Percolation transitions in compressed SiO₂ glasses

Abstract

Amorphous–amorphous transformations under pressure are generally explained by changes in the local structure from low- to higher-fold coordinated polyhedra^1,2,3,4. However, as the notion of scale invariance at the critical thresholds has not been addressed, it is still unclear whether these transformations behave similarly to true phase transitions in related crystals and liquids. Here we report ab initio-based calculations of compressed silica (SiO₂) glasses, showing that the structural changes from low- to high-density amorphous structures occur through a sequence of percolation transitions. When the pressure is increased to 82 GPa, a series of long-range (‘infinite’) percolating clusters composed of corner- or edge-shared tetrahedra, pentahedra and eventually octahedra emerge at critical pressures and replace the previous ‘phase’ of lower-fold coordinated polyhedra and lower connectivity. This mechanism provides a natural explanation for the well-known mechanical anomaly around 3 GPa, as well as the structural irreversibility beyond 10 GPa, among other features. Some of the amorphous structures that have been discovered mimic those of coesite IV and V crystals reported recently^5,6, highlighting the major role of SiO₅ pentahedron-based polyamorphs in the densification process of vitreous silica. Our results demonstrate that percolation theory provides a robust framework to understand the nature and pathway of amorphous–amorphous transformations and open a new avenue to predict unravelled amorphous solid states and related liquid phases^7,8.

Main

Understanding the physical mechanisms controlling the transformation from one amorphous 'phase' to another is an open fundamental issue in materials science^1,2,8,9. The short-range structures of amorphous solids such as SiO₂, GeO₂, Si, Ge and the chalcogenides are very similar to those of their crystalline counterparts and it is the random nature of the inter-unit connections that produces disorder at long length scales. In such systems, it has been argued that amorphous states can be described in terms of changes in the electronic and structural properties at short- and medium-range order. These properties include electronic bonding^2,4,10, the coordination number^1,3,4,11 and the ring distribution^11,12. However, to describe the passage from one amorphous 'phase' to another at the critical threshold, an explicit scale-invariant quantity (that is, an order parameter) needs to be defined. This concept has not been considered for pressure-driven amorphous solid transformations thus far.

When an amorphous solid is pressurized, it is commonly assumed that transformations from low to high density occur gradually, with coexisting low- and high-fold coordinated polyhedra. This structural evolution, sometimes referred to as polyamorphism, differs from the polymorphism of crystals, in which transitions occur from a specific phase to another at a critical pressure. The change from tetrahedral (Si–O coordination number Z = 4) to octahedral (stishovite-like structure, Z = 6) structures in vitreous silica (v-SiO₂) is accompanied by a well-known mechanical anomaly at 3 GPa (refs. ^13,14), a percolation phenomenon at about 7 GPa (ref. ¹⁵) and a two-step elastic-to-plastic transformation, one around 10 GPa (refs. ^12,14,16) and a second around 20 GPa (ref. ¹⁴).

A recent ab initio machine learning method applied in combination with empirical force fields in compressed amorphous silicon highlights a three-step transformation sequence for amorphous silicon under increasing pressure⁴. However, the passage from one 'phase' to another was not addressed. For v-SiO₂ at ambient pressure, ab initio methods complement experimental data^17,18, but, despite many efforts^12,19,20, application to the above-mentioned issues has been limited due to the prohibitive calculation times required for a reliable thermodynamic sampling^21,22.

We were thus motivated to use an ab initio-based approach, the self-consistent-charge density functional-based tight-binding (SCC-DFTB) method²³. Combined with the molecular dynamics (MD) technique, it is able to reproduce many features of crystalline silica (c-SiO₂) and v-SiO₂ with an accuracy similar to those of ab initio methods, but at least three orders of magnitude faster. This allowed us to simulate more than forty SiO₂ glasses and explore the full pressure range up to 82 GPa at room temperature. We also decompressed the samples at 8 GPa and 10 GPa (for details, see Methods). Our samples contain 1,008 atoms (336 SiO₂ units) and are large enough to observe and describe the percolation critical phenomena governing the aforementioned transformations.

Polyhedra coordination and connectivity

Figure 1a illustrates the v-SiO₂ transformation from Z = 4 to 6 with increasing pressure p. At ambient temperature and 0 < p < 3 GPa, similar to c-SiO₂, the Si–O bond of v-SiO₂ is characterized by an sp³ hybridization that favours the formation of SiO₄ tetrahedra. For 3 < p < 10 GPa, the densification of v-SiO₂ includes changes in the electronic structure, with an increase in the Fermi energy and of the atomic charges (Extended Data Figs. 1 and 2). These changes produce an increase in the Si–O bonding ionicity (Extended Data Fig. 1c), boosting the formation of higher-fold coordinated polyhedra (Fig. 1a, b). As a consequence, beyond 10 GPa, the inter-polyhedra connectivity changes from purely corner-sharing (CS) tetrahedra to a more complex connected network (Extended Data Figs. 3 and 4). The latter involves corner-sharing and edge-sharing (ES) SiO₅–SiO₆ and SiO₆–SiO₆ (Fig. 1a, c) and a lower extent of face-sharing (FS) polyhedra (Extended Data Fig. 5).

