The effect of replacing magnesia by alumina on the pressure-dependent structure of amorphous enstatite was investigated by applying in situ high-pressure neutron diffraction with magnesium isotope substitution to glassy (MgO)0.375(Al2O3)0.125(SiO2)0.5. The replacement leads to a factor of 2.4 increase in the rate-of-change of the Mg–O coordination number with pressure, which increases from 4.76(4) at ambient pressure to 6.51(4) at 8.2 GPa, and accompanies a larger probability of magnesium finding bridging oxygen atoms as nearest-neighbors. The Al–O coordination number increases from 4.17(7) to 5.24(8) over the same pressure interval at a rate that increases when the pressure is above ∼3.5 GPa. On recovering the glass to ambient conditions, the Mg–O and Al–O coordination numbers reduce to 5.32(4) and 4.42(6), respectively. The Al–O value is in accordance with the results from solid-state 27Al nuclear magnetic resonance spectroscopy, which show the presence of six-coordinated aluminum species that are absent in the uncompressed material. These findings explain the appearance of distinct pressure-dependent structural transformation regimes in the preparation of permanently densified magnesium aluminosilicate glasses. They also indicate an anomalous minimum in the pressure dependence of the bulk modulus with an onset that suggests a pressure-dependent threshold for transitioning between scratch-resistant and crack-resistant material properties.

Magnesia and alumina are essential components in a broad range of silicate materials, where important examples include commercial display glasses1,2 and crustal and mantle minerals.3,4 Here, there is great interest in the structural response of these systems to pressure, and how this response affects material properties, such as density, elastic constants, and the ability to flow.

Pressures in the gigapascal regime are readily generated on sharp-contact loading when glass is indented or scratched, and it is advantageous to design glass products that can resist surface deformation and also have a high fracture toughness, i.e., an ability to resist fracturing once a crack is formed.5–7 These requirements are, however, difficult to achieve simultaneously because of incompatible structural features: a high atomic packing density promotes hardness, whereas a low atomic packing density facilitates deformation by densification that can militate against the buildup of the high local stresses that stimulate crack propagation.8 Magnesium is reported to be a key ingredient for making glasses that overcome this trade-off in aluminosilicate9 and aluminoborosilicate10 systems.

High pressures are also generated in the Earth’s interior, where the CaO–MgO–Al2O3–SiO2 (CMAS) system is often used as a model for crustal and mantle minerals,3 and the glassy forms of these materials can be used as proxies for probing, e.g., the pressure-dependent behavior of dry basaltic melts.11,12 The magnesium end member enstatite (MgSiO3) is a common pyroxene with pressure- and temperature-dependent properties that are important for understanding the evolution of the Earth’s interior.13 It is one of the few silicate minerals that have been observed outside the Solar System, and its terrestrial abundance may well originate from interstellar dust grains, where the crystalline phase likely emerges from initially amorphous structural conformations.14 

Definitive experimental information on the glass structure under high pressure conditions is essential for establishing the structure–property relationships. There have, however, been few, if any, in situ experiments to investigate the effect of pressure on the structure of amorphous magnesium aluminosilicates. Most information on the coordination environment of aluminum emanates from solid-state 27Al nuclear magnetic resonance (NMR) experiments performed ex situ on densified glasses recovered to ambient from high pressure conditions, i.e., the experiments were performed on materials that had undergone structural relaxation during the recovery process.15–17 In general, solid state 25Mg NMR experiments yield little or no structural information on the coordination environment of magnesium in amorphous aluminosilicates because of the low natural abundance (10%), low gyromagnetic ratio, and significant quadrupole moment (nuclear spin I = 5/2) of the 25Mg isotope,18 and the broad distribution of electric field gradients at the different sites of the Mg2+ ion that originates from structural disorder.19,20 Recently, however, high field solid-state 25Mg NMR experiments have been performed on isotopically enriched CaMgSi2O6 (diopside), Na2MgSi3O8, and K2MgSi5O12 glasses.21 The results were interpreted in terms of distinct four- and six-coordinated magnesium sites, although the reported mean Mg–O coordination number n̄MgO 5.1 for diopside is 16% larger than the value n̄MgO = 4.40(4) found from experiments using neutron diffraction with magnesium isotope substitution.20 The effect of pressure on the 17O, 25Mg, and 29Si NMR parameters for glassy MgSiO3 has recently been investigated by density functional theory.22 Conventional neutron or x-ray diffraction experiments on magnesium aluminosilicate glasses show a considerable overlap of the nearest-neighbor Mg–O and Al–O peaks in the measured pair-distribution functions because of the similarity between the Mg–O and Al–O bond distances.

In this paper, we apply in situ high-pressure neutron diffraction with magnesium isotope substitution to investigate the structure of (MgO)0.375(Al2O3)0.125(SiO2)0.5 (or MgAS_37.5_50) glass over the pressure range from ambient to 8.2 GPa. This technique was chosen because it delivers site-specific information on the coordination environment of magnesium,20 allowing for the removal of overlap between the nearest-neighbor Mg–O and Al–O correlations. The technique is also combined with solid-state 27Al NMR spectroscopy to measure the structure of the glass recovered to ambient conditions. The MgAS_37.5_50 glass composition corresponds to enstatite or (MgO)0.5(SiO2)0.5 after 25 mol. % of the magnesia is replaced by alumina. Enstatite is the alumina-free end-member composition along the 50 mol. % silica tie-line in the ternary MgO–Al2O3–SiO2 system, and the pressure-dependent structure of the glass has been studied extensively.11,23–25 It, therefore, provides a reference system for investigating the effect on the glass structure of replacing magnesia by alumina. Under ambient conditions, enstatite forms a partially polymerized glass network in which the number of non-bridging oxygen (NBO) atoms per tetrahedron NNBO/NT = 2, as verified by 29Si magic angle spinning (MAS) NMR experiments.2,19,26,27 The substitution of magnesia by alumina leads to an increased network connectivity, where NNBO/NT ≃ 0.85 is expected for MgAS_37.5_50 from the GYZAS model of Ref. 28 with parameter p = 0.77, which takes into account the presence of aluminum ions with a coordination number greater than 4. This decrease in the NBO content reflects the need for some of the Mg2+ ions to adopt a charge compensating role when alumina is added.

This paper is organized as follows: A précis of the diffraction theory described in Ref. 20 is provided in Sec. II, and the experimental methods are described in Sec. III. Although in situ high-pressure neutron diffraction with isotope substitution has successfully been applied to several liquid and glassy systems,29–31 its application to glassy MgAS_37.5_50 presents a challenge because of the small coherent scattering length contrast between the isotopes of magnesium.20 The results are presented in Sec. IV, where “recovered” will refer to the densified glass recovered to ambient conditions from a pressure of 8.2 GPa. The findings are discussed in Sec. V, and the conclusions are drawn in Sec. VI.

The total structure factor measured in a neutron diffraction experiment is given by32 
(1)
where k is the magnitude of the scattering vector, cα is the atomic fraction of chemical species α, bα is the bound coherent neutron scattering length of chemical species α, and Sαβ(k) is the Faber–Ziman partial structure factor for the chemical species α and β.
Let natF(k) and 25F(k) denote the structure factors measured in diffraction experiments on two glasses that are identical in every respect, except that one contains magnesium of natural isotopic abundance natMg and the other is isotopically enriched with 25Mg. The difference function
(2)
where ΔbMg=bMgnatbMg25, eliminates all the pair-correlation functions not involving magnesium. Similarly, the difference function
(3)
eliminates all the Mg–α correlations with α ≠ Mg.
The real-space functions D(r), ΔDMg(r), and ΔD(r), corresponding to F(k), ΔFMg(k), and ΔF(k), respectively, are obtained by Fourier transformation. For example, the total pair-distribution function is given by
(4)
where r is a distance in real space and ⊗ is the one-dimensional convolution operator. The normalization factor b2=(αcαbα)2 is given by the modulus of Eq. (1) after all the Sαβ(k) functions are set to zero. The window function M(k) is given by M(k) = 1 for kkmax and M(k) = 0 for k > kmax, where kmax is the maximum k-value and M(r) is its real-space manifestation. A step window function was employed because it leads to sharper peaks in D′(r) as compared, for example, to a Lorch33,34 window function. If the oscillations in F(k) do not extend beyond kmax, Eq. (4) will deliver the unmodified total pair-distribution function,
(5)
where ρ is the atomic number density and gαβ(r) is the partial pair-distribution function for the chemical species α and β. For the difference function involving only the magnesium correlations,
(6)
where
(7)
in which the normalization factor B is given by the modulus of Eq. (2) after all the Sαβ(k) functions are set to zero. If only the Mg–O correlations contribute toward the first peak in ΔDMg(r), the Mg–O coordination number can be found from
(8)
where r1 and r2 define the overall extent of the peak. For the other difference function,
(9)
where
(10)
in which the normalization factor C is given by the modulus of Eq. (3) after all the Sαβ(k) functions are set to zero. If only the Si–O and Al–O correlations contribute toward the first peak in ΔD(r) and the Si–O coordination number n̄SiO is known, the Al–O coordination number can be found from
(11)

The partial pair-distribution function gαβ(r) is zero at r-values below the distance of closest approach between two atoms of chemical species α and β. In consequence, D(r) = ΔDMg(r) = ΔD(r) = −4πρr at small r-values, i.e., all three functions follow the so-called density line. For normalized datasets, oscillations around this density line will originate from the window function M(r) and the finite counting statistics. If these oscillations are replaced by the density line values and the function is Fourier-transformed to k-space, the measured dataset and back-Fourier transform should be in accord at all k-values, indicating the absence of major systematic errors.35 

In the data analysis procedure, each peak i in rgαβ(r) was represented by the Gaussian function,
(12)
where rαβi, σαβi, and n̄αβ(i) are the peak position, standard deviation, and coordination number of chemical species β around α, respectively. A measured r-space function was fitted to a suitable sum of these Gaussian peaks, convoluted with M(r), following the protocol described in Ref. 36. The parameter Rχ was used to assess the goodness-of-fit.37 

The MgAS_37.5_50 glasses containing either natMgO (Aldrich, ≥99.99%) or 25MgO (Isoflex, 0.32% 24Mg, 99.38% 25Mg, 0.30% 26Mg) with Al2O3 (Sigma-Aldrich, 99.998%) and SiO2 (Alfa Aesar, 99.9%) were prepared using the method described in Ref. 20. The procedure led to a negligible mass loss, so the glass composition was taken to be the batch composition. Helium pycnometry gave the same number density ρ = 0.0804(1) Å−3 for both uncompressed samples, and 27Al MAS NMR experiments gave the same aluminum speciation for both samples (Sec. IV A). Isotope substitution will not change the electronic structure of the glass, and, accordingly, the measured x-ray diffraction patterns for both uncompressed samples were identical within the experimental error.20 Helium pycnometry gave ρ = 0.0853(1) Å−3 for the densified glasses as measured ∼31 days after the samples were recovered from 8.2 GPa to ambient conditions.

