Relationship between topological order and glass forming ability in densely packed enstatite and forsterite composition glasses

Abstract

The atomic structures of magnesium silicate melts are key to understanding processes related to the evolution of the Earth’s mantle and represent precursors to the formation of most igneous rocks. Magnesium silicate compositions also represent a major component of many glass ceramics, and depending on their composition can span the entire fragility range of glass formation. The silica rich enstatite (MgSiO₃) composition is a good glass former, whereas the forsterite (Mg₂SiO₄) composition is at the limit of glass formation. Here, the structure of MgSiO₃ and Mg₂SiO₄ composition glasses obtained from levitated liquids have been modeled using Reverse Monte Carlo fits to diffraction data and by density functional theory. A ring statistics analysis suggests that the lower glass forming ability of the Mg₂SiO₄ glass is associated with a topologically ordered and very narrow ring distribution. The MgO_x polyhedra have a variety of irregular shapes in MgSiO₃ and Mg₂SiO₄ glasses and a cavity analysis demonstrates that both glasses have almost no free volume due to a large contribution from edge sharing of MgO_x-MgO_x polyhedra. It is found that while the atomic volume of Mg cations in the glasses increases compared to that of the crystalline phases, the number of Mg-O contacts is reduced, although the effective chemical interaction of Mg²⁺ remains similar. This unusual structure-property relation of Mg₂SiO₄ glass demonstrates that by using containerless processing it may be possible to synthesize new families of dense glasses and glass ceramics with zero porosity.

Glasses can be synthesized by making use of a network former, such as B₂O₃, As₂O₃, SiO₂, GeO₂, or P₂O₅ which form continuous three-dimensional networks, according to Zachariasen’s theory (1). On the basis of this theory many ideas on glass formation have been proposed. One such idea by Cooper and Gupta proposed that a topologically disordered network is the key to understanding the origin of glass formation (2, 3). Indeed, it is well known that the silica glass network exhibits a broad ring distribution which is topologically disordered, and is made up of a connection of SiO₄ tetrahedra sharing oxygen atoms at the corners (4, 5). In contrast the corresponding crystalline phase has only 6-fold rings.

Recently the use of the aerodynamic levitation technique combined with a laser heating system has expanded the glass forming range of many oxides, due to the avoidance of contact between the container walls and high-temperature melt (6). Tangeman and coworkers succeeded in synthesizing MgO-SiO₂ glasses over a wide compositional range (7), covering the enstatite composition (MgSiO₃) to the forsterite composition (Mg₂SiO₄), which can be considered as analogues of quenched melts from the Earth and Lunar mantle (8–10). The Mg₂SiO₄ composition glass has been formerly synthesized using a rapid quench method and studied using vibrational spectroscopy (11, 12), however, the advantage of the levitation technique is that it allows us to synthesize a high-purity bulk sample. The structure of these glasses was recently studied using theoretical simulation (13), spectroscopic measurements (7, 13–15), diffraction measurements (16–19), and atomistic modeling on the basis of diffraction data (17–19). Using Reverse Monte Carlo (RMC) modeling (20) Kohara, et al. discovered that SiO₄ tetrahedra formed both monomers (isolated SiO₄) and dimers (Si₂O₇) in Mg₂SiO₄ glass, and that the network structure of the glass largely comprises of MgO_x polyhedra with both corner and edge-sharing oxygens (17). Sen and Tangeman confirmed from the results of molecular dynamics simulation that the presence of a large excess of bridging O atoms associated with the Si₂O₇ dimers in Mg₂SiO₄ glass and melt, is in sharp contrast with their complete absence in the crystalline phase (13). By combining neutron and X-ray diffraction data Wilding et al. reported a relatively small but abrupt change in the average Mg-O coordination number from 4.5 to 5.0 over the narrow composition range 38 mol% SiO₂ to 33 mol% SiO₂ (Mg₂SiO₄); both coordination numbers being significantly smaller than the octahedral Mg coordination which occurs in the corresponding crystalline phases (16). This result is in agreement with the RMC Mg₂SiO₄ model by Kohara et al. which yielded an average Mg-O coordination number of 4.95. However, the diffraction and RMC results are in disagreement with recent NMR measurements that suggest MgO₆ octahedra are dominant in both MgSiO₃ and Mg₂SiO₄ glasses (14, 15), leading to two vastly different structural models.

