Direct Observation of Particle Interactions and Clustering in Charged Granular Streams

LETTERS PUBLISHED ONLINE: 13 JULY 2015 | DOI: 10.1038/NPHYS3396

Direct observation of particle interactions and clustering in charged granular streams

Victor Lee*, Scott R. Waitukaitis, Marc Z. Miskin and Heinrich M. Jaeger

Clustering of fine particles is of crucial importance in settings distribution is the result of a very large number of collisions and ranging from the early stages of planet formation1–3 to the sliding/rubbing events among particles during sample preparation coagulation of industrial powders and airborne pollutants4–7. and outflow from the vessel20. It arises from tribocharging between Models of such clustering typically focus on inelastic deforma- grains of identical material, and thus statistically similar surface tion and cohesion1,4,6,8. However, even in charge-neutral particle density of transferable charges, but slightly different surface systems comprising grains of the same dielectric material, area9,21,22. Overall charge is conserved and the distribution is centred tribocharging can generate large amounts of net positive or around zero. As expected for nearly monodisperse grains, P(q) is negative charge on individual particles, resulting in long-range narrowly peaked. However, the tails of P(q) extend to magnitudes electrostatic forces9–11. The eects of such forces on cluster of several million elementary charges e (1.6 × 10−19 C) per grain. formation are not well understood and have so far not been Interactions involving these highly charged grains are the ones most studied in situ. Here we report the first observations of individ- easily detected and tracked, and in the following we focus on them. ual collide-and-capture events between charged submillimetre Figure1 c,d introduces the complex behaviours that arise when particles, including Kepler-like orbits. Charged particles can particle collisions involve charge, in particular the sequence become trapped in their mutual electrostatic energy well and of elliptical orbit fragments between successive bounces (see aggregate via multiple bounces. This enables the initiation of Supplementary Movie 1, Part 1). The fact that the particles separate clustering at relative velocities much larger than the upper hundreds of micrometres before re-approaching each other is a limit for sticking after a head-on collision, a long-standing clear indicator of long-range electrostatic forces, and being able issue known from pre-planetary dust aggregation1,12. Moreover, to observe the orbits over several successive bounces implies that Coulomb interactions together with dielectric polarization are collisional energy losses are small, at least for binary collisions. found to stabilize characteristic molecule-like configurations, The electrostatic Coulomb force F(r) between two particles providing new insights for the modelling of clustering dynamics with net charges q1 and q2 separated by a (time-varying) distance in a wide range of microscopic dielectric systems, such as r(t) gives rise to the equation of motion µd2r/dt 2 = F, where 13–16 µ /( ) charged polarizable ions, biomolecules and colloids . = m1m2 m1 + m2 is the reduced mass. For non-polarizable ( ) ( / ε )( / 3) One of the key difficulties in studying the interplay between particles, the electrostatic force is F r = 1 4π 0 q1q2r r , where ε −12 2 −1 −2 repulsive contact forces, short-range cohesion and long-range 0 = 8.85 × 10 C N m is the permittivity of free space. The ( ) electrostatic forces during cluster formation has been to obtain solution r t is a Kepler orbit. The sum E0 of the translational kinetic sufficiently detailed experimental data. Seeing how this process energy (in the centre-of-mass reference frame) and electrostatic unfolds demands in situ observation of the collision trajectories potential energy determines whether r(t) forms an elliptical < > among charged grains to extract quantitative information about (E0 0), parabolic (E0 = 0), or hyperbolic (E0 0) trajectory. their interactions. This requires the grains to be freed from gravity For dielectric particles, important corrections arise from induced and tracked with high spatial and temporal resolution to capture polarization15,16,23. These polarization forces are always attractive, individual collision events17,18. becoming increasingly important at close approach. We overcome these obstacles with the set-up shown in Fig.1 a In Fig.2 we plot examples of observed trajectories together (refs 19,20). The granular material, in our experiments fused zirco- with best fits. In fitting the data, the unknown parameters are: the nium dioxide–silicate grains a few hundred micrometres in diame- charges q1 and q2, the initial relative positions in the z direction ter, is contained in a vessel mounted inside a 3.0-m-tall cylindrical (as the camera images only the x–y plane), and the initial relative chamber. We evacuate this chamber to mitigate air drag. When a velocities. We calculate µ by measuring the grain diameters in the shutter covering a small orifice at the bottom of the vessel is opened, videos and taking ρ = 3,800 kg m−3 as the grain density. For the particles fall out freely, forming a dilute stream. Outside the cham- fits we simulate the trajectories with forces including the full set of ber, a high-speed video camera falls alongside the grains, guided by polarization terms calculated by a re-expansion method23. We note low-friction rails. In the co-moving frame seen by the camera, the that, although the trajectories are also fitted well by simple Kepler effect of gravity is eliminated, making it possible to track particle orbits, inclusion of the polarization contributions is important to interactions in detail for about 0.2 s until the camera is decelerated obtain the correct charge values (see Supplementary Information). by a foam pad. The same apparatus also allows determination of For the collision sequence in Fig.1 e, best fits indicate that q1 and the net charge on individual grains: during free fall a horizontal q2 possess opposite polarities and stay near their initial values, electric field can be applied and the resulting horizontal acceleration approximately +5.2×106 e and −1.5×106 e, respectively (Fig.2 b). / observed by the camera gives the charge to mass ratio, q m. The error bars on q1 and q2 from the fits are, however, too large Using particles with a narrow size distribution gives rise to to meaningfully constrain the amount of charge, 1q, transferred the distribution of net grain charge, P(q), shown in Fig.1 b. This during a single collision. The actual 1q is probably much smaller.

