute to lithologies in Earth's lower mantle. Stishovite, the tetragonal rutile-type polymorph of SiO2, distorts to an orthorhombic CaCl2-type structure at pressures and temperatures of the lower mantle. We compressed sintered polycrystalline stishovite and determined the unit cell parameters as a function of pressure using synchrotron X-ray diffraction. The compression behavior of sintered polycrystalline stishovite deviates systematically from previous results on stishovite powder and single crystals. Following an initial stiffening, the bulk modulus of sintered polycrystalline stishovite drops at the ferroelastic phase transition. We analyzed the observed spontaneous strains using Landau theory to predict the complete elastic behavior of sintered polycrystalline silica across the phase transition. The reduction in bulk modulus at the phase transition as derived here from the compression curve of sintered polycrystalline silica may have implications for the seismic detection of silica-rich materials in Earth's lower mantle.

Key Points

  • We compressed sintered polycrystalline stishovite across the ferroelastic tetragonal-orthorhombic phase transition
  • The compression behavior of sintered polycrystalline stishovite deviates systematically from powder
  • The bulk modulus of sintered polycrystalline stishovite drops at the ferroelastic phase transition

Plain Language Summary

When rocks from Earth's surface sink into Earth's deep interior, the component minerals change in response to the high pressures and high temperatures. Among others, a mineral called stishovite forms. Stishovite has the same chemical formula as quartz but a higher density. At depths beneath 1,500 km, the stishovite crystals distort spontaneously to adapt to higher pressures. Because of this inherent tendency of the crystals to distort, they become elastically softer close to the pressure at which they distort spontaneously. We measured the compressibility of stishovite at high pressures and found that stishovite is less compressible when many stishovite crystals are firmly connected to each other. When the crystals distort, however, the polycrystalline material suddenly becomes more compressible again. Because seismic waves travel slower in elastically soft and compressible materials, we conclude that seismic waves might be slowed down by stishovite close to those pressures or depths where the stishovite crystals distort. The change of the elastic properties of stishovite might even result in seismic waves bouncing off rocks that contain stishovite. Such seismic waves can be detected on Earth's surface and may be used to infer the presence of stishovite in Earth's lower mantle.

1 Introduction

The physical properties of crystalline high-pressure phases of silica, SiO2, are of importance for geophysics and material sciences (e.g., Hemley et al., 1994). Along a typical geotherm, stishovite would be the stable SiO2 polymorph at pressures above 10 GPa (Zhang et al., 1996). At pressures above 120 GPa and temperatures around 2,500 K, SiO2 crystallizes in the α-PbO2 crystal structure as seifertite (El Goresy et al., 2008; Grocholski et al., 2013; Murakami et al., 2003; Tsuchiya et al., 2004). Both diamond inclusions and high-pressure high-temperature experiments indicate that stishovite contributes to basaltic lithologies at conditions of Earth's lower mantle (Hirose et al., 1999; Joswig et al., 1999). The results of recent experiments and calculations suggest that silica might exsolve from Earth's outer core (Hirose et al., 2017) and be dispersed throughout the lower mantle (Helffrich et al., 2018). Besides its relevance to geophysics, polycrystalline stishovite is considered the hardest known oxide that remains metastable at ambient conditions (Léger et al., 1996; Nishiyama et al., 2014).

At ambient temperature and around 50 GPa, stishovite (space group P42/mnm; Stishov & Belov, 1962) undergoes a ferroelastic phase transition from the tetragonal rutile crystal structure to an orthorhombic phase with CaCl2-type structure (space group Pnnm; Andrault et al., 1998; Karki, Warren, et al., 1997; Kingma et al., 1995), hereafter referred to as CaCl2-type SiO2. This phase transition has received considerable attention as it possibly entails a strong reduction of the shear modulus at or close to the transition pressure (Asahara et al., 2013; Carpenter et al., 2000; Karki, Stixrude, & Crain, 1997; Karki, Warren, et al., 1997; Shieh et al., 2002). Consequently, the presence of stishovite-bearing rocks such as subducted oceanic crust in Earth's lower mantle has been inferred from seismic observations of low shear wave velocities (Helffrich et al., 2018; Kaneshima & Helffrich, 2010; Niu et al., 2003). Previous experimental work on the elastic properties of stishovite and CaCl2-type SiO2 was either limited to low pressures (Jiang et al., 2009) or focused on the equation of state (EOS) of single crystals (Hemley et al., 2000) and powders (Andrault et al., 1998, 2003; Grocholski et al., 2013; Nisr et al., 2017). The effect of nonhydrostatic compression on the phase transition from stishovite to CaCl2-type SiO2 has been investigated on powders (Shieh et al., 2002; Singh et al., 2012) and on sintered polycrystalline material (Asahara et al., 2013). In the absence of direct measurements of elastic constants of stishovite to relevant pressures, the elastic properties can be estimated by analyzing the evolution of unit cell parameters across the ferroelastic phase transition using Landau theory (Carpenter et al., 2000).

In contrast to single crystals or unconsolidated powders, compression of a sintered polycrystalline material composed of elastically anisotropic crystals may give rise to considerable internal stresses between grains. The complex stress field may alter the elastic response of the polycrystal and, in the case of stishovite, interact with the ferroelastic phase transition. For example, deviatoric stresses were shown to shift the transition from stishovite to CaCl2-type SiO2 to lower pressures (Asahara et al., 2013; Shieh et al., 2002; Singh et al., 2012). We compressed sintered polycrystalline stishovite using neon as quasi-hydrostatic pressure-transmitting medium to investigate the effect of internal stresses arising from grain-grain interactions on the elastic response and the ferroelastic phase transition of stishovite. Based on the results of our X-ray diffraction experiments, we derive EOS for stishovite and CaCl2-type SiO2 as well as a Landau theory description for the arising spontaneous strains. We further use Landau theory to evaluate the changes in elastic properties of polycrystalline silica across the ferroelastic phase transition and discuss potential consequences for the seismic detection of stishovite-bearing materials in Earth's lower mantle.

