Based on free-energy functionals, phase-field modeling enables the simulation of microstructural pattern formation and dynamics while providing spatially resolved information on the underlying local thermodynamics. In the same spirit, free-energy functionals for atomic systems can be used for analogous purposes at atomic resolution; we refer to this approach as atomic-scale phase-field modeling. However, previous attempts have been limited to classical pairwise interatomic potentials because of the high computational cost and limited scalability associated with evaluating free-energy functionals involving many-body interactions. In this work, we generalize the atomic-scale phase-field methodology to many-body systems, whose efficient evaluation is facilitated by modifying the stochastic self-consistent harmonic approximation (order-$N$ SSCHA) and by machine-learning potentials. We apply this method to bulk Cu and Si in the NPT ensemble, where the predicted equilibrium lattice constants show agreement with molecular-dynamics simulations, validating the theoretical framework for these relatively simple systems. Furthermore, we apply the method to the $\mathrm{\Sigma }5\left(310\right)\left[001\right]$ grain boundary in Cu using a simulation cell containing several hundred atoms, yielding atomically resolved maps of local free-energy distributions. The results reveal a pronounced concentration of free energy at the grain-boundary core, suggesting a possible coupling between tensile stress and entropy. We then applied the developed methodology to more complex materials, namely, the thermal expansion of ${\mathrm{SrTiO}}_3$ and the polarization-dependent free-energy surfaces of ${\mathrm{PbTiO}}_3$. For these systems, the agreement with other methods was less satisfactory than for the simpler benchmarks, revealing the current limitations of the framework and identifying directions for further development. This study represents a step toward the accurate free-energy functionals applicable to large-scale atomic systems, thereby advancing the atomic-scale phase-field concept.

Close

PrevNext

View figure in article

• Grain boundaries
• Crystal structures
• Deep potential molecular dynamics
• Phase-field modeling