Fig. 1: v-SiO₂ local structures and connectivities.

a, Snapshots of the simulation boxes showing the evolution from SiO_n to SiO_n+1 polyhedra, and from CS to ES connectivity (insets). The glass samples are pressurized by instantaneous reduction of the simulation box followed by a relaxation using MD with periodic boundary conditions. b, Fraction of Si polyhedra (top) and coordination numbers (bottom) as a function of pressure. Small symbols and arrows nearby correspond to unloadings from 8 GPa and slightly above 10 GPa. c, Shared corners and shared edges per polyhedron. The bold vertical arrows at the top of b and c mark the onset of the different regimes (see text) and the dashed vertical line indicates the elastic-to-plastic transition. Note that the scale changes at 20 GPa in the x axis. See Methods for calculation details.

On increasing the pressure from ambient conditions, c-SiO₂ changes from a tetrahedral local structure to an octahedral one through structural transformations including α-quartz to coesite I (Z = 4) at 3 GPa, then coesite I to stishovite (rutile structure, Z = 6) at 9 GPa, with a higher ionic bonding contribution¹⁰. Regarding the local environment of the O atoms, CS tetrahedra yield to divalent OSi₂ structures, and ES octahedra to trivalent OSi₃ structures. In the crystalline phase at ambient pressure, each CS tetrahedron is connected to four neighbouring tetrahedra, leading to Z_SiSi = 4 and Z_OO = 6. In stishovite, each SiO₆ is connected to ten octahedra (Z_SiSi = 10), two of which are ES and eight CS, and Z_OO = 12. In v-SiO₂, the coordination numbers exhibit similar trends with pressure (Fig. 1b), as well as jumps that allow us to identify different regimes at intermediate pressures (Fig. 1b, c, top arrows).

For 13 < p < 20 GPa, the compressed glasses contain similar fractions of SiO₄, SiO₅ and SiO₆, with a total number of shared edges per polyhedron of n_ES ≈ 1.4 and a small contribution of FS polyhedra. This is similar to the crystalline polymorph discovered recently, the so-called coesite IV⁶, which is rarely observed for high-valence and low-coordinated cations in crystals, as it is at odds with the fifth Pauling rule²⁴. For 20 < p < 45 GPa, the fraction of SiO₆ octahedra increases with a jump, while the fraction of SiO₄ tetrahedra decreases to almost zero. In parallel, n_ES increases to about 1.7 and a small fraction of FS polyhedra persist, which looks similar to coesite V, the second recently discovered c-SiO₂ polymorph⁶. For 45 < p < 70 GPa, the network connectivity is fully dominated by SiO₆ octahedra, either CS or ES, and n_ES ≃ 2, which reveals a stishovite-like 'phase'. For p > 70 GPa, n_ES > 2 due to the emergence of SiO₇ polyhedra, suggesting the onset of a post-stishovite-like 'phase'^20,25.

Finally, we calculated the fraction of Si polyhedra and the coordination number for unloadings from 8 GPa and slightly above 10 GPa. In the former case, the decompression is reversible, but it is not in the latter (Fig. 1b, small symbols). The pressure threshold at which irreversibility occurs agrees with experimental observations¹⁴. When unloading from 10 GPa to ambient pressure, a hysteresis is clearly observed due to persisting residual SiO₅ pentahedra (hence Z ≠ 4, Fig. 1b), confirming previous expectations and pointing out the key role of five-fold coordinated polyhedra in the plastic-to-elastic transition of v-SiO₂ (refs. ^12,14,16).

Percolation transitions

The above observations only include an evaluation of the local and medium-range order. However, the nature of the structural transformations from one ‘phase' to another in compressed v-SiO₂ should also be substantiated by the structure at a long length scale. Indeed, we found that clusters built from SiO_n polyhedra (including their mixtures, (SiO_n–SiO_m) eventually emerge. One of these becomes dominant and percolates by spanning its structure from one side of the simulation box to the other (Fig. 2a). This effect has important implications for the glass rigidity^15,26 and can be monitored by calculating the percolation probability, P_∞. The latter is often used as an order parameter for the analysis of a percolation transition. It tends to 1 (or 0) if the largest cluster percolates (or not), and the percolating cluster is representative of the new 'phase'²⁷. The notation (SiO_n–SiO_m)_∞ used in the following corresponds to a percolating cluster composed of alternating n- and m-fold coordinated polyhedra.

Fig. 2: Sequence of percolation transitions in v-SiO₂.

a, Snapshots of the spanning clusters in the simulation box, for pressures corresponding to P_∞ = 0 up to P_∞ = 1, and back to P_∞ = 0 for the first one, for (from left to right) CS (SiO₄–SiO₅)_∞ and (SiO₅–SiO₅)_∞ clusters and ES (SiO₆–SiO_5,6)_∞ and (SiO₆–SiO₆)_∞ clusters. b, Percolation probability, P_∞, versus pressure for the different four-, five-, six- and seven-fold coordinated Si and their combinations. Dashed and dotted magenta lines correspond to P_∞ for the cluster (SiO₄–SiO₅)_∞ for decompression from 8 GPa and 10 GPa, respectively. c, Correlation length ξ of the percolating clusters in b. d, Bulk modulus K as a function of pressure compared to high-frequency experimental data¹⁴. The inset is a magnification of the low-pressure region. Error bars are estimated by error propagation. Note that the scale changes at 20 GPa in the x axis. See Methods for calculation details.