2 Experimental

Sintered polycrystalline stishovite was synthesized from a pure silica glass rod at 15 GPa and 1,573 K using a Kawai-type multianvil apparatus. After annealing the silica rod for 30 min, the temperature was decreased to around 723 K and the sample decompressed at this temperature to prevent cracking of the sintered polycrystalline product. Purity and crystallinity of the synthesized stishovite were verified using synchrotron X-ray diffraction at beamline P02.1 at PETRA III/DESY in Hamburg and electron microscopy. Details of sample synthesis and characterization have been reported elsewhere (Nishiyama et al., 2014). A plane-parallel double-sided polished thin section with a final thickness of 13(1) μm was prepared from the synthesis product by mechanical polishing. Circles with diameters of 40 μm were cut from the thin section using a focused beam of Ga+ ions (Marquardt & Marquardt, 2012; Schulze et al., 2017). A sintered polycrystalline stishovite platelet was loaded in a BX90 diamond anvil cell (Kantor et al., 2012) along with a ruby sphere for pressure determination (Dewaele et al., 2008). We used diamond anvils with 200-μm culets that were glued to tungsten carbide seats and a rhenium foil as gasket after preindenting it to a thickness of about 40 μm and cutting a hole with 120-μm diameter in the center of the indentation using an infrared laser. Precompressed neon was loaded as pressure-transmitting medium (Kurnosov et al., 2008) to create a quasi-hydrostatic stress environment upon compression.

X-ray diffraction patterns were recorded at beamline ID15B of the European Synchrotron Radiation Facility at 30 pressures between 9 and 73 GPa and at ambient temperature. Monochromatic X-rays with a wavelength of 0.410768 Å were focused on the sample and the diffracted radiation detected by a Mar555 image plate in transmission geometry. The exact diffraction geometry was calibrated using a powdered Si standard. Exposure times were 2 to 3 s. Two-dimensional diffraction patterns were integrated to intensity-2θ profiles using the FIT2D software (Hammersley, 2016). Lattice parameters of stishovite and CaCl2-type SiO2 were extracted from Le Bail refinements performed with the program MAUD (Lutterotti et al., 2007). We used the structural model for stishovite reported by Ross et al. (1990) and refined lattice parameters and microstrains at every pressure after adjusting a polynomial baseline to account for background intensity. Uncertainties on lattice parameters were rescaled as proposed by Bérar and Lelann (1991). At the lowest pressure, we also refined the average crystallite size of the sintered polycrystalline stishovite. The result of 165 nm agrees with estimates based on electron microscopy (Nishiyama et al., 2014). At pressures clearly above the phase transition, the crystal symmetry was reduced accordingly and the appropriate parameters added to the refinement procedure. For pressures close to the phase transition, we performed two separate refinements assuming first tetragonal and then orthorhombic symmetry.

3 EOS of Sintered Polycrystalline Stishovite

Typical diffraction patterns of stishovite and CaCl2-type SiO2 are shown in Figure 1. Following previous studies on the tetragonal-to-orthorhombic phase transition (Nomura et al., 2010; Ono et al., 2002), we used the splitting of the 121 reflection to identify CaCl2-type SiO2. Due to the continuous nature of the phase transition, the orthorhombic 121 and 211 reflections clearly separate only after an initial broadening of the joint peak corresponding to the tetragonal 121 reflection (Figure 1b). From the inspection of diffraction patterns alone, we locate the phase transition between 40 and 50 GPa. As we would expect different grains of the sintered material to be subjected to different stress conditions, we cannot exclude that both tetragonal and orthorhombic structures coexist for a range of pressures as observed by Nisr et al. (2017).

Details are in the caption following the image
Segments of two-dimensional diffraction patterns (a) and integrated diffraction patterns (b) of sintered polycrystalline stishovite (P42/mnm) and CaCl2-type SiO2 (Pnnm). In (b), solid circles show background-corrected observed diffracted intensities; red lines show refined intensities; black lines show residuals; black bars indicate refined diffraction angles. Black arrows indicate splitting of the tetragonal 011 and 121 reflections into the pairs of orthorhombic 011 and 101 and 121 and 211 reflections. Note the initial broadening of these reflections close to the phase transition around 45 GPa. D marks reflections from diamond anvils.

Unit cell edge lengths a, b, and c and unit cell volumes V are listed in Table 1 for both stishovite and CaCl2-type SiO2. Their evolution with pressure is shown in Figure 2 together with results of compression experiments on single-crystal stishovite and silica powder (Andrault et al., 2003). In their high-pressure powder diffraction experiments, Andrault et al. (2003) used NaCl as pressure-transmitting medium and heated the silica powder with an infrared laser to reduce nonhydrostatic stresses. We recalculated the experimental pressures of Andrault et al. (2003) based on revised EOS for B2-structured NaCl and platinum (Fei et al., 2007) as these standard materials were used for pressure determination in the experiments of Andrault et al. (2003). The revised EOS from Fei et al. (2007) gave slightly higher pressures for B2-structured NaCl than those reported by Andrault et al. (2003) with a maximum difference of 1.4 GPa and lower pressures for platinum with a maximum difference of 9.3 GPa at the highest pressure. To better assess the effect of the stress environment imposed by the pressure-transmitting medium, we also compare our results to compression of stishovite powder in helium (Nisr et al., 2017).

