COLLECTIVE HYDRODYNAMICS OF SOFT MICROPARTICLES IN
QUASI-TWO-DIMENSIONAL
CONFINEMENT
by
WILLIAM ERIC USPAL
B.Phil., Engineering Physics, Mathematics, and Philosophy, University of Pittsburgh
(2007)
Submitted to the Department of Physics
in partial fulfillment of the requirements for the degree of
MASSACHuSETTS
INS1T!
OF TECHNOLOGY
Doctor of Philosophy in Physics
JUL 0 1 2014
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
LIBRARIES
February 2014
@
Massachusetts Institute of Technology 2014. All rights reserved.
__Signature redacted
Author__
/
Certified by.
Department of Physics
October 7, 2013
Signature redacted
Patrick S. Doyle
Singapore Research Professor of Chemical Engineering
Thesis Supervisor
/
II ,
Certified by
Signature redacted
Francis Fr edman
Signature redacted
Mehran Kardar
rofessor of Physics
Thesis Supervisor
Accepted by
Krishna Rajagopal
Associate Department Head for Education
E
Abstract
3
Collective Hydrodynamics of Soft Microparticles in Quasi-two-dimensional
Confinement
by
William Eric Uspal
Submitted to the Department of Physics
on October 7, 2013, in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy in Physics
Abstract
Flow of microparticles through geometrically confined spaces is a core element of most microfluidic technologies. Flowing particles are typically ordered and manipulated with external forces or
coflowing streams, but these methods can be limited in generality and scalability. New techniques
to control particle trajectories would enable new applications in such areas as materials assembly,
optofluidics, and miniaturized "on-chip" bioassays and cytometry.
Recently, researchers have sought to understand the conditions under which particles can organize themselves through interactions generic to the flow of suspensions through microchannels.
In particular, a particle moving through a viscous fluid will create a disturbance flow, affecting
the motion of distant particles. These hydrodynamic interactions (HI) are sensitive to particle
shape and the presence of confining boundaries. This sensitivity presents a powerful opportunity:
particle trajectories could be "programmed" into particle morphology and channel design. These
could chosen so that many-body hydrodynamic interactions drive self-organization of the desired
particle motions. Even a single particle could be designed to "self-steer" to a desired position in
the channel cross-section through its hydrodynamic self-interaction.
In this thesis, we present a series of studies exploring new possibilities for achieving selforganization, self-steering, and other flow-driven collective phenomena via design of particle shape
and channel geometry. We focus on a particular setting: quasi-two-dimensional (q2D) confinement,
in which particles are tightly "sandwiched" between parallel plates, free to move in only two dimensions. In this confinement regime, hydrodynamic interactions take a unique dipolar form. This form
had been shown to sustain novel collective phenomena with much greater spatiotemporal coherence
than can be achieved in unconfined or weakly confined suspensions. However, self-organization of
q2D suspensions had not been demonstrated prior to our studies.
Starting from a two-body problem, we progressively consider larger numbers of particles and
more complex particle shapes. In our first study, we develop model equations for the coupled
motion of two discs in a quasi-two-dimensional channel. Numerically, we find that a pair can
form a hydrodynamic bound state with complex oscillatory motion. We demonstrate that this
"quasiparticle" can be manipulated via patterning of confining boundaries. In the following
study,
we consider larger clusters of discs. We provide symmetry principles for the a priori construction
of "flowing crystals": configurations of particles that maintain their relative positions as they are
carried by the flow. The crystalline states generalize the two-body bound state to more complex
configurations and collective modes. We also consider the wider dynamical landscape, finding
4
Abstract
metastable states with new, exquisitely coordinated particle motions. However, neither flowing
crystals nor metastable states spontaneously form from a disordered configuration of discs.
In pursuit of self-steering and self-organization, we turn to particle shape, and study the dynamics of a single "dumbbell" comprising two connected discs. We find that a fore-aft asymmetric
dumbbell will reliably align with the flow and focus to the channel centerline. In contrast, a
symmetric particle will oscillate between the channel side walls indefinitely. Through theoretical
arguments, we isolate three viscous hydrodynamic mechanisms that together produce self-steering,
and which generically occur for asymmetric particles in q2D. We carry out experiments with Continuous Flow Lithography (CFL), finding qualitative and semi-quantitative agreement with our
theoretical predictions. Obtaining statistics from hundreds of particle trajectories, we provide a
convincing experimental demonstration of self-steering for device applications. To our knowledge,
this study provides the first demonstration that rigid particles can focus to the centerline in a
channel flow.
This progression culminates in our final study. Inspired by the mobility formalism of polymer
dynamics, we develop a theoretical and numerical framework that can recover the collective dynamics of many particles with complex shape. We find that small clusters of dumbbells can self-organize
from disorder into one-dimensional flowing crystals. However, dumbbells can also pair as undesirable "defects." This two-body effect frustrates self-organization in large suspensions of dumbbells,
driving formation of particle aggregates. To tame this aggregation, we rationally redesign particle
shape, tailoring hydrodynamic interactions to promote chaining of particles in the flow direction.
The redesigned "trumbbell" particles self-organize into large, two-dimensional flowing crystals. We
reveal how crystal self-organization occurs through a multistage process. One, two, several, and
finally many-body interactions become implicated in successive stages. This study is the first to
demonstrate that flowing lattices can be stabilized purely by viscous hydrodynamic interactions.
Thesis Supervisor: Patrick S. Doyle
Title: Singapore Research Professor of Chemical Engineering
Thesis Supervisor: Mehran Kardar
Title: Francis Friedman Professor of Physics
Acknowledgments
Earning a Ph.D. has been the toughest and most rewarding experience of my life. Like so many
other graduate students, I struggled initially. Callow dreams of Nature papers and scaling the
dizzying heights of theory were frustrated by my own callow approach to research. Then, at some
point - or perhaps it came as a slowly dawning realization - I decided I really wanted a Ph.D. I
wanted a Ph.D. not in some idle sense (like I want, as I write this, to learn Boeotian Greek), but
in the sense that I perceived just what it would take and I accepted the full measure of work it
would require. I embraced my project as more than just a means of support, or just one dilletantish
interest among many. I took it as a rare opportunity to do work of some enduring significance and
to put my own stamp on it.
Many people have supported me during this process. Among them, my advisor, Pat Doyle,
deserves the greatest thanks. Pat instilled in me a desire to obtain the most universal and farreaching results - to sift the essence of a problem from the mere details, and to approach it from
physical fundamentals. He could always point me in the right direction when I was lost, while giving
me the autonomy to explore and learn on my own. Pat knew when to push me to achieve more
and when to hold back - especially during the trying early years of my Ph.D. I would recommend
him as a graduate advisor with the highest enthusiasm.
H. Burak Eral has been a patient, kind, and talented experimental collaborator. The full impact
of this work really owes to his tireless efforts in the laboratory. The summer we spent working to
achieve the results of Chapter 5 was intense and exhilarating - it was truly the high point of my
graduate career. I have benefited from many stimulating research conversations with him. More
generally, I have learned from his healthy and balanced approach to life in research.
My interest in soft matter was first stirred by my undergraduate thesis research advisor, Anna C.
Balazs. My aspiration of achieving complex and adaptive materials behavior via creative application
of physical law is shamelessly cribbed from her. This aspiration will drive my research so long as
I work in academia - possibly for life. In her group, I had the pleasure of closely working with
Dr. Kurt Smith and Dr. Alexander Alexeev. I look forward to seeing Alik at future research
conferences, and avidly follow the work of his own recently formed group at Georgia Tech. I am
also grateful to my other undergraduate research sponsors - Riidiger Dieckmann, Seong H. Kim,
and especially Wolfgang J. Choyke - for their mentorship and the opportunity to work in their
laboratories.
I am privileged to have a tight-knit group of friends at the Institute: Jeremy Green, Christopher
Leon, Matthew Schram, Alexander Soane, and Arturs and Nora Vrublevskis. My spirits were kept
high by the daily patter of email (Gmail lists 4,500 threaded conversations), visits to the gym, and
many nights of interesting (and sometimes "interesting") discussion over beer and pizza.
The Doyle group has been a fun and supportive environment throughout the years. I specifically
would like to thank the graduate students and postdocs who most overlapped with me: Harry An
(bearer of many nicknames), Rathi Srinivas (who answered Bill's panicked midnight calls, assuring
him he hadn't destroyed the SFL setup), Ben Renner (for many wide-ranging discussions), Ki
Wan Bong (kind neighbor), Daniel Trahan (the undisputed U.S. Senate trivia champion), Charles
Mitchell, Matt Helgeson, and Jeremy Jones.
Finally, I would like to thank my family for their love and support. I note that, like Dr. Neil
Uspal and Dr. Julie Uspal Zarnoch, I've done my part, and when Jennifer Uspal finishes med school,
we'll at last be able to call David Uspal, M.S. the Howard Wolowitz of the family. (Additionally,
I'd like to thank David Uspal for being a good sport.)
My first year at the Institute was generously supported by a Whiteman Fellowship administered
by the Department of Physics. The work in this thesis was supported by a grant from the Institute
for Collaborative Biotechnologies through contract no. W911NF-09-D-0001 from the U.S. Army
Research Office. The content of the information does not necessarily reflect the position or the
policy of the Government, and no official endorsement should be inferred.
Table of Contents
Abstract
3
Chapter 1 Introduction
1.1 M otivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.1 Microfluidic technologies . . . . . . . . . . . . . . . . . . . . . . .
1.1.2 Fundamental microhydrodynamics. . . . . . . . . . . . . . . . . . .
1.1.3 Non-equilibrium self-organization and self-steering . . . . . . . . . .
1.2 Objectives and overview of studies . . . . . . . . . . . . . . . . . . . . . . .
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Chapter 2 Background
2.1 Particle-laden flows: General considerations . . . .
2.2 The Stokes equation . . . . . . . . . . . . . . . . .
2.2.1 Linearity . . . . . . . . . . . . . . . . . . .
2.2.2 Instantaneity . . . . . . . . . . . . . . . . .
2.2.3 Reversibility . . . . . . . . . . . . . . . . .
2.2.4 Additional properties of the Stokes equation
2.3 Flow singularities and hydrodynamic interactions .
2.3.1 The Stokeslet . . . . . . . . . . . . . . . . .
2.3.2 Otherfundamental flow singularities . . . .
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2.3.3 Geometric confinement and the method of images . . . . . . . . .
Sedimentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 Two coupled Stokeslets . . . . . . . . . . . . . . . . . . . . . . .
2.4.2 Many-body sedimentation . . . . . . . . . . . . . . . . . . . . . .
Particle motion in microchannels . . . . . . . . . . . . . . . . . . . . . . .
2.5.1 Self-steering and self-organizing particles . . . . . . . . . . . . . .
2.5.2 Quasi-two-dimensionalconfinement . . . . . . . . . . . . . . . . .
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Chapter 3 Two discs flowing in a quasi-two-dimensional ,hannel
3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Introduction.. ............
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3.3 Equations of motion and numerical method . . . . . . . . . . . . . . . . . . . .
3.4 Unbounded q2D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5 Confining side walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6 Behaviors and phase map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.7 Characterization of nonlinear oscillations . . . . . . . . . . . . . . . . . . . . . .
3.8 Bound state manipulation through patterned side walls
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3.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Chapter 4 Collective dynamics of multiple discs
4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Introduction . . . . . . . . . . . . . . . . . . . . . . .
4.3 Theoretical model . . . . . . . . . . . . . . . . . . .
4.4 Lattice Boltzmann Method . . . . . . . . . . . . . .
4.4.1 Validation: Torque and drag on a single disc .
4.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.1 Two discs, revisited . . . . . . . . . . . . . .
4.5.2 Fixed points and oscillatory modes . . . . . .
4.5.3 Metastable states and stochastic dispersion . .
4.5.4 Cyclical dynamical motifs . . . . . . . . . . .
4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . .
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Chapter 5 Engineering the trajectory of a single particle via particle shape
5.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Model equations and numerical method . . . . . . . . . .
5.3.1 Force-free equation for a single disc . . . . . . . . .
5.3.2 Additional equations and numerical method . .
5.4 Experimental method . . . . . . . . . . . . . . . . . . . .
5.5 Self-alignment of asymmetric particles under flow . . . . .
5.5.1 Experimental observation of self-alignment . . . . .
5.5.2 Derivation of equation for self-alignment . . . . . .
5.6 Dynamics of dumbbells in a microchannel: The complete picture . . . .
5.6.1 Derivation of numerical and theoretical phase boundaries . . . .
5.6.2 Experimental observation of focusing . . . . . . . . . . . . . . .
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Additional results . . . . . . . . . . . . . . . . . . .
5.7.1 Focusing is rapid near the critical boundary
5.7.2 Reversibility of focusing dynamics . . . . .
5.7.3 Conditions for global assembly . . . . . . .
Conclusions . . . . . . . . . . . . . . . . . . . . . .
Chapter 6 Self-organization of flowing crystals
6.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3 Theory and numerical method . . . . . . . . . . . . . . . . . . . .
6.3.1 Motion of a single disc . . . . . . . . . . . . . . . . . . .
6.3.2 Systems of multiple discs . . . . . . . . . . . . . . . . . .
6.3.3 Mobility tensor . . . . . . . . . . . . . . . . . . . . . . . .
6.3.4 Particle architecture and conservative forces . . . . . . . .
6.3.5 Numerical integration scheme . . . . . . . . . . . . . . . .
6.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4.1 Small cluster of dumbbells . . . . . . . . . . . . . . . . . .
6.4.2 Dumbbell suspension . . . . . . . . . . . . . . . . . . . . .
6.4.3 Engineering hydrodynamic interactions via particle shape .
6.4.4 Trumbbell suspensions . . . . . . . . . . . . . . . . . . . .
6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chapter 7
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Summary and Outlook
137
Appendix A
Single disc flow field
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Appendix B
Hydrodynamic interaction tensor
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Appendix C
Effect of disc rotations
145
Appendix D
Order parameters
149
I
List of Figures
2.1
2.2
2.3
A point force (green vector) applied to a suspended sphere creates a disturbance flow
in the surrounding fluid. We show the leading order, far-field contribution to this
flow, known as the "Stokeslet."
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
The Green's function for a point force is changed by geometric confinement: the
flow must satisfy the no-slip and no-penetration conditions on each solid boundary.
For a plane wall, both the Stokes equation in the fluid domain and the boundary
conditions on the wall can be satisfied with a system of images. A point force (green
vector) at height y = h above the wall creates at y = -h an oppositely directed
image force, a force dipole, and a mass dipole. The blue vector represents all three
im age singularities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
Reversibility implies that a sphere sedimenting near a wall experiences no lift force.
In (a), we assume the sphere has a non-zero velocity component in the wall normal
direction. In (b), we reverse the driving force. Reversibility requires that the sphere
velocity also be reversed. In (c), we rotate the coordinate system of the sphere/wall
system, obtaining the same physical situation as in (a). However, the normal component of sphere velocity is negated relative to (a). Therefore, the assumption of a
non-zero normal component is contradictory. . . . . . . . . . . . . . . . . . . . . . . 42
2.4
A polymer can be represented as a "dumbbell": two beads connected by a spring.
In a shear flow, the spring is stretched, introducing point forces that disturb the
background flow. We show the disturbance field created by the force on the left
bead. This disturbance drives the right bead away from the wall. Likewise, the right
bead drives the left bead away from the wall by the flow disturbance it creates.
. . 44
2.5
Illustration of quasi-2D hydrodynamics. A disc is tightly confined between parallel
plates and subject to an external flow (black vectors). The particle is advected
downstream (blue vector) by the flow. However, due to strong friction from the
confining plates, the particle lags the external flow, and moves upstream relative
to it (green vector). The particle therefore creates a characteristic dipolar flow
disturbance field; fluid mass is pushed away from its upstream edge and drawn into
its downstream edge. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.1
Two hydrodynamically coupled discs in an unbounded quasi-two-dimensional geometry are driven by a uniform external flow. The discs interact via dipolar flow
disturbance fields. For 012 $ 0' and 012 $ 900, the discs can have a non-zero velocity component perpendicular to the external flow field. However, they cannot move
relatively for any configuration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.2
The lateral drift velocity of two flow-driven hydrodynamically coupled discs in unbounded q2D as a function of pair angle. The red curve is given by the theoretical
expression in Eq. 3.17. The black data points were obtained numerically. We vary
angle for fixed disc separation r = 5R, B = 2.12, and a = 0.796. . . . . . . . . . . . 54
3.3
System of real and image discs used to obtain the "dressed" or effective hydrodynamic
interaction tensor in a quasi-two-dimensional channel. A discs' images split into sets
designated "near" and "far," generated as periodic copies of the two closest images
with periodicity 2W . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.4
(a) Particles of length L are confined to two dimensional motion in a channel of
width W and height H, where H < L < W, and subject to an external flow. (b)
Top down view of the system of images used to impose the no-mass flux condition
at the channel side walls. The real particles (dark red and dark blue) are dressed
by an infinite set of images (light colors). The particles lag the external flow and
are therefore coupled by dipolar flow disturbance fields. Gray vectors are particle
velocities in a frame moving with the x component of the particles' center of mass,
Xcen. (c) Particle trajectories in the xen frame for oscillation around a 00 fixed point,
as described in the text. (d) Particle trajectories around a 90' fixed point. (e) A
scattering event. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.5
Phase map indicating behavior for the initial condition (yi, y2, Ax). Yellow (light)
squares indicate oscillation around a 00 fixed point; green (light) triangles, a 90' fixed
point; blue (dark) squares, scattering; and red (medium) circles, particle-particle or
particle-wall overlap. For the oscillatory trajectories, the inset figures show the
distributions of mean angle and frequency, where fL is found by taking the spatial
Fourier transform as described in the text. . . . . . . . . . . . . . . . . . . . . . . . 57
3.6
(a) Position of a particle in y with xe, for initial separation Ay = 3L and Ax = 0
and various initial displacements of ye, from the centerline, where the fixed point
has yi/L = 2.5. (b) Matched by color, the power spectra of the trajectories in
(a), where f has units of inverse length. fo is predicted by linear theory. Arrows
indicate the shift of peaks with increasing amplitude. The appearance of second
harmonics is linked to the breaking of the half-wave symmetry yi(xem+A/2) -2.5L =
-(yi (xm) -2.5L), where A is the signal wavelength. The inset shows both the largest
amplitude trajectory from (a) and the result of performing the symmetry operation
on it; the curves do not coincide. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.7
Effective potentials in Ax (a) and y, (c) for the trajectories in Fig. 3.6, matched
by color. The potentials are shifted and rescaled for characterization in the Chebyshev basis. (b) For motion in Ax, a negative coefficient of T 4 for large amplitude
oscillations indicates a softening nonlinearity. (d) For motion in yi, large amplitude oscillations have skewed potentials, consistent with the half-wave symmetry
breaking. Arrows indicate the effect of increasing amplitude. . . . . . . . . . . . . . 59
3.8
(a) Phase portrait of two trajectories with A/L = 0.2 and f"/fo = 1 initially separated by a noise vector in phase space with magnitude E = 10-4. The trajectories
diverge exponentially (inset). (b) Distribution of scattering length x( , for A/L = 0.2
and f,/fo = 1. (c) Number of scattering and overlap trajectories for various amplitudes and spatial frequencies of a sinusoidal pattern. The results for irrational
frequency ratios are similar.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.1
(a) In this chapter, a cluster of N particles (here N = 3) is tightly confined in a
gap of height H between plates normal to the z direction. They are free to move in
x and y between side walls, where W is the width of the channel. The position of
particle i is labeled by xi and yi in a frame fixed to the channel walls. The particles
are driven by an external flow. (b) System of real and virtual particles used to derive
the thin channel hydrodynamic interaction tensor. The real particles (dark colors)
are subject to an external flow (black vectors) and are dressed by an infinite set of
images (light colors) that are constructed iteratively, via mirror reflections across
the real and virtual channel boundaries. Due to friction from the confining plates,
each particle lags its own local flow field; gray vectors show the velocities of the
real particles in frames moving with local flow. This relative motion gives rise to
hydrodynamic disturbance fields (black streamlines) that couple the particles, and
is dominated by motion in the direction opposed to that of external flow. We also
show particle velocity in a frame moving with the particle cluster's center of mass
for two of the virtual particles (green vectors). . . . . . . . . . . . . . . . . . . . . . 65
4.2
Dimensionless drag forces and torques vs. dimensionless channel height H/L for
a disc translating or rotating in a quiescent fluid for various disc sizes L, where L
characterizes the level of spatial coarse-graining. For L = 10, the disc size used in
this study, there is negligible gain in accuracy with further improvement in resolution. 70
4.3
(left) Oscillation of a particle pair with initial Ay = 3L, initial Ax = 0 and initial
center of mass position y,, = W/2 + L. The positions of the two particles in
y are shown as a function of center of mass position x,,. The solid black curve
shows LBM simulation results at Re = 0.2, and the dashed red curve shows the
result of numerically integrating the theoretical model. These curves closely agree,
though very slight attenuation in amplitude can be seen in the LBM results. (right)
Simulation results for the particle pair, shown after advection by xe,,/L = 859
particle lengths. The red and blue curves show particle positions at prior times, and
crosses indicate initial particle positions. Colors indicate the magnitude of the fluid
velocity field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.4
Oscillation of a particle pair with initial separation Ay = 2L, Ax = 0 and initial
center of mass position yc.. = W/2 + L for various values of Re. Particle positions in
y are shown as a function of center of mass position x,, in the flow direction. As Re
decreases, there is less decay of amplitude per wavelength. For clarity, we omitted
the theoretical curve for one of the particles . . . . . . . . . . . . . . . . . . . . . . . 72
4.5
The two particle configuration of Figure 4.4 has an characteristic wavelength A/L
with which yi and Y2 oscillate as xcer increases. This wavelength A/L scales with
the hydrodynamic interaction parameter 3 with a fitted exponent of -0.963, close
to the predicted value of -1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.6
Fixed points obtained a priori via symmetry considerations, depicted in top down
view. The first column shows particles in a frame moving in the center of mass when
the theoretical model is integrated. In this frame, particles remain in fixed positions.
Side walls are indicated by black lines. The second column shows particles in the
center of mass frame for the corresponding Lattice Boltzmann simulations. Colors
indicate the magnitude of the fluid velocity field. In the simulations, particles move
slightly, but remain within one radius of their initial positions. Due to this motion,
the fluid velocity field can be slightly asymmetric. (a) A "dimer column" for channel
width W/L = 9. The LBM simulation is shown after the particles were advected
downstream by xe,,/L = 241 particle lengths at Re = 0.2, where xc,, is the position
of the center of mass in the flow direction. (b) A "column" fixed point and LBM
simulation after advection by x,,/L = 833 particle lengths at Re = 0.2. (c) A
"double column" fixed point and LBM simulation after advection by xe,,/L = 524
particle lengths at Re = 0.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.7
Geometric construction of the "dimer column" fixed point. In (a), three real particles
are accompanied by an infinite set of virtual particles, the closest of which are at
y = -a/2, y = W + b/2, and y = W + b/2 + a. These quantities are related
by 3(a + b) = 2W. Each of the real and virtual particles is identical, resembling
the particle shown in (b), moving in -x with respect to the local flow field and
contributing components of velocity in positive x to the local flow fields of the other
particles. The gray vector shows the velocity of a particle with respect to the local
flow, while the black streamlines illustrate the dipolar disturbance field thus created.
Because this configuration is one dimensional, the angular dependence of the dipolar
form is not relevant here. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.8
Oscillatory modes of a three particle column fixed point with W/L = 8 and lattice
length a/L = 8/3. The top row shows trajectories found via numerical integration
of the theoretical model, starting from an initial condition in which the particles are
displaced from the fixed point along an eigenvector. In the bottom row we show the
corresponding LBM simulations. Particles are shown in their final positions, while
the crosses indicate initial positions. The red, blue, and green curves are the "tracks"
showing particle positions over time. Arrows indicate the direction of particle motion.
In the simulations, the oscillations in (a) slowly grow with time, while those in (b)
slowly decay. As discussed in the text, this effect diminishes as Re is decreased. (a)
Theory and simulation results after x,/L
= 482 advected particle lengths. For the
simulations, Re = 0.2. The particles were initially displaced from the fixed point
by Ay 1 = -0.34L, where particle 1 is the green (bottom) particle; Ay2 = -0.68L,
for the blue (middle) particle; and Ay3 = -0.34L for the red (top) particle. (b)
Results after x,,/L = 205 advected particle lengths. For the simulations, Re = 0.05.
The initial displacements from the fixed point are Ax 1 = -0.22L, Ax 2 = -0.43L,
and Ax 3 = -0.22L. In contrast with (a), here Re and x,,/L are too small for a
discernible phase difference between theory and simulations.
. . . . . . . . . . . . . 77
4.9
Snapshots of simulation results for a metastable steady "triangle" configuration with
Re = 0.2 and W/L = 8 at four values of the cluster center of mass x,. Particle
motion, initially limited to small excursions from the initial positions, grows in magnitude until the magenta and green particles pair. Ultimately, the red and green
particles escape together. Colors indicate the magnitude of the fluid velocity. . . . . 78
4.10 Dispersion c.2/L 2 with dimensionless time Uot/L for four trajectories of the metastable
"triangle" configuration of Figure 4.9, simulated with the Lattice Boltzmann method.
For each trajectory, the initial particle positions are spatially perturbed by a displacement vector with magnitude 0.025L and random angle. The four trajectories break
up at different times. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.11 (a) Particle motion in the center of mass frame for two realizations of a metastable
three particle configuration with W/L = 8, as determined by integration of the theoretical model. The two realizations differ by slight noise in the initial particle positions. A random perturbation uniformly distributed over the interval [-0.0125L, 0.0125L]
is applied to the x and y positions of each particle. The green (middle) particle pairs
and escapes with either the red (left) particle, as shown in the first panel, or the blue
(right) particle, as shown in the second panel. (b) Distribution of escape pathways
for the three particle configuration. Bins are labeled by which two particles pair.
For each trajectory, the initial particle positions are given a random perturbation, as
in (a). The escape outcome is sensitive to this perturbation. (c) Euclidean distance
A2 (t) = EZ[(Xi,A(t) - xiB(t))2 + (yi,A(t) - Yi,B(t)) 2 ] between the two realizations
(trajectories in phase space) in (a) as a function of dimensionless time, where i indexes the three particles, and A and B label the two trajectories. For some initial
transient period, both trajectories are bound as three particle configurations and
diverge exponentially in phase space, A ~ eAUot/L, with a Lyapunov exponent of
A = 0.00185. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.12 Cyclical motifs for N = 3 particles discovered via a "brute force" search with the
theoretical model and confirmed with LBM simulations. In a search, we sweep over
initial spatial configurations of N particles, integrating the model forward in time for
a specified time span and identifying candidate cycles as those with small Euclidean
distance between the initial and final spatial configurations. The Euclidean distance
d is defined as d2 = Ei[(Xi(tfinai) - X,(0)) 2 + (y,(tfnal) - yi(0)) 2]. The simplicity
of the model makes this approach computationally tractable. Colors indicate the
magnitude of the fluid velocity field. (a) In the "juggling" motif, the three particles
move clockwise, as indicated by the black arrow, cyclically exchanging positions.
Particles pause in the bottom position, recalling how a juggler will momentarily
have a ball in hand. Each exchange of particle positions occurs after about 115
advected particle lengths, so that the entire cycle takes about xm/L = 345 lengths.
(b) The "bowtie" motif. At first glance, it might appear that this motif is associated
with a fixed point in which particles are positioned on the centerline, aligned with
the flow. However, such a configuration would quickly disperse. Particles return
roughly to their initial positions after approximately xc/L = 960 particle lengths.
5.1
5.2
5.3
5.4
5.5
Schematic diagram of the model system. A particle comprising two rigidly connected
discs is confined in a thin microchannel of height H and width W and driven by an
external flow. The flow is approximately uniform in the channel midplane, and has
depth averaged velocity U0 . The disc radii are R 1 and R2 , with R 1 > R 2 , and the
disc centers are separated by distance s. Two thin lubricating fluid layers of height h
separate the discs from the confining plates (i.e. the channel "ceiling" and "floor.")
The instantaneous particle configuration is specified by two coordinates: the location
in y of the midpoint between disc centers, yc = (Y1 + y2)/ 2 , and the angle 0 between
the external flow and the particle axis.
. . . . . . . . . . . . . . . . . . . . . . . .
A symmetric particle oscillates between side walls. When the symmetry is slightly
broken, this oscillation is damped, and the particle aligns with the flow as it focuses
to the centerline. A very asymmetric particle is "overdamped," and rapidly aligns
before slowly focusing. The trajectories were obtained numerically for the parameters
given in the caption of Fig. 5.12b. The x axes are scaled by a factor of 1/40 to show
the full range of particle behaviors.
