The 2/3 Rule of Glass Physics Implies Universalities in Crystal Melting

Since more than 100 years, melting is thought to be governed by the Lindemann criterion. It assumes that a crystal melts when, upon heating, the growing atomic vibration amplitudes become sufficiently large to destabilize its crystalline lattice. However, it is unclear why the viscosities eta or the related relaxation times tau of the resulting liquids, measured directly at the melting point Tm, differ by up to nine decades, depending on the material. Based on the empirical rule that the ratio of the glass-transition temperature and Tm is about 2/3, here we show that this strong variation is due to differences in the liquid’s fragilities, a property associated with pronounced non-Arrhenius behavior and often ascribed to cooperative motions. We propose that, without cooperativity, all crystals would melt into liquids with a universal viscosity value and relaxation time. Hence, the real melting point is only partly determined by the Lindemann criterion and strongly enhanced by the cooperativity of the resulting liquid. Our findings are corroborated by the determination of the idealized, fragility-free melting temperatures, and of the corresponding eta and tau values for various example materials.

💡 Research Summary

The paper revisits the long‑standing Lindemann criterion for crystal melting and asks why the viscosity (ηₘ) and the corresponding relaxation time (τₘ) measured at the melting temperature (Tₘ) can differ by up to nine orders of magnitude among different substances. The authors argue that this huge spread is not a failure of the Lindemann picture per se, but rather a consequence of the varying “fragility” of the liquids that emerge upon melting.

A central empirical observation in glass physics is that the glass‑transition temperature (T_g) is roughly two‑thirds of the melting temperature, i.e. T_g ≈ (2/3) Tₘ. The authors adopt this “2/3 rule” as a bridge between crystal thermodynamics and liquid dynamics. Fragility is quantified by the Angell slope m, defined as the derivative of log η (or log τ) with respect to T_g/T near T_g. High‑m liquids display strong non‑Arrhenius (Vogel‑Fulcher‑Tammann, VFT) behavior, whereas low‑m liquids are close to Arrhenius.

Using the VFT expression
 η(T) = η₀ exp