OVERVIEW
Silica structure
Silica, the simplest oxide of silicon, is abundantly found in the Earth as white sand or quartz Its chemical composition, SiO2, plays a crucial role in semiconductor technology, materials science, and geoscience Extensive research has been performed on silica under ultrahigh pressures and temperatures to simulate extreme conditions akin to those found on Earth These simulation methods are advantageous, enabling the exploration of various pressure and temperature scenarios without limitations.
The radial distribution function (RDF) is essential for analyzing the microstructure of SiO2 and other materials In experimental techniques, the RDF is derived from the total structure factor, which is measured using methods like neutron scattering or X-ray diffraction The scattering factor is calculated using a specific formula, enabling a deeper understanding of material properties.
৷ݥݛݠ৯৲, where ৲ represents the wavelength and the scattering angle is denoted
5 as ৯ [8] In atomistic simulations, analyzing the RDF only requires the atomic coordinates
The phase transition temperatures for the melt-crystal system range from 1673 to 1823 K, while for the melt-glass system, they are between 1247 and 1533 K These phase transitions lead to changes in structural order, resulting in variations in density Melted silica has a density of approximately 2.2 g/cm³, but upon solidification, vitreous silica has a density between 2.2 and 2.5 g/cm³, whereas crystalline silica exhibits a density ranging from 2.3 to 4.6 g/cm³.
Glass silica features a continuous random-network structure, as first described by Zachariasen This atomic arrangement distinguishes glass's short-range order (SRO) from the long-range order seen in crystals and the disordered nature of amorphous or liquid states Silica glass is primarily composed of structural units where a silicon (Si) atom is surrounded by three to five oxygen (O) atoms, with the most common configuration being tetrahedrons formed by one Si atom and four O atoms, resulting in a mean coordination number (CN) of approximately 4 Deviations from this CN introduce asymmetry, increasing the system's total energy To achieve stability, it is essential to minimize this energy.
Through the RDF, we can determine the SRO and IRO of the structure The results measured by previous experimental and simulated methods are similar
The molecular dynamics (MD) results from S K Mitra reveal that the spatial structure of SiO2 glass consists of fundamental SiO4 units that connect to form a continuous random network The O-Si-O bond angle distribution peaks at approximately 109.5°, indicative of a regular tetrahedral configuration The average distance between a Si atom and its nearest O atoms within this tetrahedron is about 1.6 ± 0.1 Å, with a coordination number (CN) for the Si-O pair of 4 ± 0.1 The internal rearrangement order (IRO) of silica glass is influenced by variations in the Si-O-Si angle and Si-Si distance, with the Si-O-Si linkage angle ranging from 120° to 180° Neutron scattering studies indicate that the peak position of these angles is located at 144°.
The experimental study utilizing X-ray diffraction indicates that the O-Si-O bond angle distribution peaks at 109.3(3)°, with a root mean square value of 4.2(3)° In contrast, the Si-O-Si bond angle distribution is notably broad, reaching a maximum value of 180°.
The study reveals that the average bond lengths for Si-O, O-O, and Si-Si are 1.59 Å, 2.61 Å, and 3.07 Å, respectively, aligning closely with the findings of T Wu et al [15].
F Mauri et al [16] The Si-O-Si bond angle distribution concentrates from 130pto
The structure features a 180p configuration with a peak angle interval of 140-155p In this arrangement, the fundamental units of SiO4 are interconnected through corner-sharing bonds, utilizing a common oxygen atom known as the bridging oxygen atom Notably, there are no edge-sharing or face-sharing linkages present in this configuration.
The IRO of SiO 2 is analyzed through the distribution of Si-Si-Si, O-O-O, O-Si-
The study of Si-O-Si angles reveals that the peak of the Si-Si-Si bond angle distribution is approximately 60° for 3-fold rings and ranges from 80° to 150° for 4-fold to 8-fold rings, as demonstrated by Kohara et al using the Reverse Monte Carlo (RMC) method J P Rino and colleagues analyzed the SiOp2-type covalent structure, defining the ring size as the shortest path between two neighboring oxygen atoms, excluding the silicon atom The angles for Si-Si-O, O-O-O, and O-O-Si exhibit peaks at 15°, 60°, and 40°, respectively They noted that ring sizes vary from 3-fold to 10-fold, with 6-fold rings being the most prevalent in both liquid and glass states, although broader sizes may also occur.
In their simulation study, R G Della Valle and H C Andersen demonstrated that the computational results are largely independent of model size They reported mean bond lengths for Si-O, O-O, and Si-Si as 1.645 Å, 2.694 Å, and 3.184 Å, respectively, with corresponding coordination numbers of 4.0, 6.0, and 4.0 The calculated O-Si-O angle of 109.5° aligns closely with previous experimental findings This angle's significance is attributed to the potential energy dependence on the O-Si-O configuration rather than the Si-O-Si angle Furthermore, while analyzing the microstructure of vitreous silica at ambient pressure is essential, exploring structural changes under extreme conditions is equally crucial The microstructure of SiO2 experiences significant alterations with pressure, transitioning from SiO4 to SiO6 through SiO5 units, with the formation of SiO6 units being most pronounced between 8 and 40 GPa.
In summary, at ambient pressure, SiO4 units connect through corner-sharing bonds to create a silica network As pressure increases, the average coordination number (CN) of Si-O rises, leading to the formation of 5-fold, 6-fold, or even higher coordinated silicon due to the increased atomic packing Additionally, previous research indicates that the size of the model has a minimal impact on the results.
7 microstructure The changes of the structure depend less on temperature and more on compression.
Structure of ternary MgO-SiO 2 models
At ambient pressure, SiO2 forms a continuous random network of SiO4 tetrahedrons linked by bridging oxygen (BO) atoms, while non-bridging oxygen (NBO) atoms connect to only one silicon atom, and free oxygen (FO) atoms are unlinked In silica glass at 0 GPa, the composition is predominantly BO, with minimal NBO and no FO present Experimental and simulation studies have shown the microstructural changes when MgO is doped into pure silica, revealing detailed composition-dependent structures in MgO-SiO2 glass through neutron and X-ray diffraction analysis The first peak of the Si-O pair distribution function (PRDF) is observed at approximately 1.64 Å, with a silicon-oxygen coordination number (CN) of 4.0, which remains consistent with pure vitreous silica Additionally, the O-Si-O bond angle in SiO4 units is about 105° for MgSiO3 and around 106° for Mg2SiO4.
The introduction of MgO into silica glass results in a minor alteration of approximately 2%, leading to a decrease in the coordination number of O-Si to 1.35 at ambient pressure, significantly lower than that in pure silica glass This reduction is due to the disruption of O-Si bonds, which facilitates the conversion of bridging oxygens (BO) to non-bridging oxygens (NBO) and the formation of Si-NBO-Mg linkages In MgSiO3 glass, Q n species, which represent the relationship between Si-O and O-Si coordination, predominantly consist of Q1, Q2, and Q3, accounting for roughly 80% of the total Q n at ambient pressure Specifically, at zero pressure, the proportions of NBO, BO, and fully coordinated oxygens (FO) are 62%, 37%, and 0%, respectively, though other studies report slightly different percentages As densification occurs, the prevalence of high-order Q n species increases, enhancing the connectivity of the Si-O network Consequently, while the Mg-O dopant has a minimal impact on the fundamental structural units, it significantly influences the linkages among them, positioning Si and O as network formers and Mg as a network modifier, or even a network former in certain scenarios.