. . . . . . . . . . . . . . . . . . . . . . . . . .
Comparison of models for flow of a disc. We show a as a function of h for four
values of H as determined by our simplified model and the more detailed analysis of
Halpern and Secomb. Our simplified model shows good quantitative agreement for
small H and captures the trends in h and H. . . . . . . . . . . . . . . . . . . . . .
This photograph of the experimental setup shows the microfluidic channel, the moving microscope stage and continuous flow lithography setup, and the camera. The
inset shows a zoomed in view of the PDMS channel with ruler markings. The scale
bar is 100 um . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Hydrodynamic self-interaction drives alignment of an asymmetric particle. (a) Illustration of the self-interaction of a symmetric particle. A disc's vector shows the
component of the flow disturbance from the other disc in 0, the direction of increasing 0. The vectors are identical: there is no rotation of the particle. (b) When the
two discs have different radii, the particle aligns with the flow. . . . . . . . . . . .
82
. 86
. 87
. 89
. 90
. 91
5.6
Experimental angle vs. time for various R with 9 = 3.3, h = 0.06, and H = 1.6.
We scale the data for each f by a fitted i, collapsing all data onto a universal curve
predicted by theory. (inset) The dependence of the experimental timescales r on
R, along with a theoretical curve for the same parameters (solid) and a theoretical
curve with h adjusted for best fit (dashed). . . . . . . . . . . . . . . . . . . . . . . . 92
5.7
Experimental images of particle self-alignment. (a) Snapshots of a symmetric particle
at various times, matched to the times in Fig. 5.6. The scale bar is 100 pm. (b)
Snapshots for R = 2 at the same times as in (a). . . . . . . . . . . . . . . . . . . . . 93
5.8
A symmetric particle oscillates via the combined effects of hydrodynamic interaction
with itself and with its own images. Self-interaction leads to cross-streamline migration ("lateral drift") when the particle angle 9 $ 0' and 0 51 900. The images rotate
the particle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.9
Portraits showing particle trajectories in the phase space (y,, 9). Portraits were
obtained numerically for W = 21, § = 3.5, H = 1.6, and h = 0.08. Arrows give
direction of motion in phase space. Dots identify the trajectories shown in Fig. 5.2..
5.10 Linearized model of an asymmetric particle. Rotation by the images is opposed by
self-alignment. The particle drifts in the y direction when 0 is displaced from the
equilibrium value 0 = 00. The lateral displacement A is defined as A =- y, - W/2.
96
.
97
5.11 Theoretical and numerical critical boundaries. For various sets of parameters H,
§, R, and h, we numerically obtain points on the critical boundary separating the
underdamped and overdamped oscillatory regimes via the method described in the
section "Numerical Phase Boundary." These points are shown as symbols in the
figure. For each parameter set, we also obtain theoretical curves via Eqs. 5.18 and
5.19, shown as solid lines and matched to the symbols by color. Each curve fits its
corresponding numerical data with the same fitted dimensionless prefactor of 1/3.
Moreover, as shown in Fig. 5.12, all curves and data can be collapsed onto a single
boundary via an empirically fitted rescaling of §-1/ 5 1/ 6 V. For clarity, not all of
the parameter sets in the collapsed Fig. 5.12 are shown. . . . . . . . . . . . . . . . . 98
5.12 Phase diagram showing the critical boundary that separates the underdamped and
overdamped regimes. The symbols are points on the boundary obtained numerically
for various parameters. The solid lines, matched by color to the symbols, are theoretical curves for the same parameters. The numerical data and theoretical curves
collapse onto one universal boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.13 Individual particle trajectories. All scale bars are 100 Am. (a) Experimental montage
showing reflection of a symmetric particle. The corresponding theoretical trajectory
is shown in the inset. (b) A strongly asymmetric particle with 0 = -10* focuses to
the centerline. (c) A strongly asymmetric particle with a large initial angle aligns
and then focuses to the centerline. (d) Position data for the trajectory in (a). The
theoretical trajectory for R = 1 was scaled in x by a factor of 0.475. A theoretical
curve with R = 1.01, for which the rescaling is 0.4, better captures the curvature of
the data. (e) For the particle in (b), the rescaling is 0.15. (f) For the two timescale
process of (c), different rescalings of 3 and 0.1 are required to capture the initial and
steady dynamics. For all trajectories, § = 3.3, h = 0.3, and H = 1.6. . . . . . . . . . 101
5.14 Statistics of particles in a flow-through device. (a) Fluorescence microscopy image of
symmetric and asymmetric particles flowing in a channel. The asymmetric particles
focus to the centerline (red). The white lines indicate the channel side walls. The
scale bar is 100 pm. (b) Distributions of transverse positions for symmetric particles
(R = 1) measured near the inlet (blue, left hatching) and outlet (red, right hatching).
Both distributions are nearly uniform across the channel width. (c). Distributions
of transverse positions for the asymmetric particles (R = 1.3). The particles begin
nearly uniformly distributed at the inlet. Most focus to the centerline near the
outlet. Statistics are gathered from over 300 symmetric and 300 asymmetric particle
trajectories. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.15 Position of a particle in a channel for various asymmetries f, initial condition
(yc,O) = (W/4,1600) at xc = 0, and parameters h = 0.08, H = 1.6, W = 21,
and 9 = 3.5. The most rapid convergence to y, = W/2 with xc occurs for R at the
critical value of Rcrit = 1.10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.16 Demonstration of the reversibility of the dynamics of an asymmetric particle via
numerical integration of the governing equations. A particle initially with y, = W/4
and 9 = 450, as indicated, is advected by the flow Uo = Uoi until time t = T,
where T 300. At this time, the particle is nearly aligned and focused. The flow is
reversed, and the particle retraces its trajectory and recovers its initial configuration
after an additional time T. Thereafter, it rotates into a configuration with 9 = 900.
The entire trajectory is mirror symmetric across this configuration. Ultimately, the
particle aligns with the reversed flow -Uo0 and focuses to the centerline. To show
the full range of particle motion, the x axis has been compressed by a factor of 10.
104
5.17 (a) For a weakly asymmetric particle at its 9 = 900 fixed point, the large disc is close
to the nearest images, interacting more strongly with them than the small disc does.
The tendency of the particle to self-align is exactly balanced by the stronger flow
field experienced by the large disc. Here, R = 1.05, h = 0.08, H = 1.6, = 3.5, and
W = 21, and the fixed point occurs at yc/W = 0.192. (b) These fixed points are
marginally stable. For a small initial displacement from the fixed point, the particle
appears to "bounce" along the wall as it is advected down the channel. The x axis
is compressed by a factor of 20 in this image. . . . . . . . . . . . . . . . . . . . . . . 106
.
6.1
6.2
(a) A single aligned and focus particle is part of an infinite lattice of real and image
particles. When one or more image particles are exchanged for real particles, the resulting configuration should also steadily translate along the channel with no relative
particle motion. Each of the real and image particles is separated by W/N, where
N is the number of real particles. (b) An infinite two-dimensional lattice should
likewise steadily translate. The lattice length a is determined by particle density.
. 111
Particle architectures considered in this work. A dumbbell comprises hydrodynamically interacting discs, with f = R 1 /R 2 = 1.5. The disc centers are connected by
a stiff Hookean spring with equilibrium length § = s/R 2 = 3.5. A trumbbell has
two "tails" separated by angle 4= 50'. The two tail discs are connected by a third
stiff spring (not shown) so that this angle remains fixed. Through hydrodynamic
self-interaction, both the dumbbell and trumbbell align under flow so that the head
disc is upstream . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
6.3
(a) Trajectories of an isolated pair of dumbbells from one hundred random initial
conditions. Blue and green curves are obtained from each run as the y positions
of the two head discs plotted against the pair center of mass x. A majority of
dumbbells focus to the centerline; these trajectories have a characteristic exponential
envelope. A substantial number focus to the doublet crystal positions y/W = 1/4
and y/W = 3/4. Other pairs form stable defects that are attracted to positions near
the side walls, or unstable but long-lived oscillatory defects that eventually break
up to form doublet crystals. The channel width is W = 30. (b) Histogram of the
final head disc positions of the trajectories in (a). Approximately 20% of dumbbell
pairs form doublet crystals. (c) Pair behaviors obtained in the simulations of (a).
The unstable defects translate back and forth across a section of the channel width
before breaking up. Singlets weakly repel each other in the flow direction; there is
no steady separation in x.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.4
Frames from a representative simulation of a dumbbell suspension. The channel
width is W = 20 and the simulation contains N = 20 dumbbells. There are periodic
boundary conditions in the flow direction, and the simulation box has length l2/W =
7.5. Particles are labeled by number. (a) The randomly seeded initial condition.
(b) A large number of particles have aggregated. The configuration of neighboring
particles in the aggregate strongly resembles the defect states obtained in Fig. 6.3.
For instance, particles 9 and 20 appear to be an unstable defect, and particles 7
and 17 resemble a stable defect. (c) Particles 2, 9, 6, 5 have escaped the central
aggregate. Relatively separated from each other and the rest of the suspension,
they behave like isolated single particles and migrate towards the channel centerline.
Downstream of the central aggregate, emerging spatial order can be discerned in
particles 11, 14, 12, 16, 8, 18, and 3. (d) Particles 14, 11, 12, and 16 have formed a
doublet crystal, which is moving upstream towards the aggregate. (e) Particles 11
and 16 have left the crystal, and particles 19 and 10 have been recruited to it. The
crystal is approaching a stable defect (particles 13 and 1) and an unstable defect
(particles 4 and 15). Further downstream, particles 3 and 18 have formed a doublet
crystal. (f) One doublet crystal has broken up from encountering defects, while
another (particles 3 and 18) approaches an aggregate. . . . . . . . . . . . . . . . . . 120
6.5
Statistics of thirty-two different runs of a dumbbell suspension with the same parameters as in Fig. 6.4. Simulations are run for time t = 5.0 x 104. (a) The pair
correlation function g(Ax, Ay) showing the probability with which two head discs
are separated by (Ax, Ay). The function is normalized by the probability function
for a suspension with uniform density #0. The sterically excluded area is indicated by
a dashed line. There is a bright ring around this region, indicating a short-range attraction responsible for defect formation and aggregation. (b) A correlation function
corrected for the variation of particle density across channel width. With this correction, peaks in pair separation at (Ax = ±W/3, Ay = 0), (Ax = ±W/2,Ay = 0),
and (Ax = t2W/3, Ay = 0), indicated by white arrows, become more clearly visible. These peaks are due to transient formation of doublet and triplet crystals. (c)
The conditional correlation function g(Ax, Y2IY1 = W/6). This function expresses
the probability of finding a head disc at position (xi + Ax, y2), given that a second
head disc is at y1 = W/6. There are triplet crystal peaks at (AX = 0, Y2 = W/2)
and (AX = 0, Y2 = 5W/6), in addition to defects surrounding the excluded volume
region. The function is normalized by particle density profile in (d). (d) The variation of particle density across the channel. Particles are depleted from the channel
center and enriched along the side walls, as the frames in Fig. 6.4 suggest. . . . . . 122
6.6
(a) Disturbance flow created by a single isolated "trumbell" in unbounded q2D, calculated numerically. Streamlines are shown in black. The total flow disturbance is
due to the superposition of dipole singularities: yellow arrows show dipoles from friction on the discs, and white arrows show dipoles from internal spring forces. Notably,
the streamlines are fore-aft asymmetric, bent in the downstream direction. The color
field indicates the x component of the disturbance velocity. To focus attention on the
far-field disturbance, we do not show the area immediately around the particle. (b)
The disturbance flow field can be regarded as the sum of multipole components. The
lowest order contribution is a point dipole. The quadrupolar correction bends the
streamlines downstream. (c) In contrast, the disturbance streamlines for a dumbbell are bent upstream. Accordingly, its quadrupolar correction has opposite sign as
the trumbbell quadrupole. (d) Effective potentials for two dumbbells (dashed red
line) and two trumbells (solid black line) aligned in the flow direction in unbounded
q2D. The quadrupolar contribution to the interaction of two dumbbells is repulsive. For two trumbbells, the quadrupolar component is attractive. Higher order,
shorter range multipole components are repulsive, stabilizing the trumbbells against
collision. As a result, two trumbbells have an equilibrium separation Aze, = 7.4. . . 123
6.7
An initially disordered suspension of trumbbells can self-organize into a two-dimensional
crystal. There are N = 20 particles in a simulation box with W = 20 and LX/W =
7.5. (a) Trumbbells are initially placed with random positions and angles. (b) The
particles have aligned with the flow. Groups of particles have locally self-organized,
either laterally, as doublets, or in the streamwise direction, as strings held together
by the quadrupolar interaction. (c) The particles have entirely partitioned into two
separate lanes located near the doublet crystal positions y = W/4 and y = 3W/4.
In this case, the two lanes have an equal number of particles. However, not every
particle has found a partner. For some particles the partner position is vacant (e.g.
particle 9.) For others, the partner is shared (e.g. particles 4, 10, 1) in a triangular
configuration. (d) The particles have spread more evenly across the channel, but vacancies and triangle formations remain. (e) The suspension now resembles a strained
crystal, and is on the threshold of relaxation to an unstrained lattice. Particle 2 will
capture particle 16, allowing 6 and 19 to partner. (f) The particles have settled into
an apparently "perfect" crystal, and have approximately the same neighbors as in
fram e (c) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
6.8
. (a) Evolution of the order parameters with center of mass position xc for the
trajectory shown in Fig. 6.7. Dashed lines indicate the values of x, for the frames
shown in that figure. (b) Long time evolution of the particle positions in the streamwise direction. Aside from a small amplitude, low frequency density wave, particles
positions in x are approximately evenly spaced and steady. . . . . . . . . . . . . . . 127
6.9
Vacancy defects propagate by a simple mechanism. (a) A cluster of three isolated
particles. Particles 1 and 2 are initially placed in a doublet crystal configuration,
and particle 3 is placed next to particle 2. Due to dipolar HI, a doublet crystal
has a greater downstream velocity than an isolated particle, since each particle in
the crystal increases the local fluid velocity of its partner ("transverse anti-drag.")
The crystal collides with the slower particle 3. Particle 3 slows down particle 2 and
speeds up particle 1, forming a triangular configuration. Particle 1 swaps partners,
leaving particle 2 upstream. (b) The same mechanism occurs in a two-dimensional
crystal. As a result of defect motion, the two rows of the crystal slide past each
other: particle 24 has exchanges particle 15 for particle 10, and particle 22 is about
to exchange particle 2 for particle 15. . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
6.10 Two vacancy defects in parallel lanes can annihilate each other on close approach.
If an equal number of particles partition to two lanes, defect annihilation precedes
relaxation to a defect-free crystal. (a) Particles 27 and 14 are associated with vacancies. (b) Particles 27 and 14 form triangular configurations alongside particles 14 and
2. (c) Particle 12 has successfully partnered with 27, pushing particle 2 upstream.
Particle 14 has not fully partnered with particle 18. (d) Instead of partnering with
particle 18, particle 14 is captured by particle 2. (e) Particle 3, instead of partnering
with 25, is likewise attracted to a downstream particle, particle 18. (f) A shear wave
propagates down the lattice. (g) The shear wave is dissipated, and the crystal relaxes
to equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
6.11 Suspensions can crystallize with two types of permanent defect. If an unequal number
of particles is partitioned between the crystal lanes, then vacancies have no means
to heal. Secondly, a stray particle on the centerline can propagate freely through
a doublet crystal. Frames (a) through (e) show both types of defect. A vacancy
switches from particle 4 to particle 9 through the mechanism discussed in Fig. 6.9,
moving upstream. Particle 7 strains the lattice as it moves down the centerline.
Particles flow around it, returning to their previous y positions upstream of the
defect. Due to transverse anti-drag, the inclusion has a higher velocity than doublet
pairs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.12 Self-organization of three lanes in a wide channel. There are N = 45 particles in a
simulation box with W = 30 and LX/W = 6. (a) Particles are seeded with a random
initial configuration. (b) Particles have aligned with the flow and organized into
strings and, at right, a quadruplet. (c) At left, particles have sorted themselves into
three lanes. Strings of particles are joining these lanes. (d) Lanes now extend over
most of the simulation box. A long string of particles flows around an "inclusion"
that is not in a lane. (e) Particles are now entirely within the three lanes. The
lanes contain different numbers of particles, and vary in density in the streamwise
direction. These density variations propagate through the lanes. . . . . . . . . . . . 132
6.13 (a) Head disc pair correlation function g(Ax, Ay) calculated over the entire trajectory of Fig. 6.12 and normalized by the probability function for uniform density
#o. The dashed white line indicates the sterically excluded area. Particles are depleted from close contact not only by steric interactions, but also by hydrodynamic
interactions. The five streaks are due to particle laning. (b) The correlation function of (a) corrected for variation of particle density across the channel width. Each
lane has localized peaks, indicating local crystalline order. In Fig. 6.12(e), particles and their neighbors in other lanes adopt crystalline order where the the local
lane densities match. (c) Correlation function normalized by #o and calculated from
the beginning of the simulation through frame (c) of Fig. 6.12. Particles have not
yet partitioned into lanes, but particle pairing by quadrupolar HI leads to peaks at
(Ax = t6.5, Ay = 0). (d) The correlation function of (c), corrected for variation of
particle density across the channel width. . . . . . . . . . . . . . . . . . . . . . . . . 133
C.1 Evolution of dumbbell angle with time with and without disc torques. We rescale
time with our theoretically estimated values of the dumbbell rotational drag coefficient. For each R, the curves obtained with and without disc torques collapse, since
the self-alignment timescale is proportional to the dumbbell rotational drag coefficient. The ratio of rotational coefficients with and without disc torques is shown in
Supplementary Figure C.2 for each value of R. The parameters h = 0.06, H = 1.6,
and 9 = 3.3 are the same as in Fig. 5.6. . . . . . . . . . . . . . . . . . . . . . . . . . 147
C.2 Comparison of dumbbell rotational drag coefficients. We show the ratio of theoretically estimated dumbbell rotational drag coefficients with and without contributions
from disc rotations as a function of R. The parameters h = 0.06, H = 1.6, and
s = 3.3 are the same as in Fig. 5.6. In Fig. C.1, these theoretically estimated coefficients were shown to collapse data from numerical simulations. Torque from the
discs does not wholly account for the quantitative discrepancy between theoretical
and experimental self-alignment timescales in Fig. 5.6. . . . . . . . . . . . . . . . . . 148
CHAPTER 1
Introduction
This thesis represents an effort to harness novel fundamental microhydrodynamic phenomena for
microfluidic device applications. This chapter will introduce the main themes of this research: The
need for new techniques to control the motion of individual flowing microparticles; the dependence
of fluid-mediated particle interactions on geometric confinement and particle morphology; and the
enticing possibility that these hydrodynamic interactions can be engineered to drive suspension selforganization, "programming" particle motion via design of particle shape and channel geometry.
1.1
1.1.1
Motivation
Microfluidic technologies
One measure of technological progress is our ability to step into worlds remote from our everyday
experience in order to manipulate, measure, and change matter [1]. Can we stretch individual
DNA molecules, like a rubber band? Can we a weigh an individual cell? Can we count exactly
how many molecules of a certain protein are expressed in that cell? These are all possibilities (see
[2], [3], and [4]) enabled by microfluidics: the science and engineering of fluid flow in microscopic
geometries. Cells and polymers are submicron and micron-sized objects that naturally occur in
aqueous environments. Such objects, when confined to fluid-filled microchannels, can be manipulated with controlled external flow or by forces created by electrodes or transducers integrated into
the device. Microchannels are typically fabricated from PDMS, an optically transparent material.
26
1.1. Motivation
This transparency allows the suspended objects to be addressed by laser and imaged with optical
microscopy. Over the past two decades, the possibilities of a platform that integrates manipulation
and measurement of individual microscale objects have driven the phenomenal growth of research
into "lab-on-a-chip" technologies [5]. These technologies represent a fundamental shift from older
bulk measurement and processing techniques that inherently average over many microscopic components of a sample.
A second measure of progress is the integration of new techniques into everyday life: translation
from the research laboratory to the factory floor, home, or hospital, for instance. For a new
technique to displace an old one, it must be competitive on cost and performance. To some degree,
these considerations have slowed wider diffusion of lab-on-a-chip technologies [6]. For instance,
PDMS channels are generally cheaply and easily fabricated, but the use of complicated device
features or external supporting apparatus can undermine this cost advantage. As another example,
devices need to operate with high throughput to meet the demands of industrial applications.
In principle, continuous flow devices can achieve high throughput through continuous operation.
However, a key challenge is to reconcile continuous flow with the performance of operations on
individual flowing objects. For instance, if the objects flow as a disordered suspension, it is difficult
to individually distinguish and scan them. Microfluidics would greatly benefit from new techniques
for controlling the motion of flowing particles in simple channel geometries and with minimal
external apparatus.
1.1.2
Pundamental microhydrodynamics
Any effort to control microparticle trajectories must contend with the peculiar physics of fluids
at the microscale, where viscous effects overwhelmingly dominate inertia, and fluid behavior is
strikingly different from our everyday experience of it. This "creeping flow" limit has important
consequences that both constrain the possible motions of suspended particles and facilitate their
mathematical description.
When a particle moves through fluid, it creates a disturbance flow that affects the motion
of distant particles. Moreover, a disturbance flow can be reflected from a confining boundary,
coupling back to the moving particle. These "hydrodynamic interactions" (HI) play an essential
role in determining the behavior of a flowing suspension. Fortunately, the properties of viscous flow
vastly simplify the description of HI. As we will discuss in the next chapter, HI can be modeled
with fundamental flow singularities reminiscent of the singularities of electrostatics. A particle
of arbitrary shape can be modeled with an appropriate spatial distribution of flow singularities.
Moreover, confining boundaries can be modeled with image singularities. The presence of these
images can dramatically change the decay law and tensorial form of the "dressed" or effective
hydrodynamic interaction between two confined particles.
The dependence of hydrodynamic interactions on particle shape and confinement raises the
possibility that HI can be harnessed to control particle motion via theoretically informed particle
and channel design. In particular, this thesis will consider particles of various shape in quasi-twodimensional (q2D) confinement, in which particles are "sandwiched" between parallel plates and
free to move in only two directions. As will be shown, the leading order flow singularity in q2D
has a unique dipolar form. This form has been shown to sustain novel collective phenomena with
much greater spatiotemporal coherence than can be achieved in unconfined or weakly confined
suspensions.
1.1. Motivation
27
On the other hand, a restrictive consequence of the creeping flow limit is time reversibility.
Suppose a viscous fluid, possibly containing suspended rigid particles, is driven by applied forces
or motion of confining boundaries. The fluid/particle system is allowed to evolve for some time
under these forces. If the forces are then reversed, both suspended particles and fluid elements
must retrace their trajectories. This property was dramatically demonstrated by G. I. Taylor in a
famous educational video, available online [7]. The space between two concentric cylinders is filled
with viscous fluid. Taylor dyes a portion of the fluid and spins the inner cylinder. The sheared
blob of dyed fluid forms a streak, and then disappears. Taylor reverses the motion. The streak
reappears, and then coalesces into the original dyed blob.
Reversibility constrains rigid particles in viscous flows to time reversible motions. (A periodic
orbit is a permissible motion, for instance.) This constraint raises a challenge for this research:
To what extent can complex particle motions be realized within this constraint? This question
is especially relevant to the design of self-organizing suspensions, and the discussion of the next
section will lead to a more incisive reformulation.
1.1.3
Non-equilibrium self-organization and self-steering
"Self-organization" is the spontaneous appearance of global order in a system of coupled components in the course of its time evolution. Order is not externally imposed, but arises through the
components' interactions and the system's governing laws. Self-organization that occurs in thermal
equilibrium is generally called "self-assembly." On one view, self-assembly is a new name for an
old subject: the foundations of the thermodynamic and kinetic theories of crystallization were laid
by Gibbs and others over a century ago. However, the term is more popularly associated with
recent developments in supramolecular chemistry, colloid science, and nanotechnology. Stimulated
by advances in synthesis techniques, and inspired by examples in biology (including viral capsids,
membranes, and microtubules), researchers have sought to design nanoscale and microscale components that spontaneously assemble into larger and more complex structures through exploration
of their free energy landscape. Self-assembly can therefore bridge the mesoscopic gap between
chemistry and the macroscale. Much effort has gone into systematically relating target structure
to component design (e.g. [8]), and the field is by no means settled.
However, an even richer array of structures and behaviors can be expected in non-equilibrium
self-organization, which is even less understood theoretically. In non-equilibrium self-organization,
also known as dynamic self-assembly [9], an external source of energy (e.g. fields, gradients, and
flows) drives some or all of the interactions between the system's components. Self-organized dynamical structures become possible, including sustained oscillations [10], synchronization of individual oscillators [11], vortex arrays [12], metachronal waves [13], and swarming [14] of self-propelled
rods [15], spheres [16], bacteria [17, 181, and artificial "ant colonies" of chemically communicating
microcapsules [19]. In these structures, continuous supply of energy is balanced by continuous
dissipation. Self-organized materials systems can even achieve some of the characteristic functions
of life, such as homeostasis [20], by instantiating feedback loops and other motifs of nonlinear
dynamics. Notably, hydrodynamic interactions are implicated in many of these examples.
A particle suspension driven by an external channel flow provides a potential setting for nonequilibrium self-organization. The external flow, driven by a difference in fluid pressure between
the inlet and outlet, provides a continuous source of energy. Particles are coupled by hydrodynamic
interactions sustained by the external flow. Even an isolated particle can hydrodynamically couple
28
1.2. Objectives and overview of studies
to itself through the reflection of its disturbance flow by confining channel boundaries. In this
connection, it is appropriate to distinguish the "self-steering" of an individual particle from the
self-organization of many coupled particles. From any initial orientation and position in the channel
cross-section, a self-steering particle achieves one of a much smaller set of orientations and positions
- possibly only one. From a dynamical systems perspective, self-steering and self-organization
are effectively identical: any initial state in phase space evolves towards one of a limited set of
attractors. Given this unified description of self-organization/self-steering, our challenge becomes
better apparent: how can the mapping of many states to few be reconciled with time reversibility?
Self-steering of deformable particles (e.g. droplets, capsules, and polymers) via hydrodynamic
interactions in channel flows, generally called "cross-stream migration," is a well-known and extensively studied phenomenon [21]. Deformable particles, stretched by an external flow, are driven
away from confining boundaries by reflected disturbance flows. The bare possibility that deformable
particles can self-steer in viscous flows is often attributed to the breaking of time reversal symmetry by deformability. Similarly, self-steering has been demonstrated for other irreversible systems,
including Brownian [22] and especially inertial particles [23, 24]. Recently, self-organization - formation of "flowing crystals" - has been achieved in systems of weakly inertial spheres [25]. This
common feature raises a second, related question for this thesis: Is irreversibility a necessary condition for achieving self-organization and self-steering in systems of flowing microparticles?
From an applications standpoint, a suspension designed for self-organization or self-steering
would be highly desirable. Particle trajectories could be "programmed" into particle morphology
and channel design, which would modulate hydrodynamic interactions to drive self-organization of
the desired motions. Particles that self-steer to the channel centerline would require no hydrodynamic sheath flows or external forces for positioning. For instance, tablet-shaped barcoded particles
used in multiplexed bioassays are typically positioned with coflowing fluid streams, which can lead
to poor alignment or deformation of the particle [26]. Self-organizing flowing crystals could be used
as dynamically programmable metamaterials, as continuously "printed" tissue sheets [27], or as
tunable diffraction gratings [28].
1.2
Objectives and overview of studies
The goal of this research is to expose new possibilities for achieving self-organization, self-steering,
and other flow-driven collective phenomena via design of particle shape and channel geometry. We
will combine theory, numerical modeling, and (via collaboration) experiment, seeking both rich
and complex behavior and deep conceptual insight into that behavior. Ideally, such insight would
have generic or universal validity, extending beyond the details of specific systems to more general
conclusions based on symmetry and dimensionality. To that end, we will present the following
studies:
" A quasi-two-dimensional two-body problem: two discs in a channel
" The collective dynamics of small clusters of discs in q2D channels
" The effects of particle shape on single particle dynamics in q2D channels
* The collective dynamics of many complex-shaped particles in q2D channels
1.2.
Objectives and overview of studies
29
This thesis is organized as follows. In Chapter 2, we present the mathematical background useful to understanding this work, and elaborate on many of the phenomena discussed here. Our own
studies start in Chapter 3 with the simplest problem to yield interesting behavior: two hydrodynamically coupled discs in a q2D channel. To address this problem, we develop the core theoretical
equations used throughout this thesis. In Chapter 4, we consider the collective dynamics of three
or more discs. We complement and validate our theoretical model by developing a Lattice Boltzmann code, which recovers q2D hydrodynamics from fundamental fluid kinetics. We find excellent
agreement between the two methods. In Chapter 5, we turn to the effects of shape, and study
the dynamics of a single rod-like particle via theory, numerics, and experiments. Theory and experiments agree qualitatively and semi-quantitatively, to our great satisfaction. In Chapter 6, we
develop a more general theoretical and numerical framework, adapting the mobility formalism of
polymer dynamics to the coupled motion of many q2D particles with complex shape. We use this
framework to investigate many-particle dynamics, including the possibility that flowing crystals
can self-organize via the effects of particle shape and confinement. Finally, in Chapter 7, we take
stock of our results, assess their impact, and identify opportunities for future work.
