Droplet and cluster formation in freely falling granular streams

Particle beams are important tools for probing atomic and molecular interactions. Here we demonstrate that particle beams also offer a unique opportunity to investigate interactions in macroscopic systems, such as granular media. Motivated by recent experiments on streams of grains that exhibit liquid-like breakup into droplets, we use molecular dynamics simulations to investigate the evolution of a dense stream of macroscopic spheres accelerating out of an opening at the bottom of a reservoir. We show how nanoscale details associated with energy dissipation during collisions modify the stream’s macroscopic behavior. We find that inelastic collisions collimate the stream, while the presence of short-range attractive interactions drives structure formation. Parameterizing the collision dynamics by the coefficient of restitution (i.e., the ratio of relative velocities before and after impact) and the strength of the cohesive interaction, we map out a spectrum of behaviors that ranges from gaslike jets in which all grains drift apart to liquid-like streams that break into large droplets containing hundreds of grains. We also find a new, intermediate regime in which small aggregates form by capture from the gas phase, similar to what can be observed in molecular beams. Our results show that nearly all aspects of stream behavior are closely related to the velocity gradient associated with vertical free fall. Led by this observation, we propose a simple energy balance model to explain the droplet formation process. The qualitative as well as many quantitative features of the simulations and the model compare well with available experimental data and provide a first quantitative measure of the role of attractions in freely cooling granular streams.

Figure 1

Snapshots from simulations of granular streams consisting of $d=$ 100 $\mu$m spheres freely falling out of a $D_0=$ 2.0 mm aperture. (a)–(d) Coefficient of restitution $e=$ 0.40 and cohesive strength $F_{\text{coh}}=$ 100 nN. (e)–(h) $e=$ 0.88 and $F_{\text{coh}}=$ 1 nN. The images follow the grains in the comoving frame and are taken just below the aperture [(a), (e)], and at distances $z=$ 0.7 cm [(b), (f)], 1.9 cm [(c), (g)], and $z=$ 3.6 cm [(d), (h)] from the aperture to the top of each frame. Grains are color coded according to their rms velocity difference with neighboring grains within a distance of 1.5$d$. Isolated grains are shown in pink. Images generated via Ref. [21]. For movies see Ref. [22].

Figure 2

Droplet shapes for $d=$ 200 $\mu$m spheres freely falling from a $D_0=$ 3.0 mm aperture. (a) Droplet height ${\lambda }_d$ vs. width $w_d$ for simulations in droplet-forming regime. Only a sampling of simulations is shown here to avoid clutter [data for all droplet-forming simulations are included in Fig. 2c]. Inset demonstrates what is meant by width $w_d$ and height ${\lambda }_d$. (b) Droplet width $w_d$ (red squares) and height ${\lambda }_d$ (blue triangles) vs. $F_{\text{coh}}$ for $e=0.61$. Error bars show standard deviation of measured ${\lambda }_d$ and $w_d$. Aperture diameter is indicated by dashed line. (c) Distribution of aspect ratios ${\lambda }_d/w_d$ for all droplet-forming streams.

Figure 3

Droplet separation dynamics. Data are for the same grain diameter and aperture size as in Fig. 2. Distance between the center of mass $\Delta z(t)$ (insets) and separation velocity $\Delta v(t)\equiv d(\Delta z)/\mathit{dt}$ for four pairs of adjacent droplets from streams with $e=0.61$ and (a) $F_{\text{coh}}=$ 400 nN and (b) $F_{\text{coh}}=$ 1100 nN. For each pair, the the time $t$ is offset so that at $t=0$ the center of the pair of droplets passes through the aperture ($z=0$). Images to the right show the pairs of droplets tracked here at a depth $z=$ 18.4 cm below the aperture ($t~$ 180 ms).

Figure 4

Phase diagram parameterized by cohesive force $F_{\text{coh}}$ and restitution coefficient $e$. Data are for the same grain diameter and aperture size as in Fig. 2 Data points are color coded by the average contact number $\langle C(z)\rangle$, computed at depth $z=$ 4 cm. Images (a)–(h) from the simulations at the same depth are shown for selected ($F_{\text{coh}},e$) pairs. Grains in these images are colored by their local contact number according to the same color scale. Data along the left vertical axis corresponds to $F_{\text{coh}}=0$. The image in the upper right highlighted by the red border corresponds to $z=0$ instead of 4 cm.