Figure 1-1 Ternary MgO-SiO2 network, circles are color-coded to determine the type of species Gray is silicon atom; white is oxygen atom; orange circles are Mg atoms
Experimental methods for investigating the molecular structure of glasses include nuclear magnetic resonance, X-ray diffraction, X-ray absorption, neutron scattering, vibrational spectroscopy, and Raman spectroscopy Research by C D Yin and colleagues indicates that the mean coordination number (CN) for Mg-O in MgSiO3 (enstatite) is 4.5 and 5.0 in Mg2SiO4 (forsterite) In MgSiO3, magnesium can exhibit either 6-fold coordination with four neighboring oxygen atoms at 2.08 Å and two at 2.50 Å, or 4-fold coordination with four neighboring oxygen atoms at 2.04 Å Neutron investigations reveal Si-O, Mg-O, and O-O bond lengths of 1.64 Å, 2.0 Å, and 2.69 Å, respectively, while X-ray methods report Si-O, Si-Si, and Si-Mg/Si distances of 1.64 Å, 2.74 Å, and 3.2 Å The first peak in the Mg-O pair radial distribution function (PRDF) occurs at 2.0 Å, characterized by an asymmetric shape and a pronounced high-r tail, indicating a broad distribution of Mg-O bonding distances and distortion in the magnesium coordination polyhedron The CN of Mg-O in MgSiO3 is estimated at 4.5 ± 0.1, suggesting a mixture of coordinated MgO3 and MgO4 configurations.
In the study of MgO4 and MgO5 tetrahedra, it is observed that both structures are dominant and exist in equal proportions For Mg2SiO4, the coordination number (CN) of the Mg-O pair is around 5, while the coordinated magnesium exhibits a CN of 4.5, although it can reach a higher CN of 5.1 Notably, within the MgOx unit, the Mg-O distance is shorter in MgO4 compared to MgO5 and MgO6.
Recent studies utilizing simulation methods have validated the findings, particularly under ultra-high pressure and temperature conditions where simulations effectively investigate microstructure Kubicki and colleagues discovered that magnesium (Mg) species combine with four nearest oxygen (O) species within a 2.00 Å distance to form MgO4 units, while connecting with two additional O species at a 2.20 Å distance leads to a distorted octahedral coordination Additionally, molecular dynamics (MD) simulations by Y Matsui and K Kawamura indicate that the Mg-O bond length in MgO4 units fluctuates between 1.92 and 1.98 Å, while the octahedral coordination in MgO6 units shows a consistent mean value.
The bond length of 2.07 Å and a coordination number (CN) of Mg at 5.7 indicate significant structural changes in MgSiO3 and Mg2SiO4 under compression Research shows that the coordination of Si and Mg is pressure-dependent, with Spera et al using Matsui and OG interatomic potentials to reveal that the mean CN of Mg-O increases from 5.0 to 7.7 within the 0-50 GPa pressure range At 50 GPa, 6-, 7-, and 8-fold coordinated MgO dominate, while MgO3, MgO4, and MgO5 units are virtually absent At ambient pressure, 4-fold coordinated Si is prevalent, constituting 90% of SiO4 units in the Matsui model and 70% in the OG model Furthermore, the CN of Si-O reaches 5.5 at the highest pressure considered, indicating a dominance of SiO5 and SiO6 units.
D B Ghosh and colleagues analyzed the structural property of Mg 2 SiO 4 glass by employing the first-principles MD [6] The system condition is at 300K and in 0-170 GPa pressure range Calculated first peak positions at ambient pressure of Si-O, Mg-O, O-O, Si-Si, Mg-Si, Mg-Mg are 1.63, 1.98, 2.68, 3.00, 3.19 and 2.92 Å respectively, average corresponding bond lengths are 1.62, 2.05, 2.71, 3.02, 3.22 and 3.42 Å and corresponding mean CNs are 4.0, 4.6, 5.8, 2.1, 4.6 and 4.5 This result is in good agreement to value reported in works [3,21,26,33,34]
The Intermediate Range Order (IRO) in materials relates to the connectivity of polyhedrons, although the difference between Short Range Order (SRO) and IRO is often ambiguous Generally, IRO is significant within a distance of 3 to 10 Å, roughly corresponding to 2π divided by the scattering vector at the First Sharp Diffraction Peak.
SiOx polyhedrons bond through corner-sharing, edge-sharing, and face-sharing linkages, with significant changes occurring under compression At low pressure, these polyhedrons primarily connect via corner-sharing linkages.
As pressure rises, new linkages like edge-sharing and face-sharing emerge, enhancing the overall network structure A similar shift in connectivity is observed in Mg-Si pairs.
The analysis of network properties in MgSiO3 glass, specifically through Q n species analysis, reveals significant changes in the distribution of Q n units as pressure increases Notably, the quantity of Q n units with 0 ≤ n ≤ 2 decreases, while those with 3 ≤ n ≤ 6 rise However, the Q n distribution related to rings in MgSiO3 liquid under compression remains underexplored To better understand the connectivity characteristics of these rings, we focus on analyzing the Q n distribution of SiOx units across various ring types, maintaining n as the representation of the number of bridging oxygens (BOs) in the SiOx units.
Studies utilizing RMC modeling and density functional theory (DFT) simulations reveal that the Voronoi volumes of magnesium (Mg) surpass the Bader volumes In MgSiO3 glass, the Mg-Voronoi volume measures around 13.36 Å in the RMC model and 12.69 Å in the DFT model, with these values decreasing to 11.83 Å and 12.52 Å, respectively.
Previous studies indicate that adding magnesium (Mg) to silicon dioxide (SiO2) in different proportions minimally impacts the fundamental SiOx unit, yet it significantly alters the Si-O network's structure In our molecular dynamics (MD) model, we conduct an in-depth analysis of the microstructural characteristics and properties of MgSiO3 glass and liquid, comprising 5000 atoms We utilize techniques such as radial distribution function (RDF) and coordination number (CN) to examine the local structural changes in the material under pressure.
O rings are analyzed to clarify the division of the second peak in the Si-Si pair distribution function (PRDF) at 200 GPa and the emergence of magnesium-rich regions under high pressure Furthermore, Q n and Voronoi analyses are performed to enhance the understanding of the ring structure.
METHODOLOGY
Construction
Simulation methods are increasingly vital in understanding complex systems, effectively linking theory with practical insights into physical phenomena Molecular Dynamics (MD) simulation is particularly noteworthy for its ability to explore the microscopic dynamics of atoms, enabling the monitoring and prediction of a system's evolution over time Research utilizing MD simulation has consistently aligned with experimental results, affirming its accuracy Additionally, MD simulation is computationally efficient and reliable, providing a cost-effective alternative to traditional experimental methods, even under extreme conditions.
In molecular dynamics (MD) simulation, atoms are represented as rigid spheres that move within three-dimensional space, adhering to classical mechanics principles The interactions between these atoms are governed by an interatomic potential that relies on their coordinates While numerous resources on MD simulation exist, we recommend the book by A Satoh for further reading.
This article utilizes the Verlet algorithm to simulate molecular motion within a system of k atoms randomly distributed in a cube with an edge length of l The particle movements adhere to Newton's laws of classical mechanics, influenced by the interaction potential The mathematical representation of these motions is derived from Newton's law.
