Encyclopedia of Glass Science, Technology, History, and Culture. Группа авторов
in analogy with the structural changes associated with the α–β transition in cristobalite [17]. In addition, it has been suggested that the abrupt rotation of Si─O─Si equivalent to torsion movement between two SiO4 tetrahedra is the cause of anomalous thermomechanical properties in silica glass [17].
5.3 Medium‐range Order
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 B2O3, 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 B3O6 + 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.
5.4 Structure‐related Properties
Important thermodynamic properties can be calculated once numerical simulations have yielded atomic configurations and velocities. The pressure is, for example, calculated from
(24)
where the bracket indicates an equilibrium time average and N, V, T, fij, and rij are as usual the number of atoms, cell volume, temperature, and pair force and distance between atoms i and j, respectively.
The internal energy (Eint) is
(25)
and the molar heat capacity at constant volume (Cv):
(26)
Alternatively, one can derive Cv from the potential energy fluctuations through
(27)
and two other interesting properties are the thermal expansion coefficient, (αp)
(28)
where H is enthalpy, and the thermal pressure coefficient (βV):
(29)
After a model of atomistic simulation is validated so that it can reproduce static structure of glass, it can be applied to investigate transport and dynamical properties such as diffusion constants, viscosity, or the Van Hove correlation function (Chapter 4.6).
5.5 Experimental and Computational Complementarity
The statically arrangement of structural units for B2O3 glass and the dynamically arrangement of structural units for SiO2 glass represent new insights on glass structure provided by MD simulations, in these cases, by the TAD, which escape any experimental determinations. These two examples thus illustrate the complementary nature of numerical simulations and experimental studies of glass structure. When the history of structural studies on glass is looked back on, it is clear that both diffraction and spectroscopic studies have made fundamental contributions to the construction of structural models. The RDF or the PDF can indeed be readily calculated from the Fourier transform of experimental X‐ray, neutron, or electron diffraction data (Chapter 2.2). Because this type of information represents averaged one‐dimensional