Glass transition of dense fluids of hard and compressible spheres

We use computer simulations to study the glass transition of dense fluids made of polydisperse repulsive spheres. For hard particles, we vary the volume fraction, $\phi$, and use compressible particles to explore finite temperatures, $T>0$. In the hard sphere limit, our dynamic data show evidence of an avoided mode-coupling singularity near ${\phi }_{\text{MCT}}\approx 0.592$; they are consistent with a divergence of equilibrium relaxation times occurring at ${\phi }_0\approx 0.635$, but they leave open the existence of a finite temperature singularity for compressible spheres at volume fraction $\phi >{\phi }_0$. Using direct measurements and a scaling procedure, we estimate the equilibrium equation of state for the hard sphere metastable fluid up to ${\phi }_0$, where pressure remains finite, suggesting that ${\phi }_0$ corresponds to an ideal glass transition. We use nonequilibrium protocols to explore glassy states above ${\phi }_0$ and establish the existence of multiple equations of state for the unequilibrated glass of hard spheres, all diverging at different densities in the range $\phi ∊\left[0.642,0.664\right]$. Glassiness thus results in the existence of a continuum of densities where jamming transitions can occur.

Figure 1

(Color online) Sketch of two possible phase diagrams for harmonic spheres. (a) In mean-field replica calculations, an ideal glass transition occurs at finite temperature for $\phi$ above that of point $G$, the ideal glass transition point for hard spheres. The pressure of the equilibrium glass diverges at larger density at glass close packing (GCP). (b) No glass transition occurs before jamming at point $J$ at $T=0$ where the equilibrium pressure also diverges, with no glass transition occurring at finite temperature at larger density. Alternative phase diagrams such as (a) with $G=\text{GCP}$ or (b) with $G\ne J$ are in principle also possible.

Figure 2

Phase diagram for harmonic spheres. Open symbols represent the investigated state points. The location of glass lines obtained by fitting to known functional form (MCT, VFT) and using activated scaling (22) is shown, while the super-Arrhenius line corresponds to volume fractions where ${\tau }_{\alpha }\left(\phi ,T\right)$ increases faster than an Arrhenius law when $T$ is reduced at constant $\phi$.

Figure 3

Activated dynamic scaling of relaxation time scales for $\phi ∊\left[0.567,0.736\right]$. The data for $\phi <{\phi }_0$ and $\phi >{\phi }_0$ collapse on two distinct branches, as described by Eq. (20). Times are rescaled by $1/\sqrt{T}$ so that the $T\to 0$ limit coincides with hard spheres thermalized at $T=1$. The error bars describe the range of values for which acceptable data collapse is obtained; $\mu =1.3$ is fixed using potential energy considerations.

Figure 4

Equilibrium pressure for hard spheres obtained from direct Monte Carlo simulations (circles) and extended to large $\phi$ using the scaling behavior of the harmonic sphere system (triangles). The free volume prediction [Eq. (16)] is presented for three locations of the divergence at ${\phi }_{\text{VFT}}=0.615$, ${\phi }_0=0.635$, and the best fit ${\phi }^{\star }=0.672$, and none of them accurately describes the data. Instead, the BMCSL equation of state from liquid state theory describes the data very well over the entire fluid range and leaves the pressure finite at ${\phi }_0$, where the relaxation time diverges.

Figure 5

“Angell plot” for hard spheres showing the pressure dependence of $\text{ln}{\tau }_{\alpha }$ where the Arrhenius behavior predicted by free volume arguments should appear as a straight line. Hard spheres instead display fragile behavior which is not well fitted by a Bässler law, Eq. (13) with $\alpha =2$, but is consistent with a finite pressure singularity. Excellent fits to Eq. (12) with $\left(\delta =1,Z_0=30.2\right)$ and $\left(\delta =2,Z_0=34.4\right)$ are shown.

Figure 6

Top: temperature dependence of the pressure of harmonic spheres at different volume fractions. Bottom: rescaling the pressure by the factor $Z\left(\phi ,T\to 0\right)$ collapses the data for all $\phi$, in agreement with Eq. (25). For $\phi >{\phi }_0$ the fluid is no longer the thermodynamically stable phase at $T=0$; these data are shown with open symbols.

Figure 7

Various pressure-volume fraction relations obtained from simulations of binary hard spheres. Equilibrium data and BMCSL line are as in Fig. 4. The nonequilibrium data from hard sphere compressions and soft spheres annealing are shown as diamonds and squares, respectively, lines being guides to the eye. They demonstrate the existence of multiple branches, all diverging at different densities, and result from the glassy behavior of hard spheres.