Divergence behavior of the interfacial dynamic length scales in the supercooled Kob–Andersen liquid near a frozen amorphous wall

When a supercooled binary Lennard-Jones liquid (the Kob–Andersen model) is confined between walls made of a frozen amorphous equilibrium configuration of the same material, the wall slows structural relaxation over a distance that grows on cooling. Two length scales characterize this influence: $\tilde z$, the crossover distance from the near-wall law $\tau(z)\propto\exp[\Delta/(z-z_p)]$ to bulk relaxation, and $\xi_{\max}$, the maximum over time of the stretched-exponential decay length of the incoherent intermediate scattering function $F_s(q,z,t)$ (a self-correlation function of single-particle motion). The originating 2002 study found that both scales grow only modestly on cooling and posed the question addressed here: do they diverge at the Kauzmann temperature $T_K\approx 0.29$ (the extrapolated temperature at which the liquid's excess entropy would vanish), only at $T=0$, at some other temperature, or not at all; why does their ratio depend on temperature; and what theory describes their growth and nature? Later work sharpened but did not settle this: the same authors' extended study found Arrhenius-compatible growth; inhomogeneous mode-coupling theory (the extension of the kinetic theory of the glass slowdown to spatially inhomogeneous states) predicts a wall-perturbation length diverging at the mode-coupling temperature $T_c$; and modern wall-probe simulations of the same model find the length saturates near $T_c$ rather than growing toward $T_K$, with behavior depending on the observable used. Determining the asymptotic temperature dependence and divergence behavior of these interfacial dynamic length scales, and providing a theoretical account of their nature and mutual relation, remains open.

Background

When a liquid is cooled below its melting point without crystallizing, it becomes a supercooled liquid whose structural relaxation time grows by many orders of magnitude before the material falls out of equilibrium as a glass. The classical picture of Adam and Gibbs attributes this slowdown to increasingly cooperative motion: rearrangement requires domains within which many particles move together, and these domains are supposed to grow as temperature decreases. Measuring the size of such cooperative regions and its temperature dependence is a central problem in glass physics.

A standard simulation model for this regime is the Kob–Andersen (KA) binary Lennard-Jones mixture, an 80:20 mixture of two particle species interacting through truncated, shifted Lennard-Jones potentials with cross-interactions chosen to suppress crystallization. Its bulk behavior is well characterized: the dynamics is described above the critical temperature of mode-coupling theory, $T_c\approx 0.435$ (in reduced units), by mode-coupling theory (MCT), a kinetic theory in which relaxation slows through a nonlinear feedback of density fluctuations onto themselves; and extrapolation of the configurational entropy — the entropy difference between the supercooled liquid and the set of its mechanically stable inherent structures — vanishes at an estimated Kauzmann temperature $T_K\approx 0.29$ (Coluzzi, Parisi and Verrocchio, 2000; Sciortino, Kob and Tartaglia, 1999). The Kauzmann temperature is the putative location of an ideal thermodynamic glass transition at which the excess entropy of the liquid would vanish; whether relaxation times and associated length scales actually diverge at such a finite temperature, or only as $T\to 0$ (the Arrhenius-like alternative to finite-temperature divergence scenarios such as Vogel–Fulcher behavior, in which relaxation times would diverge at a finite temperature), is a long-standing unresolved issue in the field.

The 2002 study of Scheidler, Kob, Binder and Parisi introduced a clean way to measure a dynamic length scale in this model: confine the liquid between walls made of a frozen equilibrium configuration of the same liquid (amorphous walls). Because the wall particles are taken from an equilibrated configuration of the identical system, the static structure of the confined liquid equals that of the bulk — only the dynamics is perturbed. Particles at distance $z$ from the wall are monitored through the incoherent intermediate scattering function $F_s(q,z,t)$, the self-correlation function that measures how far particles initially located at distance $z$ have moved by time $t$. Near the wall the relaxation is slower by orders of magnitude, with a relaxation time growing like $\tau(z)\propto\exp[\Delta/(z-z_p)]$ as $z$ decreases (with $\Delta\approx 7.5$ and $z_p\approx -0.5$), while bulk behavior is recovered far from the wall. Two length scales characterize how far the wall's influence extends: $\tilde z$, the crossover distance at which the near-wall law gives way to the bulk relaxation time, and $\xi_{\max}$, the maximum over time of the decay length $\xi(t)$ in the empirical description $F_s(q,z,t)=F_s^{\mathrm{bulk}}(q,t)+a(t)\exp[-(z/\xi(t))^{\beta(t)}]$ (a stretched-exponential decay in the distance from the wall).

Between $T=1.0$ and $T=0.5$ — the lowest temperature then accessible, limited by the extremely slow relaxation of particles close to the wall — both scales grow only modestly (by a factor of about 4 for $\tilde z$ and 2.5 for $\xi_{\max}$) while bulk relaxation times grow by a factor of about 500. The growth is compatible with a power-law divergence at $T_K$ with exponent near 1, but equally compatible with divergence only at $T=0$ or with no divergence at all; the ratio $\tilde z/\xi_{\max}$ is itself temperature-dependent, suggesting the two scales may have different critical behavior; and in films thin enough that both walls influence the fluid, the slowdown exceeds any linear superposition of single-wall effects, indicating a strongly nonlinear mechanism (such as the nonlinear feedback of MCT). Subsequent research has produced competing theoretical predictions and new measurements in the same geometry without settling the question.

Problem Statement

Determine the temperature dependence and divergence behavior of the interfacial dynamic length scales $\tilde z(T)$ and $\xi_{\max}(T)$ of the supercooled Kob–Andersen binary Lennard-Jones liquid confined by frozen amorphous walls: establish whether these scales diverge at a finite temperature (such as the Kauzmann temperature $T_K\approx 0.29$), grow without bound only as $T\to 0$, saturate near the mode-coupling temperature $T_c\approx 0.435$, or follow some other scenario — and provide a theoretical account of (i) the functional form of the growth and (ii) the relation between the two scales, as encoded in the temperature dependence of their ratio $\tilde z/\xi_{\max}$. The temperature range the question concerns extends substantially below $T=0.5$, the lowest temperature reached by the originating study, and includes the region near the mode-coupling temperature $T_c\approx 0.435$ and the approach to the Kauzmann temperature $T_K\approx 0.29$.

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