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

摘要

We demonstrate how the matrix-valued mode-coupling theory of the glasstransition and glassy dynamics in planar confinement converges to thecorresponding theory for two-dimensional (2D) planar and the three-dimensionalbulk liquid, provided the wall potential satisfies certain conditions. Sincethe mode-coupling theory relies on the static properties as input, theemergence of a homogeneous limit for the matrix-valued intermediate scatteringfunctions is directly connected to the convergence of the corresponding staticquantities to their conventional counterparts. We show that the 2D limit ismore subtle than the bulk limit, in particular, the in-planar dynamicsdecouples from the motion perpendicular to the walls. We investigate thefrozen-in parts of the intermediate scattering function in the glass state andfind 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 andlateral force kernels.