The glass susceptibility: growth kinetics and saturation under shear

We study the growth kinetics of glassy correlations in a structural glass by monitoring the evolution, within mode-coupling theory, of a suitably defined three-point function $\chi_C(t,t_w)$ with time $t$ and waiting time $t_w$. From the complete wave vector-dependent equations of motion for domain growth we pass to a schematic limit to obtain a numerically tractable form. We find that the peak value $\chi_C^P$ of $\chi_C(t,t_w)$, which can be viewed as a correlation volume, grows as $t_w^{0.5}$, and the relaxation time as $t_w^{0.8}$, following a quench to a point deep in the glassy state. These results constitute a theoretical explanation of the simulation findings of Parisi [J. Phys. Chem. B {\bf 103}, 4128 (1999)] and Kob and Barrat [Phys. Rev. Lett. {\bf 78}, 4581 (1997)] and are also in qualitative agreement with Parsaeian and Castillo [Phys. Rev. E {\bf 78}, 060105(R) (2008)]. On the other hand, if the quench is to a point on the {\em liquid side}, the correlation volume grows to saturation. We present a similar calculation for the growth kinetics in a $p$-spin spin glass mean-field model where we find a slower growth, $\chi_C^P \sim t_w^{0.13}$. Further, we show that a shear rate $\gdot$ cuts off the growth of glassy correlations when $t_w\sim 1/\gdot$ for quench in the glassy regime and $t_w=\min(t_r,1/\gdot)$ in the liquid, where $t_r$ is the relaxation time of the unsheared liquid. The relaxation time of the steady state fluid in this case is $\propto \gdot^{-0.8}$.