Table 1. Refined Unit Cell Parameters of Sintered Polycrystalline Stishovite and CaCl2-Type SiO2 at High Pressures
Pressure Unit cell parameters
P(GPa) a(Å) b(Å) c(Å) V3) wR(%)a
Stishovite –P42/mnm
9.67(48) 4.1319(4) 4.1319(4) 2.6501(7) 45.24(1) 1.08
11.38(10) 4.1251(7) 4.1251(7) 2.6468(11) 45.04(2) 1.27
13.99(70) 4.1156(9) 4.1156(9) 2.6422(15) 44.75(3) 1.50
15.77(79) 4.1093(8) 4.1093(8) 2.6395(13) 44.57(3) 1.63
17.74(4) 4.1030(7) 4.1030(7) 2.6365(11) 44.38(2) 1.47
19.59(98) 4.0961(4) 4.0961(4) 2.6340(6) 44.19(1) 1.14
21.40(10) 4.0910(4) 4.0910(4) 2.6307(6) 44.03(1) 1.06
23.15(3) 4.0848(4) 4.0848(4) 2.6288(7) 43.86(1) 1.07
24.99(9) 4.0794(4) 4.0794(4) 2.6257(6) 43.70(1) 1.10
27.16(41) 4.0738(4) 4.0738(4) 2.6229(7) 43.53(1) 1.11
30.45(152) 4.0685(3) 4.0685(3) 2.6203(5) 43.37(1) 1.11
31.02(3) 4.0615(4) 4.0615(4) 2.6168(7) 43.17(1) 1.10
33.27(9) 4.0559(5) 4.0559(5) 2.6135(9) 42.99(2) 1.17
35.70(3) 4.0494(5) 4.0494(5) 2.6101(10) 42.80(2) 1.15
38.05(8) 4.0433(6) 4.0433(6) 2.6071(11) 42.62(2) 1.17
40.39(5) 4.0365(6) 4.0365(6) 2.6031(12) 42.41(2) 1.15
42.89(8) 4.0303(7) 4.0303(7) 2.5982(14) 42.20(3) 1.27
45.27(1) 4.0241(8) 4.0241(8) 2.5943(15) 42.01(3) 1.26
47.59(6) 4.0169(7) 4.0169(7) 2.5903(14) 41.79(3) 1.23
49.87(13) 4.0117(6) 4.0117(6) 2.5872(11) 41.64(2) 1.31
CaCl2 -type SiO2– Pnnm
45.27(1) 3.9973(19) 4.0510(19) 2.5946(9) 42.01(3) 0.96
47.59(6) 3.9884(23) 4.0458(23) 2.5905(11) 41.80(4) 0.94
49.87(13) 3.9824(24) 4.0417(25) 2.5874(12) 41.65(4) 1.05
52.16(12) 3.9712(22) 4.0406(22) 2.5834(12) 41.45(4) 1.03
54.38(10) 3.9629(24) 4.0362(25) 2.5801(12) 41.27(4) 1.00
56.40(6) 3.9530(28) 4.0345(29) 2.5763(14) 41.09(5) 0.97
58.51(301) 3.9444(27) 4.0331(28) 2.5727(15) 40.93(5) 0.97
62.63(12) 3.9267(24) 4.0282(25) 2.5654(13) 40.58(4) 0.95
64.64(1) 3.9181(25) 4.0262(26) 2.5620(14) 40.41(4) 0.97
66.59(341) 3.9082(28) 4.0240(30) 2.5597(15) 40.26(5) 0.95
66.96(343) 3.9056(27) 4.0233(29) 2.5584(15) 40.20(5) 0.95
70.99(23) 3.8937(31) 4.0186(31) 2.5520(17) 39.93(5) 0.97
72.89(375) 3.8880(31) 4.0131(31) 2.5505(17) 39.80(5) 1.00
  • Note. Numbers in parentheses are standard errors on the last digit.
  • a Weighted R factor.
Details are in the caption following the image
Unit cell edge lengths (a, b) and unit cell volumes (c) of stishovite and CaCl2-type SiO2 as a function of pressure. Curves show finite-strain equations of state. Open symbols show data excluded from analysis. Close to the tetragonal-orthorhombic phase transition, unit cell parameters were refined for both symmetries. This study: sintered polycrystalline silica in neon; Nisr et al. (2017): powder in helium; Andrault et al. (2003): powder in NaCl or without medium (single-crystal data for P < 10 GPa).

While all three data sets are consistent around 10 GPa, sintered polycrystalline stishovite is less compressible along the a axis leading to an increasing deviation of the a axis length and the unit cell volume from those of stishovite powder up to the transition pressure. Above the transition pressure, however, the orthorhombic a and b axes as well as the volume compression curve of the sintered polycrystalline material evolve in parallel to the powder compression curves of Andrault et al. (2003) with increasing pressure. The c axes of powdered and sintered polycrystalline stishovite and CaCl2-type SiO2 essentially follow the same compression curve.

