Self-assembly and non-equilibrium phase coexistence in a binary granular mixture

TL;DR

This work demonstrates that a vibrated binary granular monolayer can form a square binary crystal (S1) and exhibit stable liquid–solid coexistence akin to equilibrium phase transitions, despite strong non-equilibrium driving. By combining long experiments with DEM and EDMD simulations, the authors show that the phase coexistence obeys a lever-rule-like organization even far from thermodynamic equilibrium. A striking non-equilibrium finding is that the crystalline phase is hotter than the coexisting fluid, explained by coupling between local structure and energy transfer, including off-plane collisions and kinetic-theory arguments for binary mixtures. The results illuminate how equilibrium-like phase behavior can emerge in driven granular matter and provide a kinetic-theory framework to understand temperature gradients at interfaces, with implications for designing homogeneous binary granular mixtures and applying non-equilibrium theories to granular phase transitions.

Abstract

We report the experimental observation of a square crystalline phase in a vibrated binary mixture of spherical grains. This structure spontaneously forms from a disordered state, consistently with predictions obtained in an equilibrium system with similar geometrical properties under conservative dynamics. By varying the area fraction, we also observe stable coexistence between a granular fluid and an isolated square crystal. Using realistic simulations based on the discrete element method and an idealized collisional model integrated via event-driven molecular dynamics, we not only reproduce experimental results but also help to gain further insights into the non-equilibrium phase coexistence. Through the direct phase coexistence method, we demonstrate that the system shows behavior highly similar to an equilibrium first-order phase transition. However, the crystal remains at a higher granular temperature than the fluid, which is a striking non-equilibrium effect. Through qualitative argument and supported by kinetic theory, we elucidate the role of the coupling between local structure and energy transfer mechanisms in sustaining kinetic temperature gradients across the fluid-solid interface.

Figures (14)

• Figure 1: Sketch of the experimental setup. A binary mixture of polyamide grains lies on a sandblasted anodized aluminium plate, which is vertically vibrated. Four aluminium spacers are screwed on the plate to confine the motion of the grains in a square area of side $L = 15$ cm through lateral walls of height $h = 6$ mm while the full lateral side of the plate is $L_p=20$ cm. The granular system is confined from above by a glass cover which allows for the direct visualization of the system through a camera. To ease the imaging of the grains, four light panels are placed around the apparatus. The measurements of the plate acceleration are made possible by a one-axis accelerometer (Brüel & Kjær, Type 4534-B-001) that can be rigidly joined to its bottom side. The base of the apparatus is held up by three leveling feet (not shown), which allow for horizontal adjustment.
• Figure 2: Sketch of the equilibrium reference system of non-additive hard disks in which the S1 phase was first observed Fayen2020Fayen2022. On the left-hand side, we show the mapping between spheres laying on a plane and non-additive disks which undergo elastic collision as they reach a distance $\sigma_{LS}=\sqrt{\sigma_L\sigma_S}$. On the right-hand side, we show a portion of the S1 crystal assembled in the equilibrium reference system.
• Figure 3: Results from a long experiment with polyamide beads on a rough surface under sinusoidal vibration. Here $q=0.5$, $x_S=0.51$, $\phi=0.858$, $\Gamma=1.79$, $f=53$ Hz. a) Time evolution of the fraction of particles belonging to an S1 environment (see the text for the definition). The red dots mark the time at which the snapshots of the system shown in panels b-e) are taken (they are in chronological order). In the upper-left inset of panel a), we show the time evolution of the average large particle displacement over a time interval of 30 seconds (smoothed using a running average with a time window of 5 minutes). We use this quantity to characterize the average mobility of the system. In the bottom-right inset of panel a), we plot the time evolution of $q_4$ and $q_6$. Also shown here is a zoom-in on the unit cell of the S1 phase, which consists of four large grains surrounding a small one.
• Figure 4: Time averaged $q_4$ and $F_{S1}$ as a function of the area fraction for experiments with polyamide beads on a rough surface under sinusoidal vibrations. In all the cases $q=0.5$, $x_S=0.51$, $\Gamma=1.79$ and $f=53$ Hz. Time averages are performed over the last 50 minutes of the experiments. The total duration $t_{tot}$ varies for different experiments, we have $t_{tot}=\{20,20,20,29,89\}$ hours corresponding to the explored area fraction $\phi=\{0.81,0.82,0.83,0.84,0.858\}$. In the two insets, we provide the final snapshots for the experiments performed at $\phi=0.83$ (top left) and $\phi=0.84$ (bottom right).
• Figure 5: Comparison between the maps of average $q_4$ and large particle displacement over 0.17 seconds for the experiments performed at $\phi=0.858$ (panel a and b) and $\phi=0.84$ (panel c and d). In the first case, the displacement field is homogeneous regardless of the local value of $q_4$, in the latter, low mobility regions correspond to the high $q_4$ cluster. The analysis is performed over a 60-second-long high-rate acquisition (60 fps) done at the end of the experiments. Each shown map results from an average of 3520 single-frame analyses (or double-frame in the case of the displacement).