Monte Carlo Simulations of Crystal Defects in Open Ensembles

Abstract

Zero- and two-dimensional crystal defects form in open statistical ensembles, such as the grand canonical, that are usually inaccessible with conventional simulation techniques. This longstanding challenge is overcome with a new Hamiltonian Monte Carlo method that samples energy-biased gradual transformations. The method enables free energy calculations for nonideal point defects and the direct prediction of finite-temperature interface structures.

Figures (3)

• Figure 1: $\mu VT$ and $\mu PT$ MC simulations of fcc Cu at 1200 K. (a) Interstitial or vacancy concentrations, represented in terms of excess atoms per site, $\langle N \rangle / N_0 - 1$, as a function of chemical potential. $\mu PT$ calculations were also fit to an analytical model of free energy (see text). (b) Gibbs free energy per atom from integrating $dG = \mu dN$ with reference to the free energy of the perfect crystal at $N_0$. The dashed line indicates that $\langle N \rangle(\mu=g_0)$ effectively coincides with minimum $g$, validating the approximation of $\mu \simeq g_0$ for calculating the equilibrium vacancy concentration of an $NPT$ system.
• Figure 2: $\mu PT$ MC equilibration of a $\Sigma27(552)[1\bar{1}0]$ tilt GB in W at three temperatures. (a) The initial structure, a $5\times 5$ tiling of the periodic unit, is a bicrystal with integer $(552)$ interface planes, i.e. $n = 0$. (b) The equilibrated structure at $T=0.25T_{\rm{m}}$, which is slightly lower energy than previous optimizations. (c) Evolution of $n$ vs the number of MC trials or MD steps. (d) Similarly, the cell length normal to the boundary, which defines volume, due to normal displacements of the upper bounding block. (e) Displacement of the bounding block parallel to the boundary.
• Figure 3: $\mu PT$ MC equilibration of a $\Sigma 5(310)[001]$ tilt GB in Cu. (a) The initial "normal kite" structure, a $5\times 5$ tiling of the periodic unit, is the 0 K ground state with $n =0$. (b) The "split-kite" structure is vibrationally stabilized at 0.5 $T_{\rm{m}}$ with $n = 0.4$, in agreement with free energy calculations freitas18. (c) Evolution of $n$ during simulations, showing a phase transformation at 0.5 $T_{\rm{m}}$ and the metastability of the original phase at 0.25 $T_{\rm{m}}$.