ABSTRACT
Title of dissertation: MODELING AND EXPERIMENTAL
MEASUREMENT OF TRIBOELECTRIC
CHARGING IN DIELECTRIC
GRANULAR MIXTURES
Dylan Carter
Doctor of Philosophy, 2020
Dissertation directed by: Professor Christine Hartzell
Department of Aerospace Engineering
Triboelectric charging, the phenomenon by which electrical charge is exchanged
during contact between two surfaces, has been known to cause significant charge
separation in granular mixtures, even between chemically identical grains. This
charging is a stochastic process resulting from random collisions between grains,
but creates clear charge segregation according to size in dielectric granular mix-
tures. Experiments in grain charging are frequently conducted with methods that
may introduce additional charging mechanisms that would not be present in air-
less environments, and often aren?t capable of measuring the precise charge of each
grain. We resolved these issues through the development of a model that predicts
the mean charge on grains of a particular size in an arbitrary mixture, and through
experiments that do offer controlled measurement of precise grain charges. These
results can be used to develop methods for electrostatic sorting to enable in situ
resource utilization of silica-based regoliths on airless extraterrestrial bodies.
Beginning from a basic collision model for a mixture of hard spheres, we de-
veloped a robust semi-analytical model for making predictions about the charge
distribution in a dielectric granular mixture. This model takes a set of assumptions
about a mixture, including the continuous size distribution, collision frequencies,
and charge transfered per collision, and calculates the mean charge acquired by
grains of each size after all charges have been exchanged. This model allows us to
explore experimental results through many different lenses. To test our predictions
and provide a repeatable and flexible method for analyzing charging in a variety of
granular mixtures, we designed and built our own experimental test stand. This
device is housed entirely in a vacuum chamber, allowing us to induce tribocharging
in dielectric grains in a controlled airless environment and measure individual charge
and diameter of a grain by dropping samples through a transverse electric field.
We observed that mixtures of zirconia-silica grains containing two primary
size fractions exhibited size-dependent charge segregation when charged in vacuum.
Unlike in other experiments with grains charged by fluidization with a gas, we con-
sistently observed that the small grains charged predominantly positive, while the
large grains were primarily negative. We considered a variety of charge transfer
mechanisms and generated predicted charge distributions for each using the mod-
eling framework we developed. Comparing these models to the collected data, we
are able to assess the viability of each potential transfer mechanism by examining
properties of its resulting distribution, including the relative charge magnitudes for
each size fraction, the point at which the polarity changes, and the polarity and
magnitude of the charge carrier density.
The results of this work provide solid supporting evidence for the role of posi-
tive charge carriers in dielectric tribocharging. While some prior work has suggested
positive ions from the atmosphere and/or adsorbed water are responsible, we have
observed that even when these environmental factors are reduced or eliminated,
silica-based materials still exhibit positive charge transfer. The modeling frame-
work developed in search of a descriptive model for this effect is a useful, adaptable
tool. The experimental apparatus itself, and especially in conjunction with these
modeling tools, overcomes some of the more difficult challenges faced by experi-
mentalists investigating granular tribocharging, enabling further investigation into
tribocharging in regolith and other dielectric materials.
MODELING AND EXPERIMENTAL MEASUREMENT
OF TRIBOELECTRIC CHARGING IN
DIELECTRIC GRANULAR MIXTURES
by
Dylan Carter
Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
2020
Advisory Committee:
Professor Christine Hartzell, Chair/Advisor
Professor David Akin
Professor Raymond Sedwick
Professor Daniel Lathrop
Professor Derek Richardson
?c Copyright by
Dylan Carter
2020
Acknowledgments
This research would not have been possible without the endless support of my
advisor, Dr. Christine Hartzell. She kept me on track when I ran away with an idea,
pushed me when I was holding back from one, and found the path when I didn?t
know where to go at all. I couldn?t have asked for a better advisor to have my back.
I thank my committee for their guidance and expertise throughout this process:
Dr. Dave Akin, Dr. Ray Sedwick, Dr. Derek Richardson, and Dr. Dan Lathrop.
I also owe thanks to Dr. Elaine Oran, previously a member of my committee.
Whether as my teacher or a professional reference, you have all taught me so much.
To my parents, Bill and Mena, who always knew I would end up here: thank
you for giving me every advantage you could to help me reach my potential.
To my fiancee? Molly Evans: there are no words that can sufficiently thank you
for putting up with the long nights and longer days. I?m sure if you could leave me
for a horse, you would, but here we are.
This work was supported by a NASA Space Technology Research Fellowship
(NNX15AP69H). In addition to their generous financial support, I am grateful for
the mentorship of my research collaborator, Dr. Carlos Calle, and his team at
the Electrostatics and Surface Laboratory (ESPL) at the Kennedy Space Center
SwampWorks facility. Special thanks to Jose Nun?ez, Jay Philips, Paul Mackey,
Mike Hogue, and Michael Johansen for their advice and experience, and for making
me feel at home. And of course, I couldn?t forget Jerry Wang and Joel Malissa for
welcoming me as an honorary intern during my summer visits.
ii
Back at the GRAINS lab in Maryland, I couldn?t have made it through the
years of research without my labmates, Thomas Leps and Dr. Anthony DeCicco.
From Monday trivia to Formal Fridays, black T-shirts to three-piece suits, your
friendship, not to mention design input, on this shared journey has made this all
possible.
iii
Table of Contents
Acknowledgements ii
Table of Contents iv
List of Figures vi
1 Introduction 1
1.1 Triboelectric Charging in Granular Media . . . . . . . . . . . . . . . 1
2 Literature Review 6
2.1 Tribocharging in Insulators . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Models for Charge Transfer . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Experimentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3 Modeling Charge Transfer in Continuous Mixtures 16
3.1 Model Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.1.1 Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.1.2 Size Distribution . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.1.3 Electron Transfer Rates . . . . . . . . . . . . . . . . . . . . . 21
3.1.3.1 Electron Loss Fraction . . . . . . . . . . . . . . . . . 22
3.1.3.2 Collision Area . . . . . . . . . . . . . . . . . . . . . . 23
3.2 Steady-State Solutions to the Charge Distribution Function . . . . . . 26
3.2.1 Solutions for Continuous Size Distributions . . . . . . . . . . . 26
3.2.2 Solutions for Discrete Size Distributions . . . . . . . . . . . . 28
3.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.3.1 Comparison of Continuous and Discrete Models . . . . . . . . 32
3.3.2 Polarity Reversal . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.3.2.1 Bidisperse Mixtures . . . . . . . . . . . . . . . . . . 34
3.3.2.2 Continuous Mixtures . . . . . . . . . . . . . . . . . . 37
3.3.2.3 Compared to Previous Experiments . . . . . . . . . . 38
3.3.3 Predicted Charge Magnitude as Compared to Experiments . . 42
3.4 Notes on this Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
iv
4 Experimentation 44
4.1 Existing Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.2 Experimental Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.2.1 Physical Setup . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.2.2 Experiment Procedure . . . . . . . . . . . . . . . . . . . . . . 48
4.2.2.1 Vacuum Chamber . . . . . . . . . . . . . . . . . . . 49
4.2.2.2 Grain Shaker . . . . . . . . . . . . . . . . . . . . . . 51
4.2.2.3 Sample Release and Measurement . . . . . . . . . . . 55
4.2.2.4 Carriage Return . . . . . . . . . . . . . . . . . . . . 60
4.3 Data Processing and Analysis . . . . . . . . . . . . . . . . . . . . . . 61
4.3.1 Image Filtering and Grain Identification . . . . . . . . . . . . 62
4.3.2 Trajectory Determination . . . . . . . . . . . . . . . . . . . . 65
4.3.3 Size Determination . . . . . . . . . . . . . . . . . . . . . . . . 67
4.3.3.1 Sizing Procedure . . . . . . . . . . . . . . . . . . . . 69
4.3.4 Calculation of Grain Charges . . . . . . . . . . . . . . . . . . 71
5 Experimental Results 75
5.1 Charge Exchange Models . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.1.1 Inelastic Deformation . . . . . . . . . . . . . . . . . . . . . . . 76
5.1.2 Constant Transfer . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.1.3 Combination of Size and Material Differences . . . . . . . . . 77
5.1.4 Exponential Decay . . . . . . . . . . . . . . . . . . . . . . . . 78
5.1.5 Total Surface Area . . . . . . . . . . . . . . . . . . . . . . . . 79
5.1.6 Collision Models . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.1.6.1 Constant Velocity . . . . . . . . . . . . . . . . . . . 80
5.1.6.2 Constant Energy . . . . . . . . . . . . . . . . . . . . 80
5.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.3 Model Fits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
6 Conclusions and Next Steps 91
6.1 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6.2 Next Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
v
List of Figures
3.1 Diagram of physical system considered. Grains with a normalized
size distribution given by g(R) are mixed in a container of volume
Vc. Mixing is performed by moving the container with average speed
vr, such that the grains inside also move with average speed vr. The
grains are then assumed to move in random directions with average
speed vr as well. When a grain of radius Ri collides with a grain of
radius Rj, some contact area is formed in region Aij. High-energy
electrons in this region are capable of transferring between the two
grains and settling into stable low-energy states. . . . . . . . . . . . . 19
3.2 Non-dimensional charge distribution Q? = Q(R)/4e??0R
2
1 for a spec-
ified size distribution function. . . . . . . . . . . . . . . . . . . . . . . 34
3.3 Effect of variation in peak width of size distribution on charge po-
larity, for non-dimensional charge Q? = Q(R)/4e??0R
2
1. (a) Normal-
ized particle size distribution functions g(R) in the form of Equation
(3.27), with k2/k1 = 8, R1 = 100?m, and R2 = 50?m. (b) Non-
dimensional charge distribution functions corresponding to the size
distributions in (a). Vertical lines at R = R1 and R = R2 provided
as reference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.4 Non-dimensional net charge Q?1 = Q1(R)/4e?? R
2
0 1 on grains of radius
R1 = 100?m for a size distribution of the form of Equation (3.27),
with a = 0.005?m. As the ratio of the peak heights (k) increases, the
size ratio (d = R2/R1) below which large grains charge negatively
decreases. Note that the charge predicted by the continuous distri-
bution is very similar to that predicted by the discrete model, with
some variation due to the effect of the nonzero peak width (as seen
in Figure 3.3b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.1 Experimentail setup. The device is designed to be placed in a vacuum
chamber and operated at high vacuum. Grains in the canister are
tribocharged via shaking and dropped through a region containing
a transverse electric field. The camera is dropped and records the
trajectories of the grains while both are in freefall. . . . . . . . . . . . 48
4.2 Vacuum chamber configuration. . . . . . . . . . . . . . . . . . . . . . 50
vi
4.3 The grain shaker and sample release subsystem of our experiment.
The central steel cylinder contains the granular sample and is at-
tached to the vertically-mounted speaker. When the stepper motor
at right turns, the lid is rotated, exposing the hole in the lid that is
not blocked and letting a 5 mm stream of grains fall. . . . . . . . . . 52
4.4 The canister in which grains are charged in our experiment. The
interior of the canister has been coated with grains of the same mate-
rial as those to be charged, so that no grain-to-wall charging occurs.
When the lid is rotated during sample collection, grains fall through
the 5 mm hole in the lid at bottom left. . . . . . . . . . . . . . . . . . 53
4.5 Shaker subsystem controller, external to the vacuum chamber. . . . . 54
4.6 View of the parallel-plate electrodes, as seen from the camera (bot-
tom). The background behind the electrodes is painted with an ultra-
flat black paint to minimize backscattered light. Note the alignment
string visible faintly between the electrodes. . . . . . . . . . . . . . . 55
4.7 The camera carriage at the top of the experimental apparatus. The
corner bracket (shown fixed to the electropermanent magnet) is at-
tached to the carriage and is released from the magnet when collecting
data, causing the carriage to fall along the low-friction guide rail. . . 56
4.8 Virtual instrument (VI) for automating data collection. The user
inputs the desired frame rate (default 1000 Hz), charge time (default
5 min), a unique name for the experiment, and the desired save folder.
When these fields are complete, the ?Collect Data? button becomes
active, as shown. When the camera is plugged in and the ?Collect
Data? button is pressed, the system proceeds through the stages of
data collection, constantly monitoring and displaying the camera?s
thermal status. If the camera begins to overheat or the user presses
?Cancel Run,? the system quickly closes the canister (if necessary)
and aborts all processes, allowing the user to unplug the camera. . . . 58
4.9 ?Forklift? device for returning the camera carriage to the top after
each experimental run. The leadscrew uses a dry lubricant to reduce
both friction and outgassing, and is driven by a stepper motor at the
top of the device. When the lift returns to the bottom, it activates a
limit switch and stops. . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.10 Virtual instrument (VI) for automating carriage return. When a
National Instruments analog input board is connected, the ?Input
Active? and top and bottom contact indicators become active, and
the ?Auto Return? function is enabled. In manual mode, the user can
input the desired return direction and speed (lower speeds provide
additional torque when necessary). . . . . . . . . . . . . . . . . . . . 62
4.11 Sample frame, with and without features overlaid. At top, the noise-
reduced frame alone. At bottom, the same frame with grain locations
(blue crosses), string center (red line) identified. Grain trajectories
(yellow arcs) are shown for the 30 frames following the one shown.
Units are pixels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
vii
4.12 Interface for editing tracks found by the tracking algorithm. Tracks
are presented sequentially in the left window, and possible grain can-
didates are presented in the right window. The user can navigate
through each frame and add or remove any grains to the track. Esti-
mates for possible missing locations are presented based on the cal-
culated trajectory. Units are pixels in source images. . . . . . . . . . 64
4.13 Large size fraction of zirconia-silica grains (diameters 200 ?m to 300
?m) under microscope. Grains are highly spherical with uniform
smooth surfaces, with a few exceptions. . . . . . . . . . . . . . . . . . 69
4.14 Results of experimental analysis of 1:1 (by mass) mixture of small
(100-200 ?m) and large (200-300 ?m) spherical silica beads. The dis-
tribution of observed grain sizes from experiment videos is compared
(at right) to the actual distribution as measured under microscope. . 73
5.1 Size distribution of small and large mixtures, as measured under mi-
croscope. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.2 Diagram illustrating the exponential decay model. For each point on
the surface of grain i with area dAi, the probability of transfer P is
multiplied by the number of charge carriers ?idAi. The total number
of transferred charges per collision is found by integrating P?idAi
over the entire surface of grain i. . . . . . . . . . . . . . . . . . . . . 79
5.3 Results for mixtures composed entirely of large grains. Note the high
variance in the measured charge-to-mass ratio compared to the mean
value. For this nearly monodisperse case, little charge segregation
occurs based on size. . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.4 Results for mixtures composed of 1:8 small to large grains, by mass.
Note the irregular shape of the observed sizes; this is due to the lower
quantity of data relative to other size fractions. When the number of
large grains increases, many small grains adhere to them while falling,
so fewer clean trajectories are observed. . . . . . . . . . . . . . . . . . 82
5.5 Results for mixtures composed of 1:4 small to large grains, by mass. . 82
5.6 Results for mixtures composed of 1:2 small to large grains, by mass. . 83
5.7 Results for mixtures composed of 3:4 small to large grains, by mass. . 83
5.8 Results for mixtures composed of 1:1 small to large grains, by mass. . 83
5.9 Results for mixtures composed of 2:1 small to large grains, by mass. . 84
5.10 Data collected for the same mixtures on different days. All exper-
iments shown high variance in charge magnitude and polarity, but
approximately the same mean. . . . . . . . . . . . . . . . . . . . . . . 85
5.11 Goodness of fit over all data for the overall best-fit value of ?0. Some
experimental runs exhibited higher variance than others, resulting
in poor fit confidence, but some fits are clearly better than others.
