Calculation of excess free energies of precipitates via direct thermodynamic integration across phase boundaries

We describe a technique for constraining macroscopic fluctuations in thermodynamic variables well-suited for Monte Carlo (MC) simulations of multiphase equilibria. In particular for multicomponent systems this amounts to a statistical ensemble that implements constraints on both the average composition as well as its fluctuations. The variance-constrained semi-grand-canonical (VC-SGC) ensemble allows for MC simulations, in which single-phase systems can be reversibly switched into multiphase equilibria allowing the calculation of excess free energies of precipitates of complex shapes by thermodynamic integration. The basic features as well as the scaling and convergence properties of this technique are demonstrated by an application to an Ising model. Finally, the VC-SGC MC simulation technique is used to calculate $\alpha /{\alpha }^{\prime }$ interface free energies in Fe-Cr alloys as a function of orientation and temperature taking into account configurational, vibrational, and structural degrees of freedom.

Figure 1

(a) Derivative of free energy with respect to concentration at $T=1.5k_B$ for system size $8\times 8\times 12$. The gray areas indicate composition regions, in which spherical precipitates and flat interfaces are formed. The bold dash-dotted green lines indicate fits to Eq. (18). The inset shows the phase diagram for the Ising model described in the text, where the temperature is measured in units of $k_B$. (b) Free energy profile obtained by integration of data in (a).

Figure 2

Interface free energies as a function of temperature and orientation for the Ising model described in the text. The inset shows a snapshot of a $\left\{100\right\}$ interface from a VC-SGC MC simulation close to 50$\%$ composition (picture rendered with ovito, Ref. 17).

Figure 3

Scaling of the maximum of excess free energy [compare with Fig. 1b] with the inverse linear dimension for the Ising model described in the text. Simulation cell vectors are oriented along $\langle 100\rangle$ directions inducing the formation of $\left\{100\right\}$ interfaces.

Figure 4

Standard deviation of concentration at a temperature of $T=2k_B$ as a function of the (a) variance constraint parameter and (b) system size. Data in (b) measured at 50$\%$ composition. The black dashed line indicates scaling with $1/\sqrt{x}$.

Figure 5

Effect of the variance constraint parameter $\overline{\kappa }$ on the (a) free energy derivative and (b) standard deviation of concentration at $T=2k_B$ for system size $4\times 4\times 8$.

Figure 6

(a) Acceptance probability as a function of concentration for different values of the variance constraint parameter at a system size of $4\times 4\times 8$. (b) Acceptance probability at 50$\%$ as a function of variance constraint parameter for different system sizes.

Figure 7

(a) Derivative of free energy with respect to concentration obtained from MC simulations in the SGC and VC-SGC ensembles at 1000 K. The inset shows the calculated phase diagram, which is in excellent agreement with the published phase diagram of the potential used in the present work.[23] (b) Free energy profile obtained by integration of the data in (a), and (c) the corresponding excess free energy. The central dark gray area indicates the region within which flat interfaces are obtained while the dark gray areas to the left and right show the regions in which compact precipitates are obtained.

Figure 8

Interface free energies for Fe-Cr model alloy for different interface orientations as well as spherical precipitates.

Figure 9

(a) Free energy, (b) internal energy, and (c) entropy of the $\left\{100\right\}$ interface in the Fe-Cr model alloy. The dashed lines represent fits to second degree polynomials.