To extract quantitative information on the axial and volume compression behavior, we analyzed axial and volume compression curves of sintered polycrystalline stishovite and CaCl2-type SiO2 with third-order finite-strain EOS (Angel, 2000; Birch, 1947). For volume compression, we have
urn:x-wiley:jgrb:media:jgrb52925:jgrb52925-math-0001(1)
with the Eulerian finite volume strain fE=[(V0/V)2/3−1]/2, the unit cell volume V0, the bulk modulus K0, and its pressure derivative urn:x-wiley:jgrb:media:jgrb52925:jgrb52925-math-0002 at the chosen reference pressure and for linear compression
urn:x-wiley:jgrb:media:jgrb52925:jgrb52925-math-0003(2)
with the linear moduli ki0, their pressure derivatives urn:x-wiley:jgrb:media:jgrb52925:jgrb52925-math-0004, and the diagonal components of the Eulerian finite-strain tensor Ei=[1 − (ai0/ai)2]/2 (i = 1, 2, 3 for a, b, and c, respectively). Equation 1 was fit to the volume compression curves of stishovite and CaCl2-type SiO2 and equation 2 to the axial compression curves of stishovite. To avoid any bias caused by a mixture of both phases, we excluded data between 40 and 50 GPa from the analysis. Based on the variation of normalized stress with finite strains (Angel, 2000), we set urn:x-wiley:jgrb:media:jgrb52925:jgrb52925-math-0005 for stishovite and urn:x-wiley:jgrb:media:jgrb52925:jgrb52925-math-0006 for CaCl2-type SiO2 implying a second-order finite-strain EOS. For each phase, we chose the lowest pressure included in the analysis as the reference pressure and, for comparison, extrapolated the results back to ambient pressure using the obtained EOS parameters. The results are listed in Table 2.
Table 2. Refined Equation of State Parameters for Stishovite and CaCl2-Type SiO2
Medium Sintered polycrystalline silica Powdera (and single crystal) Powderb
Neon NaCl/none with laser annealing Helium
Stishovite –P42/mnm
P0 (GPa)c 9.67 0 0 0
V0 (Å3) 45.23(3) 46.43(10) 46.51(2) 46.569
K0 (GPa) 401(15) 344(25) 320(2) 312(2)
urn:x-wiley:jgrb:media:jgrb52925:jgrb52925-math-0007 5.8(11) 6.0(11) 4 4.59
a0 (Å) 4.1319(8) 4.1737(27) 4.1767(8)
c0 (Å) 2.6498(2) 2.6667(4) 2.6666(3)
k10 (GPa) 1060(41) 867(44) 808(10)
k30 (GPa) 1584(15) 1467(15) 1459(17)
urn:x-wiley:jgrb:media:jgrb52925:jgrb52925-math-0008 19.3(31) 20.5(31) 12
urn:x-wiley:jgrb:media:jgrb52925:jgrb52925-math-0009 12 12.0(3) 12
CaCl2 -type SiO2– Pnnm
P0 (GPa)c 52.16 0 62.99 0
V03) 41.46(3) 48.22(44) 39.98(2) 47.54(18)
K0 (GPa) 467(17) 241(18) 514(6) 238(6)
urn:x-wiley:jgrb:media:jgrb52925:jgrb52925-math-0010 4 4.72(4) 4 4.82(2)
  • Note. Values in italics were fixed during refinement. Numbers in parentheses are standard errors on the last digit.
  • a Compression data from Andrault et al. (2003); single-crystal data in water-ethanol-methanol for P < 10 GPa.
  • b Equation of state from Nisr et al. (2017).
  • c Reference pressure.

To see the effect of the revised EOS for B2-structured NaCl and platinum (Fei et al., 2007) on the EOS parameters, we reanalyzed the combined data set of single-crystal stishovite and silica powder (Andrault et al., 2003). In contrast to the original EOS with urn:x-wiley:jgrb:media:jgrb52925:jgrb52925-math-0011 for stishovite as reported by Andrault et al. (2003), we set urn:x-wiley:jgrb:media:jgrb52925:jgrb52925-math-0012 for both stishovite and CaCl2-type SiO2 based on virtually constant normalized stresses for increasing finite strains (Angel, 2000). Similarly, urn:x-wiley:jgrb:media:jgrb52925:jgrb52925-math-0013 were fixed in the analysis of axial compression curves. For stishovite, we excluded data above 40 GPa to avoid any influence of the phase transition on the derived EOS. The results are included in Table 2 together with the EOS parameters for compression of stishovite in helium (Nisr et al., 2017).

A comparison of EOS parameters for stishovite in Table 2 shows that sintered polycrystalline stishovite has the highest bulk modulus at ambient conditions and also the highest pressure derivative of the bulk modulus. Hence, sintered polycrystalline stishovite is less compressible and stiffens faster with pressure than stishovite powder. The differences between the EOS for laser-annealed stishovite powder compressed in NaCl or without pressure-transmitting medium (Andrault et al., 2003) and the EOS for stishovite powder compressed in helium (Nisr et al., 2017) may arise from different stress environments such as deviations from hydrostaticity imposed by the different pressure-transmitting media. Taking into account their uncertainties, both EOS for stishovite powder in Table 2 are consistent with the range of bulk moduli and their pressure derivatives of earlier studies as compiled by Fischer et al. (2018).

The finite-strain curves in Figure 2c reveal a kink in the volume compression curve for sintered polycrystalline silica associated with a drop in the bulk modulus at the phase transition. This behavior differs from the smooth volume variation observed for stishovite powder. After initial stiffening, sintered polycrystalline silica becomes more compressible at pressures above the phase transition to CaCl2-type SiO2. Our reanalysis of the EOS of silica powder (Andrault et al., 2003) across the phase transition from stishovite to CaCl2-type SiO2 reveals a small drop in bulk modulus at the phase transition as well. Except for the unit cell volumes at ambient conditions, the EOS for CaCl2-type SiO2 powder and sintered polycrystalline CaCl2-type SiO2 are identical within uncertainties.

In a powder, individual grains are free to distort and to adapt to the prevailing stress environment. Laser annealing further stimulates the relaxation of nonhydrostatic stresses (Andrault et al., 2003; Singh et al., 2012). The compression experiments on stishovite powder by Andrault et al. (2003) and in helium by Nisr et al. (2017) therefore capture the compression behavior of a mechanically relaxed material. In contrast, grain boundaries are locked to each other in sintered polycrystalline stishovite. As a result, compression of elastically anisotropic crystals with different orientations will inevitably lead to internal stresses as grains get clamped by their neighboring grains. The effect of clamping depends on the number of grain-grain interactions. In comparison to previous studies (Andrault et al., 2003; Shieh et al., 2002; Singh et al., 2012), the small grain size of the here-used sample might therefore enhance stresses between grains. When grain sizes get substantially smaller than in our polycrystalline stishovite sample, however, the elastic properties of grain boundaries themselves significantly contribute to the overall elastic response (Marquardt, Gleason, et al., 2011; Marquardt, Speziale, et al., 2011).

