Angular rigidity in tetrahedral network glasses with changing composition

A set of oxide and chalcogenide tetrahedral glasses is investigated using molecular dynamics simulations. We show that the changes in the Ge composition affect mostly bending around germanium in binary Ge-Se systems, leaving Se-centered bending almost unchanged. In contrast, the corresponding Se twisting (quantified by the dihedral angle) depends on the Ge composition and is reduced when the system becomes rigid. It is also shown that angles involving the fourth neighbor around Ge is found to change when the system enters the stressed rigid phase. The same analysis reveals that unlike stoichiometric selenides such as GeSe${}_2$ and SiSe${}_2$, germania and silica display large standard deviations in the bond angle distributions. Within bond-bending constraints theory, this pattern can be interpreted as a manifestation of broken (i.e., ineffective) oxygen bond-bending constraints, whereas the silicon and germanium bending in oxides is found to be similar to the one found in flexible and intermediate Ge-Se systems. Our results establish the atomic-scale foundations of the phenomenological rigidity theory, thereby profoundly extending its significance and impact on the structural description of network glasses.

Figure 1

Neighbor distribution functions (NDF) (red: neighbors 1–4, green: neighbors 5–9) around a central Ge atom in glassy GeSe${}_4$. The sum of the nine distributions yields the pair distribution function at low distance (black curve). The insert shows corresponding neighbor (NDF) peak positions as a function of the neighbor number for SiSe${}_2$ (filled circles), SiO${}_2$, GeO${}_2$ (filled boxes), and Ge${}_x$Se${}_{1-x}$ (open circles).

Figure 2

From top to bottom: oxygen, selenium, and germanium partial bond angle distributions (PBAD) in GeO${}_2$ and GeSe${}_2$ for an arbitrary $N=6$. The colored curves correspond to PBADs having the lowest standard deviation(s) ${\sigma }_{\theta }$. The sharp peaks at $\theta \simeq$ 40${}^{°}$ correspond to the hard-core repulsion. Labels defined in the bottom panel are used throughout the text.

Figure 3

Standard deviation ${\sigma }_{\text{Ge}}$ and ${\sigma }_{\text{Se}}$ extracted from the partial bond angle distributions (PBAD) for five selected compositions in glassy Ge${}_x$Se${}_{1-x}$.

Figure 4

Standard deviations ${\sigma }_{\text{Ge}}$ as a function of Ge composition, split into a contribution involving the fourth neighbor (red line, average of 104, 204, and 304) and the other contributions (black line, average of 102, 103, and 104). The shaded area corresponds to the Boolchand intermediate phase.[16, 18]

Figure 5

Definition of the dihedral angle $\mathrm{\delta ̄}$ used in the text. Note that the upper black-filled atom can be $X(r_X=2)$ or $A(r_A=4)$.

Figure 6

Pair distribution function ${\mathrm{g}}_{\text{Ge}W}$(r) expressing the correlation between the Ge atomic sites and the position of the Wannier centers in the glassy Ge${}_x$Se${}_{1-x}$ systems.

Figure 7

Pair distribution function $g_{\text{Se}W}(r)$ expressing the correlation between the Se atomic sites and the position of the Wannier centers in the glassy Ge${}_x$Se${}_{1-x}$ systems.

Figure 8

Representative subsets of Ge, Se, and Wannier centers extracted from the configurations of glassy Ge${}_x$Se${}_{1-x}$ systems. Ge atoms appear in blue, Se atoms in yellow, and Wannier centers in red. If a bond AB exists between an atom A and an atom B, the bond is colored as follow: $\mathrm{A}\to$AB/2 using the color of atom A, AB/2$\to$B using the color of atom B.

Figure 9

Partial bond angle distributions around Ge (top) and Se (bottom) for densified GeSe${}_2$. The curves in color (six in the Ge PBAD and one in the Se PBAD) correspond to distributions with a low standard deviation. The broken red curve in the bottom panel serves for comparison and represents the 102 distribution of the ambient GeSe${}_2$ glass (same as Fig. 2).

Figure 10

Standard deviation ${\sigma }_{\text{Ge}}$ and ${\sigma }_{\text{Se}}$ extracted from the partial bond angle distributions (PBAD, Fig. 9) in GeSe${}_2$ (black line), and densified GeSe${}_2$ (red).

Figure 11

Standard deviation ${\sigma }_{{\theta }_{\mathit{ij}}}$ of partial bond angle distributions (PBADs) in oxides (SiO${}_2$, GeO${}_2$, top panel) and chalcogenides (SiSe${}_2$, GeSe${}_2$, bottom panel) as a function of the angle number (see definition in Fig. 2).