In this paper we model and analyze the structure of silica glass (SiO₂), enstatite glass (MgSiO₃), and forsterite (Mg₂SiO₄) glass employing the RMC modeling technique and the density function theory (DFT) of electronic structure using measured and published synchrotron X-ray and neutron diffraction data (16–18). Special attention is paid to ring statistics, cavity volumes, and the connectivity of short-range structural units in the glasses. We also analyze the chemical nature of the Mg-O interaction in these systems to explain the different atomistic models arrived at on the basis of diffraction and NMR data to reveal the relationship between “topological order-disorder” and “glass forming ability.”

RMC Modeling

RMC modeling was performed using the RMCA code (20). Because the structural model for SiO₂ glass and Mg₂SiO₄ glass have been reported previously (5, 17), an RMC simulation was performed on an ensemble of 4,800 particles for MgSiO₃ glass. The starting configuration was created using hard sphere Monte Carlo (HSMC) simulations with constraints applied to avoid a physically unrealistic structure. Two kinds of constraints were used: closest atom-atom approach and connectivity. The choices of the closest atom-atom approaches were determined to avoid unreasonable spikes in the partial-pair correlation functions. The constraint on the Si-O connectivity was that all silicon atoms were coordinated to four oxygen atoms, but no constraint on the Mg-O connectivity was applied. After the HSMC simulations, RMC simulations with both X-ray and neutron S(Q) were performed. The atomic number density used was 0.08218 Å^-3 (16).

DFT Simulations.

The geometry optimizations and electronic structure calculations of MgSiO₃ and Mg₂SiO₄ glasses are performed using the CP2K program (21, 22) in DFT mode. The starting configurations of the 510 (MgSiO₃) and 511-atom (Mg₂SiO₄) models of glass phases have been generated by RMC modeling using the experimental densities (Fig. S1). Central in the CP2K method is the use of two representations of the electron density: localized Gaussian and plane wave (GPW) basis sets. This representation allows for an efficient treatment of the electrostatic interactions, and leads to a scheme that is formally linearly scaling as a function of the system size. The valence electron—ion interaction is based on the norm-conserving and separable pseudopotentials of the analytical form derived by Goedecker, Teter, and Hutter (23). We considered the following valence configurations: O (2s²2p⁴), Mg (2s²2p⁶3s²), and Si (3s²3p²). For the Gaussian based (localized) expansion of the Kohn-Sham orbitals we use a library of contracted m-DZVP basis sets (24), and the complementary plane wave basis set has a cut-off of 600 Ry [1 Rydberg (Ry) equals to 13.6 eV] for electron density (this equals to 150 Ry for wave functions in standard plane wave schemes). The molecularly optimized m-DZVP functions result in highly accurate results with less computational cost as experienced with the traditional basis sets that are fitted to atomic properties. Together with the GPW basis set this enables DFT simulations of systems up to 1,000 atoms or more. The increased performance is crucial for amorphous (glassy) materials as it is desirable to reduce the effects of the finite simulation cell (periodic boundary conditions) by increasing the system size and improving the statistics by including more atoms.

The generalized gradient-corrected approximation of Perdew, Burke, and Ernzerhof is adopted for the exchange-correlation energy functional (25). The average cohesive energies of the DFT-optimized crystalline structures of MgSiO₃ (320 atoms) and Mg₂SiO₄ (336 atoms) are 5.19 and 5.08 eV/atom, respectively, and the crystalline phase is 0.30 and 0.36 eV/atom more stable than glass. The optimized crystalline geometries are very close to the experimental structures of MgSiO₃ (26) and Mg₂SiO₄ (27), and the average Mg-O distances are within 0.01 Å from the experiment (maximum individual deviation 0.05 Å) (Fig. S2). Effective charges of individual atoms have been evaluated from electron density (28) and chemical bond orders between atomic pairs have been computed from the overlaps of the atomic orbital components (with a projected completeness 97.5%).

Ring Statistics and Q_n Calculations.