Many indicate a negative charge carrier density, suggesting a positive
species given our previous assumption of a negatively charged carrier. 87
viii
Chapter 1: Introduction
1.1 Triboelectric Charging in Granular Media
Charge transport in dielectric granular media has long been the subject of
study by the electrostatic community. The separation of charge in granular mixtures
due to triboelectric charging can cause dramatic and potentially dangerous electrical
discharges in both natural [1, 2, 3, 4] and man-made [5, 6, 7, 8] granular mixtures
and environments. Improving our understanding of this effect is critical for both
ensuring safety of persons and equipment in highly charged environments, and for
utilizing this understanding to more efficiently conduct powder-handling operations
like electrospraying [9] and pneumatic transport [5, 6, 7, 8, 10]. Where the granular
mixture consists of only one material, such that no apparent chemical difference
drives the charge exchange, it is observed that charge transfer is stochastic but
follows trends according to the sizes of the grains.
The exact mechanism of charge transfer and the distribution of charge accord-
ing to mixture properties remain uncertain. In part, this is due to the wide variety
of possible candidates for charging mechanism and charge carrier among different
materials, including trapped high-energy electrons [11, 12, 13, 14, 15], hydroxide
ions or other adsorbed ions in the surface water layer [8, 16, 17, 18, 19], or broken
1
polymer chains [20]. Most experiments and existing models for charge exchange
predict that larger grains will tend to acquire a positive charge and smaller grains
will become negatively charged, on average. The degree of charge separation is in-
fluenced primarily by the size differences and mass fractions of each discrete grain
size [11, 12, 13, 21, 22]. Existing models for charge exchange in granular mixtures
of a single material neglect a number of important effects, especially the influence
of contact area during collisions and the effect of non-discrete size distributions
on charge separation, and typically underestimate the magnitude of the charge ex-
change [11, 21, 23]. The difficulty in obtaining detailed measurements of individual
grain charge and charge distribution within the mixture, while controlling for the en-
vironmental conditions necessary to isolate a particular charging mechanism, make
a consistent predictive charging model especially elusive for dielectric grains.
Triboelectric charging is also believed to occur in the dusty surface layer of air-
less extraterrestrial bodies like the Moon. High-energy electrons from the solar wind
plasma are deposited into this insulating dusty surface material, known as regolith.
This charge may be readily exchanged when the surface is agitated in future manned
or robotic surface operations [24]. Not only is it important to accurately predict
the magnitude and mechanism of this transfer to mitigate dangerous electrical dis-
charge, we propose that this phenomenon may also be used as an electrostatic sieve,
extracting both material components and size fractions from the regolith in situ for
a variety of applications. Identifying an accurate tribocharging model for complex
dielectric mixtures under high-vacuum conditions will be necessary to enable this
application, but is difficult to accomplish using existing techniques for tribocharging
2
measurement.
This work improves existing tools for measuring and understanding tribocharg-
ing in granular dielectric materials. Many charge transfer mechanisms have been
proposed for dielectric grains, but the nuances of how these mechanisms manifest in
the distribution of charge throughout the mixture have not been fully explored. This
is largely due to the difficulties inherent in measuring surface charge on micron-scale
grains without influencing the charge through contact. We have created a model-
ing framework for predicting the mean charge distribution arising in a tribocharged
mixture based on a set of assumptions about the charging process, and developed
an experimental method to test these predictions. The main contributions of this
work are:
? A flexible semi-analytical model for charge transfer in dielectric granular mix-
tures with arbitrary continuous size distributions;
? An experiment for charging dielectric granular mixtures under a controlled
environment and measuring the individual charge and size distributions within
the mixture in a non-invasive way;
? Evidence for the role and density of positive charge species in tribocharging
in vacuum based on experiments conducted with zirconia-silica grains; and
? The foundations for a path toward understanding the relationship between cer-
tain proposed charge transfer mechanisms and their effect on the distribution
and segregation of charge in a mixture.
3
Publications
All publications and presentations to date that resulted from this research are
listed here for reference.
Journal Articles
? D. Carter and C. Hartzell, ?Effect of mixture properties on size-dependent
charging of same-material dielectric grains,? Physical Review E, submitted
Jan 2019.
? D. Carter and C. Hartzell, ?Experimental methodology for measuring in-
vacuum tribocharging,? Review of Scientific Instruments, 90(125105), Dec
2019.
? D. Carter and C. Hartzell, ?Extension of discrete tribocharging models to
continuous size distributions,? Physical Review E, 95(012901), Jan 2017.
Conference Papers
? D. Carter and C. Hartzell, ?A method for measuring charge separation in gran-
ular mixtures in vacuum,? 2017 Annual Meeting of the Electrostatics Society
of America, Jun 2017.
? D. Carter and C. Hartzell, ?A model of granular tribocharging for dielec-
tric mixtures with continuous size distributions,? 2016 Annual Meeting of the
Electrostatics Society of America, Jun 2016
4
? D. Carter and C. Hartzell, ?An extension of discrete tribocharging models to
continuous size distributions,? Society of Satellite Professionals International
(SSPI) Engineering Student Competition, Apr 2016.
Presentations and Posters
? D. Carter and C. Hartzell, ?Polarity Reversal and Charging Model for Tribo-
electrically Charged Silica Mixture,? 2019 Annual Meeting of the Electrostatics
Society of America, Jun 2019.
? D. Carter and C. Hartzell, ?Measurements of Same-Material Tribocharging in
Granular Silica,? 2018 AGU Fall Meeting, Dec 2018. Poster.
? D. Carter and C. Hartzell, ?Size Beneficiation of Lunar Regolith via Tribo-
electric Charging for In Situ Resource Utilization,? NASA Exploration Science
Forum 2018, Jun 2018. Poster.
? D. Carter and C. Hartzell, ?Measurements of Granular Tribocharging by High-
Speed Videography,? 2018 Annual Meeting of the Electrostatics Society of
America, Jun 2018.
? D. Carter and C. Hartzell, ?A model for tribocharging of regolith grains,?
Dust, Atmosphere, and Plasma Environment of the Moon and Small Bodies
Workshop, Jan 2017.
? D. Carter and C. Hartzell, ?An Analytical Model of Tribocharging in Re-
golith,? 2015 AGU Fall Meeting, Dec 2015. Poster.
5
Chapter 2: Literature Review
2.1 Tribocharging in Insulators
Tribocharging occurs between nearly every combination of materials, including
conductors, semiconductors, and insulators. There are many different mechanisms
and models for the charge exchange depending on properties of the materials. For
example, in conductive materials in contact, mobile electrons in the bulk are able to
easily transfer to materials with lower surface energy, allowing predictable exchange
of charge that disperses evenly throughout the materials [25].
Insulators, however, exhibit far more complex charging patterns, with charge
exchange attributed to many causes with various degrees of success. For contact
of metals with insulators, an empirically-determined effective work function admits
treatment of the charge exchange in a similar fashion to the high-conductivity case
[25, 26]. However, the contact of insulators with other insulators is far less well
understood. While the degree and cause of charge transfer is difficult to identify in
many such cases, the direction is fairly predictable, with a particular charge polarity
frequently arising after contact between certain pairs of materials.
For this reason, researchers have developed a ?triboelectric series? that lists
materials in order of polarity, used to predict the direction of charge transfer when
6
two materials from the series come into contact [27, 28]. These lists are qualitative
and often unreliable due to the wide variety of possible mechanisms causing insulator
charging: charge exchange has been attributed to such factors as the settling of
trapped high-energy electrons [11, 12, 13, 14, 15, 21], the release of adsorbed ions
[8, 19], breaking of polymer chains [20], and interactions with atmospheric ions [29],
with varying degrees of accuracy.
It has also been shown that tribocharging occurs even between identical ma-
terials, despite the lack of chemical differences between the surfaces. Lowell and
Truscott observed that when a dielectric material was rubbed across another iden-
tical surface, the moving surface tended to acquire a more negative charge [14, 15].
Shinbrot, Komatsu, and Zhao demonstrated that rubbing two identical insulating
surfaces together in air produces an increasingly large potential difference, with the
surfaces acquiring a random polarity in each experiment [29]. In general, the sym-
metry of same-material tribocharging causes the direction and magnitude of charge
exchange to vary between experiments, suggesting that a stochastic model (e.g.,
high-energy electron transfer) will be more successful than previous rigidly deter-
ministic models (e.g., the work function difference). Additionally, the transferred
charge from a surface appears to be proportional to the surface area in contact, as
expected for dielectric materials with low charge transport in their bulk.
Granular mixtures of a single insulating material are just as susceptible to
same-material tribocharging as flat surfaces. In many cases, granular mixtures in-
clude a variety of materials, and their charge distribution is governed by these ma-
terial differences; for example, pneumatic powder transport often induces significant
7
triboelectric exchange between powders and metal surfaces [5, 6, 7, 8]. In addition,
humidity in the air and adsorbed onto grain surfaces has been shown to significantly
alter charging characteristics [8, 18]. However, even granular mixtures of only a sin-
gle insulating material have been shown to readily develop charge separation through
same-material particle-particle collisions [13, 21, 30].
2.2 Models for Charge Transfer
When metals are involved in the charge exchange, the quantity of charge is
much easier to predict than for solely insulator-to-insulator transfer. A number
of studies have shown that for two conductive materials in contact, the quantity
of charge transferred can be determined by the difference in surface energy of the
materials [25]. Near the point of contact, electrons from the bulk of the materials
rearrange themselves to equalize the energy levels of the two materials. When the
materials are then separated, one material retains additional electrons acquired from
the other material, which then redistribute throughout the bulk. The quantity of
transferred charge can be approximated as ?qc = ?C0 (?1 ? ?2) [25], where C0 ise
the capacitance between the surfaces and ? is the material?s work function.
Even when only one of the contacting materials is conductive, one can typi-
cally estimate the transferred charge by empirically determining the ?effective work
function? for that material. In 1969, Davies conducted experiments in which non-
conductive polymeric materials were rubbed against a variety of metals with known
work function [31]. The surface charge density was then measured, and they found
8
that the quantity of transferred charge was indeed linear with the metals? work func-
tions. They were then able to determine an effective work function through which
the transferred charge could be predicted when charging with other conductive ma-
terials. In 1976, Gallo and Lama would further propose a model through which a
polymer?s effective work function could be estimated based on the ionization energy
of its monomer [26].
Still, the electron transfer model for metal-to-insulator charging has its prob-
lems, especially the lack of mobile electrons and a robust theoretical method to pre-
dict an insulator?s work function. Other researchers have investigated the possibility
that surface ions may be responsible for charge transfer in both metal-to-insulator
as well as insulator-to-insulator charging, where the lack of mobile electrons makes
it more likely that ions are responsible for charge exchange. In 1980, Lowell and
Rose-Innes suggested that, when metals rub against a polymer, material from either
surface can be deposited on the other surface, causing charge transfer [32]. In 2012,
Burgo, et al. [20] measured the charge distribution on the surfaces of polymers in
contact and observed patches of charge, which they attributed to the breaking of
weakly bound layers and impurities in the polymer structure near the surface.
Other non-electron models for charge transfer are based on the observation that
most materials in atmosphere have a thin water layer adsorbed to their surfaces.
When two such surfaces come into contact, it is primarily this water layer, not
the actual material in question, that is in contact. Ions dissolved here form a
?double layer,? with the ions near the surface being balanced by oppositely-charged
ions near the surface of the double layer. Shinbrot proposed that the potential
9
step created between the material surface and the charged outer layer could cause
charge exchange, and developed a quantitative model for this effect [33]; however,
he noted that this model likely cannot fully capture the charging process on its own.
Future experiments on this phenomenon, discussed in the next section, have yielded
conflicting results.
Lowell and Truscott proposed a model based on their experiments into same-
material tribocharging in which a number of electrons on insulator surfaces are in
unfavorably high-energy states, but cannot reach low energy states due to the low
conductivity of the material. During contact with another surface, these electrons
are exposed to low-energy states on the other surface and can be transferred [14, 15].
Expanding upon this model, Lacks and Levandovsky proposed a mechanism for
grain charging due to size differences in a mixture alone [11]. During agitation of a
granular mixture, grains repeatedly collide and slide against each other, allowing all
trapped electrons to achieve lower-energy states. Lacks and Levandovsky developed
a model for predicting the average steady-state charge on grains of a particular size
in a bidisperse mixture of a single material, demonstrating that larger grains will
always acquire a positive charge in such a case.
They later expanded upon this model to include the more general case of mul-
tiple size species and the possibility for low-energy electron exchange, although they
neglected the latter in simulations and analysis [12]. Particle dynamics simulations
of this ?population balance? model compare favorably with experimental results,
with larger grains acquiring a more positive charge on average than smaller grains
[11, 12, 22]. In 2009, Lacks and Kok extended the model to include a dependence of
10
the quantity of transferred charge on the sizes of the grains in contact [23]. In their
model, electrons transfer due to tunneling during collisions, with the probability of
tunneling a function of the distance between the electron on one grain and the clos-
est point on the surface of the other grain. Assuming ?0 is the maximum distance
an electron in the ground state could tunnel, one can calculate the surface area on
each grain over which electrons are able to transfer.
2.3 Experimentation
Cartwright, et al. [8], developed a full-scale experiment to investigate danger-
ous conditions that arise in pneumatic conveying of powders. In their experiments
with polyethylene powders, they observed bipolar charging of the grains, such that
the smallest grains charged negatively while the larger grains charged positively.
In 1994, Pence, Novotny, and Diaz [16] measured the effect of relative humidity
on the charge transfer between polymer powders and steel beads. They noted that
the charge magnitude increased as relative humidity rose from 0% to approximately
40%, then decreased again at higher humidities. Wiles, et al. [17] induced similar
charging between steel spheres and a polystyrene surface, noting that the rate of
charging increased linearly with relative humidity over the entire range of humidities
investigated.
In 2002 and 2003, Zhao, Castle, and Inculet [30, 34] measured the distribution
of charge in a variety of powders used as paint pigments in aerosols and electrostatic
painting guns. They charged these grains by fluidizing a bed of these powders with
11
air, then dropping them through a vertical array of Faraday pail sensors. As the
charged grains fell, they were accelerated by image charges toward the pails, which
captured them and measured their total charge. Small grains were able to accelerate
faster and therefore collected in the upper pails, while large grains collected in the
lower pails. By measuring the average size and total charge of grains collected in
each pail, they were able to very roughly estimate the charge distribution by size,
observing again that the small grains charged negatively.
Watanabe, et al. [35] measured the charge transferred in a single collision
between a stainless steel plate and non-conductive particles common in the pharma-
ceutical industry: ?-lactose monohydrate, aspirin, sugar granules, and ethylcellulose
granules. Grains were dropped onto the plate, with their charged measured imme-
diately before and after the impact. The experiment was conducted in air inside a
vacuum chamber, but the collision speed was varied by changing the vacuum level.
Humidity and temperature were kept constant for each experiment. They observed
that the transferred charge was directly proportional to the normal speed on im-
pact, but that the equilibrium charge after repeated contacts was constant for each
material.
