Time-Temperature-Transformation (TTT) Diagrams to rationalize the nucleation and quenchability of metastable $\alpha$-Li${}_3$PS${}_4$

$\alpha$-Li${}_3$PS${}_4$ is a promising solid-state electrolyte with the highest ionic conductivity among its polymorphs. However, its formation presents a thermodynamic paradox: the $\alpha$-phase is the equilibrium phase at high temperature and transforms to the stable $\gamma$-Li${}_3$PS${}_4$ polymorph when cooled to room temperature; however, $\alpha$-Li${}_3$PS${}_4$ can be synthesized and quenched in a metastable state via rapid heating at relatively low temperatures. The origin of this synthesizability and anomalous stability has remained elusive. Here, we resolve this paradox by establishing a comprehensive time-temperature-transformation (TTT) diagram, constructed from a computational temperature-size phase diagram and experimental high-time-resolution isothermal measurements. Our density functional theory calculations reveal that at the nanoscale, the $\alpha$-phase is stabilized by its low surface energy, which drastically lowers the nucleation barrier across a wide temperature range. This size-dependent stabilization is directly visualized using in-situ synchrotron X-ray diffraction and electron microscopy, capturing the rapid nucleation of nano-sized $\alpha$-phase and its subsequent slow transformation. This work presents a generalizable framework that integrates thermodynamic and kinetic factors for understanding nucleation and phase transformation mechanisms, providing a rational strategy for the targeted synthesis of functional metastable materials.