In the context of the motion of a specific atom, the equation \( F = m \cdot a \) describes the relationship between mass (\( m \)) and acceleration (\( a \)) Here, \( F \) represents the total force acting on the atom, while the distance between the atom and a selected point is denoted as \( d \) The total force (\( F \)) can be calculated using the formula \( F = \sum F_i \), where \( F_i \) represents the individual forces acting on the atom.
2.2 where ݎ ݛݜ is the interaction potential between pair ݛݜ ݤ ݛݜ is the distance between them In this work, we apply OG potential (ݎ ݛݜ ) that is presented in detail in the next section MD simulation technique uses the iterative method to solve the system of motion equations of atoms in model
The Taylor expansion for the positions is written as: ݤݦ éݦ ݤݤݗݤݦ ݗݦ éݦ ݗ ݤݦ ݗݦ éݦ
Add the two ݤݦ éݦ ݤݦ ˲éݦ ݤݦ ݗ ݤݦ ݗݦ éݦ ݈൬éݦ ൯ 2.4 derive the Verlet algorithm: ݤݦ éݦ ݤݦ ˲ ݤݦ ˲ éW ݙݦéݦ ݟ ݈éݦ 2.5
The Velocity of Verlet algorithm is expressed as: \( \mathbf{v}_i(t) = \frac{\mathbf{r}_i(t + \Delta t) - \mathbf{r}_i(t - \Delta t)}{2 \Delta t} \) This algorithm is utilized in molecular dynamics (MD) programs to compute the position of an atom at time \( t \) based on its coordinates at previous times \( t - \Delta t \) and \( t + \Delta t \).
13 ݤ ݛ ݦ ݗݦ ݤݛݦ ˲ ݤݛݦ˲ ݗݦ ݗݦ ܿݛݦ ݟݛ
Velocity at time ݦ is calculated through coordinates at times ݦ ˲ ݗݦ and ݦ ݗݦ by the expression: ݨݛݦ ݤ ݛ ݦ ݗݦ ˲ ݤ ݛ ݦ ˲ ݗݦ
ݗݦ 2.8 ܿݛݦ is the value of ܿ Ǚ ݦ : ܿǙݦ ܿ ே ܿ ݪ ே ܿ ݫ ெ ക ܿ ݬ ேݪ ও ݜ
ക ܿேݬ ও ݜ 2.9 in which, with ݪெ ୗ , ݫெ ୗ , ݬெ ୗ are unit vectors ˱ ܿ ݜ ெ ও , ˱ ܿ ݜ ே ݫ ও and ˱ ܿ ݜ ே ݬ ও are given by: ക ܿேݪ ও ݜ ݪெ ക൮ୗ ˲ݎ ݛݜ
2.1.1.2.Controlling the temperature of the molecular dynamics model
In molecular dynamics (MD) simulations, the temperature calculated from particle velocities may differ from the target system temperature due to the interplay between kinetic and potential energy Therefore, conducting an equilibrium procedure is essential before initiating the main simulation iterations The temperatures derived from the translational and angular velocities of particles are typically represented as T_trans and T_rot.
In this study, the total number of atoms is represented by \( N \), with \( N \) being a variable The parameters \( P_{vib} \) and \( P_{rot} \) are calculated using \( N \) and the desired temperature, denoted as \( T \) Typically, \( P_{vib} \) and \( P_{rot} \) are not equal to \( P \) The equilibration procedure modifies \( P_{vib} \) to align with \( P \) during the simulation by utilizing the ratio of the translational and angular velocities of each particle The scaling factors \( \alpha_v \) and \( \alpha_r \) are determined through specific calculations involving these parameters.
The averaged values of \( p_{v1} \) and \( p_{v2} \) are calculated from \( p_{v1} \) and \( p_{v2} \) respectively, after a specified number of steps (denoted as \( n \)), which can range from several tens to several hundreds, depending on the number of main loops in the process.
With the scaling factors determined ݔ ୗ ݦ and ݔ ୗ ݤ , the translational and angular velocities of all particles in a system are scaled as: ݨݛ Ɲ ݔ ୗ ݦ ݨݛ 2.13
This treatment effectively achieves the desired system temperature with high precision The process is repeated to ensure accurate temperature control, resulting in the updated temperature values for the systems.
Note: if a system has a macroscopic velocity, the scaling process must be slightly modified
2.1.1.3.Controlling the pressure of molecular dynamics model
The pressure in the MD simulation model can be adjusted by modifying the scale of the models using an adjustment factor, denoted as ৲ This factor is applied by multiplying the coordinates of all atoms in the model If the new pressure exceeds the previous pressure, an adjustment factor less than 1 is selected; conversely, if the new pressure is lower, a factor greater than 1 is chosen The adjustment factor is defined as follows: for new pressure greater than the previous pressure, ৲ equals ݗ݉a ˲, and for new pressure less than the previous pressure, ৲ equals ݗ݉a ˲ The new coordinates of all particles are calculated using the formula: ݪ Ɲ ݛ ݪݛ ৲; ݫ Ɲ ݛ ݫݛ ৲; ݬ Ɲ ݛ ݬݛ ৲ Additionally, the new proportion of the model is represented as ݅ Ɲ ݅ ৲.
2.1.1.4.The energy in molecular dynamics simulation
The stability of a simulated model is directly related to the system's energy levels, with lower energy indicating greater stability If the monitored energy increases progressively during the simulation, adjustments to the simulated system are necessary The energy can be expressed mathematically as \( E = PE + KE \), where \( PE \) and \( KE \) represent potential and kinetic energy, respectively These energy values can be calculated using the formulas: \( PE = mgh \) and \( KE = \frac{1}{2} mv^2 \).
Researchers select specific models based on the properties they are investigating, such as the ݇ ݏ ݇ݏݍ ܾ݁݉݇ ݇݉ ݍ and others These models involve various parameters including the number of atoms, total energy, volume, temperature, pressure, enthalpy, and chemical potential For example, in the ݇ ݏݍ model, these parameters play a crucial role in understanding the system being studied.
16 the numbers of atoms, the value of volume and temperature are constant throughout the simulation time
The MD calculation involves a vector sum of all forces acting on each particle This study utilizes the OG potential due to its simplicity and proven effectiveness in prior structural and dynamical simulations The Buckingham function for the OG potential between two atoms, ݛ and ݜ, is pairwise and is expressed in the following form: ݏ(ݤݛݜ) ݣ ݛ ݣ ݜ ݘ .
The effective charges of atoms \( r \) and \( s \) are represented as \( q_r \) and \( q_s \), respectively, with the distance between these atoms denoted as \( d_{rs} \) The electron charge is indicated by \( e \), while \( C_{rs} \) and \( D_{rs} \) refer to the parameters associated with van der Waals repulsion and attraction Additionally, \( L_{rs} \) is defined as the e-folding length that characterizes the radially-symmetric decay of electron repulsive Born energy between species \( r \) and \( s \) The specific values for these parameters can be found in Table 2-1.
Table 2-1 Parameters of OG potential
Model analysis
The structural transformations of MgSiO3 glass and liquid are analyzed through computational results, emphasizing the examination of radial distribution functions (RDF), coordination numbers (CN), ring statistics, and Voronoi diagrams.