James Franck Institute and Department of Physics, The University of Chicago, Chicago, Illinois 60637, USA. *e-mail: [email protected]

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a y bc0.6

To pump )

−1 −1

x e 10 z )

Grain −6 Magnet −1 0.4 −2 hopper e 10

−6 ) (10 −3

q 10 (

µ P ) (10 200 m −4

q 10 ( 0.2 −10 −5 0 5 10 P Nozzle q (106 e) High-speed g camera 0.0 −10 0 10 2 mm 6 Low-friction Falling q (10 e) rails grains d y 0 ms 5 ms 28 ms 51 ms 65 ms 81 ms e 400 Vacuum x 2 chamber z 1

300 µm m) 0 µ 3 ( y

103 ms 114 ms 130 ms 177 ms 186 ms 193 ms r

Foam cushions −400 4 −400 0 400 µ rx ( 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 o. e, Horizontal (rx) and vertical (ry) 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.

abc8 0 ms 53 ms 108 ms Model 1,500 200 6 Experimental data ) e 4

6 µ Ellipse ﬁt 2 500 m (10

q 0 1,000 −2 Model −4 Experimental data 0 1234 Trajectory segment 500 m) m) µ µ

( d ( y y r 0 ms 77 ms 160 ms r −200 1,000 0 500 µm

m) Model µ 500 −500 (

y Experimental data −400 r

0 −1,000

−200 0 200 −1,000 −500 0 500 −1,000 −500 0 µ µ µ rx ( m) rx ( m) rx ( m)

Figure 2 | Kepler-like trajectories. a, Relative position of the two grains from trajectory segment 4 in Fig.1 d,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 eects is shown by the red diamonds. b, Charges q1 (red diamonds) and q2 (blue circles) on the two particles in Fig.1 d 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.

Extrapolating kinetic-energy-based results12 for micrometre-sized acts as a slingshot for the other, and trajectories between grains of SiO2 particles hitting a large fixed target gives a rough estimation equal polarity that repel each other. Examples are shown in Fig.2 c,d of 1q≈1,000 e per collision, which at less than 1/1,000 of the total (see Supplementary Movie 1, Parts 2 and 3). grain charge is too small to noticeably affect the trajectories. Returning to the issue of aggregation, the relative velocity of ( /µ 2)1/2 We also encounter collisionless interactions, such as hyperbolic colliding particles has to stay below vstick = 2Wcoh eeff in 7,8 Kepler orbits, where one of two oppositely charged grains effectively order for them to stick . Here Wcoh is the work required to break

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a 0 ms 12 ms 24 ms 42 ms 60 ms 74 ms

