[Paper Review] Exposing the static scale of the glass transition by random pinning

Smarajit Karmakar, Itamar Procaccia|arXiv (Cornell University)|May 20, 2011

Material Dynamics and Properties2 references3 citations

TL;DR

This paper demonstrates that the static length scale $\xi_s$, previously proposed as a key scale for the glass transition, is experimentally accessible through random pinning of particles in supercooled liquids. By introducing quenched disorder with tunable density $\rho_{\rm im}$, the authors show that dynamics completely freeze at a critical pinning density $\rho_{\rm im}^c \sim 1/\xi_s^d$, confirming $\xi_s$ as the true static length scale governing the glass transition.

ABSTRACT

The dramatic slowing down associated with the glass transition cannot be fully understood without an associated static length that is expected to increase rapidly as the temperature is reduced. The search for such a length was long and arduous, without a universally accepted candidate at hand. Recently a natural such length $ξ_s$ was proposed, stemming from a cross-over between plastic and elastic mechanical responses of the material. In this Letter we show that supercooled liquids in which there exists random pinning sites of density $ρ_{ m im}\sim 1/ξ_s^d$ exhibit complete jamming of all dynamics. This is a direct demonstration that the proposed length scale is indeed {\em the} static length that was long sought-after.

Motivation & Objective

• To identify and validate the true static length scale $\xi_s$ governing the glass transition, which has long eluded definitive experimental or theoretical confirmation.
• To test whether $\xi_s$ is the dominant static length scale by introducing an external, tunable length scale via quenched disorder (random pinning sites).
• To determine if the crossover from $\xi_s$-dominated dynamics to disorder-dominated dynamics occurs at the predicted critical pinning density.
• To establish a practical, experimentally accessible method to measure $\xi_s$ using relaxation time measurements under quenched disorder, bypassing complex eigenvalue analysis.

Proposed method

• Simulate a 3D binary Lennard-Jones mixture (Kob-Andersen potential) at six temperatures between 0.45 and 1.00 to access the supercooled regime.
• Introduce quenched disorder by randomly freezing a fraction $\rho_{\rm im}$ of particles and measuring the structural relaxation time $\tau_\alpha$ via the overlap function $Q(t)$.
• Define $\tau_\alpha$ as the time when $Q(t) = 1/e$, using a distance cutoff of 0.30 for particle displacement correlation.
• Rescale the relaxation time $\tau_\alpha(\rho_{\rm im})$ using the product $TS_c \log[\tau_\alpha(\rho_{\rm im})/\tau_\alpha(0)]$ to test for data collapse under the scaling ansatz.
• Apply a scaling ansatz $\rho_{\rm im} \sim \xi_s^{-d}$ to collapse data across temperatures, confirming $\xi_s$ as the relevant static length scale.
• Derive a critical pinning density $\rho_{\rm im}^c \sim 1/|g_0'\xi_s^d|$ from the scaling, predicting an ideal glass transition at finite disorder density.

Experimental results

Research questions

• RQ1 Does the proposed static length scale $\xi_s$ govern the dynamics of supercooled liquids under quenched disorder?
• RQ2 Can the crossover from $\xi_s$-dominated to disorder-dominated dynamics be observed experimentally via random pinning?
• RQ3 Is the scaling of relaxation time with pinning density $\rho_{\rm im}$ consistent with $\xi_s$ being the true static length scale?
• RQ4 Can $\xi_s$ be extracted from relaxation time measurements under quenched disorder, offering a viable experimental alternative to eigenvalue analysis?
• RQ5 Does the critical pinning density $\rho_{\rm im}^c$ scale as $1/\xi_s^d$, confirming the theoretical prediction?

Key findings

• The relaxation time $\tau_\alpha$ increases dramatically with pinning density $\rho_{\rm im}$, and the system exhibits complete dynamical arrest at a critical $\rho_{\rm im}^c \sim 1/\xi_s^d$.
• Data collapse of $TS_c \log[\tau_\alpha(\rho_{\rm im})/\tau_\alpha(0)]$ versus $\rho_{\rm im}$ across all temperatures confirms the scaling ansatz $\rho_{\rm im} \sim \xi_s^{-d}$ with high precision.
• The critical pinning density $\rho_{\rm im}^c(T)$ scales inversely with $\xi_s^d(T)$, validating the theoretical prediction $\rho_{\rm im}^c \sim 1/|g_0'\xi_s^d|$.
• The collapse of data using $\xi_s$ as the scaling parameter confirms that $\xi_s$ is the dominant static length scale governing the glass transition dynamics.
• The relaxation time under disorder follows a universal scaling function in the dilute disorder regime, supporting the robustness of the scaling ansatz.
• The results demonstrate that $\xi_s$ can be experimentally accessed via quenched disorder, offering a practical alternative to computationally intensive eigenvalue analysis of the Hessian matrix.