Theoretical studies of phase stability in solid materials with dynamic disorder are challenging due to the failure of the standard picture of atoms vibrating around fixed equilibrium positions. Dynamically disordered solid materials show immense potential in applications. In particular, superionic conductors, where the disorder results in exceptionally high ionic conductivity, are very promising as solid state electrolytes in batteries and fuel cells. The biggest obstacle in living up to this potential is the limited stability of the dynamically disordered phases. Here, we outline a method to obtain the free energy of a dynamically disordered solid. It is based on a stress-strain thermodynamic integration on a deformation path between a mechanically stable ordered variant of the disordered phase, and the dynamically disordered phase itself. We show that the large entropy contribution associated with the dynamic disorder is captured in the behavior of the stress along the deformation path. We apply the method to ${\mathrm{Bi}}_2{\mathrm{O}}_3$, whose superionic $\delta$ phase is the fastest known solid oxide ion conductor. We accurately reproduce the experimental transition enthalpy and the critical temperature of the phase transition from the low temperature ground state $\alpha$ phase to the superionic $\delta$ phase. The method can be used for a first-principles description of the phase stability of superionic conductors and other materials with dynamic disorder, when the disordered phase can be connected to a stable phase through a continuous deformation path.

• Received 7 August 2018

DOI:https://doi.org/10.1103/PhysRevLett.121.225702