The Radial Distribution Function (RDF) illustrates how density varies with distance from a reference atom, indicating the likelihood of locating atoms within a specific spherical layer It also reveals the local structural characteristics of the system The cutoff radius, essential for analysis, is identified at the minimum value of the RDF following its first peak, with cutoff radii for Si-O and Mg-O pairs measured at 2.2 Å and 2.6 Å, respectively, under ambient pressure Additionally, structural factors can be employed to experimentally determine the RDF R D Oeffner demonstrated a method for calculating the average coordination number, mean bond length, and mean bond angle using the RDF.
Figure 2-1 The illustration of ݚݤ PRDF
When considering the model of N atoms inside the volume V, the PRDF is described by function: ݚݤ ৸ݤ৸ ୗ 2.20
19 where ৸ ୗ is the average particle density in the entire model; ৸ݤ the particle density within a distance of ݤ and ݤ éݤ away from the reference atom ৸ ୗ and ৸ݤ are given by: ৸ ୗ ݇ ݏ 2.21 ৸ݤ ݇ ˱ ݠ ݇ ݛ ݛ ݤ éݤ
The RDF between pairs ২ ˲ ২ and ২ ˲ ৩ is determined in following expression: ݚ২২ݤ ݏ ݇ ২ ˱ ݠ ݇ ݛ ২ ২ݛ ݤ éݤ
2.24 where ݇ ২ and ݇ ৩ are the numbers of ২-atom and ৩-atom in the model
The structure factors can be represented as the Fourier transform of the RDFs
The value can be more easily compared with experimental data obtained from neutron diffraction and X-ray methods The structure factor for type-2 and type-3 atoms is expressed as follows: ২৩ݣ ৫২৩ ৷৸൬ݖ২ݖ৩൯ ݗݤݤ VLQݣݤ ݣݤ ൧ݚ ২৩ ݤ ˲ ൪ ˾ ୗ.
2.25 in which ৫ ২ ৩ is the ৫ உஆஅஂஉ , ܼ ݔݞݢǾݔ is the concentration of type-২ species Summing the partial structure factors can derive the structure factor as follow: ݣ ˱ ݕ২৩ ২ ݕ৩൬ݖ২ݖ৩൯ ൦২৩ݣ ൩
˱ ݖ ২ ২ ݕ ২ 2.26 where ݕ ২ and ݕ ৩ are X-ray > ݑ ݣ@ diffraction and neutron > ݇ ݣ@ scattering lengths for type-২ and type-৩ atom, respectively
2.2.2.Coordination number and bond length
The CN ݓ ২৩ is determined by the integral expression of the RDF in the first peak interval [44]
2.27 in which, ݤ ݖ is the cutoff distance The ݓ ২৩ value illustrates how many type-৩ species are within the sphere of radius ݤ ݖ with type-২ atoms in the center The first peak position in component RDF deduces the value of bond length between the pairs of elements In other words, from the first peak position of PRDF ݚ ݛ˲ݛ , we can derive the distance between two nearest vicinity Si atoms Similarly, the bond lengths of Si-O, Mg-O, Mg-Mg, O-O are calculated from corresponding PRDFs
Figure 2-2 Ring visualization a) 7-fold Si ring at 0 GPa b) 8-fold
Si ring at 0 GPa The atoms are color-coded Black rigid spheres are O species, cyan ones represent Si atoms
In analyzing ring statistics, the -Si-O- network is represented as an undirected graph, with each atom functioning as a node In this graph, a path is defined as a sequence of interconnected nodes and links that do not overlap There are several established definitions of rings, including the classic King's criterion, the irreducible or primitivity criteria, and the shortest paths criterion It is crucial to recognize that varying definitions of rings can yield different results.
This study utilizes the no shortcut path criterion and the algorithm developed by M Matsumoto et al to analyze the formation of n-fold rings, which consist of n T atoms (where T denotes either Mg or Si) and n O atoms Typical illustrations of silicon rings are depicted in Figure 2-2.
The process is expressed as below steps:
- Select three consecutive nodes (i.e., two adjacent edges without cycles) in the network according to King's criteria [46]
- Identify the smallest possible rings that include these three nodes
To ensure that rings in a graph do not contain shortcuts, it is essential that any path connecting two vertices within the ring is not shorter than the path along the ring itself This can be accomplished by implementing Dijkstra's algorithm in accordance with Franzblau's SP ring criteria.
- Add the identified ring to the list
- Repeat the first 4 steps until all combinations of three successive nodes have been tested
- Remove any duplicate permutations of a ring from the list
- (Optional) Exclude any "crossing rings" from consideration
Voronoi diagrams are a computational technique for dividing three-dimensional space into distinct regions based on the proximity of points or objects This method, named after mathematician Georgy Voronoi, effectively illustrates how space can be organized according to the nearest neighbors.
Voronoi diagrams are mathematical constructs that partition three-dimensional space into regions, known as Voronoi cells, based on a set of points or objects called seeds or generators Each cell represents the area closest to its corresponding seed, facilitating spatial analysis and understanding of proximity relationships in various applications.
Voronoi cells are defined by boundaries that consist of surfaces equidistant from neighboring seeds, representing points in space that maintain equal distance to two adjacent seeds This arrangement results in a tessellation of 3D space, creating a complex network of interconnected polygons that delineate the boundaries of the Voronoi cells.
Voronoi diagrams have diverse applications across multiple disciplines, including physics, biology, computer graphics, and spatial analysis They are essential for studying spatial patterns, analyzing the distribution of objects, understanding proximity relationships, and exploring geometric properties.
In three-dimensional space, Voronoi cells reveal the arrangement and organization of points or objects, facilitating deeper analysis and interpretation of spatial data This study focuses specifically on Voronoi diagrams constrained to rings.
RESULT AND DISCUSSION
Local structure of MgSiO 3
Si-Si Si-O O-O Si-Mg Mg-O Mg-Mg
Figure 3-1 The PRDFs of MgSiO3 glass
Figure 3-2 The PRDFs of MgSiO 3 liquid
To examine atomic connections, the value of ݤ ݖݧݦ is determined by the location of the first notch after the initial peak in the pair radial distribution functions (PRDFs), with adjustments made for specific pressures.
In this study, the Si-O distance for the MgSiO3 glass is determined to be 2.30 Å, while for the MgSiO3 liquid, it is 2.50 Å at 0 GPa, with both values decreasing under compression The positions of the first peaks in the Si-O and Mg-O Pair Distribution Functions (PRDFs) for MgSiO3 glass and liquid at normal pressure align with previous research, as detailed in Table 3-1 Additionally, the PRDFs, overall Radial Distribution Function (RDF), and structure factor of MgSiO3 liquid and melt at various pressures show strong agreement with earlier studies on MgSiO3 glass and liquid Notably, MgSiO3 glass exhibits a higher degree of Short-Range Order (SRO) compared to MgSiO3 liquid.
Under compression, the arrangement of atoms in materials changes, resulting in the formation of densely packed atomic domains This alteration significantly affects the coordination numbers (CNs) of silicon (Si) and magnesium (Mg), which are highly pressure-dependent At a pressure of 200 GPa, the CNs for Si and Mg in MgSiO3 glass are approximately 6.5 and 7.8, respectively, while in MgSiO3 liquid, these values increase to 6.8 and 8.4 This data highlights the relationship between pressure and atomic arrangement in different states of MgSiO3.
