# Percolation transitions in compressed SiO2 glasses

Anwar Hasmy, Simona Ispas, Bernard Hehlen

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Anwar Hasmy, Simona Ispas, Bernard Hehlen. Percolation transitions in compressed SiO2 glasses. Nature, 2021, 599 (7883), pp.62-66. 10.1038/s41586-021-03918-0. hal-04856778

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# Percolation transitions in compressed SiO2 glasses

**Anwar Hasmy**

University Simon Bolivar

**Simona Ispas**

Montpellier University https://orcid.org/0000-0002-8537-8461

**Bernard Hehlen**

<figure><img src="image_2.png" /></figure>

Bernard.hehlen@umontpellier.fr

Montpellier University

## Physical Sciences - Article

**Keywords:** SiO2 glasses, amorphous-amorphous transformations under, percolation transitions

**Posted Date:** April 19th, 2021

**DOI:** https://doi.org/10.21203/rs.3.rs-379312/v1

**License:** cc BY This work is licensed under a Creative Commons Attribution 4.0 International License.

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**Version of Record:** A version of this preprint was published at Nature on November 3rd, 2021. See the published version at https://doi.org/10.1038/s41586-021-03918-0.
Percolation transitions in compressed SiO$_{2}$ glasses

A. Hasmy$^{1,2}$, S. Ispas$^{1}$, and B. Hehlen$^{1}$

$^{1}$Laboratoire Charles Coulomb (L2C), CNRS - Univ. Montpellier, 34095 Montpellier, France and

$^{2}$Departamento de Física, Universidad Simón Bolívar, Valle de Sartenejas, Caracas, Venezuela

(Dated: March 31, 2021)

Amorphous-amorphous transformations under pressure are generally explained by changes in the local structure from low to higher fold coordinated polyhedra [1–4]. However, as the notion of scale invariance at the critical thresholds has not been addressed, it is still unclear whether these transformations could be associated to true phase transitions. Here we report *ab initio* based calculations of compressed silica (SiO$_{2}$) 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 up to 82 GPa, a series of long range ('infinite') percolating clusters built up by corner- or edge-shared tetrahedra, pentahedra, and eventually octahedra, emerge at some 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 for the structural irreversibility beyond 10 GPa, among others. 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 SiO$_{5}$ pentahedra-based polyamorphs in the densification process of vitreous silica. Our observations demonstrate that the percolation theory provides a robust framework to understand the nature and the pathway of the amorphous-amorphous transformations, and open a new avenue to predict unraveled amorphous solid phases and related liquids [7, 8].

The understanding of the physical mechanisms con-
trolling the transformation from one amorphous phase
to another is an open fundamental issue in materials sci-
ence [1, 2, 7, 9]. In amorphous solids such as SiO₂, GeO₂,
Si, Ge and chalcogenides, short-range structures are very
similar to those of the cristalline counterparts and it is the
random nature of the inter-unit connections which makes
the disorder at long length scale. In these systems, it has
been argued that amorphous phases can be described in
terms of changes of electronic and structural properties
at short and medium range order. These properties are
for example the electronic bonding [2, 4, 10, 11], the co-
ordination number [1, 3, 4, 10, 12, 13] and the ring dis-
tribution [13, 14]. However, to describe the passage from
one amorphous phase to another at the critical thresh-
old, an explicit scale-invariant quantity (i.e. an order pa-
rameter) should be defined. This concept has yet never
been considered for the pressure-driven amorphous solid
transformations.

When an amorphous solid is pressurized, it is com-
monly assumed that the transformations from low to
high density materials occur gradually, with coexisting
low and high fold coordinated polyhedra. Such gradual
change, sometime referred as *polyamorphism*, contrasts
with the polymorphism in crystals, in which transitions
occur from a specific phase to another at a critical pres-
sure. In addition, it opposes to behaviors observed at
rather well defined pressures. The change from tetrahe-
dra (Z=4) to octahedra (stishovite-like structure, Z=6)
structures in vitreous silica (*v*-SiO₂) is accompanied by
the well known mechanical anomaly at 3 GPa [15, 16],
a percolation phenomenon at ≈ 8 GPa [17], and a
two-steps elastic-to-plastic transformation, one around

10 GPa [14, 16, 18] and a second one around 20 GPa [16].

*Ab initio* methods have provided a key tool to clarify
the pathway of the structural transformations in crys-
talline silicas (*c*-SiO$_{2}$) [5, 6, 19–21] and in amorphous ma-
terials. Recent artful machine learning *ab initio* method
combined with empirical force field on compressed amor-
phous silicon highlights subsequent local changes that
suggest a finite number of amorphous phases [4], but
the description of the passage from one phase to an-
other was not addressed. For *v*-SiO$_{2}$ at ambient pres-
sure, *ab initio* methods have also nicely complemented
experimental data [22, 23]. However, despite many ef-
forts [14, 24, 25], its application to the above-mentioned
issues has been limited due to the prohibitive calculation
times required for a reliable thermodynamic sampling (in
terms of system sizes, pressure steps as well time equili-
bration) [26, 27].