In 2008, Shinbrot, Komatsu, and Zhao [29] investigated same-material tri-
bocharging by rubbing together latex balloons and polycarbonate disks. The sur-
faces were rubbed together, then separated, measured, and neutralized by spraying
with ionized air. They observed that, in each experiment, the surfaces had patches
of both positive and negative charges, with one surface charging significantly more
positively and the other charging more negatively. The polarity appeared to be ran-
12
dom in each iteration, with no correlation to prior polarities. They noted that the
initial charge exchange appeared to be random, but once a dominant charge began
to develop, the surface never neutralized or changed sign, only increasing in magni-
tude. They repeated these experiments in helium and noted that the surfaces did
not significantly charge in an inert atmosphere, concluding that patches of charge
recruited ions from the air to catalyze the formation of new charged patches as the
surfaces moved against each other.
Forward, Lacks, and Sankaran developed a more rigorous methodology for iso-
lating particle-particle interactions through the use of a bed fluidized with inert ni-
trogen [36]. In a variety of experiments, they demonstrated that the size-dependent
charge polarity predicted by the high-energy electron model is independent of ma-
terial and occurs in any mixture with a wide size distribution, including lunar and
Martian regolith simulants [13, 37, 38]. They further showed that the degree of
charge separation is highly dependent on the relative masses of each size species, as
this value influences the rate at which grains collide with other species.
In 2010, Pa?htz, Herrmann, and Shinbrot [1] presented a potential explanation
for how non-conductive grains could charge in response to an externally-applied
electric field. They proposed that the grains could become partially polarized in the
direction of the field, so that collisions occurring in the direction of the electric field
cause adjoining hemispheres to neutralize. They conducted an experiment using
1.6 mm glass beads in a glass container, with electrodes at the top and bottom
to create a vertical electric field. Pressurized air fed in from the bottom fluidized
the grains, mixing them, and they measured the number of grains lofted at various
13
heights. They compared this to the same quantities from a simulation based on
their transfer mechanism, showing good agreement between the two.
Apodaca, et al. [39], conducted experiments with a range of flat polymers
to determine the relationship between surface area and charge transfer in a single
contact between identical polymer surfaces. They observed that the total quantity
of charge transferred over repeated contacts approached asymptotic values, but that
the quantity was actually proportional to the square root of the contact area, not
the area itself. They concluded that properties of surface roughness and molecular-
scale fluctuations in composition play a signficant role in charge transfer, possibly
more so than the contact area.
In 2012, Sow, et al. [40] observed that certain conditions could cause the
direction of charge transfer between identical materials to reverse. Using balloons
made from latex rubber and inflated to varying degrees, they noted that by varying
the surface strain through inflation and deflation, they could control the charge
polarity on each surface. These experiments were conducted in an environment
purged with nitrogen.
In pursuit of a method for electrostatic beneficiation of regolith, Trigwell, et
al. [41, 42, 43] conducted experiments at Kennedy Space Center to assess the fea-
sibility of separating particular chemical components of regolith by allowing them
to tribocharge against a static charger. Materials tested for this charger were alu-
minum, copper, and polytetrafluoroethylene. They demonstrated that such a device
could indeed be capable of enriching a regolith sample to higher concentrations of
a desired component (ilmenite, in their case).
14
Jaeger and Waitukaitis, et al, developed a more sophisticated method for mea-
suring the actual charge on each grain to further investigate the predictions of Lacks
and Levandovsky?s model [21, 44]. Their results also agreed with the polarity pre-
dicted by the original model, but they observed that the magnitude of the charge
was too large to be simply caused by trapped electrons. They suggested that al-
ternative mechanisms such as ion transfer or interactions with the atmosphere may
play a role in charging.
2.4 Summary
Throughout the last few decades, many researchers have attempted to identify
a consistent model for insulator tribocharging. The results of their experiments
vary dramatically, agreeing consistently on some results, such as the role of surface
roughness on the charging process. However, frequently their results appear to be
in conflict; some show that charge exchange is proportional to contact area, other
times to its square root. Some experiments suggest that charging continues up to the
Paschen limit with repeated contact, only ceasing when electrical discharge occurs;
others identify an apparent asymptote or even eventual reversal of polarity. The
issues with same-material tribocharging are even greater due to the lack of a clear
driving factor like surface energy. In the case of granular materials, the difficulty
in measuring individual grain charge to obtain a consistent charge profile makes
progress slow. In this thesis, we develop techniques that may provide a solution to
many of these challenges.
15
Chapter 3: Modeling Charge Transfer in Continuous Mixtures
This chapter contains material published in D. Carter and C. Hartzell, ?Ex-
tension of discrete tribocharging models to continuous size distributions,? Physical
Review E, 95(012901), Jan 2017 [45].
In this chapter, we expand existing models of tribocharging to make predictions
about the charge distribution in mixtures with a continuous size distribution. To
date, analytical models for charge exchange typically apply only to very simple
mixtures of grains in which each grain is assumed spherical and has one of two
allowed sizes [1, 11, 12, 13, 21, 23]. Many of these models describe charge exchange
events as identical for all pairs of grains regardless of grain size, despite the fact that
the final charge is observed to be size-dependent [11, 12, 13, 21, 22].
In their initial discussion, Lacks and Levandovsky note that they have ignored
a variety of phenomena, including the effect of aspherical shapes, sliding contact,
and electrostatic forces between grains, which may alter the charge exchange rate
[11]; Kok and Lacks similarly leave out these factors [23]. We have modified these
models by including a new method for calculating contact area differences between
grains, which drive the amount of charge exchanged during a collision. When two
real objects collide, they deform slightly such that they develop an approximately
16
flat contact area between them, across which trapped electrons are able to move. By
assuming each collision exchanges a size-independent number of electrons from each
grain, Lacks and Levandovsky neglect the effect of grain size differences and assume
electron transfer rate is independent of collision area. This area is determined by the
size and collision energy of the grains, and therefore increases with increasing grain
size. Experiments on sphere-to-surface contact charging have demonstrated that
the exchanged charge is indeed proportional to contact area, and that contact area
can be varied through changes in collision speed and sphere size [35, 46, 47]. While
the contact area formulation in Kok and Lacks? work also suggests a proportional
relationship between collision area and transferred charge, it requires that grains
are treated as hard spheres, so that changes in collision energy have no effect on the
charge [23].
Because the magnitude of insulator tribocharging is shown to be highly depen-
dent on the area in contact [14, 15, 22, 29, 46], we introduce an additional term Aij,
the contact area between grains of radii Ri and Rj during a collision. This causes
the amount of transferred charge during a collision to depend upon the relative
sizes of the grains in contact, changing the properties of the charge distribution and
making new predictions about charging trends for various size distributions. While
we initially assume that the carrier species consists of electrons, this model provides
a generic, flexible framework for describing how a particular charge transfer mech-
anism manifests as a charge distribution function. From this standpoint we will
determine which transfer models most accurately represent real granular mixtures
and assess the implications for our understanding of granular tribocharging as a
17
whole.
3.1 Model Assumptions
For the purposes of deriving this model, we will adopt the charge transfer
mechanism proposed by Lowell and Truscott [14, 15] and further elaborated by
Lacks and Levandovsky [11, 12]. Each grain is assumed to be a solid sphere of
radius R; its surface area is therefore 4?R2. The surface area density of trapped
high-energy electrons is ?H and is initially the same for all grains. Its value at
time t = 0 (before charge exchange due to mixing) is given by ?0, at which time
all grains are electrically neutral. According to the trapped electron model, each
collision exposes some number of high-energy electrons that each have a random
probability of being transferred; we will denote this probability as fH . In addition,
we will assume that each collision involves some characteristic contact area Aij,
where the colliding grains have radii Ri and Rj. We will further assume that the
relative speed between grains is size-independent and equal for all grains, and they
are all composed of the same material; therefore, the contact area is a function of
the grains? radii only. This allows us to express the number of high-energy electrons
transferred from a grain of radius Ri to a grain of radius Rj as fH?H,iAij. Note
that the average surface density of electrons ?H,i is a function of time, and therefore
varies throughout the mixing process.
18
Figure 3.1: Diagram of physical system considered. Grains with a normalized size
distribution given by g(R) are mixed in a container of volume Vc. Mixing is per-
formed by moving the container with average speed vr, such that the grains inside
also move with average speed vr. The grains are then assumed to move in random
directions with average speed vr as well. When a grain of radius Ri collides with a
grain of radius Rj, some contact area is formed in region Aij. High-energy electrons
in this region are capable of transferring between the two grains and settling into
stable low-energy states.
3.1.1 Collisions
In order to estimate the collision rates between grains, we assume that all
grains move at approximately the same speed, and therefore that the average relative
speed between two grains is a constant, here called vr. For very small particles, their
kinetic energy is frequently governed by thermal effects and the electromagnetic
force, resulting in size- and/or charge- dependencies for the particle speed. However,
we will assume for this model that the motion of the grains is dominated by some
other external force that gives each grain an average speed of vr relative to the other
grains in the bulk, e.g. a uniform acceleration applied to a bed of loose grains. See
Figure 3.1 for an illustration of this collision process.
We can use the relative speed between grains to estimate the collision rates
19
as grains move through the mixure. Consider a single grain of radius Ri moving
against a background of grains of radius Rj. The first grain moves with speed vr
relative to the background grains. In some time ?t, the grain moves a distance
vr?t. In this time, it collides with any grains whose centers are a distance Ri +Rj
from the axis of its motion. Therefore, the moving grain collides with any grains
within the volume ? (Ri +Rj)
2 vr?t. If our control volume is Vc and there are nj
grains with radius Rj in the mixture, then the rate at which our grain of radius Ri
collides with grains of radius Rj is ?ij, where:
?vrnj
? = (R +R )2ij i j (3.1)
Vc
3.1.2 Size Distribution
Previous tribocharging models have been developed specifically for application
to size distributions consisting of two discrete sizes [11, 13]. These models are
restrictive, as unsieved granular mixtures in nature are more accurately represented
by continuous size distribution functions. While bidisperse mixtures are much easier
to manipulate analytically and are relevant to simple tribocharging experiments,
they cannot explain or predict phenomena in natural granular mixtures. Thus, we
will consider charge exchange in an arbitrary continuous grain size distribution.
Consider a mixture of grains with a probability distribution of radii given by
g(R); that is, the fraction of grains in the mixture with a radius within dR of R is
?
g(R)dR. The distribution is normalized such that ?0 g(R)dR = 1. As a result, the
20
number of grains within dRj of radius Rj (defined as nj) is equal to n0g(Rj)dRj,
where n0 is the total number of grains in the mixture. To find the value of n0, we
define the total mass of the mixture as M0; we use this parameter because of the
ease with which this is directly measured in experiments. Using this definition, and
defining the mass density of grains as ?M , we can determine n0 by integrating the
mass of all grains in the mixture as follows:
4 ? ?
M = ?? 30 Mn0 R g(R)dR (3.2)
3 0
The use of a continuous size distribution also has important implications for the
collision rate. Recall that the collision rate ?ij depends upon the number of grains
nj of radius Rj in the mixture. When we expand this term, we get a differential
term dRj in the definition of ?ij. This will be important when we calculate the rate
at which high-energy electrons are transferred to the mixture background.
3.1.3 Electron Transfer Rates
We begin by considering the rate at which a single grain of radius Ri loses
electrons to low-energy states on grains of radius Rj during mixing. We have pre-
viously derived an expression for the frequency of such collisions ?ij, as well as an
expression for the number of electrons transferred per collision at a given time t.
Suppose that the charge and number of acquired electrons on a grain does not affect
the rate at which it continues to donate or acquire electrons during collisions. We
can then write an expression for the rate at which high-energy electrons on a grain
21
of radius Ri are lost:
d
(?H,i) | ?
?ijfH?H,iAij
R = (3.3)
dt j 4?R2i
This expression gives only the rate at which electrons are lost to grains in a
narrow band of radii around Rj. To obtain the rate at which the grain loses electrons
to all other grains in the mixture background, we integrate this expression over the
range of all grain sizes Rj:
d?H,i
= ??i?H,i (3.4)
dt
v ? ?rn0fH
? 2i = Aij (Ri +R2 j) g(Rj)dRj (3.5)4RiVc 0
For the time being, since we have not defined an expression for Aij, we cannot
evaluate this integral. We shall find that for most size distributions, it will be
necessary to evaluate this integral numerically.
3.1.3.1 Electron Loss Fraction
In the previous section, we defined the average rate at which a single grain of
radius Ri, representating the average grain of that radius, loses high-energy electrons
to grains of radius Rj. As we saw, this rate is given as a continuous function
(averaged over the discrete collision events that happen over much smaller time
scales) and is directly proportional to the mobile electron density ?H,i(t). Suppose
22
that we wished to find the fraction of all electrons currently being transferred from
a grain of radius Ri that were going to grains of a specific size Rj. This quantity is
critical to understanding the nature of size-dependent charge separation. We note
that this quantity, here called fij, can be equivalently described as the ratio between
the rate of electron transfer to size species Rj, given by Equation (3.3), and the rate
of overall electron transfer to grains of all sizes, given by Equation (3.4):
? Aij (Ri +R 2j) g(Rj)dRjfij = ? (3.6)
0 Aik (Ri +Rk)
2 g(Rk)dRk
We have replaced the variable of integration Rj in the denominator with the
dummy variable Rk, to distinguish it from the actual variable Rj in the numerator,
the size of the grain band with which our single grain is colliding. Note that the
electron density cancels out, and the fraction of electrons transferred is a constant
fraction with a dependence on grain radii and the size distribution only. Importantly,
we note that we can now express the rate of change of the high-energy electron
population in the following simple ways:
d?H,i | d?H,iR = fij = ??ifij?j H,i (3.7)dt dt
3.1.3.2 Collision Area
Now that we have identified all the required relationships to make predictions
about the average final charge, we turn our attention to the collision area. For the
analysis up to this point, we have assumed that each grain is a hard sphere of a
23
particular radius R. The contact area between two perfectly spherical hard grains
is simply a point, but such a model neither contributes to our understanding of
granular tribocharging nor fits experimental data regarding tribocharging in general.
In experiments on rubbing flat surfaces together, the exchanged charge is directly
proportional to the area exposed to contact, and collisions between small grains slide
against each other rather than simply rebound at a point contact [35, 46, 47]. From
this observation, we may conclude that the exchange of charge in real collisions
between grains will be heavily dependent on the size of the grains. Specifically,
because electron transfer in other systems tends to depend on the surface area, we
expect to see a similar area dependence in sliding grain collisions.
The inclusion of a collision area term is a significant improvement over the
typical approach used in many granular tribocharging models, and allows us to ex-
plore a variety of possible transfer mechanisms. The model developed by Lacks and
Levandovsky, and frequently employed to make predictions in various tribocharging
experiments, assumes that a constant number of electrons (frequently one) is trans-
ferred during each collision. This is analogous to the assertion that 1 = fHAij?H,i(t),
using the framework developed for our model. While this appears to create prob-
lems due to the fact that ?H,i loses its dependence on time, we note that in the
loss fraction fij (which we will later see is the most important term in predicting
the final charge), the electron density divides out. Therefore, in calculating the
final charge on the grains, the result is identical to the assumption that Aij is a
constant and also divides out. Other models have also been developed to include a
contact area term, especially that of Kok and Lacks [23]; however, we believe that
24
our implementation better represents what is known about the relationship between
collision speed and transferred charge. In using a ?maximum tunneling distance? as
the effective boundary of the contact area, Kok and Lacks have developed a frame-
work in which the effective contact area is different for the two grains, due to the
difference in surface curvature [23]. In our model, the contact area is the same for
the two grains, which agrees with previous experiments and models exploring the
effect of collision energy on contact area and transferred charge [14, 15, 29, 46].