Mechanical clamping and the related internal stresses can change the elastic response of sintered polycrystalline stishovite and give rise to elastic stiffening as observed here (Figure 2). At the tetragonal-orthorhombic phase transition, the silica crystals gain an additional degree of freedom due to the symmetry reduction and the related possibility to distort accordingly. By distorting, they adapt to their individual stress environments and partially release the stresses accumulated during compression. As internal stresses relax, the extent of mechanical clamping decreases, and the material becomes more compressible until internal stresses build up again due to further compression. We therefore attribute the stiffening of sintered polycrystalline stishovite followed by the compressional softening at the phase transition to CaCl2-type SiO2 to the buildup of internal stresses in sintered polycrystalline stishovite and their partial relaxation at the phase transition.

4 Landau Theory Analysis

Further insight into the elastic behavior of sintered polycrystalline stishovite can be obtained by an analysis of the spontaneous strains ei arising from the ferroelastic phase transition (Carpenter & Salje, 1998; Carpenter, 2006). Here we follow the Landau theory approach of Carpenter et al. (2000) with some small modifications. The Landau free energy expansion for the pseudoproper ferroelastic phase transition (Wadhawan, 1982) from tetragonal stishovite to orthorhombic CaCl2-type SiO2 ( urn:x-wiley:jgrb:media:jgrb52925:jgrb52925-math-0014) was given by Carpenter et al. (2000) as
urn:x-wiley:jgrb:media:jgrb52925:jgrb52925-math-0015(3)
with the driving order parameter Q, the critical pressure Pc, the Landau coefficients a and b, the coupling coefficients λ1λ6, and the bare elastic constants urn:x-wiley:jgrb:media:jgrb52925:jgrb52925-math-0016. Due to coupling between spontaneous strains and the driving order parameter, the structure distorts at a transition pressure urn:x-wiley:jgrb:media:jgrb52925:jgrb52925-math-0017 that is different from the critical pressure (Carpenter et al., 2000):
urn:x-wiley:jgrb:media:jgrb52925:jgrb52925-math-0018(4)
The order parameter evolves with pressure as (Carpenter et al., 2000)
urn:x-wiley:jgrb:media:jgrb52925:jgrb52925-math-0019(5)
with the renormalized fourth-order coefficient
urn:x-wiley:jgrb:media:jgrb52925:jgrb52925-math-0020(6)
Following the steps laid out by Carpenter and Salje (1998), we find for the symmetry-adapted spontaneous strains that arise from the tetragonal-orthorhombic phase transition:
urn:x-wiley:jgrb:media:jgrb52925:jgrb52925-math-0021(7)
urn:x-wiley:jgrb:media:jgrb52925:jgrb52925-math-0022(8)
urn:x-wiley:jgrb:media:jgrb52925:jgrb52925-math-0023(9)
These expressions are identical to those given by Carpenter (2006) except for a typo correction of urn:x-wiley:jgrb:media:jgrb52925:jgrb52925-math-0024 to urn:x-wiley:jgrb:media:jgrb52925:jgrb52925-math-0025 in the numerator of the formula for e3. By symmetry, e4=e5=e6=0 in mechanical equilibrium. It can be seen from equation 3, however, that any imposed strain along the crystal axes or shear strain within the a-b plane may displace the transition pressure from its value under mechanical equilibrium. Nonequilibrium strains may arise from nonhydrostatic compression or from stresses due to grain-grain interactions. Further definitions and equations can be found in Carpenter et al. (2000), including the expressions for the variation of individual elastic constants across the phase transition.
Previous analyses of the elastic behavior across the ferroelastic phase transition of stishovite to CaCl2-type SiO2 assumed a linear pressure dependence of bare elastic constants (Carpenter, 2006; Carpenter et al., 2000). In contrast, we express the variation of bare elastic constants in terms of finite strain for a self-consistent treatment of thermodynamic quantities at high pressures (Stixrude & Lithgow-Bertelloni, 2005):
urn:x-wiley:jgrb:media:jgrb52925:jgrb52925-math-0026(10)
where urn:x-wiley:jgrb:media:jgrb52925:jgrb52925-math-0027 and urn:x-wiley:jgrb:media:jgrb52925:jgrb52925-math-0028 are the bare elastic constants and their pressure derivatives at ambient conditions in full index notation, respectively, and urn:x-wiley:jgrb:media:jgrb52925:jgrb52925-math-0029. To keep equation 10 internally consistent, the bulk modulus is calculated from the elastic constants as the Voigt bound urn:x-wiley:jgrb:media:jgrb52925:jgrb52925-math-0030 and accordingly urn:x-wiley:jgrb:media:jgrb52925:jgrb52925-math-0031 (Stixrude & Lithgow-Bertelloni, 2005). The actual elastic response of a polycrystalline material depends on the extent to which individual grains can adapt to the external stress field (Hill, 1952; Watt et al., 1976). The Reuss bound would be appropriate if every grain experiences the same stress field and can freely deform according to its orientation relative to the stress tensor. In contrast, the Voigt bound applies when grains are mechanically clamped and forced into an overall strain state. For quasi-hydrostatic compression of sintered polycrystalline stishovite, the Voigt bound might appear to be more appropriate. We note, however, that we used equation 10 merely to describe the dependence of elastic constants on finite volume strain.
The elastic constants of stishovite and their pressure derivatives have previously been determined by Brillouin spectroscopy (Brazhkin et al., 2005; Jiang et al., 2009; Weidner et al., 1982) and computed from first principles (Karki et al., 1997; Yang & Wu, 2014). Since the experimentally determined elastic constants c11 and c12 (in contracted index notation) are affected by elastic softening even at pressures below the phase transition (Carpenter et al., 2000; Jiang et al., 2009), the corresponding bare elastic constants at ambient conditions (P = 0) were calculated by combining the two relations (Carpenter et al., 2000)
urn:x-wiley:jgrb:media:jgrb52925:jgrb52925-math-0032(11)
urn:x-wiley:jgrb:media:jgrb52925:jgrb52925-math-0033(12)
Linking equations 4-12 to the EOS 1 of stishovite (Table 2), the spontaneous strains can be calculated as a function of finite strain, that is, volume. Our approach is similar to previous formulations based on the Lagrangian definition of strain (Tröster et al., 2002, 2014). The Landau and coupling coefficients can be obtained by a least squares fit of calculated to experimentally observed spontaneous strains. We used the unit cell edge lengths for CaCl2-type SiO2 at pressures above 50 GPa (Table 1) to derive spontaneous strains as (Carpenter et al., 2000)
urn:x-wiley:jgrb:media:jgrb52925:jgrb52925-math-0034(13)
urn:x-wiley:jgrb:media:jgrb52925:jgrb52925-math-0035(14)
urn:x-wiley:jgrb:media:jgrb52925:jgrb52925-math-0036(15)
where the unit cell edge lengths of stishovite, a0 and c0, were extrapolated to the experimental pressures using the respective axial EOS 2 (Table 2). Including the EOS analysis described in section 3, we treated the combined data set of single-crystal stishovite and silica powder (Andrault et al., 2003) in the same way. The spontaneous strains calculated from both data sets are shown in Figure 3.
Details are in the caption following the image
Spontaneous strains in orthorhombic CaCl2-type SiO2 as a function of pressure (a) and unit cell volume (b). In (a), the intersection of the extrapolated squared symmetry-breaking strains (e1e2)2 with the pressure axis gives the indicated transition pressures. In (b), curves show spontaneous strains predicted by Landau theory. Open symbols show data excluded from analysis. This study: sintered polycrystalline silica; Andrault et al. (2003): powder; Asahara et al. (2013): sintered polycrystalline silica (nonhydrostatic).

