First-principles study of the effect of lattice vibrations on Cu nucleation free energy in Fe-Cu alloys

The effects of lattice vibration on Cu nucleation in Fe-Cu alloys have been investigated by a first-principles technique. We find that the vibrational effect is comparable $(\sim 36\%)$ to the configurational entropy. Lattice vibration increases the activation barrier of Cu precipitates by 0.53 eV, where the resulting activation barrier is estimated to be 0.62 eV at critical number $n^{*}$ of 12 atoms, which is in satisfactory agreement with the previous experimental prediction. Within the description of classical treatment, this increase is interpreted in terms of the associated decrease of driving force due to lattice vibration.

Figure 1

Schematic illustration of the nucleation process. Gray spheres represent the solvent Fe atoms, and black ones the solute Cu atoms.

Figure 2

(Color online) Atomic configurations for the Cu precipitate clusters.

Figure 3

Calculated bulk modulus for configurations ${\sigma }_i$ as a function of number of Cu atoms $n$ in a supercell of 128 atomic sites.

Figure 4

Distribution of the vibrational free energy $F_{\mathrm{vib}}$ per atom for configurations ${\sigma }_i$.

Figure 5

Total free energy change $\Delta F_{\mathrm{tot}}$ of Cu precipitate nucleation in Fe-Cu system at $T=773\mathrm{K}$ and the Cu concentration of 0.014. To see the impact of lattice vibration, the purely configurational contribution $\Delta F_{\mathrm{config}}$ of Eq. (3) is also shown.