Direct observation of particle interactions and clustering in charged granular streams

Transcription

1 Direct observation of particle interactions and clustering in charged granular streams Victor Lee, Scott R. Waitukaitis, Marc Z. Miskin & Heinrich M. Jaeger James Franck Institute and Department of Physics, The University of Chicago, Chicago, IL 667 Supplementary Information I Air drag At low gas pressure, the drag force F d on a particle with diameter d moving at speed v can be described by F d = F s ( + K(a + be c/k )), () where F s =πηdv is the Stokes drag force, η =.8 5 Pa s is the viscosity of air at ambient pressure kpa, a =.864, b =.9, c =.5 are empirical constants [], and K =λ/d is the Knudsen number, which is the ratio of the mean free path of the gas molecules λ to the particle radius d/. For our experiments we have d µm and at a pressure of mtorr inside the vacuum chamber the mean free path is λ.6 cm. This gives K 7, which already indicates that air drag is negligible. The maximum vertical speed of the free-falling particles is v max = gh 5.5 m/s, leading to a maximum F d nn, much smaller than the electrostatic force on the order of -5 nn and the particle weight mg 4 nn. Thus, air drag can be neglected in our experiments. II Trajectory fitting and particle charge determination To investigate trajectories between two particles, we only considered pairs of particles at least 8 µm away from surrounding grains in order to minimize multibody effects. In addition, the relative acceleration (in the x-y plane) between the surrounding grains and the center of mass of the two investigated particles was less than % of the relative acceleration between the two investigated particles. From the raw videos, interacting particles were tracked and the positions of their centers obtained. Each particle corresponded to at least pixels in diameter and the uncertainty in determining its center was. pixels. From these data, the relative distance between the particles was calculated. Fixing one of the particles at the NATURE PHYSICS

2 origin then produced the trajectories shown in the figures. Since these experimentally observed trajectories correspond to projections of the actual trajectories into the x-y plane imaged by the camera, a fitting procedure was used to obtain the unknown parameters such as the inclination along the z-axis and the charges on the particles. This fitting procedure consisted of two parts: (a) generating model trajectories by using the electrostatic interactions including polarization contributions, and (b) optimizing the input parameters for these model trajectories in order to obtain the best fit and thereby the best estimate for the particle charges. To generate model trajectories a leapfrog integration was employed that numerically integrated the equation of motion. The integration was performed with a time-step size of 5 s. To insure that this time step was sufficiently small, we performed test integrations also using a time step of 7 s. The difference between these two time steps was less than. pixels in the position of an endpoint of a typical trajectory segment lasting about 5 ms between bounces, well within the. pixel uncertainty of our experimental position measurements. A Nelder-Mead simplex algorithm was used for multidimensional optimization of the model trajectories to get them as close as possible to the observed trajectories. Five hundred fitting trials were performed for each trajectory segment. For each fitting trial, random initial guesses of the fitting parameters (the charges q and q on particle and, the initial relative positions in the z direction, and the initial relative velocities) led, via the algorithm, to a local minimum of the absolute median deviation between the model and the experimental trajectory data. We used the reduced chi-square χ re to measure the goodness of fit: χ re = r data,i r model,i ν σ, () i= where ν is the number of degrees of freedom (the number of data points minus the number of fitted parameters minus one), r data,i and r model,i are the ith observed and model trajectory data point in x-y plane, and σ is the estimated variance of position measurement. In Fig. Sa we plot the χ re versus q for trajectory segment in Figs. d&e in the main text as an example. Note it is possible for two particles with the same polarity (q q > ) to attract each other with polarization forces as long as there is sufficiently large difference in charge magnitude. Since the expression for the electrostatic force, including all polarization terms, is symmetric with respect to q and q when the particles have the same size [], we are not able to unambiguously assign charges q and q to particle and in every fitting trial without further assumptions. Furthermore, the force will not change when the polarities of q or q are switched. To extract and assign charge values, we therefore set q > and assumed q > q, which is justified as long as there is no large amount of charge (on the order of the mean grain charge value, i.e., millions of e) transferred during a single collision. As noted in the main text, this NATURE PHYSICS