500 µm

b 0 ms 32 ms 69 ms 77 ms 85 ms

500 µm

95 ms 114 ms 127 ms 163 ms 208 ms

c d Capturing Escaping Fragmentation 0 ms 1 ms

100

) 1 mm −1 10−1 (m s v 500 µm e

10−2

10−3 1234567891011 N

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. the bond formed by short-range cohesive forces, including van However, when single particles collide with clusters of particles, der Waals forces or capillary forces due to absorbed molecular we find that the effective coefficient of restitution is significantly 19 layers , and eeff is an effective coefficient of restitution (the ratio less than unity, probably because now energy can also be dissipated of relative velocity magnitude before and after a collision). We find via intra-cluster rearrangements. Figure3 b illustrates this with eeff ≈ 0.94 from analysis of head-on binary collisions at velocity snapshots (see Supplementary Movie 1, Part 5) in which two 1.4 m s−1 (see Supplementary Information), close to the value of 0.97 additional particles, one after the other, aggregate onto a cluster reported24 for small soda lime glass spheres impacting at 0.5–1 m s−1. comprised of N = 5 particles. There are three possible outcomes Grains typically will lose some translational kinetic energy in binary when an incident particle strikes a cluster: the incident particle collisions, but particle rotation can have a significant effect. An is trapped in the mutual electrostatic potential well and after one example is shown in Fig.3 a (see Supplementary Movie 1, Part 4), or more bounces sticks to the cluster, as in Fig.3 b (capture); the where, after two collisions with elliptical trajectories in between, a particle bounces and escapes from the potential well (escape); or the third collision makes one grain suddenly take off on a hyperbolic particle kicks out one or more different particles from the potential trajectory. In this case, the fast-rotating grain was slightly non- well (fragmentation). Figure3 c shows that escape is suppressed for spherical, making it possible to determine its rotational kinetic N =2, and for N >2 the incident particle is always captured unless energy, which during this last collision decreased by ∼10 pJ, an it fragments the cluster. amount that matches the measured increase in total translational We can use fragmentation events in which only a single particle is kinetic energy. Thus, the conversion of rotational into translational kicked out (arrows in Fig.3 c) to estimate the depth of the potential > µ 2/ kinetic energy during impact can lead to eeff 1. well that binds grains to the cluster. The kinetic energy v 2 of

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© 2015 Macmillan Publishers Limited. All rights reserved LETTERS NATURE PHYSICS DOI: 10.1038/NPHYS3396 the incident particle must overcome the short-range cohesive energy a Wcoh plus the initial electrostatic potential energy U stored in a 0.3 bond. In Fig.3 c, the minimum incident velocity for fragmentation is − )

1 −1 e

approximately 0.02 m s and of similar magnitude to the maximum 0.2 velocity vcapture for capture, corresponding to a kinetic energy −6

of approximately 4 pJ. This is three orders of magnitude larger ) (10 μ

q 300 m

−15 ( than the value Wcoh ≤ 10 J found for copper and glass grains P 0.1 of similar size19. Thus U clearly dominates the binding energy. Equating the incident kinetic energy with the Coulomb energy, µ 2/ ( / ε ) / ( )1/2 6 0.0 vcapture 2= 1 4π 0 |q1q2| d, we estimate |q1q2| ≈ 2 × 10 e. −10 0 10 This large amount of net charge per particle confirms the notion q (106 e) that at least some of the grains in a cluster are drawn from the tails b 500 μm of the charge distribution P(q) in Fig.1 b. 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 L1S1 L1S2 L2S1 L2S2 L2S2 L1S3 more grains all in direct contact, such as particles inside the cluster in Fig.3 b or the triangular configuration in Fig.3 e. For particles with high dielectric constant such configurations can be stabilized through the always attractive polarization forces, even L3S1 L2S3 L1S4 L5S1 L3S3 L3S3 if contacting grains have the same polarity, provided there is c 20

significant contrast in charge magnitude25. Seeing such dynamics σ in situ highlights the particularly important role polarization can net ( play in dust aggregation, initiating cluster formation from a single e

0 μ charged particle and potentially leading to runaway growth14. m −2

Going further, we can use the shape of the charge distribution ) to control the aggregation outcomes. In particular, for a bimodal −20 P(q) with positive and negative peaks roughly symmetric around zero, we would expect typical cluster configurations in which d 0 ms 3 ms 18 ms 50 ms 103 ms contacting grains exhibit alternating polarity. Recent experiments ( ) have demonstrated that such a P q can be achieved by mixing 500 μm bidisperse same-material grains22. The P(q) for such a mixture using 32 ms µ µ

diameters 326 10 m and 251 10 m is shown in Fig.4 a. Because ef20 σ tribocharging will transfer charge in a manner that the larger grains net 0 ms 26 ms 33 ms 42 ms54 ms 230 ms (