Herein lies our motivation to use an *ab initio* based
approach, the Self-Consistent-Charge Density Functional
based Tight-Binding (SCC-DFTB) method [28]. Com-
bined with the Molecular Dynamics (MD) technique, it
is able to reproduce many features of *c*-SiO<sub>2</sub> and *v*-SiO<sub>2</sub>
with similar accuracy as *ab initio* methods, but being
at least three order of magnitudes faster in terms of
CPU time. This allowed us to simulate more than 40
SiO<sub>2</sub> glasses and to explore the full pressure range up
to 82 GPa at room temperature. We also decompressed
one of the resulting samples from 8 GPa and another
from just above 10 GPa (for details see Methods). Our
samples contain 1008 atoms (336 SiO<sub>2</sub> units) which is
large enough to observe and describe the percolation crit-
ical phenomena governing the aforementioned transfor-
mations.
<figure><img src="image_3.png" /></figure>

FIG. 1: (a) Snapshots of the simulation boxes showing the evolution of the short range structures from SiO$_{n}$ to SiO$_{n+1}$ polyhedra, and of their connectivity from corner- to edge-sharing connectivity (circular insets), upon pressure. The glass samples are pressurized by instantaneous reduction of the simulation box followed by a relaxation using periodic boundary conditions. (b) Fraction of Si polyhedra and Si-O and Si-Si coordination number 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 on top in panels (b,c) mark the onset of the different regimes (see text) and the dashed line indicates the elastic to plastic transition. Note that the scale changes at 20 GPa in the abscissa axis.

**Polyhedra coordination and connectivity.**

Figure 1a illustrates the *v*-SiO$_{2}$ transformation from
$Z = 4$ to 6 as the pressure $P$ is increased. At ambient
temperature and $0 < P < 3$ GPa, similarly to *c*-SiO$_{2}$,
the Si-O bond of *v*-SiO$_{2}$ is characterized by a sp$^{3}$ hy-
bridization which favors the formation of SiO$_{4}$ tetrahe-
dra. However, for $3 < P < 10$ GPa, the densification
of *v*-SiO$_{2}$ includes changes in the electronic structure,
with an increase of the Fermi energy and of the atomic
charges of O and Si (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 con-
nectivity change from purely corner-sharing tetrahedra
to a more complex connected network (Extended Data
Figs. 3 and 4). The latter involves corner- and edge-
sharing (Fig. 1a,c) SiO$_{5}$-SiO$_{6}$ and SiO$_{6}$-SiO$_{6}$ and to a

lower extent face-sharing polyhedra (Extended Data Fig. 5).

We recall that on increasing the pressure from ambient
conditions, c-SiO₂ passes from tetrahedral to octahedral
(Z = 6) local structure through structural transforma-
tions including α-quartz to coesite I (Z = 4) at 3 GPa,
then coesite I to stishovite (rutile structure, Z = 6) at
9 GPa. It is well known that the bonding is more ionic
in the later that in the former case [11]. Regarding the
connectivity of the polyhedra, since the oxygen atoms are
divalent when Z = 4, each SiO₄ shares one oxygen atom
 corne
 (corner-sharing) with its four closest neighboring tetrahe-
dra. This is the cases of v-SiO₂ and c-SiO₂ for P < 3 GPa
 and P < 9 GPa, respectively. In c-SiO₂, for Z = 6 and
 P > 9 GPa, the oxygen atoms are trivalent, hence each
SiO₆ shares two couples of oxygen atoms with two differ-
 ent nearest neighboring octahedra (edge-sharing), while
the other connectivities are corner-shared with the eight
octahedra nearest neighbors remaining. In v-SiO₂, the
changes of polyhedra coordination and connectivity al-
low to identify different regimes (top arrows in Fig. 1b-c).
For 13 < *P* < 20 GPa, the compressed glasses contain al-
most similar fractions of SiO₄, SiO₅ and SiO₆, with a to-
tal number of shared edges per polyhedron *n*<sub>es</sub> ≃ 1.4 and
a small connectivity contribution of face-sharing polyhe-
dra. This recalls the crystalline polymorph discovered
recently, the so-called coesite IV [6], rarely observed for
high-valence and low-coordinated cations in crystals, as it
is at odds with the Pauling rule. For 20 < *P* < 45 GPa,
the fraction of SiO₆ octahedra increases with a jump,
while the fraction of SiO₄ tetrahedra falls down to almost
zero. Also, *n*<sub>es</sub> increases to ≃ 1.7 and the presence of a
small fraction of face-sharing polyhedra persists, which
looks similar to coesite V, the second newly discovered c-
SiO₂ polymorph [6]. For 45< *P* < 70 GPa, the network
connectivity is fully dominated by SiO₆ octahedra, ei-
ther corner-shared and edge-shared, and *n*<sub>es</sub> ≃ 2, which
suggests the emergence of a stishovite like phase. On
increasing the pressure beyond 70 GPa, the total num-
ber of edge-sharing per polyhedra becomes greater than
2 due to the contribution of SiO₇ polyhedra, suggesting
the emergence of a post-stishovite like phase [25, 29].

