Influence of short-range order on diffusion in multiprincipal element alloys from long-time atomistic simulations

Short-range order (SRO) is an important structural characteristic in multiprincipal element alloys (MPEAs) that plays a crucial role in their superior mechanical and physical properties. The development of SRO is usually tangled with structural evolution associated with diffusion-induced atomic transport, making it challenging to accurately determine the diffusion coefficients in MPEAs based on conventional modeling techniques. In this study, we combine machine learning with kinetic Monte Carlo to compute the self-diffusion coefficients in NiCoCr and NiFeCr MPEAs with distinct SRO tendencies under long-time structural evolution. Our results show that the long-term diffusion process governed by thermodynamic and kinetic factors can promote SRO formation due to chemically biased exchange between vacancy and specific components in the alloy. We further demonstrate that accurate diffusion coefficients in MPEAs can only be obtained through long-term simulations allowing for sufficient structural evolution. This study underlines the shortcomings of the presently widely employed techniques to study diffusion in MPEAs and highlights the intricate coupling between SRO and diffusion.

Figure 1

Prediction performance of the ML model for energy barriers and site energy differences as measured by the mean absolute error (MAE) and Pearson correlation coefficient ($R$) from tenfold cross validation for NiCoCr and NiFeCr, respectively. (a), (b) NiCoCr (c), (d) NiFeCr.

Figure 2

Time evolution of mean squared displacements (MSD) at different temperatures and the corresponding diffusion coefficients in NiCoCr and NiFeCr. (a)–(c) The MSD evolution at 550, 950, and 1350 K, respectively, in NiCoCr. (d)–(f) The MSD evolution at 550, 950, and 1350 K, respectively, in NiFeCr.

Figure 3

The first to third SRO changes in NiCoCr and NiFeCr due to vacancy-mediated diffusion at 950 K. (a)–(c) NiCoCr. (d)–(f) NiFeCr.

Figure 4

Determination of the diffusion coefficients in random solid solution states using the symmetric migration barrier $E_m^{{}^{\prime }}$. (a)–(c) NiCoCr under 550, 950, and 1350 K. (d), (e) NiFeCr under 550, 950, and 1350 K.

Figure 5

The diffusivity and absolute SRO change with ML kMC steps in NiCoCr and NiFeCr at 950 K for the random structures and the structures with SRO equilibrated by the VCSGC MC algorithm at 950 K. (a) NiCoCr. (b) NiFeCr. Note that the absolute SRO is the sum of the absolute value of SRO parameters for all the atomic pairs.

Figure 6

The first NN Warren-Cowley SRO and the distributions of migration energy barriers at 1350 K by ML kMC. (a) Time-dependent SRO changes with simulation starting from an initial random structure. (b) The initial stage of SRO change, which is the enlarged part of the red dot frame of (a). (c), (d) The distributions of energy barriers for all first NN V-X exchanges and the chosen migration barriers within the initial 10 000 ML kMC steps, respectively. (e), (f) The distribution of energy barriers for all first NN V-X exchanges and the chosen migration barriers within the equilibrium state (sampling from the final 10 000 steps from a 1 200 000 ML kMC simulation), respectively.

Figure 7

Analysis of atomic transport during ML kMC simulations. (a) Atom configurations at 1350 K. (b) The counting statistic of vacancy-reaching frequency in each subregion in continuous 10 000 ML kMC steps ($a_0$ is the lattice constant). The red frame corresponds to the same position in the $x\text{−}y$ plane between (a) and (b).