Encyclopedia of Glass Science, Technology, History, and Culture. Группа авторов
advantage of the RMC method is that knowledge of interatomic potentials is not required, but its drawback is that it is not applicable to novel glass systems for which no experimental data can be compared with model values. Besides, there is a risk of arriving at an incorrect structure if the iterative procedure leads to a local, and not to the true minimum of χ 2. A simple way to avoid such a pitfall is to add a set of effective constraints on bond lengths, bond angles, or coordination numbers (CN) that will prevent spurious results from being obtained.
4 Molecular Dynamics Simulations
The main advantage of MD simulation is to provide important structural information that complements conclusions drawn from experimental studies. One such important result was the demonstration that the structure of sodium silicate glasses fits the modified random network model because the spatial distribution of sodium ions showed a clustering tendency [10]. The result was especially significant as the extremely high fictive temperatures of glasses quenched in MD simulations strongly favor a much more random distribution.
For such reasons, MD simulations have become the most popular method to study theoretically glass and liquid structures (e.g. [11, 12]). Their main advantage is to yield from the three‐dimensional coordinates calculated for all atoms a variety of structural information that can often be checked against experimental data. In addition, they also provide information that escape experimental determinations and may thus point to the existence of unknown structural features.
Since they deal with the instantaneous state of a system, MD simulations rest on the Lagrangian function L(r,
(12)
and on the Lagrangian equations of motion
(13)
This leads to
(14)
where m(i) is the mass of atom i and
where f(i) is the force exerted on atom i.
If the coordinates of all atoms are known at time t0, all the forces exerted and the resulting velocities are calculated with Eq. (15) and then Eq. (14). All the velocities and coordinates after a time step of Δt are updated by the time integration of the equations of motion (14).
When MD methods are applied to glass or disordered systems, several important points should be noted:
1 As the proper choice of the interaction potential model is extremely important, a model with complex interaction potentials may be required if any dynamic structure or property is to be calculated after along with the static structure.
2 The particular starting configuration is not important as long as equilibration time steps are numerous enough at high temperature. Almost the same structural information should be obtained from different initial configurations. If not, the calculated results are unreliable.
Advanced techniques may be used to calculate structure and properties more efficiently. To omit unimportant contributions to the dynamics, one can, for example, keep constant bond lengths such as O─H during the MD calculation. As an alternative to this dynamic constraint method, one can use nonequilibrium MD, which is especially efficient to calculate transport properties such as viscosity. Unlike with conventional MD, a continual friction force can be imposed on the system and its response be monitored.
5 Modeling: Simulation Techniques and Examples
5.1 Overall Glass Structure
In atomistic simulations the positional correlation of atoms is easily investigated within a radius of half of simulation cell size (~25 Å). The most widely derived results are the PDF, the RDF, or total correlation function, T(r), which can be readily compared with those obtained in diffraction studies.
In simulations, the PDF and the RDF are derived as follows. The single and pair (two‐body) probability density PN (1), PN (2) are defined as
(16)
(17)
where N is again the number of atoms and ri is the coordinate of atom i.
The value of PN (1)(r) in homogeneous system turns out to be the number density ρ, which is defined as N/V, where V is the volume.
Moreover, PN (2)(r) is expressed in terms of the PDF, g(r, r′), and number density ρ as:
(18)
As to the RDF, J(r), it is defined as the number of atoms between r and r + dr from the center of an arbitrary origin atom:
(19)
An alternative function called total distribution function, T(r), is calculated as:
(20)
The information directly obtained from diffraction experiments is the intensity I(Q), which is related to J(r) by
(21)
Finally, the frequently used structure factor, S(Q), is the Fourier transform of the number density ρQ first calculated in atomistic simulation:
(22)
Then, S(Q) is calculated from ρQ:
(23)
It is quite important to reproduce the experimental J(r) in the real space domain or I(Q) in the wave‐number domain to validate the calculated three‐dimensional structure.
Depending on the atoms considered, X‐ray and neutron diffraction experiments can yield different profiles so that both kinds of profiles should be calculated and compared with the relevant data as done in