Encyclopedia of Glass Science, Technology, History, and Culture. Группа авторов. Читать онлайн. Newlib. NEWLIB.NET

Автор: Группа авторов
Издательство: John Wiley & Sons Limited
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Жанр произведения: Техническая литература
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isbn: 9781118799499
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is that the timescale of microscopic processes is ultimately controlled by the 10−14–10−15 second period of atomic vibrations. In atomistic simulation, a time integration with similar discrete steps thus is required to describe accurately the time evolution of a system. Even with today's most powerful computers, this constraint restricts simulations to low‐viscosity conditions at which relaxation times (cf. Chapter 3.7) are lower than about 10−6 seconds to be consistent with the calculational time steps.

      It follows that the glass transition cannot be investigated as a number of calculation steps of the order 1018 would be required to deal with a relaxation time of ~103 seconds. Likewise, crystal nucleation and growth or liquid–liquid phase separation take place too slowly to be subjected to atomistic simulations. Besides, simulated melts are quenched at cooling rates at least six orders of magnitude faster than the highest rates of ~106 K/s achievable practically. The fictive temperatures of the simulated glasses that then be up to 1000 K higher than those of real glasses, which one should keep in mind when making any kind of comparison between both kinds of materials.

Schematic illustration of the periodic boundary conditions.

      2.2 The Importance of Interatomic Potentials

      Strictly speaking, in numerical simulations the Hamiltonian should be calculated from the quantum wave‐function equation, = , where ψ and E are the wave‐function and energy of the system, respectively. As expounded in Chapter 2.9, such calculations require so much computing work that they are currently restricted to smaller systems typically made up of a few tens of atoms. In the present chapter, simulations made within a classical framework will thus be considered instead. They rely on the fact that the differences between vibrational and electronic energies and frequencies are so large that atomic vibrations may be considered to take place within a fixed electronic configuration. This is the celebrated Born–Oppenheimer approximation whereby the Hamiltonian of a system is expressed as the sum of kinetic and potential energies, which are functions of the selected set of coordinates q(i) and momenta p(i):

      where E0, k, and r0 are the dissociation energy, a measure of the bond strength, and the equilibrium interatomic distance of the molecule, respectively, three parameters that are determined from vibrational spectroscopy data.

      In a condensed phase, potentials are much more complicated since a given atom interacts with a great many others over distances that can be large. To keep the number of parameters as small as possible in the expression of potential energies, one thus groups into the same term all interactions between given pairs of like or unlike atoms regardless of their mutual distances. Although the Morse potential remains a good starting point for systems where bonding is covalent, other kinds of analytical expressions are generally used for potential energies in the MC and MD simulations dealt with in this chapter. As borne out by the variations with composition of macroscopic properties, atomic interactions have the simplifying feature that they are primarily pairwise in oxide or salt systems. This feature is embodied in the most popular potentials used for these systems, namely, the Buckingham,

      (5)equation

      and the Born–Mayer–Huggins potentials,

      (6)equation

      In both potentials, the first, second, and third terms represent repulsive interaction, dipole–dipole dispersion, and dipole–quadrupole dispersion, respectively. In more precise formulations, ternary and higher‐order effects must be accounted for so that the potential energy is made up of terms depending on the coordinates of individual atoms, pairs, triplets, etc.

Graph depicts the examples of potential energy models: Morse and Buckingham potentials used in B2O3 simulations for B-O and for O-O and B-B, respectively.