In Gugan?s treatment of Hertz?s theory for collisions between two spheres
[48], the collision area is given in terms of the collision speed U , reduced radius
3 3
R = RiRj
4??MR R
? , reduced mass M =
mimj i j
? = 3 3 , and effective elastic coefficientRi+Rj mi+mj 3(Ri+Rj)
1??2 1??2
X j? = i + = constant. Because we are here assuming that all grains are ofEi Ej
the same material, they all have the same density, modulus E, and Poisson?s ratio
?. We have also assumed that the relative speed between any two grains is vr, so
all factors involving only these terms can be dropped for simplicity:
A2.5 = 16.4v2M R2X ?M R2ij r ? ? ? ? ? (3.8)
Now we expand these quantities and solve for Aij:
Aij ? rijRiRj (3.9)
( )RiRjrij = 0.4 (3.10)
R3 +R3i j (Ri +R )
0.8
j
25
Because we are considering an area term, we have rearranged the expression
so that it can be written as the product of the radii of the involved grains and a
non-dimensional term rij. Note that this looks very similar to the square of the
reduced radius defined above. This makes logical sense, as real collisions are not
simply point contacts and will involve an amount of contact area highly dependent
on the size and shape of the grains.
3.2 Steady-State Solutions to the Charge Distribution Function
3.2.1 Solutions for Continuous Size Distributions
To predict the grain charge at some time t after the initiation of mixing,
we must first calculate the rate at which a single grain accumulates low-energy
electrons. Consider now that each grain of radius Rj gives a fraction fji of its high-
energy electrons to grains of radius Ri. The rate at which electrons are given in this
d?
way after some time t is ?4?R2jnjf H,jji . Therefore, if we divide this quantity bydt
ni, the number of grains of radius Ri, we get the rate at which low-energy electrons
settle on the average grain of size Ri. Using Equation (3.4), the change in charge
QL,ij(t) contributed by these low-energy electrons from grains of radius Rj is:
dQL,ij nj
= ?4e??jR2jfji ?H,j (3.11)dt ni
To obtain the overall contribution to the charge from all other grain sizes,
we simply integrate this expression over all sizes Rj. Meanwhile, the grain is also
26
losing high-energy electrons due to each of these collisions. The rate at which it
loses electrons can be given as the rate of change of surface electron density (found
in Equation (3.4)) multiplied by the surface area and electron charge ?e:
dQH,i ? 2d?H,i= 4e?Ri = 4e??iR2i?H,i (3.12)dt dt
The overall rate of change of the charge of a single grain of radius Ri will be
given by the sum of these rates:
dQ ?i dQH,i dQL,ij
= + (3.13)
dt dt dRj dt
Recall from Equation (3.4) that the surface electron density can be written as
an exponential decay function, with ?H,i(t) = ? e
??it
0 . We can make this substitution
and integrate over time to obtain an expression for the actual charge at time t:
[ ( ) ? ? ]n ( )j
Qi(t) = 4e??0 R
2
i 1? e??it ? R2jfji 1? e??jt (3.14)
0 ni
The above expression accounts for the fact that, at time t = 0, the grains
are electrically neutral. By taking the time to infinity, we get an expression for
the charge once all high-energy electrons have settled into low-energy states. This
causes the exponential terms to die out, leaving only the following expression:
? ? ?? R2A 2ij (Ri +Rj) g(Rj)dRj
Qi,final = 4e?? ?0 R2 ? ? j ?i ? (3.15)
0 0 Ajk (Rj +Rk)
2 g(Rk)dRk
Equation (3.15) is an expression for the final charge on a grain of arbitrary
27
size Ri in a mixture of grains with size distribution g(R) after all mobile charges
have settled into their preferred states. From the definition of the contact area, we
can see that these integrals do not have an obvious closed-form solution (although
the complexity is significantly reduced by applying Lacks and Levandovsky?s equal
contact area assumption). In addition, the use of an arbitrary size distribution
function also prevents us from directly solving the integral in the general case.
Therefore, solutions to the final charge equation must be obtained through numerical
integration. This will be especially useful in the case of unusual size distribution
functions, such as Taylor series curve fits to granular mixtures found in nature,
where analytical solutions introduce unnecessary complexity.
3.2.2 Solutions for Discrete Size Distributions
Although a continuous size distribution is a better approximation of a real
granular mixture than a set of discrete sizes, it is still instructive to explore how
this model behaves in the case of a discrete size distribution, in order to allow
comparison to both Lacks and Levandovsky?s original model and to experimental
observations. As discussed previously, most existing models to predict charging in
laboratory mixtures employ a discrete size distribution, typically of only two primary
grain radii, which we will here call R1 and R2, where R1 is the larger size. This size
distribution can be represented by a sum of two Dirac delta functions, which can be
thought of as infinitely narrow normal distributions such that only two grain sizes
are represented. Therefore, our size distribution will look like:
28
k1?1 + k2?2
g(R) = , ?i = ? (R?Ri) (3.16)
k1 + k2
The weights k1 and k2 are related to the mass of each size species. The mass
of a single species i is given by the following expression:
4??Mn0k R
3
i
M = ii (3.17)
3 (k1 + k2)
We can define a set of non-dimensional constants d = R2 < 1 and m = M2 > 0.
R1 M1
We will find that this greatly simplifies our analysis. Plugging the mass relation-
ships and non-dimensional constants into our expression for the size distribution,
we obtain the following:
d3?1 +m?2
g(R) = (3.18)
d3 +m
Since we have already obtained an expression for the final charge, we need
only plug this size distribution into Equation (3.15) and solve. Consider first the
charge on grains of radius R1, the larger size species. The grain charge expression
contains two nested integrals. The integral in the denominator of Equation (3.15),
here represented by I0, is:
? ?
I 20(Rj) = Ajk (Rj +Rk) g(Rk)dRk (3.19)
0
Note that this expression is independent of Ri, the grain species size that we
are investigating (which here is R1). Now consider the remaining integral:
29
? ? R2 2jAij (Ri +Rj) g(Rj)I(Ri) = dRj (3.20)
0 I0(Rj)
For the case of discrete sizes, we can analytically solve this integral using the
definition of an integral of Dirac delta functions ?k over a domain containing Rk and
plug this into the expression in Equation (3.15) for charge on grains of size Ri:
( )
3 2 2
Q = e?? 4R2 ? d Ai1 (Ri +R1) ? mAi2 (Ri +R2)i,final 0 i (3.21)d3A11 +msA12 dsA21 +mA22
Before substituting for Ri, we can first expand Aij in terms of the grain sizes Ri
and Rj. Recall that this term is symmetric in i and j; that is, Rij = Rji. Therefore,
we can be certain that R12 = R21. Furthermore, we can simplify the algebra by
defining r = r12 and s =
1 (1 + d)2 < 1, which are terms that appear frequently
4
in the reduced expression. The expression reduces significantly in the case where
i = j:
21.2R2
r = i 2ii 0.4
(2R3) (2R )0.8
= 1, Aii = Ri (3.22)
i i
( )0.4
2
r = d , A12 = rR1R2 (3.23)
s (1 + d3)
Finally, we can substitute the grain radii R1 and R2 for Ri in Equation (3.21)
to find the net charge on grains of each species:
30
R21msr (m? d) (d? sr)Q1,final = 4e??0 (3.24)
(d2 +msr) (dsr +md)
R2d31 sr (m? d) (d? sr)Q2,final = ?4e??0 (3.25)
(d2 +msr) (dsr +md)
3.3 Analysis
We can demonstrate that these charge equations obey the law of charge con-
servation; that is, the total charge on grains of size R1 is equal and opposite to
the total charge on grains of size R2, as we have assumed that all grains start out
electrically neutral. The net total charge is Q0 = n1Q + n
3Mi
1 2Q2, where ni = 4?? 3MRi
is the number of grains in the mixture with radius Ri. Performing this calculation,
we get the following conservation condition:
3M ( )1
Q0 = d
3Q1 +mQ3 2 = 0 (3.26)4??MR2
From Equations (3.24) and (3.25), we can see that d3Q1 = ?mQ2. The can-
cellation of terms resulting in net charge neutrality supports the validity of the
assumptions leading to this model; although real mixtures will have a wide spread
in charge within grain sizes, our average charge simplification has clearly not resulted
in a violation of charge conservation. In the following sections, we will explore ad-
ditional differences between our new charge distribution function and the model
proposed by Lacks and Levandovsky.
31
3.3.1 Comparison of Continuous and Discrete Models
The original models for granular tribocharging considered only a finite number
of discrete grain radii. However, a far more realistic distribution when the mixture
is composed primarily of specific sizes is a sum of normal distributions. While even
this may not be sufficient to properly model the size distribution found in most
naturally-occuring granular mixtures, it is an instructive example in the differences
and similarities between our continuous model and the discrete model. We will
consider a grain mixture composed of a sum of normal distributions centered around
two primary sizes R1 and R2, given below:
g(R) = k e?a1(R?R1)
2 2
1 + k e
?a2(R?R2)
2 (3.27)
Note that, although it is best practice to normalize this distribution function
?
to ?0 g(R)dR = 1, the fact that the distribution only ever appears in both the
numerator and denominator of a ratio suggests that the distribution need only be
normalizable, but not actual normalized, as the normalization factor will divide out.
We will also specify that R k22 < R1 and define k = for the sake of conveniencek1
and consistency. Here ki is a factor determining the height of the Gaussian peak
corresponding to the distribution of grains of size Ri, so that k is the height of the
R2 peak relative to that of R1. The coefficients a1 and a2 in the exponents are
related to the standard deviation of the distributions, a measure of the width of the
peaks. Each of these properties can be calculated from the experimentally measured
32
size distribution of a sample of the mixture.
Consider now the case for which R1 = 100?m, R2 = 50?m, a1 = a2 =
0.005?m?2, and k = 8. The size distribution is shown in Figure 3.2a for a non-
dimensional form of Qfinal, where:
? QfinalQ = (3.28)
4e?? R20 1
We can calculate the charge distribution for two cases: the constant contact
area assumption made by Lacks and Levandovsky?s model, and the size-dependent
contact area model using the definition for Aij described above. The final charge
distribution is given in Figure 3.2b. Note that the constant contact area assumption
produces a parabolic charge distribution, in which smaller grain sizes acquire an
average negative charge while larger grains become more positively charged. On
the other hand, the charge distribution for the size-dependent contact area model
appears more closely related to a cubic function, with an additional peak near the
small end of the grain size distribution. This means that the smallest grains actually
acquire a positive charge, while many large grains are also negatively charged.
3.3.2 Polarity Reversal
The apparent reversed charge polarity in the case of size-dependent colli-
sion area goes against conventional predictions for granular insulator tribocharging.
Lacks and Levandovsky?s model predicts that mixture properties have no effect on
the polarity, and experiments to date seem to support this assertion [11, 12, 13, 21].
33
(a) Normalized particle size distribution (b) Non-dimensional charge distribution
function g(R) in the form of Equation (3.27), function corresponding to size distribution in
with k /k = 8, a = a = 0.005?m?22 1 1 2 , (a) (solid line, magnified x10 for clarity). For
R1 = 100?m, and R2 = 50?m. comparison, the distribution usingAij = con-
stant is overlayed (dashed line).
Figure 3.2: Non-dimensional charge distribution Q? = Q(R)/4e?? 20R1 for a specified
size distribution function.
In the following sections, we dissect the final charge expression given in Equation
(3.15) to explore the parameters that determine charge polarity. We find that the
definition of the area term Aij, as well as the mixture parameters d and m, play
a significant role in determining the final charge polarity in a bidisperse (or nearly
bidisperse) mixture.
3.3.2.1 Bidisperse Mixtures
Recall that each of the non-dimensional terms m, d, r, and s is positive for all
sizes R1 and R2. Therefore, in Equations (3.24) and (3.25), while the denominator
is necessarily always positive, the negative terms in the numerator make the sign
ambiguous at first glance. In previous models, the larger grain size in a bidisperse
mixture always acquires a positive charge, a prediction supported by experimental
34
data; however, these models did not include an area term Aij. For our model, we
will explore the sign of the numerator of the fraction in Equation (3.24):
sgn(Q1,final) = sgn [(m? d) (d? sr)] (3.29)
We can expand the expression d? sr in terms of only d as follows:
?? [ ?(1 + d)3 ]0.4d? sr = d 1? ? > 0 (3.30)
4 (1 + d3)
It can be trivially shown that the bracketed expression ranges between 1/4
(as d approaches 0) and 1 (as d approaches 1); we can be sure then that d ? sr is
positive for all values of d. Therefore, the sign of the charge on grains of size R1
is entirely determined by the simple expression m ? d, the difference between the
ratio of masses of the two species and the ratio of their radii. Specifically, the larger
grains will only achieve the traditionally-predicted positive polarity if the ratio of
their mass to the mass of the smaller grains is larger than the ratio of their radius
to the radius of the smaller grains. Unfortunately, experiments up to this point
could neither confirm or refute this prediction. This may be due to a general lack of
reporting of mass ratios in granular tribocharging experiments, as this value affects
only the relative magnitude of average grain charge in the models used and is of
significantly less interest than the size ratio d. In the following section, we will find
that this polarity reversal also appears in the continuous distribution model, but
the exact conditions required to elicit this behavior are more difficult to describe in
closed form.
35
We have already shown (in Section 3.1.3.2) that Lacks and Levandovsky?s
simplification of a single electron transfer per collision is functionally equivalent to
the assumption that all contact areas are equivalent. To better understand the
influence of the area term on the charge polarity, we can once again calculate the
charge on a grain of size R1, with the area term left as a variable. We are particularly
interested in the ratio of contact areas, so let us define an additional non-dimensional
term aij =
Aij . From Equations (3.21) and (3.28):
A11
( )
? ma22 ? d3 ? sa12 (m? d)Q1 = msa12 (3.31)(d3 +ma12s) (da12s+ma22)
Although we have not yet defined Aij in this example, we will make the very
basic assumption that, due to the geometry of the grains, collisions between two
larger grains cannot (on average) have smaller contact areas than collisions between
two smaller grains. Thus we will state only that a22 ? a12 ? 1. If the contact areas
are equal, then a22 = a12 = 1 and we obtain the expression found by Lacks and
Levandovsky:
? d (1 + 3d) +m (3 + d)Q1 = ms (1? d) > 0 (3.32)4 (d3 +ms) (ds+m)
Because d < 1 by definition, we see that Equation (3.32) must always be
positive when the contact area term is neglected. When the contact area is not
neglected, we can determine what properties Aij must have in order to result in a
polarity reversal. Consider now the sign of Equation (3.31):
36
[ ( )
2 ( )]d a22
sgn (Q?1) = sgn d s? +m ? s (3.33)a12 a12
Now, both a22 and s have values less than 1, and s > d2 for all d < 1. Therefore,
a12
2
if a12 < 1 for d < 1 and approaches 1 more slowly than
d , then the first term will
s
be negative for all values of d greater than some critical value. Furthermore, the
second term will be negative above a different critical value for d if a22 approaches
1 more slowly than sa12. The relationship between these rates, and the value of
m in the mixture, will determine the final value of d for which the entire quantity
Q1 is negative. Because we are dealing with areas here, we expect that a22 will
have a d2 dependence and a12 will have approximately a d dependence (if any at
all). Since a 112 is approximately of order d , we do in fact expect that the first
parenthetical expression goes as s?d while the second goes as d?s, suggesting that
Q1 is proportional to m? d. This reaffirms our observation of the polarity reversal
derived above in Equation (3.29).
3.3.2.2 Continuous Mixtures
It is important to note that, because we are working with a continuous distri-
bution, the concept of polarity in the sense used in the discrete model is ambiguous.