As a further note to the reader: Chapters 3 through 6 are adapted from articles prepared for
journal publication. Since the model initially presented in Chapter 3 was subsequently refined and
further developed in the later studies, some repetition of material is inevitable. We apologize for
any boredom thus incurred, although - to make a virtue out of necessity - we note that these
chapters are largely self-contained and can be read independently.
CHAPTER 2
Background
In this chapter, we will present the basic theoretical and experimental background necessary to
understand this thesis. We will begin by discussing the unique properties of highly viscous flows,
which include the vast majority of flows encountered in microscopic settings. These properties
allow the reduction of the complete dynamics of particles and suspending fluid to the dynamics of
particles coupled by fluid-mediated effective interactions, or "hydrodynamic interactions."
Next, we consider the mathematical description of these hydrodynamic interactions (HI). The
linearity of the governing equations permits a Green's function approach. In analogy with electrostatics, the perturbation of the flow field by a particle can be described by a superposition of
fundamental flow singularities, each with a characteristic disturbance field. We show how to obtain
the typical leading order disturbance flow field (the "Stokeslet") by solving a Green's function problem, and discuss other important flow singularities. We stress the physical interpretation of these
singularities: disturbance flows are determined by the conservation and transport of momentum
and fluid mass. We discuss how confining boundaries, mathematically represented by the method
of images, can significantly change the functional form of hydrodynamic interactions.
Having established the basic formal framework, we turn to sedimentation for examples of classic
single, two, and many-body microhydrodynamic phenomena. These examples concretely illustrate
the unique properties of viscous flows and provide useful points of comparison for the collective
phenomena investigated in later chapters.
In most microfluidic device applications, particles are not directly driven by external forces.
32
2.1.
Particle-laden flows: General considerations
Rather, particles are carried by an external flow, which itself is driven by a pressure drop between
the channel inlet and outlet. We review rigid particle dynamics in channel flows. Nearly all rigid
particles simply translate in the flow direction with no migration towards or away from the channel
centerline. Researchers have sought to characterize and even engineer mechanisms by which flowdriven particles can individually "self-steer" or collectively self-organize into definite transverse
positions and multiparticle configurations. As we discuss, these efforts have generally considered
particle deformability and finite inertia, which break time reversal symmetry.
Finally, we discuss the setting of the work in this thesis: "quasi-two-dimensional" (q2D) confinement. In q2D, particles are tightly "sandwiched" between parallel walls, restricted to motion
in a plane. This geometry is increasingly important for microfluidic device applications. Crucially,
the leading order fundamental flow singularity has a dipolar form - significantly different from the
Stokeslet. As we detail, novel collective phenomena sustained by dipolar HI in collections of flowing
droplets and particles have been the subject of much recent investigation.
2.1
Particle-laden flows: General considerations
We start with some general comments about the motion of particles in a suspending fluid. For
simplicity, suppose the particles are rigid. Clearly, their motion can be calculated from the equations
for rigid body dynamics, provided we can determine the forces and torques on them at each moment
in time. Of course, these forces can include familiar conservative interactions between particles,
e.g. long-range electrostatic forces and excluded volume forces, depending on the details of the
particles. However, we must also consider the stresses exerted by the fluid on the particle/fluid
interface. In turn, the particles couple back to the fluid, driving fluid motion. We therefore have a
problem of fluid/structure interaction,and the equations for particle motion must be supplemented
with equations modeling the fluid, as well as equations capturing how the solid and fluid domains
are coupled at the interface.
Since a typical fluid particle (e.g. a water molecule) is much smaller than a solid particle, the
fluid can be regarded as a continuum, governed by partial differential equations. The rigid body
equations express conservation of momentum for solid particles. Likewise, the Cauchy momentum
equation expresses conservation of linear momentum in the fluid:
P(
v
-Vv)=V-o-+f
(2.1)
In these equations, p is the density of the fluid, v is the fluid velocity field in a fixed frame of
reference, f represent any body forces on the fluid (e.g. gravity), and a is a quantity called the
stress tensor. The meaning of the left hand side of 2.1 becomes more clear when it write it as the
material derivative -: it is the acceleration "following the fluid." Even if the flow field is steady,
so that gat =0, a dyed parcel of fluid will speed up and slow down according to the spatial variation
of the velocity field v.
The stress tensor captures forces that act on surfaces. For a surface with normal vector n, the
force per unit area on that surface is a stress vector n - a. The stress tensor can be separated into
the viscous stress tensor s and a component from the pressure P:
o-= -PI+s,
(2.2)
(2
2.1.
Particle-laden flows: General considerations
33
where I is the identity tensor. Pressure is a familiar quantity: it is a scalar field that exerts normal
forces on surfaces due to elastic collisions of fluid particles. Viscous stresses arise from the internal
deformation ("strain") of the fluid. As such, they are only present when the fluid is in motion.
For most flows, the fluid can be regarded as incompressible, so that
V -v = 0.
(2.3)
Application of the divergence theorem to Eq. 2.3 leads to a simple physical interpretation:
J
JS
pv - ndS = 0.
(2.4)
The mass flux into a control volume V through a bounding surface S must be zero. For an
incompressible fluid, the density p is constant throughout the fluid, and pressure P is no longer
related to it by a equation of state. Instead, pressure acts a Lagrange multiplier field ensuring that
the flow field v satisfies the incompressibility constraint.
The form of the viscous stress tensor s is given by a material-dependent constitutive equation.
Throughout this work, we will consider Newtonian fluids, for which
s = 2pF,
(2.5)
where y is the dynamic viscosity of the fluid, and the rate of strain tensor F is
r = 1 (Vv + (VV)T).
(2.6)
The rate of strain tensor is the symmetric, deformational component of the velocity gradient Vv.
The antisymmetric, rotational component is given by the vorticity tensor f,
=
-
2
(Vv
-
(Vv)T),
(2.7)
and we have Vv = F + 0. Therefore, a linearly varying flow field ("simple shear") can be decomposed into straining motion and rigid body rotation. Moreover, a more generally varying flow field
can be expressed as a Taylor expansion and therefore locally so decomposed. This decomposition
clarifies the interpretation of Eq. 2.5. In a rigid body motion, fluid parcels maintain their relative
positions, but in a straining motion, they slide past each other. This sliding is opposed by molecular
friction. This friction allows the transfer of momentum between adjoining fluid parcels. For this
reason, the quantity v =a /p, the kinematic viscosity, is also called the diffusivity of momentum.
As will be shown in subsequent sections, conservation of momentum and its transport by viscosity
plays a major role in determining microhydrodynamic flows.
Finally, substituting our expression for a, we obtain
P
- +v.Vv
( t
=-VP+ 1pV 2 v+f,
(2.8)
which is the celebrated Navier-Stokes equation.
We now consider how fluid and solid are coupled at the interface. For the fluid, the solid provides
two boundary conditions. Since the fluid cannot penetrate the solid, the normal component of the
34
2.2. The Stokes equation
fluid velocity at the interface must be equal to that of the solid at each interfacial point. Secondly,
the solid provides a "no-slip" or "stick" boundary for the fluid. The tangential velocity of the fluid
must equal the tangential velocity of the solid at each point. (This famous boundary condition is
empirically highly successful. Its theoretical justification has been the subject of much investigation
and occasional controversy.) If a solid particle has a translational velocity UP and a angular velocity
WP, then the no-slip and no-penetration conditions give
v(r) Is = UP + wP x (r - r,),
(2.9)
where r, is the center of mass of the particle, and r is a point on the particle surface S.
For a solid particle, the fluid exerts a stress on the particle surface. The stress can be integrated
to find the force
F = fc o- ndA
(2.10)
and the torque
T=i
r x o- - ndA.
(2.11)
In principle, the foregoing equations, along with initial conditions and boundary conditions on
the fluid, can be solved for particle motion. However, the Navier-Stokes equation is notoriously
difficult to solve, largely owing to the nonlinear term v - Vv in Eq. 2.8. Via physically motivated
scaling arguments, Eq. 2.8 can be reduced to a linear equation, bringing great gains in analytical
and numerical tractability. In the following section, we will obtain the Stokes equation and examine
its mathematical and physical properties.
2.2
The Stokes equation
Since the various terms in the Navier-Stokes equation originate in different physical effects, their
relative significance can depend on the time and length scales of interest. Through a scaling analysis
of Eq. 2.8, we can rigorously justify the neglect of the terms on the left hand side of the equation
for most microflows.
Consider a single sphere with velocity U and radius a. These provide characteristic time and
length scales a/U and a over which the hydrodynamic fields should vary. We nondimensionalize v
as v = v/U, and P by an as yet unspecified pressure scale Po, so that P P/Po. Assuming that
the body force f = 0, we obtain
pU2
rlPo~
a
(9i
p
++ - ()
(P )
a
a2
2
(2.12)
We rearrange this equation to obtain
p
y
--- +
( 0t
(aP
AU
V
V29
(2.13)
The left hand side is scaled by the Reynolds number, defined as
Re =
pUa
= -V
A
_Ua
(2.14)
2.2. The Stokes equation
35
This dimensionless group captures the relative strength of inertia and viscosity, and is generally
very low for microscopic objects. For instance, in Chapter 5, we calculate a Reynolds number of 6
x 10-5 from experimental parameters. Therefore, the acceleration terms can be neglected, giving
SaPo
t2
) + 9
= 0.
(2.15)
Since the dimensionless quantities are 0(1), the quantity in parentheses must likewise be 0(1) for
the two terms of the equation to balance. Therefore, the viscous pressure scale is Po = pU/a.
Restoring dimensions, we obtain the Stokes equation:
--VP +
V 2 v + f = 0.
(2.16)
The Stokes equation and the incompressibility condition V -v = 0 govern highly viscous or "creeping" flows, which include the vast majority of microscopic flows.
Some rather peculiar properties of the Stokes equation are discussed in the following subsections. Following [29], we emphasize linearity, instantaneity,and reversibility. These properties both
simplify the mathematical solution of microflows and complicate their conceptual understanding:
they are physically profound. Our intuitive grasp of fluids, acquired from our everyday experience
of macroscopic flows, cannot be applied to the microscopic without revision.
2.2.1
Linearity
Our scaling analysis was motivated by the difficulty presented by the nonlinear term v -Vv; without
it, Eq. 2.16 is linear in both v and P. Linearity permits the use of the superposition principle,
which generally proceeds as follows: A complicated problem is broken into simpler subproblems.
The subproblems completely decouple. The solution to the complete problem is determined as the
sum of the individual subproblem solutions.
Superposition can be applied to boundary conditions. For instance, a translating and rotating
sphere can be broken into two subproblems: the problem of a translating sphere, and the problem
of a rotating sphere. The complete flow field v is the sum of the flow fields obtained for the
individual subproblems. The complete force F and torque T on the particle are likewise obtained
by summation of forces and torques from the two subproblems.
Furthermore, linearity permits the use of Green's functions. Disturbances introduced in the
flow field, such as by the motion of solid particles, can be represented by the superposition of
fundamental flow singularities,which are themselves determined as the response of the fluid to
impulsive or point-like perturbations. As will be discussed below, these singularities include the
response of the fluid to a point force, to a point source of fluid mass, and to higher order derivatives of
these. An analogy may be drawn with electrostatics: just as the electric monopole is a fundamental
source of electric field lines, so is the "Stokeslet" a fundamental source of hydrodynamic disturbance
via the diffusion of momentum.
2.2.2
Instantaneity
We interpreted the Reynolds number as capturing the relative importance of inertia and viscosity
in the fluid. It can also be interpreted as a ratio of the timescale for particle motion to the timescale
for relaxation of the fluid via viscous momentum transport.
36
2.2. The Stokes equation
Once again, consider a sphere with velocity U and radius a. It moves a distance equal to its own
radius over a time a/U. The time needed for a flow disturbance to propagate the same distance by
viscous momentum transport is related to the kinematic viscosity v, known as the "diffusivity of
momentum." As the dynamic viscosity has units of [L] 2 [TI-1, the viscous timescale must be a 2/v.
The ratio of viscous to particle timescales is therefore Ua/v - the Reynolds number.
In the limit Re -+ 0, the entire flow field instantaneously adjusts to particle motion. The flow
field is quasi-static, slaved to the dynamics of the particles. The fast hydrodynamic degrees of
freedom can be "integrated out" as fluid-mediated effective interactions, or "hydrodynamic interactions," between suspended particles.
Another argument for instantaneity can be provided upon examination of Eqs. 2.16 and 2.3.
The two equations do not depend on time. At any particular instant, the pressure field P and flow
field v are completely determined by boundary conditions on the fluid, including the instantaneous
configuration of the particle suspension. A change in these boundary conditions (e.g. via particle
motion) is instantaneously communicated to every point in the fluid.
2.2.3
Reversibility
Creeping flows are time reversible. Consider Eq. 2.16. Fluid flow is driven by either a body force
f, or by conditions (e.g. pressure) specified on the boundary of the domain. Reversibility implies
the following: Suppose PS and vs are solutions of Eq. 2.16 for the pressure and velocity fields. If
the driving body force or boundary conditions are reversed, then -PS and -vs are the solutions
for the reversed problem by linearity of Eq. 2.16. The flow pattern remains the same; the arrows
on the streamlines are simply reversed.
In principle, if a periodic driving force or boundary condition is applied to a suspension of
particles, the particles should acquire no net displacement over one cycle. In practice, the manybody hydrodynamics of a suspension can be chaotic. Chaotic dynamics depend sensitively on the
location of the system in phase space, and a small amount of noise - inevitable in experiments - can
break time reversibility. Small perturbations to particle positions grow exponentially. Chaoticity
originating in hydrodynamic interactions has been demonstrated in numerical simulations of three
sedimenting spheres [30]. Experimentally, Pine et al. found that a periodically sheared suspension
of spheres exhibits a transition to irreversible motion at a finite strain amplitude [31]. Above
this amplitude, particles follow random walk statistics. However, this transition may be due to
near-contact interactions between particles, rather than hydrodynamic interactions [32].
Another example of reversibility is Edward Purcell's "no-scallop theorem." Time reversibility
restricts viable swimming strategies at the microscale. A time reversible motion, such as the opening
and closing of a scallop's hinge, cannot propel a microorganism or artificial microswimmer [33].
In some circumstances, reversibility can be invoked to qualitatively predict particle motion. An
example will be given in a following section on sedimentation.
2.2.4
Additional properties of the Stokes equation
We note some mathematical details which will be useful later in this chapter. In a force-free region
(f = 0), taking the divergence of Eq. 2.16 gives a Laplace equation for the pressure:
V 2 p = 0.
(2.17)
2.3. Flow singularities and hydrodynamic interactions
37
Taking the Laplacian of Eq. 2.16, we obtain a vectorial biharmonic equation:
V 4 v = 0.
2.3
(2.18)
Flow singularities and hydrodynamic interactions
Suspended particles are coupled by disturbances they create in the surrounding fluid. From our
everyday experience of fluids, this should not be so surprising; it is not difficult to imagine how
a swimmer can feel the passage of another swimmer, for instance. However, from the preceding
subsections, we expect the form and properties of flow disturbances at the microscale to be rather
different from those at the macroscale.
2.3.1
The Stokeslet
As a concrete example of microscopic flow disturbance, we will consider the "Stokeslet," or the
response of a fluid to a point force. The Stokeslet flow pattern is shown in Fig. 2.1. The point
force is shown as a green vector. This force could be the force of gravity on the red sphere, for
instance. It is important to note that the flow pattern shown is not the complete response of the
fluid to an externally driven sphere. Rather, it is the leading order, far-field component of the flow
field. If we imagine a second heavy sphere placed somewhere in the Stokeslet field, we can see that
the interaction between bodies can drive lateral motion - motion perpendicular to the direction of
the driving force - which does not occur for an isolated sphere.
I
/
(
/
/
2
Fig. 2.1: A point force (green vector) applied to a suspended sphere creates a
disturbance flow in the surroundingfluid. We show the leading order, far-field
contribution to this flow, known as the "Stokeslet."
In a following section, we will derive the equations of motion for two coupled sedimenting
spheres. However, we must first obtain a function describing the Stokeslet flow field of Fig. 2.1.
38
2.3. Flow singularities and hydrodynamic interactions
We write Stokes' equation with a point force,
-VP + pIV 2 v + f6(r) = 0,
(2.19)
where f is now a constant vector. Solution of this Green's function problem can be mathematically
involved. We will take an approach adapted from [34], of which we first took note via [35]. We will
first Fourier transform Eq. 2.19:
-ikP + yIk 2 iT + f = 0,
(2.20)
where the tilde denotes a transformed variable. The continuity equation gives k P(k) = -
= 0, so that
(2.21)
.
The solution for P(r) is a non-trivial integral:
P(r) =
(27r)3
f
-ik k2feik-rd3k.
(2.22)
Recall that P and v obey a Laplace equation and a vectorial biharmonic equation, respectively.
From electrostatics, we know that the solution to the inhomogeneous equation V 2 = -6(r) is
= -1D
(2.23)
47r
Since the Fourier transformed equation is -k 2 >
we have
-1,
=
feikr
=k- 1
- dk.
(2r)3] k2
47rr
11-
_
(2.24)
We take the gradient of this equation, and then the inner product with f:
V(
1
4wr
)
f=
ik -feik~r 3
k
d k.
1i
k2
(27r)3
(2.25)
Comparing against Eq. 2.22, we have the solution for the pressure:
P(r) = f -r
(2.26)
Now we would like to obtain v by a similar strategy. Substituting Eq. 2.21 into Eq. 2.20, we obtain
, =( -1
I+ kk)
k2-
f.
(2.27)
We can easily determine the inverse transform of the first term via Eq. 2.24:
v(r) =
[ '
4Aro
r
-
1
p(2th )3
J
(kk
k
4
) d3k] - f
(2.28)
Again, we would like to avoid performing the integral by manipulating the Fourier representation
2.3. Flow singularities and hydrodynamic interactions
39
of a fundamental solution. In this case, we need the fundamental solution to the inhomogeneous
biharmonic equation V 44D = -6(r), which is
(2.29)
<D = r
87r
Again, looking at the Fourier transformed inhomogeneous biharmonic equation, we have
-1J eik-r
3
- r-d
k.
r
r =
-13
87r
(27r)3
(2.30)
k4
Comparing Eq. 2.28 and Eq. 2.30, we need to obtain two factors of k by taking two derivatives.
At first glance, one might think to operate on Eq. 2.30 with the Laplacian V 2 , but this would give
a scalar, not a dyadic tensor. Instead, we will use a dyadic product of gradient operators V 9 V.
The first gradient on Eq. 2.29 gives
V<D =r
(2.31)
87rr
By the chain rule, the second gradient gives
V
V®V~ =-rr
I
r + 87rr' .
(V(D) = 87rr2
(2.32)
Note that the outer product V ®r is, in Cartesian coordinates, (1'a+
:Lx + 99 + 22, or the identity tensor I. We now have
I
87rr
We substitute into Eq. 2.28 to obtain
rr
1
kkeik-r
87rr2
Sir2
rr
(27r)3
k4
v(r) =
d3 k.
2r)
I +
-
- f.
9a
8+2
z)(XI + y + z2)
=
(2.33)
(2.34)
We have at last obtained the velocity field of Fig. 2.1. The quantity 0(r), defined via v(r)
9(r) - f, is known as the Oseen tensor. Notably, the field v decays as 1/r. Furthermore, if the
Stokeslet is used to represent the flow field created by an externally driven sphere, it clearly cannot
account for the no-slip and no-penetration conditions at the sphere surface, since v does not depend
on the sphere radius. In the following subsection, we will discuss other flow singularities needed for
the full solution.
2.3.2
Other fundamental flow singularities
When a sphere is driven by gravity, momentum is continuously added to the system. However,
motion of the sphere is overdamped: it quickly achieves a steady settling velocity for which the
forces of gravity and drag balance. Momentum, a conserved quantity, must diffuse away from the
sphere via viscosity, creating the Stokeslet flow field.
However, momentum is not the only conserved quantity. As the sphere falls, it must displace
fluid mass from its leading edge and draw fluid mass into the space vacated by its trailing edge.
The transport of fluid mass contributes to the flow disturbance, albeit at higher order than the
40
2.4. Sedimentation
Stokeslet. As with the Stokeslet, the transport of mass can be modeled with fundamental flow
singularities.
We can show that the flow disturbance due to mass transport is subleading via scaling arguments. Consider a point-like source of fluid mass. Mass is added to the system at a rate Th. If
the flow field v is steady, then conservation of mass requires that 47rr 2 pv = rh, so that v
-
1/r
2
.
However, a sphere does not change the amount of mass in the system. We can model the displacement of mass with a source and a sink of equal strength. This mass dipole can be obtained as
a derivative of the mass source. Therefore, its flow field decays as 1/r 3 - much faster than the
Stokeslet. However, it can be shown that no-slip and no-penetration boundary conditions on the
surface of a sedimenting sphere are satisfied by the combination of a Stokeslet and mass dipole,
both placed at the sphere center [36]. Alternatively, both the Stokeslet and mass dipole can be
obtained by solving the Stokes equation for a sphere translating through quiescent fluid, e.g. by
using a stream function approach.
Other higher order flow singularities can be obtained as derivatives of the Stokeslet. For instance, in the absence of an external driving force, a force dipole is often the leading order flow
singularity representing a suspended particle. Moreover, both the force dipole and mass dipole are
needed to represent the flow field created by a point force near a plane wall, as will be discussed in
the next subsection. Details of the various higher order singularities can be found in [36].
2.3.3
Geometric confinement and the method of images
The Stokeslet is the response to a point force in an unbounded three-dimensional space. The flow
field does not necessarily satisfy conditions on confining boundaries. If a point force is exerted on
a fluid in the vicinity of a plane wall, we expect the flow field to look quite different from Fig.
2.1. The same is true in electrostatics: the field created by a charge near a conducting surface is
quite different from the field of an isolated charge. As with electrostatics, we can use the method
of images to construct a solution which satisfies both the governing equations in the domain of
interest and the boundary conditions at the surface.
In electrostatics, the image of a single charge near a plane wall is simply another charge of
opposite sign, reflected across the wall. In microhydrodynamics, a point force near a wall has a
more complicated set of images (Fig. 2.2). In addition to an oppositely directed point force, there
is also a mass dipole and a force dipole [37]. The resulting flow field can be complicated: in Fig.
2.2, a sphere is forced away from a wall, creating circulating regions of flow. Moreover, the plane
wall can screen the flow disturbance, changing the ~ 1/r decay into ~ 1/r 2 or even ~ 1/r 3 [37, 38].
The resulting "dressed" Green's function for a point force near a plane wall is often called the
Blakeslet after its discoverer.
Screening by confinement can be even more dramatic in other confining geometries. For instance,
the flow disturbance created by a point force in a cylindrical tube decays exponentially.
2.4
Sedimentation
Sedimentation provides many classic examples of single and many-body phenomena sustained by
hydrodynamic interactions. Even a single settling sphere illustrates the properties of Stokes flow
introduced above. For instance, if we consider a single sphere settling alongside a wall, we can
qualitatively predict its motion by invoking time reversibility. If we suppose it has a velocity
2.4. Sedimentation
41
Fig. 2.2: The Green's function for a point force is changed by geometric confinement: the flow must satisfy the no-slip and no-penetration conditions on
each solid boundary. For a plane wall, both the Stokes equation in the fluid
domain and the boundary conditions on the wall can be satisfied with a system
of images. A point force (green vector) at height y = h above the wall creates
at y = -h an oppositely directed image force, a force dipole, and a mass dipole.
The blue vector represents all three image singularities.
component normal to the wall (i.e. perpendicular to the direction of gravity), as
in Fig. 2.3(a), we
can obtain a contradiction. In Fig. 2.3(b), we reverse the driving force g and the
particle velocity.
In (c), we have rotated coordinates so that the sphere/wall configuration and
the gravitational
force are the same as in (a). The velocity of the sphere should likewise be the same
as in (a), but
the normal component has opposite sign. Therefore, the sphere can only move
in the direction of
gravity.
Quantitatively, the sphere sediments more slowly than a sphere in unbounded fluid.
In one
view, the particle experiences a disturbance flow directed against the direction of
gravity from the
image singularities. In another view, the particle experiences friction from the plane
wall: the
particle must drive flow past a no-slip surface.
2.4.1
Two coupled Stokeslets
Having derived the Stokeslet, we return to the dynamics of two coupled sedimenting
spheres. The
simplest model takes a sphere to be a "bare" Stokeslet, neglecting the mass dipole singularities
and the boundary conditions on the sphere surface [36]. This model has error 0
, where r is
the vector from the center of sphere A to the center of sphere B. At zeroth order, the
two spheres
are non-interacting, and each has a velocity determined by a force balance 67ryaU
= f, where a
42
2.4. Sedimentation
(a)
(b)
velocity
r
(c)
coordinates
g
Fig. 2.3: Reversibility implies that a sphere sedimenting near a wall experiences
no lift force. In (a), we assume the sphere has a non-zero velocity component
in the wall normal direction. In (b), we reverse the driving force. Reversibility
requires that the sphere velocity also be reversed. In (c), we rotate the coordinate
system of the sphere/wall system, obtaining the same physical situation as in
(a). However, the normal component of sphere velocity is negated relative to (a).
Therefore, the assumption of a non-zero normal component is contradictory.
is the sphere radius, 67rpa is the drag coefficient for a sphere, and f is an external force, such as
gravity. The zeroth order velocity is therefore UO = f . At first order, the velocity of a sphere
is perturbed by the flow disturbance created by the other sphere. We obtain the velocity of sphere
Bas
UB
f
6-7rpta
+
[ (iI+±
r2
87rpr
f ±0
(3.r3
(2.35)
Notably, this expression is invariant under r -+ -r, and therefore UA - UB. The symmetry of
the two-body interaction rules out relative motion of the spheres. Suppose f is in the -Z direction.
If we take the dot product of Eq. 2.35 and , we find that the lateral component of the velocity is
U
=
U"
sin(0) cos(0),
(2.36)
where 0 is the angle between the vector r and . If the spheres are aligned horizontally (6 = 0) or
vertically (6 = r/2), there is no "lateral drift." We have written Eq. 2.36 in terms of Uo in order
to make explicit that lateral drift is first order in (a).
Further, horizontal and vertically aligned spheres have different settling velocities. If we take
the dot product of Eq. 2.35 with the 2 direction, we obtain
U
= -Uo
u - UO
(1+ sin2 (O)).
(2.37)
Both the vertical and horizontal arrangement of spheres fall faster than an isolated sphere, but the
2.5. Particle motion in microchannels
43
speed of a vertical pair is
U0 1+
,
(2.38)
.
(2.39)
while the speed of a horizontal pair is
U0 [I +
a)
Similarly, a falling rod will fall faster when vertically oriented than when horizontally oriented.
2.4.2
Many-body sedimentation
Passing from two settling spheres to three or more, we find a rich array of collective phenomena
sustained by hydrodynamics. Three spheres situated on the vertices of an isosceles triangle in a
horizontal plane (i.e. with normal in the direction of gravity) can exhibit oscillatory motion [39].
For three spheres placed in a vertical plane, two spheres will pair and a third escape after a transient
mixing period [30]. The escape time and the identity of the escaping sphere depend sensitively on
initial conditions: the three-body problem is chaotic.
Collective phenomena in suspensions are comprehensively reviewed in [40] and [29]. Suspensions of sedimenting spheres form "swirling" regions with correlation lengths of many sphere radii.
Particle shape anisotropy introduces new behavior. Sedimenting rods tend to form clusters. These
clusters organize into long streamers that settle together, while regions of clarified fluid move upward.
In general, sedimenting suspensions exhibit large velocity fluctuations, owing to the long-ranged
1/r decay of the Stokeslet, and do not self-organize into crystalline arrays [41]. For instance, a
sedimenting row of spheres is unstable to clumping [41, 42].
2.5
Particle motion in microchannels
Sedimentation provides many fascinating examples of phenomena sustained by hydrodynamic interactions. However, in most microfluidic applications, particles are not subject to an external force,
but are instead carried by an external flow. The external flow is driven by boundary conditions (e.g.
pressure) at the inlet and outlet of a microchannel. The solid surfaces bounding the microchannel
determine the spatially varying profile of the external flow field. For instance, flow driven between
two parallel plates ("slit-like" confinement) will assume a Poiseuille or parabolic profile. If a sphere
is suspended in this flow, reversibility implies that it will translate in the streamwise direction with
no motion towards or away from the channel center, as with the sedimenting sphere.
