Encyclopedia of Glass Science, Technology, History, and Culture. Группа авторов

is the scattering‐factor function (also called form factor). Figure 2 shows that even for the same glass sample, S_N(q) and S_X(q) are rather different since both are a weighted sum of the partial structure factors, each of which has a multitude of positive and negative peaks. Hence, the resulting sum may show a peak for S_N(q) that is basically absent in S_X(q). Since interpretation of experimental data is further complicated by the fact that one has k(k + 1)/2 partial structure factors in systems with k species, computer simulations can help to interpret correctly the various peaks if they do reproduce the partial structure factors in terms of peak positions and heights. As a matter of fact, the necessary accuracy is usually ensured only by ab initio simulations.

      Another important situation is that of surfaces, for which experimental scattering techniques involving grazing‐ray geometries are much less powerful to obtain microscopic information. Because effective potentials are usually developed for bulk systems, their reliability is not guaranteed for surfaces for which the local structure can be very different. One example are two‐membered rings in SiO₂ that are present at the surface but absent in the bulk. Hence, ab initio simulations are to be preferred for investigating surfaces since they can deal with such heterogeneous geometries.

One interesting example of surface effects is the difference between the interatomic distance and bond‐angle distributions of water molecules in the bulk and adsorbed at the surface of amorphous silica (Figure 3). Although there are no significant differences in the positions of the O─O and O─H peaks (Figure 3a), close to the surface the fluid is less structured than in the bulk as signaled by smaller peaks. Likewise, differences concerning the angle between water dipole (i.e. the bisector of the HOH angle) and the surface normal (Figure 3b) indicate a strong change of the local geometry. Only accurate interaction potentials, such as ab initio simulations will be able to pick up these effects.

Figure 3 Structural differences between bulk (BW) and surface (SW) water in SiO₂ glass [15]. (a) Water–water radial distribution function. (b) Distribution of the angle θ between the water dipole (bisecting the HOH angle) and the surface normal.

4 Vibrational Properties

      The vibrational density of states (vDOS) g(ω) is the normalized distribution function of the eigenfrequencies ω_p of the dynamical matrix of the system which is directly related to the second derivative of the potential with respect to the coordinates R _I and R _J, i.e. ∂ ²Φ/(∂R _{I, i} ∂R _J,j) for i,j ∈{x, y, z}, thus

(12)

      where 3N − 3 is the number of eigenmodes with nonzero frequency. Associated with each eigenfrequency ω_p of the dynamical matrix is an eigenvector e _p that gives detailed information regarding the particles that oscillate with the frequency ω_p. Studies of e _p have allowed to gain insight into the nature of the vibrational modes of various materials such as silica and germania glasses [12, 16–18] and more complex systems ([6, 10, 11, 19], Chapter 3.4).

      The structure of a solid is related to a balance of the forces between its atoms, i.e. to the first derivative of the potential since these forces must mutually compensate to ensure a mechanically stable equilibrium. In contrast, vibrational properties are related to the curvature of the potential, i.e. to its second derivatives. As a consequence, it is quite hard to find effective potentials that reproduce correctly the experimental vDOS, even for systems as simple as pure silica [20].

The true vDOS given by Eq. (12) is not directly accessible in experiments, in which only an effective vDOS, G(ω) = C(ω)g(ω), can be measured. The function C(ω) describes how the modes with frequency ω are coupled to the probing radiation (photons, neutrons, etc.). For inelastic neutron‐scattering experiments, this function can be derived, within approximations, from the eigenvectors e _p of the dynamical matrix [21].

In practice, the vibrational properties of a system can be calculated in two different ways. The first is to determine the local potential minimum of the glass structure (i.e. to quench the sample to T = 0 K) and then to calculate directly the dynamical matrix, for instance, by a finite‐difference scheme for the forces. By diagonalizing this matrix, one can then obtain its eigenfrequencies and eigenvectors and, hence, g(ω). The second method consists in running a simulation in which one solves Newton’s equations of motion at low temperatures where the harmonic approximation is valid. If one measures the time autocorrelation function of the velocity of a particle, its time Fourier transform is directly proportional to the vDOS. This approach is from the computational point of view cheap, but it has the drawback of not giving any information on the eigenvectors, i.e. on the nature of the motion of the atoms.

The advantages of ab initio over effective pair‐potential simulations are illustrated by the neutron‐effective vDOS for silica (Figure 4). The effective potentials give a surprisingly good description of the structural properties of real silica [23] and also of the high‐frequency vDOS where modes concern intra‐tetrahedral excitations [18]. However, the vDOS calculation with the effective potential gives a broad, flat band between 200 and 600 cm⁻¹ whereas the ab initio simulations show, in agreement with the experimental data, a marked peak at around 400 cm⁻¹ [17].

      A further technique used to get insight into the vibrational properties of glasses are measurements of dielectric properties (Chapter 3.4), which are directly related to the local polarizability of the material. Because ab initio simulations do give a good description of polarizability, they allow the high‐frequency (ɛ_∞) and static dielectric constant (ɛ₀) to be accessed directly. Also called relative permittivity, the former quantity can be estimated as one third of the trace of the purely electronic dielectric tensor , which describes the reaction of the electrons to the presence of an external electric field if the ions were kept at fixed positions. For a glass, this tensor is basically isotropic and diagonal. The static dielectric constant ɛ₀ reflects the ionic displacement contributions to the dielectric constant and it can be expressed as [24]:

Figure 4 Effective