3 assumption is reasonable given the published experimental results from collisions with fixed targets. The values q, q, q, and q shown in Fig. b in the main text were determined by the average and the standard deviation of those q and q data that corresponded to the % smallest χ re values (Fig. Sb). Note that even with the largest χ re value within this % range (.8 in Fig. Sb), the observed trajectory can be fitted well with the model trajectory (Fig. Sc). The standard deviation of residuals ( r data,i r model,i ) from the fit is of the order of. pixels (Fig. Sd). (a) (b) χ re χ re q ( 6 e) q ( 6 e) (c) (d). model residuals of r (pixel) t (frames) FIG. S: Charge determination from trajectory fitting. (a) Reduced chi-square χ re vs charges on the two grains q (open red diamonds) and q (solid blue circles). (b) Zoomed-in data from (a) gives the % best trials that are used for q and q determination. (c) Comparison between experimentally observed trajectory (blue circles) and the model (red diamonds) for the circled datum in (b). (d) Residuals from fitting the trajectory to the model in (c), which gives a standard deviation of. pixels. NATURE PHYSICS

4 (a) (b). model without polarization model with polarization model without polarization model with polarization residuals of r (pixel) t (frames) FIG. S: (a) Comparison between experimentally observed trajectory (blue circles) and the best fits with polarization contributions (red diamonds, χ re =.), and without polarization contributions (yellow circles, χ re =.8). (b) Residuals from fitting the trajectory to the models in (a). We also simulated the trajectories with electrostatic forces without considering polarization effect (Fig. Sa). Over the limited range in particle separation available and given the uncertainties of our, both types of fit seem similarly valid in terms of the fitting residuals and the best χ re values (e.g., Fig. Sb). However, the charge product q q based on forces without polarization terms is larger than the one based on full polarization forces by about 4%. If polarization terms are not included, there is only a term containing q q in the force, and there would be more freedom in picking widely different q and q. With polarization terms, the fitting constrains the range of possible charge magnitudes. Figure S presents the observed trajectories from other single-particle collision events (Supplementary Movie, Part 4; Supplementary Movie, Parts -) and compares them to the best-fit model with polarization forces. As the trajectories are all curved toward the origin, there is no obvious transition from attractive to repulsive trajectories during collisions. Some parts of trajectories during collisions are not shown in the figure because of difficulties to extract them with the particletracking procedure [] when two particles touch or when one grain blocks another one within the field of view. The insets to Fig. S show that the corresponding q and q values in each case stay, within error bars, at their initial values. 4 NATURE PHYSICS

5 (a) 5 q ( 6 e) (b) model origin q ( 6 e) trajectory segment 5 4. trajectory segment 5 model origin (c) 5 model origin 5 q ( 6 e) trajectory segment (d) trajectory segment q ( 6 e) 5 5 model origin FIG. S: (a)-(d) Comparison between experimentally observed trajectories (blue circles) and model trajectories (red diamonds). The trajectories correspond to Supplementary Movie, Part 4 (panel a), and Supplementary Movie, Parts - (panels b-d). Insets: Charges on the two grains q (open red diamonds) and q (solid blue circles) for each model trajectory segment as indicated by the numbers. The length of the error bars corresponds to one standard deviation above and below the average. III Nearly head-on collision Figure S4 shows an essentially head-on collision that allowed us to measure the effective coefficient of restitution e eff (the ratio of relative velocity magnitude before and after a collision) of grains. Before the collision, the target particle (Particle ) moves slowly relative to the other nearby particles, and the incoming particle (Particle ) is approaching at high impact velocity (.4 m/s in the x-y plane). After the collision, Particle shoots off at high velocity while Particle moves slowly, similar to the behavior due to momentum transfer in a -particle Newton s cradle. The incoming and outgoing trajectories are approximately parallel (no more 5 NATURE PHYSICS 5

