Database-driven semigrand canonical Monte Carlo method: Application to segregation isotherm on defects in alloys

The application of existing semigrand canonical ensemble Monte Carlo algorithms to alloys requires the chemical potential difference values between pairs of atomic species in the alloys as inputs. However, finding the appropriate values for a target system at a desired temperature and bulk composition is a time-consuming task consisting of multiple test runs to determine the chemical potential differences. This problem becomes more serious when dealing with systems containing three or more atomic species, such as medium- and high-entropy alloys, due to the increase of the number of chemical potential differences that need to be calculated. Here we propose a method for sampling from the semigrand canonical ensemble that relies on energy databases acting as an external atomic reservoir at the desired temperature and composition. Given these energy databases, the desired bulk composition and corresponding chemical potential differences can be satisfied in a “single” Monte Carlo simulation. Moreover, the energy databases shed light on the underlying energetics of alloys, reflecting their local chemical ordering. We demonstrate the validity of this method using analyses of segregation isotherms at grain boundaries and dislocations in two alloy systems: Fe–1-at.-$\%$-Si and NiCoCr medium-entropy alloy. We also discuss the possibly relevant information contained in such energy databases.

Figure 1

(a) Particle exchange trial move. (b) Flowchart of the energy change sampling algorithm. (c) Flowchart of the MC run in the energy change sampling algorithm.

Figure 2

Energy database information. Each energy database is built from the set of energy change values $\left\{\mathrm{\Delta }{U}_{i\to j}\right\}$ output from the energy change sampling; these energy databases contain the probability distribution functions (PDFs) and the cumulative distribution functions (CDFs) of $\left\{\mathrm{\Delta }{U}_{i\to j}\right\}$.

Figure 3

(a) Flowchart of the DD-SGCMC algorithm. (b) Schematics of the particle identity change trial move. (c) Flowchart of the MC run in the DD-SGCMC algorithm. (d) Calculation of $\mathrm{\Delta }{U}_{\text{rsv}}$ from the energy database using a random number $x$.

Figure 4

Energy databases for the Fe–1-at.-$\%$-Si system. (a) Fe to Si energy databases. (b) Si to Fe energy databases.

Figure 5

Temperature-dependent probabilities of finding Si atoms out to the third neighbor shell around Fe and Si atoms.

Figure 6

(a) A $\mathrm{\Sigma }3\left(112\right)\left[1\overline{1}0\right]$ symmetric tilt grain boundary system containing 4320 atoms. Atoms ordered in bcc structure are colored in blue, while atoms in another type of structure, namely, the atoms on the grain boundary, are colored in white. The atomic system was created by using atomsk [24], and the figure was obtained by using OVITO [25]. (b) Si concentration along the direction perpendicular to the GB, as obtained using a DD-SGCMC method (starting position of the center of the GB shown by dashed lines). The dimensions of the system are $165.430\times 31.837\times 19.496\mathring{\text{A}}$, and $L$ is equal to 165.43 $\mathring{\text{A}}$.

Figure 7

(a) Si concentration along the direction perpendicular to the GB, as obtained using an SGCMC method. (b) Comparison between the Si concentrations obtained with database-driven SGCMC and SGCMC algorithms near the GB. The total length of the system $L$ is equal to $165.43\mathring{\text{A}}$

Figure 8

GB energy of the Fe–1-at.-$\%$-Si alloy at 300, 500, and 1100 K. The calculated GB energy of pure Fe is also included for comparison.

Figure 9

Energy databases for the NiCoCr MEA system at (a) 850 K and (b) 1050 K.

Figure 10

Comparison of the short-range order parameters for the NiCoCr MEA system at (a) 850 K and (b) 1050 K, as obtained by using database-driven SGCMC and SGCMC methods and the short-range order parameters calculated by Li et al. [12].

Figure 11

NiCoCr MEA system with stacking fault containing 9216 atoms. The atoms in HCP structure are highlighted in red, and the atoms in fcc structure are shown in gray. The atomic system was created by using atomsk [24], and the figures were obtained by using OVITO [25]. The dimensions of the system are $30.176\times 98.554\times 34.844\mathring{\text{A}}$.

Figure 12

Elemental distribution in a NiCoCr MEA stacking fault system according to database-driven SGCMC simulations at (a) 850 K and (b) 1050 K. Elemental distribution in a NiCoCr MEA stacking fault system according to SGCMC simulations at (c) 850 K and (d) 1050 K. The total length of the system $L$ is equal to $98.55\mathring{\text{A}}$.

Figure 13

(a) Extrinsic stacking fault energy of NiCoCr MEA at 850 and 1050 K before segregation and after segregation. (b) Segregation energy in NiCoCr MEA systems at 850 and 1050 K containing an extrinsic stacking fault.