## Main

One of the key difficulties in studying the interplay between repulsive contact forces, short-range cohesion and long-range electrostatic forces during cluster formation has been to obtain sufficiently detailed experimental data. Seeing how this process unfolds demands in situ observation of the collision trajectories among charged grains to extract quantitative information about their interactions. This requires the grains to be freed from gravity and tracked with high spatial and temporal resolution to capture individual collision events17,18.

We overcome these obstacles with the set-up shown in Fig. 1a (refs 19, 20). The granular material, in our experiments fused zirconium dioxide–silicate grains a few hundred micrometres in diameter, is contained in a vessel mounted inside a 3.0-m-tall cylindrical chamber. We evacuate this chamber to mitigate air drag. When a shutter covering a small orifice at the bottom of the vessel is opened, particles fall out freely, forming a dilute stream. Outside the chamber, a high-speed video camera falls alongside the grains, guided by low-friction rails. In the co-moving frame seen by the camera, the effect of gravity is eliminated, making it possible to track particle interactions in detail for about 0.2 s until the camera is decelerated by a foam pad. The same apparatus also allows determination of the net charge on individual grains: during free fall a horizontal electric field can be applied and the resulting horizontal acceleration observed by the camera gives the charge to mass ratio, q/m.

Using particles with a narrow size distribution gives rise to the distribution of net grain charge, P(q), shown in Fig. 1b. This distribution is the result of a very large number of collisions and sliding/rubbing events among particles during sample preparation and outflow from the vessel20. It arises from tribocharging between grains of identical material, and thus statistically similar surface density of transferable charges, but slightly different surface area9,21,22. Overall charge is conserved and the distribution is centred around zero. As expected for nearly monodisperse grains, P(q) is narrowly peaked. However, the tails of P(q) extend to magnitudes of several million elementary charges e (1.6 × 10−19 C) per grain. Interactions involving these highly charged grains are the ones most easily detected and tracked, and in the following we focus on them.

Figure 1c, d introduces the complex behaviours that arise when particle collisions involve charge, in particular the sequence of elliptical orbit fragments between successive bounces (see Supplementary Movie 1, Part 1). The fact that the particles separate hundreds of micrometres before re-approaching each other is a clear indicator of long-range electrostatic forces, and being able to observe the orbits over several successive bounces implies that collisional energy losses are small, at least for binary collisions.

The electrostatic Coulomb force F(r) between two particles with net charges q1 and q2 separated by a (time-varying) distance r(t) gives rise to the equation of motion μd2r/dt2 = F, where μ = m1m2/(m1 + m2) is the reduced mass. For non-polarizable particles, the electrostatic force is F(r) = (1/4πɛ0)(q1q2r/r3), where ɛ0 = 8.85 × 10−12 C2 N−1 m−2 is the permittivity of free space. The solution r(t) is a Kepler orbit. The sum E0 of the translational kinetic energy (in the centre-of-mass reference frame) and electrostatic potential energy determines whether r(t) forms an elliptical (E0 < 0), parabolic (E0 = 0), or hyperbolic (E0 > 0) trajectory. For dielectric particles, important corrections arise from induced polarization15,16,23. These polarization forces are always attractive, becoming increasingly important at close approach.

In Fig. 2 we plot examples of observed trajectories together with best fits. In fitting the data, the unknown parameters are: the charges q1 and q2, the initial relative positions in the z direction (as the camera images only the xy plane), and the initial relative velocities. We calculate μ by measuring the grain diameters in the videos and taking ρ = 3,800 kg m−3 as the grain density. For the fits we simulate the trajectories with forces including the full set of polarization terms calculated by a re-expansion method23. We note that, although the trajectories are also fitted well by simple Kepler orbits, inclusion of the polarization contributions is important to obtain the correct charge values (see Supplementary Information). For the collision sequence in Fig. 1e, best fits indicate that q1 and q2 possess opposite polarities and stay near their initial values, approximately +5.2 × 106 e and −1.5 × 106 e, respectively (Fig. 2b). The error bars on q1 and q2 from the fits are, however, too large to meaningfully constrain the amount of charge, Δq, transferred during a single collision. The actual Δq is probably much smaller. Extrapolating kinetic-energy-based results12 for micrometre-sized SiO2 particles hitting a large fixed target gives a rough estimation of Δq 1,000 e per collision, which at less than 1/1,000 of the total grain charge is too small to noticeably affect the trajectories.

We also encounter collisionless interactions, such as hyperbolic Kepler orbits, where one of two oppositely charged grains effectively acts as a slingshot for the other, and trajectories between grains of equal polarity that repel each other. Examples are shown in Fig. 2c, d (see Supplementary Movie 1, Parts 2 and 3).