The 27Al MAS NMR experiments on the MgAS_37.5_50 samples recovered to ambient conditions from 8.2 GPa were performed ∼33 weeks after the samples were decompressed. The spectra were collected at a magnetic field strength of 16.4 T using a commercial spectrometer and a 3.2 mm MAS NMR probe (Agilent Technologies). The resonance frequency for 27Al at this field strength was 182.3374 MHz. The glass samples were crushed and loaded into 3.2 mm zirconia rotors, which were spun at a computer-controlled spinning frequency of 22.0 kHz. The 27Al frequency was referenced to an external standard of aqueous aluminum nitrate at 0.0 ppm.

The single pulse, or Bloch decay, experiments were performed using a π/12 pulse width of 0.6 µs, a recycle delay of 4 s, and signal averaging of 1000 scans. The spectra were Fourier-transformed and phase-corrected using a commercial software application, without apodization. The spectral fitting was performed in DMfit,39 making use of the Czjzek function to reproduce second-order quadrupolar line shapes for the four-coordinated Al(IV), five-coordinated Al(V), and six-coordinated Al(VI) aluminum species. A small mixed Gaussian/Lorentzian peak was included in the fits to account for a minor Al(VI) signal originating from aluminum in the zirconia rotor.

The triple-quantum magic-angle spinning (3QMAS) NMR data were collected using a hypercomplex 3QMAS pulse sequence with a Z filter.40 The solid 3π/2 and π/2 pulse widths were optimized to 3.0 and 1.1 µs, respectively. A lower power π/2 pulse width of 15 µs was used as the soft reading pulse of the Z filter, following a storage period of 45.5 µs (one rotor cycle). The 27Al 3QMAS NMR spectra were typically obtained using 480 acquisitions at each of 48–128 t1 points, with a recycle delay of 2 s. The datasets were Fourier-transformed and phase-corrected (sheared) using a commercial software application, incorporating line broadening of 100 and 50 Hz in the MAS and isotropic dimensions, respectively. Plots of the two-dimensional data were generated using positive intensity contour maps and with projections of the data onto the MAS and isotropic shift axes (top and left, respectively).

The neutron diffraction experiments were performed using the diffractometer D4c at the Institut Laue-Langevin.41 This instrument was chosen because of its high incident flux and its ability to measure diffraction patterns with excellent reproducibility and count-rate stability over a wide k-range.42 

The diffraction patterns for the uncompressed samples were measured at room temperature (≃298 K) and ambient pressure using an incident neutron wavelength λ = 0.4958(1) Å with the samples contained in a cylindrical vanadium container. Full experimental details are given in Ref. 20, and the data analysis followed the procedure described in Ref. 35.

The in situ high-pressure diffraction experiments were performed at ambient temperature using a type VX5 Paris-Edinburgh press equipped with cubic BN anvils having a single toroid profile.43 The incident neutron wavelength was λ = 0.4964(1) Å. Finely powdered samples, of the mass required to ensure an ambient pressure packing fraction of 100%, were shaped for the anvils by using a pellet press. Each sample was supported by gaskets made from the alloy Ti0.676Zr0.324, and the equation of state (EOS) for this alloy is given in Ref. 44. The volume of an uncompressed sample defined by the anvil and gasket geometry was ≃91.2 mm3.45 The pressure exerted on the sample and the associated error was found from the load applied to anvils with an identical single toroid profile mounted in the same type of Paris-Edinburgh press using a calibration curve that has been checked extensively.45 For this setup, the pressure gradient in the volume of the sample illuminated by the neutron beam is expected to be ≲2% at a pressure of 6 GPa.46 

Diffraction patterns were measured for the natMg containing sample in its gasket at pressures of 1.7(5), 3.0(5), 3.9(5), 5.4(5), 7.1(5), and 8.2(5) GPa and for the 25Mg containing sample in its gasket at pressures of 3.9(5) and 8.2(5) GPa. Diffraction patterns were also measured for both samples in the anvils of the press at ambient pressure after decompression from 8.2 GPa. In addition, diffraction patterns were measured for an uncompressed empty gasket and several empty gaskets recovered from different high pressure conditions to estimate the gasket scattering under load, the empty anvils to help estimate the background scattering, and different sized vanadium pellets in the anvils of the press to aid in the data normalization. The sample-in-gasket counting time was ∼22 h for each glass at 3.9 GPa, ∼21 h for each glass at 8.2 GP, ∼9 h for each of the recovered glasses, and ∼7–8 h for each glass at the other pressure points. The relative sample-in-gasket and empty gasket counting times were optimized to minimize the statistical error on the gasket-corrected intensity.47 

In general, it was found that a simplification of the data analysis scheme described in Ref. 48 led to a reduction in the residual slope on a corrected F(k) function, thus minimizing the Fourier transform artifacts associated with the corresponding D′(r) function. Essentially, a combination of empty gasket intensities Ig(θ) was subtracted from the measured sample-in-gasket intensity Isg(θ), and the resultant dataset was then scaled by a factor a such that it oscillates about the calculated self-scattering limit given by the second term on the right-hand side of the equation,
(13)
where 2θ is the scattering angle, k = (4π/λ)sin θ, limk→∞F(k) = 0, binc,α is the bound incoherent neutron scattering length of chemical species α, and Pα(k) originates from inelastic scattering from chemical species α.49 On forming a difference function ΔFMg(k), subtle refinements to Ig(θ) led to real-space functions ΔDMg(r) that oscillate about the low-r density-line limit with minimal deviation.

The coherent neutron scattering lengths are bAl = 3.449(5) fm, bSi = 4.1491(10) fm, bO = 5.803(4) fm,50, bMgnat = 5.375(4) fm,51 and, taking into account the isotopic enrichment, bMg25 = 3.729(12) fm. The latter was obtained from a re-evaluation of the coherent scattering length of the 25Mg isotope using neutron powder diffraction.20 

The fitted single pulse 27Al MAS NMR spectra for the recovered samples are shown in Fig. 1. The fitted parameters are listed in Table I, where they are compared with those measured for the uncompressed glass.20 The results for either the uncompressed or recovered glasses are in agreement within the experimental error. Relative to the uncompressed material, there is a smaller population of Al(IV) species and an enhanced population of Al(V) species in the recovered glass. The densified glass also contains Al(VI) species, which are absent in the uncompressed material.

FIG. 1.

27Al MAS NMR spectra (black curves) measured for the recovered MgAS_37.5_50 glasses containing natMg or 25Mg. In each panel, the orange curve represents the background measured for the empty rotor, and the Czjzek fit to the spectral components is represented by the blue curve for Al(IV), the green curve for Al(V), and the magenta curve for Al(VI). The sum of the fitted functions is represented by the red curve, which overlays the black curve at most values of δMAS.

FIG. 1.

27Al MAS NMR spectra (black curves) measured for the recovered MgAS_37.5_50 glasses containing natMg or 25Mg. In each panel, the orange curve represents the background measured for the empty rotor, and the Czjzek fit to the spectral components is represented by the blue curve for Al(IV), the green curve for Al(V), and the magenta curve for Al(VI). The sum of the fitted functions is represented by the red curve, which overlays the black curve at most values of δMAS.

Close modal
TABLE I.

Parameters obtained from Czjzek fits to the 27Al MAS NMR spectra for the uncompressed and recovered glasses. The parameters correspond to the average isotropic chemical shift δiso, full width at half maximum (FWHM) of a Gaussian distribution of these shifts Δδ(27Al), average quadrupolar coupling constant |CQ| (defined in the sense of a modulus as per Ref. 38), relative integrated peak area I, and average coordination number n̄AlO. The latter was calculated using the full precision of the values found for the speciation from the data analysis.

GlassConditionSiteδiso (±1 ppm)Δδ(27Al) (±1 ppm)|CQ| (±0.1 MHz)I (±2%)n̄AlOReference
natMgAS_37.5_50 Uncompressed Al(IV) 63 15.8 7.9 84 4.16(7) 20  
Al(V) 38 23.5 8.1 16 
Al(VI) ⋯ ⋯ ⋯ ⋯ 
25MgAS_37.5_50 Uncompressed Al(IV) 64 16.2 7.9 83 4.17(7) 20  
Al(V) 38 20.6 8.1 17 
Al(VI) ⋯ ⋯ ⋯ ⋯ 
natMgAS_37.5_50 Recovered Al(IV) 66 16.5 9.7 67 4.39(6) This work 
Al(V) 36 14.2 9.1 28 
Al(VI) 12.4 6.3 
25MgAS_37.5_50 Recovered Al(IV) 66 17.2 9.4 63 4.44(6) This work 
Al(V) 36 15.6 8.9 31 
Al(VI) 12.5 6.4 
GlassConditionSiteδiso (±1 ppm)Δδ(27Al) (±1 ppm)|CQ| (±0.1 MHz)I (±2%)n̄AlOReference
natMgAS_37.5_50 Uncompressed Al(IV) 63 15.8 7.9 84 4.16(7) 20  
Al(V) 38 23.5 8.1 16 
Al(VI) ⋯ ⋯ ⋯ ⋯ 
25MgAS_37.5_50 Uncompressed Al(IV) 64 16.2 7.9 83 4.17(7) 20  
Al(V) 38 20.6 8.1 17 
Al(VI) ⋯ ⋯ ⋯ ⋯ 
natMgAS_37.5_50 Recovered Al(IV) 66 16.5 9.7 67 4.39(6) This work 
Al(V) 36 14.2 9.1 28 
Al(VI) 12.4 6.3 
25MgAS_37.5_50 Recovered Al(IV) 66 17.2 9.4 63 4.44(6) This work 
Al(V) 36 15.6 8.9 31 
Al(VI) 12.5 6.4 

The high-resolution 27Al 3QMAS NMR spectra provide additional evidence for the identity of the Al polyhedra present in these glasses (Fig. 2). The uncompressed glass contains clear resonances from Al(IV) and Al(V) and only a very weak signal with coordinates corresponding to Al(VI). As indicated by the number of contours and by the relative intensities of the peaks in the isotropic projection to the left of these data, aluminum in the uncompressed glass is largely found in a tetrahedral coordination environment, consistent with the analysis of the 27Al MAS NMR data. As discussed in Sec. III B, measurement of an empty 3.2 mm zirconia rotor confirmed the weak Al(VI) peak to originate from the sample holder and not from aluminum in the glass sample.

FIG. 2.

27Al 3QMAS NMR spectra measured for (a) the uncompressed (natMgO)0.3837(Al2O3)0.1177(SiO2)0.4987 glass investigated in Ref. 28 and (b) the recovered natMgAS_37.5_50 glass investigated in this work. The contours originating from the different aluminum species are distinguished from those originating from the spinning side bands and, in (a), the zirconia rotor.

FIG. 2.