The ring statistics were calculated using the shortest-path analysis (29, 30) and included up to 12-fold rings. The number of atoms were counted that existed in a loop from an atom at a defined starting point, by returning to the same atom through the shortest-path length. This analysis was performed in order to avoid the counting of a large-number of n-fold rings that could be divided into smaller-number m-fold rings, where m < n . Furthermore, multiple counting was avoided by checking the linkage in individual rings. In this study we define an n-fold ring as a ring consisting of n polyhedra and define the maximum coordination distance for Si-O correlation to be 1.8 Å for both glasses and the maximum Mg-O correlation distance to be 2.5 Å (MgSiO₃ glass) and 2.8 Å (Mg₂SiO₄ glass). It is confirmed that the normalized fraction of ring statistics are insensitive to the distances chosen. To analyze the structural and chemical order within the network in glasses, we characterized the concentration of various types of Si environments. Silicon oxide environments are typically denoted as Q_n species, where n varies between 0 and 4 and denotes the number of bridging oxygens, using atomic configurations obtained by RMC modeling.

Cavity Analysis.

The cavity analysis has been performed as described in ref. 31. The system is divided into a cubic mesh with a grid spacing of 0.20 Å, and the points farther from any atom at a given cutoff (here 2.5 Å) are selected and defined as “cavity domains.” Each domain is characterized by the point where the distance to all atoms is a maximum. If there are no maxima closer than the di-vacancy cutoff (here 2.0 Å), we locate the center of the largest sphere that can be placed inside the cavity. This point can be used for calculating partial-pair distribution functions, including vacancy-vacancy correlations. Around the cavity domains we construct cells analogous to the Voronoi polyhedra in amorphous phases (cf. the Wigner-Seitz cell) and analyze their volume distribution.

Results and Discussion

The partial structure factors, S_SiSi(Q), S_SiO(Q), and S_OO(Q) of SiO₂, MgSiO₃, and Mg₂SiO₄ glasses obtained from the RMC model are shown in Fig. 1. It is well known that SiO₂ glass exhibits the first sharp diffraction peak (FSDP) at approximately Q = 1.52 Å^-1. It is of note that the strong positive contribution of the S_OO(Q) in SiO₂ glass to the FSDP becomes smaller and the peak position shifts to higher Q in both MgSiO₃ and Mg₂SiO₄ glasses. The S_SiO(Q) shows a similar behavior, but the contribution of S_SiSi(Q) to the FSDP in Mg₂SiO₄ glass is larger than those in both SiO₂ and MgSiO₃ glasses, although the fraction of silicon in Mg₂SiO₄ glass is the smallest. This unusual behavior cannot be observed in total X-ray and neutron data, because the weighting factor of the Si-Si correlation is very small in both cases. However, this behavior suggests significant modification of the silicate network occurs in Mg₂SiO₄ glass in comparison with SiO₂ glass and MgSiO₃, and also in the formation of chemical ordering recently argued by Salmon et al (32). Indeed, according to the RMC simulation reported by Kohara, et al. (17), the SiO₄ network has disappeared in Mg₂SiO₄ glass, which is manifested by significant peak in the S_SiSi(Q) as a signature of chemical ordering of silicon. Also, the arrangement of SiO₄ tetrahedra has fewer steric constraints in Mg₂SiO₄ glass due to the deformation of the network. Similar behavior can be seen in 64 mol% PbO-34 mol% SiO₂ glass (33), which can be synthesized by a conventional melt-quench technique.

Fig. 1.

Partial structure factors, S_ij(Q), for S_SiSi(Q), S_SiO(Q), and S_OO(Q) of SiO₂, MgSiO₃, and Mg₂SiO₄ glasses obtained from the RMC model. Black line, SiO₂ glass; Red line, MgSiO₃ glass; Blue line, Mg₂SiO₄ glass. The experimental X-ray and neutron total structure factors, S(Q), of MgSiO₃ and Mg₂SiO₄ glasses together with total S(Q) and partial structure factors, S_ij(Q), obtained from RMC modeling are shown in Fig. S3. Partial-pair distribution functions, g_ij(r), of MgSiO₃ and Mg₂SiO₄ glasses obtained from the RMC model are shown in Fig. S4.

