On the Boroxol Ring Fraction in Melt-Quenched B$_2$O$_3$ Glass

TL;DR

This work addresses the long-standing challenge of atomistically modeling melt-quenched B2O3 glass with a high boroxol-ring content. By developing a DFT-accurate ML potential (ML-31) trained on boroxol-rich configurations and using a large descriptor range, the authors reveal the critical role of intermediate-range order in stabilizing boroxol rings and demonstrate that boroxol content increases as the quench rate decreases, approaching experimentally inferred levels. An energy analysis shows a minimum around 70–75% boroxol, suggesting an energetically favorable amorphous state in this region, though density constraints and sampling limitations hinder reaching the full experimental fraction of 75%. The results point to promising strategies, including density-guided quenching, negative-pressure protocols, and advanced sampling techniques, to realize ultrastable boroxol-rich glasses and to refine theoretical frameworks for glass topology.

Abstract

An atomistic structural model for melt-quenched B$_2$O$_3$ glass has eluded the simulation community so far. The difficulty lies in the abundance of the six-membered boroxol rings - an intermediate-range order motif suggested through Raman and NMR spectroscopy - which is challenging to obtain in atomistic molecular dynamics simulations. Here, we report the development of a DFT-accurate machine-learned potential for B$_2$O$_3$ and employ quench rates as low as 10$^{9}$ K/s to obtain B$_2$O$_3$ glasses with more than 30% of boron atoms in boroxol rings. Also, we show that the pressure, and consequently the boroxol fraction, in the deep potential molecular dynamics (DPMD) simulations critically depends on the range of the geometry descriptor used in the embedding neural network, and at least a 9 $\unicode{x212B}$ range is required. The boroxol ring fraction increases with decreasing quench rate. Finally, amorphous B$_2$O$_3$ configurations display a minimum in energy at a boroxol fraction of 75%, intriguingly close to the experimental estimate in B$_2$O$_3$ glass.

Figures (6)

• Figure 1: Comparison of boron trioxide with the common glass, silicon dioxide.
• Figure 2: Performance of ML-31 MLP with different geometry descriptor cut-offs used in the embedding network. (a) RMSE of energy and the magnitude of force on the atom of the validation dataset versus cut-off. The size of the validation dataset is about 10% of that of the training dataset. (b) Boroxol fraction and mean pressure of B$_2$O$_3$ melt simulated by ML-31 MLP-based DPMD at 2400 K vs. cut-off, modeled under constant-NVT conditions at a density of 1.49 g/cc, post an equilibration time of 10 ns. The results pertain to a system containing 1,700 atoms. Error bars represent the standard deviation calculated from four independent runs, using different initial configurations and velocity distributions. The pressure data for configurations generated through MD simulations with a model of a specific cut-off, but evaluated with a model of a different cut-off are presented in Figure \ref{['fig:virial_validaiton']}.
• Figure 3: Dependence of boroxol ring fraction in the melt-quenched glass at 300 K on the quenching rate. Simulations were initiated from the melt at a temperature of 2000 K and a density of 1.49 g/cc, and followed the NVT-$\boldsymbol{\rho}_{\text{EXP}}$ procedure. Four independent simulations were conducted, each beginning with a different initial set of atom coordinates and velocities. Error bars represent the standard deviation calculated from these four runs. Star: Boroxol fraction in B$_2$O$_3$ melt at 2000 K, following a 10 ns equilibration period prior to quench. System sizes of either 1700 or 480 atoms were considered. The star in green colour is from a trajectory generated with the MACE mace_arxiv MLP trained on the ML-31 dataset. The single blue circle represents a single independent run with ML-31/R6, involving 1700 atoms and a quenching rate of $1\times10^{9}$ K/s, yielding a boroxol fraction of 50% at 1200 K at the experimental density (1.537 g/cc). The data point for ML-31/R9 with 480 atoms at 4$\times10^9$ K/s is at 1400 K. Similarly, the data point for ML-26/R9 with 1700 atoms and quench rate $4\times10^{10}$ K/s is at a temperature of 1000 K. These values are unlikely to change by further cooling to 300 K. NVT-ML-31/R9, 1700 was obtained under constant-NVT quench, i.e., the melt at 2000 K and the liquid/glass during quench were maintained at the glass density of 1.834 g/cc. The blue circle took 90 days, while the rest of the data presented in this figure took an aggregate of 250 days of compute time on one A100 GPU card.
• Figure 4: Arbitrarily chosen sequence of events during the formation of boroxol rings from different DPMD (ML-31/R9) trajectories containing 1700 atoms. Configurations in Panels (a) and (b) are from trajectories at 2000 K and 1.49 g/cc, while that in Panel (c) is from a trajectory at 1400 K and 1.515 g/cc. Green: Boron, Red: Oxygen. The numbers in the lower left corner of each configuration indicate the timestamp of the trajectory in femtoseconds. The zero in the timestamp is not the initial configuration, but rather a frame in a well-equilibrated trajectory. Estimates of the duration between consecutive boroxol formation events (checked for every 20 ps) are 10 ps and 300 ps at 2000 K and 1400 K, respectively. Additional events are displayed in SI Figure \ref{['fig:boroxol_formed2']}.
• Figure 5: Vibrational density of states of amorphous B$_2$O$_3$ configurations containing different boroxol ring fractions. Inset displays the region around 800 cm$^{-1}$, whereas an intense Raman peak is observed experimentally at 808 cm$^{-1}$. The VDOS data has been smoothened using a Gaussian function with a half-width of 10 cm$^{-1}$.