One might hope for more interesting behavior from a particle with more complicated shape - a
rod or ellipsoid, for instance. The problem of an axisymmetric rigid particle in unbounded simple
shear, a linearly varying flow, was first considered by Jeffery [43] and generalized by Bretherton [44].
A particle translates with velocity equal to external velocity evaluated at the particle center, and
tumbles in one of a continuous family of periodic trajectories called Jeffery's orbits. In a channel
flow, a rigid rod or ellipsoid will tumble in a modified Jeffery's orbit with no lateral migration,
despite the additional complications of hydrodynamic images and a quadratically varying flow
profile [45].
44
2.5. Particle motion in microchannels
2.5.1
Self-steering and self-organizing particles
The motion of a particle tumbling in a Jeffery's orbit is periodic. Lateral migration is ruled out
by reversibility. However, additional physical effects can break time reversal symmetry, leading to
lateral migration. For instance, a single deformable particle, such as a droplet or polymer, located
near a plane wall and driven by shear flow will be repelled from the wall [21]. The mechanism
for this migration is illustrated in Fig 2.4. A polymer can be modeled as a dumbbell: two beads
connected by a spring. When the spring is stretched by the flow, the spring forces on the two beads
introduce two Blakeslet flow disturbances. The Blakeslet associated with the left bead drives the
right bead away from the wall, and vice versa. Therefore, deformability, hydrodynamic images,
and shear flow all play a necessary role in driving lateral migration. In a channel, repulsion from
the four bounding walls drives a polymer chain towards the centerline. The attraction of a single
particle to a definite transverse position or finite set of positions can be regarded as "self-steering."
Fig. 2.4: A polymer can be represented as a "dumbbell": two beads connected
by a spring. In a shear flow, the spring is stretched, introducing point forces
that disturb the background flow. We show the disturbance field created by the
force on the left bead. This disturbance drives the right bead away from the
wall. Likewise, the right bead drives the left bead away from the wall by the flow
disturbance it creates.
Finite inertia also breaks time reversibility - recall the v - Vv term in Eq. 2.8. It has been
known for several decades that spheres driven by flow through a tube of radius R will form a ring
of radius Rring = 0.6R at finite Reynolds number [46]. However, this effect has been exploited in
microdevices only recently. Finite Reynolds number can be achieved in a microchannel with high
flow rate (typically ~ cm/s) and low viscosity of the suspending fluid. A single inertial microparticle
in a square microchannel will focus to one of four definite positions in the channel cross-section
[23, 24].
In combination with hydrodynamic interactions, deformability or inertia can drive self-organization
2.5. Particle motion in microchannels
45
of suspensions or small clusters of flowing particles. For instance, trains of spherical particles at
finite Reynolds number self-assemble into a lattice ordered both perpendicular to and along the
direction of external flow, as determined by the balance of inertial lift and viscous hydrodynamic
forces [25]. This inertial ordering effect was exploited to efficiently encapsulate cells in droplets
[47] and for high throughput cytometry [48]. In another study, simulations predict clustering, axial
symmetry breaking, and alignment of deformable particles driven by fluid pressure drop in a tube
[49].
2.5.2
Quasi-two-dimensional confinement
0
4.
X
oil
R
side view
Fig. 2.5: Illustration of quasi-2D hydrodynamics. A disc is tightly confined
between parallel plates and subject to an external flow (black vectors). The
particle is advected downstream (blue vector) by the flow. However, due to
strong friction from the confining plates, the particle lags the external flow, and
moves upstream relative to it (green vector). The particle therefore creates a
characteristicdipolarflow disturbancefield; fluid mass is pushed away from its
upstream edge and drawn into its downstream edge.
As we have seen with the example of the Blakeslet, flow disturbance fields can change dramatically in the presence of confining boundaries. Consider hydrodynamic interactions when the size
of a particle is comparable to the height of a confining slit, such that particles are constrained to
"quasi-two-dimensional" (q2D) motion (Fig 2.5). The tightly confined particles experience strong
friction from the confining plates, and will therefore lag a pressure-driven external flow. Due to this
46
2.5. Particle motion in microchannels
lag, the particles create flow disturbances with a characteristic dipolar structure: moving upstream
relative to the fluid, particles push fluid mass away from their upstream edges and draw fluid mass
into their downstream edges [50, 51].
That the leading order far-field flow disturbance is determined by the conservation of fluid mass
is due to the confining plates. The plates exert friction on the fluid, removing momentum from
the system, and screening long-range momentum transport. In contrast with three dimensions,
conservation of momentum plays no role in determining the far-field disturbance.
The strength of a dipolar flow disturbance is proportional to the particle area and decays as the
inverse square distance 1/r 2 from the particle center. As with the mass dipole in 3D, this scaling
can be obtained by first considering the flow field generated by a point mass source with rate rh.
Conservation of mass requires 27rrpv = rh, so that v ~ 1/r. Therefore, the flow field of a mass
dipole decays as 1/r 2 .
Quasi-two-dimensional microchannels have recently proven to be a rich setting for collective
phenomena involving flowing droplets or solid particles [52]. Arrays of q2D particles form large
scale patterns when driven by an external flow [53] and sharp interfaces in sedimentation [54]. Onedimensional flowing crystals of pancake-shaped droplets, ordered in the streamwise direction, can
sustain transverse and longitudinal acoustic waves, or "microfluidic phonons." [55] The dispersion
relations of two-dimensional flowing crystals have also been obtained via theory [56]. When the
channel width becomes comparable to the particle size, the effects of side walls become important.
Side walls screen the hydrodynamic interaction in the flow direction, modulating the dipolar form
by an exponential decay, and modifying phonon dispersion relations for droplet trains [57]. Other
efforts have examined jams and shock waves occurring in one-dimensional droplet trains [58] and
disordered two-dimensional droplet suspensions [59] in q2D channels.
However, researchers have yet to demonstrate self-organization of flowing crystals in q2D. The
one-dimensional and two-dimensional flowing crystals considered to date are only marginally stable:
the amplitude of a collective mode neither grows nor decays in time. Consequently, crystals do not
spontaneously form from disorder, and have no "restoring force" against perturbation by channel
defects. A natural question is how to introduce an effective attraction to the crystalline states,
causing particles to assemble from disorder, and providing a "restoring force" against perturbations.
One indication is provided by a recent study which demonstrated stable pairing of droplets via the
higher flow disturbance multipoles induced by shape deformation [60]. This finding suggests a key
role for particle shape in achieving self-steering and self-organization.
CHAPTER 3
Two discs flowing in a
quasi-two-dimensional channel
The two-body problem provides a logical starting point for collective dynamics. In this chapter, we
study the dynamics of two hydrodynamically coupled discs in two quasi-two-dimensional geometries:
"unbounded q2D," in which the discs are free to move in an infinite plane between two confining
plates, and the "q2D channel," in which side walls additionally bound particle motion in an in-plane
direction. We develop model equations for disc dynamics which can be integrated numerically. The
key quantity in these equations, the "hydrodynamic interaction tensor," captures the functional
form of hydrodynamic interactions between discs, including modifications introduced by confining
side walls. These equations provide an important foundation for the rest of this thesis.
We apply the equations to the two-body problem in unbounded q2D. Due to the pair exchange
symmetry of the hydrodynamic interaction tensor in this geometry, there cannot be relative motion
of the two discs, regardless of their spatial configuration. In more physical language, two identical
discs "push" on each other identically. However, the discs can have a component of velocity
perpendicular to the direction of the external flow. This "lateral drift" is analogous to the drift of
two sedimenting spheres, discussed in Chapter 2, and will play an important role in Chapter 5.
With the addition of confining side walls, pair behavior becomes rather more complex. If the
discs are initially close together, they form an oscillatory bound state; far apart, they break up and
scatter to infinity. The bound states comprise a continuous family of orbits in phase space, and
48
3.1. Overview
states on the edge of this basin have large amplitude oscillations which are characterized by a softening spring nonlinearity. We demonstrate that confining side walls can be patterned to manipulate
particle pairs. For a sinusoidal pattern with resonant frequency and sufficiently large amplitude,
a bound state can break apart; likewise, initially free particles can form a bound state. Patterning introduces chaoticity into pair dynamics, allowing a pair to explore its dynamical landscape
diffusively. The transition between bound state and scattering therefore occurs stochastically.
The results presented in this chapter have been published in reference [61]. This chapter was
reproduced in part with permission from Uspal, W.E. and Doyle, P.S., Phys. Rev. E, 85, 016325
(2012), copyright 2012 by the American Physical Society.
3.1
Overview
We model a pair of hydrodynamically interacting particles confined in a channel with thin rectangular cross section. We find that the particles have a finite region of attraction, which arises from
the screening of dipolar hydrodynamic interactions by the side walls. Outside this region, the two
particles break apart and scatter; inside, they oscillate together as an effectively free quasiparticle.
We demonstrate that modulation of channel geometry provides a means to irreversibly manipulate
bound pairs.
3.2
Introduction
Control of flowing suspensions of particles is central to many emerging microfluidic applications,
including cell sorting [62], information processing [63], and assembly of complex structures [64].
If the desired colloidal manipulations could be encoded directly into microchannel geometry, they
could be performed sequentially, continuously, and with high throughput. Consider a solution of
microparticles injected into a flow-through microchannel. As a suspended particle travels from the
inlet to the outlet, it encounters the various stages of the device in a definite sequence. Particles
are continuously injected into the device. Finally, continuous flow permits continuous operation,
and therefore high throughput.
Complex two-dimensional extruded features - including posts or obstacles, side wall curvature,
and channel constrictions and expansions - can be fabricated cheaply and easily with soft lithography. As one approach, these features can be patterned to shape the flow field of the carrier
fluid, affecting particle motion. For instance, abrupt changes in channel width create recirculating vortices, which can be used to selectively trap flowing particles [65]. Recently, Amini et al.
demonstrated how to sculpt fluid streams via a simple "library" of cylindrical pillars [66].
This approach focuses on controlling the background external flow field experienced by flowing
particles, and not on interactions between the particles themselves. However, as discussed in the
previous chapter, hydrodynamic interactions between suspended particles can change significantly
in the vicinity of a rigid wall. Further, the presence of a wall can sustain complex, nonlinear
behavior in even a two body problem. For instance, in a recent work, Drescher et al. demonstrated that two bottom-heavy Volvox algae colonies swimming near a wall form bound pairs with
complex oscillatory motion [10]. Patterning geometry to modulate particle-particle hydrodynamic
interactions and, in particular, manipulate hydrodynamically bound pairs is an approach that has
remained largely unexplored.
3.3. Equations of motion and numerical method
49
In this chapter, we consider quasi-two-dimensional two body problems. First, we consider a
pair of discs in unbounded q2D: the "thin slit," for which side walls are neglected. This problem
is trivial but instructive. We then consider the dynamics of a pair of discs in a "slot" or channel
geometry. We develop theoretical model equations for disc dynamics and a scheme for their numerical integration. We find two classes of behavior: scattering, in which pair distance grows without
bound, or nonlinear oscillations, for which pair distance remains finite. We show that these behaviors arise from the interplay of the dipolar form and the screening of hydrodynamic interactions
by the side walls. The oscillations take place far from equilibrium, as the energy provided by the
external flow is dissipated by viscosity in the overdamped dynamics of this system. Nevertheless,
they retain surprising similarity to free oscillations in a conservative system, owing to the time
reversal symmetry of the underlying equations [67]. We characterize the observed behavior as due
to an effective softening spring nonlinearity, and, drawing on this analogy to a finite potential well,
show that a resonant, long wavelength perturbation to the channel boundaries can allow bound
pairs to break stochastically.
Our results demonstrate that irreversible particle manipulations can be performed through
patterning of confining boundaries. Furthermore, the theoretical equations developed in this chapter
- in particular, the hydrodynamic interaction tensor for a channel - will provide the necessary
foundation for studies in subsequent chapters.
3.3
Equations of motion and numerical method
Consider a particle confined between parallel plates. If the characteristic particle size L is comparable to the height of the slit H, its motion is confined to a plane. When the particle moves with
respect to the surrounding fluid, it creates an in-plane disturbance field with a long-range dipolar
form [51]. This form arises from mass conservation: because of friction from the confining walls,
the flux of momentum from a force multipole is exponentially screened over a length scale set by
the slit height H. Lack of momentum conservation fundamentally distinguishes this "quasi-two
dimensional" (q2D) system from genuine 2D Stokes flow [50]. The velocity field will have an approximate parabolic dependence in the z (plate normal) direction, determining the areal density of
the force of friction on the fluid as -yHU(r), where -y, = 8p/H 2 , i is the bulk dynamic viscosity,
and U(r) is the fluid velocity relative to the channel walls at a point r in the midplane z = H/2.
The friction on the particle, determined by the details of the lubricating fluid that separates the
particle from the walls, can be associated with a parameter
This coefficient will in general be
higher than ye, so that in an external flow the particle will not be freely advected, but lag the
surrounding fluid and create a dipolar disturbance field.
Now consider a collection of N such particles. We can approximate the velocity field at particle
i, U(ri), as the external field plus the disturbance created by the other particles and, neglecting
velocity gradients, use the drag coefficient ( for a cylindrical particle of radius R = L/2 in a q2D
uniform flow, to be obtained shortly. This gives a system of 2N equations for force-free particles:
'y,.
((U(ri) - Uf) + yc7rR 2 HU(ri) -
-yp7rR 2 HUP =
0,
(3.1)
where U' is the velocity of particle i relative to the channel walls, and ri = (xi, yi) is its position
in the fixed coordinate system of Fig. 3.4(a). Sensibly, the particle will have the same velocity as
the local flow if the particle friction coefficient -y, is equal to the fluid friction coefficient -yc. If the
50
3.4. Unbounded q2D
thickness of the lubricating layers is known, the coefficient -, can be calculated numerically via
the model developed in Halpern and Secomb [68]. However, we note that the particle velocity is
directly related to the local flow velocity by a parameter a:
U=
*C
U(ri) = aU(ri).
_
(3.2)
+ 7rR2 H-y
Experimentally, a can determined from the velocity of a single particle driven by external flow
through a slit or wide channel. This parameter is bounded by 0 < a < 1. For small (large) values
of a, a particle significantly (barely) lags the local flow field.
Assuming a uniform external flow Uo = Uo., the local field is evaluated to leading order in rij:
U(ri) = Uo +
G('i)(rij) - (U - Uo),
where G(t)(rij) is a tensor determining the contribution of particle
(3.3)
j
to the local field at i, and
rij -= r - ri. These equations can be rearranged into matrix form, AUP = B, where UP is a vector
containing the 2N particle velocities. The off-diagonal terms in the resistance matrix A represent
the coupling between particles, while terms involving the external flow Uo are collected in the
vector B. Numerically, we can solve the N particle problem by forming A and B at each timestep,
inverting A to obtain UP, and integrating forward in time via a Runge-Kutta routine. While A is
populated only by single and two body terms, its inversion solves a many-body problem.
Our model is a minimalistic representation of hydrodynamically coupled particles as coupled
dipolar flow singularities, reducing the full set of PDEs for the flow field to a set of 2N coupled
ODEs. Such "singularity" models have been applied to systems of vortices, swimmers, and sedimenting particles [30, 67, 69, 70]. The simplicity of such models permits rapid identification of
dynamical motifs sustained by hydrodynamics, which can then be studied in detail via experiments
or more fully featured simulations. For instance, the main results for sedimentation of coupled
Stokeslets [30] were recovered in multipole simulations that included finite size effects and lubrication interactions [71].
However, before the model can be applied to study motion of coupled discs, we must derive
the hydrodynamic interaction tensor. There will be two forms of this tensor, corresponding to two
q2D geometries. In unbounded q2D, particles are free to move in an infinite plane between the two
confining plates. In a q2D channel, side walls put finite bound on motion in one of the in-plane
directions.
3.4
Unbounded q2D
To obtain the hydrodynamic interaction tensor in unbounded q2D, we consider a simple problem.
A single cylindrical particle with radius R and velocity Uf is confined in a slit and subject to an
external flow U which is uniform in the midplane. The flow field u can be modeled with the 2D
Brinkman equation,
(3.4)
-VP 2 D + /2D V2U - pHa2 u = 0
with V - u = 0, where a 2 = 8/H 2 , p2D - pH, and u is the fluid velocity in the midplane. P2D is
the 2D pressure and has units of surface tension. We solve for the flow field u in Appendix A in
3.4. Unbounded q2D
51
order to determine the drag coefficient ( [72] and and the hydrodynamic interaction tensor G
The latter can be determined from the disturbance field (u - U).
.
The drag coefficient is
47D 2 a 2 R2 +aRK1(aR)(35
(= 4wjip
D
R +
Ko (aR)
(4
The force on the particle from the flow is
(3.5)
F = ((U - Ut') + 7rR 2 _eHU.
(3.6)
The second term in Eq. 3.6 breaks Galilean invariance, and arises from the external pressure
gradient required to drive the flow against friction from the walls. Eq. 3.1 is obtained after
including the friction on the particle.
The hydrodynamic interaction tensor G(ij) should be read as "the disturbance at particle i in
direction a due to motion of particle j in direction 6." Retaining only the far-field, dipolar term,
the hydrodynamic interaction tensor is xy-symmetric, and non-zero only for i # j:
B-=1+2K1(aR) R2
aRKO(aR))
X = (Xi - Xj), Y
(yi - yj)
2
2
G$W = B(X _ y )/r!-, GOP = 2BXY/r!G(23) = G(13)
G(1) = -G3)
yx
xy,
yy
xx
Now we consider the two-body problem, shown schematically in Fig. 3.1. The HI tensor is
invariant under a transformation from X and Y to -X and -Y, so that there is no relative
motion: r 12 is fixed. In more physical language, the identical discs exert identical "push" on each
other. However, the particles have a component of velocity perpendicular to the external flow
Ufy - UO sin(0 12 ) cos(0 12 ), where 012 is the angle between r 12 and UO. The two discs can "push"
each other across the streamlines of the external flow. This "lateral drift" strongly resembles
the lateral motion of sedimenting spheres coupled via Stokeslet disturbance flows, as discussed in
Chapter 2.
We can obtain this result analytically by exploiting a feature of the hydrodynamic interaction
tensor in cylindrical coordinates. The radial and azimuthal directions completely decouple:
+
G
G
-
2K1(ai) )2
aRjKo(aRj)
Bjlrj,
= 0,
G
Gj = -Bj/r?.
) = 0.
The subscripts of G(23) now refer to directions ? and 0#. P is a unit vector in direction of the vector
from the center of disc j to the center of disc i. # is defined as the angle between i' and the i
direction. The unit vector 4 is orthogonal to i' and in the direction of increasing #.
We consider the influence of particle 1 on particle 2, so that
#
= 012
(Fig. 3.1). We can write
52
3.4. Unbounded q2D
particle 2
12
particle 1
Fig. 3.1: Two hydrodynamically coupled discs in an unbounded quasi-twodimensional geometry are driven by a uniform external flow. The discs interact via dipolarflow disturbancefields. For 012 $ 00 and 012 5 900, the discs
can have a non-zero velocity component perpendicularto the external flow field.
However, they cannot move relatively for any configuration.
two decoupled equations for the flow field at disc 2:
UO(r 2 ) = Uo,4 + G
Ur(r 2 ) = UO,r + Gr
We can use UP"
=
r(Uf
- Uo,4 ),
(3.7)
- UO,r).
(3.8)
aUr(ri) and Up" = aU 4(ri):
U4 (r 2 ) = Uo,4 + G(2 (aUO(ri) - Uo,4 ),
(3.9)
Ur(r 2 ) = UO,r + G( 1) (aUr(ri)- UO,r).
(3.10)
Due to the pair exchange symmetry, we can rewrite these equations as:
U4 (r 2 ) = Uo,4 + G$(2) (aUo(r2 ) - Uo, 4 ),
Ur(r 2 )
=
UO,r + G
1
)(aUr(r 2 ) - UO,r).
(3.11)
(3.12)
3.5.
Confining side walls
53
Rearranging:
Ur(r 2 ) = U0,r( - Gr)
(1 - aGr)
Uo(r2) =
The uniform external flow is Uo = Uo:,
We can find the drift velocity of disc 2 as
- G ()
Uo,(
'00(
(1 - aG(2))
U
(3.13)
(3.14)
.
so that U0,0 = - sin(012 )Uo, and Uo,r
=
cos(0
UY,2 = aU(r 2 ) cos(012) + aUr(r2 ) sin(012).
12 )Uo.
(3.15)
Using these facts, we obtain:
U,
2
1 - G(21)
=aUo sin(012) cos(01 2 ) [_1)
1 - aGrr
1 - G(
(21)
I - aG.,
(3.16)
Substituting for the components of the hydrodynamic interaction tensor, we finally obtain:
2
-2aBUo sin(012 )cos(012 )(1 - a)
r2
(i
(3.17)
2B2
By pair exchange symmetry, U 'I = Up,2. In Fig 3.2, we compare this theoretical expression with
numerical results, finding excellent agreement between them.
3.5
Confining side walls
Clearly, with an additional set of confining side walls, a pair cannot indefinitely maintain constant
velocity in y. To introduce side walls, we draw on an analogy with 2D electrostatics and enforce the
no-mass flux boundary condition via a set of images. [Fig. 3.4(b)] The side walls are separated by
a distance W, and each particle is dressed by an infinite set of reflections. Consequently, particle
i will not only interact with other particles, but also with its own images and the images of other
particles. By performing the appropriate summations, we obtained the dressed self- and two particle
contributions to the hydrodynamic interaction tensor.
A particle and its images can be divided into two sets (Fig. 3.3). The first set, designated "far,"
includes the original particle, as well as periodic images displaced from the original particle in the
y direction with periodicity 2W. The "near" set is seeded from the original particle's mirror image
across the closest side wall, and includes periodic copies of this image. The "near" set is so named
because it includes the image closest to the original particle, and the y component of velocities in
this set are negated relative to the original particle. Summing over the images in these two sets,
54
3.5. Confining side walls
0.02
0.01
-0.01 -
-0.02
0
30
60
90
120
.
150
00
180
Fig. 3.2: The lateral drift velocity of two flow-driven hydrodynamically coupled
discs in unbounded q2D as a function of pair angle. The red curve is given
by the theoretical expression in Eq. 3.17. The black data points were obtained
numerically. We vary angle for fixed disc separation r = 5R, B = 2.12, and
a = 0.796.
the self-interaction (i
=
j) is determined to be:
G()
xxfar
xyfar
-
-
-C13,
a
xxnear
=
G()
yy,far
G'near
2
2W)
-"~
-
ar =
=,
eari
yynear
R7r
+a2K1(aR)
aRKo(aR))
C-
-
xxfar
xxnear
-
-G~)=0Gi)
xynear =
yxnear
sin
-
0
a,ar
For i - j,
X-- a7r(xi - xj)/2W,
YL = r(yi ± yj)/2W
3.5. Confining side walls
55
near
far
image channel
real channel
E -+
**JAY
Ax
near
image channel
far
Fig. 3.3: System of real and image discs used to obtain the "dressed" or effective
hydrodynamic interaction tensor in a quasi-two-dimensionalchannel. A discs'
images split into sets designated "near" and "far," generated as periodic copies
of the two closest images with periodicity 2W.
G('j
xx,f ar
2 cos 2 YC
C(cosh
2
2
cosh 22X- - cosh2 X- 2- cos y-
X-
- cos y-)
X- sin Y- sinh X~
=2C cos Y- cosh
G(')
2
xy,far
(cosh 2 X- - cos 2
y-)
G(f
yx,far
xx,near
Gaxy,far,
G-yy,far =-G
-%(arxx,far
cos 2 Y+ cosh 2 X- - cosh 2 X- - cOS2 y+
(cosh 2 X- - cos 2 y+) 2
X- sin Y+ sinh Xcos Y+ cosh
G(j)
xy,near = -2C
(cosh 2 X- - cos 2 y+)2
G(O
yx,near
G(tj)
ceo =
xy,near
yy,near
-
xx,near
anear %afar
For a given channel width W, Y+ and Y- are bounded from above and below. As |X-f -+ oc
56
3.6. Behaviors and phase map
for fixed Y+ and Y-, the components of G(23) decay exponentially with screening
length W/ir or
2W/7r. This screened dipolar field has been analogized to the field from a
"leaky" capacitor, with
the "leakiness" arising from the inherently discrete nature of
the charge distribution [57]. Moreover,
the tensor now depends on particle position rj, since the disturbance field
created by a particle
now depends on its distance from the channel walls.
We note that our expression for the disturbance velocity created by motion in
the x direction
reduces to an expression obtained for a quasi-one-dimensional channel, Eq. (1)
in [57], in the limit
that -y -+ 0. This limit represents the neglect of fluid incompressibility, i.e. the
effect of channel
blockage by a finite sized particle. However, this effect is not significant for the
channel sizes we
consider. The thin channel approximation allows us to neglect the effects of
particle rotation and
shear layers near solid boundaries. We have explored including all reflections in
evaluation of the
local field, i.e. replacing the second UO in Eq. 3.3 with U(rj), but the quantitative
effect was
insignificant.
For ease of reference in later chapters, we also provide G~ij) in Appendix B.
3.6
Behaviors and phase map
(b)
(a)
top view
L
W
image channel
(d)
H,
v
side view
Y X
-
real channel
(e)
image
channel
Fig. 3.4: (a) Particles of length L are confined to two dimensional motion in a
channel of width W and height H, where H < L < W, and subject to an external
flow. (b) Top down view of the system of images used to impose the no-mass
flux condition at the channel side walls. The real particles (dark red and dark
blue) are dressed by an infinite set of images (light colors). The particles lag
the external flow and are therefore coupled by dipolar flow disturbance fields.
Gray vectors are particle velocities in a frame moving with the x component of
the particles' center of mass, xc,
(c) Particle trajectories in the xem frame
for oscillation around a 00 fixed point, as described in the text. (d) Particle
trajectories around a 90' fixed point. (e) A scattering event.
3.6. Behaviors and phase map
57
Numerically, we address the N = 2 problem with parameters W = 8L, -y,/7 = 25, and C
obtained from H/L = 2/3. Variation of these parameters did not significantly affect the dynamics
we report. We sweep over initial particle configurations (Yi, Y2, Ax), where Ax is separation in the
flow direction, Ax
x2 - x1. The angles 612 = 00 and 012 = 90' still constitute fixed points when
ycm = W/2, where ycm
(Y1 + y2)/ 2 , consistent with the symmetry in y. Otherwise, depending on
the initial configuration, the two particles either oscillate around a 00 or 90' fixed point, remaining
always together, or break apart and scatter, with IAx| growing without bound. (Fig. 3.5) Owing
to the time reversal symmetry [67] of the underlying equations, the oscillations are closed loops in
phase space. When particle trajectories are plotted in the frame moving with xCM
(x1 + x 2 )/2,
they generically resemble Figs. 3.4(c) and 3.4(d): two figure eights for 00 and two loops (which do
not necessarily cross) for 90'. We distinguish scattering trajectories as having a final JAx > 2W.
Invoking time reversal symmetry, we see that these trajectories are pieces of longer trajectories,
symmetric around Ax = 0, for particles that start with Ax = ±oo, approach and interact near
Ax = 0, and scatter to Ax = -Foo. Robust bound states are possible only for particles that are
initially close together, |AxJ < 2W. Finally, we reject trajectories in which particles overlap with
each other or with the side walls. While we could eliminate overlap via the inclusion of lubrication
or contact forces, we are primarily interested in the effect of far-field hydrodynamics.
8
2000
-1500
6
1000
m
4
5t
4
0
0 Mean45angle90
1000
2\
750
500
0
250
0
5
2/L
0
0
2
y/L 4
6
0.005
0.01
0.015
8
Fig. 3.5: Phase map indicating behavior for the initial condition (y1, Y2, Ax).
Yellow (light) squares indicate oscillation around a 0' fixed point; green (light)
triangles, a 90' fixed point; blue (dark) squares, scattering; and red (medium)
circles, particle-particleor particle-walloverlap. For the oscillatory trajectories,
the inset figures show the distributions of mean angle and frequency, where fL
is found by taking the spatial Fourier transform as described in the text.