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 while not in the latter (small
symbols in Fig. 1b). The pressure threshold at which
irreversibility occurs nicely agrees with experimental
observation [16]. When unloading from 10 GPa to
ambient pressure, an 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 5-fold coordinated polyhedra in the
plastic-to-elastic transition of *v*-SiO₂ [14, 16, 18].

**Percolation transitions.** The above observations only include an evaluation of the local and medium range order. However, the nature of the phase transformations from one regime to another in compressed *v*-SiO₂ should also be substantiated by the structure at long length scale. This is the purpose of the forthcoming discussion. Indeed, we found that clusters built up of SiO*n* polyhedra (including their mixtures SiO*n*-SiO*m*) eventually emerge and one of them becomes dominant and percolates, by spanning its structure from one side to the other side of the simulation box (Fig. 2a). This effect has important implications on the glass rigidity [17], and can be monitored by calculating the percolation probability, *P*∞. The latter is frequently used as an order parameter for the analysis of a percolation phase transition. It tends to 1 (or 0) if the largest cluster percolates (or not) and the percolating cluster is representative of the new phase [30]. Accordingly, the notation (SiO*n*-SiO*m*)∞ used in the following corresponds to a percolating clus-

ter built of an alternation of *n*- and *m*-fold coordinated
polyhedra.

Figure 2b shows the percolation probability, $P_\infty$, of the biggest cluster for connected networks of SiO₄, SiO₅ and SiO₆ polyhedra, and their mixtures, as a function of pressure in compressed *v*-SiO₂. For the (SiO₄-SiO₄)₄∞ cluster, $P_\infty$ drops from 1 to 0 at 13 GPa. In addition, between 3 and 8 GPa, a new percolating cluster built up by a connected skeleton of alternating tetrahedra and pentahedra, (SiO₄-SiO₅)₄∞, emerges. This coincides with the percolation transition observed at 8 GPa [17]. With increasing pressure, two more percolating clusters appear at 10 GPa and coexist with the first one: one built up by connected SiO₅-SiO₅ pentahedra and the second by connected SiO₅-SiO₆ polyhedra. 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 phases reported previously [31, 32]. Then, at 13 GPa, a (SiO₆-SiO₆)₄∞ cluster appears and percolates. The coexistence of all these percolating clusters beyond 13 GPa is similar to the coesite IV, whose crystalline structure combines SiO₄-SiO₅, SiO₅-SiO₅, SiO₅-SiO₆ and SiO₆-SiO₆ planes in different crystallographic directions. By analogy, the coesite IV-like vitreous phase is labeled *v*-coesite IV in Fig. 2b.

Slightly above 20 GPa, SiO₄ tetrahedra vanish and ac-
cordingly, P∞ of the (SiO₄-SiO₅)∞ cluster falls down to
0. Simultaneously, a percolating cluster (SiO₆-SiO₅,₆)∞
connected by edge-sharing SiO₅-SiO₆ and SiO₆-SiO₆
polyhedra emerges, together with some face-sharing con-
tribution. In that pressure range, the amorphous phase
(labeled *v*-coesite V in Fig. 2b) possesses the same
dominant edge-sharing polyhedra connectivity as well as
equivalent alternations of SiO*n*-SiO*m* polyhedra as in
coesite V [6]. It also associates to the delayed revert
structural transformation observed experimentally above
≈ 20 GPa in *v*-SiO₂ [16], suggesting that edge-shared
structures remain stable during decompression.

The analyses of the percolation probability of a clus-
ter built up of pure edge-shared octahedra, similar to the
stishovite situation evidences that such cluster percolates
around 40 GPa and, accordingly, a *v*-stishovite phase re-
places 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 [29]. Therefore, our results demonstrate that in-
stead 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 phases (Fig. 2).

The correlation length ξ has been estimated in the
pressure range where the different percolating clusters
appear. This quantity accounts for the maximum length
at which the scale invariance exists. At the thermody-
namics limit, it is expected that such quantity diverges
<figure><img src="image_4.png" /><figcaption>FIG. 2: (a) Snapshot of the spanning clusters in the simulation box for pressures corresponding to $P_{\infty} = 0$ up-to $P_{\infty} = 1$, and back to $P_{\infty} = 0$ for the first one. From left to right: corner-shared $(\text{SiO}_4\text{-SiO}_5)_{\infty}$ and $(\text{SiO}_5\text{-SiO}_5)_{\infty}$ clusters, and edge-shared $(\text{SiO}_6\text{-SiO}_{5,6})_{\infty}$ and $(\text{SiO}_6\text{-SiO}_6)_{\infty}$ clusters. (b) Percolation probability, $P_{\infty}$, versus pressure for the different 4-, 5-, 6- and 7- folded coordinated Si, and their combinations. Dashed and dotted magenta lines correspond to $P_{\infty}$ of the cluster $(\text{SiO}_4\text{-SiO}_5)_{\infty}$ for decompression from 8 GPa and 10 GPa, respectively. (c) Correlation length $\xi$ of the percolating clusters in panel (b). (d) Bulk modulus $K$ as function of pressure, and comparison with high frequency experiments (unrelaxed bulk modulus) [16]. The inset zooms on the low pressure region. Note that the scale changes at 20 GPa in the abscissa axis.</figcaption></figure>