CN and the formation of high-fold polyhedrons, we visualize the model in the Figure 3-5 and Figure 3-6
Figure 3-3 comparison between this study and previous work
[33,55] The structure factor of Si-O and O-O for MgSiO 3 glass
(top), the overall RDF and structure factor for MgSiO3 liquid
Table 3-1 The first peak position of PRDFs, the CN of MgSiO3 glass (a) and MgSiO3 liquid (b), at 0 GPa a)
First peak position (Å) Coordination number
First peak position (Å) Coordination number
Figure 3-4 The dependance of Si and Mg CNs on pressure
Figure 3-5 Visualization of Mg and Si CNs of the models at different pressures of MgSiO3 glass The Mg atoms (big spheres),
In the depicted structure, three coordinated silicon (Si) atoms, represented as small spheres, form a polyhedron with silicon-oxygen (Si-O) coordination The color coding indicates the coordination number (CN), with black/gray denoting a 3-fold coordination, cyan for 4-fold, green for 5-fold, and dark blue for higher coordination levels.
6-fold and magenta for 7-fold or higher
Figure 3-6 Visualization of Mg and Si CNs of the models at different pressures of MgSiO3 liquid The Mg atoms (big spheres),
In the structure, three coordinated silicon (Si) atoms are represented as small spheres, while silicon-oxygen (Si-O) polyhedra are color-coded to indicate their coordination numbers (CN) The color scheme includes black/gray for 3-fold coordination, cyan for 4-fold coordination, green for 5-fold coordination, and dark blue for higher coordination states.
6-fold and magenta for 7-fold or higher
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WĂŝƌƌĂĚŝĂůĚŝƐƚƌŝďƵƚŝŽŶĨƵŶĐƚŝŽŶ'K ͲK;ƌͿ ƌ;Ϳ ϭϱϬ'WĂ ϮϬϬ'WĂ ϳϬ'WĂ ϭϬϬ'WĂ ϯϬ'WĂ ϱϬ'WĂ ϮϬ'WĂ Ϯϱ'WĂ ϭϬ'WĂ ϭϱ'WĂ Ϭ'WĂ ϱ'WĂ
Figure 3-7 The PRDFs of Si-O and O-O in MgSiO 3 glass (top) and liquid (bottom)
Table 3-2 The number of corner-(Nc), edge-(Ne), face-sharing
(Nf) network per the number of Si atoms at different pressure, corresponding with the mean bond length of them, corner-, edge-
, face-sharing bond length of MgSiO3 glass (a) and MgSiO3 liquid
(GPa) Nc Ne Nf CSBL ESBL FSBL
(GPa) Nc Ne Nf CSBL ESBL FSBL
Figure 3-7 illustrates the PRDFs of Si-O and O-O Under compression, the initial peaks of Si-O in both glass and liquid states experience a rightward shift followed by a leftward shift, whereas the O-O peaks consistently shift to the left.
The behaviors of local structural changes in MgSiO3 glass and liquid exhibit significant similarities Due to time constraints in research, we will henceforth concentrate exclusively on the properties of MgSiO3 in its liquid state.
The average bond lengths of O-O and Si-Si decrease significantly, while the Si-
As pressure increases, the bond length of O shows a gradual decrease The pair distribution functions (PRDFs) for O-O and Si-Si in MgSiO3 liquid reveal that the initial peaks of both distributions shift to lower values, signifying a contraction in bond lengths under pressure.
As pressure increases, the contraction of O-O and Si-Si bonds is linked to the formation of SiOx polyhedrons, such as SiO5, SiO6, and SiO7 This pressure-induced transformation enhances the -Si-O- network, leading to a rise in edge-sharing and face-sharing Si-Si bond linkages Notably, the average bond length of edge-sharing Si-Si bonds is shorter than that of corner-sharing bonds but longer than that of face-sharing bonds As a result, the mean Si-Si bond length significantly decreases with increasing pressure.
WĂŝƌƌĂĚŝĂůĚŝƐƚƌŝďƵƚŝŽŶĨƵŶĐƚŝŽŶ'^ŝ Ͳ^ ŝ;ƌͿ ƌ;Ϳ ϭϱϬ'WĂ ϮϬϬ'WĂ ϳϬ'WĂ ϭϬϬ'WĂ ϯϬ'WĂ ϱϬ'WĂ ϮϬ'WĂ Ϯϱ'WĂ ϭϬ'WĂ ϭϱ'WĂ Ϭ'WĂ ϱ'WĂ
Figure 3-8 The PRDF of Si-Si in MgSiO3 liquid
Under high pressures, SiOx units with x ≥ 6 become predominant, resulting in a reduced mean bond length of Si-O within these units The first peak of the O-Si-O bond angle distribution in SiO7 units is smaller than in SiO6 units but larger than in SiO8 units Additionally, due to the strong correlation between the pair radial distribution function (PRDF) and bond angle distribution, the mean O-O bond length also significantly decreases under compression.
Figure 3-9 The bond angle distribution (left) and bond length distribution within each type of SiOx ( = 6, 7, 8) units x
At 0 GPa, the composition of SiOx units reveals that approximately 75% are SiO4, 20% are SiO5, and SiOx units with x ≥ 6 are nearly absent The presence of SiO3 and SiO5 in MgSiO3 at this pressure is attributed to MgO doping As pressure increases, the ratios of SiOx units fluctuate significantly, with SiO4, SiO5, SiO6, and SiO7 reaching their maximum proportions at pressure ranges of 0 to 6 GPa, 6 to 27 GPa, 27 to 125 GPa, and above 125 GPa, respectively Notably, the peaks for SiO5, SiO6, and SiO7 occur at 14, 62, and 175 GPa By 200 GPa, SiO4 and SiO5 nearly disappear, while SiO7 units dominate Additionally, at 200 GPa, MgOy units with y ≥ 6 become the predominant structural units in the system.
Figure 3-10 The change in the fraction of SiOx units under compression
At 200 GPa, atomic arrangements exhibit increased order, indicating the presence of a crystalline phase between 150 and 200 GPa after transitioning from liquid to solid under compression Notably, at this pressure, the second peak of the Si-Si Pair Distribution Function (PRDF) splits into two subpeaks, which will be analyzed further through ring statistics in the subsequent section.
Ring analysis
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Figure 3-11 Ring statistics at different pressures a) n-fold Si-O ring, b) n-fold T-O ring
The ring analysis aims to enhance our understanding of atomic arrangements in MgSiO3 at intermediate distances Figure 3-11 presents the distribution of ring sizes, focusing on the connections between silicon (Si) and oxygen (O) in Si-O rings, as well as the interactions involving magnesium (Mg) or silicon (Si) with oxygen.
The analysis examines O rings under a pressure range of 0 to 200 GPa To determine the fraction of specific O rings, the count of those rings is divided by the total number of rings, which varies in size from 2 to 11.
MgO doping causes distortion and disruption in the Si-O network, leading to the formation of larger Si-O rings While T-O rings show a consistent Gaussian-like distribution across all pressures, Si-O rings only exhibit this pattern at 0 GPa, suggesting that T-O rings have a greater degree of topological order than Si-O rings.
Under pressure, the size distribution of Si-O rings changes due to the influence of Mg atoms on SiOx polyhedrons and the Si-O network At ambient pressure, Mg acts as both a network-forming and network-modifying agent, playing an intermediate role However, this influence becomes more significant at high pressure, leading to the disruption of short paths between Si atoms As a result, the ring statistics deviate from a Gaussian distribution when pressures exceed 0 GPa.