58
3.7
3.7. Characterization of nonlinear oscillations
Characterization of nonlinear oscillations
In order to examine the nonlinear oscillations in detail, we consider a pair of particles with initial
separation Ax = 0 and Ay = 3L, where Ay = Y2 - Yi. When ym is on the channel centerline, the
pair is at a fixed point and only translates in the flow direction. If the initial y,, is displaced from the
centerline for fixed initial separation, the pair will oscillate around a 900 fixed point with amplitude
in y identical to the magnitude of the initial displacement. Since time can be arbitrarily rescaled
when Re = 0, we consider the variation of y1/L with xzc/L instead of with time. [Fig. 3.6(a)]
The small amplitude signals are sinusoidal, well described by linearization about the fixed point,
while large amplitude signals are nearly triangular. The power spectra, determined by spatially
Fourier transforming the trajectories in Fig. 3.6(a), reveal a shift in the fundamental frequency
with amplitude, as well as growth in odd and, eventually, even harmonics. [Fig. 3.6(b)] As shown
in the inset, the appearance of even harmonics is the signature of broken half-wave symmetry. This
symmetry breaking can be attributed to the strong interaction of a particle with its nearest image
in the vicinity of a wall. The nearest image retards motion in y, since the component of its velocity
in y is opposite that of the original particle.
4
increasing
amplitude
3
.j
20
200
400
x
(a)
Xm
A-
2
600
/L
4
2
02
second
harmonics
-10
1
100L
(b)
0.005
0.01
(b
0.02
Fig. 3.6: (a) Position of a particle in y with xcm for initial separation Ay = 3L
and Ax = 0 and various initial displacements of y,, from the centerline, where
the fixed point has y1/L = 2.5. (b) Matched by color, the power spectra of
the trajectories in (a), where f has units of inverse length. fo is predicted
by linear theory. Arrows indicate the shift of peaks with increasing amplitude.
The appearance of second harmonics is linked to the breaking of the half-wave
symmetry y1(xeM
+ A/2) - 2.5L = -(yl(xm) - 2.5L), where A is the signal
wavelength. The inset shows both the largest amplitude trajectory from (a)
and the result of performing the symmetry operation on it; the curves do not
coincide.
3.8. Bound state manipulation through patterned side walls
59
Since the oscillations are closed loops in phase space, they resemble free motion in a conservative
potential. We define an effective potential in coordinate q as Vq = -2.
For 900 oscillations,
potentials defined via coordinates yi and Ax are single valued. In Fig. 3.7, we shift and rescale
the effective potentials for the trajectories of Fig. 3.6 in order to characterize their shapes via
projection onto a basis set of Chebyshev polynomials. For the potential defined in Ax, a softening
nonlinearity is indicated by growth in a negative coefficient of the fourth Chebyshev polynomial
T 4 (x) = 8x 4 - 8x 2 + 1 with increasing amplitude of oscillation. All oscillatory trajectories have
V/&|Axj > 0. If a trajectory could explore the region where OV/iJAx| < 0, then lAx| would grow
without bound, which would be observed as a scattering event. This softening can be attributed
to the weakening of the pair interaction in |AxJ by the side walls.
1
1
0.5
E
CN%
0 .5
0
increasing
-0.5[
-1L
(a)
ai)
0
a
0-
amplitude
-0.5
0
2 A x / A max
0.5
I
0 1 2 3 4 5 6 7 8 9
(b)
-1
n
1
1
0.5X
E
CN
-0
0
-1
(C)
0.5
0-
-0.5
0
2 Y, / ymax _-
0.5
1
(d)
0 1 2 3 4 5 6 7 8 9
n
Fig. 3.7: Effective potentials in Ax (a) and yi (c) for the trajectoriesin Fig. 3.6,
matched by color. The potentials are shifted and rescaledfor characterizationin
the Chebyshev basis. (b) For motion in Ax, a negative coefficient of T 4 for large
amplitude oscillations indicates a softening nonlinearity. (d) For motion in yi,
large amplitude oscillations have skewed potentials, consistent with the half-wave
symmetry breaking. Arrows indicate the effect of increasing amplitude.
3.8
Bound state manipulation through patterned side walls
The finitude of the effective potential suggests that if a particle pair could explore its effective
potential diffusively, it might unbind stochastically. Since the fixed point of oscillations is at the
60
3.8. Bound state manipulation through patterned side walls
channel centerline, a sinusoidal pattern that displaces the side walls (varying the walls' position
in y with x, but not with time, and fixing W) recalls parametric variation of a spring tether
point, a route to chaotic escape for finite potential oscillators. For a configuration with initial
separation Ay = 3L and Ax = 0, as previously considered, and y, initially on the the centerline,
y,,
= W/2, we vary the amplitude A/L and spatial frequency f"/fo of a wall perturbation.
Without a wall perturbation, this configuration is a fixed point. fo is the fundamental frequency
of small amplitude oscillations, shown in Fig. 3.6. The wavelength of the perturbation A, = fjl
is always large, A, > W > L, such that we can retain the model developed for straight walls. For
each set of parameters, we perform ten simulations for a dimensionless time 7.5 x 10 4 L/Uo, with
each trajectory initially perturbed in phase space by noise with magnitude e = 10-4. We calculate
the Euclidean distance of each trajectory with respect to a reference trajectory. Taking scattered
trajectories to have a final |Axi > 2W, we define the scattering distance x,
as the minimum x,
for which this criterion is satisfied.
10
0
5-
60
11 10
52
540
tU
L
4
10
200
0
l0
18
(b)
0.
5
(a)
y 2 /L
0.3
2
78 43
y/L
1
5
0.0
(C)
w
Fig. 3.8: (a) Phase portrait of two trajectories with A/L = 0.2 and f"/fo = 1
initially separated by a noise vector in phase space with magnitude f = 10-4.
The trajectories diverge exponentially (inset). (b) Distribution of scattering
length x3 for A/L = 0.2 and f,/ fo = 1. (c) Number of scattering and overlap trajectoriesfor various amplitudes and spatial frequencies of a sinusoidal
pattern. The results for irrationalfrequency ratios are similar.
We find that while most trajectories remain bound states, a finite amplitude perturbation at a
resonant wavelength can lead to chaotic scattering. Scattering and overlap trajectories are clustered
around f,/fo = 1. [Fig. 3.8(c)] The scattering trajectories are chaotic, diverging exponentially from
the reference trajectories. [Fig. 3.8(a)] For A/L = 0.1, A/L = 0.2 and A/L = 0.5 at fe/fo = 1,
we run an additional set of one thousand trajectories to probe stability and the distribution of
scattering length. None of the most weakly perturbed trajectories scattered. This suggests that
experimental realizations of bound pairs would be robust against imperfections in channel geometry.
3.9. Conclusions
61
The distribution of scattering lengths for A/L = 0.2 is shown in Fig. 3.8(b). Although the particles
are non-Brownian, both the abrupt rise on the left and the long tail on the right are typical
features of a distribution of first passage times for a diffusive particle with absorbing boundary.
For A/L = 0.5, particles scatter quickly, accumulating little distance in phase space, so that xd
is strongly peaked around x*/Ao = 5.3, where A0 = fo . We note time reversing the above
described scattering trajectories produces solutions in which particles are initially widely separated
but induced to approach and form bound states, whereas robust bound states in the straight wall
system required an initial lAxi < 2W. Therefore, patterned walls can function either to release
bound pairs or trap initially free particles.
3.9
Conclusions
In summary, we have shown that particles driven by an external flow in a narrow channel can
form bound states when coupled through boundary-mediated hydrodynamic interactions. Owing
to time reversal symmetry, the bound state resembles a free "quasiparticle," oscillating in an effective potential constituted by the confining boundaries and the colloids' own motion. The softening
nonlinearity of this potential arises from the screening of interactions in the flow direction by the
side walls. This softening limits pair binding to a finite region, outside of which the particles
scatter to infinity. Patterning the confining boundaries for modulation of the hydrodynamic interaction provides a means to irreversibly trap, manipulate, and release colloidal particles without the
application of external forces.
CHAPTER 4
Collective dynamics of multiple discs
In this chapter, we generalize and extend the results of Chapter 3 by considering larger clusters of
discs in quasi-two-dimensional channels. We first develop a q2D Lattice Boltzmann Method (LBM)
code that recovers quasi-two-dimensional hydrodynamics from "the ground up," via a minimalistic
model of fundamental fluid kinetics. Results obtained with LBM validate the model equations
developed in Chapter 3.
We then provide symmetry principles for the a priori construction of "flowing crystals": configurations of particles that maintain their relative positions as they are carried by the external
flow downstream. One particular class of flowing crystal subsumes and generalizes the two-body
bound state of Chapter 3. We confirm our theoretical findings via numerical simulation. The flowing crystals have oscillatory collective modes, which we investigate numerically and relate to the
dipolar form of the underlying hydrodynamic interactions.
We widen our scope to the larger dynamical landscape, using the model equations to efficiently
explore this landscape and identify metastable configurations in which particles remain close for
large advected distances. These states are then simulated in detail with LBM. The metastable
states introduce new, exquisitely coordinated collective motions with large amplitude particle displacements.
The results presented in this chapter have been published in reference [73], and are reproduced
by permission of The Royal Society of Chemistry.
64
4.1
4.1.
Overview
Overview
Spatially ordered equilibrium states - crystals - and their excitations - phonons - are the mainstay
of condensed matter physics. Flowing, nonequilibrium crystalline states of microparticles and
droplets are desirable for microfluidic logic, assembly, and control, and have been achieved in
recent work via exploitation of viscous hydrodynamic interactions in geometric confinement. For
the most part, these studies considered large ensembles of particles and, accordingly, large scale
collective modes arising from small displacements of individual particles. Via theoretical modeling
and computational simulations, we show that for small clusters of flowing particles tightly confined
in a shallow, "quasi-two-dimensional" microchannel, new types of ordered behavior emerge, varying
from steady states in which particles maintain their relative positions, to exquisitely coordinated
collective motion with large particle displacements. These new collective behaviors require a thin
channel geometry: strong confinement in one spatial direction and weak confinement in another.
We elucidate principles and techniques for the a priori construction or rapid numerical discovery
of these states, which could be exploited for the orchestration of particle motion in lab-on-a-chip
devices and other applications.
4.2
Introduction
Imposing spatial and temporal order on flowing streams of particles is growing in practical significance for microfluidic applications. For bioanalysis, including on-chip flow cytometry [74] and
multiplexed assays with functionalized particles [26], the suspended objects must be individually
distinguishable and addressable as they flow through a scanning region. Flowing, tunable lattices of
particles are desirable for optofluidics [75] and continuous fabrication of metamaterials or cell-laden
microtissues [76]. Order can be achieved by via hydrodynamic focusing with sheath flows [77] or
by positioning with external fields [78]. However, these methods can be limited in generality and
scalability. Recently, researchers have sought to understand how particles can organize themselves
through forces generic to the flow of suspended objects through microchannels, such as viscous hydrodynamic interaction forces. For instance, trains of spherical particles at finite Reynolds number
self-assemble into a lattice ordered both perpendicular to and along the direction of external flow,
as determined by the balance of inertial lift and viscous hydrodynamic forces [25]. This inertial
ordering effect was exploited to efficiently encapsulate cells in droplets [47] and for high throughput
cytometry [48]. In another study, simulations predict clustering, axial symmetry breaking, and
alignment of deformable particles driven by fluid pressure drop in a tube [49]. Particles optically
driven around a ring will pair via an effective attraction that arises from the hydrodynamic interaction between them and the curvature of their trajectories [79]. The flowing ordered states of these
studies occur far from thermodynamic equilibrium, sustained by energy provided by the external
forces or flow. While still lacking a settled body of theory, nonequilibrium self-organization offers
a promising framework for engineering systems with new classes of programmable complexity [9].
The dipolar interaction of q2D confinement lends itself to realization of crystalline states, since
ensemble summations of velocity fluctuations converge even in the limit of infinite system size
[52]. In contrast, the - 1/r decay of the Stokeslet diverges for three-dimensional summations,
leading to large velocity fluctuations in unconfined or weakly confined sedimenting suspensions
[80]. Microfluidic crystals have been realized in q2D as flowing, ordered trains of droplets driven
by external flow, exhibiting transverse and longitudinal acoustic waves ("phonons") and nonlinear
4.2. Introduction
65
instabilities [55]. Arrays of q2D particles form large scale patterns when driven by an external
flow
[53] and sharp interfaces in sedimentation [54]. The addition of side walls distinguishes the thin
channel geometry, shown schematically in Figure 4.1 (a), from the slit, which is unbounded
in the
xy plane. The side walls screen the hydrodynamic interaction in the flow direction, modulating
the
dipolar form by an exponential decay, and modifying phonon dispersion relations for droplet
trains
[57]. Recent efforts have examined jams and shock waves occurring in one-dimensional
droplet
trains [58] and disordered two-dimensional droplet suspensions [59] flowing in the
thin channel
geometry. A comprehensive review of this work is provided by Beatus et al. [52]
(a)
(b)
-top
i
Ii
image channel
view
___v
I.
Z
]
side view
~
0
real channel
Image channel
0
ma
'Pb
L
~
0
1
Fig. 4.1: (a) In this chapter, a cluster of N particles (here N = 3) is tightly
confined in a gap of height H between plates normal to the z direction. They
are free to move in x and y between side walls, where W is the width of the
channel. The position of particle i is labeled by xi and yi in a frame fixed to the
channel walls. The particles are driven by an external flow. (b) System of real
and virtual particles used to derive the thin channel hydrodynamic interaction
tensor. The real particles (dark colors) are subject to an external flow (black
vectors) and are dressed by an infinite set of images (light colors) that are
constructed iteratively, via mirror reflections across the real and virtual channel
boundaries. Due to friction from the confining plates, each particle lags its
own local flow field; gray vectors show the velocities of the real particles in
frames moving with local flow. This relative motion gives rise to hydrodynamic
disturbancefields (black streamlines) that couple the particles, and is dominated
by motion in the direction opposed to that of external flow. We also show
particle velocity in a frame moving with the particle cluster's center of mass for
two of the virtual particles (green vectors).
These recent studies examined large ensembles of particles, whether flowing in a linear train
or a two dimensional swarm. Due to the size of these systems and the long-range nature
of the
dipolar interaction, the details of spatial microstructure average out of the description of collective
behavior, permitting coarse-grained modeling via continuum approximations and mean-field
theory.
66
4.3. Theoretical model
Moreover, for typical amplitudes of collective modes in these systems, the spatial displacement of
an individual particle is generally small. For the dynamics of small clusters, on the other hand,
we anticipate sensitive dependence on spatial configuration and larger, individualistic, and more
complex excursions of single particles. Accordingly, cluster dynamics must be resolved at the
single particle level. In this chapter, we study the emergent dynamics of clusters of multiple
flowing rigid particles in a thin channel via theory and Lattice Boltzmann simulations. We find
a rich variety of dynamical behaviors, including stable and metastable configurations in which
particles maintain their relative positions, collective modes with relative particle displacements in
two dimensions, cycles in which particles exchange positions, and stochastic cluster dispersion. We
provide symmetry principles for the a priori construction of stable configurations and demonstrate
techniques for rapid identification of more complex dynamical motifs. These findings could be used
for control of highly confined particles in lab-on-a-chip devices, especially where complex motion of
individual particles is desired. Furthermore, we suggest implications for the role of hydrodynamic
interactions in hydrodynamic diffusion and irreversibility.
4.3
Theoretical model
In a previous chapter, we developed a minimal theoretical model that treated the particles as
coupled dipolar flow singularities, dressed by a set of virtual particles in order to impose the
boundary conditions at the side walls. We offer a brief restatement of the model equations, and
mention one modification we make in this chapter.
We obtained a force balance equation for disc i,
((U
- U(ri)) + y,7rR 2HU(ri) - yp7rR 2H U' = 0
(4.1)
where U(ri) is the local flow field at particle i, evaluated at the disc position ri in the midplane
z = H/2 of the confining slit; Uf is the velocity of the disc; R is the disc radius; and the channel
friction coefficient -c = 8p/H 2 . The particle friction coefficient y, depends on the thickness of the
discs' lubricating layers, but it can be lumped into a parameter a, where
U
=
( + 7rR2 H-YP
)
UR =oaU(ri).
U(r)
(4.2)
This parameter is easily determined experimentally.
For a system of N particles subject to a uniform external flow Uo, the local flow field at particle
i is determined through an implicit equation
U(ri) = Uo +
G@i)(rij, rj) - (U' - U(rj)),
(4.3)
where rij = rj - ri, and G(W)(rij, rj) is the hydrodynamic interaction tensor, which determines
the contribution of particle j to the local field at i. This tensor couples the particles and encodes
information about the system geometry. Throughout this chapter, we use the tensor for a q2D
channel, given in detail in Appendix B. For the work of this chapter, U(rj) replaces a second
occurrence of Uo on the righthand side of Eq. 4.3.
As in Chapter 3, the equations can be rearranged into matrix form AUP = B, where UP is
4.4. Lattice Boltzmann Method
67
a vector containing all 2N particle velocities, A is a resistance matrix that includes all pairwise
interactions, and B collects terms involving the external flow UO. This system can be integrated
numerically.
Substituting Equation 4.2 into Equation 4.3 and using the thin channel interaction tensor, we
can identify two dimensionless parameters that govern particle dynamics. These are WIL, the
dimensionless channel width, and a parameter 3 that characterizes the strength of hydrodynamic
coupling between the particles. This parameter is
/3
(1-- a) 1+
(2K 2K(aR)
1 (aR)
aRKo(aR)
(4.4)
'
where a 2 = 8/H 2 , and KO and K 1 are modified Bessel functions. The first term in parentheses
immediately arises from making the substitution. The second term scales G() (ri, r) and accounts
for fluid entrained in the viscous boundary layer of a particle, which increases the particle's effective
hydrodynamic radius. Since the boundary layer thickness is order H, this term asymptotes to one
as the channel height is decreased. We demonstrate the validity of 6 as a governing dimensionless
parameter when reconsider the two-body problem via recovery of a predicted scaling and collapse
of data onto a single curve.
Finally, we note that by approximating the local flow as uniform, we have neglected velocity
gradients, even though the typical particle separation for our system is only a few particle diameters.
However, the Fax6n correction determining the contribution of velocity gradients to the force on
a cylinder in a Brinkman medium was shown to be proportional to V 2U(ri) [81]. Since the far
field flow disturbance is the gradient of a potential # that obeys Laplace's equation, V 2 0 = 0, this
correction vanishes.
4.4
Lattice Boltzmann Method
The Lattice Boltzmann Method (LBM) simulates hydrodynamics from the "bottom up," through a
coarse-grained model of populations of particles colliding and streaming on a grid. In the collision
step, the populations at a fluid node relax to an equilibrium distribution that maximizes local
entropy while conserving the collision invariants, density and momentum. These two macroscopic
fields are computed for each node by taking moments of the local fluid populations. In the streaming
step, populations are shifted to neighboring nodes along lattice links. For the correct choice of lattice
architecture, this model recovers hydrodynamics for length scales larger than the grid spacing and
time scales above the step size. Whereas with the singularity model, we took the dipolar form
of hydrodynamic interactons as our starting point, in LBM, this form should emerge from the
underlying lattice dynamics. Moreover, LBM naturally includes hydrodynamic near fields and the
effects of finite inertia and easily handles complicated geometries. While in this work we consider
rigid discs, deformable particles or particles with complicated shape can also be coupled to LBM.
We use a D2Q9 grid with the popular single relaxation time (BGK) model, which is detailed
extensively elsewhere [82, 83] . However, modifications are required to simulate q2D flow. We
include the effect of the walls normal to z via a drag term linear in the local velocity. Such a
term had been used in several previous LBM studies of Hele-Shaw flow [84, 85]. However, for the
large flow domains and small channel heights we simulate, the pressure drop in the flow direction
is substantial. In typical, weakly compressible BGK models, pressure and density are related by
68
4.4. Lattice Boltzmann Method
an equation of state, P
=
p/3. Large pressure drops introduce compressibility error. Moreover,
continuity requires an increase in flow velocity between the inlet and outlet, (PVx)left = (Pvx)right-
This undesirable unidirectional extensional flow breaks the fore-aft symmetry of the velocity field
in the Stokes regime, and would tend to align particle clusters with the flow. Therefore, we adopt
the incompressible, "pressure-based" BGK model of Guo et al. [86]
Hereafter we apply the method of that work for a thin channel of height H. By construction,
the numerical model is two dimensional, but we are interested in simulating a three dimensional
system; therefore, we must take care with quantities that depend on spatial dimension. The mass
density is fixed as P2D = 1. Therefore, P3D = 1/H. Dynamic viscosities P3D and p2D also depend
on spatial dimension, but the kinematic viscosity v does not: v = p3D/P3D = p2D/P2D. From
above, the force of friction on a column of fluid of height H, area A, and midplane velocity u
is 7yAHu, where -y, = 83D/H2 . (Recall that we assume a Poiseuille profile in z, and u is the
maximum of this profile.) Substituting p3D = VP3D and P3D = 1/H, we obtain A8v/H 2 u. The
area of a fluid node is, in lattice units, A = 1, so that 8v/H 2u is the frictional force we need to
apply on a fluid node with local velocity u.
The fluid populations are designated gi, with equilibrium distributions given by
P
(eq)
9i
=
e - u + uu: (eie - cI)1,
+
I csP2D
C
+(45
(4.5)
2Jcs
where wi and ei are the usual D2Q9 weights and lattice vectors, and c2 - 1/3.
In this model, the macroscopic fields are velocity and pressure, not velocity and density. We
define A = 8v/H 2 . The velocity at a node is computed as
U =
gie
1 + A/2
(4.6)
and the pressure as
P
C p
gi.
(4.7)
The collision step is
gi(x, t)
= gi(x,t)
-
[gi(x, t) _ geq)(x, t) + F
(4.8)
+ u (eiei
(4.9)
where
Fi = -wiP
2D
1- -
A [u
-
c)
is the contribution of the force of friction.
In comparison with weakly compressible BGK models, the incompressible BGK model introduces a new error term for unsteady flow that is O(Ma2 ), where the Mach number Ma = u/c.
However, as we simulate low Reynolds number flow, i.e. the quasi-steady regime, this term is small.
For all simulations, we use relaxation time T = 1 timesteps, so that the kinematic viscosity v = 1/6
in LBM units.
Particles are included as a rigid discs. The discs are coupled to LBM fluid via transfer of
momentum in the "bounce-back" boundary condition, using a first-order boundary interpolation
4.5.
Results
69
method. This coupling determines the drag forces and drag torques on the disc. In Appendix B,
we show that the coupling recovers the theoretical translational and rotational drag coefficients
determined by Evans and Sackmann [72]. LBM nodes within the boundaries are taken to be solid.
As the discs move over the grid, LBM nodes are transferred between the solid and fluid domains,
requiring removal or refill of fluid populations. In order to conserve momentum, this transfer
requires calculation of additional forces on the discs, although we find that these forces do not
significantly affect particle dynamics.
The discs are also subject to frictional forces and torques from the walls. The discs follow
Newton's equations of motion, which are integrated via the DPD-VV scheme in Nikunen et al.
[87] This scheme adapts the familiar velocity Verlet numerical integration method for velocity
dependent forces. We use the iterative version of this scheme, recalculating disc velocities, as well
as the components of outgoing fluid populations that depend on the disc velocities, until a specified
tolerance in the disc velocities is satisfied. However, we note that even a single pass seems accurate
and numerically stable for rigid particles.
The discs are advected by the flow, and we are interested in dynamics over hundreds of advected
particle lengths. For computational efficency, we move the computed flow domain as a window containing the particles. In order that flow remain fully developed, a buffer of length W is maintained
between the leftmost and rightmost particles and the boundaries. At the boundaries, we impose
the velocity profile for steady flow in a thin channel via the method of Zou and He [88]. The profile
is given by:
U=UO
~
cosh(-(y - W/2)/H)
cosh(v W/2H )
(4.10)
This profile is approximately uniform for most of the channel. Boundary layers at the channel
walls satisfy the no-slip condition. When fluid nodes are added to the right edge of the domain,
we extrapolate the local pressure and fill the nodes with equilibrium populations. We did not find
variation of this method to have significant effect.
4.4.1
Validation: Torque and drag on a single disc
In order to test our q2D Lattice Boltzmann model quantitatively, we measure dimensionless drag
forces and torques for comparison with the theoretical values derived in Evans and Sackmann [72].
In the following, we use a square domain of side length 25L, while we vary disc size L and dimensionless height H/L. Larger values of L resolve more spatial detail, improving numerical accuracy,
at the cost of greater computation time. We measure the dimensionless drag force FX/4-7ry2DU on
a disc translating with fixed velocity U in a quiescent fluid. We fix ReP = UPL/v as ReP = 0.1.
Likewise, we measure the dimensionless torque r/47rR2 I2DW on a disc rotating with fixed angular
velocity w in a quiescent fluid, keeping Re' = 0.1, where Re' = 2wR 2/ v. The size of the domain
allows us to neglect boundary effects. As Figure 4.2 demonstrates, there is negligible improvement
in accuracy when L is increased beyond the value L = 10 used throughout this work.
4.5
Results
The geometry of our quasi-two-dimensional system is shown in Figure 4.1. A channel of width
W and height H contains N discs of diameter L. The position of particle i is denoted by xi and
70
4.5. Results
Theory, translational
Theory, rotational
LBM, L = 15
E3 LBM, L =10
A LBM, L = 5
-
10-
-
CJ
-
6-- V
L
2I
I
I
0.4
0.6
0.8
H/L
-
1
Fig. 4.2: Dimensionless drag forces and torques vs. dimensionless channel
height H/L for a disc translating or rotating in a quiescent fluid for various
disc sizes L, where L characterizes the level of spatial coarse-graining. For
L = 10, the disc size used in this study, there is negligible gain in accuracy with
further improvement in resolution.
yi. In the theoretical model, L is taken as the fundamental length scale, and Uo as the velocity
scale. In simulations, we set L = 10 lattice lengths, and define a Reynolds number Re = uoH/v,
where uO is the maximum velocity at the channel boundaries, appearing in Equation 4.10. In both
the theoretical model and the simulations, we fix H/L = 2/3 for simplicity. We also fix the ratio
of particle and channel friction coefficients as y,/y, = 25. The parameter f is therefore fixed as
f3 = 1.82. In what follows, we generally consider the evolution of the cluster with the position of the
center of mass in the flow direction, XcM = I Ej xi, instead of with time t. In the limit of Stokes
flow, velocity can be arbitrarily rescaled (cf. Equation 4.3), and therefore so can the dimensionless
time tUo/L.
4.5.1
Two discs, revisited
The theoretical model was developed in the previous chapter, where it was applied to a system of
two particles. It was found that, depending on the initial conditions, the two particles will either
break apart and scatter to infinity, or remain together as a oscillatory bound state. These bound
states are marginally stable orbits around two types of fixed point. The first type of fixed point
has particle separation in the flow direction Ax = 0, where Ax
x 2 - x 1 , and Ay 0 0, where
Ay
y2 - yi. The center of mass position is on the channel centerline, ym = W/2. For this type
of fixed point, the particle separation vector r 12
r2 - ri is oriented at 900 with respect to the
external flow. The second, 0' fixed point has y. = W/2, Ax $ 0, and Ay = 0.
We quantitatively compare Lattice Boltzmann and the theoretical model for two initial conditions and channel geometries that lead to oscillations around a 900 fixed point. For initial Ax = 0,
4.5. Results
71
Ay = 3L, ycm = W/2 + L, and WIL = 9, we find that LBM results at Re = 0.2 closely correspond
to theoretical predictions, with very slight attenuation of amplitude over several wavelengths. This
can be seen in Figure 4.3 (left), which show particle positions yi and Y2 with center of mass position
Xcm.
91
8
2uo
Lattice Boltmn
-- Theory
--
7-
6 -J 5 -
01
0
U0
400
800
1200
0
xcm
Fig. 4.3: (left) Oscillation of a particle pairwith initial Ay
3L, initialAx
0
and initial center of mass position ycm = W/2 + L. The positions of the two
particles in y are shown as a function of center of mass position x,.
The
solid black curve shows LBM simulation results at Re = 0.2, and the dashed red
curve shows the result of numerically integrating the theoretical model. These
curves closely agree, though very slight attenuation in amplitude can be seen in
the LBM results. (right) Simulation results for the particle pair, shown after
advection by xcm/L = 859 particle lengths. The red and blue curves show
particle positions at prior times, and crosses indicate initialparticle positions.
Colors indicate the magnitude of the fluid velocity field.
On the other hand, decay of amplitude with xc is much faster at Re = 0.2 for another, less
dilute configuration with initial Ay = 2L, initial Ax = 0, and initial y', = W/2 + L. Figure 4.4
shows results for this trajectory and for other values of Re. Since yi is plotted against xcm instead
of with time, these curves should overlap if the velocity scale is unimportant, i.e. in the absence of
inertial effects. As Re is reduced, the effect diminishes in significance, and the trajectories approach
overlap. We conclude that the marginal stability of the theoretical model can be recovered in the
limit of Re -+ 0, although the inertial effect is more significant for configurations with closer contact
between surfaces.
