Transport Properties from Molecular Dynamics
Equilibrium molecular dynamics gives transport coefficients from fluctuations in a trajectory. Self-diffusion is usually taken from the long-time slope of the mean squared displacement (the Einstein method), often with a finite-size correction such as that of Yeh and Hummer for periodic cells. Shear viscosity is usually obtained by Green–Kubo integration of the autocorrelation of an off-diagonal stress tensor element, or from the equivalent Einstein expression for the growth of the integrated stress. Viscosity is the noisier of the two and needs long runs, many time origins and care with the slowly decaying tail of the autocorrelation at low temperature. These quantities underpin tests of the Stokes–Einstein relation.
Adatom surface diffusion from neural network MD (2018)
Gao and Kitchin ran 10 to 15 ps MD of a Pd adatom on Pd(111) at several temperatures with a neural network potential, took diffusion coefficients from the linear (Brownian) region of the mean square displacement via the two-dimensional Einstein relation, after the short-time ballistic regime, and fit them to an Arrhenius law. The resulting activation energy of 0.132 eV agreed with NEB bridge-site barriers of 0.133 to 0.136 eV from the same potential, showing that the dynamics and the static barriers are consistent (Gao 2018). Summary: gao2018.
Fitted stress autocorrelation and per-atom decompositions (2022)
For liquid Al–Si, Zhan and Kitchin collected the stress autocorrelation only out to 1 to 2 ps, fitted it with an analytic oscillating plus stretched-exponential decay, corrected the residual with a small neural network, and integrated the fit to 6 ps, which saved computation while keeping viscosity variance low. They also decomposed both diffusion and viscosity into per-atom contributions, the latter using per-atom stresses and Voronoi volumes in an Einstein-type formula. The per-atom viscosity matched Green–Kubo when all atoms were included and made it possible to measure how a subset of atoms (those in icosahedral clusters) changes the bulk transport properties (Zhan 2021). Summary: zhan2022b.
Vacancy diffusion in a solid oxide with a neural network potential (2018)
Transport in solids can be treated the same way. Wang, Tharval and Kitchin ran 10 ps constant-temperature MD of tetragonal ZrO2 containing one oxygen vacancy between 1000 and 1800 K with a DFT-trained neural network potential. They obtained oxygen diffusivities from the slope of the averaged mean squared displacement, and an Arrhenius fit gave an effective activation energy of 0.272 eV for vacancy diffusion. Before the production runs, they checked energies along a short 600 K trajectory against DFT (maximum deviation 8 meV/atom), which shows a practical way to validate a learned potential for MD (Wang et al. 2018). Summary: wang2018.