For example, it can be shown that for a size distribution composed of two normal
distributions like the one used Figure 3.2a, the resulting charge distribution always
takes on the approximately-cubic form seen in Figure 3.2b. However, the contribu-
tion to the charge from each Gaussian peak varies in a way that reflects the polarity
37
reversal seen in the discrete case. This manifests through a shift in the zeroes of the
continuous charge distribution; for example, as the parameters of the size distribu-
tion approach the conditions required for a polarity reversal, the negative region of
the charge distribution shifts toward the larger grain sizes. This causes larger grains
to become more negatively charged, while smaller grains become more positively
charged overall. Figure 3.3 demonstrates the effect of this phenomenon as the size
peaks vary in width. This illustrates the importance of using a continuous grain
size distribution rather than a simple discrete size model: the charge distribution is
highly dependent on many parameters ignored by the discrete model, especially the
width of the peaks in the distribution.
We have also seen how the charge polarity depends primarily on m and d in a
primarily two-species distribution when the area term is included. We can plot the
non-dimensional charge Q?1 on grains of radius R1 against the mixture parameters k
and d to better understand their effect. Recal that we can write m?d as d (kd2 ? 1).
Therefore, the polarity reverses (Q?1 becomes negative) with decreasing k and d. In
Figure 3.4, we see this trend: the critical value of R2 below which Q
?
1 is negative
occurs at larger values for decreasing k.
3.3.2.3 Compared to Previous Experiments
The charge transfer model proposed here predicts that, for certain grain size
distributions, larger grains charge negatively and smaller grains charge positively.
This phenomenon has not been reported in experiments to date, but the lack of
38
(a)
(b)
Figure 3.3: Effect of variation in peak width of size distribution on charge polarity,
for non-dimensional charge Q? = Q(R)/4e?? R20 1. (a) Normalized particle size distri-
bution functions g(R) in the form of Equation (3.27), with k2/k1 = 8, R1 = 100?m,
and R2 = 50?m. (b) Non-dimensional charge distribution functions corresponding
to the size distributions in (a). Vertical lines at R = R1 and R = R2 provided as
reference.
39
Figure 3.4: Non-dimensional net charge Q?1 = Q
2
1(R)/4e??0R1 on grains of radius
R1 = 100?m for a size distribution of the form of Equation (3.27), with a = 0.005?m.
As the ratio of the peak heights (k) increases, the size ratio (d = R2/R1) below
which large grains charge negatively decreases. Note that the charge predicted by
the continuous distribution is very similar to that predicted by the discrete model,
with some variation due to the effect of the nonzero peak width (as seen in Figure
3.3b).
existing experimental evidence could be attributed to a number of factors. The new
polarity distribution is a result of the inclusion of area-dependent charge transfer
because, when the area dependence is removed, the predictions made by the con-
tinuous distribution model closely match those of the discrete model. The fact that
the polarity reversal in bidisperse mixtures often appears in between size peaks,
where the number of grains present is comparatively very small, and that mass
ratios are often unreported in experiments, make the polarity reversal difficult to
observe. However, experiments on collisions of spheres suggest that the transferred
charge is indeed proportional to the contact area [35, 46, 47], suggesting that our
implementation of a contact area dependence is accurate. Additionally, small grains
may have more irregular surfaces that do not transfer charge as we expect. Many
grain charging experiments occur in atmosphere; some experiments have shown that
40
charging in vacuum or a neutral gas produces vastly different charge patterns [29],
while others have suggested that humidity from the air adsorbed onto grains cre-
ates conductive paths during collisions that may alter the process of charge transfer
[8, 16, 17, 18, 49]. These possibilities must be explored to determine what conditions
are necessary in order to correctly predict grain charging.
In particular, the role of surface water in obfuscating experiment results cannot
be understated. Prior experiments indicate that the presence of an electric field
in a granular mixture can lead to charge exchange during collisions, leading to
a self-reinforcing phenomenon in which the charge separation continues to grow
with continued mixing [1, 18]. Zhang, et al, proposed a charge transfer mechanism
entirely based on the motion of dissociated ions in the surface water layer on the
grains, whereby the contact of two grains creates a conductive path across which
these ions can move and leave net charge on each grain after separation [18]. This
phenomenon may obscure the effect of contact-area-dependent charging processes
that would otherwise dominate the charge transfer in dry environments like deserts
and dusty airless bodies like the Moon. The fact that charging is known to occur in
these dry environments, however, suggests that these models may still be accurate,
although care must be taken to eliminate surface water when conducting experiments
to compare to the models.
41
3.3.3 Predicted Charge Magnitude as Compared to Experiments
In Figure 3.2b, we saw that our new contact-area-dependent charge transfer
model predicts a much lower charge magnitude than existing models for the same size
distribution. In fact, it has been noted that even existing models underestimate the
charge magnitude compared to grain charging experiments, leading to speculation
regarding the validity of the trapped electron model in general [21]. We believe that
the large difference in charge magnitude and the polarity reversal may be related
to a single phenomenon ignored by the models. In particular, we suspect that the
influence of atmospheric ions or adsorbed humidity encourages charge transfer in a
manner that neglects the effect of contact area, as discussed in Section 3.3.2.3. This
effect has been connected to increased charge separation in a number of experiments
[8, 16, 29]. The fact that many experiments have been conducted in atmosphere
or in the presence of other gases and have not involved pretreatment of the grains
to eliminate adsorbed water suggests that this confounding factor may indeed be
present, causing the results to diverge from the idealized case.
3.4 Notes on this Model
In the pursuit of a more realistic model for same-material granular insula-
tor tribocharging, we have built upon Lacks and Levandovsky?s original model and
added size-dependent charge exchange and the ability to consider arbitrary size
distributions, rather than discrete, bidisperse mixtures. The predictions made for
continuous size distributions are similar to those made for discrete distributions of
42
a similar form. In the limit where the continuous distribution approaches a discrete
distribution, the charge predictions converge. We have also modified the underlying
model for charge transfer by including a dependence on contact area in determin-
ing the number of electrons transferred in each collision. This model predicts that
mixtures of two size species with a smaller population of larger grains may dis-
play a reversed charge polarity (i.e., large grains charge negatively and small grains
charge positively) compared to Lacks and Levandovsky?s model. Leading theories
on insulator charging suggest that charging in vacuum or under different humid-
ity conditions may lead to different charging behavior, especially charge separation
magnitude and polarity of final charge. We are designing an experiment to test
the applicability of our model to granular tribocharging under various conditions,
particularly in vacuum where the absence of humidity may cause contact area to
play a more significant role. Future experiments will include pre-experiment baking
of the grains to remove the adsorbed surface water, to ensure charging events are
due to grain surface contact only. By experimentally testing for polarity reversal
in a humidity-controlled environment, we hope to gain additional evidence in sup-
port of, or contradicting, the trapped high-energy model for triboelectric charging
in insulators.
43
Chapter 4: Experimentation
This chapter contains material published in D. Carter and C. Hartzell, ?Ex-
perimental methodology for measuring in-vacuum granular tribocharging,? Review
of Scientific Instruments, 90(125105), Dec 2019 [50].
4.1 Existing Techniques
Many common experimental tribocharging measurement techniques are un-
able to resolve individual grain charges, or correlate precise grain sizes to the mea-
sured charge. Instead, techniques using charged probes [13, 36, 51] or Faraday cups
[30, 34] measure bulk charge polarity and/or magnitude of a particular subset of
the tribocharged sample, then measure the size distribution of that subset. These
methods are useful in measuring the degree to which charge is segregated by size,
but do little to characterize the magnitude of that charge or its distribution within
particular regions of the size distribution.
Another common limitation of existing experimental methods is the mecha-
nism by which grains are charged. Some experiments fail to control for particle-to-
container charging, causing the charge distribution to be significantly less represen-
tative of particle-to-particle charging, particularly when the container is made of a
44
different material [1, 30, 34]. In other cases, to avoid particle-to-container charging,
the mixture is agitated by fluidization with a gas, sometimes air [1, 21, 30, 34, 44]
or a presumably neutral gas like nitrogen [13, 36, 37, 38] or helium [29] to avoid
influencing the charging process. However, experiments have suggested that the
ambient atmosphere plays a large role in the magnitude and mechanism of charge
separation, even for gases like helium and nitrogen [29]. Although little data exists
on whether this influences the direction or relative distribution of charge within the
mixture, it is known that the humidity of the ambient atmosphere does affect the
charging process [8, 16, 17, 18].
There are experimental methods that provide the ability to correlate charge
with grain size in a non-invasive way [44], but many still use charging methods that
are influenced by atmospheric effects. We have adapted the high-speed videography
methodology described by Waitukaitis and Jaeger [44] to conduct the tribocharging
process in vacuum, minimizing the effects of container contact and atmospheric in-
fluence on the charging process. This method sends falling charged particles through
a uniform electric field and observes their trajectories from a co-moving camera. As
the experiment is conducted in vacuum, the effects of air resistance are eliminated,
leaving only gravity and the electric force to act on the grains.
Our experiment differs from that of Waitukaitis and Jaeger in a few key ways.
Primarily, the grains are kept in high-vacuum conditions throughout the charging
and measurement processes. Although our operational pressure is not low enough
here to ensure total elimination of adsorbed water from the atmosphere, we mitigate
the effect of humidity by keeping the chamber sealed and under vacuum when not
45
in use, storing the grains in a sealed container with desicant packs, and maintaining
an overall low-humidity laboratory environment. Future efforts could also include
using lower pressures and/or preliminary high-vacuum baking of the grains before
experimentation.
Additionally, we have designed our measurement and analysis software to mea-
sure individual grain diameters, rather than simply identifying large vs. small grains,
to work for arbitrary size distributions. Although this experimental teststand was
designed with silica beads on the 10?4 meter scale, it is applicable to a variety of
granular mixtures with arbitrary size distributions with diameters as small as 10?5
meters, or even smaller with the use of higher magnification lenses. This also allows
us to investigate the charging of complex mixtures like lunar regolith simulant.
4.2 Experimental Design
In our experiments, we have measured tribocharging in mixtures of zirconia-
silica (ZrO2:SiO2) beads from Glenn Mills, Inc. Silica is known to undergo strong
triboelectic charging and is the primary component of JSC-1A lunar regolith sim-
ulant. An accurate model for how zirconia-silica charges with itself will provide a
strong foundation for a regolith charging model and for understanding triboelectric
charge exchange in ceramics and other dielectrics in general. Most of the mixtures
used consist of approximately 80g total of two sizes of beads, with radii sieved to
100?m ? R1 ? 200?m and 200?m ? R2 ? 300?m. The two size fractions are mixed
in various mass ratios of small to large grain species, denoted by m. Note that the
46
two sizes used are not from the same fabrication batch, and that subtle differences
between them may contribute to charge exchange. Although this has significant
implications for the accuracy of our modeling, the results of the experiment still
validate the experiment?s effectiveness as a measurement tool.
4.2.1 Physical Setup
A photograph of the experimental apparatus is shown in Figure 4.1. The
design of this device was inspired by a similar mechanism by Jaeger, et al [21, 44],
which was developed to enable individual grain charge measurements. However, we
identified a few areas in which we could improve on this technique. First, the only
place where they used a vacuum was in creating a 10 mTorr vacuum in the grain
drop region, to eliminate air resistance on the falling grains. They charged their
grains separately by fluidization with air, then poured them through a funnel into
the measurement region. This additional handling, and especially the process of
charging in atmosphere, likely influences the charging in ways that are difficult to
predict and quantify. For this reason, we sought to conduct the entire experiment
in vacuum, and to minimize handling or otherwise disturbing the grains between
charging and measurement.
The other main way that we have improved the experiment is by reducing its
overall footprint and automating much of the process. Their original experiment was
2.5 m tall and required weighting the camera carriage to overcome air resistance,
using substantial padding to cushion the heavy carriage, and returning the device
47
Figure 4.1: Experimentail setup. The device is designed to be placed in a vacuum
chamber and operated at high vacuum. Grains in the canister are tribocharged via
shaking and dropped through a region containing a transverse electric field. The
camera is dropped and records the trajectories of the grains while both are in freefall.
to the top using a hoist. We sought a solution that could be implemented on nearly
any lab desktop, and that could be operated with minimal manual oversight and
intervention. The automation of the data collection and carriage return procedures
expedites the process of data collection once a mixture has been brought to pressure.
4.2.2 Experiment Procedure
Experimental data collection is conducted in a sequence of four phases: sam-
ple preparation and vacuum pumpdown, introduction of triboelectric charge into
the mixture, sample extraction and measurement, and reset of the experiment for
further measurement. The inclusion of this final phase allows us to take as many
measurements of the same prepared mixture as desired without breaking vacuum.
The primary limiting factor in continued sample collection is the camera temper-
48
ature, as there is little to no heat dissipation from the camera by conduction or
convection. For this reason, we are typically able to collect 5-8 videos of data for a
single experimental run, each containing anywhere from 20-200 trajectories. In the
following sections, we outline the design and operating procedure of our experiment.
4.2.2.1 Vacuum Chamber
Our experiment was designed to be compact and fit within our laboratory vac-
uum chamber. The influence of atmosphere on the charging process has been noted
in prior experiments, so properly controlling this variable is essential to understand-
ing the mechanism of charge transfer. The vacuum chamber used in this experiment
is rectangular, with internal dimensions of 24? wide by 24? deep by 30? tall, and is
shown in Figure 4.2a. The experimental apparatus is placed inside this chamber di-
agonally to allow the camera cables space to flex in the corner space without creating
friction when the camera is dropped. A 21-pin electrical feedthrough allows signals
to be passed into the chamber to the various motors and switches that control the
experiment. The camera receives and transmits data through a separate multipin
electrical feedthrough, and the high-voltage electrodes are also powered through a
separate high-voltage feedthrough. These connections are all shown in Figure 4.2b.
When a granular mixture has been prepared and placed in the experimental
apparatus, the chamber is sealed and evacuated. Pumpdown occurs in two stages.
First, an Edwards XDS 15i scroll pump brings the pressure down to approximately
0.3 Torr, taking up to 30 minutes to reach this pressure. At this point the Leybold
49
(a) The 24? x 24? x 30? Kurt J. Lesker vacuum (b) Sensors and electrical
chamber housing our experiment. feedthroughs for the vacuum chamber
in which our experiment is housed.
Figure 4.2: Vacuum chamber configuration.
Turbovac 450i turbopump can be activated. Considering the humid environment in
the Maryland area, and the inherent outgassing of many of the components in our
setup, we have found that this pump can consistently reach pressures as low as 9
?Torr after one week of continuous operation, or 27 ?Torr after approximately 16
hours. When operating this experiment, we prepare the chamber at the end of the
workday and are able to collect data at less than 30 ?Torr the following morning.
These low pressures not only reduce the influence of atmosphere on the charging
process, they are necessary to eliminate the drag force on the falling grains.
50
4.2.2.2 Grain Shaker
When the chamber has been evacuated to the desired pressure, the shaker can
be activated. A picture of the shaker mechanism is shown in Figure 4.3. The grains
are placed within a cylindrical steel canister made from a modified salt shaker. We
have fitted a circular aluminum insert with a single hole into the canister, extending
out through one of the many holes in the lid of the canister. As long as the swiveling
lid covers this small hole, it presses against the hole to seal the grains in while shaking
the canister, but when the lid is rotated, a narrow stream of grains falls through the
opening. The diameter of the opening was chosen to be 5 mm to ensure that the
? 300 ?m grains would fall without arching at the aperture, but would not pour so
densely that the grains could not be easily distinguished or tracked.
