Roles of Energy Dissipation in a Liquid-Solid Transition of Out-of-Equilibrium Systems

Self-organization of active matter as well as driven granular matter in nonequilibrium dynamical states has attracted considerable attention not only from the fundamental and application viewpoints but also as a model to understand the occurrence of such phenomena in nature. These systems share common features originating from their intrinsically out-of-equilibrium nature, and how energy dissipation affects the state selection in such nonequilibrium states remains elusive. As a simple model system, we consider a nonequilibrium stationary state maintained by continuous energy input, relevant to industrial processing of granular materials by vibration and/or flow. More specifically, we experimentally study roles of dissipation in self-organization of a driven granular particle monolayer. We find that the introduction of strong inelasticity entirely changes the nature of the liquid-solid transition from two-step (nearly) continuous transitions (liquid-hexatic-solid) to a strongly discontinuous first-order-like one (liquid-solid), where the two phases with different effective temperatures can coexist, unlike thermal systems, under a balance between energy input and dissipation. Our finding indicates a pivotal role of energy dissipation and suggests a novel principle in the self-organization of systems far from equilibrium. A similar principle may apply to active matter, which is another important class of out-of-equilibrium systems. On noting that interaction forces in active matter, and particularly in living systems, are often nonconservative and dissipative, our finding may also shed new light on the state selection in these systems.

Popular Summary

Granular matter is ubiquitous in our daily lives and also in nature. Unlike in atomic and molecular systems, interparticle collisions between granular matter dissipate energy in inelastic collisions. How these collisions affect the self-organization of driven granular matter is an important fundamental question. Using a driven granular monolayer system as a model, we show that energy dissipation alters the nature of the liquid-solid transition from continuous to discontinuous.

Our experimental setup consists of spherical particles held between two parallel plates. The particles—either steel or rubber—are 3 mm in diameter and are held in a single layer; the formation of a bilayer is inhibited by the small separation between the plates (4 mm). We assume a continuous, global energy input, and we use high-speed (100–500 frames per second) cameras to track the positions of the particles when vibrations are applied at a frequency of 50 Hz. For the steel balls, we recover the behavior observed for the thermal counterpart. For the rubber balls, on the other hand, we uncover the coexistence of a solid and liquid phase with different effective temperatures, which does not persist in thermal systems.

Our results provide new insights into the roles of dissipative interactions in self-organization of out-of-equilibrium systems such as driven granular matter and active matter. We expect that our findings will motivate additional studies of living systems that can be thought of as active matter with dissipative interactions.

Figure 1

Phase behavior of 2D hard disks predicted by the KTHNY theory. In this table, we summarize how the spatial correlations of the positional and bond-orientational order parameter should increase with an increase in $\varphi$. The other characteristics of each phase are also shown.

Figure 2

$\varphi$ dependence of $g(r)$ and $g_6(r)/g(r)$ for a steel ball system for $\mathrm{\Gamma }=3.3$. To visually confirm a decay slower than the asymptotic decay for ordered phases, we plot the data in the log-log plots, where only the points having positive values are displayed. (a) $\varphi$ dependence of $g(r)$. The symbols are $\varphi =0.70$ (dark blue), 0.71 (yellow), 0.72 (green), 0.73 (orange), and 0.74 (red) from the bottom to the top. The straight line has a slope of $-1/3$. (b) $\varphi$ dependence of $g_6(r)/g(r)$. The symbols are $\varphi =0.65$ (bright blue), 0.70 (dark blue), 0.71 (yellow), 0.72 (green), 0.73 (orange), and 0.74 (red) from the bottom to the top. The straight line has a slope of $-1/4$. From these results, we identify $\varphi =0.65$, 0.70, and 0.71 as a liquid state, $\varphi =0.72$ and 0.73 as a hexatic state, and $\varphi =0.74$ as a solid state.

