Understanding the ideal glass transition: Lessons from an equilibrium study of hard disks in a channel

Abstract

We use an exact transfer-matrix approach to compute the equilibrium properties of a system of hard disks of diameter $\sigma$ confined to a two-dimensional channel of width $1.95\sigma$ at constant longitudinal applied force. At this channel width, which is sufficient for next-nearest-neighbor disks to interact, the system is known to have a great many jammed states. Our calculations show that the longitudinal force (pressure) extrapolates to infinity at a well-defined packing fraction ${\varphi }_K$ that is less than the maximum possible ${\varphi }_{\max }$, the latter corresponding to a buckled crystal. In this quasi-one-dimensional problem there is no question of there being any \emph{real} divergence of the pressure at ${\varphi }_K$. We give arguments that this avoided phase transition is a structural feature -- the remnant in our narrow channel system of the hexatic to crystal transition -- but that it has the phenomenology of the (avoided) ideal glass transition. We identify a length scale ${\tilde{\xi }}_3$ as our equivalent of the penetration length for amorphous order: In the channel system, it reaches a maximum value of around $15\sigma$ at ${\varphi }_K$, which is larger than the penetration lengths that have been reported for three dimensional systems. It is argued that the $\alpha$-relaxation time would appear on extrapolation to diverge in a Vogel-Fulcher manner as the packing fraction approaches ${\varphi }_K$.

17 pages, 16 figures

References in corpus (5)

• The role of local structure in dynamical arrest
• Why is Random Close Packing Reproducible?
• Strong geometric frustration in model glassformers
• Simulation-based equation of state of the hard disk fluid and prediction of higher-order virial coefficients
• Interface free energies in p-spin glass models

Cited by in corpus (4)

• Beyond the single-file fluid limit using transfer matrix method: Exact results for confined parallel hard squares
• Resolving the two-dimensional ANNNI model using transfer matrices
• Highly confined mixtures of parallel hard squares: A Density Functional Theory study
• Numerical transfer matrix study of frustrated next-nearest-neighbor Ising models on square lattices