^{1}and H. M. Jaeger

^{1}

### Abstract

We present the design and performance characterization of a new experimental technique for measuring individual particle charges in large ensembles of macroscopic grains. The measurement principle is qualitatively similar to that used in determining the elementary charge by Millikan in that it follows individual particle trajectories. However, by taking advantage of new technology we are able to work with macroscopic grains and achieve several orders of magnitude better resolution in charge to mass ratios. By observing freely falling grains accelerated in a horizontal electric field with a co-falling, high-speed video camera, we dramatically increase particle tracking time and measurement precision. Keeping the granular medium under vacuum, we eliminate air drag, leaving the electrostatic force as the primary source of particle accelerations in the co-moving frame. Because the technique is based on direct imaging, we can distinguish between different particle types during the experiment, opening up the possibility of studying charge transfer processes between different particle species. For the ∼300 μm diameter grains reported here, we achieve an average acceleration resolution of ∼0.008 m/s^{2}, a force resolution of ∼500 pN, and a median charge resolution ∼6× 10^{4} elementary charges per grain (corresponding to surface charge densities ∼1 elementary charges per μm^{2}). The primary source of error is indeterminacy in the grain mass, but with higher resolution cameras and better optics this can be further improved. The high degree of resolution and the ability to visually identify particles of different species or sizes with direct imaging make this a powerful new tool to characterize charging processes in granular media.

The authors gratefully acknowledge the contributions to the project made by Gustavo Castillo, Estefania Vidal, Suomi Ponce Heredia, and Alison Koser. We benefited a great deal from discussions with Dan Lacks, Troy Shinbrot, Ray Cocco, and Ted Knowlton. We thank Mrs. Joan Winstein for her recognition and support. This work was supported financially by the National Science Foundation (NSF) through its Materials Research Science and Engineering Center (MRSEC) program (DMR-0820054) and by the US Army Research Office through Grant No. W911NF-12-1-0182. S.R.W. acknowledges support from a University of Chicago Millikan Fellowship.

I. INTRODUCTION

II. EXPERIMENTAL SETUP

III. DATA AND ANALYSIS

A. Particle identification and tracking

B. Trajectory filtering

C. Relating accelerations to charges

IV. CONCLUSIONS

### Key Topics

- Electric measurements
- 13.0
- Cameras
- 11.0
- Acceleration measurement
- 7.0
- Faraday cups
- 6.0
- Charge transfer
- 5.0

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##### H04N5/30

## Figures

The experimental setup. Grains freely fall inside a vacuum chamber from the hopper/nozzle through a region between two parallel copper plates held at a potential difference *V* ( points into the page). A camera connected to a carriage-rail system falls with the grains while simultaneously recording video. A vertical reference string hangs in front of the chamber to correct for camera yaw and roll.

The experimental setup. Grains freely fall inside a vacuum chamber from the hopper/nozzle through a region between two parallel copper plates held at a potential difference *V* ( points into the page). A camera connected to a carriage-rail system falls with the grains while simultaneously recording video. A vertical reference string hangs in front of the chamber to correct for camera yaw and roll.

Particle tracking. (a) Color-inverted still from experiment. Circled particles are positive particle identifications, while the few remaining particles are rejected on basis of size, shape, proximity to the string, or being far out of focus. The fuzzy vertical black strip is the string, and the thin white line is its calculated center position. (b) Example uncorrected particle position (black line), string position (dotted black line), and corrected particle position (red points) vs. time (particle and string are measured relative to their initial positions). The green data are the residuals from fitting the corrected trajectory to a parabola, which have a rms deviation of ∼0.5 pixels. (c) Sampling of the corrected horizontal deflection (i.e., *x* − *v* _{0} *t* − *x* _{0}) for a few particles of different charges at *V* = 2250 V ( V/m).

Particle tracking. (a) Color-inverted still from experiment. Circled particles are positive particle identifications, while the few remaining particles are rejected on basis of size, shape, proximity to the string, or being far out of focus. The fuzzy vertical black strip is the string, and the thin white line is its calculated center position. (b) Example uncorrected particle position (black line), string position (dotted black line), and corrected particle position (red points) vs. time (particle and string are measured relative to their initial positions). The green data are the residuals from fitting the corrected trajectory to a parabola, which have a rms deviation of ∼0.5 pixels. (c) Sampling of the corrected horizontal deflection (i.e., *x* − *v* _{0} *t* − *x* _{0}) for a few particles of different charges at *V* = 2250 V ( V/m).

