Diffusion via molecular dynamics

Molecular dynamics examines the temporal evolution of a collection of atoms on the basis of an explicit integration of the equations of motion. From the point of view of diffusion, this poses grave problems. The time step demanded in the consideration of atomic motions in solids is dictated by the periods associated with lattice vibrations. Recall our analysis from chap. 5 in which we found that a typical period for such vibrations is smaller than a picosecond. Hence, without recourse to clever acceleration schemes, explicit integration of the equations of motion demands time steps yet smaller than these vibrational periods. 

An essentially equivalent argument can be made by noting that the hopping rate is given by F = and hence if we use v = 10 s and the activation 

Despite the reservations set down above, to carry out a molecular dynamics study of the diffusion process itself one resorts to a computational cell of the type described earlier. The temperature is assigned and maintained via some scheme such as the Nose thermostat (Frenkel and Smit, 1996), and the atomic-level trajectories are obtained via a direct integration of the equations of motion. In fig. 3.22, we showed the type of resuiting trajectories in the case of surface 

What this formula tells us is to evaluate the excursion undertaken by each atom during the time t and to average over all N particles. As a result of this expression, by measuring the mean excursion taken by a diffusing atom in a known time, the diffusion constant can be inferred. 

A more sophisticated version of basically this same idea is associated with the Green-Kubo formalism in which the relevant transport coefficient is seen as an average of the appropriate velocity autocorrelation function. In particular, the diffusion coefficient may be evaluated as 

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Inherent structure analysis of diffusion via molecular dynamics of a deeply supercooled binary Lennard-Jones fluid have provided renewed impetus to the decisive role played by thermodynamic factors [52,53]. The location of the mode crossover temperature and the onset of super-Arrhenius behavior were related to the static structure of the liquid via the potential energy hypersurface [52,53],... 

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