Figure 2b shows the percolation probability, P_∞, of the largest cluster for connected networks of SiO₄, SiO₅ and SiO₆ polyhedra, and their mixtures, as a function of pressure in v-SiO₂ (density-dependent P_∞ is shown in Extended Data Fig. 6a). For the (SiO₄–SiO₄)_∞ cluster, P_∞ decreases from 1 to 0 at 13 GPa. In addition, between 3 and 8 GPa, a percolating cluster emerges that is composed of a connected skeleton of alternating tetrahedra and pentahedra, (SiO₄–SiO₅)_∞. This coincides with the percolation transition observed at 7 GPa (ref. ¹⁵). With increasing pressure, two more percolating clusters appear at 10 GPa, (SiO₅–SiO₅)_∞ and (SiO₅–SiO₆)_∞, coexisting with the first one. The percolating cluster structures containing a mixture of SiO₄ and SiO₅ between 8 and 10 GPa, as well SiO₆ between 10 and 13 GPa, recall the pressure-induced post-quartz amorphous states²⁸. At 13 GPa, a (SiO₆–SiO₆)_∞ cluster appears and percolates. The coexistence of all these percolating clusters beyond 13 GPa is similar to coesite IV, the crystalline structure of which combines SiO₄–SiO₅, SiO₅–SiO₅, SiO₅–SiO₆ and SiO₆–SiO₆ planes in different crystallographic directions. By analogy, the vitreous state for 13 GPa < p < 20 GPa is labelled v-coesite IV in Fig. 2b.

Slightly above 20 GPa, SiO₄ tetrahedra vanish and, accordingly, the P_∞ of the (SiO₄–SiO₅)_∞ cluster decreases to 0. Simultaneously, a percolating cluster, (SiO₆–SiO_5,6)_∞, emerges, which is composed of ES SiO₆–SiO₅ and SiO₆–SiO₆ polyhedra, together with some FS contribution. In this pressure range, the amorphous state (labelled v-coesite V in Fig. 2b) possesses the same dominant ES polyhedra connectivity as well as alternations of SiO_n–SiO_m polyhedra equivalent to those in coesite V⁶. Our result correlates with the irreversible structural behavior observed experimentally above about 20 GPa in v-SiO₂ (ref. ¹⁴), suggesting that ES structures remain stable during decompression.

Analysis of the percolation probability of a cluster composed of pure ES octahedra, similar to stishovite, provides evidence that such a cluster percolates around 40 GPa and, accordingly, a v-stishovite state replaces the v-coesite V one (Fig. 2a, b). The pressure at which this occurs is in good agreement with the value reported for compressed v-SiO₂ when Z becomes equal to 6 (ref. ²⁵). Therefore, our results demonstrate that, instead of a single and gradual transition, the mechanism of the structural transformation from Z = 4 to 6 includes a series of percolation transitions between well-defined amorphous states (Fig. 2). Moreover, estimation of P_∞ as a function of the SiO_n fraction shows that the critical fractions lie in the range expected from percolation theory, that is, around 2/Z (Methods and Extended Data Fig. 6b). Finally, an analysis from the viewpoint of O atoms also reveals that OSi₂–OSi₃ and OSi₃–OSi₃ clusters percolate around 3 GPa and 10 GPa, namely, at pressures similar to those for the SiO_n–SiO_m structures (Methods, Fig. 2b and Extended Data Fig. 7c).

The correlation length ξ has been estimated in the pressure range at which the different percolating clusters appear. This quantity accounts for the maximum length at which scale invariance exists. At the thermodynamic limit, this quantity is expected to diverge at the critical pressure and it should reach a maximum for finite size systems, which is indeed the case for the different situations described above (Fig. 2c). Beyond 70 GPa, a large cluster of ES (SiO₆–SiO₇)_∞ polyhedra starts to grow; this, together with the fact that at this pressure the total number of shared edges per Si polyhedron exceeds 2, supports the post-stishovite 'phase' suggested previously. The jump in the percolation probability P_∞ from 0 to 1 provides evidence of the dominant character of a well-defined amorphous state in a given pressure range. However, the non-negligible partial overlap between the ξ curves beyond 3 GPa may indicates a 'phase' coexistence between the dominant percolating cluster and a ‘metastable’ emerging state (Fig. 2c) when approaching the critical pressures. This is similar to the metastability observed in coesite IV and V polymorphs^5,6.

When unloading from 8 GPa, the percolating (SiO₄–SiO₅)_∞ cluster vanishes at a pressure similar to that at which it percolates during compression (Fig. 2b). This confirms the reversible character suggested above by analysis of the local and medium-range structure. Conversely, when the sample is decompressed from 10 GPa (dashed curve), P_∞ exhibits a hysteresis, that is, the (SiO₄–SiO₅)_∞ cluster continues to percolate at pressures much lower than for the percolating compression route (Fig. 2b). A similar plastic behaviour is observed for the other percolating clusters, that is, (SiO₅–SiO₅)_∞ and (SiO₅–SiO₆)_∞ (data not shown). This demonstrates that the formation of percolating clusters containing SiO₅ pentahedra for p ≳ 10 GPa prevents recovery of the pristine glass structure and probably explains the irreversible behaviour.

