Inhomogeneous Mode-Coupling Theory and Growing Dynamic Length in Supercooled Liquids

We extend mode-coupling theory (MCT) to inhomogeneous situations, relevant for supercooled liquid in an external field. We compute the response of the dynamical structure factor to a static inhomogeneous external potential and provide the first direct evidence that the standard formulation of MCT is associated with a diverging length scale. We find that the so-called cages are, in fact, extended objects. Although close to the transition the dynamic length grows as $|T-T_c{|}^{-1/4}$ in both the $\beta$ and $\alpha$ regimes, our results suggest that the fractal dimension of correlated clusters is larger in the $\alpha$ regime. We derive inhomogeneous MCT equations valid to second order in gradients.

Figure 1

Numerical solution of the schematic IMCT equations for $T>T_c$ (see 23). Main plot: ${\chi }_{\mathbf{q}}(t)$ for different $\mathbf{q}$ as a function of time and for $\epsilon ={10}^{-6}$. From top to bottom: $q=0$, 0.06, 0.2, 0.4, and 1. Note that the shape of ${\chi }_{\mathbf{q}}(t)$ in the $\alpha$ regime is independent of $\mathbf{q}$, as predicted by Eq. (6). We have, in fact, checked that the predicted scaling is very well obeyed in that region. Inset: $\epsilon {\chi }_{\max }(\mathbf{q})\equiv \epsilon {\chi }_{\mathbf{q}}(t={\tau }_{\alpha })$ as a function of $q{\epsilon }^{-1/4}$ in log-log, for different $q$’s and $\epsilon$’s. Note the $q^{-4}$ behavior for large $q{\epsilon }^{-1/4}$, as indicated by the dashed line.