A replica theory for the dynamic glass transition of hardspheres with continuous polydispersity

Glassy soft matter is often continuously polydisperse, in which the sizes or various properties of the constituent particles are distributed continuously. However, most of the microscopic theories of the glass transition focus on the monodisperse particles. Here, we developed a replica theory for the dynamic glass transition of continuously polydisperse hardspheres. We focused on the limit of infinite spatial dimension, where replica theory becomes exact. In theory, the cage size $A$, which plays the role of an order parameter, appears to depend on the particle size $\sigma$, and thus, the effective free energy, the so-called Franz-Parisi potential, is a functional of $A(\sigma)$. We applied this theory to two fundamental systems: a nearly monodisperse system and an exponential distribution system. We found that dynamic decoupling occurs in both cases; the critical particle size $\sigma^{\ast}$ emerges, and larger particles with $\sigma \geq \sigma^{\ast}$ vitrify, while smaller particles $\sigma < \sigma^{\ast}$ remain mobile. Moreover, the cage size $A(\sigma)$ exhibits a critical behavior at $\sigma \simeq \sigma^{\ast}$, originating from spinodal instability of $\sigma^{\ast}$-sized particles. We discuss the implications of these results for finite dimensional systems.