at the critical pressure, while it should reach a maximum for finite size systems as in our computer model. Indeed, for the different situations described above a maximum is always reached just before the corresponding critical pressure (Fig. 2c). Beyond 70 GPa a big cluster built up

of edge-sharing $(\text{SiO}_6\text{-SiO}_7)_{\infty}$ polyhedra starts to grow, which, coupled to the fact that for such pressure the total number of shared edges per Si polyhedron exceeds 2, supports the post-stishovite like phase suggested above. The jump in the percolation probability $P_{\infty}$ from 0 to 1
evidences the dominant character of a well-defined amorphous phase in a given pressure range. However, the non-negligible partial overlap between the ξ-curves beyond 13 GPa may indicates a phase coexistence between the dominant percolating cluster and the emerging one (Fig. 2c). This suggests that this phase acquires a metastable character when approaching the incoming critical pressure. Such behavior recalls the metastability observed in coesite IV and V crystalline polymorphs [5, 6].

When unloading from 8 GPa, the percolating $(SiO_4-SiO_5)_\infty$ cluster vanishes at a similar pressure at which it percolates during compression (Fig. 2b). This confirms the reversible character already suggested by the local and medium range structural analysis above. Conversely, when the sample is decompressed from 10 GPa (dashed curve), $P_\infty$ exhibits a hysteresis, i.e. the $(SiO_4-SiO_5)_\infty$ cluster continues to percolate at pressures much lower than for the percolating compression route (Fig. 2b). A similar plastic behavior is observed for the other percolating clusters, i.e. $(SiO_5-SiO_5)_\infty$ and $(SiO_5-SiO_6)_\infty$ ones (data not shown). This shows that the formation of percolating clusters containing $SiO_5$ pentahedra for $P \gtrsim 10$ GPa prevents recovering the pristine glass structure, and likely explains the irreversible behavior.

To explore the impact of these percolation transitions on the glass rigidity, we calculated the bulk modulus $K$ of our pressurized silicas (Fig 2d). As expected, the increasing connectivity at short and long length scales yields to a strong increase of $K$, i.e. from about 30 GPa at ambient conditions up-to $\simeq 375$ GPa at $P = 82$ GPa. Its evolution nicely parallels that of the crystalline counterparts: $SiO_4$-polymorphs (except coesite I) below 10 GPa, coesite IV ad V at intermediate pressures, and stishovite above 35-40 GPa [6, 33]. Our calculations also reproduce the well known minimum around 3 GPa (inset of Fig. 2d). Following the percolation mechanisms discussed above, we associate the increased rigidity starting above 4 GPa to the series of the percolating clusters emerging upon pressure. Finally, our estimation of $K$ is very close to high frequency experimental data except maybe around 10 GPa where the simulations exhibit a minimum, when $(SiO_5-SiO_5)_\infty$ and $(SiO_5-SiO_6)_\infty$ clusters emerge.

Additionally, for the different connectivity situations, we have analyzed the structure of the percolating cluster at the critical pressure thresholds. For this, we spanned these clusters taking into account the periodic boundary conditions before extracting them out of the box. Then, the structure of the isolated percolating clusters was analyzed by computing the structure factor $S(q)$. As expected from the percolation theory, at the critical thresholds, the cluster structures have fractal shapes, i.e. $S(q) \propto q^{-D_f}$ at intermediate wave-vector $q$-values, with $D_f \simeq 2.5$, which is also in agreement with the theoretical predicted value (Fig. 3).

<figure><img src="image_5.png" /><figcaption>FIG. 3: Structure factors $S(q)$ for the isolated and extended percolating clusters at the critical pressures (indicated in the snapshots) in log-log scale.</figcaption></figure>

## CONCLUSIONS AND OUTLOOK

Our results show that in compressed $v-SiO_2$ the criticality of the amorphous-amorphous transformations can be described within the percolation theory by considering the percolation probability $P_\infty$ as the order parameter. The long-range percolating clusters built up of different polyhedra coordination $(SiO_n-SiO_m)_\infty$ and connectivity (corner- or edge-shared) have the expected scale invariance at the critical threshold. The structure of these clusters beyond 13 GPa are similar to those of coesite IV, V and stishovite. The bulk moduli of the glass samples and the corresponding polymorphs are also similar. This suggests that the mechanical properties relate primarily to the peculiar connectivity describing the system at all length scales, rather than by the crystalline or amorphous nature of the network.