27Al 3QMAS NMR spectra measured for (a) the uncompressed (natMgO)0.3837(Al2O3)0.1177(SiO2)0.4987 glass investigated in Ref. 28 and (b) the recovered natMgAS_37.5_50 glass investigated in this work. The contours originating from the different aluminum species are distinguished from those originating from the spinning side bands and, in (a), the zirconia rotor.

Close modal

Upon pressurization and subsequent recovery to ambient conditions, the 27Al 3QMAS NMR spectra change considerably. In addition to the Al(IV) and Al(V) peaks seen in the uncompressed glass, a much more intense peak from Al(VI) is evident in the 2D contour plot and the isotropic projection. In both recovered glasses, the intensity of this peak is much larger than the rotor background and can be attributed to an overall increase in the aluminum coordination number at high pressures. Although the isotropic projections from these 3QMAS NMR data are not strictly quantitative due to differences in the 27Al quadrupolar coupling (see Table I) and the inability to uniformly excite all aluminum environments with this type of NMR experiment, the relative peak intensities for Al(IV), Al(V), and Al(VI) are in general agreement with the fitted results listed in Table I, confirming a permanent increase in the aluminum coordination number under the pressure used in this study.

1. Total structure factors

The pressure dependence of the measured F(k) functions is shown in Figs. 3 and 4. With increasing pressure, there is a notable shift in the position of the first sharp diffraction peak (FSDP) from kFSDP ≃ 1.8 Å−1 under ambient conditions toward a larger k-value and a reduction in the peak height. On recovery of the glass to ambient conditions from 8.2 GPa, the FSDP shifts to a smaller k-value and regains some of its intensity. These changes indicate alterations to the intermediate range order of the glass network as the material is first compacted and then allowed to relax.52 

FIG. 3.

Pressure dependence of the measured F(k) functions for the natMgAS_37.5_50 glass. For each dataset, the black vertical error bars represent the measured function and the red curve represents the back-Fourier transform of the corresponding D′(r) function (Figs. 1214) after the low-r oscillations, below the distance of closest approach between two atoms, are set to the theoretical density-line limit. The green dashed line represents the position of the FSDP at ambient pressure.

FIG. 3.

Pressure dependence of the measured F(k) functions for the natMgAS_37.5_50 glass. For each dataset, the black vertical error bars represent the measured function and the red curve represents the back-Fourier transform of the corresponding D′(r) function (Figs. 1214) after the low-r oscillations, below the distance of closest approach between two atoms, are set to the theoretical density-line limit. The green dashed line represents the position of the FSDP at ambient pressure.

Close modal
FIG. 4.

Measured F(k) functions for the MgAS_37.5_50 glass containing either natMg or 25Mg at ambient pressure, 3.9 GPa, 8.2 GPa, or recovered to ambient conditions from 8.2 GPa. For each dataset, the black vertical error bars represent the measured function, and the red curve represents the back-Fourier transform of the corresponding D′(r) function (Figs. 12 and 13) after the low-r oscillations, below the distance of closest approach between two atoms, are set to the theoretical density-line limit.

FIG. 4.

Measured F(k) functions for the MgAS_37.5_50 glass containing either natMg or 25Mg at ambient pressure, 3.9 GPa, 8.2 GPa, or recovered to ambient conditions from 8.2 GPa. For each dataset, the black vertical error bars represent the measured function, and the red curve represents the back-Fourier transform of the corresponding D′(r) function (Figs. 12 and 13) after the low-r oscillations, below the distance of closest approach between two atoms, are set to the theoretical density-line limit.

Close modal

2. Equation of state

In x-ray and neutron diffraction experiments on amorphous materials, it is found that the scaled position of the FSDP kFSDP/kFSDP0 increases with the reduced density ρ/ρ0 = V0/V, where V denotes the volume and the superscript/subscript “0” refers to a parameter value at ambient pressure.68, Figure 5 shows the results for glassy SiO2,63–66 CaSiO3,69 (MgO)0.62(SiO2)0.38,70 GeO2,29,48,71,72 and B2O373,74 under compression at room temperature and for molten CaSiO3 and MgSiO3 under compression at temperatures in the range ≃1873–2390 K,75 for density regimes in which the ambient pressure tetrahedral SiO4 or GeO4 or planar triangular BO3 structural motifs are retained. The datasets are found to lie on a common curve described by53 
(14)
The pressure-dependent density of the MgAS_37.5_50 glass was estimated from the measured kFSDP values using Eq. (14).
FIG. 5.

Reduced density dependence of the scaled FSDP position for glassy and liquid network forming oxides in which the ambient pressure tetrahedral SiO4 or GeO4 or planar triangular BO3 structural motifs are retained over the plotted density range. The black solid curve shows the quadratic fit given by Eq. (14) with a goodness-of-fit parameter R2 = 0.999. The plot is adapted from Ref. 53.

FIG. 5.

Reduced density dependence of the scaled FSDP position for glassy and liquid network forming oxides in which the ambient pressure tetrahedral SiO4 or GeO4 or planar triangular BO3 structural motifs are retained over the plotted density range. The black solid curve shows the quadratic fit given by Eq. (14) with a goodness-of-fit parameter R2 = 0.999. The plot is adapted from Ref. 53.

Close modal

The resultant pressure–volume EOS is shown in Fig. 6(a), where it is compared to the equations of state for the aluminosilicate glasses (MgO)0.4878(Al2O3)0.0244(SiO2)0.4878 (MgAS_48.8_48.8) and (MgO)0.4454(Al2O3)0.1091(SiO2)0.4454 (MgAS_44.5_44.5) from the Brillouin scattering experiments of Wei et al.54 In these experiments, the glass density was obtained from the measured sound velocities by assuming elastic deformation. Given their compositional similarity, the combined datasets for MgAS_48.8_48.8 and MgAS_44.5_44.5 were fitted to a common third order Birch–Murnaghan EOS,76 which gave the zero-pressure bulk modulus K0 and pressure derivative K0 values listed in Table II. The large errors on the fitted parameters may be related to the absence of data points in the ∼2.2–7.7 GPa regime. Here, and in the following text, a distinction is not made between the adiabatic bulk modulus KS, as measured in sound velocity experiments, and the isothermal bulk modulus KT, i.e., it is assumed that γKS/KT ≃ 1. The EOS for MgAS_37.5_50 is in accordance with the Brillouin scattering results at low pressures but deviates markedly with increasing pressure, approaching the EOS measured for MgSiO3 in x-ray absorption experiments.55 

FIG. 6.

Pressure–volume EOS at room temperature for glassy (a) magnesium aluminosilicates, (b) MgSiO3 and Mg2SiO4, and (c) SiO2. In (a), the blue triangles for MgAS_48.8_48.8 and MgAS_44.5_44.5 are from Brillouin scattering experiments54 and the blue dashed curve shows a fit to both datasets using a third order Birch–Murnaghan (BM) EOS. The magenta triangles for MgAS_37.5_50 are from the measured FSDP positions (the present work), and the magenta dashed curve shows a fit using a fourth order BM EOS. The black solid and dashed curves show the BM EOSs for MgSiO3 shown in (b).55 In (b), the red squares and black circles for MgSiO3 are from Brillouin scattering56 and x-ray absorption55 experiments, respectively. The red solid curve shows a fit using a third order BM EOS, and the black solid and dashed curves show fits using a fourth order BM EOS over the pressure ranges of 0–127 and 0–6 GPa, respectively. The magenta triangles for MgSiO3 are from the measured FSDP positions,11 and the magenta dashed curve shows a fit using a fourth order BM EOS. For Mg2SiO4, the green squares show the EOS obtained from the measured area of a very thin piece of glass57 and the green solid curve shows a fit using a fourth order BM EOS. In (c), the black squares are from the linear strain measured using an optical method,58 the black circles are from x-ray absorption experiments,59,60 the black open triangles are from ultrasonic interferometry experiments,61 and the black dashed curve is from Brillouin scattering experiments.62 The magenta symbols are from the measured FSDP positions in x-ray63–65 and neutron66 diffraction work. The parameters for the fitted BM EOSs are listed in Table II.

FIG. 6.

Pressure–volume EOS at room temperature for glassy (a) magnesium aluminosilicates, (b) MgSiO3 and Mg2SiO4, and (c) SiO2. In (a), the blue triangles for MgAS_48.8_48.8 and MgAS_44.5_44.5 are from Brillouin scattering experiments54 and the blue dashed curve shows a fit to both datasets using a third order Birch–Murnaghan (BM) EOS. The magenta triangles for MgAS_37.5_50 are from the measured FSDP positions (the present work), and the magenta dashed curve shows a fit using a fourth order BM EOS. The black solid and dashed curves show the BM EOSs for MgSiO3 shown in (b).55 In (b), the red squares and black circles for MgSiO3 are from Brillouin scattering56 and x-ray absorption55 experiments, respectively. The red solid curve shows a fit using a third order BM EOS, and the black solid and dashed curves show fits using a fourth order BM EOS over the pressure ranges of 0–127 and 0–6 GPa, respectively. The magenta triangles for MgSiO3 are from the measured FSDP positions,11 and the magenta dashed curve shows a fit using a fourth order BM EOS. For Mg2SiO4, the green squares show the EOS obtained from the measured area of a very thin piece of glass57 and the green solid curve shows a fit using a fourth order BM EOS. In (c), the black squares are from the linear strain measured using an optical method,58 the black circles are from x-ray absorption experiments,59,60 the black open triangles are from ultrasonic interferometry experiments,61 and the black dashed curve is from Brillouin scattering experiments.62 The magenta symbols are from the measured FSDP positions in x-ray63–65 and neutron66 diffraction work. The parameters for the fitted BM EOSs are listed in Table II.

Close modal
TABLE II.

Parameters for the fitted third or fourth order Birch–Murnaghan equations of state for vitreous silica, magnesium silicates, and magnesium aluminosilicates. The datasets are from optical measurements of the linear strain (LS), x-ray absorption (XRA), Brillouin scattering (BS), or the FSDP position via Eq. (14). The bulk modulus K0,BS obtained directly from the sound velocities measured in Brillouin scattering experiments is also listed. The majority of the fitted functions are shown in Figs. 6 and 7.