Our model for grain-to-grain tribocharging requires that all charge transfer
occurs between pairs of colliding grains, with minimal influence from collisions with
the walls of the container. Other experiments and simulations have shown that
grains allowed to mix within a container will experience significant charge segrega-
tion between the container walls and the grains, so electrically isolating the grains
from the container walls is essential. To ensure that the only collisions in our can-
ister occur between grains of the same material, we have coated the interior of the
canister with a layer of zirconia-silica grains, essentially treating the mixture bulk
as having an infinitely continuous boundary condition. Previous experiments using
the same technique have shown this to be an effective method of preventing grain-
to-wall charge transfer [44]. The grains are attached to the canister using a thin
51
Figure 4.3: The grain shaker and sample release subsystem of our experiment. The
central steel cylinder contains the granular sample and is attached to the vertically-
mounted speaker. When the stepper motor at right turns, the lid is rotated, exposing
the hole in the lid that is not blocked and letting a 5 mm stream of grains fall.
52
Figure 4.4: The canister in which grains are charged in our experiment. The interior
of the canister has been coated with grains of the same material as those to be
charged, so that no grain-to-wall charging occurs. When the lid is rotated during
sample collection, grains fall through the 5 mm hole in the lid at bottom left.
layer polyurethane adhesive, onto which a fresh sample of grains is pressed until all
gaps have been filled.
When the chamber has been pumped down to the operating pressure, as dis-
cussed in the previous section, the canister is vigorously shaken vertically to induce
triboelectric charge exchange. The speaker is powered through a Boss 1100W ampli-
fier, as shown in Figure 4.5a, to which we apply a waveform generated in LabVIEW.
The interface for operating the speaker is shown in Figure 4.5b. Using this VI, we
are able to control the frequency, amplitude (through the volume input), shaking
duration, and shape of the waveform. In benchtop tests with a transparent con-
53
(a) Amplifier that interfaces with the (b) Interface for controlling the shaker. User in-
shaker. Waveform data received as high- puts the desired frequency, time, shape, and vol-
level inputs from LabVIEW UI via Na- ume (amplitude) for the shaing waveform, then
tional Instruments Data Acquisition (NI initiates shaking with the ?Start Shaker? button.
DAQ) board and passed to the speaker. The indicators at top right track the total elapsed
shaking time, persisting through stopping and
starting where necessary. Pressing ?Close Con-
troller? aborts shaking and the VI entirely.
Figure 4.5: Shaker subsystem controller, external to the vacuum chamber.
tainer of the same size and weight, we have identified that with a square waveform
at approximately 14 Hz, the height to which the grains are launched during shaking
reaches a maximum, suggesting that this may be near the natural frequency of the
system. At amplitudes that loft these grains near the top of the canister, the grains
appear to be thoroughly mixed after cessation of shaking, exhibiting no apparent
size segregation. To minimize the heat losses in the speaker during shaking, which
can cause problems due to the vacuum environment, we set the volume slider to
10%, which limits the current through the speaker to approximately 1.5 A. The
grains are then shaken continuously for 5 minutes, thoroughly mixing them and
causing them to exchange charge.
54
Figure 4.6: View of the parallel-plate electrodes, as seen from the camera (bottom).
The background behind the electrodes is painted with an ultra-flat black paint to
minimize backscattered light. Note the alignment string visible faintly between the
electrodes.
4.2.2.3 Sample Release and Measurement
Once the grains are charged, the camera is turned on and the copper electrodes
are charged to their operating potential of 4000 V. A head-on view of the electrodes
as seen from the camera is shown in Figure 4.6 The electrodes are rectangular copper
plates, each 110 mm wide, 400 mm tall, and spaced 60 mm apart. The electrodes are
reinforced from behind with rigid copper bars to prevent bowing. The placement
and height of these electrodes provide 170-200 ms per video in which the grains
experience a consistent lateral electric force. Samples are taken from the canister
by activating a stepper motor, as seen on the right of Figure 4.3.
The camera is a PCO Dimax CS3 high-speed camera, shown on its carriage
in Figure 4.7. This camera is compact enough to fit in our chamber and is rated
55
Figure 4.7: The camera carriage at the top of the experimental apparatus. The
corner bracket (shown fixed to the electropermanent magnet) is attached to the
carriage and is released from the magnet when collecting data, causing the carriage
to fall along the low-friction guide rail.
for impacts up to 150 Gs. At the bottom of the carriage is a pair of circular rubber
bumpers, which compress when the falling carriage hits the base of the vacuum
chamber to absorb its kinetic energy and reduce the deceleration to an acceptable
20 Gs. The carriage is mounted on a low-friction rail, using a low-outgassing dry
lubricant to minimize friction drag as the carriage falls. Extending horizontally
over the top of the carriage is a steel corner bracket, which sits in contact with an
electropermanent magnet at the top of the experimental apparatus, which operates
as a reverse electromagnet. The magnet holds the carriage suspended at the top
of the low-friction rail until a 12 V signal is applied, creating an opposing induced
magnetic field that cancels that of the permanent magnet and allowing the carriage
to fall.
We record video with a Tamron 1:1 macro lens at 1000 frames per second,
56
yielding a resolution of 11?m per pixel. Many lenses that give higher magnification
are significantly longer, and although further magnification would improve the res-
olution of grain sizes, it would also dramatically reduce the length and number of
trajectories captured. The frame rate was chosen to maximize blur reduction and
trajectory length while ensuring sufficient illumination for resolving the images. At
1000 frames per second, most grains move no more than 1 mean grain diameter at
any point along their trajectory, making it easier to track their motion across frames.
This frame rate is also low enough that the grains can be illuminated sufficiently in
this confined space to calculate their centers and approximate diameters. Illumina-
tion of the falling grains is performed by two banks of LED strips mounted on either
side of the camera carriage track. Due to the compact nature of our test stand, there
is limited available space for mounting lights; the lens and closely-spaced electrodes
obscure the grains from most directions.
Sample collection and measurement is a time-sensitive process, and is auto-
mated entirely through the LabVIEW VI interface shown in Figure 4.8. Although
the canister aperture is small, we would like to minimize the size of the extracted
sample so that many runs can be conducted sequentially without significantly af-
fecting the mixture. The canister must be closed shortly after data collection to
keep from wasting samples. Furthermore, the camera does not have a low-power
standby mode; once powered, the camera takes up to 30 seconds to boot up and
then continuously reads data from the sensor until triggered to stream this data to
storage. Because of the high-vacuum environment, this causes the camera, among
other components like the LEDs, to heat up quickly unless depowered soon after
57
Figure 4.8: Virtual instrument (VI) for automating data collection. The user inputs
the desired frame rate (default 1000 Hz), charge time (default 5 min), a unique name
for the experiment, and the desired save folder. When these fields are complete, the
?Collect Data? button becomes active, as shown. When the camera is plugged in
and the ?Collect Data? button is pressed, the system proceeds through the stages of
data collection, constantly monitoring and displaying the camera?s thermal status.
If the camera begins to overheat or the user presses ?Cancel Run,? the system
quickly closes the canister (if necessary) and aborts all processes, allowing the user
to unplug the camera.
data collection is complete.
The VI prompts the user to input the desired frame rate and a descriptive name
for the sample to be taken, as well as the time for which the sample was shaken. Once
these fields have been completed, the ?Run? button is enabled. Before beginning
data collection, the user must plug in the camera to begin the bootup process and
power on the high-voltage electrodes. When the ?Run? button is pressed, LabVIEW
polls the camera for its status for 30 seconds before aborting. If the camera becomes
ready during this time, the system proceeds to prepare for data collection. First, the
58
camera is sent a set of commands to create the appropriate data buffers and set the
region of interest, frame rate, and corresponding exposure time. When the camera
is prepared for operation, the sample extraction stepper motor is activated, rotating
90 degrees so that the canister aperture is exposed and grains begin to stream down
between the electrodes. As soon as the stepper motor stops, LabVIEW applies
power to the LEDs and triggers the camera to begin recording, then applies power
to the electropermanent magnet to release the carriage. The carriage falls alongside
the grains, filming them in an approximately co-moving frame, before reaching the
base of the chamber after approximately 0.6 seconds. After 1 second elapses from
activation of the electropermanent magnet, LabVIEW tells the camera to cease
recording, the sample extraction motor rotates the canister lid back into place, and
the LEDs and electropermanent magnet are depowered. The camera continues to
read out captured images to a single binary file until all files have been read, at
which point the camera can be depowered.
In the final step of data collection, LabVIEW reopens the binary file contain-
ing all captured images and begins reading them out into individual TIFF images.
Using the known image dimensions from the specified region of interest, segments
of the binary file are read out into images until all images are created, then a brief
readme file is created containing the region of interest, frame rate, and exposure
time used to generate the data. In order to uniquely identify each video, a specific
name is created for the folder and as a prefix for each image. This name consists
of the parameters used to create the video appended together, such as ?ThreeQuar-
ter4kV 300s 1000Hz Oct11 1307,? which indicates that this video labeled Three-
59
Figure 4.9: ?Forklift? device for returning the camera carriage to the top after each
experimental run. The leadscrew uses a dry lubricant to reduce both friction and
outgassing, and is driven by a stepper motor at the top of the device. When the lift
returns to the bottom, it activates a limit switch and stops.
Quarter4kV was created for a mixture shaken for 300 seconds and filmed at 1000
frames per second on October 11 at 1:07pm. This allows us to identify key param-
eters of the mixture at a glance and when reading the data in for analysis.
4.2.2.4 Carriage Return
Because the camera must fall to the bottom of the device to collect data, we
must return it to the top without breaking vacuum to collect another set of data. We
have accomplished this with a forklift-like mechanism attached to a leadscrew, which
is moved up and down by a stepper motor controlled through a third LabVIEW VI.
The forklift mechanism is shown in Figure 4.9. The forklift raises the carriage until
the steel bracket reconnects with the electropermanent magnet, then moves back
down to the bottom of the device where it won?t interfere with the falling carriage.
60
This process can be automated using limit switches at either end of the lead-
screw to put the carriage in place and return the forklift to the bottom of the device
with a single button press. The bottom switch acts as a single pole double throw
(SPDT) switch in the ?normally closed? configuration, pulling the output voltage
up only when actuated by the forklift. The top switch is closed when the corner
bracket on the carriage makes contact with the electropermanent magnet, pulling
its output voltage down when the camera carriage reaches the top; the conductive
corner bracket is wired to the logic board?s 5V rail, and the outer casing of the
electropermanent magnet is also conductive, effectively creating a single pole single
throw (SPST) switch between the two. These outputs are connected to the motor
driver?s direction pin through a logical OR gate, with a digital high moving the
leadscrew carriage down; the leadscrew will always move upward unless only the
top switch has been pressed. If using a LabVIEW analog or digital input board,
the states of these switches can also be displayed, and the VI will check for the
condition where either both switches are pressed or the bottom switch is being re-
peatedly opened and closed; these conditions indicate that the carriage has reached
the bottom of the chamber.
4.3 Data Processing and Analysis
The collected data is processed in two phases. First, the images are passed
through an algorithm to identify grain locations relative to the reference string,
and manually inspected to add back missing locations and remove clumped and
61
Figure 4.10: Virtual instrument (VI) for automating carriage return. When a Na-
tional Instruments analog input board is connected, the ?Input Active? and top
and bottom contact indicators become active, and the ?Auto Return? function is
enabled. In manual mode, the user can input the desired return direction and speed
(lower speeds provide additional torque when necessary).
poorly-resolved grains. These positions are linked together across frames, giving us
an estimate for the acceleration, and charge-to-mass ratio, due to the electric field.
Next, we determine and subtract out the background brightness from each image,
leaving only the pixel values representing light reflected off the grains. The total of
these values for a single grain is proportional to its cross-sectional area, giving us
an estimate of the grain diameter, and therefore its mass.
4.3.1 Image Filtering and Grain Identification
Figure 4.11 shows a sample video frame obtained by applying a noise-reduction
filter to the raw data. The method we use for identifying and tracking particles was
developed by Crocker and Grier [52]. We apply a band-pass filter to identify objects
62
Figure 4.11: Sample frame, with and without features overlaid. At top, the noise-
reduced frame alone. At bottom, the same frame with grain locations (blue crosses),
string center (red line) identified. Grain trajectories (yellow arcs) are shown for the
30 frames following the one shown. Units are pixels.
of a specified size, which we will refer to as a ?standard diameter.? For the sample
mixture discussed here, grains have diameters ranging between 100 and 300?m, so
we use a standard diameter of 25 pixels, or approximately 280?m. This value was
chosen because the algorithm works best when the standard diameter is only slightly
larger than the objects to be found, and the distribution of observed grain sizes trails
off at approximately 270?m. After applying this filter, the result is then rescaled to
a value out of 255.
63
Figure 4.12: Interface for editing tracks found by the tracking algorithm. Tracks
are presented sequentially in the left window, and possible grain candidates are
presented in the right window. The user can navigate through each frame and add
or remove any grains to the track. Estimates for possible missing locations are
presented based on the calculated trajectory. Units are pixels in source images.
Two prominent features remain in the filtered image: the alignment string,
and the grains themselves. To find the string, we locate its center on the image?s
central horizontal row of pixels, then move toward the top and bottom of the image,
estimating the center along each row of pixels. These locations are used to calculate
a linear fit to the string, providing a reference to the camera?s minute variations
in roll and yaw as it falls. The grains are found by identifying all local brightness
maxima within a radius of a standard diameter. These estimated centers are refined
by finding the centroid of the pixel values in the vicinity of each point. This reduces
the influence of residual noise or asphericity on the identified center, but can cause
additional errors when two grains overlap or collide.
64
4.3.2 Trajectory Determination
The grains are accelerated laterally by the electric field, and many also have
some small non-zero vertical velocity relative to the camera. The trajectories formed
by these grains from image frame to image frame can be analyzed to determine the
charge-to-mass ratio of each grain. To construct these trajectories from the raw
image data, we find the least-mean-square displacement of all pairings of grains
with their nearest neighbors across each pair of frames, assigning a unique track
number to each set of positions. From the sample video described in Section 4.2,
this process identifies 402 trajectories, many of which are simply individual unpaired
grains; 125 of these contain at least 10 frames, for which the average length is 42
frames.
The filters in the previous section have a tendency to miss a small percentage
of the grains that do not meet the brightness threshold, causing some tracks to
be divided in two or more segments. This is exacerbated in the later frames as
grains gain speed, and may move so fast that their motion between frames exceeds
the maximum for tracking. These effects inflate the number of valid trajectories
found. Furthermore, while actual collisions between grains appear to be rare, close
crossings do happen on occasion. When they do, only the brighter grain is identified,
and one trajectory sometimes incorrectly continues with the wrong grain. Even if
these events are correctly flagged and rejected, the remaining tracks are primarily
incomplete, and it is advantageous to manually inspect and correct them to obtain
accurate trajectories. Frequently, two or or more grains adhere together, typically
65
a small and large grain but occasionally creating a large clump of many grains.
The algorithm often identifies this clump as one or more grains, but the size and
acceleration of its components are not usable data. These points also must be
rejected before processing the data.