Our analysis is based on two central ideas that deserve to be considered in many other pressure- or temperature-driven amorphous-amorphous transformations, including oxide glasses, chalcogenides, metallic glasses, as well
as H₂O. The first one refers to the formation of long-range percolating clusters with a specific local coordination at a critical thermodynamical variable. Indeed, the temperature-driven amorphous transformations in liquid H₂O from the high to low density has been explained in these terms. The same has been argued for the glass transition in model systems [35–37] and seems also compatible with recent experimental observations [38]. The second one states that with pressure or temperature, the local coordination of materials built of polyhedra can evolve successively from low to high coordination through a series of percolation transitions. By providing an unified structural picture from short to long length scales of amorphous phases, these concepts could help to unveil affinities with crystalline polymorphs as well as to predict new phases appearing under extreme conditions.
# METHODS

## Glass preparation

The initial glass configuration containing 1008 atoms (336 SiO₂ units) placed in a cubic simulation box was prepared by carrying out classical MD simulations, using periodic boundary conditions. We used the melt-and-quench approach, the constant volume-constant temperature (NVT) ensemble and the atomic interactions were modelled using the BKS pair potential [39]. Then, we carried out MD simulations within the framework of the Self-Consistent Charge Density Functional based Tight-Binding (SCC-DFTB) method for the electronic structure calculations [28]. These calculations were also done using the NVT ensemble. The Andersen thermostat was used in order to maintain the room temperature and the samples were relaxed for more than 40 ps, with a time step equal to 2 fs. Within the SCC-DFTB approximation, the electronic integrals corresponding to the Hamiltonian matrix elements are replaced by analytical functions. The coefficients of these functions are obtained by fitting Density Functional Theory (DFT) calculations corresponding to systems obtained at different physicochemical conditions. We used the PBC-0-3 set of parameters for SiO₂ [40]. This results in an electronic structure calculation method at least three orders of magnitude faster than DFT, currently considered the most 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 the 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

The glass sample at ambient pressure prepared using the above procedure had a density of 2.3 g/cm³. In order to mimic a quasi-statically compression, the box length was reduced by step of 1% (and atoms positions rescaled) after each 40 ps, until reaching a pressure of ≈ 27 GPa. A second series of glasses was obtained by instantaneous volume reduction of the sample obtained at 15 GPa yielding thereby seven samples at target pressures up to 82 GPa. Two successive volume reductions of 0.125% followed by a 40 ps relaxation were performed to 6 samples from this second series. The largest compression rate used here is almost equivalent to 0.02 GPa/ps. This corresponds to a compression velocity at least one order of magnitude slower than the one used in a previous DFT study of compressed SiO₂ at T=300 K [14], while the number of atoms in our calculation box is almost 7 times larger. In order 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% after each 40 ps, until reaching the ambient pressure. 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 have analyzed 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 $\kappa_{Si-O}$ calculated as done elsewhere [41]. For the latter, the relation $\kappa_{Si-O} = |Q_{Si}/\nu_{Si} - Q_{O}/\nu_{O}|/2$ is used, where $\nu_{Si}$ and $\nu_{O}$ are the valence of Si and O atoms, respectively. The total and the projected O-2p DOS of the simulated glass at ambient pressure are in good agreement with experimental and DFT data [23] (Extended Data Fig. 1a). The effect induced by the sample densification in the total DOS of v-SiO₂ also agrees [11, 24] (Extended Data Fig. 1c,d). Similarly, the Si and O Mulliken atomic charges and the Si-O bond ionicity change is consistent with previous experimental and DFT findings [11, 14] (Extended Data Fig. 2). This favors the formation of pentahedra and higher fold coordinated Si polyhedra [42] (Fig. 1). The maxima reached in the pressure range 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

In order to compare the structural properties of our samples to experimental data, we computed the neutron and X-ray total static structure factors. We firstly computed the partial structure factors $S_{\alpha\beta}(q)$ using the definition [43]:

$$S_{\alpha\beta}(q) = \frac{f_{\alpha\beta}}{N} \sum_{j=1}^{N_{\alpha}} \sum_{k=1}^{N_{\beta}} \langle \exp(i\vec{q} \cdot (\vec{r}_j - \vec{r}_k)) \rangle \quad \alpha, \beta = \text{Si, O.}$$

Here $f_{\alpha\beta} = 1$ for $\alpha = \beta$ and $f_{\alpha\beta} = 1/2$ otherwise, $N_{\alpha}$ is the number of particles of species $\alpha$ 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 relation [43].

$$S_N(q) = \frac{N}{\sum_{\alpha=\text{Si,O}} N_{\alpha} b_{\alpha}^2} \sum_{\alpha,\beta=\text{Si,O}} b_{\alpha} b_{\beta} S_{\alpha\beta}(q),$$

with the neutron scattering length $b_{\alpha}$ given by $b_{\text{Si}} = 4.1491$ fm and $b_{\text{O}} = 5.803$ fm, respectively [44]. The X-ray total structure factor $S_X(q)$ is given by [45]:

$$S_X(q) = \frac{N}{\sum_{\alpha} N_{\alpha} f_{\alpha}^2(q/4\pi)} \sum_{\alpha,\beta} f_{\alpha}(q/4\pi) f_{\beta}(q/4\pi) S_{\alpha\beta}(q).$$

Here $f_{\alpha}(s)$ is the scattering-factor function (also called form factor), computed as a linear combination of five Gaussians using the parameters derived elsewhere [46].
The calculated X-Ray $S_X(q)$ and neutron $S_N(q)$ structure factors are compared to experimental data [3, 13]. 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 $vSiO_2$ at intermediate length scales (Extended Data Fig. 3).

