Modeling vitreous silica bilayers

Theoretical modeling is presented for a freestanding vitreous silica bilayer which has recently been synthesized and characterized experimentally in landmark work. While such two-dimensional continuous random covalent networks should likely occur on energetic grounds, no synthetic pathway had been discovered previously. Here the bilayer is modeled using a computer assembly procedure initiated from a single layer of a model of amorphous graphene, generated using a bond-switching algorithm from an initially crystalline graphene structure. Each bond is decorated with an oxygen atom and the carbon atoms are relabeled as silicon, generating a two-dimensional network of corner-sharing triangles. Each triangle is transformed into a tetrahedron, by raising the silicon atom above each triangular base and adding an additional singly coordinated oxygen atom at the apex. The final step in this construction is to mirror-reflect this layer to form a second layer and attach the two layers to form the bilayer. We show that this vitreous silica bilayer has the additional macroscopic degrees of freedom to form easily a network of identical corner-sharing tetrahedra $\mathit{if}$ there is a symmetry plane through the center of the bilayer going through the layer of oxygen ions that join the upper and lower monolayers. This has the consequence that the upper rings lie exactly above the lower rings, which are tilted in general. The assumption of a network of perfect corner-sharing tetrahedra leads to a range of possible densities that we characterize as a flexibility window, with some similarity to flexibility windows in three dimensional zeolites. Finally, using a realistic potential, we have relaxed the bilayer to determine the density and other structural characteristics such as the Si-Si pair distribution functions and the Si-O-Si bond angle distribution, which are compared with experimental results obtained by direct imaging.

Figure 1

Ring size distributions (fraction of rings containing $n$ silicon atoms) obtained experimentally for vitreous SiO${}_2$ bilayers formed on Ru(0001) (Ref. 3; red dashed line) and graphene (Ref. 4; black dashed line). Also shown are ring size distributions of computer-generated a-G seeds used in this study (with different generating protocols but labeled by sample size $N$).

Figure 2

The freestanding vitreous silica bilayer is modeled by first creating an amorphous graphene layer from a crystalline graphene sheet through a bond-switching algorithm (top panel), where the carbon atoms are shown as solid black dots. An oxygen ion is then placed at the center of each carbon-carbon bond, forming a network of corner-sharing triangles (second panel), and the carbon is $\mathit{replaced}$ by a silicon ion. The silicon atom is then lifted out of the plane and an oxygen ion (shown as a solid red dot) is placed above this silicon to form a tetrahedron. This silica monolayer is mirror inverted and placed directly above the first monolayer (third panel) shown in plan view. Finally these two monolayers are brought together to form the silicon bilayer (fourth panel) in which the central oxygen ions are $\mathit{combined}$ to make single bridging oxygen ions between the two monolayers.

Figure 3

Showing the structure of the bilayer made of perfect corner-sharing tetrahedra from various perspectives, relaxed with the harmonic potential and with a dimensionless area of $A^{*}=0.9$. This puts the structure into an arbitrary place in the flexibility window at some density (not restricted to the experimental density). This illustration is a piece from a bilayer with a periodic supercell with 864 silicon atoms. Notice that the central oxygen atoms all lie in the symmetry plane. Each tetrahedron has four oxygen ions shown as small red spheres at the vertices and a silicon ion at the center.

Figure 4

The energies of the relaxed configurations shown as a function of the density (upper abscissa) and the reduced area, $A^{*}=A/A_0$ (lower abscissa), obtained for the 400 (upper panel) and 864 (lower panel) SiO${}_2$ molecule system, using both harmonic and TS potentials. The distances indicated are for the parameter $\sigma$ in the box in the upper panel. The minima for the TS potentials for the 400 and 864 molecule configurations are at densities of $\sim$18.4 SiO${}_2$ nm${}^{-2}$ and $\sim$20.0 SiO${}_2$ nm${}^{-2}$, respectively. The yellow lines and arrows in both panels highlight the density range observed from experiment (Ref. 16).

Figure 5

Radial distribution functions $g\left(r\right)$ calculated for the silicon sub-network using the Si-Si separations projected onto the $xy$ plane. The RDFs are calculated for the 400 and 864 SiO${}_2$ molecule systems (respectively ${\mu }_2=0.500$ and $1.046$) using the harmonic (black lines) and TS potential (blue lines). Successive functions are offset along the ordinate axes for clarity. In each case the lower curve is for the 400 molecule configuration and the upper for the 864 molecule case. The original carbon RDF determined from the a-G sample is also shown (red). This function has been scaled along the abscissa in terms of the first peak positions for comparison. The harmonic potential functions are obtained at a density of $A^{*}=0.81$ (at which the energy can be quenched) while the TS functions are obtained at the respective energy minima. The uppermost curves are the amorphous (magenta) and crystalline (cyan) functions obtained from experiment (Ref. 16).

Figure 6

(a) Si-O-Si bond angle distributions determined at a density of $A^{*}=0.90$ for the harmonic potential (black) and at the densities corresponding to the respective energy minima for the TS (blue) for the 400 molecule system. The light blue plot shows the corresponding function for the bulk glass at ambient pressure using the TS potential (Ref. 17). (b) Si-Si-Si bond angle distributions obtained under the same conditions as for panel (a). For reference the additional red line shows the C-C-C distribution from the original two-dimensional a-G configuration.

Figure 7

The low-density end of the flexibility window (for the harmonic potential case) for the two amorphous (green and red; sample sizes as indicated) and one crystalline (black) sample, shown as lines ending in a $+$ symbol. Note that the high density end of the window for harmonic potentials is to the left and determined by repulsive forces between neighboring ions. The figure also shows the corresponding energy minima from the TS potential ($\times$) colored as for the corresponding flexibility windows. The blue line highlights the experimental density range.

Figure 8

Showing a tetrahedral pair, one from the upper layer and one from the lower layer. Each tetrahedron has four oxygen ions shown as small red spheres at the vertices and a silicon ion (not shown) at the center. Note that the Si-O-Si link between the two tetrahedra is, in general, not straight.