Microscopic model for a granular solid-liquid-like phase transition

TL;DR

The authors develop a microscopic model for the solid–liquid phase transition in a quasi-2D vibrated granular bed, treating the crystalline solid cluster as a synchronized, completely inelastic particle (with $r_{\rm eff}=0$) that collides collect-and-collide with the confining walls. Mechanical pressure balance between coexisting liquid and solid phases, along with an energy-budget constraint, yields a closed equation linking the solid fraction $n$ to the driving amplitude $A$, frequency $f$, and microstructural parameters ($\lambda$, $\beta$, $\kappa$, $\delta$, $\alpha$). The model, validated against three experimental configurations, captures both first- and second-order transitions and their dependence on dissipation and forcing, and it aligns with prior experimental and simulation results. These findings illuminate how synchronization and dissipation govern phase coexistence in vibrated granular systems and offer a quantitative framework for predicting the onset and nature of the transition. All key quantities are expressed with explicit $ $math$ $ notation to support precise reproduction and searchability.

Abstract

Forced granular matter in confined geometries presents phase transitions and coexistence. Depending on the system and forcing parameters, liquid-vapor and liquid-solid co-existing states are possible. For the solid-liquid coexistence that is observed in quasi-two-dimensional vibrated systems, both first- and second-order transitions have been reported. Experiments show that particles in the solid cluster move collectively, synchronized with the cell's vibration, in a similar way to the collect-and-collide regime observed in granular dampers. Here, we present a model that proposes a microscopic origin of this granular phase transition and co-existence. Imposing synchronicity, we model the solid cluster as an effective particle of zero restitution coefficient. In addition, we use the mechanical equilibrium between the two phases, with an equation of state validated for hard spheres relating the horizontal velocities in each phase. Balancing energy input and dissipation per unit time we obtain a global power equation, which relates the characteristic vertical and horizontal velocities to the microscopic relevant parameters (geometric and dissipation coefficients) as well as to the vibration amplitude and solid cluster's size. The predictions of the model compare quite well with our experimental results and with the experimental and dynamic simulation results reported elsewhere.

Figures (12)

• Figure 1: Typical image of the solid bi-layered square symmetry crystal in coexistence with the liquid phase ($f = 80$ Hz, $h = 1.94a$, $a = 1$ mm and $A = 0.24a$). The color circles indicate the degree of order, quantified through the absolute value of the four-bond orientational parameter, $|Q_4^j|$. Only the central sector of size $L/2\times L/2$ is shown.
• Figure 2: (a) Schematic representation of two square interlaced layers for which particles of diameter $a$ are packed with a mean free path $\lambda$. The center of particles in the bottom (top) layer are shown with solid (open) black circles. The 2D projected square lattice has a unit cell length $(a+\lambda)/\sqrt{2}$, its Voronoi area is $\beta = (a+\lambda)^2/2$. (b) Schematic representation of a side view of two particles in the lowest layer supporting another particle on the top layer. The distance $2(a+\lambda)/\sqrt{2}$ is twice the unit cell length of the 2D projected square lattice.
• Figure 3: Vertical position of 10 particles that have been detected and tracked within the solid phase during 10 oscillation cycles. Data is obtained from images captured from an inclined side view of the experimental cell.
• Figure 4: (a) Example of quasi-particle trajectories computed numerically, with $r_{\rm eff}~=~0$, $\Gamma = 3$, $f = 80$ Hz, $\lambda/a~=~0.1$ and $h/a = 1.83$ ($A/a = 0.1165$ and $h'/a=0.2015$). From this kind of trajectory, the collision velocity $v_{\rm coll} = v_s - v_w$ can be computed at each impact. (b) Normalized $v_{\rm coll}^2$ as function of $A/a$, for the same set of parameters of figure (a). For $A<A^* = 0.1013a$ ($\Gamma^* = 2.61$) only one collision per period occurs. (c) Normalized dissipated energy per oscillation cycle, $\Delta E_{\rm diss}/(mN_s A^2\omega^2)$ (continuous line) as function of $A/a$ for the same parameters as in (b). The dissipation is maximum at $A_{\rm max} = 0.1099a$ ($\Gamma_{\rm max} = 2.83$), indicated by the vertical dashed line. The dashed-dotted line (dashed line) shows as comparison the same quantity in absence of gravity, given by the exact Eqn. (\ref{['EDiss1']}) (approximated Eqn. (\ref{['Diss2']})).
• Figure 5: Average normalized mean free path $\lambda/a$ as function of the solid phase fraction $n = N_s/N$. Error bars are computed form the standard deviation. Data for the three configurations are presented: C1 ($\diamond$), C2a ($\circ$) and C2b ($\square$). Data is presented for $\Gamma>\Gamma_c$.