At ambient pressure, 4- and 5-fold T-O rings make up about 24% of the total ring distribution, while 6-fold, 7-fold, and 8-fold rings account for approximately 15.2%, 6.2%, and 2.2%, respectively As pressure increases, the distribution shifts towards smaller rings, with the 3-fold ring becoming the most prevalent at pressures of 50 GPa and above, comprising over 50% of the rings at 100 GPa or higher This trend indicates a significant transformation in ring size distribution under increased pressure conditions.
≥ 6) become nearly absent at high pressures At 200 GPa, the fraction of 5-fold T-
The proportion of O rings has increased by approximately 3%, primarily due to the rise in mean coordination number (CN) for silicon (Si) and magnesium (Mg) under compression This change leads to tighter atomic packing, resulting in a higher percentage of small tetrahedral-octahedral (T-O) rings and a corresponding decrease in larger T-O rings.
The Si-O ring distribution undergoes a notable transformation under compression, particularly at varying pressures At 0 GPa, the distribution is Gaussian, with the 5-fold ring being the most prominent However, as pressure increases to 20 GPa, the 5-fold ring's proportion decreases significantly, while the previously least common 11-fold ring becomes dominant Further increases in pressure continue to alter the ring statistics, with 4-fold and 5-fold rings maintaining the smallest proportions This illustrates the substantial impact of pressure on the Si-O network's structure.
At 200 GPa, the 9-fold ring fraction in the -Si-O- network exceeds other configurations, significantly influencing the intermediate-range order (IRO) This effect is crucial for interpreting the splitting peak noted in the second peak of the Si-Si pair radial distribution function (PRDF).
Figure 3-12 Si-Si-Si distance distribution on different types of rings at 200 GPa
This article analyzes the Si-Si distance distribution among silicon atoms in various ring structures, highlighting the influence of ring size on bond lengths Specifically, 4, 5, and 6-membered rings show Gaussian-shaped distributions with peak distances of 4.1, 4.8, and 4.7 Å, respectively Notably, the distribution in 4-fold rings is narrower than that in 5-fold and 6-fold rings, as indicated by the full width at half maximum (FWHM) In contrast, larger 10-fold and 11-fold rings exhibit significant deviations from the smaller rings' distributions, suggesting greater flexibility in their topology The formation of these larger rings involves multiple Si-O bonds, contributing to their distinct structural characteristics.
The arrangement of SiOx units within high-fold rings allows for greater freedom while reducing the overall cohesive energy, in contrast to low-fold rings like 2-fold and 3-fold, which require stretching Si-O bond lengths and adjusting bond angles to manage increased energy The energy constraints significantly affect smaller rings more than larger ones, leading to a wider variety of shapes in higher-fold rings Consequently, the distribution of Si-Si-Si distances in larger rings deviates from a Gaussian shape.
The 9-fold rings exhibit a notable peak in the Si-Si-Si distance distribution at approximately 5.3 Å, despite their asymmetric nature At 200 GPa, these rings are significantly more common, comprising nearly 24% of the total ring population, while the 11-fold rings, the second most prevalent type, account for only around 15% This dominance in both proportion and the high peak position of the Si-Si-Si distance distribution in the 9-fold rings plays a crucial role in shaping the Si-O pair distribution function (PRDF).
Under compression, the second peak of the Si-Si Pair Distribution Function (PRDF) shifts to lower values, with a notable split into two subpeaks at 200 GPa The first subpeak appears at approximately 4.6 Å, while the second is around 5.3 Å This splitting is attributed to the extensive formation of 9-fold rings, which leads to a reduction in the average bond length between interacting atoms, akin to the behavior observed at pressures below 200 GPa The higher subpeak corresponds to the pronounced peak seen in the Si-Si-Si distance distribution.
To gain a better understanding of the rings, Figure 3-13 presents the distributions of Q n (where n represents the number of BOs) under compression
At 0 GPa, the Q n distribution of 2-fold and 3-fold rings exhibits peaks at n = 4 (as shown in Figure 3-13.a) In 2-fold rings, the value of Q 3 is approximately equal to the value of Q 5 , at around 27% The fraction of Q 3 in 2-fold rings is higher compared to other ring sizes For rings with 4 or more members, Q 3 represents the highest proportion Across different rings at the same pressure, the distribution of
Q n is generally similar, except for the Q n distribution on the 11-fold ring at 100 GPa (as depicted in Figure 3-13.c) The shift of the peak in the Q n distribution,
38 from 3 or 4 at ambient pressure to 6 or 7 at pressures of 50 GPa or higher, indicates that rings have a tendency to merge or fuse together under compression ϭ Ϯ ϯ ϰ ϱ ϲ ϳ ϴ ϵ ϭϬ Ϭ ϭϬ ϮϬ ϯϬ ϰϬ ϱϬ ƌŝŶŐϮ
Figure 3-13 Q n distribution of different ring sizes at different pressures.
Mg-rich region
The analysis of ring statistics reveals the heterogeneity in MgSiO3, highlighting that large rings within the Si-O network lack silicon atoms due to the absence of shortcut paths This absence leads to a high density of oxygen atoms within these large rings, creating a negatively charged area To neutralize this charge, Mg 2+ ions migrate inward, resulting in the formation of an Mg-rich region.
As pressure rises, the non-uniformity within the material leads to increased heterogeneity, characterized by a greater proportion of large Si-O rings, which enhances the material's ability to capture Mg atoms Thus, the heterogeneity of MgSiO3 intensifies with elevated pressure.
Figure 3-14 Cyan, black and red spheres are Si, O and Mg atoms respectively, the yellow path is 10-fold ring a) Ring with surrounding oxygen atoms b) Mg-rich region inside the ring
Voronoi diagrams
An analysis of Si- and O-Voronoi volume distributions was performed to clarify the atomic arrangement within the Si-O network The results, illustrated in Figure 3-15, show how Voronoi volumes on specific rings change with pressure Figure 3-16 exemplifies the Si- and O-Voronoi volumes on a 6-fold ring at pressures of 50 and 200 GPa, while Figure 3-17 depicts the spatial distribution of these Voronoi volumes on the same 6-fold ring.
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^ŝsŽƌŽŶŽŝ KsŽƌŽŶŽŝ s Ž ůƵ ŵ Ğ Ž Ĩ K s Ž ƌŽ Ŷ Ž ŝƉ Ž ůLJ Ś ĞĚ ƌŽ Ŷ Ɛ ; ϹͿ
Figure 3-15 Characteristic of Voronoi polyhedrons of rings: average volume of Si-Voronoi (a) and O-Voronoi (b)
Voronoi on rings under compression a) b)
Figure 3-16 Voronoi on rings a) 6-fold ring at 50 GPa b)
6-fold ring at 200 GPa Cyan and gray polyhedrons correspond for Voronoi centered by Si and O atoms
At 0 GPa, the Si-Voronoi volume ranges from 7.50 to 7.76 Å In contrast, O-Voronoi volumes at ambient pressure strongly depend on the ring type and are significantly larger than Si-Voronoi volumes At 0 GPa, the O-Voronoi volume increases from approximately 13 Å to nearly 16 Å as the ring size expands from
The O-Voronoi volume varies with ring size, particularly decreasing for rings larger than size 6, with the 11-fold ring exhibiting an O-Voronoi volume of approximately 14.3 ų This variation is linked to the distortion of SiOx units and their arrangement within the ring structure Additionally, at ambient pressure, the O-Si-O bond angles predominantly peak at around 109°.