## Local connectivity

We computed the pair distribution function (PDF) $g_{\alpha\beta}(r)$ for all compressed $v$-SiO$_2$ samples using the definition [43]:

$$g_{\alpha\beta}(r) = \frac{V}{N_{\alpha}(N_{\beta} - \delta_{\alpha\beta})} \left\langle \sum_{i=1}^{N_{\alpha}} \sum_{j=1}^{N_{\beta}} \frac{1}{4\pi r^2} \delta(r - |\vec{r}_i - \vec{r}_j|) \right\rangle,$$

where $\langle \cdot \rangle$ represents the thermal average, $V$ is the volume of the simulation box, and $\delta_{\alpha\beta}$ is the Kronecker delta. For each of the three pairs, the first peak of the corresponding PDF corresponds to the distribution of the first neighbors shell distances 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 rather 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 neighbors and to estimate the evolution of the polyhedra coordination number with pressure (Fig. 1b). The identification of the Si-O and Si-Si first neighbors also allowed to estimate the number oxygen atoms shared by two Si polyhedra neighbors, i.e, one, two, three or more, corresponding to corner-, edge- and face-shared polyhedral connectivity, respectively (Fig. 1c and Extended Data Fig. 5). The computed SiO distances are compared with experimental data in Extended Data Fig. 4a. The agreement is remarkable in particular in view of recent X-Ray measurements [3]. O-O and Si-Si distances are also shown for information.

The Si-O-Si and O-Si-O bond-angle distributions (BAD) and their pressure dependence are plotted in Extended Data Fig. 4b-c. The above mentioned up-shift of the FSDP translates into a fast reduction of the angles up to the first plastic transformation around 10 GPa when corner shared SiO$_5$- and SiO$_6$-based percolating clusters emerge. With increasing pressure above 10 GPa, and in addition to the primary peak at $\approx 130^\circ$, we notice the occurrence of two other peaks pointing at $\approx 98^\circ$ and $\approx 68^\circ$, respectively, in the bond angle distribution of Si-O-Si. The position of the three peaks shows a weak pressure dependence, while 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 $\sim 10$ GPa and 20-25 GPa corresponding to the two stages of plastic transformation, and the O-O bond length presents a maximum at pressure around 3 GPa

corresponding to the minimum of the bulk modulus.

In order to give more insights in the medium range of the silica network with increasing pressure, we computed the ring distributions. No changes occurs for pressures below 4 GPa. Then, for pressures below 8 GPa, we observed a mild increase of the small ring sizes (2-, 3- and 4-membered ones) accompanying by the decrease of the proportion of large ring sizes (7- and 8-membered ones). When approaching the threshold the plastic regime at $\approx 10$ GPa, the same pressure dependences were observed but presenting stronger trends. This increase of small rings at the expense of large ones at high pressure confirms previous DFT calculations [14].

## Cluster analysis

To identify the different clusters with a specified polyhedra coordination and connectivity, as well the biggest one, we use a strategy inspired by the Friends-of-Friends algorithm which is widely used to characterize dark matter halos from N-body simulations [47]. In our implementation the first nearest neighbors correspond to the specific Si-O coordination and polyhedra connectivity we want to analyze. To determine if the biggest cluster percolates or not, we first extract all clusters from the box excepting the biggest one. Then we replicate the box containing this cluster along the three spatial directions. The Friends-of-Friends algorithm is applied again, and if the resulting biggest 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 to each SiO$_n$-SiO$_{n+1}$ possible connectivities, that is corner-, edge-, and face-shared polyhedra.

The percolation probability $P_\infty$ is defined as the number of times the resulting biggest cluster percolates during the last 10 ps of the relaxation time, normalized by the total number of explored configurations. The correlation length $\xi$ was calculated using the definition [30]:

$$\xi^2 = \frac{\sum_s 2R_{g,s}^2 s^2 n_s}{\sum_s s^2 n_s},$$

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 biggest one if it percolates. The gyration radius $R_{g,s}$ was estimated using the relation:

$$R_{g,s}^2 = \frac{1}{2s^2} \sum_{i,j} r_{ij}^2,$$

where the sums run over pairs of Si atoms belonging to each cluster of size $s$. The correlation length $\xi$ 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, and its
error bar is calculated by error propagation while the pressure error was estimated from the standard deviation. To minimize errors on 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 compressibility [15], while high frequency experiments measure the volumetric variations at the time scale of the relaxational processes probed by the instrument [16]. In compressed v-SiO$_2$, both 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 [16, 48, 49]. Intriguingly, above 12 GPa, our K calculated values are much closer to the high frequency data than to the static ones, a behavior which possibly reflects the typical time relaxation ($\simeq 10$ ps, corresponding to GHz frequencies) and the small volumetric variation ($<0.4$ nm$^3$) we used for the estimation of K in our MD simulations. For $P > 55$ GPa, the experimental value (static and high frequency) is $\simeq 420$ GPa [48, 49] in good agreement with our calculations.

