A Kinetic Criterion for Stokes-Einstein Relation Breakdown Based on Effective Collisional Geometry

Abstract

Here we propose a kinetic framework for interpreting the Stokes-Einstein (SE) relation breakdown in supercooled liquids by introducing an effective collision diameter, $d_{\mathrm{eff}}$, derived from transport data. Numerical simulation of a model CuZr alloy reveal that $d_{\mathrm{eff}}$ increases upon cooling but saturates near the first peak of the radial distribution function just before SE breakdown. This saturation defines a geometric upper bound for the collisional cross-section beyond which further slowdown is governed by cooperative, heterogeneous motion rather than local collisional transport. Our analysis yields a compact criterion for SE breakdown in a mean-field perspective and provides physically interpretable inputs for future data-driven models of glassy dynamics.

Figures (2)

• Figure 1: Effective collision diameter vs. temperature. Squares: transport-inferred $d_{\mathrm{eff}}(T)$. Circles: nearest-neighbor spacing $r^{\mathrm{max}}_{1}(T)$ from global $g(r)$. Upon cooling, $d_{\mathrm{eff}}$ increases and approaches a ceiling set by $r^{\mathrm{max}}_{1}$. The approach of $r^{\mathrm{max}}_{1}(T)$ to this ceiling coincides with the onset of Stokes–Einstein behavior breakdown; further slowdown of $D$ and $\tau_\alpha$ is not captured by a single local length, and the inversion yields a turnover of $d_{\mathrm{eff}}$, reflecting the dominance of heterogeneous, activated dynamics.
• Figure 2: Temperature dependence of the dimensionless kinetic–geometric quantity $\mathcal{S}\ast=3\pi n\, (r_1^{\max})^2 \sqrt{D\tau_\alpha}$ for two metallic glass-forming alloys, CuZr and NiAl. For both systems, upon cooling from the high-temperature liquid, $\mathcal{S}\ast$ develops a minimum close to unity, $\mathcal{S}\ast \simeq 1$, at the temperature where the transport-inverted effective diameter $d_{\mathrm{eff}}$ saturates at the structural neighbor spacing.