21,22 e become on average positive and the smaller grains negative , this 0 μ m has the added benefit that polarity can be identified directly from the −2 ) images by particle size. Figure4 b shows a taxonomy of the resulting, 500 μm −20 experimentally observed cluster configurations. These structures resemble molecules or fragments of self-assembled, electrostatically Figure 4 | Granular molecules formed from bimodal particle-charge stabilized lattices13,26. distribution. a, Charge distribution resulting from mixing grains with Modelling these ‘granular molecules’ by taking many-body diameters 32610µm and 25110µm in equal number. The large grains polarization effects into account (see Methods) we find that most of (red solid line) are predominantly (80%) positive whereas the small grains the observed configurations correspond to energetic ground states (blue dashed line) are predominantly (95%) negative. Inset: Optical (Fig.4 c). Some combinations of large (L) and small (S) particles can micrograph of large and small grains. b, Still images of granular molecules. have more than one stable state. As an example, Fig.4 d shows a Their structure consists of alternating large and small grains (labelled LmSn, square L2S2 molecule reconfiguring into a linear shape after being where m+n is the total number of particles in the molecule), giving rise to impacted, and then back to a square (see Supplementary Movie 1, string-like (L1S1,L1S2,L2S1,L2S2), square (L2S2,L3S3), trigonal planar (L1S3, Part 6), suggesting that the square shape is the ground state whereas L3S1), tetrahedral (L1S4) and triangular bipyramid (L5S1) geometries. the linear configuration is metastable (calculation shows that the c, Stable granular molecule configurations corresponding to b, obtained by energy for the square shape is 1 pJ lower). minimizing the total electrostatic energy (including polarization eects) for Because like charges of similar magnitude repel each other, the grains with net charge equalling the average positive (+1.8×106 e) or configurations in Fig.4 b, which are all based on a bimodal P(q), negative (−2.3×106 e) charge of the bimodal P(q) shown in a. Colour tend to be less densely packed than clusters formed from a P(q) represents the net surface (free and bound) charge density σnet. with a single peak around q = 0 (Fig.1 b). This is exemplified by d, Sequence of stills showing conformation change of an L2S2 molecule the star-shaped configurations L3S1 and L1S3, the tetrapod L1S4, after impact from the right (arrow). The square structure breaks open, and the L3S3 structure. However, forces due to polarization become becomes linear, and returns to square. e, Sequence showing the formation important at close approach and, as mentioned before, for large of a triangular molecule from two large positively charged grains and one differences in charge magnitude these forces can pull same-polarity small negatively charged grain. The closing of the large, visible gap grains into contact. Figure4 e demonstrates this with an event (see (t=42 ms) between the large particles cannot be explained by short-range Supplementary Movie 1, Part 7) in which a dimer captures another sticking and directly implies attractive forces from polarization eects. particle to form a stable triangular configuration (Fig.4 f). f, One of many possible charge assignments that can produce stable These results have implications for a wide range of situations triangular molecules from two grains with similar charge magnitude but where collision-induced particle aggregation is important. In opposite polarity and one grain of lesser net charge. In this example the particular, our results show at the single particle level how long- charges are +1.8×106 e and +0.2×106 e for the large grains, and range electrostatic forces can capture grains whose kinetic energy −2.3×106 e for the small grain.

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is orders of magnitude larger than Wcoh. Multiple bounces enabled 14. Ivlev, A. V., Morfill, G. E. & Konopka, U. Coagulation of charged microparticles by the electrostatic potential well very effectively dissipate kinetic in neutral gas and charge-induced gel transitions. Phys. Rev. Lett. 89, energy, an effect that is further enhanced in N > 2 clusters by 195502 (2002). internal reconfigurations, all of which increases the likelihood of 15. Barros, K. & Luijten, E. Dielectric effects in the self-assembly of binary colloidal aggregates. Phys. Rev. Lett. 113, 017801 (2014). capture and aggregation. Already a very small size dispersion, 16. Freed, K. F. Perturbative many-body expansion for electrostatic energy and such as in our nearly monodisperse sample, suffices to generate field for system of polarizable charged spherical ions in a dielectric medium. highly charged particles, an effect likely to become amplified for J. Chem. Phys. 141, 034115 (2014). larger dispersions. 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Competing financial interests Nature 437, 235–240 (2005). The authors declare no competing financial interests.