As mentioned in section 3, the splitting of diffraction lines gives only a rough estimate for the transition pressure (Figure 1). The transition pressure can be precisely located by observing structural properties that are directly linked to the transition mechanism such as the frequency of the soft optic mode (Carpenter et al., 2000; Kingma et al., 1995). The symmetry-breaking spontaneous strain (e1e2) reflects the main structural distortion of the phase transition. (e1e2)2 should increase linearly with pressure, and the intersection of the linear trend with the pressure axis at zero spontaneous strain gives a reliable estimate of the transition pressure (Carpenter et al., 2000).

The symmetry-breaking spontaneous strains observed on sintered polycrystalline CaCl2-type SiO2 follow the behavior predicted by Landau theory (Figure 3a). Figure 3a shows that the transition pressures of sintered polycrystalline stishovite (44.7 ± 2.9 GPa) and stishovite powder (48.3 ± 1.7 GPa) are almost identical when considering their uncertainties as derived from the linear fits. Moreover, spontaneous strains observed on sintered polycrystalline CaCl2-type SiO2 are consistent with the spontaneous strains of CaCl2-type SiO2 powder at pressures close to the phase transition. Both Singh et al. (2012) and Asahara et al. (2013) observed the phase transition at substantially lower pressures upon nonhydrostatic compression (Figure 3a). We therefore conclude that our experiments were not strongly affected by nonhydrostatic compression.

To determine all relevant coefficients of equation 3 by least squares fits of calculated to experimentally observed spontaneous strains, we fixed the transition pressures to the values derived from Figure 3a. To reduce correlations between parameters, we further fixed the difference between the critical pressure and the transition pressure to the value given by Carpenter et al. (2000), that is, urn:x-wiley:jgrb:media:jgrb52925:jgrb52925-math-0037 GPa and set b = 11 (Carpenter, 2006; Carpenter et al., 2000). Since the pressure derivatives of the bare elastic constants urn:x-wiley:jgrb:media:jgrb52925:jgrb52925-math-0038 and urn:x-wiley:jgrb:media:jgrb52925:jgrb52925-math-0039 should be affected by elastic softening (Jiang et al., 2009) but cannot be simply obtained by combining the experimentally observed pressure derivatives, we fixed urn:x-wiley:jgrb:media:jgrb52925:jgrb52925-math-0040 to the value given by Jiang et al. (2009) and treated urn:x-wiley:jgrb:media:jgrb52925:jgrb52925-math-0041 as free parameter during the fitting procedure. Due to the limited pressure range of spontaneous strains observed on sintered polycrystalline CaCl2-type SiO2, however, urn:x-wiley:jgrb:media:jgrb52925:jgrb52925-math-0042 was fixed to the value obtained by fitting the spontaneous strains of CaCl2-type SiO2 powder. All other relevant elastic constants and pressure derivatives were also fixed to their experimentally determined values (Jiang et al., 2009). Table 3 lists the final combinations of coefficients, and the predicted evolution of spontaneous strains is shown in Figure 3b. We note that our approach to derive Landau and coupling coefficients relies on a minimum of assumptions and is internally consistent.

