Granular cooling of ellipsoidal particles in microgravity.

A three-dimensional granular gas of ellipsoids is established by exposing the system to the microgravity environment of the International Space Station. We use two methods to measure the dynamics of the constituent particles and report the long-time development of the granular temperature until no further particle movement is detectable. The resulting cooling behavior can be well described by Haff's cooling law with time scale τ. Different analysis methods show evidence of particle clustering towards the end of the experiment. By using the kinetic theory for ellipsoids we compare the translational energy dissipation of individual collision events with the overall cooling time scale τ. The difference from this comparison indicates how energy is distributed in different degrees of freedom including both translation and rotation during the cooling.

Introduction

The ideal gas equation as the relationship between pressure, volume, and temperature is a well-known law for thermal gases. Although there are many analogies between the statistical physics of thermal and granular systems, granular particles exhibit a unique cooling behavior when left without agitation. This is because the interaction in granular systems happens exclusively over particle contacts. In a granular gas these collisions between particles are inelastic and the system will dissipate energy over time, i.e., it will cool. Haff was the first to look at the cooling of granular flows in 1983[1]. He predicts that the mean velocity of the constituent particles decreases algebraically for dilute systems, i.e., granular gases. This prediction has been tested and confirmed several times[2-6]. Another consequence of the inelastic nature of the particle interactions has been proposed to be the clustering of the particles, as granular gases with large amount of particles can separate into dense and dilute regions[7-10]. Apart from theoretical ideas and numerical simulations, some experiments have also been performed to validate this prediction. Experimentally, granular gases can be established by placing granular particles into weightlessness. Therefore, one either has to counter the gravitational force in the framework of a levitation experiment[11-13] or by providing a microgravity environment such as in a droptower experiment or a parabolic flight[14-16]. Most of the experiments were conducted in two dimensions. Only recently also three-dimensional systems were investigated[6,13,17,18], which are expected to show clustering already at rather low-volume fractions[19,20]. Obviously, the driving protocol as well as the detailed environmental parameters to establish weightlessness have a strong influence on the dynamics of the system[17,21].

In this work, we investigate the cooling behavior of a three-dimensional granular gas constituted of ellipsoidal particles. The investigation was conducted as an educational experiment on board the International Space Station (ISS, cf. Fig. 1), which provides a high-quality microgravity environment for long times (~80 s for one experiment). Among the existing experimental works, Harth et al.[18] investigated similar non-spherical particles in low-gravity conditions and verified the theoretical prediction of the collision rate[22] during homogeneous cooling. The relatively short low-gravity time (~9 s) limited by the low-gravity platform prevents analysis of long-term cooling behavior. To our knowledge there is no other experimental investigation of non-spherical particle systems under low-gravity before the current one.

Fig. 1

Experiment in ISS.

Astronaut Samantha Cristoforetti conducting the experiment on board the ISS during Expedition 42/43.

Our granular system consists of 96 commercially available Mars Inc. M&M candies, which have an ellipsoidal shape with a short axis of radius a ≃ 3.50 mm and two identical long axes of radius b ≃ 6.75 mm (i.e., an oblate ellipsoid). The mass of one such ellipsoid m is about 0.91 g. The container has a spherical shape and is made of synthetic material and has a diameter of about 150 mm. The packing fraction of our system is thus about 3.6%. The video footage is taken by a Canon XF305 at a frame rate of 29.97 fps. The container is shaken up randomly by hand for several seconds to reach an observable homogeneous spatial distribution of the particles, and then placed onto a holder where the video camera records the dynamics of the cooling particles from above. We adopt two image-processing methods to analyze the videos and obtain the mean translational velocity 〈v(t)〉 in two dimensions. These results are then qualitatively and quantitatively compared with the cooling theory and help us detect the signatures of clustering.

Methods

There are in total five experiments performed during the mission. Four of them have problems of minor misplacement and/or unstable positioning of the sample cell under the camera view, which prevent us from applying image processing for quantitative analysis. They are nevertheless useful for our qualitative observation of the clustering behavior as shown in Fig. 2.

Fig. 2

Particle clustering.

In four out of the total five experiments we observe first a homogeneous distribution of the particles as shown by a, and then a separation of the particles into dense and dilute regions as shown by b.

Two image-processing approaches

We implemented two methods to evaluate the cooling dynamics of the one experiment with the best quality. A first approach is based on the optical flow method of Lucas and Kanade[23]. The algorithm tracks so-called features that are unique regions in two successive images determined by high-intensity gradients. We subtract successive images with a distance of Δ = 2, 3, and 8 frames, with the former two parameters yielding very similar results and thus only Δ = 2 presented in the Results section. The resulting difference images provide the advantage of reducing the static reflections on the container walls (e.g., as can been observed in Fig. 2). This rather robust algorithm is easy to implement and gives a good estimate of the mean velocity of the particles if the only changing pixels in the images are caused by particle movement. A similar method was employed for the evaluation in a dense granular system under X-ray radiography[24]. In our evaluation, however, the artifacts such as the reflections can not be completely removed by subtracting images. More importantly, the changing features related with the particle movement themselves are partially caused by particle rotation. The resulting calculation of the translational velocities of the particles thus overestimates the real values.

In a second approach, we try to reconstruct the trajectory of each particle in the video footage. We miss depth information as we have a two-dimensional projection of a three-dimensional experiment. As a further consequence, we lose track of particles that move behind each other in the projection. We reduced this effect by performing the analysis for the different colors of the particles separately. In each frame the particles are identified by fitting an ellipse to connected objects of the same color. By means of a proximity-based tracking procedure the trajectories can be reconstructed among images (cf. Fig. 3a). Whenever particles get out of sight, their positions are linearly interpolated.

Fig. 3

Particl tracking.

a Proximity-based particle tracking: the ellipsoidal particles are tracked by fitting an ellipse to the individual objects in the image. The trajectories are reconstructed by linking the positions of nearest neighbors in two successive images. If particles get lost in the procedure, their trajectories are interpolated. b Collision tracking: a snapshot of one collision event: following the trajectory tracking, collisions fitting with two-dimensional measurement are detected and analyzed. See supplementary videos collision1.avi and collision2.avi for complete events.

Collision characterization

During the particle tracking, some collision events between the particles are used for measurement of energy dissipation rate (cf. Fig. 3b). We manually select those events with the motions of both particles before and after the collisions approximately within a plane vertical to the camera view. In other words, we consider in our selected collision events that the changes of the depths of two particles are small compared to their changes of speed in this plane, and the measured translational kinetic energies from the two-dimensional image analysis are close to their three-dimensional values. We measure the translational speeds (v1, v2) and of the two colliding particles before and after the collision and calculate the translational kinetic energy T and before and after collisions. The ratiois used along with the kinetic theory in the previous sections to calculate the cooling time scale. Note that r is not the coefficient of restitution ϵ that is commonly used in such calculations. Discussions about its usage can be found in the previous section.

As discussed in the previous section, a three-dimensional diagnostics of both the translational and rotational motion of the particles shall be the major goal of our future improvement of the setup.