GlassK0 (GPa)K0K0 (GPa−1)Fit range (GPa)MethodK0,BS (GPa)References
SiO2 37.0(5.5) −5.6(6.2) ⋯ 0–2.29a LS 36.8 58 and 67  
32.4(2.8) −1.2(2.4) 0.51(21) 0–6 LS, XRA ⋯ 58–60 and the present work 
MgSiO3 69.1(1.1) 5.1(1.2) ⋯ 0–14.7 BS 78.4(6) 11 and 56  
16.9(3.2) 5.9(1.3) −0.004(0.77) 0–127 XRA ⋯ 55  
50.8(7.6) −13.7(1.5) −2.5(5) 0–6 XRA ⋯ 55 and the present work 
68.3(8.9) −6.1(2.3) −0.47(21) 0–8.2 FSDP ⋯ 11 and the present work 
Mg2SiO4 33.3(15.8) −1.7(8.8) 1.4(1.7) 0–12.9 LS ⋯ 57  
MgAS_37.5_50 71(3) −7.0(5) −0.72(6) 0–8.2 FSDP ⋯ The present work 
MgAS_48.8_48.8/ 43(18) 46(26) ⋯ 0–23.6 BS 85.4(3)–89(1) 54  
MgAS_44.5_44.5        
GlassK0 (GPa)K0K0 (GPa−1)Fit range (GPa)MethodK0,BS (GPa)References
SiO2 37.0(5.5) −5.6(6.2) ⋯ 0–2.29a LS 36.8 58 and 67  
32.4(2.8) −1.2(2.4) 0.51(21) 0–6 LS, XRA ⋯ 58–60 and the present work 
MgSiO3 69.1(1.1) 5.1(1.2) ⋯ 0–14.7 BS 78.4(6) 11 and 56  
16.9(3.2) 5.9(1.3) −0.004(0.77) 0–127 XRA ⋯ 55  
50.8(7.6) −13.7(1.5) −2.5(5) 0–6 XRA ⋯ 55 and the present work 
68.3(8.9) −6.1(2.3) −0.47(21) 0–8.2 FSDP ⋯ 11 and the present work 
Mg2SiO4 33.3(15.8) −1.7(8.8) 1.4(1.7) 0–12.9 LS ⋯ 57  
MgAS_37.5_50 71(3) −7.0(5) −0.72(6) 0–8.2 FSDP ⋯ The present work 
MgAS_48.8_48.8/ 43(18) 46(26) ⋯ 0–23.6 BS 85.4(3)–89(1) 54  
MgAS_44.5_44.5        
a

Corresponds to the Eulerian strain range 0fE=[V/V02/31]/20.039.58 

In Fig. 6(b), the EOS for MgSiO3 glass, found from the kFSDP values measured in the neutron diffraction experiments,11 is compared to the equations of state found from the Brillouin scattering experiments of Sanchez-Valle and Bass56 and the x-ray absorption measurements of Petitgirard et al.55 The parameters obtained from the fitted third or fourth order Birch–Murnaghan EOSs are listed in Table II, where K0 denotes the double pressure derivative of K0. As for glassy MgAS_37.5_50, the EOS for MgSiO3 found from the measured kFSDP values is in accord with the Brillouin scattering results at low pressures but deviates markedly with increasing pressure, approaching the EOS found from the x-ray absorption experiments.55 According to the Brillouin scattering experiments, the addition of magnesia to silica to produce enstatite increases the bulk modulus K0,BS of the glass, and the replacement of magnesia by alumina to produce MgAS_37.5_50 may result in a further increase (Table II). In both cases, the additions lead to an increased stiffness to initial deformation as compared to silica.

In Fig. 6(b), the EOS for glassy Mg2SiO4 is also presented, where the sample volume was found from the measured area of a very thin piece of gold coated glass in a diamond anvil cell.57 The results show that the densification of this material is larger than that observed for enstatite glass in the Brillouin scattering work.

In Fig. 6(c), the EOS for silica glass estimated from the measured kFSDP values in x-ray63–65 and neutron66 diffraction experiments is compared to the EOSs found from optical measurements of the linear strain,58 x-ray absorption,60 Brillouin scattering,62 and ultrasonic interferometry.61 For the investigated pressure range, the results from all the experimental methods are in overall accord: The deformation in silica glass is elastic up to a pressure around 9–10 GPa.58,67 At larger pressures, plastic deformation will result in permanent densification, leading to a breakdown of the assumptions in which reliable density values can be obtained from Brillouin scattering or ultrasonic interferometry experiments.61,62 As illustrated in Fig. 7, a gradual transition is observed between the EOS for elastically deformed silica glass, in which the silicon atoms are tetrahedrally coordinated,66 and the EOS found for plastically deformed six-coordinated silica glass, in which the silicon atoms are octahedrally coordinated.60 

FIG. 7.

Pressure–volume EOS at room temperature for glassy SiO2 over an extended pressure range. The black squares are from the linear strain measurements of Meade and Jeanloz,58 and the black circles are from the x-ray absorption experiments of Sato and Funamori.59,60 The black solid curve shows a fit using a fourth order Birch–Murnaghan EOS for the pressure range 0–6 GPa in which the deformation is elastic (Table II). The fitted EOS delivers a broad minimum in the pressure dependence of the bulk modulus at ≃1.5 GPa (see Sec. V B for a discussion). The blue solid curve shows a fit to a third order Birch–Murnaghan EOS for the plastically deformed glass when the Si–O coordination number n̄SiO = 6: K0 = 190 GPa, K0 = 4.5, and the zero-pressure density ρ0 = 3.88 g cm−3.60 

FIG. 7.

Pressure–volume EOS at room temperature for glassy SiO2 over an extended pressure range. The black squares are from the linear strain measurements of Meade and Jeanloz,58 and the black circles are from the x-ray absorption experiments of Sato and Funamori.59,60 The black solid curve shows a fit using a fourth order Birch–Murnaghan EOS for the pressure range 0–6 GPa in which the deformation is elastic (Table II). The fitted EOS delivers a broad minimum in the pressure dependence of the bulk modulus at ≃1.5 GPa (see Sec. V B for a discussion). The blue solid curve shows a fit to a third order Birch–Murnaghan EOS for the plastically deformed glass when the Si–O coordination number n̄SiO = 6: K0 = 190 GPa, K0 = 4.5, and the zero-pressure density ρ0 = 3.88 g cm−3.60 

Close modal

Discrepancies between the equations of state obtained from different experimental techniques can originate from the frequency dependence of the bulk modulus and the measurement timescale associated with each technique. The bulk modulus of a material is dependent on the frequency because the response of its volume to an applied pressure will depend on the timescale on which the material can respond. The measurement timescale associated with a given probe will determine whether that response can be observed.

In the case of enstatite, the x-ray absorption experiment gives a large pressure-induced volume change and K0 = 16.9(3.2) GPa,55 whereas the Brillouin scattering experiment gives a smaller pressure-induced volume change and a larger value K0 = 78.4(6) GPa56 [Fig. 6(b)]. In the x-ray absorption work, the material is given a long time to respond to the applied pressure, so plastic deformation (via densification and/or shear flow) can be observed. By contrast, the measurement timescale associated with the Brillouin scattering experiments is too short to observe such relaxation.25 At the lowest pressures, the data points from the x-ray absorption experiment are closer to those found from Brillouin scattering than suggested by the EOS fitted over the extended pressure range 0–127 GPa [Fig. 6(b)]. A revised fit over the reduced pressure range 0–6 GPa provides a value K0 = 50.8(7.6) GPa closer to the Brillouin scattering result (Table II). The EOS for enstatite from the measured kFSDP values captures the transition between elastic deformation at low pressures and plastic deformation at high pressures. We note that in the technique used to measure the EOS for glassy Mg2SiO4,57 the material is given time to relax on the measurement timescale.

Similarly, the Brillouin scattering experiments on glassy MgAS_48.8_48.8 and MgAS_44.5_44.5 are likely to underestimate the pressure-induced volume change at high pressures. There is, however, an overall accord with the results for glassy MgAS_37.5_50 in the lowest pressure regime, where the deformation will be predominantly elastic and, therefore, less sensitive to the longer observation timescales necessary to observe plastic deformation [Fig. 6(a)]. In the case of silica, the deformation mechanism is elastic until a threshold pressure around 9–10 GPa is achieved58,67 [Fig. 6(c)].

3. Pressure-dependent glass structure

The difference functions ΔFMg(k) obtained from the total structure factors of Fig. 4 are shown in Fig. 8. The datasets for (i) the compacted glasses at 3.9 and 8.2 GPa and (ii) the recovered glass are less precise than their uncompressed counterpart, as befits the measurement of diffraction patterns for small samples in a Paris-Edinburgh press. For these pressure points, there were originally oscillations that extended beyond k ≃ 10 Å−1, with a periodicity matching that of the uncompressed Ti0.676Zr0.324 gasket under ambient conditions. These features may have originated from a small deviation in the scattering observed from the gaskets used for the natMg and 25Mg samples and were removed by subtracting a scaled version of the empty gasket diffraction pattern. Such features did not exist for the uncompressed sample measured in the vanadium container. As shown in Fig. 8, each of the measured ΔFMg(k) functions has the same overall features. There is also good agreement between a measured dataset and the back-Fourier transform of its corresponding ΔDMg(r) function (Fig. 9) after the low-r oscillations are set to the density-line limit, which indicates the absence of major systematic errors.20 The datasets measured using the Paris-Edinburgh press were truncated at kmax ≲ 13 Å−1 (Table III) because of the small statistical precision at larger k-values. This truncation also has the effect of minimizing any parasitic contribution from gasket scattering. We note that the Ti0.676Zr0.324 alloy exists as a hexagonal close-packed α-phase in the pressure regime ≲12 GPa44 and that the Ti–Ti and Zr–Zr bond lengths for the α-phase in metallic Ti and Zr are typically 2.895–2.95177 and 3.179–3.231 Å,78 respectively. Hence, if any residual gasket scattering is present, its pair-distribution function will not contribute to the first peak region of ΔDMg(r).

FIG. 8.

Measured ΔFMg(k) functions for glassy MgAS_37.5_50 at ambient pressure, 3.9 GPa, 8.2 GPa, or recovered to ambient conditions from 8.2 GPa. In each panel, the black vertical error bars represent the measured function and the red curve represents the back-Fourier transform of the corresponding ΔDMg(r) function (Fig. 9) after the low-r oscillations, below the distance of closest approach between two atoms, are set to the theoretical density-line limit.

FIG. 8.

Measured ΔFMg(k) functions for glassy MgAS_37.5_50 at ambient pressure, 3.9 GPa, 8.2 GPa, or recovered to ambient conditions from 8.2 GPa. In each panel, the black vertical error bars represent the measured function and the red curve represents the back-Fourier transform of the corresponding ΔDMg(r) function (Fig. 9) after the low-r oscillations, below the distance of closest approach between two atoms, are set to the theoretical density-line limit.

Close modal
FIG. 9.

Fitted ΔDMg(r) functions for glassy MgAS_37.5_50 at ambient pressure, 3.9 GPa, 8.2 GPa, or recovered to ambient conditions from 8.2 GPa. In each panel, the black solid circles represent the measured function, the black solid curve represents the fitted function, and the other curves show the contributions from the Mg–O (magenta solid curves) and Mg–β (β ≠ O) (cyan dashed curve) correlations. The latter was introduced to constrain the peaks fitted at smaller r-values. The displaced green solid curve shows the residual.

FIG. 9.