Figure 5

Average local velocity deviations as a function of depth. Data are for the same grain diameter and aperture size as in Fig. 2. The average axial velocity at the aperture was $v_{z,0}\approx 10$ cm/s. (a) Mean-square velocity deviations $\langle (\Delta \mathbf{v}){}^2\rangle$ inside the hopper ($z<0$) and in the falling stream ($z>0$). (b) Radial velocity deviations $\langle (\Delta v_r){}^2\rangle =\langle (\Delta v_x){}^2+(\Delta v_y){}^2\rangle$. (c) Axial velocity deviations $\langle (\Delta v_z){}^2\rangle$. In (a) through (c), solid curves correspond to five different spraying and clustering streams, while dotted curves correspond to two different droplet-forming streams. (d) Same as in (c) with droplet-forming streams (dotted curves) excluded. Dashed line: Fit to $\Delta v_z^2$ [Eq. (2)], giving $\Delta z=$ 200 $\mu$m.

Figure 6

Evolution of contact number and network size. Data are for the same grain diameter and aperture size as in Fig. 2. (a) Normalized average contact number vs. distance below aperture. Data shown for $F_{\text{coh}}=$ 0 nN and $e=$ 0.04 (open blue squares), 0.4 (open red circles), 0.88 (open green triangles), and for $F_{\text{coh}}=$ 100 nN, $e=$ 0.88 (solid blue squares); 300 nN, 0.61 (solid green triangles); 1000 nN, 0.40 (solid red circles). Inset: Initial contact number vs. restitution coefficient for $F_{\text{coh}}=$ 0 (triangles), 100 nN (squares), 500 nN (circles), and 1000 nN (diamonds). (b) Average network size vs. distance below aperture for $e=$ 0.61 streams. From bottom to top along right-hand edge: $F_{\text{coh}}=$ 0, 30, 50, 70, 100, 150, 200, 600, 1000, 1300, 2000. (c) Large-$z$ saturation value of average network size vs. cohesive force for $e=$ 0.61 streams.

Figure 7

Initial and final separation velocity varying $F_{\text{coh}}$. Data are for the same grain diameter and aperture size as in Fig. 2. (a) Average initial separation velocity $\Delta v_0$ (black triangles) and final separation velocity $\Delta v_f$ (red squares) vs. $F_{\text{coh}}$. (b) $\Delta v_0$ for pairs of adjacent droplets from streams with $e=$ 0.61 and $F_{\text{coh}}$ from 400 nN to 2000 nN plotted against the mean number of grains in each pair $N_{\text{pair}}=(N_i+N_j)/2$ for droplets $i$ and $j$. (c) $\Delta v_0$ vs. droplet separation at the aperture $\Delta z_0=\Delta z(t=0)$ scaled by the grain diameter $d$. Solid line is a fit to $\Delta v_0=\alpha (\Delta z_0/d)$ with $\mathit{gd}/v_{z,0}=$ 1.4 cm/s.

Figure 8

Contact number vs. $F_{\text{coh}}$ and $v_{\text{cap}}$. Data are for the same grain diameter and aperture size as in Fig. 2. (a) Mean contact number $C$ measured $4$ cm below the aperture plotted against the cohesive force $F_{\text{coh}}$ for $e=$ 1.0 (green diamonds), 0.88 (blue triangles), 0.73 (orange squares), 0.61 (red circles), and 0.40 (magenta squares). (b) Mean contact number at $4$ cm from the aperture plotted against the capture velocity $v_{\text{cap}}$ with same symbols as in (a).

Figure 9

Droplet length scale at the aperture. Data are for the same grain diameter and aperture size as in Fig. 2. Measured droplet separation at aperture (red circles) and toy model fit (solid line). The single-fit parameter has the value $s=(2.05\pm 0.03)\times {10}^{-2}$ nN/mm${}^2$.