Direct observation of particle interactions and clustering in charged granular streams

Abstract

Clustering of fine particles is of crucial importance in settings ranging from the early stages of planet formation^1,2,3 to the coagulation of industrial powders and airborne pollutants^4,5,6,7. Models of such clustering typically focus on inelastic deformation and cohesion^1,4,6,8. However, even in charge-neutral particle systems comprising grains of the same dielectric material, tribocharging can generate large amounts of net positive or negative charge on individual particles, resulting in long-range electrostatic forces^9,10,11. The effects of such forces on cluster formation are not well understood and have so far not been studied in situ. Here we report the first observations of individual collide-and-capture events between charged submillimetre particles, including Kepler-like orbits. Charged particles can become trapped in their mutual electrostatic energy well and aggregate via multiple bounces. This enables the initiation of clustering at relative velocities much larger than the upper limit for sticking after a head-on collision, a long-standing issue known from pre-planetary dust aggregation^1,12. Moreover, Coulomb interactions together with dielectric polarization are found to stabilize characteristic molecule-like configurations, providing new insights for the modelling of clustering dynamics in a wide range of microscopic dielectric systems, such as charged polarizable ions, biomolecules and colloids^13,14,15,16.

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 events^17,18.

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.

Figure 1: Free-fall video imaging of particle interactions.

a, Sketch of experimental set-up. Charged ZrO₂:SiO₂ grains fall freely inside a vacuum chamber, while their motion is captured by a co-falling high-speed camera. b, Particle-charge distribution P(q) for nearly monodisperse grains. Inset right: The same data plotted on a log-linear graph. Inset left: Optical micrograph of the grains. c, Representative still frame from high-speed video, showing the dilute particle stream in the full field of view. d, Sequence of zoomed-in still frames tracking the interaction of two oppositely charged grains. The image at t = 0 ms corresponds to the area inside the yellow frame in c; subsequent frames have been re-centred to keep the slightly smaller grain in the middle of the image. Blue dashed lines indicate the path of the other grain as it repeatedly approaches and bounces off. e, Horizontal (r_x) and vertical (r_y) components, in the x–y imaging plane, of the relative position vector of the grains in d. The time interval between adjacent data points is 1 ms, the arrow indicates the direction of motion, and the numbers denote the four distinct trajectory segments between bounces.

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 area^9,21,22. 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 polarization^15,16,23. 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 x–y 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.

Figure 2: Kepler-like trajectories.

a, Relative position of the two grains from trajectory segment 4 in Fig. 1d, e (blue circles). The data approximately follow an elliptical Kepler orbit (green dashed line) with focus at the origin (black cross). A fit including corrections for dielectric polarization effects is shown by the red diamonds. b, Charges q₁ (red diamonds) and q₂ (blue circles) on the two particles in Fig. 1d for each trajectory segment, extracted from 10% best fits with polarization contributions. The length of the error bars corresponds to one standard deviation above and below the average. c, Example of a hyperbolic trajectory due to attractive electrostatic interaction. d, Hyperbolic trajectory due to repulsive interaction. Insets to c and d: Still images from the videos from which the data were extracted. The arrows in a,c and d indicate the direction of the relative motion at the beginning of the video.

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 v_stick = (2W_coh/μe_eff²)^1/2 in order for them to stick^7,8. Here W_coh 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 e_eff is an effective coefficient of restitution (the ratio of relative velocity magnitude before and after a collision). We find e_eff ≍ 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 e_eff > 1.

Figure 3: Separation and aggregation of charged dielectric particles.

a, Time sequence of particles bouncing and re-approaching over two collisions but separating after the third. b, Time sequence of two particles (coloured green and yellow) aggregating onto an already formed five-particle cluster. c, Collision outcomes for a single particle colliding with relative velocity v (in the x–y plane) with a cluster comprised of N particles: capture (blue triangles), escape (green circles) and fragmentation (red stars). Arrows indicate fragmentation events where only one particle is kicked out from the cluster, which we can use to estimate the binding strength (see text). d, Example of a fragmentation event, showing a N = 9 cluster right before and 1 ms after being hit by a fast moving particle, whose direction is indicated by the yellow arrow. e, Polarization forces can stabilize close-packed arrangements of contacting grains.

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 W_coh 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 v_capture for capture, corresponding to a kinetic energy of approximately 4 pJ. This is three orders of magnitude larger than the value W_coh ≤ 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, μv_capture²/2 = (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 negative^21,22, 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 lattices^13,26.

Figure 4: Granular molecules formed from bimodal particle-charge distribution.

a, Charge distribution resulting from mixing grains with diameters 326 ± 10 μm and 251 ± 10 μm in equal number. The large grains (red solid line) are predominantly (80%) positive whereas the small grains (blue dashed line) are predominantly (95%) negative. Inset: Optical micrograph of large and small grains. b, Still images of granular molecules. Their structure consists of alternating large and small grains (labelled L_mS_n, where m + n is the total number of particles in the molecule), giving rise to string-like (L₁S₁, L₁S₂, L₂S₁, L₂S₂), square (L₂S₂, L₃S₃), trigonal planar (L₁S₃, L₃S₁), tetrahedral (L₁S₄) and triangular bipyramid (L₅S₁) geometries. c, Stable granular molecule configurations corresponding to b, obtained by minimizing the total electrostatic energy (including polarization effects) for grains with net charge equalling the average positive (+1.8 × 10⁶ e) or negative (−2.3 × 10⁶ e) charge of the bimodal P(q) shown in a. Colour represents the net surface (free and bound) charge density σ_net. d, Sequence of stills showing conformation change of an L₂S₂ molecule after impact from the right (arrow). The square structure breaks open, becomes linear, and returns to square. e, Sequence showing the formation of a triangular molecule from two large positively charged grains and one small negatively charged grain. The closing of the large, visible gap (t = 42 ms) between the large particles cannot be explained by short-range sticking and directly implies attractive forces from polarization effects. f, One of many possible charge assignments that can produce stable triangular molecules from two grains with similar charge magnitude but opposite polarity and one grain of lesser net charge. In this example the charges are +1.8 × 10⁶ e and +0.2 × 10⁶ e for the large grains, and −2.3 × 10⁶ e for the small grain.

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 W_coh. 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 1–4 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 mechanisms^10,27,28 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 x–y 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 U_n at iteration n changed by less than |(U_n − U_n−1)/U_n−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 (d_A + d_B)/2 < r_AB was used, where d_A and d_B are the diameters for any two particles A and B, and r_AB is the centre-to-centre distance between them.