Phase-field modeling, which is based on the Landau-Ginzburg-Devonshire theory, has achieved great success in explaining and predicting the behavior of ferroelectrics. However, such phase-field modeling is based on continuum mechanics, and thus cannot be applied directly to atomic systems, even though interest in ferroelectrics moves to smaller and smaller scales. Here, we have developed phase-field modeling of ferroelectrics from the atomic scale; that is, we derived the free energy of ferroelectrics by combining the bond-valence molecular dynamics potential and the variational Gaussian theory. By solving the phase-field equation based on this free energy, we successfully reproduced the temperature dependence of electric polarization in ${\mathrm{PbTiO}}_3$. Furthermore, local thermodynamic analysis based on our modeling suggests that this phase transition occurs because the heavy atoms such as Pb do not have enough space for vibration in the ferroelectric state, and this increases free energy locally at high temperatures. Our study expands the application of phase-field modeling to the atomic scale and expands our understanding of phase transitions.

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• Ferroelectricity
• Phase transitions
• Ferroelectrics
• Molecular dynamics
• Phase-field modeling