We also show a trajectory at Re = 1, for which decay is rapid. For this trajectory, it can be
easily seen that yl and y2 drift towards focusing positions. This is reminiscent of the focusing effect
exploited in inertial microfluidics [24]. Inertia could provide another axis of control for particles
flowing in q2D confinement, although the design rules, theory, and experimental results developed
for weakly or moderately confined spheres may not be directly transferable to q2D particles.
Finally, we use this initial particle configuration and channel geometry to demonstrate the uni-
72
4.5. Results
versality of our theoretical model and the effect of varying the parameter 3. Since the hydrodynamic
interaction term in Equation 4.3 is order 3 relative to the external flow U , then the
wavelength of
0
oscillation of yi or Y2 with xcm should vary as /--1, since the particles' motion in
y arises strictly
from hydrodynamic interactions. For H/L = 2/3 and H/L = 1/3, we vary the value
of y, to tune
0, and find that A/L, the wavelength of oscillation, does follow the predicted scaling,
and that the
data for both values of H/L collapse onto one curve.
6-
r
+--4 Re==0.2
9e Re= 0.1
ilaRe
5-
o
=0.05-
3
20
100
200
xC/L
Xcm
300
400
Fig. 4.4: Oscillation of a particle pair with initial separationAy = 2L, Ax = 0
and initial center of mass position yc, = W/2 + L for various values of Re.
Particle positions in y are shown as a function of center of mass position xcm
in the flow direction. As Re decreases, there is less decay of amplitude per
wavelength. For clarity, we omitted the theoretical curve for one of the particles.
4.5.2
Fixed points and oscillatory modes
It is possible to construct special "fixed point" particle configurations on the basis
of symmetry
and the functional form of hydrodynamic interactions. These configurations are fixed
points of the
dynamical system constituted by Equations 4.2 and 4.3. Particles in these configurations
maintain
their relative positions as the cluster flows down the channel.
Recall that our theoretical model imposed boundary conditions at the channel side walls
via the
method of images, in which each real particle is dressed by an infinite set of virtual particles.
When
the distinction between real and virtual particles is disregarded and all are considered together,
they
can be taken to be coupled through simple dipolar interactions, as in a slit, rather than
through
screened dipolar interactions, as restricting our attention to the real particles would require.
If each
particle is subject to the same local flow field, then there is no relative motion, and
the particles
maintain their spatial configuration in the center of mass frame. This is possible if
each particle
"looks like" every other particle through translational symmetry; relative motion would
break this
symmetry. In Figure 4.6, we show three classes of fixed point that can be constructed through
this
principle: the "dimer column," the "column," and the "double column." Each of these
fixed points
can be obtained for any number N of particles. As shown, Lattice Boltzmann simulations
confirm
4.5. Results
73
0 H/L=2/3
* H/L = 1/3
1000-
OL963
--.
100
0.25
0.5
1
2
Fig. 4.5: The two particle configuration of Figure 4.4 has an characteristic
wavelength A/L with which yi and Y2 oscillate as x,, increases. This wavelength
A/L scales with the hydrodynamic interactionparameter 3 with a fitted exponent
of -0.963, close to the predicted value of -1.
that these are fixed points.
In order to illustrate the symmetry principle, in Figure 4.7(a) we show the "dimer column"
geometry for three particles, including nearby virtual particles. Each of the infinite set of real and
virtual particles resembles the dipole in Figure 4.7(b), moving in the negative x direction with
respect to the local flow, and therefore contributing a component of velocity strictly in positive x
to the other particles' local flow fields, which are identical to its own. By the image construction
we obtain N(a + b) = 2W. Therefore, while rows (a) and (b) in Figure 4.6, showing only real
particles, appear quite distinct, we see that the "column" configuration is only the "dimer column"
configuration with a = b. In the "double column," particles now contribute components in y and
-y to the local flow fields of other particles, due to the angular dependence of the dipolar form, but
these components cancel because of the symmetry of the arrangement. (On the other hand, this
would not be true of a "double dimer column," which is therefore not a fixed point.) In view of the
translational symmetry of the set of real and virtual particles, these fixed points can be regarded
as "flowing crystals."
As with crystals, these fixed points are associated with characteristic oscillatory modes, which
can be obtained in the model via numerical calculation and diagonalization of the Jacobian matrix.
Figure 4.8 shows the two oscillatory modes of a three particle column. The eigenvalues of the
Jacobian for these modes are strictly imaginary: the modes are marginally stable in linear theory. To
further probe stability, we displace the particles from their fixed point positions along an eigenvector
with finite amplitude and integrate the equations of motion. The initial displacement neither grows
nor decays in time. This marginal stability, in which we obtain a nested set of closed orbits, recalls
our earlier study of two particle dynamics. When we simulate the eigenmodes in Lattice Boltzmann,
we find that the modes can either slowly grow or decay with time, i.e. the eigenvalues can have a
small real part. This is not surprising, as Lattice Boltzmann is inherently a finite Reynolds number
technique. Inertia increases the order of the differential equations governing particle
dynamics,
74
4.5. Results
(a)
2u0
(b)
U0
(C)
Fig. 4.6: Fixed points obtained a priori via symmetry considerations, depicted
in top down view. The first column shows particles in a frame moving in the
center of mass when the theoretical model is integrated. In this frame, particles
remain in fixed positions. Side walls are indicated by black lines. The second
column shows particles in the center of mass frame for the correspondingLattice
Boltzmann simulations. Colors indicate the magnitude of the fluid velocity field.
In the simulations, particles move slightly, but remain within one radius of
their initialpositions. Due to this motion, the fluid velocity field can be slightly
asymmetric. (a) A "dimer column" for channel width W/L = 9. The LBM
simulation is shown after the particles were advected downstream by xem/L =
241 particle lengths at Re = 0.2, where xcm is the position of the center of mass
in the flow direction. (b) A "column" fixed point and LBM simulation after
advection by xem/ L = 833 particle lengths at Re = 0.2. (c) A "double column"
fixed point and LBM simulation after advection by xem/ L = 524 particle lengths
at Re = 0.2.
4.5. Results
75
(a)
Image channel
I
(b)
bt
b
$1W
rea channel
imago channel
Fig. 4.7: Geometric construction of the "dimer column" fixed point. In (a),
three real particles are accompanied by an infinite set of virtual particles, the
closest of which are at y = -a/2, y = W + b/2, and y = W + b/2 + a. These
quantities are related by 3(a+ b) = 2W. Each of the real and virtual particles is
identical, resembling the particle shown in (b), moving in -x with respect to the
local flow field and contributing components of velocity in positive x to the local
flow fields of the other particles. The gray vector shows the velocity of a particle
with respect to the local flow, while the black streamlines illustrate the dipolar
disturbance field thus created. Because this configuration is one dimensional,
the angular dependence of the dipolarform is not relevant here.
potentially affecting the stability of the dynamical states found for Re = 0. The effect of decreasing
Re in the simulations is to decrease the significance of the effect, as we had shown quantitatively
for the case of two particles.
There are also fixed points for unbounded q2D flow (i.e. without side walls.) The case of N = 2
is trivial; the pair simply translates with fixed particle separation, and there are no oscillatory eigenmodes. At the other extreme are infinite one dimensional or two dimensional lattices. The lattices
include those constructed above, with virtual particles replaced by real particles, as well as lattices
that are periodic in the flow direction, as in Tlusty et al. [89] These lattices do have eigenmodes,
such as the "microfluidic phonons" of that work. However, infinite lattices can be destabilized by
nonlinear instabilities, whereas the cluster fixed points constructed here are sustained for hundreds
of advected particle lengths. We do not know of any fixed points for N =
3 or N = 4. For N = 5,
there is a fixed point with particles arranged at the vertices of a regular pentagon. In the theoretical model, its Jacobian has only two eigenvalues of any significance: a pair of real numbers.
There are no oscillatory collective modes, and the cluster is linearly unstable. The magnitude of
the eigenvalue depends on the length scale of the pentagon. For instance, when the radius of the
circumscribing circle is 3L, the cluster breaks up after approximately xcm/L = 240 particle lengths
with even a very modest amount of noise applied to the initial particle positions. (Two separate
noise terms were applied to the x and y positions of each particle, where the noise was uniformly
distributed over the interval [-0.00025L, 0.00025L].) Experimentally, the unbounded q2D geome-
76
4.5. Results
try can only be approximately realized as a very wide channel. When the cluster is positioned in
the center of a channel with W/L = 50, the residence length of the cluster is reduced even further,
to XC/L = 160. We suggest that the effect of the side walls should be considered for nearly all
practical channel sizes.
4.5.3
Metastable states and stochastic dispersion
Within the theoretical model, fixed points constructed via symmetry considerations are marginally
stable. There are still other, "metastable" configurations for which particles remain together for
many advected lengths xn/L before eventual break-up of the configuration, both in the theoretical
model and in simulations. For instance, Figure 4.9 shows LBM results for a "triangle" configuration.
The cluster only disperses after xn/L = 1047 advected particle lengths. Moreover, for much of
this initial transient period, relative positions are roughly maintained, as in the first two panels.
We find such steady metastable states by setting the relative particle velocities found via Equations
4.2 and 4.3 to zero and solving the resulting algebraic equations numerically. These states provide
new steady particle configuration geometries beyond the linear configurations of Figure 4.6.
We characterize the eventual break-up and dispersion of the configuration via the quantity
N
(t)/L
(xi(t)/L - xc(t)/L)2 ,
=
(4.11)
i=1
recalling previous studies of collective diffusion of confined Brownian colloids [90]; however, the
particles we consider are non-Brownian. The dispersion u 2 /L 2 is shown for four realizations of this
metastable triangle in Figure 4.10, with each trajectory subject to a slight random perturbation
to its initial particle positions. (Each individual particle is spatially displaced by a vector with
magnitude 0.025L and random angle.) Break-up occurs for various values of dimensionless time
Uot/L. The cluster disperses stochastically, owing to sensitivity to initial conditions.
To determine whether chaotic three body hydrodynamics can account for this sensitivity, we
examine the three particle configuration of Figure 4.11 (a) by integrating the theoretical model.
Recall that, by construction, this model contains only far-field hydrodynamics. We integrate several
thousand trajectories, again with each starting with a small, random perturbation to the initial
particle positions. (In this case, we apply two separate noise terms to the x and y positions of each
particle, where the noise is uniformly distributed over the interval [-0.0125L,0.0125L].) We find
two modes of cluster break-up: either the blue particle escapes from the red and green particles,
(Figure 4.11 [a], top panel) or the blue and green particles escape together as a dimer and leave the
red particle behind (Figure 4.11 [a], bottom panel). Qualitatively, this break-up can be understood
on the basis of the so-called transverse "anti-drag" inherent in the dipolar form of hydrodynamic
interactions. (Figure 4.1[b]) The blue particle leaves the red and green particles behind when it
is close to the lower wall, so that it is significantly sped up by interaction with its nearest image.
Similarly, when two particles form a pair oriented perpendicular to the external flow - such as when
the green and blue particles escape together - "anti-drag" increases the speed of the pair relative
to that of a single particle. Once particles are separated by a distance comparable to W, their
interaction is screened, and break-up is complete. The distribution of these escape pathways is
shown in Figure 4.11 (b). This stochastic break-up process arises from sensitivity to initial particle
positions, which can be quantified by measuring the characteristic time for exponential divergence
4.5. Results
77
(a)
(b)
theory
theory
simulation
simulation
Fig. 4.8: Oscillatory modes of a three particle column fixed point with W/L 8 and lattice length a/L = 8/3. The top row shows trajectories found via
numerical integrationof the theoretical model, startingfrom an initial condition
in which the particles are displaced from the fixed point along an eigenvector.
In the bottom row we show the correspondingLBM simulations. Particles are
shown in their final positions, while the crosses indicate initial positions. The
red, blue, and green curves are the "tracks" showing particle positions over
time. Arrows indicate the direction of particle motion. In the
simulations,
the oscillations in (a) slowly grow with time, while those in (b) slowly decay.
As discussed in the text, this effect diminishes as Re is decreased. (a) Theory
and simulation results after xcm/L = 482 advected particle lengths. For the
simulations, Re = 0.2. The particles were initially displaced from the fixed
point by Ayi = -0.34L, where particle 1 is the green (bottom) particle; Ay
2 =
-0.68L, for the blue (middle) particle; and Ay 3 = -0.34L for the red (top)
particle. (b) Results after x,/ L = 205 advected particle lengths. For the
simulations, Re = 0.05. The initial displacements from the fixed point are
Axi = -0.22L, Ax 2 = -0.43L, and Ax 3 = -0.22L. In contrast with (a), here
Re and xcm/ L are too small for a discernible phase difference between theory
and simulations.
78
4.5. Results
2uo
UO
0
Fig. 4.9: Snapshots of simulation results for a metastable steady "triangle"
configuration with Re = 0.2 and W/L = 8 at four values of the cluster center
of mass xe,.
Particle motion, initially limited to small excursions from the
initial positions, grows in magnitude until the magenta and green particles pair.
Ultimately, the red and green particles escape together. Colors indicate the
magnitude of the fluid velocity.
of initially neighboring trajectories, as in Figure 4.11 (c).
Our results on chaotic dispersion of three particle clusters recall a pioneering numerical
study
of the chaotic dynamics of three sedimenting spheres, represented mathemtically
as Stokeslets [30].
This might suggest that metastable states are generic to three body dynamics of hydrodynamically
coupled particles, if unbounded q2D did not once again provide a point of constrast.
Clusters of
three particles in the unbounded geometry immediately break up as a dimer and
a single particle.
The bare dipolar form is not sufficient for three body chaotic dynamics. This does
not rule out
metastable states and chaotic dispersion for higher N in unbounded q2D, which could
be explored
with the techniques described here.
4.5.4
Cyclical dynamical motifs
The techniques discussed above are used to find configurations in which particles maintain
their
relative positions as the cluster flows down the channel, whether indefinitely, as with
the marginally
stable fixed points,-or for many advected particle lengths, as with the metastable steady
states. For
the marginally stable fixed points, there are collective modes in which particles oscillate
around
their equilibrium positions. However, these are not the only ordered motions possible.
There
are metastable dynamical motifs in which particles follow cyclical paths. These
configurations
4.6. Conclusions
79
40
30
20-
10 -
0
2000
4000
6000
Uet/L
Fig. 4.10: Dispersion .2 /L 2 with dimensionless time Uot/L for four trajectories of the metastable "triangle" configurationof Figure 4.9, simulated with the
Lattice Boltzmann method. For each trajectory, the initial particlepositions are
spatiallyperturbed by a displacement vector with magnitude 0.025L and random
angle. The four trajectories break up at different times.
cannot be found by minimizing the particles' relative velocities, as above, since the particles are
constantly in motion. Instead, we scan for these dynamical motifs by directly integrating the
equations of motion for various initial particle configurations and sorting out trajectories for which
the final configuration is close to the initial configuration in phase space. The simplicity of the
theoretical model permits computation of thousands of trajectories overnight on a single processor.
Candidate motifs can then be studied in greater detail via Lattice Boltzmann simulations. Via
direct integration and LBM simulations, we find the "juggling" motif of Figure 4.12(a), in which
three particles cyclically exchange positions, and the "bowtie" configuration of Figure 4.12(b). As
metastable states, these configurations also eventually disperse.
4.6
Conclusions
Using a theoretical approach and Lattice Boltzmann simulations, we revealed new classes of nonequilibrium ordered states for small particle clusters flowing in quasi-two-dimensional channels. For
several of these classes, particles maintain their relative configurations either indefinitely (the
marginally stable configurations, which are fixed points of a dynamical system) or for hundreds
of advected particle lengths (the steady metastable configurations.) The marginally stable configurations can be regarded as "flowing crystals," since they are constructed by exploiting the
translational symmetry of an infinite set of real and virtual particles. As with crystals, they have
collective modes in which particles oscillate around their equilibrium positions. The oscillations
are not strictly transverse or longitudinal, but occur in two dimensions. Moreover, these "crystals"
may, in fact, be more stable than the one-dimensional trains and two-dimensional arrays examined
in other studies, since the virtual particles are completely slaved to the real particles.
Metastable states provide new configuration geometries in which particles remain in steady
80
4.6. Conclusions
relative positions. They are sensitive to initial particle configuration, and their eventual break-up
is characterized by individualistic particle motion. This sensitivity affects even a gross and discrete
outcome, how three particles in a "triangle" split into a bound pair and an isolated particle.
Moreover, sensitivity requires only three hydrodynamically interacting particles. Our system offers
a facile experimental and theoretical platform for the study of irreversible behavior that arises
from reversible equations of motion. For example, a possible line of investigation is the effect
of number of particles N on the statistics of dispersion. For small N, the dynamically "sticky"
metastable states could contribute to anomalous diffusion. For large N, the dimensionality of the
particles' configuration space is large, and mean-field theory should be applicable. Furthermore, the
metastable states can be harnessed for lab-on-a-chip devices. Consider Figure 4.9. If a steady state
with little relative motion is desired, the particles scarcely move as the cluster flows x,,/L = 582
particle lengths downstream, as shown by the second panel. For a HeLA cell with L = 20 pm,
this corresponds to an advected length - 1.1 cm, which is a typical microchannel length. For a
channel twice as long, the three particles come to "mix" and closely interact through individualistic
trajectories.
For other classes of metastable states - the cyclical dynamical motifs - particle motion is
complex but still spatially and temporally ordered, with particle excursions occuring over many
particle diameters in two dimensions.
We elucidated the principles for a priori construction of fixed points, as well as numerical
techniques for rapid numerical discovery of metastable steady and cyclical states. These can be
applied to any number N of particles; for the sake of brevity, we omitted results on N = 4, N = 5,
and so on. Moreover, our findings and techniques suggest future directions of research incorporating
other physical effects. Lattice Boltzmann can be coupled to deformable particles instead of rigid
discs. For deformable particles, distant segments move relatively and interact hydrodynamically,
and the collective behaviors revealed here could couple to the particles' internal elastic modes.
Likewise, spring forces can be incorporated in the theoretical model. The particles can be made
non-identical, either through varying particle size, or through varying the friction coefficient -Yp.
Practically speaking, the latter might vary through the course of an experiment, either in a time
invariant manner, through particle polydispersity, or transiently, through fluctuations in the z
direction for weakly confined particles. If particles are non-identical, they no longer have the same
interaction parameter /, and hydrodynamic interactions are no longer symmetric on the two particle
level [79]. For instance, when 7p varies for three particles by ±10%for a "column" fixed point in
the theoretical model, they no longer maintain steady relative positions, but adopt complicated
quasi-periodic orbits. A limitation of the marginally stable fixed points constructed above is that
they are not attractors for particle dynamics; that is, disordered suspensions of particles will not
spontaneously "crystallize," and access to the fixed points is limited by initial particle configuration.
It is conceivable that particles could be induced to assemble when time reversal symmetry is broken,
as when particles are deformable, and/or when particle symmetry is broken, as when particles have
dissimilar size or shape. Our theoretical framework can also incorporate Brownian noise, and
potentially be applied to the dynamics of macromolecules in slit-like confinement. Experimental
evidence and theoretical arguments indicate that hydrodynamic interactions can be neglected in
the mean-field limit for q2D confined macromolecules, although this still a subject of discussion
[89, 91]. Our work suggests that hydrodynamics could significantly affect chain dynamics when the
number of interacting segments N is small. We anticipate that this work will provide a starting
point for further discovery of rich physics in q2D particle-laden flows.
4.6. Conclusions
81
(a)
(b)
0
Blu.
and Blue and
red
green
Red and
green
(c)
10100
100
1
1000
0
~O
30
Fig. 4.11: (a) Particle motion in the center of mass frame for two realizations
of a metastable three particle configuration with W/L = 8, as determined by
integration of the theoretical model. The two realizations differ by slight noise
in the initial particle positions. A random perturbation uniformly distributed
over the interval [-0.0125L, 0.0 125L] is applied to the x and y positions of
each particle. The green (middle) particle pairs and escapes with either the
red (left) particle, as shown in the first panel, or the blue (right) particle, as
shown in the second panel. (b) Distribution of escape pathways for the three
particle configuration. Bins are labeled by which two particles pair. For each
trajectory, the initial particle positions are given a random perturbation, as
in (a). The escape outcome is sensitive to this perturbation. (c) Euclidean
distancea (t ) T
[(xiA(t ) - xi,Bt))2 ± (yi,A )
ya,B )2 between the two
realizations (trajectories in phase space) in (a) as a function of dimensionless
time, where i indexes the three particles, and A and B label the two trajectories.
For some initial transient period, both trajectories are bound as three particle
configurations and diverge exponentially in phase space,
~ eAUot/L, with a
Lyapunov exponent of A = 0.00185.
-
82
4.6. Conclusions
(a)
(b)
2uO
U0
0
Fig. 4.12: Cyclical motifs for N = 3 particles discovered via a "brute force"
search with the theoretical model and confirmed with LBM simulations. In a
search, we sweep over initial spatial configurations of N particles, integrating
the model forward in time for a specified time span and identifying candidate
cycles as those with small Euclidean distance between the initial and final spatial
configurations. The Euclidean distance d is defined as d 2 =
f[(x (tfinal) x,(0))2 + (yi(tfinal) - y(O)) 2]. The simplicity of the model makes this approach
computationally tractable. Colors indicate the magnitude of the fluid velocity
field. (a) In the 'juggling" motif, the three particles move clockwise, as indicated
by the black arrow, cyclically exchanging positions. Particlespause in the bottom
position, recalling how a juggler will momentarily have a ball in hand. Each
exchange of particle positions occurs after about 115 advected particle lengths,
so that the entire cycle takes about xcm/L = 345 lengths. (b) The "bowtie"
motif. At first glance, it might appear that this motif is associated with a fixed
point in which particles are positioned on the centerline, aligned with the flow.
However, such a configuration would quickly disperse. Particles return roughly
to their initial positions after approximately xcm/ L = 960 particle lengths.
CHAPTER 5
Engineering the trajectory of a single
particle via particle shape
In the previous chapter, we identified and characterized "flowing crystals" of discs in quasi-twodimensional channel confinement. Disc motion is orderly and predictable in these collective states,
fulfilling a major goal of this thesis. On the other hand, these flowing crystals are not self-organizing.
Their realization is limited by the initial spatial configuration of the discs. In the language of
dynamical systems, flowing crystals of discs are "marginally stable": the amplitude of a crystal
collective mode neither grows nor decays in time. Consequently, a crystal has no ability to "heal"
a perturbation to disc positions, e.g. after disruption by an encounter with a channel defect.
In pursuit of self-steering and self-organization, we turn to consideration of particle shape.
Anisotropy provides the most basic point of departure from the simple disc shape. Rods and
ellipses, for instance, are anisotropic, and a rod can be easily modeled as a "dumbbell" of two
rigidly connected discs. A second level of complexity is fore-aft asymmetry: variation of particle
shape along the major axis. Likewise, we can easily model fore-aft asymmetry by making one disc
larger than the other.
In this chapter, we consider the dynamics of dumbbell particles driven by flow in a q2D channel. We find that symmetric dumbbells oscillate indefinitely between channel walls, recalling the
behavior of two discs in Chapter 3. For asymmetric dumbbells, however, we obtain a surprising
behavior: self-steering. Asymmetric dumbbells reliably align with the flow, rotating so that the
84
5.1.
Overview
larger "head" disc is upstream of the "tail" disc, and focus to the channel centerline. We reproduce
this behavior experimentally with Continuous Flow Lithography, a technique which allows in situ
fabrication of q2D particles. Via experiments, we both make detailed comparisons of individual
trajectories with numerical predictions, and build a statistical picture of particle dynamics from
hundreds of trajectories.
Theoretically, we isolate three viscous hydrodynamic mechanisms that combine to produce selfsteering. These mechanisms generically occur for fore-aft asymmetric particles in q2D channel
confinement. They arise from a particle's hydrodynamic with itself and with its own images across
confining side walls. We show that alignment and focusing is time reversible - contrary to intuition
- by providing both an explicit demonstration of reversibility, and a discussion of the system's
underlying dynamical structure.
These results provide the first demonstration that a rigid particle can self-steer in a channel
flow. As we demonstrate experimentally, self-steering q2D particles can be used in microfluidic
device applications, eliminating the need for sheath flows or external forces. Moreover, our results
present a major step towards the achievement of realization of q2D suspensions, which we consider
in Chapter 6.
The results in this chapter have been accepted for publication in reference [92], and are reproduced with permission from the Nature Publishing Group, Copyright 2013. The experimental data
in this chapter were obtained through the tireless efforts of H. Burak Eral.
5.1
Overview
Recent advances in microfluidic technologies have created a demand for techniques to control the
motion of flowing microparticles. We consider how the shape and geometric confinement of a rigid
microparticle can be tailored for "self-steering" under external flow. We find that an asymmetric
particle, weakly confined in one direction and strongly confined in another, will align with the
flow and focus to the channel centerline. Experimentally and theoretically, we isolate three viscous
hydrodynamic mechanisms that contribute to particle dynamics. Through their combined effects,
a particle is stably attracted to the channel centerline, effectively behaving as a damped oscillator.
We demonstrate the use of self-steering particles for microfluidic device applications, eliminating
the need for external forces or sheath flows.
5.2
Introduction
In slow viscous flows, suspended particles are coupled by the flow disturbances they create in
the surrounding fluid. These hydrodynamic interactions (HI) can drive spatial organization of a
microparticle or system of microparticles in geometric confinement. Specific examples include the
cross-stream migration of a single polymer near a wall [21], the clustering of red blood cells in a
tube [49], and the crystallization of rigid spheres with finite inertia in a square channel [23, 25].
Both practical and theoretical considerations motivate interest in hydrodynamic "self-steering" (of
a single particle) and self-organization (of multiple interacting particles.) In microfluidic devices,
control over particle position allows the high throughput performance of operations on individual
flowing objects, e.g. in on-chip cytometry [77] and multiplexed assays with functionalized particles
[26]. While particles can be directly positioned with external fields or sheath flows [26, 78], these
methods can require cumbersome apparatus or complex channel structure. An elegant alternative
5.3. Model equations and numerical method
85
is to tailor particle and channel design for self-steering or self-organization. Moreover, if the selfsteered position of an object depends on a certain property of the object, a heterogeneous suspension
can be separated by that property. For instance, both the stiffness [62] and shape [93] of blood
components are of interest for microfluidic separations. From a theoretical perspective, a unifying
framework for non-equilibrium self-organization and self-steering is highly sought after [9]. Specific
mechanisms for cross-streamline migration and focusing in channel flow have been extensively
investigated for Brownian [22], inertial [23, 25], and deformable [21, 49] particles. In these cases,
migration arises from the interplay of viscous hydrodynamics near a channel boundary and another
physical effect that breaks the reversibility of viscous flow. Conceptually, it seems difficult to
reconcile self-organization and self-steering, in which any initial state will evolve towards one of a
limited set of dynamical attractors, and reversibility, which requires particle behavior to make no
distinction between two possible directions of time.
As discussed in the previous chapter, the unique features of the q2D dipolar flow disturbance
allow the realization of "flowing crystals" with novel collective modes [52, 55, 56, 73]. These are
configurations of particles that maintain spatial order as they are advected by an external flow.
They are marginally stable: the amplitude of a collective mode neither grows nor decays in time.
Consequently, realization of crystals - including those of the previous chapter - is limited by initial
configuration, and they are sensitive to break-up via nonlinear instabilities and channel defects.
A natural question is how to introduce an effective attraction to the crystalline states, causing
particles to assemble from disorder, and providing a "restoring force" against perturbations. One
indication is provided by a recent study which demonstrated stable pairing of droplets via the
higher flow disturbance multipoles induced by shape deformation [60]. This finding suggests a key
role for particle shape in achieving self-steering and self-organization.
In this chapter, we combine theoretical and experimental approaches to investigate how particle
shape can be tailored to induce self-steering under flow in q2D microchannels. Our main finding is
that a single rigid, asymmetric particle (Fig. 5.1) will spontaneously align with the external flow and
focus to the channel centerline (Fig. 5.2). This self-steering can be tuned via channel and particle
geometry. Moreover, it is time reversible; to our knowledge, all previous instances of hydrodynamic
self-steering have been irreversible. Via a simple theoretical model, confirmed by experiments, we
demonstrate how assembly arises from the interplay of three viscous effects: rotation and crossstreamline migration, via a particle's hydrodynamic self-interaction, and rotation via a particle's
interaction with hydrodynamic images. Each effect has an analogue in bulk sedimentation, but not
in bulk channel flow. We demonstrate application of these findings in a device setting. Finally, we
discuss their implications for the design of self-organizing "swarms" of interacting particles.