To explore the impact of these percolation transitions on glass rigidity, we calculated the bulk modulus K of our pressurized silicas (Fig. 2d). The increasing connectivity at short and long length scales yields a strong increase in K, that is, from approximately 30 GPa at ambient conditions up to approximately 375 GPa at p = 82 GPa. Its evolution parallels those of its crystalline counterparts: SiO₄ polymorphs (except coesite I) below 10 GPa, coesite IV and V at intermediate pressures and stishovite above 35–40 GPa (refs. ^6,29). Our calculations also reproduce the well-known minimum around 3 GPa (Fig. 2d, inset) and we associate the increased rigidity starting above 4 GPa to the series of percolating clusters that emerge with increasing pressure. Finally, our estimation of K is very close to high-frequency experimental data except perhaps around 10 GPa, where the simulations exhibit a minimum, when (SiO₅–SiO₅)_∞ and (SiO₅–SiO₆)_∞ clusters emerge. These results suggest that the mechanical properties relate primarily to the peculiar connectivity describing the system at all length scales, rather than to the crystalline or amorphous nature of the network.

We have also analysed the structure of the percolating clusters at the critical pressure thresholds. For this, we expanded these clusters out of the box, taking into account the periodic boundary conditions, and computed their structure factor S(q). As expected from percolation theory, at the critical thresholds the clusters are fractals, that is, $S(q)\propto {q}^{-{D}_{{\rm{f}}}}$ at intermediate wavevector q values, with D_f ≃ 2.5, in agreement with the value predicted theoretically (Fig. 3).

Fig. 3: Fractal percolating clusters at threshold.

Structure factors S(q) for the isolated and extended v-SiO₂ percolating clusters near the critical pressures (indicated in the snapshots) in log–log scale. See Methods for calculation details.

Conclusions and outlook

The structural transformation from low- to high-density amorphous SiO₂ solid occurs through a sequence of percolation transitions. Each emerging amorphous phase (polyamorph) actually corresponds to a state of a megabasin of the configurational energy landscape³⁰, and each megabasin is associated with a specific crystalline polymorph with the same local structures and connectivities as those of the percolating cluster. When the pressure is increased and the energy barrier separating two megabasins is overcome, a transformation is induced to an amorphous state with higher connectivity. Other pressure- or temperature-driven amorphous transformations in oxide glasses, chalcogenides and metallic glasses could be explained in these terms. A percolation phenomenon has been proposed to explain the transition between low- and high-density liquid phases of water³¹. The same has been argued for the glass transition in model systems^32,33,34. Changes in the local connectivity similar to those observed here have been demonstrated in pressurized SiO₂, GeO₂ and H₂O liquids, but for different pressure windows and interpolyhedra distances^35,36,37,38. The question whether amorphous states are related to underlying and unidentified liquid phases can now be addressed by considering a scale-invariant quantity as the percolation probability. This will help to catalogue amorphous solids and unveil their affinities with liquid and crystalline phases.

Methods

Glass preparation

The initial glass configuration of 1,008 atoms (336 SiO₂ units) in a cubic simulation box was prepared by carrying out classical MD simulations using periodic boundary conditions. We used the melt-and-quench approach and the constant volume–constant temperature (NVT) ensemble, and the atomic interactions were modelled using the Van Beest–Kramer–VanSanten (BKS) pair potential³⁹. We then carried out MD simulations within the framework of the SCC-DFTB method for the electronic structure calculations²³. These calculations were also performed using the NVT ensemble. An Andersen thermostat was used to maintain room temperature and the samples were relaxed for more than 40 ps, with a time step of 2 fs. Within the SCC-DFTB approximation, the electronic integrals corresponding to the Hamiltonian matrix elements were replaced by analytical functions. The coefficients of these functions were obtained by fitting DFT calculations corresponding to systems obtained at different physico-chemical conditions. We used the Periodic Boundary Conditions (PBC)-0-3 set of parameters for SiO₂ (ref. ⁴⁰). This resulted in an electronic structure calculation method at least three orders of magnitude faster than DFT, which is currently considered the best-performing ab initio method in materials science.

The orbitals were occupied according to a Fermi–Dirac distribution with a temperature of 300 K. We included Broyden charge mixing in the SCC cycle with a mixing parameter of 0.1. The Γ-point approximation was used for the electronic band structure energy and the projected electronic density of states (DOS) calculations. For the latter, a Gaussian broadening of 0.4 eV was considered.