Fitted ΔDMg(r) functions for glassy MgAS_37.5_50 at ambient pressure, 3.9 GPa, 8.2 GPa, or recovered to ambient conditions from 8.2 GPa. In each panel, the black solid circles represent the measured function, the black solid curve represents the fitted function, and the other curves show the contributions from the Mg–O (magenta solid curves) and Mg–β (β ≠ O) (cyan dashed curve) correlations. The latter was introduced to constrain the peaks fitted at smaller r-values. The displaced green solid curve shows the residual.

Close modal
TABLE III.

Mg–O peak parameters obtained from the fitted ΔDMg(r) functions. The error associated with rMgO,i is typically ±0.004 Å (i = 1) or ±0.008 Å (i = 2). The error associated with σMgO,i is typically ±0.004 Å (i = 1) or ±0.007 Å (i = 2). The error associated with n̄Mg,iO (i = 1 or 2) is typically ±0.03. The weighted mean Mg–O distance r̄MgO was obtained from the fitted peaks by applying Eq. (15) and has a typical error of ±0.008 Å. The overall Mg–O coordination number n̄MgO (sum) was obtained using the full precision of the measurements and has a typical error of ±0.04. The Rχ values for the fitted range ≃1.70–3.00 Å are also given, along with the kmax values used for Fourier transformation.

ConditionFirst peakSecond peakOverallOther parameters
rMgO,1 (Å)σMgO,1 (Å)n̄Mg,1OrMgO,2 (Å)σMgO,2 (Å)n̄Mg,2Or̄MgO (Å)n̄MgO (sum)Rχkmax−1)
Uncompressed 2.003 0.092 3.17 2.186 0.152 1.59 2.051 4.76 0.042 23.65 
3.9 GPa 2.000 0.083 3.49 2.189 0.200 1.97 2.052 5.46 0.078 12.90 
8.2 GPa 2.000 0.148 3.95 2.189 0.231 2.57 2.051 6.51 0.060 12.70 
Recovered 2.017 0.040 3.42 2.189 0.200 1.90 2.066 5.32 0.072 11.55 
ConditionFirst peakSecond peakOverallOther parameters
rMgO,1 (Å)σMgO,1 (Å)n̄Mg,1OrMgO,2 (Å)σMgO,2 (Å)n̄Mg,2Or̄MgO (Å)n̄MgO (sum)Rχkmax−1)
Uncompressed 2.003 0.092 3.17 2.186 0.152 1.59 2.051 4.76 0.042 23.65 
3.9 GPa 2.000 0.083 3.49 2.189 0.200 1.97 2.052 5.46 0.078 12.90 
8.2 GPa 2.000 0.148 3.95 2.189 0.231 2.57 2.051 6.51 0.060 12.70 
Recovered 2.017 0.040 3.42 2.189 0.200 1.90 2.066 5.32 0.072 11.55 
The first peak in the ΔDMg(r) functions of Fig. 9 was attributed to Mg–O correlations and was fitted using two Gaussian functions to allow for an asymmetrical distribution of Mg–O bond distances, which is particularly evident for the uncompressed glass.20 In the measurements using the Paris-Edinburgh press, the overall Mg–O coordination number for these fitted Gaussian functions was constrained to give the value found from direct integration over the first peak in the ΔDMg(r) function obtained by the application of a Lorch33,34 window function [Eq. (8)]. Table III lists the fitted parameters and the weighted mean bond distance,
(15)
where gMgO(r) was found by summing the Gaussian functions fitted to the first peak and r1 and r2 define the overall extent of this peak.

The difference functions ΔF(k) obtained from the total structure factors of Fig. 4 are shown in Fig. 10. The functions are larger in magnitude than their ΔFMg(k) counterparts (Fig. 8), leading to an improved signal-to-noise ratio over an extended k range. The results show a pressure-induced reduction in the height of the FSDP positioned at kFSDP ≃ 1.72 Å−1 as it shifts toward a larger k-value and a sharpening of the principal peak positioned at kPP ≃ 2.72 Å−1 as it also shifts toward a larger k-value. On recovering the glass to ambient conditions, the FSDP regains height as the principal peak loses height, and both features shift toward a smaller k-value.

FIG. 10.

Measured ΔF(k) functions for glassy MgAS_37.5_50 at ambient pressure, 3.9 GPa, 8.2 GPa, or recovered to ambient conditions from 8.2 GPa. In each panel, the black vertical error bars represent the measured function and the red curve represents the back-Fourier transform of the corresponding ΔD′(r) function (Fig. 11) after the low-r oscillations, below the distance of closest approach between two atoms, are set to the theoretical density-line limit.

FIG. 10.

Measured ΔF(k) functions for glassy MgAS_37.5_50 at ambient pressure, 3.9 GPa, 8.2 GPa, or recovered to ambient conditions from 8.2 GPa. In each panel, the black vertical error bars represent the measured function and the red curve represents the back-Fourier transform of the corresponding ΔD′(r) function (Fig. 11) after the low-r oscillations, below the distance of closest approach between two atoms, are set to the theoretical density-line limit.

Close modal

The first peak in ΔD′(r) is attributed to the Si–O and Al–O correlations (Fig. 11) and was, therefore, fitted using two Gaussian functions. In silica and in silicate glasses, such as MgSiO3, there is no indication of the Si–O coordination number changing from its ambient pressure value n̄SiO = 4 over the investigated pressure range,11,66 so the Si–O coordination number for the first Gaussian function was fixed at 4. For the uncompressed sample, the Al–O coordination number for the second Gaussian function was fixed at the value found from the 27Al MAS NMR experiments (Table I). For the other samples, the Al–O coordination number was fixed at the value found from direct integration over the first peak in the ΔD′(r) function obtained after the application of a Lorch33,34 window function [Eq. (11)]. As compared to the step window function defined in Sec. II, the Lorch function reduces the amplitude of small r-spaced Fourier transform artifacts that lead to excursions below the low-r density line limit. Its application to the datasets for the uncompressed and recovered glasses led to n̄AlO values in accordance with those found from the 27Al MAS NMR experiments (Table I). In fitting ΔD′(r), the Si–O and Al–O peak widths at each pressure point were constrained to avoid small r-spaced oscillations in the fitted peak shapes that are incommensurate with the kmax value used to define the step window function. The same fitting procedure has previously been used in diffraction work to measure the composition dependent structure of magnesium and zinc aluminosilicate glasses at ambient pressure. It led to Al(IV)–O and Al(V)–O bond lengths in agreement with those found from crystallography experiments and bond valence theory.2,20,79 Table IV lists the fitted parameters. The Si–O bond lengths r̄SiO are typical of tetrahedral SiO4 units.2 

FIG. 11.

Fitted ΔD′(r) functions for glassy MgAS_37.5_50 at ambient pressure, 3.9 GPa, 8.2 GPa, or recovered to ambient conditions from 8.2 GPa. In each panel, the black solid circles represent the measured function, the black solid curve represents the fitted function, and the other curves show the contributions from the Si–O (blue dashed curve), Al–O (red solid curve), and O–O (green dashed curve) correlations. The displaced green solid curve shows the residual. The O–O correlations were introduced to constrain the peaks fitted at smaller r-values.

FIG. 11.

Fitted ΔD′(r) functions for glassy MgAS_37.5_50 at ambient pressure, 3.9 GPa, 8.2 GPa, or recovered to ambient conditions from 8.2 GPa. In each panel, the black solid circles represent the measured function, the black solid curve represents the fitted function, and the other curves show the contributions from the Si–O (blue dashed curve), Al–O (red solid curve), and O–O (green dashed curve) correlations. The displaced green solid curve shows the residual. The O–O correlations were introduced to constrain the peaks fitted at smaller r-values.

Close modal
TABLE IV.

Si–O and Al–O peak parameters obtained from the fitted ΔD′(r) functions. The fixed parameters are marked with an asterisk, where the n̄AlO value for the uncompressed glass was obtained from 27Al MAS NMR spectroscopy (Table I). The errors associated with r̄SiO and σSiO are typically ±0.003 and ±0.005 Å, respectively, and the errors associated with r̄AlO, σAlO, and n̄AlO are typically ±0.005 Å, ±0.005 Å, and ±0.08, respectively. The Rχ values for the fitted range 1.30–2.70 Å are also given, along with the kmax values used for Fourier transformation.

Conditionr̄SiO (Å)σSiO (Å)n̄SiOr̄AlO (Å)σAlO (Å)n̄AlORχkmax−1)
Uncompressed 1.618 0.045 4.00* 1.765 0.051 4.17* 0.021 23.65 
3.9 GPa 1.600 0.059 4.00* 1.794 0.072 4.46 0.059 15.00 
8.2 GPa 1.633 0.063 4.00* 1.805 0.157 5.24 0.061 15.00 
Recovered 1.623 0.049 4.00* 1.784 0.066 4.48 0.028 14.65 
Conditionr̄SiO (Å)σSiO (Å)n̄SiOr̄AlO (Å)σAlO (Å)n̄AlORχkmax−1)
Uncompressed 1.618 0.045 4.00* 1.765 0.051 4.17* 0.021 23.65 
3.9 GPa 1.600 0.059 4.00* 1.794 0.072 4.46 0.059 15.00 
8.2 GPa 1.633 0.063 4.00* 1.805 0.157 5.24 0.061 15.00 
Recovered 1.623 0.049 4.00* 1.784 0.066 4.48 0.028 14.65 

To check on the self-consistency of the results, the D′(r) functions obtained from the total structure factors were fitted with the nearest-neighbor Si–O, Al–O, and Mg–O coordination numbers fixed at the values corresponding to the measured difference functions. The fitted functions are shown in Figs. 12 and 13, and the fitted parameters are summarized in Tables V and VI. The datasets could be fitted with only minor adjustments to the Gaussian peak positions.

FIG. 12.

Fitted D′(r) functions for glassy MgAS_37.5_50 at ambient pressure or recovered to ambient conditions from 8.2 GPa for the samples containing either natMg (left column) or 25Mg (right column). In each panel, the black solid circles represent the measured function, the black solid curve represents the fitted function, and the other curves show the contributions from the Si–O (blue dashed curve), Al–O (red solid curve), Mg–O (magenta solid curves), and O–O (green dashed curve) correlations. The displaced green solid curve shows the residual. The O–O correlations were introduced to constrain the peaks fitted at smaller r-values.

FIG. 12.

Fitted D′(r) functions for glassy MgAS_37.5_50 at ambient pressure or recovered to ambient conditions from 8.2 GPa for the samples containing either natMg (left column) or 25Mg (right column). In each panel, the black solid circles represent the measured function, the black solid curve represents the fitted function, and the other curves show the contributions from the Si–O (blue dashed curve), Al–O (red solid curve), Mg–O (magenta solid curves), and O–O (green dashed curve) correlations. The displaced green solid curve shows the residual. The O–O correlations were introduced to constrain the peaks fitted at smaller r-values.

Close modal
FIG. 13.