Figure 3

Examples of the analysis of the spatial decay of $g_6(r)/g(r)$ for a steel ball system for $\mathrm{\Gamma }=3.3$. The blue points are used for the fittings. The blue curves are exponential fittings, whereas the orange lines have a slope of $-1/4$, which is expected at ${\varphi }_h$. From the above, we can judge that the liquid-to-hexatic transition takes place between $\varphi =0.715$ and 0.72. The accuracy of this determination is limited by the discreteness of the area fractions we employ.

Figure 4

Continuous and discontinuous nature of the phase transition, respectively, in steel and rubber ball systems. Blue curves and symbols are for a steel ball system, whereas red ones are for a rubber ball system. (a) The divergence of the correlation length of the hexatic order ${\xi }_6$ (filled circle) towards the hexatic ordering point ${\varphi }_h$ and that of the translational order $\xi$ (open circle) towards the solidification point ${\varphi }_s$ for a steel ball system. The solid and dashed curves are the prediction of the KTHNY theory (see Fig. 1). (b) The $\varphi$ dependence of ${\xi }_6$ (filled circle) and $\xi$ (open circle) for a homogeneous liquid state of a rubber ball system. (c) The susceptibility of the hexatic order parameter ${\chi }_6$ for steel (blue filled circle) and rubber ball systems (red filled circle). The steep increase near ${\varphi }_h$ is observed for a steel ball system, reflecting the nearly continuous nature of the transition, whereas there is no such behavior for a rubber ball system, reflecting the strong discontinuous nature of the transition. (d) The area fraction of the solid phase to the total area, $A_S/A$, as a function of $\varphi$. It starts to increase from ${\varphi }_L$ until ${\varphi }_S$ in proportion to $\varphi -{\varphi }_L$, indicating the validity of the lever rule. We also confirm the similar relation between the number of particles of the solid phase $N_S$ and the total number of particles $N$. This means that each coexistence phase retains the same number densities irrespective of $\varphi$ in the coexistence region. The error bars in (a)–(c) represent the standard deviations of the fittings. For all the data, $\mathrm{\Gamma }=3.3$.

Figure 5

Liquid-solid coexistence for rubber ball systems observed at $\varphi =0.70$, 0.72, and 0.74, which are between ${\varphi }_L(=0.695)$ and ${\varphi }_S(=0.775)$, as $\mathrm{\Gamma }=3.3$. The two-phase coexistence can be seen by binarization by using the following three quantities. (a) The hexatic order parameter ${\psi }_6$ of each particle [here, the threshold is chosen as ${\psi }_6^{\text{th}}=0.6$, and the pattern is insensitive to the choice for $\varphi =0.6–0.7$ (see Fig. 6)]. Yellow (solid) and black (liquid) particles correspond to particles having ${\psi }_6\ge {\psi }_6^{\text{th}}$ and ${\psi }_6<{\psi }_6^{\text{th}}$, respectively. (b) The local density $\rho$, which is calculated from the Voronoi area of each particle (the threshold $1/{\rho }^{\text{th}}=9.4{\mathrm{mm}}^2$). Blue-green (solid) and pink (liquid) particles correspond to particles having $\rho \ge {\rho }^{\text{th}}$ and $\rho <{\rho }^{\text{th}}$, respectively. (c) The displacement over 10 s of each particle (the threshold ${\delta }^{\text{th}}=3.4\mathrm{mm}$). Blue (solid) and green (liquid) particles correspond to particles having $\delta \le {\delta }^{\text{th}}$ and $\delta >{\delta }^{\text{th}}$, respectively. We can see that the solid regions identified by these three quantities are well correlated with each other. We can also see the monotonic increase of the solid fraction with an increase in $\varphi$.

Figure 6

$\varphi$ dependence of the probability distribution function of the hexatic order parameter ${\psi }_6$, $P({\psi }_6)$, at $\mathrm{\Gamma }=3.3$. Blue curves are for a steel ball system, whereas red curves are for a rubber ball system. For a steel ball system, $P(\psi )$ always has a unimodal shape, whereas for a rubber ball system it has a clear bimodal shape for $\varphi$ between ${\varphi }_L$ and ${\varphi }_S$. The long tails of $P({\psi }_6)$ toward low ${\psi }_6$ in homogeneous solid phases come from defects.