Glass compression

At ambient pressure, the glass sample prepared using the above procedure had a density of 2.3 g cm⁻³. To mimic quasi-static compression, the box length was reduced in steps of 1% (and atom positions rescaled) every 40 ps, until a pressure of approximately 27 GPa was reached. A second series of glasses was obtained by instantaneous volume reduction of the sample obtained at 15 GPa, thus yielding seven samples at target pressures of up to 82 GPa. Two successive volume reductions of 0.125% followed by a 40-ps relaxation were performed on six samples from this second series. The largest compression rate used here is almost equivalent to 0.02 GPa ps⁻¹. This corresponds to a compression velocity at least one order of magnitude slower than that used in a previous DFT study of compressed SiO₂ at T = 300 K (ref. ¹²), while the number of atoms in our calculation box is almost seven times larger. To distinguish between the elastic and plastic regime, we also decompressed the samples from 8 and 10 GPa. This was done by increasing the box length and scaling the atom positions by 1% every 40 ps, until ambient pressure was reached. The reported structural quantities were averaged over all configurations generated in the last 10 ps of the sample relaxation time.

Electronic structure

For the simulated compressed glasses we analysed the total and projected DOS, the Fermi energies and the Mulliken atomic charges (Q_Si for Si and Q_O for O), as well the average Mulliken ionicity κ_Si–O, calculated as done elsewhere⁴¹. For the latter, the relation κ_Si-O = |Q_Si/v_Si – Q_O/v_O|/2 was used, where ν_Si and ν_O are the valences of Si and O atoms, respectively. The total and projected O-2p DOSs of the simulated glass at ambient pressure are in good agreement with the experimental and DFT data¹⁸ (Extended Data Fig. 1a). The effect induced by sample densification in the total DOS of v-SiO₂ also agrees^10,19 (Extended Data Fig. 1c, d). Similarly, the Si and O Mulliken atomic charges and the Si–O bond ionicity change are consistent with previous experimental and DFT findings^10,12 (Extended Data Fig. 2). This favours the formation of pentahedra and higher-fold coordinated Si polyhedra⁴² (Fig. 1). The maxima reached in the pressure range of 10–20 GPa where the SiO₅ fraction is dominant (Fig. 1) and the slight decreases beyond 20 GPa highlight the large Si–O bonding ionic character of SiO₅ pentahedra compared to octahedra and tetrahedra.

Structure factor

To compare the structural properties of our samples to experimental data, we computed the neutron and X-ray total static structure factors. We first computed the partial structure factors S_αβ(q) using the definition⁴³

$${S}_{\alpha \beta }(q)=\frac{{f}_{\alpha \beta }}{N}\mathop{\sum }\limits_{j=1}^{{N}_{\alpha }}\mathop{\sum }\limits_{k=1}^{{N}_{\beta }}\langle \exp (i\overrightarrow{q}\cdot ({\overrightarrow{r}}_{j}-{\overrightarrow{r}}_{k}))\rangle ,\,\,\,\alpha ,\beta ={\rm{S}}{\rm{i}},{\rm{O}}\cdot $$

Here, α, β = Si, O. Here, f_αβ = 1 for α = β and f_αβ = 1/2 otherwise, N_α is the number of particles of species α and N is the total number of atoms. The total structure factors are combinations of partial structure factors. For the neutron structure factor, we used the relation⁴³

$${S}_{{\rm{N}}}(q)=\frac{N}{{\sum }_{\alpha ={\rm{Si}},{\rm{O}}}{N}_{\alpha }{b}_{\alpha }^{2}}\sum _{\alpha ,\beta ={\rm{Si}},{\rm{O}}}{b}_{\alpha }{b}_{\beta }{S}_{\alpha \beta }(q),$$

with the neutron scattering length b_α given by b_Si = 4.1491 fm and b_O = 5.803 fm, respectively⁴⁴. The X-ray total structure factor S_X(q) is given by⁴⁵

$${S}_{{\rm{X}}}(q)=\frac{N}{{\sum }_{\alpha }{N}_{\alpha }{f}_{\alpha }^{2}(q/4{\rm{\pi }})}\sum _{\alpha ,\beta }{f}_{\alpha }(q/4{\rm{\pi }}){f}_{\beta }(q/4{\rm{\pi }}){S}_{\alpha \beta }(q)\,.$$

Here f_α(s) is the scattering-factor function (also called the form factor), computed as a linear combination of five Gaussians using parameters derived elsewhere⁴⁶.

The calculated X-ray S_X(q) and neutron S_N(q) structure factors are compared to experimental data^3,11. The calculated S_X(q) reproduces the decrease and shift of the first sharp diffraction peak (FSDP) due to the collapse of the open structure of v-SiO₂ at intermediate length scales (Extended Data Fig. 3).

Local connectivity

We computed the pair distribution function g_αβ(r) for all compressed v-SiO₂ samples using the definition⁴³

$${g}_{\alpha \beta }(r)=\frac{V}{{N}_{\alpha }({N}_{\beta }-{\delta }_{\alpha \beta })}\langle \mathop{\sum }\limits_{i=1}^{{N}_{\alpha }}\mathop{\sum }\limits_{j=1}^{{N}_{\beta }}\frac{1}{4{\rm{\pi }}{r}^{2}}\delta (r-|{\overrightarrow{r}}_{i}-{\overrightarrow{r}}_{j}|)\rangle ,$$