5.3
Model equations and numerical method
We consider a simple model geometry that captures the generic effects of asymmetry. A particle
comprises two discs of radius R 1 and R 2, with R 1 :> R 2 , which are rigidly connected with distance
s between their centers. It is confined in a shallow channel of height H in the z direction and
width W in y. Two lubricating gaps of height h separate each of the discs from the confining
walls in z (Fig. 5.1). There is a pressure-driven flow in x with a parabolic profile in z and
an approximately uniform depth averaged velocity Uo (i.e. Hele-Shaw flow.) In the absence of
inertia, the governing dimensionless parameters are strictly geometric: h = h/H, H = H/R2,
W = W/R 2, R = R 1 /R 2, and § = s/R 2 [68]. We define dimensionless time as i
tUo/s. The
86
5.3. Model equations and numerical method
UO
0
e*
2
W
R,
yC
side view
zt
J
E
H
Fig. 5.1: Schematic diagram of the model system. A particle comprising two
rigidly connected discs is confined in a thin microchannel of height H and width
W and driven by an external flow. The flow is approximately uniform in the
channel midplane, and has depth averaged velocity UO. The disc radii are R 1 and
R 2 , with R 1 ;> R 2 , and the disc centers are separated by distance s. Two thin
lubricatingfluid layers of height h separate the discs from the confining plates
(i.e. the channel "ceiling" and "floor. ") The instantaneousparticle configuration is specified by two coordinates: the location in y of the midpoint between
disc centers, y=
(y1 + y2)/2, and the angle 9 between the external flow and
the particle axis.
instantaneous particle configuration is defined by the location in y of the midpoint between disc
centers, yc
(Y1 + y2)/2, and angle 9 between the external flow and the particle axis, as shown in
Fig. 5.1. Due to translational symmetry, the position in the flow direction xc = (Xi + x2)/2 does
not affect particle dynamics.
We develop the governing equations and numerical integration method in the following subsections. In the model, we write a force balance equation for each disc. Each disc experiences drag
from the local flow, friction from the confining plates, and a rigid constraint force. The local flow
at each disc is determined self-consistently as the external flow plus contributions from the other
disc and the discs' hydrodynamic images. The images impose a no mass flux boundary condition
on the confining side walls.
5.3. Model equations and numerical method
87
R = 1.0
symmetric
R
1.05
slightly asymmetric
R =1.5
Fig. 5.2: A symmetric particle oscillates between side walls. When the symmetry
is slightly broken, this oscillation is damped, and the particle aligns with the flow
as it focuses to the centerline. A very asymmetric particle is "overdamped," and
rapidly aligns before slowly focusing. The trajectorieswere obtained numerically
for the parametersgiven in the caption of Fig. 5.12b. The x axes are scaled by
a factor of 1/40 to show the full range of particle behaviors.
5.3.1
Force-free equation for a single disc
As in the work of previous chapters and Appendix A, we write a force-free equation for disc i
moving with velocity Up' in a q2D flow with velocity U(ri) as
(i(U(ri) - Ut') -
7rRypUf + 7rRycU(ri) + Fi
=
0.
(5.1)
Here, U(ri) is the depth-averaged velocity at the disc position ri = (xi, yi); 7c = 12p/H; y is the
fluid viscosity, which ultimately drops out of the equations; and (i is the drag coefficient, derived in
Appendix A. In comparison with the previous chapters, we have added a new term: Fi is the force
of rigid constraint if disc i is connected to other discs. The friction coefficient 7p is determined
by the details of flow through the lubricating gaps of height h that separate the particle from the
confining plates. Whereas we previously left yp unspecified, here we assume simple shear in the
gaps, taking -y, = 2p/h, in order to facilitate comparisons with experimental data.
88
5.3. Model equations and numerical method
For a free single disc, Fj = 0, resulting in U = aiU(ri), where
ci
((++
c'7rRi'
(i + -p7rRil
.
(5.2)
The velocity of the disc is directly proportional to the depth averaged local flow velocity. The
constant of proportionality ai depends on the degree of confinement through the dimensionless
parameters H and h. Eq. 5.2 for ai invites comparison with the more detailed lubrication analysis
of Halpern and Secomb, which considers a more realistic shape, a disc with rounded edges [68].
For unbounded q2D, Halpern and Secomb determine a 2 as a function of H and h via a system
of three equations. One of these equations is given by the force-free condition, and the other two
by matching pressures at the boundaries i.) between the gaps and the rounded edges, and ii.)
between the rounded edges and the region external to the disc. In principle, our model could
be extended to incorporate the results of Halpern and Secomb by incorporating four additional
equations for each disc (two for the x direction and two for the y direction.) However, direct
comparison reveals that our simplified model shows qualitative and some quantitative agreement
with Halpern and Secomb, and can be expected to capture the trends in the system's dependence
on the dimensionless parameters. In particular, Fig. 5.3 shows good quantitative agreement in ai
for small H, i.e. when the radius of a disc is much larger than the channel height.
5.3.2
Additional equations and numerical method
A rigid constraint k between discs i and j is associated with a constraint equation rij -(Ut' - U ) = 0
and a force F(k), where ri = ri - rj. In our disc-rod model of a dumbbell particle, the rigid
constraint and the four disc force balance equations realize one torque balance and two force
balance conditions. To simplify the analysis, we neglect the effects of lubrication forces and the
rotation of individual discs. In Appendix C, we show that including disc rotations would improve
the quantitative accuracy of our model, but not change our qualitative findings.
The local flow field at disc i is determined through an implicit equation
U(r;) = U 0 +
G(2)(rij, rj) - (U - U(rj)),
(5.3)
where G2) (rij, rj) is a tensor containing the leading order, far-field contribution of disc j to the
local field at i. This tensor, given in Appendix B, includes the effect of the hydrodynamic images
needed to impose boundary conditions at the side walls. The hydrodynamic strength of disc j is
characterized by a quantity B3 - R? that scales G(1). The constraint equations and Eqs. 6.4
and 5.3 can be arranged into matrix form AUP = B, where Up is a vector containing all 2N disc
velocities, 2N local fields, and n constraint forces. We consider a single particle, so that N = 2
and n = 1. A is a matrix constructed from disc interactions, and B collects terms involving U 0 .
This system can be solved and integrated numerically. The only free parameters are h, H, R, W,
and 9, defined above.
5.4. Experimental method
89
0.8
C1
0.6 -
Halpern/Secomb,
i = 0.2
Simplified model,
Halpem Secomb,
- Simplified model,
Halpern/Secomb,
- - Simplified model,
-Halpern/Secomb,
Simplified model,
H = 0.2
A = 0.4
- -
0.4-
0.2
0
0
0.05
0.1
R = 0.4
A = 0.8
A = 0.8
H = 1.6
H = 1.6
0.15
0.2
h
Fig. 5.3: Comparison of models for flow of a disc. We show a as a function of h
for four values of H as determined by our simplified model and the more detailed
analysis of Halpern and Secomb. Our simplified model shows good quantitative
agreement for small H and captures the trends in h and H.
5.4
Experimental method
In the experiments, we use continuous flow lithography (CFL) to fabricate particles with desired
shape and initial configuration in situ under q2D channel flow. With this technique, described in
detail in Dendukuri et al. [94], particle with predefined geometries can be synthesized in situ at
the desired initial position and orientation. In brief, an acrylate oligomer (poly(ethylene glycol)
diacrylate) mixed with a photoinitiator is pumped through the poly(dimethylsiloxane) (PDMS)
channel depicted in Fig. 5.1 using external pressure. The channel is mounted on an inverted
microscope. Particles are polymerized with short pulses of UV light (50 ms for Fig. 5.7 and 100
ms for Fig. 5.13.) The geometry of the particle in the xy plane is imposed by a lithographic
mask placed between the microscope objective and the UV source. The height of the particle
inside the channel is dictated by the UV exposure time [95]. Importantly, the well-known oxygen
inhibition effect in the CFL method [94, 95] provides uniformly thin, unpolymerized, lubricating
layers between the microparticle and the top and bottom PDMS walls.
Channels are 30 Am in
depth, 500 pm in width and 2.4 cm in length. Using the hydraulic diameter 2HW/(H + W) as a
length scale, 55 cp as the prepolymer viscosity, 50 pm s- as a typical flow speed, and 1.12 x 10 3
kg/m 3 as the prepolymer density, a typical Reynolds number is Re = 6 x 10-5. Inertial effects are
90
5.5. Self-alignment of asymmetric particles under flow
therefore negligible.
Fig. 5.4: This photograph of the experimental setup shows the microfluidic channel, the moving microscope stage and continuous flow lithography setup, and
the camera. The inset shows a zoomed in view of the PDMS channel with ruler
markings. The scale bar is 100 pm.
As the particles move along the channel, the microscope stage is translated by a homemade
linear motor, ensuring that the particles remain in the field of view. Movies of particle trajectories
are recorded using a CCD camera and analyzed offline to determine particle position and angle.
We confirm that the applied pressure and the flow speed remain constant throughout the course of
an experiment by synthesizing a disc and tracking its motion along the channel. The flow speed is
determined by tracking 1.6 pm fluorescent tracer beads mixed with the flowing solution [96].
For experiments presented in Fig. 5.14, we synthesize fluorescent particles using CFL. To
covalently bind fluorescent dye to the synthesized particles, we add acrylate-modified rhodamine
to acrylate oligomer and photoinitiator solution. The synthesized particles are collected in a TrisEDTA (TE) buffer containing 0.1% vol/vol surfactant Tween-20 in an 1.7 mL Eppendorf tube.
The collected particles are resuspended in an approximately density matched solution containing
25% vol/vol poly(ethylene glycol) (molecular weight 400 g/mol) in TE buffer. Fluorescent particles
are pumped through the detection channel. Particles are imaged with an appropriate UV light
source and filter set for Rhodamine. The synthesis and detection channels are 30 Pm in depth, 300
pm in width and 2.4 cm in length. We have analyzed over 300 symmetric and asymmetric particles.
5.5
Self-alignment of asymmetric particles under flow
As the first step in building a complete picture of particle dynamics, we neglect the effect of
side walls, isolating a particle's self-interaction. For an identical pair of discs, the interaction is
symmetric: disc 1 pushes on disc 2 just as much as disc 2 pushes on disc 1 (Fig. 5.5). The interaction
cannot lead to relative motion of the discs, including rotation of the entire particle [79]. However,
5.5. Self-alignment of asymmetric particles under flow
91
when 9 5 0' and 0 # 90', it introduces a component to the particle velocity perpendicular
to the
direction of the external flow [61]. When 00 < 9 < 900, the particle migrates in the direction
of
decreasing y; when 900 < 9 < 1800, it migrates with increasing y. This "lateral drift," occurring
for both symmetric and asymmetric particles, also occurs for a rod or pair of spheres sedimenting
in bulk.
(a)
(b)
Fig. 5.5: Hydrodynamic self-interaction drives alignment of an asymmetric particle. (a) Illustration of the self-interaction of a symmetric particle. A disc's
vector shows the component of the flow disturbancefrom the other disc in 9, the
direction of increasing 9. The vectors are identical: there is no rotation of the
particle. (b) When the two discs have different radii, the particle aligns with
the flow.
When the discs are dissimilar, the particle aligns itself with the external flow, such that the larger
disc is upstream of the smaller disc (Fig. 5.5b). The principal cause of self-alignment is that one disc
is hydrodynamically stronger than the other: in q2D, the magnitude of the dipolar flow disturbance
created by a disc scales as the disc area. In a following subsection, we derive an exact expression
for 9 as a function of time. Taking t = 0 when 9 = 900, we obtain i = -fr ln(csc(9) +cot(0)), where
the timescale fr (R, 9, H, h) depends on particle geometry. Notably, it diverges for R = 1.
Hydrodynamic self-orientation has not been observed for a rigid particle in bulk channel flow.
Bretherton considered bodies with axial and fore-aft symmetry in slow unidirectional
shear flows,
which include pressure-driven bulk channel flows. He found that nearly all particles tumble in
Jeffery orbits with no equilibrium orientation and no cross-streamline migration, except for certain
"extreme," high aspect ratio shapes [44]. To our knowledge, these shapes
have not been realized
experimentally. Subsequently, flow-driven doublets of unequal spheres, analogous to the dumbbells
we consider, were studied by Nir and Acrivos [97] and Adler [98]. These also tumble with no net
migration. However, self-alignment has recently been predicted for asymmetric objects in bulk
sedimentation, with a dynamical equation similar to ours when the object is initially oriented in a
vertical plane [99].
5.5.1
Experimental observation of self-alignment
We recover these predictions experimentally. We polymerize particles with various R and measure
how 9 evolves with i. We fit a timescale - to the data of each A. When 9 is plotted
against f/r,
all data collapses onto a universal curve (Fig. 5.6). We leave the data for R = 1 unscaled; for
92
5.5. Self-alignment of asymmetric particles under flow
this singular case, the particle maintains its initial angle. The curve asymptotes to 6 = 00 and
6 = 1800, and is manifestly time reversible. In the inset of Fig. 5.6, we plot the dependence of the
experimental timescales
on P alongside a theoretical curve predicted for the same parameters.
The theoretical and experimental timescales have the same order of magnitude and the same trend
with R. Moreover, by adjusting h, we generate a theoretical curve with good fit to the data. The
effect of the neglected near-field physics is simply to renormalize h.
180
II III
.4'
1
4
I:
IV
I
I
I
V VI
150
.II
-
-
1 61
1 -
I
U.LLLL.LJJ_.ULLA-.,
100
120 -
* Experiment
Predicted theory, h=2pm.
- - - Best fit theory, h=4pm
So
-
-
60-
0
90
40-
60
30
-
SR=2.
R=2. oO
R=1. 5
N1
3
R=1. 0
Theo ry
-2
-1
*
-
A
v
o
01
20
1.
0
1
2
2.0
3. 0
2.5
3
4
5
Fig. 5.6: Experimental angle vs. time for various R with = 3.3, h = 0.06, and
H = 1.6. We scale the data for each A by a fitted r, collapsing all data onto a
universal curve predicted by theory. (inset) The dependence of the experimental
timescales i on R, along with a theoretical curve for the same parameters (solid)
and a theoretical curve with h adjusted for best fit (dashed).
.
5.5.2
Derivation of equation for self-alignment
We now consider in detail how to obtain the self-alignment equation for a single dumbbell in
unbounded q2D. We define directions 9 and 6, where 8^ is a unit vector parallel to the vector from
the center of disc 1 to the center of disc 2, and 6 is a unit vector orthogonal to 8 in the direction of
increasing 6. The instantaneous configuration of the dumbbell is specified by 6, the angle between
r and X.
Conveniently, the dynamics of a dumbbell in the A and 6 directions completely decouple, due to
the form of the hydrodynamic interaction tensor in cylindrical coordinates. Motion of disc 1 in 6 is
not affected by motion of disc 2 in s, and similar statements hold when we swap the two directions
5.5. Self-alignment of asymmetric particles under flow
93
(a)
(b)
Fig. 5.7: Experimental images of particle self-alignment. (a) Snapshots of a
symmetric particle at various times, matched to the times in Fig. 5.6. The
scale bar is 100 pim. (b) Snapshots for R = 2 at the same times as in (a).
or disc indices. Moreover, since the forces of constraint are directed along s, we find from writing
a force balance equation in b that the b component of the velocity of disc i is directly proportional
to the 6 component of the local fluid velocity:
(i(U(ri) - UTP)
upo=
-77rRi2U'
+7~~
'\
0
+ 7,7rRUo(ri) = 0,
(5.4)
Ue (ri) = aiUo(ri).(.)
+ -YyrR?
We can therefore write
9 = (a 2 UO(r 2 ) - a1Uo(ri))/s,
(5.6)
where s is the length of the rod connecting the centers of discs 1 and 2. Now the problem is simply
one of finding U0 (ri). We write the local field of disc 1 as
B2
Uo(ri) = -Uo sin(O) - B2
(U T2- 2,
U(r 2 ))
-Uo sin(0) +
B2
(5.7)
(1- a 2 )U(r 2 ).
The first of these equations expresses that the 0 component of the local flow velocity at disc 1 is
given by the projection of the external flow in the 0 direction and the 0 component of the flow
disturbance from disc 2. The latter depends on the difference between the 6 component of disc 2's
velocity and the 9 component of its local flow field, and not at all on any components in S. This is
an instance of the decoupling of S and 9 discussed above. In passing from the first equation to the
second equation, we invoked Eq. 5.6.
We have a similar equation for disc 2:
Uo(r2) = -Uo sin(6) +
B1
2
(1- ai)UO(ri).
(5.8)
94
5.6. Dynamics of dumbbells in a microchannel: The complete picture
Substituting Eq. 5.7 into Eq. 5.8 and rearranging,
Uo(r 2)[1 - B1B 2 (1 - ai)(1 - a2)/s4] = -U[1 + B 1 (1 - a1 )/s2] sin(9).
(5.9)
By symmetry, we have
Uo(ri)[1 - B 1B 2 (1 - ai)(1 - a2)/s4] = -Uo[1 + B 2 (1 - a 2 )/S 2] sin(9).
(5.10)
Therefore, Eq. 5.6 becomes
= -rJ 'sin(9)
where the prefactor
1
7r-
r
-
(5.11)
is
_ Uo[a 2 (1 + Bi(1 - ai)/s2 ) - a1(1 + B 2 (14 - a2)/s2 )1
s(1 - B1B 2 (1 - ai)(1 - a 2 )/s )
Eq. 5.11 can be linearized as exponential approach near 9
velocity near 0 = 7r/2. It integrates to
-Tr ln
csc()+cot()
csc(9 0 ) + cot(o) )
(5.12)
0 and 9 = 7r, and constant angular
t.
(5.13)
We define t = 0 for 00 = 7r/2 in order to best demonstrate the reversibility of the rotational
dynamics.
5.6
Dynamics of dumbbells in a microchannel: The complete picture
Having isolated a particle's self-interaction, we consider how it combines with image interactions to
produce the behaviors of Fig. 5.2. Consider the symmetric particle in Fig. 5.8. To leading order, the
translation in y arises from self-interaction. The chief effect of the images is to rotate the particle.
The particle in Fig. 5.8 begins by migrating towards the lower wall. It is rotated into 9 = 00, for
which the lateral velocity is zero. This configuration is an extremum of the oscillation. The particle
is rotated further and migrates away from the wall. The mirror symmetry of the particle at the
extremum ensures that the outgoing trajectory is mirror symmetric with the incoming trajectory.
After crossing the centerline, the particle will reflect from the upper wall. Moreover, an oscillation
with a 0 = 900 extremum can be produced with a different initial condition. Again, we can find
an analogue in bulk sedimentation; a rod falling between vertical walls will oscillate between them
with 0 = 00 and 9 = 90' modes of reflection [100]. Numerically, we construct a phase portrait for
a symmetric particle (Fig. 5.9), showing trajectories in the space of particle configurations (yc, 0).
Owing to the properties of viscous flow, the spatial configuration of a particle completely specifies
the state of the system. We find that there are marginally stable fixed points at (yc, 9) = (W/2, 00)
and (ye, 0) = (W/2, 900), each of which is associated with a continuous family of periodic orbits.
For an asymmetric particle, self-alignment changes the fixed point (yc, 0) = (W/2, 00) into an
attractor, as we demonstrate with a linearized model (Fig. 5.10). We define A = yc - W/2, and
model lateral drift as A = -a9, where a > 0 depends on the dimensionless parameters. We model
the rotational dynamics by 9 = bA - c9, with coefficients b > 0 and c > 0 that respectively capture
the strength of the images and self-alignment. When the particle is displaced from the centerline
5.6. Dynamics of dumbbells in a microchannel: The complete picture
95
rotation
by image
latera
drift
Images
Fig. 5.8: A symmetric particle oscillates via the combined effects of hydrodynamic interaction with itself and with its own images. Self-interaction leads to
cross-streamline migration ("lateral drift") when the particle angle 9 $ 0' and
9 , 90'. The images rotate the particle.
(A # 0), the effect of the images is to rotate the particle away from 9 = 0', which is opposed
by selfalignment. These equations can be combined into A = -abA
- cA. Without self-alignment (c = 0),
the particle oscillates around the fixed point. When c 5 0, the particle is attracted to the fixed
point
via either a decaying oscillation or an "overdamped" approach. These regimes are
separated by a
critical boundary in parameter space v/
~ c. Numerically, we construct a boundary by finding the
critical Rcrit as a function of W for various sets of the parameters 9, H, h, as described
in the next
subsection. We obtain expressions for a, b, and c via heuristic arguments, detailed below,
yielding
a function W = F(H, 9, R, h) that fits the numerical data for each individual parameter
set, as
shown in Fig. 5.11. We collapse the numerical data and theoretical curves with the empirically
fitted scaling fcrit = -1/5ftl/6& in Fig. 5.12, exposing the universal shape of the curve.
This phase diagram can guide the design and optimization of self-steering particles. For a
given
set of parameters 9, H, and W, focusing occurs over the shortest streamwise
travel distance at Rrit,
as the analogy with a "critically damped" oscillator suggests. The critical boundary occurs
when
the timescale for self-alignment is comparable to the timescale for a particle to migrate across
the
channel width. Along the boundary, decreasing W while increasing R or decreasing § is an
effective
design strategy to reduce streamwise travel distance by decreasing lateral migration distance
and
enhancing self-alignment. Strikingly, the diagram does not depend on h, the dimensionless
lubricating gap thickness, which can be independently tuned. Decreasing
h slows down the particles,
96
5.6. Dynamics of dumbbells in a microchannel: The complete picture
w
ww
o
-180
-90
0
R =1
90
180
0
-180
-90
0
90
180
0
-180
-90
R =1.05
0
90
180
R =1.5
Fig. 5.9: Portraitsshowing particle trajectories in the phase space (y ,6). Por0
traits were obtained numerically for W = 21, 9 = 3.5, H = 1.6, and h = 0.08.
Arrows give direction of motion in phase space. Dots identify the trajectories
shown in Fig. 5.2.
strengthening hydrodynamic interactions and reducing the travel distance for focusing.
Having considered small displacements from (yc, 6) = (W/2, 0'), we construct phase
portraits for
R = 1.05 and R = 1.5 (Fig. 5.9). The slightly asymmetric particle approaches
(yc, 6) = (W/2, 00)
via a decaying oscillation, but there are marginally stable fixed points with 6
= ±900. These
"bouncing states" are due to interaction of a particle with a nearby image, discussed
later in the
chapter (subsection "Conditions for global assembly.") For R = 1.5, any point in
the phase space
is along a trajectory connecting the unstable fixed point (Yc, 6) = (W/2,1800)
with the stable
fixed point. For a highly asymmetric particle, there is a separation of timescales
between rapid
self-alignment and slow lateral focusing. This separation can be seen in the convergence
of all
trajectories to a slow manifold, outlined in red. Since the attractor is asymptotic and
accompanied
by a repeller, it is compatible with reversibility: if the flow is reversed, the fixed points
exchange
stability, and a particle retraces its trajectory in phase space, attracted to the other
fixed point.
We explicitly demonstrate reversibility in the subsection "Reversibility of focusing dynamics."
5.6.1
Derivation of numerical and theoretical phase boundaries
We combine numerical and theoretical approaches to obtain the phase boundary in Figs.
5.11 and
5.12. We first discuss our numerical approach. For various sets of the parameters 9,
h, and H,
we vary W. For each W, we vary R from 1? = 1 to R = 2, beginning
with R = 1. At each R,
we numerically construct the Jacobian of the aligned and focused fixed point. We diagonalize
the
Jacobian to obtain eigenvalues and eigenvectors. For a particular set of 9, h, and H,
we define
a critical point for each W as the first R for which the imaginary components of the
eigenvalues
vanish. We therefore obtain a boundary relating Reit and W. We collapse the data for
the various
sets of 9, h, and H with empirically fitted scalings, obtaining a collapsed numerical
boundary
relating -1/5l/6WT
to R.
Now we consider how to obtain a collapsed boundary via theoretical arguments. We recall
the
5.6. Dynamics of dumbbells in a microchannel: The complete picture
97
rotation by
image
lateral
drift
rotation
by self
Fig. 5.10: Linearized model of an asymmetric particle. Rotation by the images
is opposed by self-alignment. The particle drifts in the y direction when 0 is
displaced from the equilibrium value 0 = 0'. The lateral displacement A is
defined as A
y, - W/2.
condition for criticality, v a'b ~ c, where a characterizes the strength of the "lateral drift" effect, c
the self-alignment, and b the strength of the real particle's interaction with its images. We adopt a
heuristic approach, driven by our understanding of the underlying physics, to obtain estimates for
these coefficients.
In light of the results obtained in the section on self-alignment, a sensible choice is to take
c = r1,
given by Eq. 5.12.
Now we seek to obtain an estimate for a. At lowest order - prior to consideration of hydrodynamic interactions - the discs are driven only by the external flow in x, so that the velocity of
disc 1 is a 1Uo, and its relative velocity is (1 - ai)Uo. Disc 1 therefore creates a flow disturbance
with strength B 1 (1 - ai)Uo. This flow disturbance is responsible for the y component of the flow
field in the vicinity of disc 2, which would otherwise be zero. To estimate this y component, we
must consider both the spatial decay of the dipolar form and its angular dependence. The spatial
decay clearly gives a factor of 1/s2. The angular dependence is such that the y component is zero
at 0 = 0. Therefore, the y component can be taken to be linear in 6 for small 6. Hence, we obtain
Uy(r 2 ) = UoB 1 (1 - ai)/s26, so that the ^ velocity of disc 2 is a 2 UoB 1 (1 - ai)/s20. A similar
expression holds for disc 1. Neglecting a factor of 2, the translational velocity of the particle is
9C = -aO, where
98
5.6. Dynamics of dumbbells in a microchannel: The complete picture
60
* 9 = 3.5, h = 0.08, R = 1.6
0 S = 5.0, h = 0.08, A = 1.6
* s = 7.0, h = 0.08, H = 1.6
S= 5.0, h =0.08, H =0.8
9 = 5.0, = 0.08, H = 0.4
S = 5.0, I = 0.08, H = 0.2
S = 5.0, fi= 0.01, H = 1.6
+ s= 5.0, h = 0.16,L = 1.6
45
30
1511.1
1.2
1.3
1.4
R
Fig. 5.11: Theoretical and numerical critical boundaries. For various sets of parameters H, , R, and h, we numerically obtain points on the critical boundary
separating the underdamped and overdamped oscillatory regimes via the method
described in the section "Numerical Phase Boundary." These points are shown
as symbols in the figure. For each parameter set, we also obtain theoretical
curves via Eqs. 5.18 and 5.19, shown as solid lines and matched to the symbols
by color. Each curve fits its corresponding numerical data with the same fitted
dimensionless prefactor of 1/3. Moreover, as shown in Fig. 5.12, all curves and
data can be collapsed onto a single boundary via an empirically fitted rescaling
5
of
/-1
f/ 6 6W. For clarity, not all of the parameter sets in the collapsed Fig.
5.12 are shown.
a = Uo[a 2B1(1 - ai)+ a1B 2 (1 - a2)]/s 2.
(5.14)
Note that we have neglected the effect of the rigid constraint, which would ensure that disc 1
and disc 2 have the same X velocity at 6 = 0.
We now consider b, which characterizes the effect of the side walls. Near the centerline, the
effect from the side walls is dominated by the two nearest images: one across the lower, y = 0 side
wall, and the other across the upper, y = W side wall. We consider the lower image first. It tends
to rotate the real particle into negative 0 via the interaction of image disc 1 with real disc 2 and the
interaction of image disc 2 with real disc 1. As with a, we take disc 1 to produce a flow disturbance
with strength Bi(1 - al)Uo, and disc 2 to produce a disturbance with strength B (1 - a )Uo. For
2
2
the dumbbell angular velocity, we obtain 6
-UO5 [a 2 B 1 (1 - ai) + a1B 2 (1 - a 2 )] /sy 2 . Here we
have included the angular dependence of the dipolar interaction via a small angle 6, which can
5.6. Dynamics of dumbbells in a microchannel: The complete picture
40
A
30
-1
damped
-
20-
A
99
* 9= 3.5, = 0.08, A= 1.6
0 N = 5.0, j=0.08, = 1.6
+ * = 7.0, = 0.08,R= 1.6
A 9= 5.0,E = 0.08,
=1.1
x
=5.0, =0.08, A=0.8
* 9= 5.0, h = 0.08, A = 0.4
- = 5.0, I = 0.08, A = 0.2
= 5.0, = 0.04, A = 1.6
l
= 5.0, 1=0.01, A = 1.6
+ S= 5.0, = 0.16, A = 1.6
overdamped
10-
underdamped
I
.
1
I
.
1.2
I
.
1.4
I
.
1.6
1.8
R
Fig. 5.12: Phase diagram showing the critical boundary that separates the underdamped and overdamped regimes. The symbols are points on the boundary
obtained numerically for various parameters. The solid lines, matched by color
to the symbols, are theoretical curves for the same parameters. The numerical
data and theoretical curves collapse onto one universal boundary.
be taken to be 6 ~ s/y for wide channels, where s/W < 1. The interaction therefore has a 1/y 3
dependence. However, near the centerline, the rotation driven by the lower image
(into negative
9) is opposed by rotation driven by the upper image (into positive 0.) We can linearize
for small
deviations A = y, - W/2 from the centerline. We obtain 9 = bA, where b is
b =- UO [a2B1 (I - al) + al B2 (I - a2)] /W4.