Returning to the issue of aggregation, the relative velocity of colliding particles has to stay below vstick = (2Wcoh/μeeff2)1/2 in order for them to stick7,8. Here Wcoh is the work required to break the bond formed by short-range cohesive forces, including van der Waals forces or capillary forces due to absorbed molecular layers19, and eeff is an effective coefficient of restitution (the ratio of relative velocity magnitude before and after a collision). We find eeff 0.94 from analysis of head-on binary collisions at velocity 1.4 m s−1 (see Supplementary Information), close to the value of 0.97 reported24 for small soda lime glass spheres impacting at 0.5–1 m s−1. Grains typically will lose some translational kinetic energy in binary collisions, but particle rotation can have a significant effect. An example is shown in Fig. 3a (see Supplementary Movie 1, Part 4), where, after two collisions with elliptical trajectories in between, a third collision makes one grain suddenly take off on a hyperbolic trajectory. In this case, the fast-rotating grain was slightly non-spherical, making it possible to determine its rotational kinetic energy, which during this last collision decreased by 10 pJ, an amount that matches the measured increase in total translational kinetic energy. Thus, the conversion of rotational into translational kinetic energy during impact can lead to eeff > 1.

However, when single particles collide with clusters of particles, we find that the effective coefficient of restitution is significantly less than unity, probably because now energy can also be dissipated via intra-cluster rearrangements. Figure 3b illustrates this with snapshots (see Supplementary Movie 1, Part 5) in which two additional particles, one after the other, aggregate onto a cluster comprised of N = 5 particles. There are three possible outcomes when an incident particle strikes a cluster: the incident particle is trapped in the mutual electrostatic potential well and after one or more bounces sticks to the cluster, as in Fig. 3b (capture); the particle bounces and escapes from the potential well (escape); or the particle kicks out one or more different particles from the potential well (fragmentation). Figure 3c shows that escape is suppressed for N = 2, and for N > 2 the incident particle is always captured unless it fragments the cluster.

We can use fragmentation events in which only a single particle is kicked out (arrows in Fig. 3c) to estimate the depth of the potential well that binds grains to the cluster. The kinetic energy μv2/2 of the incident particle must overcome the short-range cohesive energy Wcoh plus the initial electrostatic potential energy U stored in a bond. In Fig. 3c, the minimum incident velocity for fragmentation is approximately 0.02 m s−1 and of similar magnitude to the maximum velocity vcapture for capture, corresponding to a kinetic energy of approximately 4 pJ. This is three orders of magnitude larger than the value Wcoh ≤ 10−15 J found for copper and glass grains of similar size19. Thus U clearly dominates the binding energy. Equating the incident kinetic energy with the Coulomb energy, μvcapture2/2 = (1/4πɛ0) |q1q2| /d, we estimate (|q1q2|)1/2 2 × 106 e. This large amount of net charge per particle confirms the notion that at least some of the grains in a cluster are drawn from the tails of the charge distribution P(q) in Fig. 1b.

The fact that P(q) is peaked around zero makes it very likely to find cluster configurations where one highly charged particle has attracted several particles with much smaller charge magnitude. A signature of this is a closely packed arrangement of three or more grains all in direct contact, such as particles inside the cluster in Fig. 3b or the triangular configuration in Fig. 3e. For particles with high dielectric constant such configurations can be stabilized through the always attractive polarization forces, even if contacting grains have the same polarity, provided there is significant contrast in charge magnitude25. Seeing such dynamics in situ highlights the particularly important role polarization can play in dust aggregation, initiating cluster formation from a single charged particle and potentially leading to runaway growth14.

Going further, we can use the shape of the charge distribution to control the aggregation outcomes. In particular, for a bimodal P(q) with positive and negative peaks roughly symmetric around zero, we would expect typical cluster configurations in which contacting grains exhibit alternating polarity. Recent experiments have demonstrated that such a P(q) can be achieved by mixing bidisperse same-material grains22. The P(q) for such a mixture using diameters 326 ± 10 μm and 251 ± 10 μm is shown in Fig. 4a. Because tribocharging will transfer charge in a manner that the larger grains become on average positive and the smaller grains negative21,22, this has the added benefit that polarity can be identified directly from the images by particle size. Figure 4b shows a taxonomy of the resulting, experimentally observed cluster configurations. These structures resemble molecules or fragments of self-assembled, electrostatically stabilized lattices13,26.

Modelling these ‘granular molecules’ by taking many-body polarization effects into account (see Methods) we find that most of the observed configurations correspond to energetic ground states (Fig. 4c). Some combinations of large (L) and small (S) particles can have more than one stable state. As an example, Fig. 4d shows a square L2S2 molecule reconfiguring into a linear shape after being impacted, and then back to a square (see Supplementary Movie 1, Part 6), suggesting that the square shape is the ground state whereas the linear configuration is metastable (calculation shows that the energy for the square shape is 1 pJ lower).