where $\langle \cdot \rangle$ represents the thermal average, V is the volume of the simulation box and δ_αβ is the Kronecker delta function. For each of the three pairs, the first peak of the corresponding pair distribution function corresponds to the distribution of the first-neighbour shell distances for Si–O, O–O and Si–Si, respectively. For the Si–O pair, a minimum after the first peak located at 2.3 Å defines well the upper limit of this distribution at all pressures. We used this value as a cutoff distance to estimate the Si–O coordination number and the fraction of SiO_n polyhedra for the considered pressures. Similarly, a cutoff of 3.5 Å was used to identify the first Si–Si neighbours and to estimate the evolution of the polyhedra coordination number with pressure (Fig. 1b). Identification of the Si–O and Si–Si first neighbours also allowed us to estimate the number of O atoms shared by two Si polyhedra neighbours, that is, one, two, three or more, corresponding to CS, ES and FS polyhedral connectivity, respectively (Fig. 1c and Extended Data Fig. 5). The computed Si–O distances are compared with experimental data in Extended Data Fig. 4a. The agreement is remarkable, in particular in view of recent X-ray measurements³. In this figure, the O–O and Si–Si distances are also shown, for information. The latter are compared to Si–Si distances in the crystalline counterparts, showing that the minima around 10 GPa arise from the formation of coesite-like structures in the glass, in particular SiO₅ pentahedra.

The Si–O–Si and O–Si–O bond-angle distributions and their pressure dependence are plotted in Extended Data Fig. 4b, c. The above-mentioned upshift of the FSDP translates into a fast reduction of the angles up to the first plastic transformation around 10 GPa when CS SiO₅- and SiO₆-based percolating clusters emerge. With increasing pressure above 10 GPa, and in addition to the primary peak at around 130°, we notice the occurrence of two other peaks pointing approximately at 98° and 68°, respectively, in the bond-angle distribution of Si–O–Si. The positions of the three peaks show a weak pressure dependence, and the two peaks located at smaller angles become prominent in the pressure range of v-coesite IV and V for the first one and v-stishovite for the second. The Si–O and Si–Si bond lengths exhibit anomalies at pressures of approximately 10 GPa and 20–25 GPa, corresponding to the two stages of plastic transformation, and the O–O bond length presents a maximum at a pressure of about 3 GPa, corresponding to the minimum of the bulk modulus.

To provide more insights into the medium range of the silica network with increasing pressure, we computed the ring distributions. No changes occur for pressures below 4 GPa. For pressures below 8 GPa, we observe a mild increase in small ring sizes (2-, 3- and 4-membered ones) accompanied by a decrease in the proportion of large ring sizes (7- and 8-membered ones). When approaching the threshold of the plastic regime at around 10 GPa, the same pressure dependences are observed, but with stronger trends. This increase in small rings at the expense of large ones at high pressure confirms previous DFT calculations¹².

Cluster analysis

To identify the different clusters with specified polyhedra coordination and connectivity, as well the largest cluster, we used a strategy inspired by the friends-of-friends algorithm, which is widely used to characterize dark-matter halos from N-body simulations⁴⁷. In our implementation, the first nearest neighbours correspond to the specific Si–O coordination and polyhedra connectivity we want to analyse. To determine whether the largest cluster percolates or not, we first extract all clusters from the box, except the largest one. We then replicate the box containing this cluster along the three spatial directions. The friends-of-friends algorithm is applied again and, if the resulting largest cluster is larger than the box size in at least two directions, we assume that the cluster percolates. For a given pressure, the above procedure is implemented for each possible SiO_n–SiO_n+1 connectivity, that is, CS, ES and FS polyhedra.

The percolation probability P_∞ is defined as the number of times the resulting largest cluster percolates during the last 10 ps of the relaxation time, normalized by the total number of explored configurations. For different cluster connectivities, Fig. 2b and Extended Data Fig. 6a, b show P_∞ as a function of pressure, sample density and SiO_n fraction, respectively. In Extended Data Fig. 6b we plot only the data for the connected polyhedra that have the same coordination, otherwise the fraction of mixed coordination cannot be properly defined. The critical fractions where CS (SiO_n–SiO_n)_∞ (n = Z = 5 and 6) percolating clusters emerge and SiO₄–SiO₄ ones disappear are consistent with the expected critical occupation probability in bond percolation (P_c ≈ 2/Z). For example, in square lattices and diamond networks (Z = 4), P_c is equal to 0.5 and approximately 0.39, respectively, and in triangular lattices and simple cubic networks (Z = 6) is equal to approximately 0.35 and 0.25, respectively. For ES (SiO₆–SiO₆)_∞, the critical fraction of SiO₆ is larger than for CS cases. This is because only a small number of SiO₆ octahedra are ES-connected. Note that ES connectivity is similar to a site percolation mechanism presenting a larger P_c than the one resulting from bond percolation.

The correlation length ξ was calculated using the definition²⁷

$${\xi }^{2}=\frac{{\sum }_{s}2{R}_{g,s}^{2}{s}^{2}{n}_{s}}{{\sum }_{s}{s}^{2}{n}_{s},}$$

where R_g,s and n_s are the gyration radius and the number of clusters of size s, respectively. The sum runs over all clusters of size s, excluding the largest one if it percolates. The gyration radius R_g,s was estimated using the relation

$${R}_{g,s}^{2}=\frac{1}{2{s}^{2}}\sum _{i,j}{r}_{ij}^{2},$$

where the sums run over pairs of Si atoms belonging to each cluster of size s. The correlation length ξ is averaged over the last 10 ps of the relaxation time.

