A growing length-scale in supercooled liquids: Cluster formation induced by local densification

In glass forming liquids close to the glass transition point, even a very slight increase in the macroscopic density results in a dramatic slowing down of the macroscopic relaxation. Concomitantly, the local density itself fluctuates in space. Therefore, one can imagine that even very small local density variations control the local glassy nature. Based on this perspective, a model for describing growing length scale accompanying the vitrification is introduced, in which we assume that in a subsystem whose density is above a certain threshold value, $\rho_{\rm c}$, owing to steric constraints, particle rearrangements are highly suppressed for a sufficiently long time period ($\sim$ structural relaxation time). We regard such a subsystem as a glassy cluster. Then, based on the statistics of the subsystem-density, we predict that with compression (increasing average density $\rho$) at a fixed temperature $T$ in supercooled states, the characteristic length of the clusters, $\xi$, diverges as $\xi\sim(\rho_{\rm c}-\rho)^{-2/d}$, where $d$ is the spatial dimensionality. This $\xi$ measures the average persistence length of the steric constraints in blocking the rearrangement motions and is determined by the subsystem density. Additionally, with decreasing $T$ at a fixed $\rho$, the length scale diverges in the same manner as $\xi\sim(T-T_{\rm c})^{-2/d}$, for which $\rho$ is identical to $\rho_{\rm c}$ at $T=T_{\rm c}$. The exponent describing the diverging length scale is the same as the one predicted by some theoretical models and indeed has been observed in some simulations and experiments. However, the basic mechanism for this divergence is different; that is, we do not invoke thermodynamic anomalies associated with the thermodynamic phase transition as the origin of the growing length scale. We further present arguements for the cooperative properties based on the clusters.