Atomistic simulations of diffusive phase transformations with non-conservative point defects

Abstract

Most of the phase transformations modifying the microstructure, thereby the materials properties, are controlled by the diffusion of atoms. The rate but also the selection of phase transformations depend on the concentration of lattice point defects (PDs), because substitutional atoms exchange with PDs to diffuse and PDs are non-conservative species. During manufacturing or in use, whenever PD diffusion and creation/annihilation reactions at extended defects in the microstructure are slower than the kinetics of the microstructure, these PDs may not have their equilibrium concentration. A departure of PDs from local equilibrium can be transient under thermal conditions, or permanent in materials driven out of equilibrium as under irradiation. Non-equilibrium PDs can have a dramatic effect on the evolution of the microstructure or even on the stationary microstructure in driven systems. We present an atomic kinetic Monte Carlo (AKMC) method, which is able to tackle the atomic-scale couplings between PD diffusion, annihilation/creation reactions and the kinetics of decomposition of a solid solution into a two-phase microstructure. By introducing PD source-and-sinks (SAS) at specific lattice sites, we control the PD reactions and highlight the role of non-equilibrium quenched-in point defects on the evolution kinetics of short-range order parameters and subsequent second-phase precipitation. Then, we open the discussion on various kinetic phenomena that require taking into account the role of non-equilibrium PDs at different scales of time and space.

Graphical abstract

Appendices

Appendix 1: AKMC parameters

The interatomic pair interactions of the A-B system are between first nearest-neighbor lattice sites only. They are equal to $\varepsilon _\mathrm{AA}=-1.05$, $\varepsilon _\mathrm{BB}=-1.05$ and $\varepsilon _\mathrm{AB}=-1.025$ eV, which gives an ordering energy equal to $v=\varepsilon _\mathrm{AA}+\varepsilon _\mathrm{BB}-2\varepsilon _\mathrm{AB}=-0.05$ eV. The atom-vacancy pair interactions are set to $\varepsilon _\mathrm{AV}=-0.40$ and $\varepsilon _\mathrm{BV}=-0.35$ eV.

When an atom is at the saddle-point, during its exchange with a vacancy, it interacts with with the 6 nearest neighbors. The corresponding pair interactions are equal to $\varepsilon _\mathrm{AA}^\mathrm{sp}=-1.500$ and $\varepsilon _\mathrm{AB}^\mathrm{sp}=-1.55$ eV for a A atom, and $\varepsilon _\mathrm{BA}^\mathrm{sp}=-1.500$ and $\varepsilon _\mathrm{BB}^\mathrm{sp}=-1.55$ eV for a B atom. These interaction parameters control the migration barriers. For instance, they give 1.15 eV for an A-V exchange in pure A, 1.025 eV for a B-V exchange in pure A, 1.25 eV for a B-V exchange in pure B, and 1.025 eV for a A-V exchange in pure B.

The saddle-point parameters that control the kinetics of vacancy creation/annihilation reactions at source/sink sites are set to $Q_\mathrm{AA}^\mathrm{sp}=Q_\mathrm{BA}^\mathrm{sp}=-6.6$, $Q_\mathrm{AB}^{sp}=Q_\mathrm{AB}^\mathrm{sp}=6.7$ eV for the simulations at 1000 K. The same parameters are all set to $-6.35$ eV for the simulations at 600 K.

Appendix 2: Chemical potentials

The chemical potentials of atoms A and B in binary alloys AB of different compositions at $T=600$ K are plotted in Fig. 10. The circles are the values extracted from Monte Carlo simulations in the semi-grand canonical ensemble (GCMC). In GCMC simulations, the difference $\Delta \mu =\mu _\mathrm{B}-\mu _\mathrm{A}$ is imposed, the values of $\mu _\mathrm{A}$ and $\mu _\mathrm{B}$ are then deduced from the Helmholtz free energy: $F=(1-X_\mathrm{B})\mu _\mathrm{A}+X_\mathrm{B}\mu _\mathrm{B}$ obtained by integration of $\mu _\mathrm{B}-\mu _\mathrm{A}$ with respect to the atomic fraction of B. The values predicted by the Bragg–Williams approximation (regular solution model, see e.g., Ref. 71) are shown for comparison.

Figure 10

Chemical potentials of A and B atoms at 600 K, as a function of the B concentration.

The GCMC simulations only give the chemical potentials of stable or metastable solid solutions ($X_\mathrm{B} \lesssim 0.05$ and $X_\mathrm{B} \gtrsim 0.95$ in Fig. 10). The values of $\mu _\mathrm{A}$ and $\mu _\mathrm{B}$ used in the NCV-AKMC simulations are set equal to the GCMC measured values as long as the composition of the local finite volume centered on the SAS site $X_\mathrm{B}^s$ (fraction of B atoms among the first and second nearest neighbors of the SAS site) is the one of a stable solid solution ($X_\mathrm{B}^s<X_\mathrm{B}^\alpha$ or $X_\mathrm{B}^s>X_\mathrm{B}^\beta$). For intermediate compositions ($X_\mathrm{B}^\alpha<X_\mathrm{B}^s<X_\mathrm{B}^\beta$), the chemical potentials are set to their equilibrium two-phase values, i.e.,: $\mu _\mathrm{A}=\mu _\mathrm{A}(X_\mathrm{B}^\alpha )=\mu _\mathrm{A}(X_\mathrm{B}^\beta )$, and $\mu _\mathrm{B}=\mu _\mathrm{B}(X_\mathrm{B}^\alpha )=\mu _\mathrm{B}(X_\mathrm{B}^\beta )$ (corresponding to the dashed-dotted line in Fig. 10). For the equilibrium compositions $X_\mathrm{B}^\alpha$ and $X_\mathrm{B}^\beta$, the differences between the Monte Carlo and the Bragg–Williams chemical potentials are relatively small ($\sim 0.008$ eV), but leads to differences of approx. 15% on the equilibrium vacancy concentration, $X_\mathrm{V}^\mathrm{eq}$, at 600 K.