Hybrid deterministic and stochastic approach for efficient atomistic simulations at long time scales

We propose a hybrid deterministic and stochastic approach to achieve extended time scales in atomistic simulations that combines the strengths of molecular dynamics (MD) and Monte Carlo (MC) simulations in an easy-to-implement way. The method exploits the rare event nature of the dynamics similar to most current accelerated MD approaches but goes beyond them by providing, without any further computational overhead, (a) rapid thermalization between infrequent events, thereby minimizing spurious correlations, and (b) control over accuracy of time-scale correction, while still providing similar or higher boosts in computational efficiency. We present two applications of the method: (a) Vacancy-mediated diffusion in Fe yields correct diffusivities over a wide range of temperatures and (b) source-controlled plasticity and deformation behavior in Au nanopillars at realistic strain rates (${10}^4/$s and lower), with excellent agreement with previous theoretical predictions and in situ high-resolution transmission electron microscopy observations. The method gives several orders-of-magnitude improvements in computational efficiency relative to standard MD and good scalability with the size of the system.

Figure 1

Schematic and flowchart of the algorithm. Shell S${}_w$ of constant potential energy and energy well W as described in the text are shown here. See Eq. (6) for a definition of ${\overline{t}}_{\mathbf{W}}$.

Figure 2

${\log }_{10}$(diffusivity) vs inverse temperature. (a) The straight lines denote Mendelev et al.'s (Ref. 18) MD calculations. These are valid only until 700 K. (b) Asterisks denote diffusivity measurements per our approach. (c) The dashed line shows experimental measurements (Ref. 19) valid between 1000 and 1200 K.

Figure 3

Simulation cell for stress-strain calculations. (a) Prior to application of any strain, (b) after yielding ($\text{strain}=12\%$) with strain $\text{rate}=5\times {10}^4/$s. In (b) the leading partial has nucleated on a {111} slip plane, leaving behind the two-layer-thick hcp region denoting an intrinsic stacking fault. Failure is thus through slip and not twinning, in agreement with Ref. 23. Atoms are identified as per bond order parameter $Q_6$ (Refs. 21 and 22). Perfect hcp atoms have been removed for clarity.

Figure 4

(a) Stress-strain plots at three different strain rates: $2.5\times {10}^6/$s (open circles), $5\times {10}^5/$s (asterisks), $5\times {10}^4/$s (pluses). The initial stress corresponding to zero strain is nonzero due to surface effects (Ref. 27). (b) ${\log }_{10}$(stress) at 11$\%$; strain (relative to surface stress at zero strain) vs ${\log }_{10}$(strain rate).