Critical phenomena in quasi-two-dimensional vibrated granular systems

The critical phenomena associated to the liquid-to-solid transition of quasi-two-dimensional vibrated granular systems is studied using molecular dynamics simulations of the inelastic hard sphere model. The critical properties are associated to the fourfold bond-orientational order parameter ${\chi }_4$, which measures the level of square crystallization of the system. Previous experimental results have shown that the transition of ${\chi }_4$, when varying the vibration amplitude, can be either discontinuous or continuous, for two different values of the height of the box. Exploring the amplitude-height phase space, a transition line is found, which can be either discontinuous or continuous, merging at a tricritical point and the continuous branch ends in an upper critical point. In the continuous transition branch, the critical properties are studied. The exponent associated to the amplitude of the order parameter is $\beta =1/2$, for various system sizes, in complete agreement with the experimental results. However, the fluctuations of ${\chi }_4$ do not show any critical behavior, probably due to crossover effects by the close presence of the tricritical point. Finally, in quasi-one-dimensional systems, the transition is only discontinuous, limited by one critical point, indicating that two is the lower dimension for having a tricritical point.

Figure 1

Shallow box system of lateral dimensions $L_x,L_y\gg \sigma$ and height in the range $\sigma <h<2\sigma$. A grain is shown as reference. The whole box is vibrated vertically with amplitude $A$ and angular frequency $\omega$ in the presence of gravity.

Figure 2

Cluster with square symmetry obtained in simulations in a system of $N=1580$ particles in a box with lateral size $40\sigma \times 40\sigma$ and height $h=1.8\sigma$. The amplitude is $A=0.2\sigma$. The color code indicates the absolute value of ${\chi }_4$ for each particle. Grains have been drawn at a smaller size, with diameter $\approx 0.8\sigma$, to appreciate the crystalline structure of the cluster. Had they been depicted with their real size, the two layers would have overlapped when projected in two dimensions [12].

Figure 3

Liquid-to-solid transition as evidenced by the order parameter $〈|{\chi }_4|〉$ when increasing the amplitude. Above the transition amplitude, stable solid clusters form. The error bars indicate the standard deviation. Top: discontinuous transition for $h=1.74\sigma$, where the inset evidences the existence of bistability. Bottom: continuous transitions for $h=1.8\sigma$. The solid line is the fit close to the transition to determine the critical exponent.

Figure 4

Amplitude-height phase space of the transition for $N=1580$ particles, where the shaded region represents the bistablilty of the system. The dashed lines denote the discontinuous transitions, whereas the solid lines the continuous ones. The tricritical point is indicated by an empty circle and the upper critical point at the end of the continuous transition by a black circle. The arrow indicates that up to the highest values of $A$ the discontinuous transition is present, without any evidence of a lower critical point. In all cases we explore the phase space until no transition was found. The positions of the tricritical and critical points for other values of $N$ are indicated in Table 2. Typical configurations for special points in the parameter space are displayed.

Figure 5

Order parameters $P_c$ to the power $\gamma$ (top), $\mathrm{\Delta }$ (middle), and $c$ (bottom) as a function of the box height $h$, for $N=1580$ particles, where $P_c$ has been averaged over the range $A=0.108$ to $A=0.148$. The discontinuous transition line, the tricritical point, and the upper critical point are identified by the vanishing of $P_c, \mathrm{\Delta }$, and $c$, respectively. The inset (top) shows the probability density function for $h=1.615$ as an example. The exponent $\gamma$ has been fitted to 1.5, for which $P_c^{\gamma }$ vanishes linearly.

Figure 6

Fourfold structure factor $S_4\left(k\right)$ for $N=3555$ and $h=1.82\sigma$, before (blue) and after (red) the transition, for increasing values of a amplitude as indicated by the arrow. The critical amplitude is $A_c\approx 0.094\sigma$.

Figure 7

Static susceptibility $S_4\left(0\right)$ and correlation length ${\xi }_4$ for $N=9875$ and $h=1.85\sigma$ as a function of the amplitude $A$ (top) and the reduced amplitude $\epsilon =(A_c-A)/A_c$ in log-log scale with $A_c=0.109\sigma$ (bottom). The vertical dashed lines indicate the position of $A_c$.

Figure 8

Typical cluster in the rectangular geometry for $h=1.8\sigma$ and $A=0.1\sigma$. The cluster consists of a rectangular strip that crosses the periodic boundary.

Figure 9

Phase space of the rectangular system. The dashed line indicates a discontinuous transition, ending in a critical point (gray circle). The arrow indicates that the discontinuous transition is present up to the highest $A$, without any evidence of a lower critical point.