Mechanical properties

We estimated the bulk modulus K of the compressed glasses by using the relation K = −Vdp/dV, where V is the sample volume and p is the pressure averaged over the last 10 ps of the relaxation time. To obtain K, we used the central difference scheme, where its error bar is calculated by error propagation and the pressure error is estimated from the standard deviation. To minimize errors in the finite difference, when two subsequent pressures lie within the estimated error bars, the second is excluded from the estimation of K. We recall that volumetric measurements give direct access to the static compressibility¹³, whereas high-frequency experiments measure the volumetric variations at the timescale of the relaxational processes probed by the instrument¹⁴. In compressed v-SiO₂, the two values are similar in the elastic regime (p < 10 GPa), but strong variations have been observed in the plastic case for 10 < p < 55 GPa (refs. ^14,48,49). Interestingly, above 12 GPa, our calculated K values are much closer to the high-frequency data than to the static ones, a behaviour that possibly reflects the typical time relaxation (approximately 10 ps, corresponding to gigahertz frequencies) and the small volumetric variation (<0.4 nm³) used for the estimation of K in our MD simulations. For p > 55 GPa, the experimental value (static and high-frequency) is around 420 GPa (refs. ^48,49), in good agreement with our calculations.

Affinities with c-SiO₂ polymorphs

To identify the SiO₂ polymorph most akin to amorphous states, we built supercells containing 582 atoms for tridymite, cristobalites, coesites and stishovite, and 432 and 486 atoms for β- and α-quartz, respectively. The unit cells of the polymorphs were obtained from crystallographic databases under AMCSD codes 0000789 for α-quartz, 0018071 for β-quartz, 0001629 for α-cristobalite, 0017646 for β-cristobalite, 0000531 for tridymite and 0001306 for stishovite⁵⁰. These supercells were relaxed using the Limited memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) minimization method implemented in the DFTB+ package²³. Only the lengths of the lattice vectors were allowed to vary during structure optimization (with the exception of coesites IV and V) to preserve the same sample densities reported in ref. ⁶.

The criterion used to define the v-coesite IV 'phase' relies on the coexistence of (SiO₄–SiO₅)_∞, (SiO₅–SiO₅)_∞, (SiO₅–SiO₆)_∞ and (SiO₆–SiO₆) _∞ percolating clusters. We found that almost all the SiO₅–SiO₆ and SiO₆–SiO₆ are ES, with a weak contribution of FS polyhedra, as in coesites IV and V. Indeed, in coesite IV we have validated the coexistence of different crystalline planes that have the above-mentioned connectivities.

The ES SiO₅–SiO₆ and SiO₆–SiO₆ polyhedra become dominant at a pressure of 20 GPa. They are in a similar proportion (around 40%) and control the network connectivity as in coesite V. Beyond 40 GPa, the fraction of SiO₅ starts to decrease and the SiO₆–SiO₆ connectivity then becomes dominant, which allows us to identify the stishovite-like 'phase'.

To validate the above observations, the total DOS, Fermi energy, Mulliken atomic charges and bond ionicity were computed and compared to the results obtained for c-SiO₂ polymorphs (Extended Data Figs. 1 and 2). The electronic properties of v-SiO₂ appear to be closer to α-quartz than to other SiO₄-based polymorphs at ambient pressure, as suggested elsewhere¹⁰. With increasing pressure, the electronic properties of the vitreous polyamorphs—v-coesite IV, v-coesite V and v-stishovite—are also compatible with those of coesite IV and V and stishovite crystals, respectively. The ionic character of the Si–O bond is also larger for the five- and six-fold Si coordinated polyhedra⁵¹. A similar conclusion holds when comparing the polyhedra coordination and connectivity with the corresponding v-SiO₂ percolating clusters, as well as for the calculated bulk moduli. The latter agree very well with those of SiO₄-tetrahedra-based polymorphs at low pressures and to those of coesite IV and V and stishovite (Fig. 2d) when the pressure is increased beyond 13 GPa.

Finally, Extended Data Table 1 presents the densities of the crystalline polymorphs and the corresponding pressure ranges^6,28,52,53. A comparison with the corresponding polyamorph states of v-SiO₂ is striking. One notices, for example, that a small pressure of around 3 GPa is enough to compact the open structure of vitreous silica up to a density very close to that of α-quartz. At high pressure, the density ranges of the polyamorphs follow those of their crystalline counterparts.

Percolating clusters at the critical threshold

We analysed the structures of the percolating clusters around the critical pressures. For this, we expanded and extracted the percolating clusters out of the box by taking into account the periodic boundary conditions. At the critical threshold, these clusters are usually larger than the box size because of their branched structure. We computed the structure factor S(q) of the isolated clusters by using the corresponding expression for a diluted system⁵⁴:

$$S(q)=1+\frac{1}{N}\sum _{i\ne j}\frac{\sin (q{r}_{ij})}{q{r}_{ij}},$$

where N is the number of Si atoms and q is the modulus of the wavevector. This expression implies that S(0)  = N_Si and S(∞) = 1, where ${N}_{{\rm{Si}}}$ is the number of polyhedra (Si atoms). For fractal structures, at intermediate q values S(q) scales as ${q}^{-{D}_{{\rm{f}}}}$, where D_f is the fractal dimension. For compressed v-SiO₂ near the critical pressures, we have found that D_f = 2.5, in agreement with the value predicted by percolation theory (Fig. 3).