6 time FIG. S4: Successive image stills showing a nearly head-on collision. The time interval between two frames is ms. Fast (higher than.5 m/s) particles are elongated as lines in the stills. than 5 off) in the x-y plane, and we measure e eff to be about.94. Out-of-plane (z-direction) motion could influence this determination of e eff. The -mm depth of field of the lens (Nikon AF Micro-Nikkor ED mm f/4 D IF with f-number and focused distance.5 m) allows us to roughly estimate the z- direction velocities v z of the slow particles. From the sharpness of the edge of the particles, we find that the out-of-plane velocity v z of Particle before the collision and Particle after the collision cannot be larger than. m/s, about orders of magnitude less than the x-y components of the impacting and leaving velocities. Therefore, the z-motion is too small to affect the above value for e eff. IV Potential influences of humidity Humidity can influence tribocharging in many aspects, with potentially different outcomes depending on the mechanism. First, by increasing the surface conductivity of dielectric particles humidity can allow charge to leak away [4]. Second, humidity can affect how much charge is being transferred between insulating surfaces during contact. In particular, the amount of contact charging can be larger at higher humidity than in dry environments [5,6]. Third, the gas vapor pressure in the environment surrounding the contact area will affect the electric breakdown strength, which will limit the amount of charge buildup that is possible. In addition, recent work [7] has shown that humidity can significantly affect the degree to which charging by induction is able to build up charge on colliding dielectric particles [8,9]. Thus, depending on the surface properties of the particles, the details of the sample preparation process, and the particular environmental conditions during an experiment one might obtain different charge distributions P (q) and, in general, humidity is likely to be a significant factor. In the experiments reported here, 6 NATURE PHYSICS

7 the focus was on describing particle interactions and clustering for given charge distributions P (q), and we strived to keep the experimental conditions as constant as possible. Specifically, only particles from the same fabrication batch were used during a given experimental run, particles were stored in a humidity controlled environment before each experiment, and the actual experiments took place inside a turbo-pumped vacuum system with pressure < mtorr, which can be viewed as a very low-humidity environment. We believe that under these conditions net charge on dielectric ZrO -SiO particles is built up by repeated collisions and/or rubbing contacts among particles before they enter the freely falling stream, and we established in prior experiments [] that the charge carriers are not electrons but likely ions from molecularly thin layers of moisture adsorbed on the particle surfaces. The tribocharging in our experiments took place in the absence of any externally applied electric field. Therefore, induction charging, irrespective of how it is affected by humidity, is unlikely to play a significant role in setting P (q) in the experiments described here. The field was turned on only in separate experiments to extract P (q) under vacuum conditions, and in those experiments it was applied only in the region where the dilute stream underwent free fall after leaving the hopper, i.e., after contact tribocharging had taken place; furthermore, any trajectories undergoing collisions inside the dilute stream were removed from the analysis leading to P (q) []. Supplementary References [] Millikan, R. A. The general law of fall of a small spherical body through a gas, and its bearing upon the nature of molecular reflection from surfaces. Phys. Rev., - (9). [] Nakajima, Y. & Sato, T. Calculation of electrostatic force between two charged dielectric spheres by the re-expansion method. J. Electrostat. 45, -6 (999). [] Crocker, J. C. & Grier, D. G. Methods of digital video microscopy for colloidal studies. J. Colloid Interface Sci. 79, 98- (996). [4] W. R. Harper. Contact and Frictional Electrification. (Laplacian Press Morgan Hill, CA, 998). [5] Wiles, J. A., Fialkowski, M., Radowski, M. R., Whitesides, G. M. & Grzybowski, B. A. Effects of surface modification and moisture on the rates of charge transfer between metals and organic materials. J. Phys. Chem. B 8, 96- (4). [6] Pence, S., Novotny, V. J. & Diaz, A. F. Effect of surface moisture on contact charge of polymers containing Ions. Langmuir, (994). NATURE PHYSICS 7

8 [7] Zhang, Y. Z. et al. Electric field and humidity trigger contact electrification. Phys. Rev. X 5, (5). [8] Siu, T., Cotton, J., Mattson, G. & Shinbrot, T. Self-sustaining charging of identical colliding particles. Phys. Rev. E 89, 58 (4). [9] Pähtz, T., Herrmann, H. J. & Shinbrot, T. Why do particle clouds generate electric charges? Nature Phys. 6, (). [] Waitukaitis, S. R., Lee, V., Pierson, J. M., Forman, S. L. & Jaeger, H. M. Sizedependent same-material tribocharging in insulating grains. Phys. Rev. Lett., 8 (4). [] Waitukaitis, S. R. & Jaeger, H. M. In situ granular charge measurement by free-fall videography. Rev. Sci. Instrum. 84, 54 (). 8 NATURE PHYSICS