Landau Theory Analysis

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 (Shieh et al., 2002; Andrault et al., 2003; 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 et al., 2011a; Marquardt et al., 2011b).

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 (Fig.

7.2). 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 ac-cordingly. 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 inter-nal 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 build-up of internal stresses in sintered polycrystalline stishovite and their partial relaxation at the phase transition.

7.4 Landau Theory Analysis

Further insight into the elastic behavior of sintered polycrystalline stishovite can be obtained by an analysis of the spontaneous strains e_i arising from the ferroelastic phase transition (Carpenter and Salje, 1998; Carpenter, 2006). Here we follow the Landau theory approach of Carpenter et al. (2000) with some small modifications. The Landau free energy expan-sion for the pseudo-proper ferroelastic phase transition (Wadhawan, 1982) from tetragonal stishovite to orthorhombic CaCl₂-type SiO₂ (P4₂/mnm P nnm) was given by Carpenter et al. (2000) as

with the driving order parameterQ, the critical pressureP_c, the Landau coefficientsaand b, the coupling coefficientsλ1–λ6, and the bare elastic constantsc_{i j}⁰. Due to coupling between spontaneous strains and the driving order parameter, the structure distorts at a transition pressure P_c^∗that is different from the critical pressure (Carpenter et al., 2000):

P_c^∗=P_c+ 2λ²₂

a(c₁₁⁰ −c₁₂⁰ ) (7.4)

The order parameter evolves with pressure as (Carpenter et al., 2000) Q²= a

b^∗(P_c^∗−P) (7.5)

(e₁−e₂) =

These expressions are identical to those given by Carpenter (2006) except for a typo cor-rection of c₃₃⁰ to c₁₃⁰ in the numerator of the formula for e₃. By symmetry, e₄ =e₅ =e₆ =0 in mechanical equilibrium. It can be seen from equation (7.3), however, that any imposed strain along the crystal axes or shear strain within thea-bplane may displace the transition pressure from its value under mechanical equilibrium. Non-equilibrium 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 con-stants (Carpenter et al., 2000; Carpenter, 2006). In contrast, we express the variation of bare elastic constants in terms of finite strain for a self-consistent treatment of thermody-namic quantities at high pressures (Stixrude and Lithgow-Bertelloni, 2005):

c_{i jkl}⁰ = (1+2f_E)^5/2

where c_{i jkl0}⁰ and c⁰⁰_{i jkl0} are the bare elastic constants and their pressure derivatives at ambi-ent conditions in full index notation, respectively, and δ^{i j}_kl =−δi jδkl−δilδjk−δjlδik. To keep equation (7.10) internally consistent, the bulk modulus is calculated from the elas-tic constants as the Voigt bound K₀ =c_iikk0⁰ /9 and accordingly K₀⁰ =c_iikk0⁰⁰ /9 (Stixrude and 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 ten-sor. 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 (7.10) merely to describe the dependence of elastic constants on finite volume strain.

7.4 Landau Theory Analysis

The elastic constants of stishovite and their pressure derivatives have previously been de-termined by Brillouin spectroscopy (Weidner et al., 1982; Brazhkin et al., 2005; Jiang et al., 2009) and computed from first principles (Karki et al., 1997a; Yang and Wu, 2014). Since the experimentally determined elastic constants c₁₁and 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)

c₁₁₀⁰ +c₁₂₀⁰ =c₁₁₀+c₁₂₀ (7.11) c₁₁₀⁰ −c₁₂₀⁰ = (c₁₁₀−c₁₂₀)P_c

P_c^∗ (7.12)

Linking equations (7.4)–(7.12) to the EOS (7.1) of stishovite (Table 7.2), the sponta-neous strains can be calculated as a function of finite strain, i.e. volume. Our approach is similar to previous formulations based on the Lagrangian definition of strain (Tröster et al., 2002; Tröster et al., 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 7.1) to derive spontaneous strains as (Carpenter et al., 2000)

e₁= a−a⁰

where the unit cell edge lengths of stishovite,a⁰andc⁰, were extrapolated to the experimen-tal pressures using the respective axial EOS (7.2) (Table 7.2). Including the EOS analysis described in section 7.3, 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 7.3.

As mentioned in section 7.3, the splitting of diffraction lines gives only a rough estimate for the transition pressure (Fig. 7.1). 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 (Kingma et al., 1995; Carpenter et al., 2000). The symmetry-breaking spontaneous strain(e₁−e₂)reflects the main structural distortion of the phase transition. (e₁−e₂)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 (Fig. 7.3a). Figure 7.3a 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 poly-crystalline 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 nonhy-drostatic compression (Fig. 7.3a). We therefore conclude that our experiments were not strongly affected by nonhydrostatic compression.

Figure 7.3: 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 (e₁−^e2)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).

To determine all relevant coefficients of equation (7.3) by least-squares fits of calculated to experimentally observed spontaneous strains, we fixed the transition pressures to the values derived from Figure 7.3a. 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), i.e. P_c−P_c^∗=50.7 GPa, and set b=11 (Carpenter et al., 2000; Carpenter, 2006). Since the pressure derivatives of the bare elastic constants c₁₁₀⁰ and c₁₂₀⁰ should be affected by elastic softening (Jiang et al., 2009) but cannot be simply obtained by combining the experimentally observed pressure derivatives, we fixedc₁₁₀⁰⁰ =4.9 to the value given by Jiang et al. (2009) and treatedc₁₂₀⁰⁰ as free parameter during the fitting procedure. Due to the limited pressure range of spontaneous strains observed on sintered polycrystalline CaCl₂-type SiO₂, however,c₁₂₀⁰⁰ 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 7.3 lists the final combinations of coefficients, and the predicted evolution of spontaneous strains is shown in Figure 7.3b. We note that our approach to derive Landau and coupling coefficients relies on a minimum of assumptions and is internally consistent.

7.5 Elasticity of Stishovite across the Ferroelastic Phase