Читать книгу Encyclopedia of Glass Science, Technology, History, and Culture - Группа авторов - Страница 316

Medium‐range order is difficult to study either experimentally or through numerical simulations. The size distribution of rings made up of cations and oxygen atoms is in particular an important parameter when investigating geometrical features in the 5–15 Å range. Usually a ring is characterized by the number of its network‐forming cations, which can be derived from the calculated atomic coordinates as shown in Figure 9 for simulated B₂O₃ and SiO₂ glasses. Compared with cristobalite, tridymite, and quartz, whose ring sizes are 6, 6, and 6 and 8, respectively, simulated SiO₂ glass shows a broad distribution around 6. It has been speculated that the existence of sizes of odd‐numbered rings are characteristic of disordered structures and might impede glass crystallization, because fivefold rotational symmetry does not exist in crystals where four‐, six‐, and eight‐membered rings are primarily observed. In B₂O₃ glass, which is extremely reluctant to crystallize, the existence of B₃O₆ units indeed causes the presence of a peak at 3 in the ring statistics.

Figure 8 Torsion angle distribution in simulated B₂O₃ glass between BO₃ and BO₃ units (a) and between B₃O₆ and B₃O₆ units (b) [6]. Data sampled at 0 K to prevent peaks from being blurred by atomic vibrations.

Figure 9 Ring size distribution in simulated B₂O₃ and SiO₂ glasses [11]. Data sampled at 0 K.

Of more general relevance is the vibrational density of states (VDOS). One can calculate it by solving the eigenvalue problem once the curvature of the energy surface around the stable configuration is obtained. The VDOS for simulated B₂O₃ glass is shown in Figure 10, where the peaks labeled A, B, and C represent the vibrations associated with independent BO₃ units observed in crystalline B₂O_3, where B₃O₆ units are absent, the breathing mode of B₃O₆ units observed in the crystal of metaboric acid comprised of B₃O₆ unit, and the vibrations of B₃O_{6 + n} units with several BO₄ tetrahedra comprised in rings [6], respectively. The VDOS can thus provide important structural information in terms of cooperative motion of structural units.

Figure 10 Vibrational density of states in simulated B₂O₃ glass [6]. See text for the assignments of the peaks labeled A, B, and C.

There is another geometrical method relying on the so‐called “Voronoi diagram” (e.g. [18]). It is largely employed for monatomic system for which partitioning three‐dimensional space is simple and easy when the calculated atomic coordinates obtained by atomistic simulations are used to delineate the portion of space assigned to every atom. These “Voronoi polyhedra” are then characterized by their numbers of faces and corners whose distributions change as positional relationships vary in the glass structure. The other geometrical method is called the analysis of “bond orientational order.” The order parameter that is rotationally invariant can be calculated with spherical harmonic functions. This order parameter has been used to investigate a local icosahedral order chiefly for monatomic system, because its value can discriminate geometrical differences between FCC, HCP, icosahedral, and BCC clusters (e.g. [18]).

In summary, atomistic simulations provide a key to explaining the concept of “modified random network theory [10]” in alkaline silicate glass [10] or the existence of “super‐structural units [19]” in B₂O₃ glass [6]. However, new analytical methods are required to understand medium‐range order in more detail.