Fig. 2 shows the distribution of SiO₄ tetrahedra and ring statistics of SiO₄ tetrahedra in SiO₂, MgSiO₃, and Mg₂SiO₄ glasses with a typical linkage of 4-fold and 6-fold Si-O rings. As can be seen in Fig. 2A, SiO₂ glass forms an SiO₄ network of corner-shared oxygens, providing a very broad ring size distribution and MgSiO₃ glass has large fraction of 3-fold and 4-fold rings, showing how the SiO₄ network is modified by the addition of Mg-O. In contrast, Mg₂SiO₄ glass does not form a SiO₄ network and the SiO₄ monomer and Si₂O₇ dimer are the predominant silicate species (17). These behaviors are reflected in the Q_n distribution for SiO₄ tetrahedra. The contribution of Q₁, Q₂, and Q₃ species are all significant in MgSiO₃ glass, while that of Q₁ is the most abundant in Mg₂SiO₄ glass. The connectivity analysis (33) also shows that about 14% oxygen in Mg₂SiO₄ glass can be assigned as “free oxygen” that does not participate in silicate fragments (13), which is in sharp contrast to MgSiO₃ where the contribution of “free oxygen” is small (∼2%). In terms of stoichiometry, the emergence of free oxygen in Mg₂SiO₄ glass is coupled to the existence of bridging oxygens in silicate fragment chains (Q₁, Q₂, and Q₃).

Fig. 2.

The distribution of SiO₄ tetrahedra (A), ring statistics (B) of SiO₄ tetrahedra in SiO₂, MgSiO₃, and Mg₂SiO₄ glasses and a typical linkage of 4-fold and 6-fold Si-O rings (C). Red sphere, O; Blue sphere, Si.

The distribution of MgO_x polyhedra in MgSiO₃ and Mg₂SiO₄ glasses and ring statistics are shown in Fig. 3. The Mg-O peak position observed in the g_MgO(r) of the RMC model at approximately 2.0 Å in both MgSiO₃ and Mg₂SiO₄ glasses (Fig. S4), is shorter than average Mg-O bond length of approximately 2.08 Å (MgSiO₃) and 2.13 Å (Mg₂SiO₄) in MgO₆ octahedra found in the corresponding crystal phases (17, 26, 27). This difference suggests that the Mg-O coordination number is ∼5 in these glasses on the basis of effective ionic radii (34). In the case of MgSiO₃ glass, the RMC model suggests that MgO₄ is the major unit, which is consistent with the coordination number derived from a combination of X-ray and neutron diffraction data (16, 17). However, the result of two recent NMR studies (14, 15) suggests that MgO₆ octahedra are the primary units in both MgSiO₃ glass and Mg₂SiO₄ glass (although 5-fold sites could not be ruled out). Attempts to include large fractions of MgO₆ polyhedra in our RMC simulations failed, even if the coordination number was calculated out to large distances i.e., 3.0 Å. Rather, in the case of Mg₂SiO₄ glass, the average Mg-O coordination number is found to be about 5 (corresponding a bond cutoff of 2.8 Å in the RMC model), which gives a very narrow ring size distribution as shown in Fig. 3B. In both glasses, a large fraction of 2-fold rings (edge-sharing) of MgO_x polyhedra are found, but MgSiO₃ glass yields a rather broad ring distribution of MgO_x.

Fig. 3.

The distribution of MgO_x polyhedra (A) and ring statistics (B) in MgSiO₃ and Mg₂SiO₄ glasses (maximum correlation distance is 2.5 Å for MgSiO₃ glass and 2.8 Å for Mg₂SiO₄ glass). Yellow sphere, magnesium; Red sphere, oxygen.

In order to shed more light onto Mg-O coordination controversy, we performed DFT calculations for MgSiO₃ crystal and glass and for Mg₂SiO₄ crystal and glass. The results for the effective charges (Q_eff) and atomic volumes (V_at) of Mg are listed in Table 1. The results show that Bader volumes are smaller than Voronoi volumes because the atomic boundary is closer to Mg than the geometric mean value (with oxygen). The resulting Q_eff values are correspondingly smaller for the Voronoi charges as there is more counterbalancing electron density inside the (larger) atomic volume of the cation. The effective charges (nominally +2) indicate a significant polarization effect between Mg and O, and some electron density has been transferred to Mg²⁺ which displays electrophilic nature.

Table 1.