Fitted D′(r) functions for glassy MgAS_37.5_50 at 3.9 or 8.2 GPa for the samples containing either natMg (left column) or 25Mg (right column). In each panel, the black solid circles represent the measured function, the black solid curve represents the fitted function, and the other curves show the contributions from the Si–O (blue dashed curve), Al–O (red solid curve), Mg–O (magenta solid curves), and O–O (green dashed curve) correlations. The displaced green solid curve shows the residual. The O–O correlations were introduced to constrain the peaks fitted at smaller r-values.

FIG. 13.

Fitted D′(r) functions for glassy MgAS_37.5_50 at 3.9 or 8.2 GPa for the samples containing either natMg (left column) or 25Mg (right column). In each panel, the black solid circles represent the measured function, the black solid curve represents the fitted function, and the other curves show the contributions from the Si–O (blue dashed curve), Al–O (red solid curve), Mg–O (magenta solid curves), and O–O (green dashed curve) correlations. The displaced green solid curve shows the residual. The O–O correlations were introduced to constrain the peaks fitted at smaller r-values.

Close modal
TABLE V.

Si–O and Al–O peak parameters obtained from the fitted D′(r) functions. The fixed parameters are marked with an asterisk, where the n̄AlO values correspond to the measured ΔD′(r) functions (Table IV). The errors associated with r̄SiO and σSiO are typically ±0.003 and ±0.005 Å, respectively, and the errors associated with r̄AlO and σAlO are typically ±0.005 Å. The Rχ values for the fitted range 1.30–2.70 Å are also given, along with the kmax values used for Fourier transformation.

GlassConditionr̄SiO (Å)σSiO (Å)n̄SiOr̄AlO (Å)σAlO (Å)n̄AlORχkmax−1)
natMgAS_37.5_50 Uncompressed 1.618 0.044 4.00* 1.773 0.052 4.17* 0.037 23.65 
25MgAS_37.5_50 1.618 0.044 4.00* 1.772 0.052 4.17* 0.026  
natMgAS_37.5_50 3.9 GPa 1.616 0.050 4.00* 1.798 0.086 4.46* 0.114 19.80 
25MgAS_37.5_50 1.617 0.060 4.00* 1.794 0.104 4.46* 0.104  
natMgAS_37.5_50 8.2 GPa 1.617 0.060 4.00* 1.818 0.101 5.24* 0.047 19.80 
25MgAS_37.5_50 1.618 0.068 4.00* 1.793 0.100 5.24* 0.042  
natMgAS_37.5_50 Recovered 1.613 0.050 4.00* 1.793 0.076 4.48* 0.091 19.80 
25MgAS_37.5_50 1.616 0.050 4.00* 1.793 0.046 4.48* 0.050  
GlassConditionr̄SiO (Å)σSiO (Å)n̄SiOr̄AlO (Å)σAlO (Å)n̄AlORχkmax−1)
natMgAS_37.5_50 Uncompressed 1.618 0.044 4.00* 1.773 0.052 4.17* 0.037 23.65 
25MgAS_37.5_50 1.618 0.044 4.00* 1.772 0.052 4.17* 0.026  
natMgAS_37.5_50 3.9 GPa 1.616 0.050 4.00* 1.798 0.086 4.46* 0.114 19.80 
25MgAS_37.5_50 1.617 0.060 4.00* 1.794 0.104 4.46* 0.104  
natMgAS_37.5_50 8.2 GPa 1.617 0.060 4.00* 1.818 0.101 5.24* 0.047 19.80 
25MgAS_37.5_50 1.618 0.068 4.00* 1.793 0.100 5.24* 0.042  
natMgAS_37.5_50 Recovered 1.613 0.050 4.00* 1.793 0.076 4.48* 0.091 19.80 
25MgAS_37.5_50 1.616 0.050 4.00* 1.793 0.046 4.48* 0.050  
TABLE VI.

Mg–O peak parameters obtained from the fitted D′(r) functions. The fixed parameters are marked by an asterisk and were obtained from the measured ΔDMg(r) functions (Table III). The error associated with rMgO,i is typically ±0.004 Å (i = 1) or ±0.007 Å (i = 2). The error associated with σMgO,i is typically ±0.004 Å (i = 1) or ±0.008 Å (i = 2). The weighted mean Mg–O distance r̄MgO was obtained from the fitted peaks by applying Eq. (15) and has a typical error of ±0.008 Å.

  First peakSecond peakOverall
GlassConditionrMgO,1 (Å)σMgO,1 (Å)n̄Mg,1OrMgO, 2 (Å)σMgO, 2 (Å)n̄Mg,2Or̄MgO (Å)n̄MgO(sum)
natMgAS_37.5_50 Uncompressed 2.009 0.086 3.17* 2.185 0.153 1.59* 2.056 4.76* 
25MgAS_37.5_50 2.010 0.086 3.17* 2.178 0.155 1.59* 2.055 4.76* 
natMgAS_37.5_50 3.9 GPa 2.013 0.100 3.49* 2.211 0.204 1.97* 2.067 5.46* 
25MgAS_37.5_50 2.020 0.103 3.49* 2.211 0.210 1.97* 2.071 5.46* 
natMgAS_37.5_50 8.2 GPa 2.000 0.111 3.95* 2.189 0.193 2.57* 2.057 6.51* 
25MgAS_37.5_50 2.000 0.097 3.95* 2.189 0.157 2.57* 2.060 6.51* 
natMgAS_37.5_50 Recovered 2.012 0.100 3.42* 2.201 0.177 1.90* 2.064 5.32* 
25MgAS_37.5_50 2.027 0.088 3.42* 2.201 0.201 1.90* 2.074 5.32* 
  First peakSecond peakOverall
GlassConditionrMgO,1 (Å)σMgO,1 (Å)n̄Mg,1OrMgO, 2 (Å)σMgO, 2 (Å)n̄Mg,2Or̄MgO (Å)n̄MgO(sum)
natMgAS_37.5_50 Uncompressed 2.009 0.086 3.17* 2.185 0.153 1.59* 2.056 4.76* 
25MgAS_37.5_50 2.010 0.086 3.17* 2.178 0.155 1.59* 2.055 4.76* 
natMgAS_37.5_50 3.9 GPa 2.013 0.100 3.49* 2.211 0.204 1.97* 2.067 5.46* 
25MgAS_37.5_50 2.020 0.103 3.49* 2.211 0.210 1.97* 2.071 5.46* 
natMgAS_37.5_50 8.2 GPa 2.000 0.111 3.95* 2.189 0.193 2.57* 2.057 6.51* 
25MgAS_37.5_50 2.000 0.097 3.95* 2.189 0.157 2.57* 2.060 6.51* 
natMgAS_37.5_50 Recovered 2.012 0.100 3.42* 2.201 0.177 1.90* 2.064 5.32* 
25MgAS_37.5_50 2.027 0.088 3.42* 2.201 0.201 1.90* 2.074 5.32* 

The results obtained from the difference functions were used to provide constraints in fitting the D′(r) functions measured for the natMg_37.5_50 glass at the remaining pressure points. For example, in the pressure regime between ambient and 3.9 GPa, the Mg–O coordination number is expected to lie in between the values found from the corresponding ΔDMg(r) functions and the Al–O coordination number is expected to lie in between the values found from the corresponding ΔD′(r) functions. The fitted functions are shown in Fig. 14, and the fitted parameters are summarized in Tables VII and VIII.

FIG. 14.

Fitted D′(r) functions for glassy natMgAS_37.5_50 at 1.7, 3.0, 5.4, and 7.1 GPa. In each panel, the black solid circles represent the measured function, the black solid curve represents the fitted function, and the other curves show the contributions from the Si–O (blue dashed curve), Al–O (red solid curve), Mg–O (magenta solid curves), and O–O (green dashed curve) correlations. The displaced green solid curve shows the residual. The O–O correlations were introduced to constrain the peaks fitted at smaller r-values.

FIG. 14.

Fitted D′(r) functions for glassy natMgAS_37.5_50 at 1.7, 3.0, 5.4, and 7.1 GPa. In each panel, the black solid circles represent the measured function, the black solid curve represents the fitted function, and the other curves show the contributions from the Si–O (blue dashed curve), Al–O (red solid curve), Mg–O (magenta solid curves), and O–O (green dashed curve) correlations. The displaced green solid curve shows the residual. The O–O correlations were introduced to constrain the peaks fitted at smaller r-values.

Close modal
TABLE VII.

Si–O and Al–O peak parameters obtained from the fitted D′(r) functions for the natMgAS_37.5_50 glass at the remaining pressure points. The fixed parameters are marked with an asterisk. The errors associated with r̄SiO and σSiO are typically ±0.003 and ±0.005 Å, respectively, and the errors associated with r̄AlO, σAlO, and n̄AlO are typically ±0.005 Å, ±0.005 Å, and ±0.08, respectively. The Rχ values for the fitted range 1.30–2.70 Å are also given, along with the kmax values used for Fourier transformation.

Condition (GPa)r̄SiO (Å)σSiO (Å)n̄SiOr̄AlO (Å)σAlO (Å)n̄AlORχkmax−1)
1.7 1.606 0.045 4.00* 1.777 0.050 4.34 0.076 21.0 
3.0 1.607 0.040 4.00* 1.784 0.050 4.35 0.052 19.8 
5.4 1.623 0.060 4.00* 1.794 0.147 4.78 0.057 19.8 
7.1 1.622 0.069 4.00* 1.803 0.151 5.09 0.066 19.8 
Condition (GPa)r̄SiO (Å)σSiO (Å)n̄SiOr̄AlO (Å)σAlO (Å)n̄AlORχkmax−1)
1.7 1.606 0.045 4.00* 1.777 0.050 4.34 0.076 21.0 
3.0 1.607 0.040 4.00* 1.784 0.050 4.35 0.052 19.8 
5.4 1.623 0.060 4.00* 1.794 0.147 4.78 0.057 19.8 
7.1 1.622 0.069 4.00* 1.803 0.151 5.09 0.066 19.8 
TABLE VIII.

Mg–O peak parameters obtained from the fitted D′(r) functions for the natMgAS_37.5_50 glass at the remaining pressure points. The error associated with rMgO,i is typically ±0.004 Å (i = 1) or ±0.007 Å (i = 2). The error associated with σMgO,i is typically ±0.004 Å (i = 1) or ±0.008 Å (i = 2). The error associated with n̄Mg,iO is typically ±0.04 (i = 1 or 2), and the error on n̄MgO (sum) is typically ±0.06. The weighted mean Mg–O distance r̄MgO was obtained from the fitted peaks by applying Eq. (15) and has a typical error of ±0.008 Å.

