Liquid-Gas Phase Separation in Confined Vibrated Dry Granular Matter

A new phase transition is observed experimentally in a dry granular gas subject to vertical vibration between two horizontal plates. Molecular dynamics simulations of this system allow us to investigate the observed phase separation in detail. We find a high-density, low temperature liquid, coexisting with a low-density, high temperature gas moving coherently. The importance of the coherent motion for phase separation is investigated using frequency modulation.

Figure 1

Snapshots showing the cell containing the glass spheres (bright) viewed from above. The volume filling fraction occupied by the particles was $\overline{\varphi }=0.059$. The cell was vibrated at $f=45\mathrm{Hz}$ and from top left to bottom right the maximum acceleration $\Gamma$ is 5.9, 7.5, 9.0, 10.6, 12.0, and 13.7 respectively; the corresponding amplitudes, $A/d$, are 1.19, 1.51, 1.81, 2.13, 2.41, and 2.76. In the first snapshot (top left) the particles are distributed in the horizontal plane homogeneously throughout the cell. As $A$ is increased a dilute region suddenly appears at the top left corner in the system (top middle). This dilute region expands as $A$ is increased further (top right, bottom left and bottom middle). Eventually at large enough $A$ the system returns to a homogeneous state (bottom right).

Figure 2

Experimental phase diagram in the space of the dimensionless amplitude, $A/d$, and volume filling fraction, $\overline{\varphi }$, for three gap heights, $h$: 0.5 cm (dotted), 0.75 cm (dashed) and 1.0 cm (solid). The lines show the approximate boundary of the coexistence region obtained by increasing (triangle up) or decreasing (triangle down) the amplitude at fixed filling fraction. We only show the hysteresis for the system with the largest gap. The upper horizontal axis shows the volume filling fraction expressed as the mean number of particle layers, assuming a hexagonal close packing, for $h=1.0\mathrm{cm}$. The inset shows the appearance and disappearance amplitude for phase separation as a function of frequency at a filling fraction of $\overline{\varphi }=0.035$. For large enough frequencies it can be seen that the appearance and disappearance of phase separation is almost independent of frequency.

Figure 3

Phase diagram in the space of $A/d$ and $\overline{\varphi }$ obtained from simulations of the large system. The orange shaded region shows where phase separation occurs, while outside that region the system remains homogeneous within the time scales simulated. For systems which phase separate, the locally coexisting densities $\varphi$ are given as black circles. They form a “binodal line”. The black triangles, confine the region of negative compressibility (the spinodal region) obtained by the simulations of the small system. The inset shows the pressure, $p$, as a function of $\overline{\varphi }$ from simulations of a small system for three different amplitudes. The intermediate amplitude shows a negative compressibility for a range of values of $\overline{\varphi }$. The snapshot shows a phase-separated system for $A/d=1.5$ and $\overline{\varphi }=0.085$, where the color indicates high (red) and low (blue) kinetic energy of each particle and the shadow underneath is a measure of the local volume filling fraction, where the dense phase appears darker. The coexisting temperatures can be found in the inset of Fig. 4 as black circles.

Figure 4

Granular temperature $T$ as a function of amplitude $A/d$ for three different volume filling fractions $\overline{\varphi }$ for a small system which does not phase separate. The curves exhibit two distinct regimes and a rapid rise in $T$ between them. The black circles show the temperature of the coexisting phases obtained in the large system. They lie on different branches of the $T(A)$ curves which correspond to different regimes. The inset shows $T$ as a function of $\overline{\varphi }$ for three different amplitudes. Only for the intermediate value, $A/d=1.5$, shown does phase separation occur due to the presence of the rapid drop in $T$ with $\overline{\varphi }$. The circles show the coexisting values of the local temperature $T$ and the local density $\varphi$. The amplitude $A/d=1.5$ is marked in the main panel as a dotted vertical line.