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Jamming in confined geometry: Criticality of the jamming transition and implications of structural relaxation in confined supercooled liquids

In marginally jammed solids confined by walls, we calculate the particle and ensemble averaged value of an order parameter, $\left<Ψ(r)\right>$, as a function of the distance to the wall, $r$. Being a microscopic indicator of structural disorder and particle mobility in solids, $Ψ$ is by definition the response of the mean square particle displacement to the increase of temperature in the harmonic approximation and can be directly calculated from the normal modes of vibration of the zero-temperature solids. We find that, in confined jammed solids, $\left<Ψ(r)\right>$ curves at different pressures can collapse onto the same master curve following a scaling function, indicating the criticality of the jamming transition. The scaling collapse suggests a diverging length scale and marginal instability at the jamming transition, which should be accessible to sophisticatedly designed experiments. Moreover, $\left<Ψ(r)\right>$ is found to be significantly suppressed when approaching the wall and anisotropic in directions perpendicular and parallel to the wall. This finding can be applied to understand the $r$-dependence and anisotropy of the structural relaxation in confined supercooled liquids, providing another example of understanding or predicting behaviors of supercooled liquids from the perspective of the zero-temperature amorphous solids.