Atomic charges and volumes of Mg cations in MgSiO₃ and Mg₂SiO₄. Bader atom concept is based on electron density and its gradient (28)

System	Qeff (Bader, e)	Qeff (Voronoi, e)	Vat (Bader, Å3)	Vat (Voronoi, Å3)
MgSiO3 glass (RMC)	-	-	-	13.36 (1.85)
MgSiO3 glass (DFT)	+1.75 (0.01)	+1.25 (0.04)	5.96 (0.53)	12.69 (1.43)
MgSiO3 crystal (DFT)	+1.76/+1.76	+1.25/+1.27	4.65/5.11	8.94/10.21
Mg2SiO4 glass (RMC)	-	-	-	11.83 (1.19)
Mg2SiO4 glass (DFT)	+1.73 (0.02)	+1.22 (0.04)	5.96 (0.56)	12.52 (1.42)
Mg2SiO4 crystal (DFT)	+1.74/+1.76	+1.19/+1.27	4.80/4.98	9.59/9.87

The values in parenthesis are standard deviations (errors). Atomic boundary is reached in a given direction once the gradient changes its sign and the electron density is integrated within the given volume in order to assign the atomic charge. Voronoi prescription of atomic volume is analogous to the Wigner-Seitz cell in crystalline materials. The crystalline structures comprise two classes of Mg, each, and values for both cases are displayed.

The chemical bond orders for Si-O and Mg-O distances (“bonds”) in MgSiO₃ and Mg₂SiO₄ are plotted in Fig. 4 A–D. The bond order concept is simple because a single covalent bond should produce a value close to unity and weaker types of chemical interaction appear as smaller values. The results for Si-O distances below 1.8 Å confirm this criterion and show that the values depend strongly on the bond length. For crystalline structures, we observe that the strong polarization interaction with Mg is reflected as a weakening of the terminal Si-O bonds in Mg₂SiO₄ (charge transfer from oxygen towards Mg), whereas the bridging Si-O bonds appear with smaller bond strenghts in MgSiO₃. There are also some defects in the silicate fragments in the glassy phase which appear as Si-O distances between 1.8–2.2 Å.

Fig. 4.

Scatter plot of the chemical bond order as a function of distance for (A) Si-O pairs and (B) Mg-O pairs in the MgSiO₃ glass. The crystalline reference values are included in red, and the separation between bridging (longer) and terminal (shorter) Si-O bonds is visible. (C and D) The same for Mg₂SiO₄ glass where the crystalline phase comprises only terminal oxygens. (E) Snapshot of the local environment around an Mg cation in the MgSiO₃ glass. The 5-fold coordination of Mg is distorted and contacts with two bridging oxygens are highlighted. (F) Snapshot of the Mg₂SiO₄ glass where several Mg cations and their local environment is included. Both 4-fold and 5-fold coordination of Mg is visible, and a “free” oxygen, which does not participate in any silicate fragment but is shared by Mg, is highlighted.

The Mg-O bond orders are smaller as the type of chemical interaction is different and they show an interesting scatter: While the data points for the crystalline (octahedral) bonds are within a small region, the Mg-O bonds appear in a broad range with a rapidly decreasing trend. The results indicate that a reasonable threshold for counting “bonds” lies around 2.5–2.6 Å, where the average value is of the same order as for some individual second-nearest-neighbor distances (maximum values for distances > 3.0 Å). The DFT result for Mg-O coordination ranges between 4.4–5.0 and 4.5–5.2 Å for MgSiO₃ and Mg₂SiO₄, respectively, using bond cutoff distances of 2.5–2.8 Å. The corresponding DFT Mg-O pair distribution functions, g_ij(r), show remarkably similar shapes for the first peak (nearest-neighbor distances) compared to that of the RMC structures (Fig. S4). The bond orders below 2.5 Å are generally larger than the corresponding crystalline values, and this is due to the fact that the Mg-O coordination is smaller in the glassy phase, where the cations compensate the smaller number of oxygen contacts by increasing the ionic bond strengths correspondingly. The compensation in bond strengths is also shown by the chemical coordination (valence) of Mg which has been counted by adding bond orders for each atom: The glassy phase results in values of 0.80 ± 0.02 (MgSiO₃) and 0.84 ± 0.02 (MgSiO₄), respectively, while the corresponding crystalline (octahedral) reference values are between 0.81–0.84.