Condition (GPa)First peakSecond peakOverall
rMgO, 1 (Å)σMgO, 1 (Å)n̄Mg,1OrMgO,2 (Å)σMgO,2 (Å)n̄Mg,2Or̄MgO (Å)n̄MgO (sum)
1.7 2.013 0.083 3.28 2.176 0.145 1.73 2.058 5.02 
3.0 2.009 0.095 3.39 2.187 0.203 1.86 2.056 5.25 
5.4 2.013 0.122 3.71 2.167 0.210 1.99 2.051 5.70 
7.1 2.000 0.123 3.88 2.178 0.215 2.43 2.049 6.31 
Condition (GPa)First peakSecond peakOverall
rMgO, 1 (Å)σMgO, 1 (Å)n̄Mg,1OrMgO,2 (Å)σMgO,2 (Å)n̄Mg,2Or̄MgO (Å)n̄MgO (sum)
1.7 2.013 0.083 3.28 2.176 0.145 1.73 2.058 5.02 
3.0 2.009 0.095 3.39 2.187 0.203 1.86 2.056 5.25 
5.4 2.013 0.122 3.71 2.167 0.210 1.99 2.051 5.70 
7.1 2.000 0.123 3.88 2.178 0.215 2.43 2.049 6.31 

We note that the diffraction data for glassy MgAS_37.5_50 were re-analyzed using the densities found from the Birch–Murnaghan EOS for glassy MgAS_48.8_48.8 and MgAS_44.5_44.5 obtained from the Brillouin scattering experiments of Wei et al.54 [Fig. 6(a)]. The resultant ΔDMg(r) functions led to n̄MgO values of 5.40(6) and 5.45(6) at pressures of 3.9 and 8.2 GPa, respectively, and the resultant ΔD′(r) functions led to n̄AlO values of 4.12(8) and 2.72(8) at pressures of 3.9 and 8.2 GPa, respectively. The Al–O values are significantly smaller than those found by using the EOS derived from the pressure-dependent position of the FSDP (Table IV), and the n̄AlO4 value at 8.2 GPa is physically unreasonable. The re-analysis, therefore, lends support to a systematic underestimation of the high-pressure dependence of the glass density in Brillouin scattering measurements.25 

The replacement of magnesia by alumina leads to polymerization of the glass network as the fraction of NBO atoms reduces from fNBO = 0.667 for enstatite to fNBO = 0.351 for MgAS_37.5_50.28 In this substitution, negatively charged aluminum-centered polyhedra are created and, in response, the fraction of Mg2+ ions taking a charge-compensating role increases from zero to 25.7%. At ambient pressure, this process is accompanied by only a small increase in the mean Mg–O coordination number from 4.46(4) to 4.76(4), which correspond to Mg–O bond distances of 2.056(8) and 2.051(8) Å, respectively.20 There is, however, a significant increase in the rate-of-change of n̄MgO with pressure, where n̄MgO=4.62(3)+0.090(3)p for enstatite compared to n̄MgO=4.65(7)+0.22(1)p for MgAS_37.5_50 [Fig. 15(a)]. In both cases, the pressure-induced increase in n̄MgO is not accompanied by an increase in r̄MgO [Fig. 15(b)], which reflects the similarity between the Mg–O bond lengths for Mg2+ in different coordination environments (Table IX) and a propensity for the bond lengths within a given polyhedron to shorten with increasing pressure.

FIG. 15.

Pressure dependence of the Mg–O (a) coordination number n̄MgO and (b) mean bond length r̄MgO for amorphous MgAS_37.5_50 and MgSiO3 under cold compression at room temperature. For MgSiO3, the results from neutron diffraction (ND)11 and x-ray diffraction (XRD)11 are compared to those obtained from the molecular dynamics (MD) simulations of either Salmon et al.11 using an aspherical ion model or Ghosh et al.25 using first-principles methods. In (a), the red dashed curve shows the linear fit n̄MgO=4.65(7)+0.22(1)p to the experimental results for MgAS_37.5_50 and the black dashed curve shows the linear fit n̄MgO=4.62(3)+0.090(3)p to the experimental results for MgSiO3, where p is in units of GPa. In the case of MgAS_37.5_50, the results are also given for the glass recovered to ambient conditions from 8.2 GPa.

FIG. 15.

Pressure dependence of the Mg–O (a) coordination number n̄MgO and (b) mean bond length r̄MgO for amorphous MgAS_37.5_50 and MgSiO3 under cold compression at room temperature. For MgSiO3, the results from neutron diffraction (ND)11 and x-ray diffraction (XRD)11 are compared to those obtained from the molecular dynamics (MD) simulations of either Salmon et al.11 using an aspherical ion model or Ghosh et al.25 using first-principles methods. In (a), the red dashed curve shows the linear fit n̄MgO=4.65(7)+0.22(1)p to the experimental results for MgAS_37.5_50 and the black dashed curve shows the linear fit n̄MgO=4.62(3)+0.090(3)p to the experimental results for MgSiO3, where p is in units of GPa. In the case of MgAS_37.5_50, the results are also given for the glass recovered to ambient conditions from 8.2 GPa.

Close modal
TABLE IX.

Ambient pressure Mg–O coordination numbers and bond distances for several magnesia containing crystalline systems.

CrystalPolyhedronDistance (Å)Reference
Ca2MgSi2O7 MgO4 1.916(5) 80  
MgAl2O4  1.923(1) 81  
MgO MgO6 2.109(1) 82  
MgSiO3  2.11(13) 83  
β-Mg2SiO4  2.09(9) 84  
Mg0.5AlSiO4  2.31(38) 85  
Mg3Al2Si3O12 MgO8 2.27(8) 86  
CrystalPolyhedronDistance (Å)Reference
Ca2MgSi2O7 MgO4 1.916(5) 80  
MgAl2O4  1.923(1) 81  
MgO MgO6 2.109(1) 82  
MgSiO3  2.11(13) 83  
β-Mg2SiO4  2.09(9) 84  
Mg0.5AlSiO4  2.31(38) 85  
Mg3Al2Si3O12 MgO8 2.27(8) 86  

In enstatite glass, molecular dynamics simulations show little change to the Qn speciation at pressures up to ∼8 GPa, where n denotes the number of bridging oxygen (BO) atoms per SiO4 tetrahedron, and an increase in n̄MgO with pressure up to 17.5 GPa [Fig. 15(a)] that originates from an increased fraction of Mg–BO connections.11 At ambient pressure, the network polymerization associated with the replacement of magnesia by alumina leads to an increase in the number of BO atoms per Mg2+ ion from NBO/NMg = 1 for enstatite to NBO/NMg = 3.027 for MgAS_37.5_50. The enhanced rate-of-change of n̄MgO with pressure in glassy MgAS_37.5_50 is, therefore, likely associated with the larger probability of Mg2+ ions finding BO atoms as nearest-neighbors. Similarly, the increase in n̄AlO with pressure is likely related to the Al3+ ions finding a larger number of BO atoms in their first coordination sphere [Fig. 16(a)].

The glass structure relaxes as the material is decompressed from 8.2 GPa to ambient pressure, and the recovered glass is 6% denser than its uncompressed counterpart (Sec. III A). On an intermediate length scale, this relaxation manifests itself in ΔF(k) by an increase in the intensity of the FSDP and shift in its position kFSDP from 1.99(2) to 1.87(2) Å−1 as the pressure is released (Fig. 10). On a local length scale, the Mg–O and Al–O coordination numbers both diminish on decompression, but the values remain larger than for the uncompressed glass (Tables III and IV). For the recovered glass, the Al–O coordination number n̄AlO = 4.48(8) found from diffraction is in agreement with the value n̄AlO = 4.42(6) found from the 27Al MAS NMR experiments (Table I), a finding that supports the efficacy of the approach used to analyze the neutron diffraction data.

The bulk modulus K = −V(∂p/∂V) = ρ(∂p/∂ρ), where ρ is the density, can be found from the measured EOS (Fig. 6) by considering consecutive pairs of data points for the pressures pi and pi+1, where i ≥ 1 is an integer. For a mean pressure p=pi+1+pi/2, the bulk modulus is given by K=Vi+1+Vi/2×pi+1pi/Vi+1Vi.89 As shown in Fig. 17, the measured dataset for glassy MgAS_37.5_50 shows only small changes to K at pressures up to ∼3.5 GPa before the appearance of an anomalous minimum. The fitted fourth order Birch–Murnaghan EOS leads to a minimum at ≃5.8 GPa, with a gradient dK/dp that becomes steeper when the pressure exceeds ∼3.2 GPa.

For enstatite glass, a minimum in the pressure dependence of K is also derived from (i) the EOS found from the kFSDP values and (ii) the EOS fitted to the dataset of Petitgirard et al.55 over the range 0–6 GPa. For the latter, a minimum is not, however, found if the EOS is fitted over the extended range 0–127 GPa, i.e., its observation depends on the extent to which the EOS is required to fit the measured data points in the low-pressure regime [Fig. 6(b)]. The origin of the shift in position of the minimum is unknown. Relevant factors may include the extent to which shear stress affects the mechanism of deformation.

In other materials, a minimum in the pressure dependence of K is also obtained from (i) the EOSs for the silica-rich glasses LiAlSi3O8,90 NaAlSi3O8 (albite),91 KAlSi3O8,92 and Na2MgSi6O1493 found from the measured refractive index,88 (ii) the EOS for glassy (Na2O)0.049(CaO)0.062(MgO)0.074(SiO2)0.806 found from optical measurements of the linear strain,58 and (iii) the EOS for silica glass measured using a variety of methods.58,87,88 The results indicate that glassy NaAlSi3O8 and KAlSi3O8 are more compressible than silica. The pressure-induced softening of the structure of silica and aluminosilicate glasses with the pyrope, albite, and jadeite compositions is indicated by the observation of a minimum at around 3–7 GPa in the pressure dependence of the measured longitudinal and shear wave velocities.94 

As in the case of enstatite glass, the appearance of a minimum in the pressure dependence of K for silica glass depends on the extent to which the EOS is required to fit the measured data points in the low-pressure regime. For example, the fourth order Birch–Murnaghan EOS fitted to the datasets of Meade and Jeanloz58 and Sato and Funamori59,60 over the range 0–6 GPa (Fig. 7) delivers a broad minimum in K vs p at ≃1.5 GPa. However, if the fitted range is extended to 10 GPa in order to encompass the entire pressure range expected for plastic deformation, a minimum is no longer observable.

Figure 16(a) shows that the initial reluctance of the MgAS_37.5_50 glass to deform on cold-compression is accompanied by a small change to n̄AlO, and the greater propensity for densification beyond a threshold pressure of ∼3.5 GPa is accompanied by a larger change to this coordination number. In comparison, n̄MgO increases linearly as the pressure is increased from ambient to 8.2 GPa [Fig. 15(a)]. On decompression, the pathway mapped by the compressional EOS is not expected to be retraced because plastic deformation will lead to hysteresis. The material is, however, anticipated to have more ability to expand in the softer regime as the pressure is reduced below ∼8 GPa but less ability to expand at pressures below ∼5.8 GPa as the material becomes stiffer and further structural reorganization is hindered (Fig. 17). In this way, a higher-density structure gets locked-in, leading to a permanently densified glass with Al–O and Mg–O coordination numbers larger than those found for the pristine material. For the densified MgAS_37.5_50 glass, the measured coordination numbers are ≃2% larger than those found for the cold-compressed material at ∼3.5 GPa [Figs. 15(a) and 16(a)], where the fractional volume change V/V0 ≃ 0.94 [Fig. 6(a)] matches the 6% densification observed for the recovered glass.