Figure 7

$\varphi$ dependence of $g(r)$ and $g_6(r)/g(r)$ for a rubber ball system at $\mathrm{\Gamma }=3.3$. (a) $\varphi$ dependence of $g(r)$. The symbols are $\varphi =0.65$ (blue), 0.69 (yellow), 0.70 (green), 0.74 (orange), and 0.80 (red) from the bottom to the top. (b) $\varphi$ dependence of $g_6(r)/g(r)$. The symbols are $\varphi =0.65$ (bule), 0.69 (yellow), 0.70 (green), 0.74 (orange), and 0.80 (red) from the bottom to the top.

Figure 8

Examples of the analysis of the spatial decay of $g(r)$ and $g_6(r)/g(r)$ for a rubber ball system at $\mathrm{\Gamma }=3.3$. Here, we show the analysis of (a) $g(r)$ and (b) $g_6(r)/g(r)$ at $\varphi =0.69$, which is very close to ${\varphi }_L=0.695$. The blue points are used for the fittings. The blue curves are exponential fittings, whereas the red and orange lines have a slope of $-1/3$ and $-1/4$, respectively. In a one-phase region, the correlation of the bond-orientational order decays exponentially and there is no indication of a power-law decay even near ${\varphi }_L$. The extracted correlation lengths in this manner are plotted as a function of $\varphi$ in Fig. 4. We note that in the two-phase coexistence region, this type of analysis is not meaningful.

Figure 9

Phase diagram of driven monolayer sphere systems. Filled circles represent state points where the measurements are made (blue, disordered liquid; orange, hexatic phase; pink, solid; green, a coexistence of liquid and solid; gray, no well-defined stationary state due to inhomogeneous excitation). (a) Steel ball systems, which obey the KTHNY-like scenario. In this case, the state behavior does not depend on $\mathrm{\Gamma }$ in the range studied. (b) Rubber ball systems, which show the distinct discontinuous transition between liquid and solid states. In this case, the dynamical steady state is realized only above $\mathrm{\Gamma }\cong 2.7$. This higher threshold value of $\mathrm{\Gamma }$ is presumably due to the inelastic nature of collisions of rubber balls with the confining plates. The images in (a) and (b) are configurations in the solid phase at $\varphi =0.78$ for steel and rubber ball systems, respectively. Green, blue, and red particles have local hexagonal, pentagonal, and heptagonal structures, respectively. The inner part surrounded by the red dashed circle has less order (i.e., more defects) for the rubber ball system than for the steel one.

Figure 10

Wall effects on the liquid-solid coexistence at $\mathrm{\Gamma }=3.3$. (a) A coexistence of the liquid and solid phase confined by a circular steel wall for $\varphi =0.74$. The liquid phase preferentially wets the wall. (b) The same as (a) for a softer wall whose surface is covered by a silicone rubber film. In this case, the solid phase partially wets the wall (see the regions surrounded by the red lines).

Figure 11

Phase behavior of mixtures of rubber and steel balls. (a) Snapshot of a $1:7$ mixture of steel and rubber balls (the total area fraction of $\varphi =0.72$) driven at $\mathrm{\Gamma }=3.3$. Here, we can see a coexistence of the solid phase purely made of rubber balls and the liquid phase which is a mixture of steel and rubber balls. (b) Snapshot of a $1:1$ mixture of steel and rubber balls (the total area fraction of $\varphi =0.72$) driven at $\mathrm{\Gamma }=3.3$. The color of the outer shell of each particle represents the type of the particle (blue, steel ball; red, rubber ball), whereas the color of the inner core represents the degree of the hexatic order (black, low ${\psi }_6$; yellow, high ${\psi }_6$).