Emergent Surface Tension in Vibrated, Noncohesive Granular Media

We describe experiments and simulations carried out to investigate spinodal decomposition in a vibrated, dry granular system. The dynamics is found to be similar to that of systems evolving under curvature-driven diffusion, which suggests the presence of an effective surface tension. By studying quasi-2D droplets in the steady state, we find behavior consistent with Laplace’s equation, demonstrating the existence of an actual surface tension. Detailed measurements of the pressure tensor in the interfacial region show that the surface tension results predominantly from an anisotropy in the kinetic energy part of the pressure tensor, in contrast to thermodynamic systems where it arises from either the attractive interaction between particles or entropic considerations.

Figure 1

Snapshots of granular spinodal decomposition taken from (a) experiment, and (b) simulation (see Supplemental Material [33]). From left to right, the images show the pattern for $ft=60$, 160, 260, and 360. The cell is subjected to vertical vibration at $\Gamma \approx 7.2$ and $f=60\mathrm{Hz}$, corresponding to $A/d=3.0$. The volume filling fraction is $\overline{\varphi }=0.05$.

Figure 2

(a) Phase diagram obtained from experiment (triangles) and simulation (shaded area). The enclosed region shows the parameters for which spontaneous phase separation occurs (the spinodal region). The conditions for the quench ($A/d=3.0$, $\overline{\varphi }=0.05$) are marked by the cross (red); (b) time evolution of the characteristic length scale, $l(t)$, during spinodal decomposition obtained from experiment (red crosses) and simulation (solid black circles). The dot-dashed line (green) has a slope of 1/3 as a guide to the eye; (c) early time evolution of $S(k,t)$ obtained from simulations for $ft$ linearly spaced in the range $4\le ft\le 20$, with the topmost curve being $ft=20$ and $f{t}_0=2$. The vertical red dashed line indicates the fastest growing wave number predicted by assuming model B dynamics, $k_c$; (d) collapse of $S(k,t)$ in the scaling regime, assuming a growth law of $t^{1/3}$. The experimental (solid) data points have been shifted vertically relative to the simulation (hollow) for clarity.

Figure 3

The main panel shows the difference in horizontal pressure, $\Delta P$, between the inside and the outside of a circular droplet as a function of the inverse of the droplet radius, $r_0$. The lower inset shows an image from simulation where the final configuration is a circular droplet ($\overline{\varphi }=0.025$). The upper inset shows the mean local filling fraction, $\varphi$, of the droplet as a function of the radial coordinate $r$. The dashed line (red) marks the position of $r_0$ at the mean filling fraction of the two phases. The width of the interface shown is independent of $\overline{\varphi }$ for large droplets.

Figure 4

The main panel shows the difference in the normal and tangential components of (a) the total pressure tensor (black line), (b) the kinetic part of the pressure tensor (red line), and (c) the collisional part of the pressure tensor (dot-dashed green line). The difference in the normal and tangential components of the total pressure tensor closely follows the anisotropy in the kinetic part of the pressure tensor. The inset shows the difference between the normal and tangential components of the kinetic energy per particle (Supplemental Material [33]).