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

is about 3. As a consequence, system sizes are smaller and accessible timescales are shorter in ab initio than in classical simulations. Therefore, the quench rates used to cool a system from its liquid to its glass state are usually on the order of 10¹³–10¹⁵ K/s. Such rates are thus significantly larger than those of classical simulations (10¹⁰–10¹⁴ K/s) and of course ways higher than experimental values (10⁻² − 10⁶ K/s). Despite these huge differences, the resulting glasses are surprisingly similar since many of their properties depend only in a logarithmic way on these rates. Therefore, it does make sense to use ab initio simulations to investigate the properties of glasses on the microscopic scale.

Table 1 Comparison of various features of large‐scale computer simulations carried out with classical and ab initio methods.

	Classical MD	Ab Initio MD
Size	1 000–500 000 atoms	100–500 atoms
Box size	∼100 Å	∼15–20 Å
Trajectory length	∼1 ns–10 μs	∼20–100 ps
Transferability	No	Yes

3 Structural Properties

      In this section we will present some examples to illustrate how ab initio simulations can help to gain a better understanding of the local structure of complex glasses. The very first ab initio MD simulations for a glass‐former were carried out by Sarnthein et al. in 1995 on silica [7]. Using the Car–Parrinello approach, they generated a glass model by equilibrating for 10 ps a liquid sample with 72 atoms (!) at 3500 K and subsequently quenching it to 300 K. Despite the smallness of the system and the high quench rate (10¹⁵ K/s!), the resulting glass structure was surprisingly similar to that of the real material with a network of SiO₄ corner‐sharing tetrahedra and a neutron structure factor compatible with the that from scattering experiments. These authors also found that their electronic density of states matched well the X‐ray photoelectron spectroscopy data although the predicted band gap of 5.6 eV underestimated the experimental value of about 9 eV, a flaw that often occurs in DFT calculations [2].

      This pioneering paper was followed by further investigations in which more complex glass‐formers were studied, such as alkali silicate glasses, calcium‐alumino‐silicates [8–11], and other glass‐formers [12, 13]. These investigations allowed to obtain detailed insight into the local arrangement of the atoms, how network modifiers like Na or Ca change these arrangements, and the connections between local structure in real space with structural features as determined in neutron or X‐ray scattering experiments.

      As an example we will briefly discuss some results obtained for sodium borosilicate of composition 30% Na₂O–10% B₂O₃–60% SiO₂ (in mol %), an important glass‐former in which boron atoms bond to either three or four oxygen neighbors, which makes its structure rather complex, Chapter 7.6, [6]. One important quantity to characterize the structure is the partial radial distribution functions g_αβ(R) which is directly proportional to the probability that two atoms of type α and β are found at a distance r from each other (Chapter 2.2). Thus, this function is defined as [14]

      (8)

      where 〈.〉 represents the thermal average, V is the volume of the simulation box, N_α is the number of particles of species α, and δ_αβ is the Kronecker delta.

For the boron‐oxygen pair (Figure 1a), the first peak of the radial distribution function is relatively large and slightly asymmetric. By considering three‐ and fourfold coordinated boron atoms, ^[3]B and ^[4]B, respectively, we can decompose this peak into two contributions and see that the broadening of the total peak is a consequence of the presence of these two populations, the ^[3]B─O and ^[4]B─O bonds giving rise to the smaller and larger distances, respectively. We can decompose these distributions further by considering the nature of the second nearest neighbor of the central boron atoms, i.e. whether the nearest oxygen is bridging connected to a Si or B atom, or whether it is a non‐bridging oxygen. We see that for the case of the ^[3]B atoms, the nature of this second nearest neighbor species influences the B─O distance since the position of the corresponding peak depends on the species, whereas this is not the case for the length of the ^[4]B─O bonds. The presence of ^[3]B and ^[4]B units is also reflected in the OBO triplet angle distribution (shown in Figure 1b) by the presence of two peaks.

Figure 1 Contributions of the various structural units of a sodium borosilicate glass to structural features [6]. (a) Distribution of B─O distances around a given B atom as given by the first peak of the partial radial‐distribution function. (b) Distribution of the angle formed by two oxygens connected to the same boron.

      Although this detailed information is valuable to understand better the local structure of the glass, it is also important to connect the results from the simulation with experimental data. Even if radial distribution functions cannot be directly accessed experimentally, it is possible to measure the static structure factor, which is directly related to the weighted sum of S_αβ(q), the space Fourier transform of g_αβ(R) [14]. For neutron scattering, one finds, for example:

      (9)

      Here, b_α is the neutron scattering length of particles of species α [14] and the partial static structure factors are given by

      (10)

Figure 2 Neutron and X‐ray structural factors (a and b panel, respectively) for a borosilicate system in the liquid and glass state.

      Source: From Ref. [6].

      where N is the total number of atoms and f_αβ = 1 for α = β and f_αβ = 1/2 otherwise. For X‐ray scattering, the relation is

      (11)

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