Connectivity of OSi_n structures

With increasing pressure, the evolution of the coordination number of the O atoms, Z′, parallels that of the Si atoms; that is, Z′ passes from 2 to 3 when Si octahedra (Z = 6) replace tetrahedra (Z = 4) (Extended Data Fig. 7a, b). The anomalous steep jumps defining the different amorphous states, as described in the main text, are also reproduced. From a strictly structural point of view, these observations suggest that the two approaches are similar. The abrupt increase in OSi₃ structures around 10 GPa leads to the percolation of (OSi₃–OSi₃)_∞ clusters, in line with the percolation of (SiO_n–SiO_m)_∞ clusters (n,m = 5,6) between 10 and 13 GPa (Fig. 2b and Extended Data Fig. 7c). Interestingly, the pressures at which OSi₂–OSi₃ and OSi₃–OSi₃ structures percolate, approximately 3 GPa and 10 GPa, are similar to the characteristic pressures corresponding to the minimum of the bulk modulus and the elastic-to-plastic transition, respectively.

The fact that these mechanical properties correlate with the percolation of (OSi_n-OSi_m)_∞ clusters probably underlines the role of the Si–Si non-bonded interactions, as stated by O’Keeffe and Hyde in their model^55,56, based on the observation that the distance between non-bonded first-neighbour cations in many non-molecular crystals is nearly independent of the bridging atom (anion). When applied to silicates, this implies that the structures are constrained by the problem of fitting large Si atoms around small O atoms. However, the model deserves to be reformulated in view of our results. Indeed, it was developed for stable crystalline minerals, whereas with pressurized v-SiO₂ we are facing unrelaxed structures with connectivities that mimic those of metastable coesites IV and V. These structures are at odds with the third and fifth Pauling rules²⁴ and show non-monotonic behaviour in their electronic and structural properties around 10–20 GPa (Extended Data Figs. 2 and 4a, respectively). These effects can hardly be accounted for by the model in its current formulation. With a view to future developments, we have noticed that the ratio between the glass density and the O–O distance is almost constant over the entire pressure range explored, suggesting that the non-bonded O–O distance could be an interesting marker to follow.

Extended data figures and tables

Extended Data Fig. 1 Electronic structure of compressed v-SiO₂.

Total DOS for a SiO₂ glass at ambient pressure, with its corresponding projected DOS (a) and compared to different SiO₄ crystalline polymorphs (b). Fermi energy (c) and total DOS (d) when the pressure increases. The results are compared with those corresponding to different SiO₂ crystalline polymorphs.

Extended Data Fig. 2 Ionic bonding of v-SiO₂.

Mulliken atomic charges for Si (a) and O (b), and the average Mulliken ionicity of the Si-O bond (c) in v-SiO₂ as a function of pressure. The results (circles) are compared with those corresponding to different SiO₂ crystalline polymorphs. The error bars in (b) correspond to the standard deviation of the average of the charges of all O atoms. Similar relative errors were estimated for (a) and (c).

Extended Data Fig 3 v-SiO₂ atomic structures.

(a) S_X(q) of our densified vitreous silicas compared to X-ray data reproduced from Prescher et al.³ and (b) Evolution of the maximum of the first sharp diffraction peak FSDP. (c) Calculated S_N(q) compared to neutron data (black lines) reproduced from Zeidler et al.¹¹.

Extended Data Fig. 4 v-SiO₂ interatomic distances and angles.

(a) Calculated Si-O, O-O, and Si-Si distances at maximum of the distribution in our densified vitreous silicas. Si-O bond length are compared to X-Ray (squares) and neutron (+) scattering data. Si-Si distances are compared to those in the crystalline polymorphs. For stishovite, the interval corresponds to pressures between 10 GPa and 30 GPa. (b) Si-O-Si bond angle distribution (BAD) and pressure dependence of the Si-O-Si BAD marked by the arrow. The average value has been calculated from 110^o to 175^o. (c) Pressure dependence of the O-Si-O and examples of bond angle distributions (BAD).

Extended Data Fig 5 Face-shared SiO_n polyhedra.

Number of face-sharing per polyhedron unit for dominant SiO_n-SiO_m connectivities as a function of pressure.

Extended Data Fig. 6 Percolation transitions.

(a) Percolation probability, P_∞, versus v-SiO₂ density for the different 4-, 5- and 6-folded coordinated Si, and their combinations. (b) P_∞ versus the fractions of SiO_n.

Extended Data Fig. 7 OSi_Z structures.

(a) Coordination numbers Z and Z′ of SiO_Z polyhedra and OSi_Z′ structures, (b) fraction of OSi_n, and (c) percolation probability of (OSi₂-OSi₂)_∞, (OSi₂-OSi₃)_∞, and (OSi₃-OSi₃)_∞ clusters.

Extended Data Table 1 Vitreous silica versus crystalline silicas