Glassy dynamics in confinement: Planar and bulk limit of the mode-coupling theory

TL;DR

This paper explores how the matrix-valued mode-coupling theory for glassy dynamics in confined systems converges to the 2D and bulk limits, highlighting subtle differences in in-plane and perpendicular dynamics and non-commuting limits.

Contribution

It provides a detailed analysis of the convergence of the mode-coupling theory in planar confinement to 2D and bulk cases, emphasizing the role of static properties and the non-commutativity of certain limits.

Findings

The 2D limit is more subtle than the bulk limit.

In-plane dynamics decouple from perpendicular motion.

Limits of infinite time and zero wall separation do not commute.

Abstract

We demonstrate how the matrix-valued mode-coupling theory of the glass transition and glassy dynamics in planar confinement converges to the corresponding theory for two-dimensional (2D) planar and the three-dimensional bulk liquid, provided the wall potential satisfies certain conditions. Since the mode-coupling theory relies on the static properties as input, the emergence of a homogeneous limit for the matrix-valued intermediate scattering functions is directly connected to the convergence of the corresponding static quantities to their conventional counterparts. We show that the 2D limit is more subtle than the bulk limit, in particular, the in-planar dynamics decouples from the motion perpendicular to the walls. We investigate the frozen-in parts of the intermediate scattering function in the glass state and find that the limits time $t\to \infty$ and effective wall separation $L\to 0$ do not commute due to the mutual coupling of the residual transversal and lateral force kernels.