### Affinities with c-SiO$_2$ polymorphs

To identify the SiO$_2$ polymorph most akin to amorphous phases, 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 the crystallographic databases [50]. These supercells are relaxed using the L-BFGS minimization method implemented in the DFTB+ package. Only the lengths of the lattice vectors were allowed to vary during the structure optimization, with the exception of coesite 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 (SiO$_4$-SiO$_5$)$_\infty$, (SiO$_5$-SiO$_5$)$_\infty$, (SiO$_5$-SiO$_6$)$_\infty$ and (SiO$_6$-SiO$_6$)$_\infty$, percolating clusters. We found that almost all SiO$_5$-SiO$_6$, SiO$_6$-SiO$_6$ are edge-shared, with a weak contribution of face-sharing polyhedra as in coesite IV and V. Indeed, in coesite IV we have validated the coexistence of different crystalline planes having the above-mentioned connectivities.

Edge-sharing SiO$_5$-SiO$_6$ and SiO$_6$-SiO$_6$ polyhedra become preponderant at $P \simeq 20$ GPa. They are in similar proportion, $\simeq 40\%$, and control the network connectivity as in coesite V. Beyond 40 GPa the fraction of SiO$_5$ starts to fall and the SiO$_6$-SiO$_6$ connectivity becomes dominant, which allows to identify the stishovite-like phase.

In order to validate the observations above, the total DOS, Fermi energy, Mulliken atomic charges and bond ionicity were computed and compared to the results obtained for compressed v-SiO$_2$ (Extended Data Figs. 1-2). The electronic properties of v-SiO$_2$ appear to be closer to α-quartz than to other SiO$_4$-based polymorphs at ambient pressure as suggested elsewhere [11]. On 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, V and stishovite crystals, respectively. The ionic character of the Si-O bond is also larger for the 5- and 6-fold Si coordinated polyhedra [20]. Similar conclusion holds when comparing the polyhedra coordination and connectivity with the corresponding v-SiO$_2$ percolating clusters, as well as for the calculated bulk moduli. The latter agrees very well with that of SiO$_4$-tetrahedra-based polymorphs at low pressures and to that of coesite IV, V, and stishovite (Fig. 2c) when pressure increases beyond 13 GPa.

Finally, Extended Data Table 1 gives the densities of the crystalline polymorphs and indicate the corresponding pressure ranges [6, 31, 51, 52]. The comparison with the corresponding polyamorph phases of v-SiO$_2$ is striking. One notices, for example, that a small pressure of $\simeq 3$ 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 that of the crystalline counterparts.

### Percolating clusters at the critical threshold

We analyzed the structure of the percolating clusters around the critical pressures. For this we expanded and extracted out of the box the percolating clusters by taking into account the periodic boundary conditions. At the critical threshold, these clusters are usually larger than the box size due to their branched structure. We computed the structure factor $S(q)$ of the isolated clusters by using the corresponding expression for a diluted system [53]:

$$S(q) = 1 + \frac{1}{N} \sum_{i \neq j} \frac{\sin(qr_{ij})}{qr_{ij}},$$

where N is the number of Si atoms and q is the modulus of the wave vector. This expression implies that $S(0) = N_{Si}$ and $S(\infty) = 1$, where $N_{Si}$ is the number of polyhedra (Si atoms). Besides, for fractal structures, at intermediate q values, $S(q)$ scales as $q^{-D_f}$, where $D_f$ is the fractal dimension (Fig. 3).

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**Acknowledgements:** The authors would like to thank the BioNano-NMRI team (L2C, UM) for computer facilities. One of us (A.H) acknowledges support from the CNRS (France) and thanks L.C. Rincón for introducing him in the SCC-DFTB method. This work was done with the support of the Centre Informatique National de l'Enseignement Supérieur (CINES, France) via a Research Allocation Panel award (project A0060910788).