Because like charges of similar magnitude repel each other, the configurations in Fig. 4b, which are all based on a bimodal P(q), tend to be less densely packed than clusters formed from a P(q) with a single peak around q = 0 (Fig. 1b). This is exemplified by the star-shaped configurations L3S1 and L1S3, the tetrapod L1S4, and the L3S3 structure. However, forces due to polarization become important at close approach and, as mentioned before, for large differences in charge magnitude these forces can pull same-polarity grains into contact. Figure 4e demonstrates this with an event (see Supplementary Movie 1, Part 7) in which a dimer captures another particle to form a stable triangular configuration (Fig. 4f).

These results have implications for a wide range of situations where collision-induced particle aggregation is important. In particular, our results show at the single particle level how long-range electrostatic forces can capture grains whose kinetic energy is orders of magnitude larger than Wcoh. Multiple bounces enabled by the electrostatic potential well very effectively dissipate kinetic energy, an effect that is further enhanced in N > 2 clusters by internal reconfigurations, all of which increases the likelihood of capture and aggregation. Already a very small size dispersion, such as in our nearly monodisperse sample, suffices to generate highly charged particles, an effect likely to become amplified for larger dispersions. Finally, the charge-stabilized granular molecules we observe highlight how intra-cluster particle configurations are controlled by dielectric polarization, an effect of much current interest within the context of self-assembled colloid and ion clusters.

## Methods

As particles we used fused zirconium dioxide–silicate grains (68% ZrO2, 32% SiO2 by volume, material density ρ = 3,800 kg m−3; Glenn Mills), sieved to average particle diameters d = 274 ± 14 μm for the monodisperse sample, and 326 ± 10 μm and 251 ± 10 μm for the binary sample (mixed in equal numbers by fluidizing with dry air for 30 min). Before the experiments, the grains were stored at 40–50% relative humidity. The grains were placed into a hopper mounted inside a transparent, cylindrical vacuum chamber that then was evacuated to <2 mtorr by a turbo pump to eliminate air drag on the falling grains. The grains flowed freely from an orifice at the bottom of the hopper. The orifice size was chosen small enough (<15 d) so that the grains formed a highly dilute stream. Video was taken by a Phantom v9.1 high-speed video camera at 1,000 frames per second, mounted on a carriage that could slide freely along low-friction rails. This apparatus could also be used for P(q) measurements, where the grains fall freely via gravity between two large vertical copper plates held at an external horizontal electric field E, causing a grain of charge q and mass m to undergo a horizontal acceleration a = qE/m. To extract the accelerations a, we tracked the horizontal trajectories and fitted with parabolas. The field was applied only to measure P(q); all trajectories and clusters shown in Figs 14 were obtained without an applied external electric field. See ref. 20 for more details about the apparatus. Note that the absence of an external electric field will also ensure that induction-based charging mechanisms10,27,28 are not likely to be significant (see Supplementary Information).

Particles in the raw videos were identified and tracked with the algorithm developed by Crocker and Grier29. The method by Nakajima and Sato23 was used to calculate the electrostatic forces between two charged dielectric particles, including the full set of higher-order polarization contributions. The net (free) charge on a grain was assumed uniformly distributed over the surface. The dielectric constant of the grain material was taken as ɛ 15, based on the volume-weighted average of ${\epsilon }_{{\text{ZrO}}_{\text{2}}}$ 22 and ${\epsilon }_{{\text{SiO}}_{\text{2}}}$ 3.9. To simulate a trajectory, a leapfrog algorithm was used for integrating the equation of motion30. As we imaged in the xy plane and a priori did not have information about the trajectory inclination with respect to the z-axis, we employed a Nelder–Mead simplex algorithm31 as a multidimensional optimization method to search for local minima of the absolute median deviation between simulated trajectories and the experimental data (see Supplementary Information). A reduced chi-squared statistic between the observed trajectory data and the model trajectory data was used for the goodness of fit. 500 search trials were performed and the 10% best-fit trials were selected to extract q1 and q2 for each trajectory. In this charge determination, we assumed q1 > 0 and |q1| > |q2| (see Supplementary Information).

For modelling granular molecules with N > 2 we used a technique recently introduced by Barros et al.32 to treat the electrostatics of many-body polarization effects, meshing each grain into 720 polarizable patches. The free charges for large (d = 326 μm) and small (d = 251 μm) grains were set to the averages for the bimodal P(q) in Fig. 4a, that is, to +1.8 × 106 e and −2.3 × 106 e, respectively (the one exception was the L1S4 structure, which required charges ±1.8 × 106 e for a stable four-armed ‘tetrapod’). The free charge on each particle was distributed uniformly over its surface. The additional surface bound charge due to polarization was then calculated iteratively until the electrostatic energy Un at iteration n changed by less than |(UnUn−1)/Un−1| < 10−4. A Nelder–Mead algorithm was used to identify, for given numbers of large and small grains, those spatial arrangements that produce local minima in the electrostatic energy, and thus are candidates for stable states. As a hard-sphere constraint in the energy minimization the condition of (dA + dB)/2 < rAB was used, where dA and dB are the diameters for any two particles A and B, and rAB is the centre-to-centre distance between them.