Table 3. Refined Landau Coefficients, Coupling Coefficients, and Bare Elastic Constants for the Stishovite–CaCl2-Type SiO2 Phase Transition
Sintered polycrystalline silica Powdera
Landau and coupling coefficients
urn:x-wiley:jgrb:media:jgrb52925:jgrb52925-math-0043 (GPa)b 50.7(50) 50.7(50)
urn:x-wiley:jgrb:media:jgrb52925:jgrb52925-math-0044 (GPa)c 44.7(29) 48.3(17)
a −0.0342(40) −0.0510(67)
b (GPa) 11 11
λ1 (GPa) 31.39(89) 8.52(100)
λ2 (GPa) 22.86(32) 27.59(104)
λ3 (GPa) 44.74(99) 33.45(139)
Bare elastic constantsd
urn:x-wiley:jgrb:media:jgrb52925:jgrb52925-math-0045 (GPa) 600(24) 589(17)
urn:x-wiley:jgrb:media:jgrb52925:jgrb52925-math-0046 (GPa) 54(24) 65(17)
urn:x-wiley:jgrb:media:jgrb52925:jgrb52925-math-0047 2.8(3) 2.8(3)
  • Note. Values in italics were fixed during refinement. Numbers in parentheses are standard errors on the last digit.
  • a Compression data from Andrault et al. (2003).
  • b From Carpenter et al. (2000).
  • c From linear fits in Figure 3a.
  • d All remaining elastic constants and pressure derivatives as given in Jiang et al. (2009).

5 Elasticity of Stishovite Across the Ferroelastic Phase Transition

The variation of individual elastic constants with pressure as calculated using the expressions given by Carpenter et al. (2000) and the parameters of Table 3 is shown in Figure 4a. Besides the small difference in transition pressures, the elastic constants predicted for sintered polycrystalline stishovite and CaCl2-type SiO2 are more perturbed at the phase transition when compared to the predictions for silica powder. Another fundamental difference becomes apparent when combining individual elastic constants to bulk and shear moduli (Figures 4b and 4c). Similar to previous results (Carpenter et al., 2000), Landau theory predicts substantial softening of the shear modulus in the vicinity of the phase transition for both sintered polycrystalline stishovite and stishovite powder (Figure 4b). While the powder data suggest only a small drop of the bulk modulus at the phase transition, however, the bulk modulus is predicted to drop significantly in the case of sintered polycrystalline silica (Figure 4c). We note that this prediction is based on the analysis of spontaneous strains only and does not involve volume compression data. The analysis of volume compression data, however, indicated a kink in the compression curves and hence a drop in the bulk modulus as well (Figure 2c).

Details are in the caption following the image
Pressure evolution of elastic constants (a), shear modulus (b), and bulk modulus (c) of stishovite and CaCl2-type SiO2 across the ferroelastic phase transition as predicted by Landau theory (a–c) and calculated from the equations of state (EOS; c). In (b) and (c), the shading indicates Voigt (upper) and Reuss (lower) bounds on the respective modulus while bold curves show the Voigt-Reuss-Hill average. In (c), vertical bars show the magnitudes of the drop in bulk modulus of sintered polycrystalline silica at the phase transition as estimated by Landau theory (LT), from the EOS, and by linear fits (LF) to pressure-volume pairs. Note the differences in pressure derivatives of the bulk moduli as derived from compression experiments and predicted by Landau theory. spx: sintered polycrystalline silica (this study); powder: compression data from Andrault et al. (2003).

The bulk moduli of stishovite and CaCl2-type SiO2 as calculated from the respective EOS are also shown in Figure 4c. For stishovite powder, the Landau theory prediction remains closer to the EOS bulk modulus when compared to sintered polycrystalline stishovite. The bulk moduli derived from the compression curves of sintered polycrystalline stishovite and CaCl2-type SiO2 are substantially higher than the Landau theory prediction. These differences arise from the low pressure derivative of the bulk modulus when computed from the pressure derivatives of individual elastic constants, urn:x-wiley:jgrb:media:jgrb52925:jgrb52925-math-0048, in comparison to urn:x-wiley:jgrb:media:jgrb52925:jgrb52925-math-0049 to 6 as derived from compression experiments (Table 2). Landau theory predicts the evolution of individual elastic constants of a (isolated) single crystal but does not capture the interactions between grains that cause the initial stiffening observed in the compression curves. The magnitudes of the drops in bulk moduli, however, are very similar to the magnitudes predicted based on single-crystal elastic constants and spontaneous strains. To estimate the drop in bulk modulus independent of any assumption, we derived the bulk moduli of sintered polycrystalline stishovite and CaCl2-type SiO2 close to the phase transition by fitting a linear trend to the three pressure-volume pairs of each phase closest to the transition pressure but outside of the potential two-phase region. The resulting drop in bulk modulus is approximately half of the drop inferred from the EOS but still amounts to about 70 GPa (Figure 4c). We attribute the high absolute values and the acute drop of the bulk modulus at the phase transition to grain-grain interactions in sintered polycrystalline silica, potentially arising from locked grain boundaries, and to the partial relaxation of internal stresses at the phase transition, respectively.

6 Sound Wave Velocities of Stishovite and Geophysical Implications

To calculate aggregate sound wave velocities, we combined bulk moduli and densities from the experimental EOS of sintered polycrystalline silica and silica powder with shear moduli predicted by Landau theory. The calculated aggregate sound wave velocities are shown as a function of pressure in Figure 5. Apart from slightly different transition pressures, shear wave (S wave) velocities predicted based on our results on sintered polycrystalline silica and on previous studies on silica powder are consistent, including the magnitude and pressure range of shear wave softening (Figure 5a). Our predictions based on compression experiments and Landau theory are also consistent with recent ab initio computations (Yang & Wu, 2014).

