Growth of Structural Lengthscale in Kob Andersen Binary Mixtures: Role of medium range order

TL;DR

The paper interrogates whether a single growing lengthscale underpins glassy dynamics in MRCO-free Kob–Andersen mixtures by contrasting dynamical lengthscales (from $S_4$ and displacement correlations) with static lengthscales derived from a mean-field caging SOP. It reveals that the bare static lengthscale grows only weakly, but a medium-range static lengthscale emerges when the SOP is coarse-grained over an optimal length $L_{\max}$, and this $\xi_{CG}$ tracks the dynamical lengthscale $\xi_4$ (and $\xi_D$) as temperature decreases. The growth is strongest for A particles in the 80:20 mixture and for both species in 60:40, with $L_{\max}$ increasing upon cooling and static correlations also showing patches correlated with mobility; this links structural order to incipient crystallisation tendencies. The results reconcile prior failures to observe static-length growth in KALJ by highlighting the necessity of intermediate-range descriptors and demonstrate a robust structure–dynamics connection across timescales, consistent with a medium-range ordering picture of glassy slowdown. Overall, the work emphasizes that long-time dynamics require medium-range structural descriptors rather than purely local order, offering a unified view of static–dynamic coupling in a canonical glass-former.

Abstract

A central and extensively debated question in glass physics concerns whether a single, growing lengthscale fundamentally controls glassy dynamics, particularly in systems lacking obvious structural motifs like the Kob Andersen binary Lennard Jones (KALJ) model. In this work, we investigate structural and dynamical lengthscales in supercooled liquids using KALJ model in two compositions: 80:20 and 60:40. We compute the dynamical lengthscale from displacement displacement correlation functions and observe a consistent growth as temperature decreases. To explore the static counterpart, we use a structural order parameter (SOP) based on the mean field caging potential. While this SOP is known to predict short time dynamics effectively, its bare correlation function reveals minimal spatial growth. Motivated by recent findings that long time dynamics reflect collective rearrangements, we perform spatial coarse graining of the SOP and identify an optimal lengthscale Lmax that maximises structure dynamics correlation. We show that the structural correlation length derived from SOP coarse grained over Lmax exhibits clear growth with cooling and closely tracks the dynamical lengthscale, especially for A particles in the 80:20 mixture and for both A and B particles in the 60:40 system. Our results reconcile the previously observed absence of static length growth in the KALJ model by highlighting the necessity of intermediate range structural descriptors. Furthermore, we find that the particles with larger structural length growth also correspond to species with latent crystallisation tendencies, suggesting a possible link between structural order, dynamics, and incipient crystallisation.

Figures (16)

• Figure 1: (a) Inverse of the four-point structure factor, $1/S_4(\mathbf{q}, \tau_\alpha)$, (Eq.\ref{['eq_s4q']}) plotted as a function of the squared wavevector, $q^2$, at various temperatures. Solid lines are Ornstein–Zernike (OZ) fits. (b) Temperature dependence of three characteristic lengthscales: the dynamical lengthscale, $\xi_{4}$ (Eq.\ref{['eq_ozs4q']}); the dynamical lengthscale, $\xi_{D}$ (Eq.\ref{['ddcr-integration']}) for A particles; and the bare static lengthcale, $\xi_{bare}$ (Eq.\ref{['bare-integration']}) for A particles. The scaling factor for $\xi_{4}$ is 2.21, $\xi_D$ is 0.31, & $\xi_{bare}$ is 0.28. We also plot the PTS lengthscale reported by Hocky et al.Hocky2012 scaled by 1.63, the PTS and finite size scaling lengthscale reported by Chakrabarty et al.Chakrabarty2017 scaled by 1.60 and the static lengthscale obtained by Zhang and Kob for the angular power spectra Zhang2020 scaled by 3.52. With a decrease in temperature, both dynamical lengthscales exhibit similar growth, while the static lengthscales show weaker growth.
• Figure 2: Spearman rank correlation, $C_{R}(\overline{\beta\phi}, \mu)$, between the coarse-grained SOP, $\overline{\beta\phi}$, and mobility, $\mu$, calculated at times scaled by the $\alpha$-relaxation time, $t/\tau_{\alpha}$, for A-type particles at different coarse-graining lengths, $L$. (a) Result at high temperature, $T = 0.70$ and (b) at low temperature, $T = 0.47$. The colour coding in (b) matches that of (a).
• Figure 3: Spearman rank correlation, $C_R(\overline{\beta\phi}, \mu(\tau_{\alpha}))$, between the coarse-grained SOP, $\overline{\beta\phi}$, and mobility, $\mu(\tau_{\alpha})$, calculated at $\alpha$-relaxation time as a function of coarse-graining length, $L$, (a) for A-type particles and (b) for B-type particles at different temperatures. The colour coding in (b) matches that of (a) (c) Temperature dependence of the average value of $L_{max}$, shows stronger growth for A particles and weaker growth for B particles. The error bars show the variation of the $L_{max}$ with configuration. Lines are a guide to the eye.
• Figure 4: Results for 80:20 KALJ system: Top panels: Snapshots at $T=0.45$ showing structural and dynamical fields for A-type particles: (a) The bare SOP, $\beta\phi$, appears spatially noisy, lacking visible large-scale structure. (b) Coarse-graining the SOP, $\overline{\beta \phi}$, with $L_{max}$, shows the emergence of patches of high and low SOP values. (c) The particle mobility field, $\mu (\tau_{\alpha})$, calculated using Eq.\ref{['eq:mobility']} over the $\alpha$-relaxation time shows low mobility domains that spatially mostly overlap with high SOP regions in (b), indicating a structure–dynamics link. Bottom Panels: Normalized excess correlation functions corresponding to the fields in top panels at different temperatures. (d) The normalised excess correlation of the bare SOP, $\Gamma_{\text{bare}}(r)$, (Eq.\ref{['eq-gamma_static_bare']}) confirms the absence of significant growth in bare static lengthscale. (e) The normalised excess correlation of the coarse-grained SOP, $\Gamma_{\text{CG}}(r)$, (Eq.\ref{['eq-gamma_static_cg']}) shows an increasing correlation length with decreasing temperature, consistent with the emergence of extended structural patches. (f) The normalized excess displacement–displacement correlation function, averaged over the configurations $\Gamma_{uu}(r,\tau_\alpha)$, (Eq.\ref{['eq_gamma']}) exhibits growth with decreasing temperature. The colour coding in (e) and (f) matches that of (d)
• Figure 5: Temperature dependence of the dynamical lengthscale, $\xi_{D}$, and coarse-grained structural lengthscale, $\xi_{CG}$, in the 80:20 KALJ system. $\xi_{D}$ is obtained from $\Gamma_{uu}(r,\tau_\alpha)$, (Eq.\ref{['ddcr-integration']}); and $\xi_{CG}$ obtained from $\Gamma_{CG}(r)$, (Eq.\ref{['CG-integration']}). For comparison, the lengthscales are all scaled. The scaling factor for $\xi_{D}$ A-type is 0.31, $\xi_{D}$ B-type is 0.22, $\xi_{CG}$ A-type is 1.00, & $\xi_{CG}$ B-type is 6.489. The dotted line is a power law fit with $\xi=\xi_{o} [(T-T_{VFT})/T_{VFT}]^{-2/3}$, where $T_{VFT}=0.319$, same as that obtained for the relaxation time. Inset: We plot $ln(\tau_{\alpha})$ against $\xi^{3/2}$ for $\xi_{4}$, $\xi_{D}$ and $\xi_{CG}$ for A particles. The linear fit confirms a power law relationship between the dynamics and the lengthscales.