FIG. 16.

Pressure dependence of the Al–O (a) coordination number n̄AlO and (b) mean bond length r̄AlO for amorphous MgAS_37.5_50 under cold compression at room temperature. The green dashed curves show (a) the linear fits n̄AlO=4.18(5)+0.06(3)p and n̄AlO=3.78(7)+0.18(1)p to the experimental results in the low and high pressure regimes, respectively, and (b) the linear fit r̄AlO=1.769(3)+0.0049(6)p to the experimental results across the entire pressure range, where the pressure p is in units of GPa. The results are also given for the glass recovered to ambient conditions from 8.2 GPa.

FIG. 16.

Pressure dependence of the Al–O (a) coordination number n̄AlO and (b) mean bond length r̄AlO for amorphous MgAS_37.5_50 under cold compression at room temperature. The green dashed curves show (a) the linear fits n̄AlO=4.18(5)+0.06(3)p and n̄AlO=3.78(7)+0.18(1)p to the experimental results in the low and high pressure regimes, respectively, and (b) the linear fit r̄AlO=1.769(3)+0.0049(6)p to the experimental results across the entire pressure range, where the pressure p is in units of GPa. The results are also given for the glass recovered to ambient conditions from 8.2 GPa.

Close modal
FIG. 17.

Pressure dependence of the bulk modulus for glassy MgAS_37.5_50 (the present work), MgSiO3,11,55 SiO2,87 and the silicates LiAlSi3O8, NaAlSi3O8, KAlSi3O8, and Na2MgSi6O14.88 The black solid curve for MgAS_37.5_50 was found from the fourth order Birch–Murnaghan EOS fitted to the data points found from the measured kFSDP values [Fig. 6(a)], and the vertical arrow indicates a regime in which there is a change to the pressure dependence of the gradient dK/dp marked by the green straight lines. For MgSiO3, the cyan solid and dashed curves were obtained from the fourth order Birch–Murnaghan EOSs found (i) from the measured kFSDP values and (ii) by fitting the x-ray absorption results of Petitgirard et al.55 over the range 0–6 GPa, respectively [Fig. 6(b)].

FIG. 17.

Pressure dependence of the bulk modulus for glassy MgAS_37.5_50 (the present work), MgSiO3,11,55 SiO2,87 and the silicates LiAlSi3O8, NaAlSi3O8, KAlSi3O8, and Na2MgSi6O14.88 The black solid curve for MgAS_37.5_50 was found from the fourth order Birch–Murnaghan EOS fitted to the data points found from the measured kFSDP values [Fig. 6(a)], and the vertical arrow indicates a regime in which there is a change to the pressure dependence of the gradient dK/dp marked by the green straight lines. For MgSiO3, the cyan solid and dashed curves were obtained from the fourth order Birch–Murnaghan EOSs found (i) from the measured kFSDP values and (ii) by fitting the x-ray absorption results of Petitgirard et al.55 over the range 0–6 GPa, respectively [Fig. 6(b)].

Close modal

The structure of the pyrope composition glass (MgO)3/7(Al2O3)1/7(SiO2)3/7 or Mg3Al2Si3O12 after cold-compression at pressures up to 24 GPa and recovery to ambient conditions has been investigated by 27Al solid state NMR experiments.17 Three regimes were reported in which the Al–O coordination number (i) changed slowly with pressure increasing from ambient to ∼5 GPa, (ii) increased rapidly with pressure increasing from ∼5 to ∼12 GPa, or (iii) showed no further change with pressure increasing to 24 GPa (Fig. 18). These results appear to map onto the in situ structural behavior shown in Fig. 16(a) for glassy MgAS_37.5_50, where (i) there is an initial regime associated with a reluctance for the Al–O coordination number to change when pressure is first applied and (ii) there is a second regime associated with an enhanced rate of change of the Al–O coordination number with increasing pressure. The highest pressure regime may be associated with a structural domain in which Al(VI) is the predominant species.

FIG. 18.

Dependence of the aluminum speciation on the maximum applied pressure in cold-compression experiments on glassy (i) pyrope in the experiments of Lee et al.17 (closed symbols) and (ii) MgAS_37.5_50 in the present work (open symbols). The yellow shaded area marks the pressure regime for pyrope glass in which n̄AlO shows its most dramatic change.

FIG. 18.

Dependence of the aluminum speciation on the maximum applied pressure in cold-compression experiments on glassy (i) pyrope in the experiments of Lee et al.17 (closed symbols) and (ii) MgAS_37.5_50 in the present work (open symbols). The yellow shaded area marks the pressure regime for pyrope glass in which n̄AlO shows its most dramatic change.

Close modal

For uncompressed glassy pyrope, detailed information on the network connectivity has been reported from solid-state 27Al and 17O NMR experiments.95 The fraction of Al(IV) is reported to be fAl(IV) = 0.90–0.93, and the fraction of NBO atoms is reported to be fNBO = 0.327. In comparison, the GYZAS model of Ref. 28 gives fNBO = 0.38–0.40 for the reported fAl(IV) values along with NNBO/NT = 0.94–1.00, NBO/NMg = 2.48–2.40, and NNBO/NMg = 1.52–1.60. The model assumes that all the four-coordinated Si and Al contribute toward the glass network along with all the oxygen atoms, which are assumed to be either one- or two-coordinated.

We note that the broad minimum in dK/dp observed for silica glass around 1.4 GPa (Fig. 17) is within the elastic deformation regime for this material and is not, therefore, associated with permanent densification.

In general, crack formation by indentation in aluminosilicate glasses is suppressed by a large material stiffness and concomitant resistance to deformation, whereas crack propagation is impeded and the fracture toughness is enhanced by the ability for the glass to deform via densification.8,96 For glassy MgAS_37.5_50, there is a change in behavior from a high bulk modulus and relatively small gradient dK/dp at low pressures (Fig. 17), which will promote scratch resistance, to a greater propensity for densification beyond a threshold pressure of ∼3.5 GPa, which will promote fracture toughness. These findings, therefore, indicate the possibility for transitioning between the scratch-resistant and crack-resistant properties of these brittle materials by engineering the EOS via the glass composition and/or varying the pressure and thermal treatment of the glass.

The structure of glassy MgAS_37.5_50 was investigated by in situ high-pressure neutron diffraction with magnesium isotope substitution. The technique delivers site-specific information on the magnesium coordination environment and also gives access to the nearest-neighbor Al–O conformations. The results show a pressure-induced increase in both the Mg–O and Al–O coordination numbers from 4.76(4) and 4.17(7) at ambient pressure, respectively, to 6.51(4) and 5.24(8) at 8.2 GPa, respectively.

The MgAS_37.5_50 composition corresponds to enstatite after 25 mol. % of the magnesia is replaced by alumina. At ambient pressure, this replacement increases the connectivity of the network structure by reducing the NNBO/NT ratio from 2 for enstatite to 0.85 for glassy MgAS_37.5_50 (Sec. I) and is accompanied by an increase in NBO/NMg by a factor of ≃3. The replacement leads to a factor of 2.4 increase in the rate-of-change of the Mg–O coordination number with pressure from 0.090(3) to 0.22(1) GPa−1 and is related to a larger probability of magnesium finding BO atoms as nearest-neighbors.

The glass recovered to ambient conditions is 6% denser than its uncompressed counterpart, and the Mg–O and Al–O coordination numbers for this material are 5.32(4) and 4.42(6), respectively. The Al–O value is in accordance with the 27Al NMR results, which show the presence of Al(VI) species that are absent in the uncompressed material.

The results offer a rationale for the process of permanent densification in magnesium aluminosilicate glasses and an explanation for the appearance of distinct pressure-dependent structural transformation regimes.17 They also indicate a pressure-dependent threshold for transitioning between the scratch-resistant and crack-resistant material properties, where the stiffness of the MgAS_37.5_50 structure below ≃3.5 GPa is related to an initial reluctance for aluminum to change its coordination environment.

The findings demonstrate the value of in situ high-pressure neutron diffraction with isotope substitution as a tool for revealing the structure of amorphous materials under extreme conditions.

We thank Alain Bertoni (Grenoble) for help with the D4c experiments and Lawrence Gammond (Bath) for valuable discussions. P.S.S. also thanks Morten Smedskjaer (Aalborg) and Wilson Crichton (ESRF) for helpful conversations. H.M. was supported by Corning Inc. (Agreement No. CM00002159/SA/01). A.Z. was supported by a Royal Society-EPSRC Dorothy Hodgkin Research Fellowship. P.S.S. and A.Z. acknowledge Corning Inc. for the award of Gordon S. Fulcher Distinguished Scholarships during which this work was conceived. We acknowledge the use of the Inorganic Crystal Structure Database accessed via the Chemical Database Service funded by the Engineering and Physical Sciences Research Council (EPSRC) and hosted by the Royal Society of Chemistry. P.S.S. and A.Z. designed the diffraction project on the glass structure. H.M. prepared the samples. R.E.Y. performed the NMR experiments and analyzed the results. H.M., P.S.S., and H.E.F. performed the neutron diffraction experiments, and H.M. analyzed the results. P.S.S. wrote the paper with input from all co-authors.

The authors have no conflicts to disclose.

Hesameddin Mohammadi: Conceptualization (supporting); Formal analysis (lead); Investigation (lead); Methodology (equal); Resources (lead); Software (equal); Validation (equal); Writing – review & editing (equal). Anita Zeidler: Conceptualization (equal); Data curation (supporting); Formal analysis (supporting); Funding acquisition (equal); Project administration (equal); Software (equal); Supervision (equal); Writing – review & editing (equal). Randall E. Youngman: Conceptualization (supporting); Data curation (supporting); Formal analysis (supporting); Funding acquisition (equal); Investigation (supporting); Methodology (supporting); Project administration (equal); Resources (supporting); Validation (supporting); Visualization (supporting); Writing – original draft (supporting); Writing – review & editing (equal). Henry E. Fischer: Investigation (supporting); Methodology (supporting); Resources (supporting); Software (supporting); Writing – review & editing (equal). Philip S. Salmon: Conceptualization (equal); Data curation (lead); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Software (supporting); Supervision (equal); Validation (equal); Visualization (lead); Writing – original draft (lead); Writing – review & editing (lead).

The datasets created during this research are openly available from the University of Bath Research Data Archive at https://doi.org/10.15125/BATH-01238.97 The diffraction datasets are available in Ref. 98.

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