We have developed an interface for examining and manually correcting trajec-
tories by adding or removing grains, searching for grains that were missed due to
overeager filtering, and deleting poorly-resolved objects like overlapping grains or
residual background noise. Each track is presented as a set of connected points, as
in Figure 4.12. The user can navigate from frame to frame using the ?Prev Frame?
and ?Next Frame? buttons, and each grain location is presented in each frame as
a candidate to be added, removed, or deleted using the corresponding buttons. In
addition, the interface allows for manually selecting regions that clearly contain
grains that were not identified by right-clicking. These grain locations can be added
back to the frame, enabling the reconstruction of long grain tracks that would have
otherwise been lost. With this method, we are able to condense the 402 trajectories
into 128, with 118 having at least 10 points and for which the average length is 54
points.
The acceleration due to the Lorentz force can now be calculated for each grain
trajectory by applying a second order polynomial fit. The forces that dominate the
motion of the grain here are the gravitational force Fg = m~g and the Lorentz force
due to the electric field, FE = qE~ . (Note that the string appears significantly angled
away from vertical in our videos; this is due to the uneven floor in our laboratory,
but can be corrected.) We define h? to be the horizontal unit vector orthogonal to the
66
direction of the gravity vector g? and parallel to the camera?s focal plane. Calculating
the angle ?vert of the string from the vertical, such that g? ? E? = sin (?vert), we find
that the net force acting in the direction of h? is F ~h = mah = qE ? h? = qEcos (?vert).
To determine the quantity q from this expression, we calculate the positions of
m
each point in a trajectory along the h? basis vector. For each point in the trajectory,
we calculate xh, the normal distance of that point to the string center. This corrects
for both the nonvertical gravitational vector and the camera?s yaw and roll, which
would otherwise introduce significant errors. We then apply a second order poly-
nomial fit to the positions, with the known elapsed time ?t = 10?3s between each
frame. The second-order coefficient afit is the acceleration orthogonal to gravity,
such that mafit = qEcos?vert. The root mean squared residual of this fit is denoted
here as ?a and recorded for each trajectory.
4.3.3 Size Determination
In order to use this experiment to measure charge segregation in arbitrary
mixtures with continuous size distributions, it is not sufficient to roughly catego-
rize grains as ?large? or ?small? based on a cursory analysis of the images, as is
frequently done in other granular tribocharging experiments. Instead, we have de-
veloped a technique for estimating an individual grain?s diameter from the images.
The method we use assumes that the total light B reflected from a grain of diameter
D onto the camera sensor is proportional to the grain?s cross-sectional area. As our
grains are nearly spherical and have diffusely reflective surfaces, this should provide
67
a reasonably accurate estimate of the size. We can then determine the diameter D
from the expression
?
D = C B (4.1)
for some empirically-determined scaling constant C. The brightness B is found
by summing the values of the pixels making up the grain in a particular image.
The constant C is measured for each material by placing a sample of grains on a
microscope slide, placing it in the experimental apparatus where the grains would
be, and taking a high-speed video of the slide under normal operating lighting
conditions. The slide is then examined under a microscope (see Figure 4.13), where
the actual spherical diameter of each grain can be determined. The background
brightness, which has a mean pixel value of approximately 20-25, is subtracted out
to give a mean-zero background level, and the pixel values in the vicinity of each
grain are summed to determine B. We can then find the average value of ?R among
B
all grains, which becomes our scaling constant C.
A major challenge we face is the prevalence of ?motion blur? in our images.
When filming these grains at 1000 frames per second, our camera has an exposure
time of ?t = 944?s. Some of our smaller grains can reach speeds that cause them to
travel upwards of 20-30 pixels between frames in that time, causing them to appear
elongated along their direction of motion. Our sizing method is resilient to blurring,
as the window around each grain is large enough to include all photons reflected onto
the sensor by the grain regardless of its speed or level of focus, but blurry grains do
68
Figure 4.13: Large size fraction of zirconia-silica grains (diameters 200 ?m to 300
?m) under microscope. Grains are highly spherical with uniform smooth surfaces,
with a few exceptions.
present a challenge for identifying and removing the non-zero-mean background.
4.3.3.1 Sizing Procedure
First, we must remove noise from the image that originates from current leak-
age in the camera sensor. Known as ?dark current?, this is a function of the camera
itself and can be found and corrected by taking a video without any illumination
(e.g., with the lens cap on). The frames of this dark video are then averaged to-
gether on a pixel-by-pixel basis to create an average frame representing the dark
current. This is then subtracted from each frame of our video. Next, we determine
the mean remaining background pixel value. A mask is applied to each frame that
selects only regions more than a standard diameter away from the string or any
grain, and the mean value of these pixels is calculated. This value is the mean value
of the background due to ambient light, and subtracting it from each frame ensures
that for any box drawn around a single grain, the total value of the contained pixels
69
represents only the light reflected by the grain itself.
We then determine an appropriate window around the grain in which to cal-
culate its brightness. The width of this window is 1.5 standard diameters and is
centered on the grain to be measured, ensuring that the entire grain is enclosed
without being so large as to include other nearby objects. This subset image is then
extracted, such that the center of the subframe (xs, ys) is centered on the grain?s
location (x, y) in the frame. Next we check that the grain fits within this window.
We identify the trajectory to which the grain belongs, and estimate its velocity
v = (x?, y?) at the location in question. The grain?s displacement is then calculated
as (?x,?y) = (x??t, y??t) pixels. If the dimensions of the window are not larger
than this displacement plus a standard diameter, we cannot be sure we have fully
captured the grain, and this location is rejected. Otherwise, sizing is done by sum-
ming the pixel values contained in the subframe and applying the conversion from
that value B to the grain diameter D defined by Equation 4.1.
Finally, because many of the grains are slightly out of focus or highly elongated,
we manually compare the average diameter of each grain trajectory to the images
of its positions. Some of these require manual correction, and some are too poorly
resolved to be confident in the results obtained. After all grains have been confirmed,
the results are saved and charge determination can begin.
To obtain an estimate for the uncertainty involved in this method, we return
to the calibration images described in Section 4.3.3. We apply this sizing algorithm
to the images taken under experimental conditions, obtaining a set of diameter
estimates Dcalc. These values are then compared to the corresponding sizes as
70
determined under microscopic analysis, and the root mean square residual is found
for each size species. We have measured a root mean square residual for diameter
measurements of 8.1 ?m and 8.4 ?m for small and large size species, respectively.
Henceforth we assume that each individual diameter measurement has an associated
uncertainty of ?D = 8.4?m. To ensure that this method is robust enough to handle
blurry and in-focus grains, we took a series of ten images using a single slide to
which grains had been attached, placing it at varying distances from the camera
lens. We found that, as long as the mean background brightness level was correctly
calculated, this method achieved the same accuracy regardless of blurriness.
4.3.4 Calculation of Grain Charges
Given the measured grain size and grain acceleration, it is possible to derive
the charge on an individual grain. From the grain?s acceleration fit a qfit = ,m
the estimated diameter Dest, the material density ?, and the electric field strength
E = V , the charge q can be expressed as follows:
L
ma 3fit ??LD
q = = est
afit
(4.2)
E 6V
In this setup, we have applied a voltage of V = 4 kV to the parallel plates, with
a spacing of L = 60 mm between them. These charge measurements are subject to
a few significant sources of error, the largest of which by far are the uncertainties in
afit and Dest. Here we assume that these uncertainties are significantly larger than
errors in the applied voltage, which we have found to be accurate to less than 1%
71
variation, and in the infinite parallel plate model for the electric field.
The uncertainty associated with the charge q can therefore be expressed as a
combination of the uncertainties ?D and ?a as follows:
???? ( ) ( )? 2 2D ?a?q = q 9 + (4.3)
D afit
A sample plot of charge measurements is given in Figure 4.14. Vertical error
bars are given by the acceleration fit uncertainty ?a, and horizontal error bars are
given by the sizing uncertainty ?R. These results include 128 trajectories collected
from four camera drops. We have plotted the charge-to-mass ratio rather than the
total charge as the former is determined from the acceleration afit and independent
of the size measurements. Note that the large grains carry a negative charge, while
the small grains are charged positively. This is a reversal from the trends observed in
most previous experiments, where the smaller grains tend to charge negatively and
the large are positive. This polarity reversal, with small grains charging positively,
agrees with predictions of our semi-analytic model [45] and is consistent across
experiments on different days and with different mixture ratios. We propose that
this charging polarity was not observed in prior experiments due to contamination of
prior experiments via atmospheric effects and/or charging other than grain-to-grain
charging.
Although the tested size distribution is not entirely discrete, it is useful to ex-
amine these results in terms of large (greater than 200?m) and small (less than
200?m) grains. We observe that the small grains have a mean charge-to-mass
72
Figure 4.14: Results of experimental analysis of 1:1 (by mass) mixture of small (100-
200 ?m) and large (200-300 ?m) spherical silica beads. The distribution of observed
grain sizes from experiment videos is compared (at right) to the actual distribution
as measured under microscope.
ratio of (3.22? 2.64) ? 10?5C/kg, whereas the large grains have a mean charge-
to-mass ratio of (?2.33? 1.21) ? 10?5C/kg. Including their calculated diame-
ters and associated uncertainties, we find that the small grains have an average
charge of qS = (1.17? 1.09) ? 10?13C and the large grains have an average charge
q = (?5.34? 2.33)? 10?13L C. Note that the large error bounds on these values are
only in part due to measurement uncertainties; granular tribocharging is a stochas-
tic process that results in a wide spread in charge for a single species, as in Figure
4.14, and this is reflected in the error bound on the mean charge.
To verify that this charging is due solely to grain-to-grain charge exchange oc-
curing in vacuum, we have conducted tests using the same mixture and experimental
conditions, but without activating the shaker. In this experiment, we found that the
small grains measured carried an average charge of qS (9.03? 68.3)?10?15C/kg, and
the large grains carried an average charge of qL (?5.35? 12.2) ? 10?14C/kg. The
73
mean charge on each species without shaking was at least an order of magnitude
less than after shaking, suggesting that the majority of charge exchange does indeed
occur in vacuum. We also note that the variance of the charge around the mean
is significantly larger than the mean itself, indicating that there is little evidence
of any significant correlation between size and charge polarity before shaking. We
attribute the presence of what little charge is present to charge exchange during
transport and preparation of the grains, but we remain confident that the dominant
mechanism creating the charge segregation observed is granular tribocharging in
vacuum.
74
Chapter 5: Experimental Results
The granular mixtures described in this chapter are composed of two size frac-
tions of zirconia-silica (ZrO2:SiO2) beads from Glenn Mills, Inc. The size distribu-
tion measured for each component is shown in Figure 5.1. Each mixture described in
this chapter is created by measuring out masses m1 and m2 of the larger and smaller
components, respectively, and combining them. This mixture is then described by
its mass ratio m = m2 .
m1
5.1 Charge Exchange Models
Before presenting the collected data, we will first outline a set of models that
are based on the continuous size distribution framework described in Chapter 3.
We will compare these models to the observed charging phenomena to assess their
relative accuracy and draw conclusions about the nature of triboelectric charge
exchange. These models represent a variety of mechanisms previously proposed for
the triboelectric charging of same-material dielectrics.
75
Figure 5.1: Size distribution of small and large mixtures, as measured under micro-
scope.
5.1.1 Inelastic Deformation
In Chapter 3 we proposed that charging was due to high-energy electron trans-
fer [45], where the charge exchanged in a single collision between grains of radii Ri
and Rj is proportional to the contact area Aij formed by Hertzian deformation of
their surfaces. We represent this mechanism in the model labeled ?Hertz.? It has
been observed that larger spheres and flat surfaces exchange charge proportional to
their surface area in contact during even a single impact [46]. This model attempts
to capture this phenomenon by describing the maximum contact area in common
between two grains that deform due to their kinetic energy at the point of impact.
5.1.2 Constant Transfer
In our initial discussion of same-material charge transfer, we noted that many
early models simplified the collision model by assuming that each collision transfers
a fixed number of electrons [11]. The Hertz model we proposed as a replacement
76
relies on the assumption that the grains collide with enough energy to experience
appreciable deformation. Theoretically, the actual energy and elastic modulus divide
out, and only the relative contact area in the space of possible grain pairs is necessary
to calculate the charge distribution. Realistically, the low collision energies and
small size of the grains may result in deformations so small as to provide very
little difference in charge transfer among sizes. In this case, the Hertz model would
simplify to the constant contact area model. We have therefore included this model,
labeled as ?Constant,? as a point of comparison to other models.
This model remains suspect for the same reasons mentioned in Chapter 3,
and most models today suggest that the quantity of transferred charge is a function
of the sizes of the two grains. However, we have included it here for the sake of
completeness. It also serves as a reference point for the degree to which collisions
cause charge to segregate by size.
5.1.3 Combination of Size and Material Differences
While we have assumed up to this point that the two size fractions of grains
that make up our mixtures are chemically identical, the possibility exists that there
may be small material differences that provide an alternative size-dependent transfer
mechanism. The two size fractions were obtained pre-sieved from Glenn Mills, Inc,
and it is unknown whether they came from the same production batch. While
batches should be roughly identical, it is prudent to include a model that takes
into account material-dependent transfer. This model is denoted as ?CombofX?,
77
where f = X is the factor by which material transfer dominates over size-based
transfer. Both mechanisms are based on the ?Constant? model described above,
but the number of transferred charges is X times larger if moving from a grain
smaller than some diameter Dmid to a grain larger than that size. Here we have
chosen Dmid = 185?m.
5.1.4 Exponential Decay
Other models for electron transfer have proposed that any electron is capable
of transferring between contacting grains through tunneling [25]. Jaeger, et al,
attempted to measure the density of high-energy electrons on zirconia-silica grains
and determined that the number of transferred charges far exceeds the available
population [21]. If ?regular? electrons are also able to transfer simply by tunneling,
this could explain the discrepancy between these observations.
Our model of this phenomenon is simplified for ease of application to our
framework. To determine the number of charges transferred from one grain to
another, we consider a single point on the surface of the first grain and find its
minimum distance r to the surface of the second grain, as shown in Figure 5.2. The
probability that an electron at the first point would transfer across to the second
grain is then proportional to e?kr. Assuming this point has area dA and contains
?dA electrons, we can then integrate this value over the surface of the first grain that
is able to ?see? the second grain, giving the average number of electrons transferred
in a single collision. While this is intended to represent electron tunneling, it can
78
Figure 5.2: Diagram illustrating the exponential decay model. For each point on
the surface of grain i with area dAi, the probability of transfer P is multiplied by
the number of charge carriers ?idAi. The total number of transferred charges per
collision is found by integrating P?idAi over the entire surface of grain i.
also represent the transfer of positive ?holes.?
5.1.5 Total Surface Area
Some experiments have suggested that the presence of an adsorbed layer of
water from the environment has a significant effect on the polarity and magnitude
of the charge distribution. In many such models, ions dissolved in the water layer
are able to transfer to the contacting surface, facilitating charge transfer on the
surfaces of dielectrics where charges would otherwise be immobile. We have created
a simplistic representation of this model, labeled ?SurfA,? in which the available
contact area is equal to the entire surface area of the grain.
5.1.6 Collision Models
When determining the rate at which a grain collides with its neighbors, we
must first determine the relative speed vr with which it travels in the mixture. If
collisions occur infrequently, and therefore the grain has only the speed imparted
79
to it when launched by the moving canister, our initial assumption may hold, but
when collisions are frequent, the grains are more likely to redistribute their energy.
Additionally, many regions of the mixture do not experience anything like the ide-
alized ballistic collision model, instead experiencing inelastic collapse in which the
grains encounter the rising canister floor and lose all their energy in a large number
of rapid collisions. We have included two models in our analysis that treat the grains
as conserving their speed, as initially assumed, and their energy.
5.1.6.1 Constant Velocity
In models denoted by the term ?ConsV,? we apply our initial assumption that
all grains move through the mixture at approximately the same relative speed vr.