Smaller rings exhibit internal O-Si-O angles in SiOx units that are less than 109°, leading to increased distortion of these units Additionally, the configuration of SiOx units on the rings is influenced by their shared connections.
Silicon (Si) atoms significantly impact the volume of O-centered Voronoi structures For smaller rings (size ≤ 5), the distortion of SiO4 units predominantly influences their volume In contrast, larger rings (size > 5) allow for greater atomic flexibility, enabling the formation of symmetric SiOx units, which effectively reduces the system's energy As a result, SiOx units in larger rings experience less distortion.
The relationship between the Si-O-Si angle and the O-Voronoi volume is significant, as illustrated in Figures 3-15.b and 3-18 Specifically, the average O-Voronoi volume correlates with the mean Si-O-Si angle, which peaks at 148.2° before declining to 137.5° when the ring size exceeds six This decline in the Si-O-Si angle is accompanied by a reduction in O-Voronoi volume, indicating that the average O-Voronoi volume in 6-fold rings is greater than that in larger rings.
At pressures of 20 GPa or higher, the Voronoi volumes of similar atoms across different rings converge to comparable values, with Si atoms surrounded by O atoms, and O atoms forming bonds with both Mg and Si Notably, the mean Si-O bond length is shorter than the mean Mg-O bond length under identical pressure conditions This leads to a shorter average distance from the center of Si-Voronoi compared to O-Voronoi, resulting in a larger O-Voronoi volume Specifically, at pressures of 50, 100, and 200 GPa, the Si-Voronoi volumes are 6.7, 6.1, and 4.8 ų, while the O-Voronoi volumes are 9.0, 7.8, and 6.0 ų, respectively.
As pressure rises, the quantity of edge-sharing and face-sharing bonds increases, leading to a reduction in model size due to compression and a heightened polymerization of the Si-O network This results in densely packed atoms, with interconnected rings that occasionally overlap, as illustrated by the overlapping 2-fold and 10-fold rings in Figure 3-14.a and the 6-fold rings shown in Figure 3-17.
42 rings merge under compression) As a result, Voronoi volumes of the same type on different rings tend to converge to an average value a) 0 GPa b) 50 GPa c) 100 GPa d) 200 GPa
Figure 3-17 The snapshot of Voronoi of 6-fold Si-O rings at a) 0 GPa, b) 50 GPa, c) 100 GPa, d) 200 GPa
ZŝŶŐƚLJƉĞFigure 3-18 The mean Si-O-Si angle on rings at 0 GPa
Our extensive study examined the structural transformations of MgSiO3 glass and liquid under compression at temperatures of 600K and 3000K The findings reveal that the structural behavior of both systems under compression is notably similar, with an evident increase in structural changes as pressure escalates.
In MgSiO3 glass and liquid, the coordination numbers (CN) for silicon (Si) and magnesium (Mg) atoms reveal distinct behaviors under compression The initial peaks of Si-O in both states shift rightward before moving left, while the O-O pair distribution functions (PRDFs) only shift left Notably, Si-Si and O-O PRDFs undergo rapid transformations under pressure, whereas Si-O PRDFs remain stable, indicating the formation of edge- and face-sharing linkages and an increase in the average CN of Si atoms At 200 GPa, the system predominantly features 9-fold rings, resulting in a pronounced peak in the Si-Si-Si distance distribution at approximately 5.3 Å Concurrently, Si-O linkages contract under high pressure, leading to the splitting of the Si-Si PRDF peak into two subpeaks at 4.6 Å and 5.3 Å This formation of large rings at elevated pressures suggests the presence of Mg-rich regions, as the absence of Si species within these rings generates highly negatively charged oxygen atoms that attract Mg²⁺ ions The non-uniform distribution of Mg contributes to the observed heterogeneity in MgSiO3 Voronoi diagrams indicate that at ambient pressure, the mean Si-Voronoi volume ranges from 7.5 to 7.8 ų for various ring types, while the mean O-Voronoi volume increases from approximately 13.0 ų for 2-fold rings to nearly 16.0 ų for 6-fold rings, before decreasing to 14.3 ų for 11-fold rings, a phenomenon attributed to the topological distortion of SiO₄ units, variations in the mean Si-O-Si angle, and the arrangement of SiOx units in larger rings.
[1] Mountjoy G, Al-hasni B M and Storey C 2011 Structural organisation in oxide glasses from molecular dynamics modelling J Non Cryst Solids 357 2522–9
[2] Pedone A, Malavasi G, Menziani M C, Segre U and Cormack A N 2008 Role of Magnesium in Soda-Lime Glasses : Insight into Structural , Transport , and Mechanical Properties through Computer Simulations J Phys Chem C
[3] Sen S, Maekawa H and Papatheodorou G N 2009 Short-Range Structure of Invert Glasses along the Pseudo-Binary Join MgSiO3 - Mg2SiO4 : Results from 29 Si and 25 Mg MAS NMR Spectroscopy J Phys Chem B 113 15243–
[4] Kohara S, Suzuya K, Takeuchi K, Loong C-K, Grimsditch M, Weber J K R, Tangeman J A and Key T S 2004 Glass Formation at the Limit of Insufficient Network Formers Science (80- ) 303 1649–52
[5] Al-hasni B M and Mountjoy G 2014 A molecular dynamics study of the atomic structure of x(MgO) 100−x(SiO2) J Non Cryst Solids400 33–44
[6] Ghosh D B, Karki B B and Stixrude L 2014 First-principles molecular dynamics simulations of MgSiO3 glass: Structure, density, and elasticity at high pressure Am Mineral.99 1304–14
[7] Anh N H, Son N H and Hong N Van 2023 Pressure-induced Glassy Networks of Enstatite (MgSiO3) and Forsterite (Mg2SiO4) VNU J Sci Math - Phys 39 53–73
[8] André G 1963 X-Ray Diffraction in Crystals Imperfect Crystals and Amorphous Bodies (W.H Freeman, San Francisco)
[9] Bucaro J A and Dardy H D 1974 High-temperature Brillouin scattering in fused quartz J Appl Phys 5324 45
[10] Zachariasen W H 1932 The atomic arrangement in glass J Am Chem Soc
[11] Tuckerman M E and Martyna G J 2000 Understanding Modern Molecular Dynamics: Techniques and Applications J Phys Chem B 104 159–78
[12] Mitra S K 1982 Molecular dynamics simulation of silicon dioxide glass Philos Mag B 45 529–48
[13] Wright A C 1994 Neutron scattering from vitreous silica V The structure of vitreous silica: What have we learned from 60 years of diffraction studies?