**Author contributions:** All authors initiated the project. A.H. and B.H. performed the MD Tight-Binding

(SCC-DFTB) calculations of the pressurized glasses, after S.I. has prepared the initial glass at ambient pressure by classical MD simulations. A.H. performed the tight binding calculations of the crystals and computed the percolation tools. All authors contributed to the data analysis: atomic structure (A.H, B.H., S.I.), electronic structure and percolation (A.H.), and inelastic structure factors (S.I.). A.H. and B.H. developed the main conclusions and wrote the paper, S.I. contributed to its final version.
<figure><img src="image_6.png" /><figcaption>Extended Data Fig. 1. 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.</figcaption></figure>

<table><caption>Extended Data Table I: Correspondence between pressure and density in c-SiO₂ polymorphs and comparison with the polyamorph phases of v-SiO₂ defined in the text.</caption><thead><tr><th colspan="3">c-SiO<sub>2</sub></th><th colspan="3">v-SiO<sub>2</sub></th></tr><tr><th>Polymorph</th><th>Pressure (GPa)</th><th>Density (g/cm<sup>3</sup>)</th><th>Pressure (GPa)</th><th>Density (g/cm<sup>3</sup>)</th><th>Polyamorph</th></tr></thead><tbody><tr><td rowspan="3">α-Quartz [31, 51]</td><td rowspan="3">0-25</td><td rowspan="3">2.6-3.6</td><td>0-3</td><td>2.3-2.7</td><td>SiO<sub>4</sub></td></tr><tr><td>8-10</td><td>3.3-3.5</td><td>SiO<sub>4</sub>, SiO<sub>5</sub></td></tr><tr><td>10-13</td><td>3.5-3.8</td><td>SiO<sub>4</sub>, SiO<sub>5</sub>, SiO<sub>6</sub></td></tr><tr><td>Coesite IV [6]</td><td>30-49</td><td>4.2-4.5</td><td>13-23</td><td>3.8-4.0</td><td>v-Coesite IV</td></tr><tr><td>Coesite V [6]</td><td>45-70</td><td>4.6-4.7</td><td>23-40</td><td>4.1-4.3</td><td>v-Coestite V</td></tr><tr><td>Stishovite [52]</td><td>10-73</td><td>4.3-4.9</td><td>40-82</td><td>4.4-4.9</td><td>v-Stishovite</td></tr></tbody></table>
<figure><img src="image_7.png" /><figcaption>Extended Data Fig. 2. Mulliken atomic charges for Si (a) and O (b), and the average Mulliken ionicity of the Si-O bond (c) as a function of pressure. The results are compared with those corresponding to different SiO₂ crystalline polymorphs.</figcaption></figure>
<figure><img src="image_8.png" /></figure>

<figure><img src="image_9.png" /></figure>

<figure><img src="image_10.png" /><figcaption>Extended Data Fig 3. $S_X(q)$ of our densified 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.* [13].</figcaption></figure>
<figure><img src="image_11.png" /></figure>

<figure><img src="image_12.png" /></figure>

<figure><img src="image_13.png" /><figcaption>Extended Data Fig. 4: (a) Calculated Si-O, O-O, and Si-Si bond length at maximum of the distribution. Si-O distances are compared to X-Ray (squares) and neutron (+) scattering data. (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° to 175°. (c) Pressure dependence of the O-Si-O and examples of bond angle distributions (BAD).</figcaption></figure>
<figure><img src="image_14.png" /><figcaption>Extended Data Fig 5. Number of face-sharing per polyhedron unit for dominant $SiO_n-SiO_m$ connectivities as a function of pressure.</figcaption></figure>
Figures

<figure><img src="image_15.png" /><figcaption>Figure 1</figcaption></figure>

(a) Snapshots of the simulation boxes showing the evolution of the short range structures from SiOn to SiOn+1 polyhedra, and of their connectivity from corner- to edge-sharing connectivity (circular insets), upon pressure. The glass samples are pressurized by instantaneous reduction of the simulation box followed by a relaxation using periodic boundary conditions. (b) Fraction of Si polyhedra and Si-O and Si-Si coordination number 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 on top in panels (b,c) mark the onset of the different regimes (see text) and the dashed line indicates the elastic to plastic transition. Note that the scale changes at 20 GPa in the abscissa axis.
<figure><img src="image_16.png" /><figcaption>Figure 2</figcaption></figure>

(a) Snapshot of the spanning clusters in the simulation box for pressures corresponding to P1 = 0 up-to P1 = 1, and back to P1 = 0 for the first one. From left to right: corner-shared (SiO4-SiO5)1 and (SiO5-SiO5)1 clusters, and edge-shared (SiO6-SiO5;6)1 and (SiO6-SiO6)1 clusters. (b) Percolation probability, P1, versus pressure for the different 4-, 5-, 6- and 7- folded coordinated Si, and their combinations. Dashed and dotted magenta lines correspond to P1 of the cluster (SiO4-SiO5)1 for decompression from 8 GPa and 10 GPa, respectively. (c) Correlation length of the percolating clusters in panel (b). (d) Bulk modulus K
as function of pressure, and comparison with high frequency experiments (unrelaxed bulk modulus) [16].
The inset zooms on the low pressure region. Note that the scale changes at 20 GPa in the abscissa axis.

<figure><img src="image_17.png" /><figcaption>Figure 3</figcaption></figure>

Structure factors S(q) for the isolated and extended percolating clusters at the critical pressures (indicated in the snapshots) in log-log scale.