Reply to "Characterizing dynamic length scales in glass-forming liquids"

Kob et al. reply

In their Correspondence, Flenner and Szamel¹ compare the temperature dependence of an alternative dynamic length scale, ξ₄, with that of ξ^dyn, which we studied². Using computer simulations of the same system, they conclude that these two length scales have a different temperature dependence. In particular, ξ₄ does not follow the striking non-monotonic temperature dependence we reported for ξ^dyn. Although both types of measurements aim at quantifying the spatial extent of dynamic correlations in supercooled liquids, the two procedures differ on essential points, which we now discuss.

Whereas we measured up to the distance at which the value of the relaxation time is affected by the presence of an amorphous wall², Flenner and Szamel quantify instead the spatial extent of spontaneous dynamic fluctuations at low wavevectors for a fixed timescale (the bulk relaxation time). These two measurements need not be directly related, although they seem to coincide at moderate temperatures (Fig. 1a of ref. 1). Although four-point functions as measured by Flenner and Szamel have played a pivotal role in previous analysis of dynamic heterogeneity, theoretical work has also revealed a number of shortcomings. Most notably, four-point functions display a strong dependence on the statistical ensemble chosen to perform measurements. As a result, they receive distinct contributions from density and energy fluctuations and show, even in idealized cases, complex scaling properties³, which complicates the direct extraction of a correlation length. A second difficulty lies in the fact that these various contributions have different temperature dependences, with a crossover taking place very close to the mode-coupling temperature where the subtle effects we reported² occur. We remark that this corresponds to the temperature scale where ξ₄ and ξ^dyn start to differ. These known weaknesses of four-point functions had in fact motivated our study of an alternative correlation length scale that is free of such ambiguities.

Given these important differences, it is not clear how a non-monotonic temperature evolution of dynamic correlations will manifest itself in the numerical data of Flenner and Szamel. Although the authors argue that the evolution of ξ₄ with the relaxation time shows a crossover (Fig.1c in ref. 1), we point out that the presented data does not clearly show such a change in behaviour. Another possibility, not explored by Flenner and Szamel, could be that the functional form of the four-point correlation function S₄(q,t) as investigated by these authors changes with temperature, in agreement with the idea that the geometry of the relaxing entities changes with temperature, as we argued². As the reported effect is small (Fig. 2b in ref. 2), one would presumably need a relative accuracy of S₄(q,t) of better than 1% at low wavevectors. The data shown by Flenner and Szamel demonstrate that at present this remains technically difficult.

Finally, the crossover we report also coincides with the emergence of non-trivial static correlations in the system, a result that cannot be obtained using purely dynamic correlations. Overall, this suggests that the approach of pinning particles, for which no a priori knowledge of the nature of the relaxing entities is needed, is extremely helpful for studying small but important effects in the relaxation properties of glass-forming systems that cannot be drawn from the measurements of Flenner and Szamel. We point out that since our results² were published, we have independently confirmed the qualitative change of transport properties in our system near the mode-coupling crossover by analysing the temperature evolution of dynamical finite size effects⁴.