The shaker is mixed by vigorously shaking it vertically using a square waveform,
causing all grains to be launched upward at the same time. At this point, they all
have the same speed: that of the bottom of the canister on the upward stroke. This
form of the model captures the collision rates as they occur at this point in time.
5.1.6.2 Constant Energy
As the launched grains collide with their neighbors, and especially as they move
side to side, they quickly redistribute kinetic energy such that each grain takes on
approximately the same energy E0. In models denoted by the term ?Co?nsE,? we cal-
culate a grain?s approximate relative speed relative to the bulk as v = 2E0r ? R?
3
2 .
m
Because constant multiplicative factors in the relative speeds divide out in calcu-
80
Figure 5.3: Results for mixtures composed entirely of large grains. Note the high
variance in the measured charge-to-mass ratio compared to the mean value. For this
nearly monodisperse case, little charge segregation occurs based on size.
lating the charge distribution function, only this relationship to R is relevant. This
form of the model may better describe the collisional behavior in this experimental
setup.
5.2 Results
The results for mixtures with mass fractions m = 0, 1 , 1 , 1 , 3 , 1, and 2 are
8 4 2 4
shown in Figures 5.3, 5.4, 5.5, 5.6, 5.7, 5.8, and 5.9. A clear segregation of polarity
according to size can be observed, with small grains charging predominantly posi-
tive and large grains charging negatively. Furthermore, as expected, the stochastic
nature of charge exchange causes a wide variation in charge magnitude, and even
polarity to an extent, at each grain diameter in the mixture. However, clear trends
in polarity and magnitude, and especially their variation with both diameter and
mass fraction, can be observed.
81
Figure 5.4: Results for mixtures composed of 1:8 small to large grains, by mass. Note
the irregular shape of the observed sizes; this is due to the lower quantity of data
relative to other size fractions. When the number of large grains increases, many
small grains adhere to them while falling, so fewer clean trajectories are observed.
Figure 5.5: Results for mixtures composed of 1:4 small to large grains, by mass.
82
Figure 5.6: Results for mixtures composed of 1:2 small to large grains, by mass.
Figure 5.7: Results for mixtures composed of 3:4 small to large grains, by mass.
Figure 5.8: Results for mixtures composed of 1:1 small to large grains, by mass.
83
Figure 5.9: Results for mixtures composed of 2:1 small to large grains, by mass.
In Chapter 3, we predicted a reversal of the traditionally observed polarity,
such that small grains would, on average, acquire a more positive charge, while
large grains would acquire a negative charge [45]. This trend is supported by our
results. There are a number of possible explanations for this effect. For one, the
charging mechanism and environment are markedly different from that utilized in
prior experiments. Other experiments frequently charge grains by fluidization a bed
of grains with air or another gas [13, 21, 30, 34, 36]. In many cases, the results of
these experiments have been examined through the lens of a collision model built
on the same foundations as ours. As previously noted, conducting our experiment
entirely in vacuum reduces the influence of recruited ions [29] and adsorbed water,
which may be responsible for the prior observed polarity.
Additionally, we suspect that the collision rates assumed in prior models are
poor representations of prior experimental methods. When the grains are fluidized
from below, collisions primarily occur when grains launched by the flowing gas
reimpact the bed. In addition to the significant rolling contacts that occur as the
84
Figure 5.10: Data collected for the same mixtures on different days. All experiments
shown high variance in charge magnitude and polarity, but approximately the same
mean.
grains settle, there is likely to be a significant segregation of launched grains by size,
with small grains launching more frequently and more energetically, which may skew
the results. In some cases, these experiments also do not insulate the walls with
the same material as the grains to prevent grain-to-wall charge transfer, allowing
some of the more energetic grains to transfer charge in ways not accounted for by
the collision model.
Each mixture presented in Figures 5.3 through 5.9 includes data taken from
multiple videos of each of multiple separately-prepared mixtures. We display these
identical mixtures, charged and measured in separate vacuum pumpdown events,
on the same plot because we have observed that the charge magnitude and polarity
crossover point are consistent in our experiments for mixtures with the same mass
ratio. All mixtures are prepared in a consistent manner, but it is prudent to verify
that each mixture still produces consistent results. In Figure 5.10 we can see the
measured grain charge distributions for individual pumpdowns of select mixtures.
85
5.3 Model Fits
Each of the models described above assumes that the surface of each grain car-
ries some finite initial density ?0 of mobile charge carriers. Recall that the charge
distribution function derived in Chapter 3 is directly proportional to this initial den-
sity. We have found the best fit of each model to the experimental data by varying
this parameter. The charge carrier density ?0 should be the same for all mixtures, so
this parameter was found with a least-mean-square fit of the corresponding model
predictions to all the data, weighted so that each mass ratio is weighted equally.
The goodness of this fit is then measured using the R2 value, which measures the
ability of the model to account for the variance in the charge-to-mass ratio with
diameter and is calculated as:
2 SSER = 1? (5.1)
SST
where SSE is the sum of squares of the residuals between the predicted and mea-
sured values, and SST is the sum of squares of the measured values about their
overall mean. The results are shown in Figure 5.11. Note that the most accurate
fits occur when ?0 < 0; that is, the transferred charge species is a positively-charged
species, rather than electrons. This has been observed in other experiments, leading
some to suggest that the transferred charge species may in fact be OH? ions or
other ions recruited from the atmosphere [29]. The lack of an appreciable ambient
atmosphere in our experiment rules out that possibility, but we cannot definitively
86
Figure 5.11: Goodness of fit over all data for the overall best-fit value of ?0. Some
experimental runs exhibited higher variance than others, resulting in poor fit confi-
dence, but some fits are clearly better than others. Many indicate a negative charge
carrier density, suggesting a positive species given our previous assumption of a
negatively charged carrier.
say that there is no adsorbed water, as substantial bake-out is typically required to
eliminate all water.
On first inspection of the goodness of fit curves, we note that many of the
models follow the same patterns, fitting the data nearly equally well at all mass
fractions. Unfortunately, we believe this to be a problem inherent to the high
variance in the data itself, as has been observed in other experiments [13, 37, 38, 53].
R2 essentially measures how well the fit explains the variance in the data compared
to simply taking the mean of the data. However, when the variance is already so
high and the mean of both the data and the fit are comparatively very small (as
we expect net charge to be approximately zero), there is a risk of both the sum of
87
squares of the residuals and the total sum of squares being very large, in which case
even reasonable models end up with R2 values near or even below zero.
While many of the models demonstrated promising results, we can draw some
conclusions from those that did not. For the Hertz ConsV and SurfA ConsV models,
and to a lesser extent the combination model as well, we see that as the mass ratio
increased, these models displayed increasingly poor performance. Following the
curves of the constant-velocity Hertz and total surface area models, we can see that
these models have zeroes at points that shift dramatically with changing mass ratio.
For many mass ratios, these zeroes move near the centers of the size distribution
peaks, suggesting a reversal occurring near that size fraction?s mean diameter. This
is not a trend that we see in the data, and the other models examined have polarity
crossover points consistently nearer the midpoint between the two sizes.
The Combo models appear to fail for other reasons. This model has a dis-
continuity at the designated point separating large and small grains; this is to be
expected, as the magnitude of charge transfer jumps dramatically for grains on either
side of that point. While this model was, by design, able to consistently predict a
polarity reversal between the two sizes, there is a great deal of inconsistency between
this model and the magnitude of the charge segregation observed in the mixtures.
Therefore, even if a material difference is present and does affect the charge dis-
tribution, we suspect that the effect of size differences still occurs to a pronounced
degree, even dominating over the material differences.
Each of these least-accurate mixtures assumed a constant relative velocity for
grains moving through the mixture during charging. Among the remaining models,
88
there is little difference in fit quality between models based on conservation of speed
or energy. This is likely because, of all models presented, only the ?Hertz? model
uses a collision area dependent on the impact energy. In an equal-energy model, the
large grains move slower than the small grains, but both still move against the same
background of target grains. The collision rates against each species are different,
but only in the Hertz model is the quantity of transferred charge in a single collision
also affected by the equal-energy assumption.
Of all the models explored here, the model with the highest average fitness was
the Hertz model [45], with the ConsE assumption. Many other models with the same
energy assumption had similar fitness, and each of them required a negative density
of charge carriers. Because our model assumes a negative charge carrier species, this
is equivalent to the conclusion that the charge carriers are positively charged, with
surface densities on the order of 8-47 charges per ?m2. Within this range, the lower
values are based on the equal-energy models, with the higher values arising from
the equal-speed models. This reveals that, when small grains have greater mobility
than larger grains, the segregation of charge throughout the mixture becomes more
pronounced. These results may also suggest that the collision energy plays a role in
determining the quantity of charge transfer in a single collision, an effect captured
by the Hertz model but not by the others presented here.
We note that, while the models presented here may not accurately capture the
charging behavior on their own, these results demonstrate a method for assessing
the viability of any arbitrary form of the collisional charge transfer. By using our
model framework to generate a predicted charge distribution based on a transfer
89
mechanism, one can directly compare properties of the distribution to that of ex-
perimental results. For example, we noted that some transfer mechanisms yielded
predicted charge distributions with two zeros, or with a single zero that shifted sig-
nificantly with mixture mass ratio. As these behaviors were not observed in the
experimental data, we are able to more confidently dismiss those proposed charge
transfer mechanisms. We believe that more detailed analysis of the relationship be-
tween transferred charge per collision and the sizes and collision rates of the grains
involved can yield quantitative measures for the validity of these models, which
should dramatically improve the ability to identify the most likely candidates.
5.4 Conclusions
Using the experimental apparatus we developed, we measured how charge is
distributed in mixtures composed of grains with two primary sizes. Despite the
high variance in charge that naturally arises from the stochastic process of charging,
which has impeded others? prior attempts to obtain such distributions, we are able
to draw some conclusions about the charging process. We noted that the results
of our experiments suggest that a positive charge species is likely to be responsbile
for the observed charging, and we have presented a framework for determining the
likelihood that a particular charge transfer mechanism is driving the charge transfer
by comparing predictions of our model with the experimental results. This procedure
can be followed to identify likely candidates based solely on the statistical properties
of the predicted charge distribution.
90
Chapter 6: Conclusions and Next Steps
6.1 Contributions
This work has explored an elusive component of same-material granular dielec-
tric tribocharging, a phenomenon which presents both challenges and opportunities
for both terrestrial industries and space exploration. The small size of these grains,
and their tendency to experience charge transfer or discharge when coming into con-
tact with any other surface, make it difficult to measure individual grain charges in
large quantities without manipulating the charge through the act of measurement
itself. While it is clear that size-dependent charge segregation trends are occurring,
the variance in the charge of a particular size fraction is significantly larger than the
mean charge of that fraction. This makes it challenging to model or even observe
the trends of the mean charge without a technique for accurately measuring the size
and charge of individual grains. Our experiment provides the capability to do just
that, in a compact mechanism that could be repeated in other laboratories with
similar-sized vacuum chambers.
The establishment of an underlying framework for examining continuously dis-
tributed size distributions is a notable improvement itself. Many of the models that
we explored through our predictive framework displayed different results for pseudo-
91
bidisperse continuous distributions, in which the continuous distribution consists of
the sum of two normal distributions, than for the discretely bidisperse mixtures
for which they were first designed. Our model can be used with any form for the
charge transfer, relative grain speed, and even collision frequency, to predict how
mean charge might be distributed in a mixture and identify better candidates to the
transfer mechanism. This model, together with our robust experimental method for
correlating charge with individual sizes, is a useful tool for exploring tribocharging
in dielectric mixtures.
We have experimentally measured tribocharging in a broad range of zirconia-
silica granular mixtures under ideal conditions. The highly spherical shape of the
grains and our commitment to conducting all charging and measurement processes
in vacuum creates conditions in which only material effects from grain-to-grain col-
lisions are responsible for the tribocharging observed. Our data presents consistent
observations that smaller grains charge positively and larger grains charge nega-
tively under these conditions, a reversal from the results of many other experiments
conducted in air and similar environments. This may be indicative of a confounding
factor like ion recruitment or adsorbed humidity that has influenced the charging in
those prior experiments. Our data can be used to identify where these effects may
be occurring and guide the direction of theories for the underlying charging behav-
ior in the absence of atmosphere, such as in airless environments like the Moon. A
model such behavior could be used for in-situ resource utilization on the Moon or
other dusty bodies for simple size and/or material beneficiation of the regolith.
Our predictive tribocharging model, when modified to assume equal kinetic
92
energy among grains and positive charge carriers, was among the most accurate
models when compared to the charging data. We also identified other candidates for
the charging mechanism that had reasonable accuracy as well. In particular, using
a positive species with the exponential transfer model, though initially suggested to
explain electron tunneling, gave a more reasonable fit compared to the other models
presented. We also observed that in most cases, fewer charge carriers were necessary
to account for the same degree of charging when assuming equally distributed kinetic
energy than equal speed. Considering one of the most difficult questions in this area
is how to reconcile the apparently low density of mobile charge carriers with the
large grain charge magnitude resulting from tribocharging, this may suggest that
collision models closer to the equal energy model are more accurate for our experient
than those closer to constant velocity. Although assessing model fitness is difficult
due to the high variance in charges, we believe that these results are promising and
provide a path toward identifying more accurate models in the future.
6.2 Next Steps
With the development and verification of the results of our experiment, there
are countless avenues for future investigation of granular mixtures using the same
methods. We have primarily focused on pseudo-bidisperse mixtures of zirconia-
silica, but the same experiment could be as easily conducted with other materials
like glass or lunar regolith simulant, in any combination of sizes. For mixtures
containing different materials or grains with irregular surface reflectivity, it may
93
be necessary to find solutions that increase the illumination on the grains, or that
provide additional magnification.
We took many precautions to reduce the amount of water adsorbed onto the
surface of our grains. The air in our laboratory was cycled through a dehumidifier,
the grains were stored in sealed plastic bags with desicant packets to collect any
residual moisture, and the chamber was kept under pressure when not in use to
reduce adhesion of water to its interior. However, we do not have the means to
verify conclusively through spectroscopic tests that water was not present on the
grains during charging. It is likely that some water remained, though it is unclear
whether this would be enough to have a significant impact on the charging process.
To eliminate this effect, the grains should be baked at temperatures above boiling
for a significant time (a day or longer) and promptly placed in the canister. Future
iterations of this experiment may benefit from implementing these preprocessing
procedures.
The experimental data we presented, combined with the predictions made by
our model, are highly suggestive of a positive transferred charge species. Additional
experimentation and analysis may provide more evidence for a particular mechanism
behind this phenomenon. We recommend exploring complex, fully continuously
distributed mixtures with broad size distributions, where the gap between the size
fractions (and typically, therefore, the polarity) is not so large.
We also believe there is significant utility in exploring precisely how a proposed
transfer species and mechanism relates to features of the predicted mean charge
distribution generated by our model framework. For example, we observed that for
94
certain forms of the collision contact area, the point of zero polarity on the predicted
mean charge changed significantly with mixture mass ratio, but for others, it did not.
It is probable that the rate of change of the quantity of transferred charge relative
to the rate of change of the size of the target grain, which is a property of the
proposed mechanism alone, can be directly correlated to parameters like the grain
size at the polarity crossover point. Developing an comprehensive understanding
of this relationship would provide a method for quickly assessing the feasibility of
any transfer mechanism. This, along with an in-depth study of other mixtures and
materials, may provide the answers to the elusive nature of size-dependent charge
segregation in granular media.
95
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