Details are in the caption following the image
Pressure evolution of S wave (a) and P wave velocity (b) of stishovite and CaCl2-type SiO2 across the ferroelastic phase transition. The shading indicates upper and lower bounds on the respective velocity calculated from the Voigt and Reuss bounds on the shear modulus, respectively, and the uncertainties of the EOS parameters. Bold curves show the aggregate velocity calculated from the Voigt-Reuss-Hill average of the shear modulus and the EOS bulk modulus. Horizontal bars indicate the pressure ranges covered by the respective compression experiments. spx: sintered polycrystalline silica (this study); powder: compression data from Andrault et al. (2003).

Predicted compressional wave (P wave) velocities, however, evolve differently with increasing pressure when using the results on sintered polycrystalline silica instead of those on silica powder (Figure 5b). At pressures below the phase transition, P wave velocities are predicted to be faster and to rise steeper with increasing pressure reflecting the initial stiffening of sintered polycrystalline stishovite on compression (Figure 2c). In addition to the softening of the shear modulus close to the phase transition, the compression curves in Figure 2c indicate a sharp drop of the bulk modulus at the phase transition (Figure 4c). This sharp drop of the bulk modulus translates into an abrupt drop in P wave velocities, which is enhanced by the use of the EOS for sintered polycrystalline silica. At pressures above the phase transition, P wave velocities rise again as the softening declines and evolve similarly at higher pressures regardless of whether the compression behavior of sintered polycrystalline CaCl2-type SiO2 or CaCl2-type SiO2 powder is considered. At pressures above the phase transition, P wave velocities as predicted here by combining experimental EOS with Landau theory are in very good agreement with P wave velocities computed from first principles (Yang & Wu, 2014).

We can think of a powder as representing a completely relaxed rock without interactions between grains and internal stresses. Such a situation would be favored by high temperatures and on long timescales. A sintered polycrystalline material, in contrast, represents a situation without relaxation and hence involves the buildup of internal stresses between elastically anisotropic grains in response to a changing external stress field. At temperatures of Earth's lower mantle, such a relaxation-free situation can only be maintained on very short time scales and after fast perturbations of the stress environment. Relaxation processes such as grain boundary movements, dislocation movements, and atomic diffusion operate on different time scales that depend on temperature (Jackson, 2007). Typical seismic frequencies (1 mHz to 1 Hz) might be high enough to probe a situation in between the totally relaxed and totally unrelaxed situations (Jackson, 2007). Grain sizes are expected to be larger in the lower mantle than in experimental samples (Solomatov et al., 2002). Differences in grain size might require additional corrections on sound wave velocities derived from experiments (Jackson, 2007). In terms of elastic interactions between grains, however, the elastic properties of a sintered polycrystalline material and an unconsolidated powder may serve as bounds for the propagation of seismic waves.

Taking the sound wave velocities as predicted based on the compression behavior of sintered polycrystalline silica and silica powder as bounds for seismic velocities, we speculate that P wave velocities of stishovite-bearing rocks might drop substantially at the ferroelastic phase transition of stishovite. In addition to elastic softening in the vicinity of the ferroelastic phase transition, P wave velocities might be further decreased by the relaxation of internal stresses at the phase transition as indicated by the compression curve of sintered polycrystalline stishovite (Figure 2c). Above the phase transition, the similar EOS of sintered polycrystalline CaCl2-type SiO2 and CaCl2-type SiO2 powder suggest a decreasing difference in the elastic response of both materials with increasing pressure as reflected in similar sound wave velocities (Figure 5). However, partial buildup and relaxation of internal stresses during the passage of a seismic wave might perturb the elastic response of stishovite-bearing rocks close to the ferroelastic phase transition beyond what would be expected based on a completely relaxed situation.

The results of our compression experiment on sintered polycrystalline stishovite and the analysis using Landau theory may have implications for detecting stishovite-bearing materials in Earth's lower mantle. At depths greater than 700 km, stishovite would contribute with 10 to 20 vol % to the mineral assemblage of subducted oceanic crust with a MORB-like composition (Hirose et al., 1999; Perrillat et al., 2006; Ricolleau et al., 2010). Rocks with much higher volume fractions of silica phases would be expected to be dispersed in the lower mantle if silica exsolved from the outer core (Helffrich et al., 2018; Hirose et al., 2017). The high viscosity of stishovite prevents stishovite-bearing rocks from mixing with the surrounding mantle (Xu et al., 2017). Scattering of S waves in the lower mantle has been related to the presence of stishovite based on low S wave velocities predicted for stishovite at conditions close to the ferroelastic phase transition (Helffrich et al., 2018; Kaneshima & Helffrich, 2010). Our findings, however, emphasize the importance of P waves since they might be affected by the phase transition in stishovite more than previously thought. P waves are indeed reflected and scattered in the lower mantle (Hedlin et al., 1997; LeStunff et al., 1995). To relate seismic scattering in the lower mantle to the presence of stishovite, a more complete understanding of how the ferroelastic phase transition interacts with seismic waves is needed. Recently, the bulk modulus of ferropericlase has been determined at seismic frequencies and pressures of the lower mantle (Marquardt et al., 2018). Similar experiments on sintered polycrystalline silica could demonstrate to which extent the bulk modulus is affected by the ferroelastic phase transition at seismic frequencies. In particular, anelastic relaxation processes and acoustic attenuation (Carpenter et al., 2000; Jackson, 2007) as well as effects arising from internal stresses in sintered polycrystalline materials as pointed out in the present study require measurements at seismic frequencies and at high pressures and high temperatures.

Acknowledgments

This work was supported by the Deutsche Forschungsgemeinschaft (DFG) grant (MA4534/3-1, INST 91/315-1 FUGG, GRK 2156/1). H. M. acknowledges support from the Bayerische Akademie der Wissenschaften (BAdW). All relevant data are compiled in Table 1 or have been published previously as cited in the text.

    Erratum

    In the originally published version of this article, there were some errors in the text and in one of the author affiliations. These errors have since been corrected, and this version may be considered the authoritative version of record.