Slow dynamics of structure and fluctuations in supercooled colloidal fluids

The importance of the dynamic singularity of the self-diffusion coefficient $D_S\sim \left[1-\Phi \right(\mathbf{x},t)/{\phi }_g{]}^2$ in the coupled diffusion equations recently proposed, the nonlinear deterministic diffusion equation for the average local volume fraction $\Phi (\mathbf{x},t)$ and the linear stochastic diffusion equation for the density fluctuations $\delta n(\mathbf{x},t),$ is emphasized for understanding the slow dynamical behavior of concentrated hard-sphere suspensions, where ${\phi }_g{=(4/3)\mathrm{}}^3/(7\mathrm{}\ln 3-8\ln 2+2)$ is the colloidal glass transition volume fraction. It is shown that there exists a crossover volume fraction ${\phi }_{\beta },$ over which $\Phi (\mathbf{x},t)$ describes the formation of long-lived, irregularly shaped domains with $\Phi (\mathbf{x},t)>~{\phi }_g,$ and $\delta n(\mathbf{x},t)$ describes two-step relaxations with time scales, $t_{\beta }\sim (1\mathrm{}-\phi /{\phi }_g{)}^{-1},$ and $t_{\alpha }\sim (1\mathrm{}-\phi /{\phi }_g{)}^{-2},$ where φ is the volume fraction of spheres. Thus the slow dynamics of a supercooled hard-sphere colloidal fluid $({\phi }_{\beta }<~\phi <{\phi }_g)$ is explored from a unified viewpoint.