The atomic charges and bond orders point out that the effective Mg-O interaction is very similar between the glassy and crystalline phase despite differences in geometrical coordination. The counter intuitive result found is that Mg in the glass occupies a larger volume with less Mg-O contacts. This feature is due to the irregular shapes of the coordination polyhedra, and is caused by the atomic disorder and misalignment of fragmented silica chains. Here, it is important to differentiate between the concepts of atomic volume [by Bader atom and Voronoi prescription (28)] and coordination polyhedron where the former can increase while the connections in the latter decrease. Selected snapshots of the Mg environment in MgSiO₃ and Mg₂SiO₄ glasses are displayed in Fig. 4 E–F, where we have highlighted Mg-coordinating bridging oxygens in the first case and a “free oxygen” in the latter case. The variation of O-Mg-O bond angles is evident in both, and Mg-O coordination ranges between 4 and 5. It is known that RMC produces the most disordered structure (35) which is consistent with a given set of diffraction data and geometrical constraints, and that DFT simulations may not achieve sufficient relaxation in the glass structure due to the long time scale. Experimentally the effect of quench rate can have an important effect on glass structure (36). The higher the cooling rate, the higher the fictive temperature and the resulting glass may correspond to a more disordered structure than the same system cooled at a slower rate. However, in this study we did not use any time scale-related calculations, so the resulting static model structure can be considered a snapshot consistent with the diffraction data. Nonetheless, the combination of methods used here provides a way of introducing energy and electronic structure into the RMC protocol (37). We note that in a similar DFT study by de Koker et al. (38) an Mg-O coordination number of 5.1 was reported for Mg₂SiO₄ melt at 3,000 K and ambient pressure.

In MgSiO₃, Mg binds to silicates (types Q₁, Q₂, and Q₃) via terminal and bridging oxygens similarly to the crystalline phase (Fig. S2). The free oxygens are absent in the crystalline Mg₂SiO₄, but they bind to Mg together with terminal oxygens (from Q₀ and Q₁ species) in the glassy phase. Typically, there are 3 to 4 Mg contacts for a free oxygen. Differences in the chemical environment of oxygen are reflected in its valence (summed bond orders), which reduces from 1.9 (bridging) to 1.6 (terminal), and to 1.1 (free oxygen). The valence of silicon is around 3.4–3.5 for all systems, and it should be compared with that of the formal covalent valence of 4.

The distribution of “-Si(Mg)-O-Si(Mg)-O-Si(Mg)-” rings in MgSiO₃ and Mg₂SiO₄ glasses are shown in Fig. 5 together with that of “-Si-O-Si-O-Si-” rings in SiO₂ glass. It is well known that glass forming ability (GFA) of Mg₂SiO₄ glass is lower than that of MgSiO₃ glass (7). As can be seen in Fig. 5, SiO₂ glass which has a high GFA, has a very broad ring distribution and is topologically disordered, whereas Mg₂SiO₄ glass which has a low GFA and a very narrow ring distribution is more topologically ordered. To further understand the glass structure and nature of the atomic packing, the atomic configurations together with the cavity volume are shown in Fig. 6. It is noted that SiO₂ glass has a cavity volume of 31.9% using the definition given in ref. 33, whereas both MgSiO₃ and Mg₂SiO₄ glasses have almost no cavities. The openness of the SiO₂ glass structure is maintained to retain the local order, and the disorder introduced by the addition of Mg-O results in a more compact structure due to the formation of MgO₅ and MgO₆ polyhedra. This trend is similar to 3Na₂O-4SiO₂ and 3Na₂O-4SiO₂ glasses (39) but is in sharp contrast to PbO-SiO₂ glass, in which more than 10% cavity volume is observed over a wide compositional range (33), which is a signature of high-glass forming ability. To understand the glass structure in detail, the connectivity of SiO₄ tetrahedra and MgO_x polyhedra were calculated and are compared to 3Na₂O-4SiO₂ and 3CaO-4SiO₂ glasses in Table 2. The most striking feature is that the connectivity of SiO₄ tetrahedra and MgO_x polyhedra are almost the same for the MgSiO₃ and Mg₂SiO₄ RMC glass models. Furthermore, it is noted that MgO_x polyhedra tend to corner-share oxygens with SiO₄ tetrahedra, and a large amount of edge-shared MgO_x polyhedra are found in both MgSiO₃ and Mg₂SiO₄ glasses. Thus, the characteristic connectivity of MgO_x in MgSiO₃ and Mg₂SiO₄ glasses is the primary reason for “topologically ordered void free glasses” in this system.

Fig. 5.