Filtering procedure to reject colliding and crossing particles. (a) Example particle trajectory (*x* − *x* _{0} vs. *t*) of a particle involved in a collision/crossing. The event leads to a perceived acceleration of ∼0.5 m/s^{2}, although it is clear from either leg of the trajectory that the actual acceleration is much smaller than this. (b) Errors on acceleration fits σ_{ a } vs. particle lifetime τ. Most measurements (blue data) fall into a band σ_{ a } ∝ τ^{−2}, while the colliding/crossing particles lie above this trend. These tracks (red data) are rejected by binning along τ, finding the average σ_{ a } in each bin by fitting to a Gaussian, and then rejecting particles whose σ_{ a } lies three sigma above their bin mean.

Filtering procedure to reject colliding and crossing particles. (a) Example particle trajectory (*x* − *x* _{0} vs. *t*) of a particle involved in a collision/crossing. The event leads to a perceived acceleration of ∼0.5 m/s^{2}, although it is clear from either leg of the trajectory that the actual acceleration is much smaller than this. (b) Errors on acceleration fits σ_{ a } vs. particle lifetime τ. Most measurements (blue data) fall into a band σ_{ a } ∝ τ^{−2}, while the colliding/crossing particles lie above this trend. These tracks (red data) are rejected by binning along τ, finding the average σ_{ a } in each bin by fitting to a Gaussian, and then rejecting particles whose σ_{ a } lies three sigma above their bin mean.

Relating acceleration distributions to charge measurements. (a) Particle acceleration distributions *P*(*a*) (weighted by ) for voltages *V* =0, 1500, and 4000 V ( = 0, 30 000, and 79 000 V/m, respectively). (b) Change in distribution width Δ_{ a } (red squares), mean (blue circles), and mean acceleration error (green dots) vs. *V*. Calculations for Δ_{ a } and are also weighted by , and the error bars are the standard deviations as calculated from *N* = 1000 bootstrap resamples of the original measurement population. Dashed line fits for and Δ_{ a } correspond to Eqs. (1) and (2) , respectively. (c) Particle size distribution *P*(*d*) as calculated from imaging particles with optical microscope. Inset: microscope image (color inverted). Using these data to calculate the mean diameter and particle mass , we can use Eqs. (1) and (2) to calculate the mean charge = −(6 ± 1) × 10^{4} *e* and width Δ_{ q } = (4.5 ± 0.4) × 10^{5} *e* of the particle charge distribution.

Relating acceleration distributions to charge measurements. (a) Particle acceleration distributions *P*(*a*) (weighted by ) for voltages *V* =0, 1500, and 4000 V ( = 0, 30 000, and 79 000 V/m, respectively). (b) Change in distribution width Δ_{ a } (red squares), mean (blue circles), and mean acceleration error (green dots) vs. *V*. Calculations for Δ_{ a } and are also weighted by , and the error bars are the standard deviations as calculated from *N* = 1000 bootstrap resamples of the original measurement population. Dashed line fits for and Δ_{ a } correspond to Eqs. (1) and (2) , respectively. (c) Particle size distribution *P*(*d*) as calculated from imaging particles with optical microscope. Inset: microscope image (color inverted). Using these data to calculate the mean diameter and particle mass , we can use Eqs. (1) and (2) to calculate the mean charge = −(6 ± 1) × 10^{4} *e* and width Δ_{ q } = (4.5 ± 0.4) × 10^{5} *e* of the particle charge distribution.

Independent measure of average charge per particle with Faraday cup. Plot of total charge in cup *Q* vs. time *t*. Data are smoothed to remove 60 Hz noise (∼10 nC peak to peak). We calculate the charge flow rate *dQ*/*dt* and mass flow rate *dM*/*dt* from this graph.

Independent measure of average charge per particle with Faraday cup. Plot of total charge in cup *Q* vs. time *t*. Data are smoothed to remove 60 Hz noise (∼10 nC peak to peak). We calculate the charge flow rate *dQ*/*dt* and mass flow rate *dM*/*dt* from this graph.

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