[14] Poulsen H F, Neuefeind J, Neumann H B, Schneider J R and Zeidler M D
1995 Amorphous silica studied by high energy X-ray diffraction Nucl Inst Methods Phys Res B 97 162–5
[15] Wu T, He S, Liang Y and Wang Q 2015 Molecular dynamics simulation of the structure and properties for the CaO-SiO2 and CaO-Al2O3 systems J Non Cryst Solids 411 145–51
[16] Mauri F, Pasquarello A, Pfrommer B G, Yoon Y G and Louie S G 2000 Si- O-Si bond-angle distribution in vitreous silica from first-principles 29Si NMR analysis Phys Rev B - Condens Matter Mater Phys 62 R4786–9
[17] Kohara S and Suzuya K 2005 Intermediate-range order in vitreous SiO2 and GeO2 J Phys Condens Matter S77–86 17
[18] Rino J P, Ebbsjử I, Kalia R K, Nakano A and Vashishta P 1993 Structure of rings in vitreous SiO2 Phys Rev B 47 3053–62
[19] Valle R G Della and Andersen H C 1992 Molecular dynamics simulation of silica liquid and glass J Chem Phys 97 2682
[20] Murakami M and Bass J D 2010 Spectroscopic Evidence for Ultrahigh- Pressure Polymorphism in SiO2 Glass Phys Rev Lett 025504 1–4
[21] Wilding M C, Benmore C J, Tangeman J A and Sampath S 2004 Evidence of different structures in magnesium silicate liquids : coordination changes in forsterite- to enstatite-composition glasses Chem Geol 213 281–91
[22] San L T, Hong N Van, Iitaka T and Hung P K 2016 Structural organization, micro-phase separation and polyamorphism of liquid MgSiO3 under compression Eur Phys J B89 73
[23] Wilding M C, Benmore C J and Weber J K R 2008 In situ diffraction studies of magnesium silicate liquids J Mater Sci 4707–13 43
[24] Yin C D, Okuno M, Morikawa H and Marumo F 1983 Structure analysis of MgSiO3 glass J Non Cryst Solids 131–41 55
[25] Wilding M C, Benmore C J, Tangeman J A and Sampath S 2004 Coordination changes in magnesium silicate glasses Europhys Lett 212 212–8
[26] Guignard M and Cormier L 2008 Environments of Mg and Al in MgO – Al2O3 – SiO2 glasses : A study coupling neutron and X-ray diffraction and Reverse Monte Carlo modeling Chem Geol 256 111–8
[27] Kubicki J D and Lasaga A C 1991 Molecular Dynamics Simulations of Pressure and Temperature Effects on MgSiO3 and Mg2SiO4 Melts and Glasses Phys Chem Miner 17 661–73
[28] Matsui Y Y and Kawamura K 1980 Instantaneous structure of an MgSiO3 melt simulated by molecular dynamics Nature 285 648–9
[29] Kapoor S, Wondraczek L and Smedskjaer M M 2017 Pressure-induced Densification of Oxide Glasses at the Glass Transition Front Mater 4 1–20
[30] Gaudio S J, Sen S and Lesher C E 2008 Pressure-induced structural changes and densification of vitreous MgSiO3 Geochim Cosmochim Acta 72 1222–
[31] Kono Y, Shibazaki Y, Kenney-Benson C, Wang Y and Shen G 2018 Pressure-induced structural change in MgSiO3 glass at pressures near the Earth’s core–mantle boundary Proc Natl Acad Sci U S A 115 1742–7
[32] Spera F J, Ghiorso M S and Nevins D 2011 Structure, thermodynamic and transport properties of liquid MgSiO3: Comparison of molecular models and laboratory results Geochim Cosmochim Acta 75 1272–96
[33] Kohara S, Akola J, Morita H, Suzuya K, Weber J K R R, Wilding M C and Benmore C J 2011 Relationship between topological order and glass forming ability in densely packed enstatite and forsterite composition glasses Proc Natl Acad Sci 14780–5 10
[34] Benmore C J, Soignard E, Guthrie M, Amin S A, Weber J K R, Mckiernan
K, Wilding M C and Yarger J L 2011 High pressure x-ray diffraction measurements on Mg2SiO4 glass J Non Cryst Solids 357 2632–6
[35] Wang Y, Sakamaki T, Skinner L B, Jing Z, Yu T, Kono Y, Park C, Shen G, Rivers M L and Sutton S R 2014 Atomistic insight into viscosity and density of silicate melts under pressure Nat Commun 1–10 5
[36] Son N H and Anh N H 2020 Structural Simulation of Mg2SiO4 under Compression VNU J Sci Math - Phys 36 18–28
[37] Yi Y S, Khim H, Kim Y H and Lee S K 2021 Spectral proxies for bonding transitions in SiO2 and MgSiO3 polymorphs at high pressure up to 270 GPa by O K -edge x-ray Raman scattering Phys Rev B 103 26–37
[38] Panero W R, Akber-Knutson S and Stixrude L 2006 Al2O3 incorporation in MgSiO3 perovskite and ilmenite Earth Planet Sci Lett.252 152–61
[39] Cormier L and Cuello G J 2011 Mg coordination in a MgSiO3 glass using neutron diffraction coupled with isotopic substitution Phys Rev B 83
[40] Liu Y, Bai C, Lv X and Wei R 2015 Molecular Dynamics Simulation on the Influence of Al2O3 on the Slag Structure at 1873 K Mater Today Proc 2 S453–9
[41] Son N H, Anh N H, Kien P H, Iitaka T, Van Hong N and Huu P 2020
Topology of SiOx-units and glassy network of magnesium silicate glass under densification: Correlation between radial distribution function and bond angle distribution Model Simul Mater Sci Eng 065007 28
[42] Satoh A 2011 2 - Outline of Methodology of Simulations Introduction to Practice of Molecular Simulation ed A Satoh (Elsevier) pp 29–47
[43] Oganov A R, Brodholt J P and Price G D 2000 Comparative study of quasiharmonic lattice dynamics, molecular dynamics and Debye model applied to MgSiO3 perovskite Phys Earth Planet Inter 122 277–88
[44] Oeffner R D 1999 A computational study of germanium dioxide (University of Cambridge)
[45] Lee M 2017 X-Ray Diffraction for Materials Research: From Fundamentals to Applications (Apple Academic Press)
[46] King S V 1967 Ring Configurations in a Random Network Model of Vitreous Silica Nature 213 1112–3
[47] Taniguchi T, Okuno M and Matsumoto T 1997 X-ray diffraction and EXAFS studies of silicate glasses containing Mg, Ca and Ba atoms J Non Cryst Solids 211 56–63
[48] Wooten F 2002 Structure, odd lines and topological entropy of disorder of amorphous silicon Acta Crystallogr Sect A Found Crystallogr 346–51 58
[49] Goetzke K and Klein H J 1991 Properties and efficient algorithmic determination of different classes of rings in finite and infinite polyhedral networks J Non Cryst Solids127 215–20
[50] Yuan X and Cormack A N 2002 Efficient algorithm for primitive ring statistics in topological networks Comput Mater Sci 343–60 24
[51] Matsumoto M, Baba A and Ohmine I 2007 Topological building blocks of hydrogen bond network in water J Chem Phys 127
[52] Franzblau D S 1991 Computation of ring statistics for network models of solids Phys Rev B44 4925–30
[53] Guttman L 1990 Ring structure of the crystalline and amorphous forms of silicon dioxide J Non Cryst Solids 116 145–7
[54] Aurenhammer F 1991 Voronoi diagrams—a survey of a fundamental geometric data structure ACM Comput Surv 23 345–405
[55] Funamori N and Yamamoto S 2004 Exploratory studies of silicate melt structure at high pressures and temperatures by in situ X-ray diffraction J Geophys Res 109 1–8
[56] Haskins J B, Stern E C, Bauschlicher C W and Lawson J W 2019
Thermodynamic and transport properties of meteor melt constituents from ab initio simulations: MgSiO3, SiO2, and MgO J Appl Phys 125
[57] Tangeman A, Phillips L, Navrotsky A, Weber J K R, Hixson A D and Key T
S 2001 Vitreous forsterite (Mg2SiO4): Synthesis, structure, and thermochemistry Geophys Res Lett 2517–20 28