The distribution of “-Si(Mg)-O-Si(Mg)-O-Si(Mg)-” rings in MgSiO₃ and Mg₂SiO₄ glasses and the distribution of “-Si-O-Si-O-Si-” rings in SiO₂ glass.

Fig. 6.

The atomic RMC configuration and cavities of SiO₂, MgSiO₃ and Mg₂SiO₄ glasses. Light gray, silicon; Red, oxygen; Green: magnesium; Cyan isosurfaces: cavities (voids).

Table 2.

The connectivity analysis of SiO₄ and MO_x (M = Mg, Na, Ca) units in silicate glasses

	MgSiO3	Mg2SiO4	3Na2O-4SiO2 (39)	3CaO-4SiO2 (39)
SiO4-SiO4	corner 100%	corner 100%	corner 100%	corner 100%
SiO4-MOx	corner 86.9%	corner 87.1%	corner 75.3%	corner 72.7%
	edge 13.1%	edge 12.9%	edge 22.8%	edge 24.3%
	face 0%	face 0%	face 1.9%	face 3.0%
MOx-MOx	corner 75.3%	corner 76.1%	corner 78.8%	corner 78.1%
	edge 23.7%	edge 23.4%	edge 19.6%	edge 20.4%
	face 1.0%	face 0.5%	face 1.6%	face 1.5%

In this paper, we have focused on the structure of the glassy state due to the availability of neutron, X-ray, and NMR data. However, we have also performed RMC modeling on X-ray data from MgSiO₃ and Mg₂SiO₄ liquids (40), and combined the results with DFT molecular dynamics (MD) simulations. The X-ray total structure factors S(Q) together with the experimental data (40) and the partial-pair distribution functions, g_ij(r) of MgSiO₃ liquid (2,153 K) and Mg₂SiO₄ liquid (2,223 K) obtained from the final DFT-MD simulation are shown in Figs. S5 and S6, respectively. As can be seen in Fig. S5, agreement between X-ray data and the DFT-MD model is good (even without any refinement). The partials g_ij(r) of Mg₂SiO₄ liquids are similar to those of MgSiO₃ liquid except for the Si-Si g_ij(r) due the network deformation of SiO₄ tetrahedra in the former. The results on the liquids give similar Mg-O coordination numbers [N_Mg-O = 4.03 in MgSiO₃ liquid (up to 2.5 Å) and 4.79 (up to 2.8 Å) while N_Mg-O = 4.19 in Mg₂SiO₄ liquid (up to 2.5 Å) and 5.02 (up to 2.8 Å)] and connectivities to those presented here for the glasses.

The DFT calculations explain the discrepancy between the NMR and diffraction results, because NMR probes chemical shifts which are very sensitive to the electronic environment of the nuclei, while diffraction is a direct probe of the average coordination number through known neutron scattering lengths or the number of electrons surrounding an atom (provided the partial functions are known). Previous studies (14, 15) concluded that for MgSiO₃ and Mg₂SiO₄ glasses, the NMR shifts are in line with the octahedral crystalline environment, although in this study we find that the Mg-O coordination is actually smaller in the glasses. This result is counter intuitive because the Mg atoms occupy larger atomic volumes in the glassy state and the MgO_x polyhedra have become irregular. The underlying reason for the behavior is due to the compensation of Mg-O bond strengths (polarization) and weak dependence on the directionality. Presumably, application of higher pressure, as in Earth’s mantle, would bring the MgO_x polyhedra closer to the regular octahedral shape, but this would require structural adjustments in the silica units as evidenced by X-ray Raman scattering measurement for MgSiO₃ glass (8). DFT computations of Mg₂SiO₄ melts under pressure have shown that the Mg-O coordination becomes 6-fold at 10–20 GPa and increases towards 8 at higher pressures (38). The structural glass model presented here at ambient pressures does however explain the low viscosity of highly mobile ultramafic lavas. The lack of polymerized silicate units and range of distorted Mg environments in the liquid state are consistent with quiescent lava flows that are more strongly affected by gravitational forces. These model glass structures are substantially different from the corresponding crystalline phases and demonstrated the powerful combination of diffraction and DFT simulation in unraveling their complicated structures. Furthermore, the atomic models describe an amorphous Mg-based network which could lead to the formation of new families of lightweight glasses, glass ceramics or cements with no porosity.