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Methods and the model trajectory data was used for the goodness of fit. 500 search trials As particles we used fused zirconium dioxide–silicate grains (68% ZrO2, 32% SiO2 were performed and the 10% best-fit trials were selected to extract q1 and q2 for ρ −3 > > by volume, material density =3,800 kg m ; Glenn Mills), sieved to average each trajectory. In this charge determination, we assumed q1 0 and |q1| |q2| (see particle diameters d =27414µm for the monodisperse sample, and 32610µm Supplementary Information). and 25110µm for the binary sample (mixed in equal numbers by fluidizing with For modelling granular molecules with N >2 we used a technique recently dry air for 30 min). Before the experiments, the grains were stored at 40–50% introduced by Barros et al.32 to treat the electrostatics of many-body polarization relative humidity. The grains were placed into a hopper mounted inside a eﬀects, meshing each grain into 720 polarizable patches. The free charges for transparent, cylindrical vacuum chamber that then was evacuated to <2 mtorr by a large (d =326µm) and small (d =251µm) grains were set to the averages for the turbo pump to eliminate air drag on the falling grains. The grains flowed freely bimodal P(q) in Fig.4 a, that is, to +1.8×106 e and −2.3×106 e, respectively 6 from an orifice at the bottom of the hopper. The orifice size was chosen small (the one exception was the L1S4 structure, which required charges 1.8×10 e enough (<15d) so that the grains formed a highly dilute stream. Video was taken for a stable four-armed ‘tetrapod’). The free charge on each particle was by a Phantom v9.1 high-speed video camera at 1,000 frames per second, mounted distributed uniformly over its surface. The additional surface bound charge due to on a carriage that could slide freely along low-friction rails. This apparatus could polarization was then calculated iteratively until the electrostatic energy Un at ( ) ( )/ < −4 also be used for P q measurements, where the grains fall freely via gravity between iteration n changed by less than | Un −Un−1 Un−1| 10 . A Nelder–Mead two large vertical copper plates held at an external horizontal electric field E, algorithm was used to identify, for given numbers of large and small grains, those causing a grain of charge q and mass m to undergo a horizontal acceleration spatial arrangements that produce local minima in the electrostatic energy, and a=qE/m. To extract the accelerations a, we tracked the horizontal trajectories and thus are candidates for stable states. As a hard-sphere constraint in the energy ( ) ( )/ < fitted with parabolas. The field was applied only to measure P q ; all trajectories minimization the condition of dA +dB 2 rAB was used, where dA and dB are the and clusters shown in Figs1 –4 were obtained without an applied external electric diameters for any two particles A and B, and rAB is the centre-to-centre distance field. See ref. 20 for more details about the apparatus. Note that the absence of an between them. external electric field will also ensure that induction-based charging mechanisms10,27,28 are not likely to be significant (see Supplementary Information). References Particles in the raw videos were identified and tracked with the algorithm 27. Siu, T., Cotton, J., Mattson, G. & Shinbrot, T. Self-sustaining charging of developed by Crocker and Grier29. The method by Nakajima and Sato23 was used to identical colliding particles. Phys. Rev. E 89, 052208 (2014). calculate the electrostatic forces between two charged dielectric particles, including 28. Zhang, Y. Z. et al. Electric field and humidity trigger contact electrification. the full set of higher-order polarization contributions. The net (free) charge on a Phys. Rev. X 5, 011002 (2015). grain was assumed uniformly distributed over the surface. The dielectric constant 29. Crocker, J. C. & Grier, D. G. Methods of digital video microscopy for colloidal of the grain material was taken as ε ≈15, based on the volume-weighted average of studies. J. Colloid Interface Sci. 179, 298–310 (1996). ε ≈ ε ≈ ZrO2 22 and SiO2 3.9. To simulate a trajectory, a leapfrog algorithm was used for 30. Hockney, R. W. & Eastwood, J. W. Computer Simulation Using Particles (CRC integrating the equation of motion30. As we imaged in the x–y plane and a priori Press, 2010). did not have information about the trajectory inclination with respect to the z-axis, 31. Lagarias, J. C., Reeds, J. A., Wright, M. H. & Wright, P. E. Convergence we employed a Nelder–Mead simplex algorithm31 as a multidimensional properties of the Nelder–Mead simplex method in low dimensions. SIAM J. optimization method to search for local minima of the absolute median deviation Optim. 9, 112–147 (1998). between simulated trajectories and the experimental data (see Supplementary 32. Barros, K., Sinkovits, D. & Luijten, E. Eﬃcient and accurate simulation of Information). A reduced chi-squared statistic between the observed trajectory data dynamic dielectric objects. J. Chem. Phys. 140, 064903 (2014).

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