Collective Drifts in Vibrated Granular Packings: The Interplay of Friction and Structure

We simulate vertically shaken dense granular packings with horizontal periodic boundary conditions. A coordinated translating motion of the whole medium emerges when the horizontal symmetry is broken by disorder or defects in the packing and the shaking is weak enough to conserve the structure. We argue that such a drift originates in the interplay between structural symmetry breaking and frictional forces transmitted by the vibrating plate. A nonlinear ratchet model with stick slips reproduces many faces of the phenomenon. The collective motion discussed here underlies phenomena observed recently with vibrofluidized granular materials, such as persistent rotations and anomalous diffusion.

Figure 1

Simulation geometries. Quasi 2D vertical layers can be random polydisperse (a)—with equal proportions of three species of radii $R_i=\{1.5,2.0,2.5\}$ mm—or ordered monodisperse (b)–(g)—$R_i=R=2$ mm—with, eventually, defects. The defect nomenclature is related to the position of the grain that breaks the symmetry. The 3D setups are a cubic (h) and cone-base cylinder (i) (see Ref. [15] for details), both monodisperse.

Figure 2

(a) $X^{\mathrm{c.m.}}(t)$ (symbols) and $x_i(t)$ for all the grains $i$ in the system (lines) for quasi-2D random packings. The grains move coherently with the c.m. Simulations with the same $\mathrm{\Gamma }$ refer to different random realizations. (b) Comparison between ${\sigma }^2(V_x^{\mathrm{c.m.}})$ and $⟨V_x^{\mathrm{c.m.}}⟩$, obtained averaging over three independent random realizations. (c) Time averaged velocity of center of mass as a function of $\mathrm{\Gamma }$ for different monodispersed packings. Each point is mediated over five independent realizations of the dynamics. Here, differently from random packings, $⟨V_x^{\mathrm{c.m.}}⟩$ vanishes for $\mathrm{\Gamma }\simeq 3$ because moderate vibrations destroy the asymmetric configuration of defects. (d) enlargement of low $\mathrm{\Gamma }\mathrm{s}$ to highlight the sign changes with fixed structure.

Figure 3

(a)–(d) Comparison between the time evolution of the c.m. velocity and the mean coordination number of the contact network $Z(t)=N^{-1}{\sum }_{ij}\mathrm{\Theta }(R_i+R_j+\delta -|{\mathit{r}}_{ij}|)$ where $0<\delta \ll R_i+R_j$ allows one to detect the nearest neighbor not in contact due to vibrations. Trajectories are smoothed with a running average. (a)–(b) random packings [cases with upside-down triangles and circles in Fig. 2]. (c) 3D cubic setup. Here $V_{xy}^{\mathrm{c.m.}}$ is two-dimensional so we plot both the modulus and the orientation ${\theta }_{xy}^{\mathrm{c.m.}}$. (d) 3D cylindrical geometry. The collective motion is a rotation around the central axis so ${\mathrm{\Omega }}_c=N^{-1}{\sum }_i|{\mathit{r}}_i{|}^{-2}({\mathit{r}}_i\times {\mathit{v}}_i{)}_z$. (e) Time averaged modulus of the normal force and compenetration between the plate and the bottom particles for two $z$-reflected configurations of defects. Site four is under the double defect [see Figs. 1]. (f)–(h) Scatter plots with bins of the total $x$-tangential force modulus exerted by the plate vs the total velocity of the bottom particles for ordered packings with defects (f) and random packings (g)–(h) shaken at $\mathrm{\Gamma }=0.7$. In the latter, ${\gamma }^{\pm }=|⟨F_x^T|v_x\gtrless 0⟩|/⟨v_x|v_x\gtrless 0⟩$ linearly interpolate the data clouds.

Figure 4

(a) Time evolution of $F_c=\mu |F^N|-|F_x^T|$. Diamonds mark the instants for which $F_c\sim 0$ corresponding to the sliding condition for the tangential force between the plate and the bottom grains. (b) Trajectory of the $X^{\mathrm{c.m.}}$ compared with the horizontal displacement of the bottom layer accumulated in the sliding instants ($\mathrm{\Delta }{X}_{\mathrm{sl}}^b$) and in all the other ones ($\mathrm{\Delta }{X}_{\mathrm{ot}}^b$). (c) Stick slip in the difference of the mean $x$ coordinates of the two lowest horizontal layers in a granular simulation. Diamonds refer to sliding instants. These three panels are obtained with BL and $\mathrm{\Gamma }=1.25$. (d) Mean velocity of the c.m. as a function of the horizontal (triangles) and vertical (diamonds) size of the system for different values of the defect concentration. The vertical size effect is shown for $c=1$, but we verified that its relevant behavior does not depend on $c>0$. (e) Scaling of $⟨V_x^{\mathrm{c.m.}}⟩$ and the diffusivity of the fluctuation around the drift $D$ (see the Supplemental Material [20]) as a function of the horizontal system size for random packings. Each point is obtained averaging over ten independent realizations of the packing; error bars are standard deviations.