(5.15)
In order to isolate W, we define b' as
b' = Uo [a 1B 2 (1 - a2) + a 2 Bi(1 - ai)]
so that W
-
(ab')1/ 4 //C,
W
or
~
(1-B1B2(1-a1)(1-
)/
4
)[a 2Bl(1-a1)+aB2(1-a
( +B2(1- 2)/2)]
2
[n2(1+B(i-ex
)/ 2)-
We now nondimensionalize this expression. We obtain Wl
f (H, 9,7
(5.16)
)3
(-
121-1(-2/4[2
~
2
)
.e(5.17)
f (H,§, R-, h), where
1(-1+
221
)
(5.18)
100
5.6. Dynamics of dumbbells in a microchannel: The complete picture
with a,, a2, B 1 and B 2 functions of R, h, and H. (Since Bi has dimensions of length squared, as
Bi ~ R?, the tildes indicate nondimensionalization by R 2 -)
For each set of parameters H, §, R, and h, this expression fits the corresponding numerical data
with a fitted dimensionless prefactor of 3:
W = 3f(f, s, N, h) = F(H, §,
R,
h),
(5.19)
where the fitted prefactor has been absorbed into the definition of F(H, §, R, h). The results
are shown in Fig. 5.11. The fit is particularly good for large W and small §, as one would
expect from the approximation s/W < 1. More rigorous estimates for a, b, and c could be
found via formal perturbation theory. However, the fit between our theoretical curves, obtained
via physical arguments, and the numerical data for a large range of individual parameter sets
indicates that we have identified the physical mechanism underlying the stability of the aligned and
focused configuration. These curves can be collapsed via fitted rescalings, exposing the universal
shape of the critical boundary. These rescalings capture the dependence of the boundary on the
dimensionless parameters in an experimentally relevant region of parameter space.
5.6.2
Experimental observation of focusing
Our complete theoretical picture predicts a wide range of experimental observations involving both
the channel side walls and particle self-interaction. We first consider three particle trajectories
in individual detail. Fig. 5.13 shows an experimental montage in which a symmetric particle is
reflected from a side wall. We obtain qualitative agreement with the theoretical trajectory generated
for the same parameters and initial conditions as the experiment, shown in the inset. The trajectory
is shown quantitatively in Fig. 5.13d. The theoretical prediction can be fitted to the experimental
data if it is rescaled in x. If we relax the assumption of perfect symmetry and take i = 1.01, the
resulting theoretical curve better captures the curvature of the data. This asymmetry corresponds
to a difference in radii of ~0.2 pm, within the uncertainty of CFL. In Fig. 5.13b, an asymmetric
particle with R = 1.3 polymerized with 0 = -10' focuses to the channel centerline. Good agreement
between theory and experiment is obtained upon rescaling. This initial condition is near the slow
manifold for overdamped dynamics. For a particle with 0 = 1350, we obtain the predicted two
timescale process of initial reorientation followed by slow focusing (Fig. 5.13c). This difference in
timescales is manifested in different rescalings needed to fit theory to data for the initial dynamics,
dominated by self-interaction, and for focusing, in which the images are important.
Finally, we apply the insights developed in this chapter to engineer a practical microfluidic
system with self-focusing particles. We can thereby build a statistical picture of particle dynamics
from hundreds of trajectories. We fabricate asymmetric and symmetric fluorescent particles in a
synthesis channel and collect them from the channel outlet in an Eppendorf tube containing a
common buffer. After rigorously washing the particles by successive steps of gentle centrifugation
and decanting, we resuspend the particles in approximately density matched solvent at the desired
concentration and flow the suspension through a detection channel. In the detection channel,
we measure the transverse position ye of each flowing particle near the inlet and the outlet with
fluorescence microscopy. The results are shown in Fig. 5.14. Starting from a broad and essentially
random distribution of transverse positions, most asymmetric particles focus to the centerline. The
finite width of the central peak is due to the finite length of the channel; with a longer channel, it
would be narrower. The two side peaks are possibly due to the high shear rate in the boundary layer
5.6.
Dynamics of dumbbells in a microchannel: The complete picture
(a)
--
161.0
-
w-
101
(d) 0.8
0.7
0.6
theory =1.0
theory =.01
- ---
0.5
0.4
0.3
1L.1
;/W=0
x/W=1
X/W2
xjW=3
xJW=4
;/W=5
;/W=6
;/W7
0.2
/W 8
0.1
(b)
1. 3
75W
(e)
4 5
-2
6
7
8
0.,....
0.4
-
p.
S
:
S
0.3-
eperimfen
O
0.2
xjW=0
j/W=3
x/W=6
x,/W=9
x/W=12
x/W-15
;/W=18
It,
45W
0.5
0.4
j
S5
2
012
4 6811214
XC/W
(c)
15W
0
;/W=21
16 18
o experiment
-theory1
- - -theory2
0.3
0
X/W=0
x /W=2
x /W=4
.L .
x,/W=6
x /W=8
x/W=10
i
_
..
_____
..
x/W=12
.
X/W=14
_
0.2
0 2
4 6 8 10 12 14 16 18 20
xj/W
Fig. 5.13: Individualparticle trajectories. All scale bars are 100 Am. (a) Experimental montage showing reflection of a symmetric particle. The corresponding
theoretical trajectory is shown in the inset. (b) A strongly asymmetric particle
with 0 = -10
focuses to the centerline. (c) A strongly asymmetric particle
with a large initial angle aligns and then focuses to the centerline. (d) Position
data for the trajectory in (a). The theoretical trajectoryfor R = 1 was scaled
in x by a factor of 0.475. A theoretical curve with A = 1.01, for which the
rescaling is 0.4, better captures the curvature of the data. (e) For the particle
in (b), the rescaling is 0.15. (f) For the two timescale process of (c), different
rescalings of 3 and 0.1 are required to capture the initial and steady dynamics.
For all trajectories,9 = 3.3, h = 0.3, and H = 1.6.
102
5.6. Dynamics of dumbbells in a microchannel: The complete picture
(a)
(b)
100
.
.
.
,
,
(c) 100
*inlet
outet
80 _
symmetric
2
60
40
40
20
20
0.1
0.2
Esmeouet
asymmetric
60 -
0.0
inlet
80
0.3
0.4 0.5 0.6
V/W
0.7
0.8 0.9
1.0
0
0.0 0.1
0.2
0.3
0.4 0.5 0.6 0.7 0.8
0.9
y/W
Fig. 5.14: Statistics of particles in a flow-through device. (a) Fluorescence
microscopy image of symmetric and asymmetric particlesflowing in a channel.
The asymmetric particlesfocus to the centerline (red). The white lines indicate
the channel side walls. The scale bar is 100 pm. (b) Distributions of transverse
positions for symmetric particles (R = 1) measured near the inlet (blue, left
hatching) and outlet (red, right hatching). Both distributionsare nearly uniform
across the channel width. (c). Distributions of transverse positions for the
asymmetric particles (R = 1.3). The particles begin nearly uniformly distributed
at the inlet. Most focus to the centerline near the outlet. Statistics are gathered
from over 300 symmetric and 300 asymmetric particle trajectories.
1.0
5.7. Additional results
103
near the walls. In contrast, the symmetric particles remain unfocused. These results demonstrate
both "self-steering" for particle-based assays, as the asymmetric particles are focused and aligned for
interrogation, and simple shape-based separation, since the centerline is enriched with asymmetric
particles at the outlet. We also observe that a small fraction of both the symmetric and asymmetric
particles (< 12%) remain close to the walls at all times.
5.7
5.7.1
Additional results
Focusing is rapid near the critical boundary
0.8
0.6
-~~
-
-
==1.10
- =1.14
-A = 1.20
- N= 1.50
=1.05-
0.4
0.2
0
50
100
150
200
250
xC/W
Fig. 5.15: Position of a particle in a channel for various asymmetries R, initial
condition (yc, 0) = (W/4, 160') at x, = 0, and parameters h
0.08, H = 1.6,
W = 21, and 9 = 3.5. The most rapid convergence to yc = W/2 with xc occurs
for R at the critical value of Rcit = 1.10.
Our analogy to a damped oscillator suggests that for a given set of parameters h, H, W, and
convergence to the centerline is most rapid for an asymmetry R near the critical boundary.
We test this for one particular initial condition. In Fig. 5.15, we show the x and y positions of
a particle with initial condition (yc, 9) = (W/4, 1350) for various asymmetries R, with the other
parameters h = 0.08, H = 1.6, W = 21, and 9 = 3.5. As discussed in the previous section, the
critical asymmetry ke for this set of parameters is cRit = 1.10. Convergence to y, = W/2 with x,
occurs most quickly for f? = Rcrit. This observation can aid the design and optimization of q2D
lab-on-chip systems.
.,
5.7.2
Reversibility of focusing dynamics
It is well-known that the low Reynolds number dynamics of a rigid particle must be
time reversible,
owing to the linearity of the Stokes equations. However, at first glance, the alignment and focusing
104
5.7. Additional results
we predict and observe for an asymmetric particle may seem to be an irreversible phenomenon. In
this section, we will show that asymmetric particle dynamics satisfy two conditions demanded by
reversibility.
First, reversibility implies the following: Suppose a particle in some initial condition at t =
0
is advected "forward" by an external flow Uo = UO^ for some time T, where X is a unit vector in
the direction of increasing x. If the flow is reversed to -UO^ at t = T, and the particle is advected
"backward" another time T, then the particle configuration at t = 2T should be identical to the
configuration at t = 0. To show that this can hold for an asymmetric particle, we note that a
particle only approaches the attractor (ye, 6) = (W/2, 00) asymptotically, remaining always on a
trajectory in configuration space connected with its initial configuration. The particle can therefore
recover its initial configuration if the flow is reversed. We note, however, that in the vicinity of
the attractor, many trajectories from disparate initial configurations converge, and noise (whether
experimental noise or numerical truncation error) can have a significant effect.
We note another, more subtle implication of reversibility. It is not difficult to see that reversibility demands mirror symmetry between forward and backward trajectories for a particle initially in
a 6 = 90' configuration. Now consider some initial configuration with 0' < 6 < 90', where in the
following discussion we restrict the definition of 6 to the forward sense (i.e. the angle between ^
and the vector from the large disc to the small disc.) The particle is attracted to (yc, 6) = (W/2, 00)
for forward advection, and (yc, 6) = (W/2,1800) for backward advection. In the latter case, it must
pass through a configuration with 6 = 900. Therefore, for forward advection, any point in configuration space is along a complete trajectory connecting a repeller (ye, 6) = (W/2, 1800) and an
attractor (yc, 6) = (W/2, 00). This complete trajectory is mirror symmetric across a configuration
with 6 = 900. The two fixed points swap stability if the flow is reversed.
initial configuration
flow reversed
Black = forward trajectory
Blue = reverse trajectory
Fig. 5.16: Demonstration of the reversibility of the dynamics of an asymmetric
particle via numerical integrationof the governing equations. A particle initially
with Ye = W/4 and 6
450, as indicated, is advected by the flow Uo = UOz
until time t = T, where T x 300. At this time, the particle is nearly aligned and
focused. The flow is reversed, and the particle retraces its trajectory and recovers
its initial configuration after an additional time T. Thereafter, it rotates into
a configuration with 6 = 900. The entire trajectory is mirror symmetric across
this configuration. Ultimately, the particle aligns with the reversed flow -Uz
and focuses to the centerline. To show the full range of particle motion, the x
axis has been compressed by a factor of 10.
We demonstrate both of the above points in Fig. 5.16 via numerical integration. A particle in
5.7. Additional results
105
the initial configuration indicated is integrated forward a time T, at which it is nearly completely
aligned and focused. The flow is then reversed. The particle recovers its initial configuration at
t = 2T. In the region of forward advection, the forward and backward trajectories overlap. (Note
that two indistinguishable sets of particle configurations are plotted in this region.) Upon further
backward advection, the particle passes through a configuration with 9 = 900. The complete
trajectory is mirror symmetric about this configuration. We note that obtaining this agreement
between forward and reverse dynamics required using a timestep a factor of 102 smaller than the
timestep used for the results in the rest of this chapter (Ji ~ 7.6 x 10-4 and 6t
0.076, respectively.)
5.7.3
Conditions for global assembly
To obtain the phase portraits of Fig. 5.9, we sweep over the configuration space of initial (ye, 0)
for various asymmetry parameters R, maintaining = 3.5, W = 21, H = 1.6, and h = 0.08. We
discard trajectories in which a disc overlaps with a side wall. (This overlap could be eliminated
via the inclusion of lubrication and repulsion forces.) Beginning with the symmetric case R = 1,
we find marginally stable oscillations around the fixed points (ye, 0) = (W/2, 00) and (y, 0) =
(W/2, 900). When particle symmetry is weakly broken at R = 1.05, the aligned and focused state
(ye, 0) = (W/2, 00) is now an attractor. The damped oscillatory approach is manifested in the
phase portrait as the spiral structure of trajectories in the vicinity of this fixed point.
However, 9 = 900 and 9 = -90* fixed points remain. The (yc, 0) = (W/2, 900) point has
bifurcated, and two marginally stable fixed points for 9 = 900 and 9 = -90* have moved off
the centerline, closer to the two side walls. These marginally stable basins are undesirable from
the standpoint of assembly, which ought to be global: ideally, any initial configuration would be
mapped to the aligned and focused state.
Momentarily deferring inquiry into these 0 = 900 and 9 = -90* fixed points, we consider the
critical asymmetry Rerit that separates the underdamped and overdamped regimes. Numerically,
we find Re,it as described in a previous section. For the parameters considered, Rcit = 1.10. At
this "critically damped" asymmetry, approach to the centerline is rapid, as we show in the next
section. However, the 0 = 900 and 9 = -90* fixed points have not yet disappeared, although they
have moved closer to the walls, and their basins of marginally stable oscillations have shrunk.
Now we consider whether we can find a R such that, for R > Rg, there are no 9 = 90' or
9
9 0
) and y- 9 0-),
= -90* fixed points. These fixed points are located at the two values of ye, y(
a
which the interaction of the real particle with the system of image particles exactly balances the
tendency of the particle to align through self-interaction. The larger, slower disc 1, closer to a side
wall than the smaller disc 2, is sped up by interaction with the closest image, such that both real
discs move at the same velocity in 1 (Fig. 5.17(a)). We can also understand why these fixed points
are marginally stable. If, for instance, the particle is displaced from the 900 fixed point away from
the wall, the tendency of the particle to self-align will be stronger than the effect of the image,
and the particle will rotate. Via the lateral drift effect, the rotation of the particle away from
0 = 900 produces a velocity in y towards ye , at which it arrives with an angle not equal to 900.
Passing through this point, the interaction with the image overtakes the self-alignment effect, and
the particle is driven to rotate in the opposite sense. The appearance of these marginally stable
oscillations suggests that the particle is "bouncing" along the wall (Fig. 5.17(b)).
On the basis of this physical understanding, we can derive the location of the fixed points,
taking g to occur at that yc900) for which disc 1 starts to overlap the side wall. Both discs are
(900)
106
5.7. Additional results
(a)
(b)
Fig. 5.17: (a) For a weakly asymmetric particle at its 6 = 90' fixed point,
the large disc is close to the nearest images, interacting more strongly with
them than the small disc does. The tendency of the particle to self-align is
exactly balanced by the strongerflow field experienced by the large disc. Here,
R = 1.05, h = 0.08, H - 1.6, 9 = 3.5, and W = 21, and the fixed point occurs
at yc/W = 0.192. (b) These fixed points are marginally stable. For a small
initial displacementfrom the fixed point, the particle appears to "bounce" along
the wall as it is advected down the channel. The x axis is compressed by a factor
of 20 in this image.
assumed to move strictly in x at the same velocity UP. We can write an equation for ux(ri):
uX(ri)
=
00
U +
B 2 (ux(r2 ) -UP)
2
(s + 2Wn)
B 2 (ux(r 2 ) -UP)
+
(2yi + s + 2Wn) 2
n=-oon=-00(5.20)
+2 00B1(ux(ri) -2 UP)
1:
n=1
(2Wn)
0CBl(ux(ri) - UP)
(2y+2Wn)2'
n=-oo
I
As this equation expresses, the contributions to the local flow field of disc 1 in - are: i.) the
external flow; ii.) disc 2, and the set of "far" images generated from disc 2 as a lattice in the y
direction with periodicity 2W; iii.) the set of disc 2's "near" images, generated as disc 2's mirror
image across the closest side wall, plus a lattice of this image's copies; iv.) the "far" lattice of disc
l's own images; and v.) the "near" lattice of disc l's images. Likewise, we have for ux(r2):
UX(r2)
=
U0 +
00
I:..
Bl(ux(ri) - UP)
±
2
(s + 2Wn)
n=-oo
B 2 (ux(r 2 ) - UP)
+2
n=1
(2Wn)2
0
B 2 (ux(r2 ) - UP)
(2yi + 2s + 2Wn) 2
=-C)O(5.21)
B1(ux(ri) - UP)
(s + 2y + 2Wn) 2 '
107
5.8. Conclusions
Moreover, since the force of constraint acts along y, we can take
UP = aiu(ri) = a2Ux(r2).
Multiplying Eq. 5.20 and Eq. 5.21 by
0!1a2)
SB (s(Oil+ 2Wn)
0p
2
UE
-
n=-oo
00
B1 (a2 - al102)
2
0102)
B1(a
(s +2 2Wn)
E
B 2 (ai -
aia2) 2
n=1
(5.23)
-Ol~o
= !ic2Uo,
B 2 (Cei -
E0 _
yj yi+
ai!f2)
2Wn)2-
n=-oon=_0
00 B1(a2 -
B1(a 2 - Cia2)
(2Wn)2
(s + 2 y, + 2Wn)2
n=-0
(2y, + 2Wn)2
E
n=-oo
up[a,
ai1a2 and rearranging, we have:
"
2
(5.22)
aia2)
(2Wn)2
0" B 2 (al -
n=1(5.24)
aia2) 2 ]=a020
+'(s2y2 + 2Wn)
Setting these equal to each other and performing the summations, we have an implicit equation
for yj:
-
7r 2 B 2 (a, - aa2)
4W 2 sin2 (s/2W)
7r2 Bi (a2 - alia2)
4W 2 sin 2 (ry1/W)
4W
7r2B22 (al - aia 2 )
sin (7r(s + 2y 1)/2W)
2
7r2 B1 (a2 12W
0i02)
2
r 2 B1(a 2 -
aia2)
7r2 B 2 (a1 - aia2)
7
4W sin (7rs/2W) 4W 2 sin2 (r(s + y1)/W)
7r 2 B 2 (Cfi - ai!e2)
7r 2 B 1 (012 - alia2)
2
2
12W
4W sin2(7r(s + 2y 1 )/2W)'
2
2
We solve this equation numerically as a function of R, taking R. to be the R at which disc 1
overlaps with the wall, or yi = R 1 . For the parameters considered, R is 1.49.
5.8
Conclusions
Experimentally and theoretically, we have shown that asymmetric particles flowing in a q2D channel self-steer - align with the flow and focus to the centerline - while symmetric particles oscillate
between side walls. Via an analogy to a damped oscillator, we isolated three contributing hydrodynamic mechanisms and exhaustively revealed the dependence of the dynamics on the governing
parameters, recovering the critical boundary between underdamped and overdamped regimes. Experiments and theory agree qualitatively and semiquantitatively.
Uniquely, focusing in q2D channel flow requires no physics beyond viscous hydrodynamics. In
contrast, an axisymmetric rigid particle in bulk channel flow tumbles in a modified Jeffery orbit
with no net migration [44, 45, 97, 98]. Chiral particles migrate across streamlines, but do not focus
[101], possibly because they have no equilibrium orientation. For the same reason, we suspect that
108
5.8. Conclusions
curved fibers, recently predicted to migrate [102], will not focus either. On the other hand, we found
intriguing connections between q2D channel flow and bulk sedimentation, which may be due to a
common feature: to leading order, the singularities that couple particles maintain fixed orientation.
In sedimentation, gravity introduces point forces oriented in the vertical direction; in q2D channel
flow, dipoles are approximately oriented upstream. In contrast, consider an axisymmetric rigid
particle in bulk and driven by flow. Due to the inextensibility of the particle, it creates a force
dipole (pair of Stokeslets) that disturbs the flow if the particle is subject to a straining field, such
as Poiseuille flow. The orientation and sign of this dipole depends on the particle orientation.
Consequently, the periodicity of a Jeffery orbit entails that the force dipole averages out, since
the particle equally samples the axes of extension and compression. With zero net force dipole,
no net lift is produced by images introduced by confining boundaries. We have also shown that
self-steering does not require irreversible physics, contrary to common intuition. Reversibility is
not violated if an asymptotic attractor in phase space is accompanied by an asymptotic repeller.
Our findings open a new direction for passive manipulation of particles flowing in microdevices:
trajectories can be engineered via particle shape and confinement. We demonstrated that dilute
suspensions of asymmetric particles entering a q2D channel in a random spatial distribution will exit
in an aligned and focused stream if the channel is sufficiently long. Symmetric particles, on the other
hand, show no focusing effect. A lab-on-a-chip system that applies these findings will not require
external forces or sheath flows to position particles, simplifying device design, manufacture, and use.
In the same demonstration, we also performed a shape-based separation, enriching the centerline
with asymmetric particles. Insights gained from this demonstration suggest future applications. We
may be able to polymerize bifunctional particles containing a fluorescent code on the downstream
edge of the particle and containing biomolecular capture probes such as DNA or antibodies in
the upstream edge of the particle. These can be used for bioassays, harnessing previously shown
advantages of using hydrogel particles for biosensing [103]. It is important to note that the particles
considered here always align in the same direction, in contrast with particles aligned with sheath
flows, such as in flow cytometry.
These results also provide the foundation for study of q2D systems in which particle-particle
interactions are important, including multiparticle clusters and concentrated suspensions. Do such
systems self-organize into the flowing crystalline states studied in Chapter 4, and discussed in this
chapter's introduction? We turn to this question in the next chapter.
CHAPTER 6
Self-organization of flowing crystals
Our thesis culminates in an investigation synthesizing the insights developed in the previous chapters. In Chapter 4, we studied the dynamics of many hydrodynamically interacting discs. We
discovered and characterized flowing crystals: configurations of particles that maintain their relative positions as they are carried by the flow. For discs, these states are marginally stable: they do
not self-organize from disorder, and their realization is limited by the discs' initial configuration. In
Chapter 5, we studied the effects of shape on the dynamics of a single q2D particle. We found that
a rod-like particle with fore-aft asymmetry will self-steer: reliably align with the flow and focus
to the channel centerline. Naturally, we are led to the following question: Can shape effects drive
self-organization of flowing crystals in many-particle systems?
To address this question, we develop a more general theoretical and numerical framework for
many-particle dynamics that can accommodate complex particle shape and any conservative interaction potentials. Particles are modeled as assemblages of discs linked by stiff springs. This
framework adapts the highly successful mobility formalism and bead-spring models commonly used
in polymer dynamics to quasi-two-dimensional confinement.
We apply this framework to two model problems: the dynamics of small clusters of particles,
and the dynamics of large suspensions. In the two problems, we anticipate that shape effects can
drive self-organization of one-dimensional and two-dimensional flowing crystals, respectively. We
find that small clusters of dumbbells can, in fact, self-organize into doublets and triplets (i.e. onedimensional crystals) predicted via the symmetry principles of Chapter 4. However, they can also
110
6.1.
Overview
form undesirable "defect" pairs. In a large suspension of dumbbells, we do not find two-dimensional
crystals; rather, large, dynamic aggregates of particles form. We quantitatively demonstrate that
this aggregation is driven by two-body defect pairing, completely dominating and frustrating the
weaker dynamics driving self-organization.
To tame this aggregation, we redesign particle shape. Specifically, we consider the effect of shape
on a particle's disturbance flow field. This flow field can be represented via a multipole expansion
which is dipolar at leading order, but has higher order terms for shapes other than discs. It was
recent shown that odd terms in this expansion can drive relative motion of two coupled particles. We
design a particle for which quadrupolar interactions drive chaining in the flow direction. Suspensions
of these two-tail "trumbells" can self-organize into nearly perfect doublet crystals. We show that
self-organization occurs over a series of stages in which one-, two-, several-, and many-body effects
are successively implicated. This study provides the first demonstration that viscous hydrodynamic
interactions alone can stabilize flowing lattices.
This chapter is adapted from a manuscript currently in preparation for submission as a journal
article.
6.1
Overview
We consider how to design a microfluidic system in which suspended particles spontaneously order into flowing crystals when driven by external pressure. Via theory and numerics, we find
that particle-particle hydrodynamic interactions drive self-organization under suitable conditions
of particle morphology and geometric confinement. Small clusters of asymmetric "tadpole" particles, strongly confined in one direction and weakly confined in another, spontaneously order in a
direction perpendicular to the external flow, forming one dimensional lattices. Large suspensions of
tadpoles exhibit strong density heterogeneities and form aggregates. By rationally tailoring particle
shape, we tame this aggregation and achieve formation of large two-dimensional crystals.
6.2
Introduction
The spontaneous appearance of order in systems of coupled particles is a durable source of intellectual fascination and practical application. Crystallization is a subject as old as equilibrium
thermodynamics, and yet renewed efforts to relate particle morphology to self-assembled structure
have been driven by the emergence of synthesis techniques that allow precise control over colloidal
shape [104, 105]. "Designer" colloids have harnessed depletion interactions [106], steric packing
[107], and "Janus" patterning [108] for assembly of novel complex materials. Self-organization
of matter driven out of equilibrium is less understood, but promises an even richer set of static
and dynamic structures [9]. For instance, Yan et al. coupled synchronization of rotating magnetic spheres to their spatial self-assembly, allowing field-tunable structural reconfigurability [109].
Suspensions that self-organize under flow into flowing crystals are particularly attractive for microfluidic device applications. Orderly flow greatly aids recognition and interrogation of suspended
objects in cytometry [48] and bioassays [26]. Flowing crystals could be used as dynamically programmable metamaterials, assembled with high throughput in continuously operating microdevices,
or as tunable diffraction gratings [28]. Researchers have achieved self-organizing flowing crystals
with acoustically excited bubbles [110] and weakly inertial spheres [25]. However, the possibility
6.2. Introduction
ill
that non-equilibrium crystallization can be induced and controlled via rational design
of particle
shape has remained largely unexplored.
(a)
o-e
0-4
0=0
4.
-0..
replace two image
..................
real
le
warithles
w
-.
"
(b)
O-n
0-*
nreal
particles
W/n
W
-
--- -------.
Fig. 6.1: (a) A single aligned and focus particle is part of an infinite lattice of
real and image particles. When one or more image particles are exchanged for
real particles, the resulting configurationshould also steadily translate along the
channel with no relative particle motion. Each of the real and image particles
is separated by W/N, where N is the number of real particles. (b) An infinite
two-dimensional lattice should likewise steadily translate. The lattice length a
is determined by particle density.
Quasi-two-dimensional microchannels have proven to be a rich setting for collective
phenomena
involving flowing droplets or solid particles, including flowing crystals, sustained by
hydrodynamic
interactions (HI) [52]. One-dimensional flowing crystals of "pancake shaped" droplets,
ordered in
the streamwise direction, can sustain transverse and longitudinal acoustic waves,
or "microfluidic
phonons." [55] As shown in Chapter 4, small clusters of discs, ordered perpendicular
to the flow direction, can maintain relative positions as they are carried by the flow [73]. Various
two-dimensional
crystal lattices are possible in unbounded q2D [56]. However, in each of these examples,
flowing
crystals are only marginally stable: the amplitude of a collective mode neither grows
nor decays
in time. Consequently, crystals do not self-organize from disorder, and have no "restoring
force"
against perturbation by channel defects.
In Chapter 5, we demonstrated via theory and experiments that a single asymmetric dumbbell
comprising two rigidly connected discs will align with the external flow and focus to the
channel
centerline. [Fig. 5.10] We isolated three viscous hydrodynamic mechanisms that together
produce
this "self-steering." The "head" and "tail" discs of a particle interact hydrodynamically.
Since
the discs are unequal in size, they have unequal hydrodynamic strength, and the larger head
disc
pushes the tail downstream. Therefore, self-interaction drives alignment
with the flow. Secondly,
when the particle is not aligned, self-interaction drives cross-streamline or lateral migration,
since
the scattered flow produced by a disc has a component perpendicular to the direction
of the
external flow. Finally, when the particle is displaced from the centerline, interaction with its
own
hydrodynamic images across the channel side walls drives rotation away from alignment.
Through
112
6.2. Introduction
the combination of these three effects, the aligned and focused configuration is an attractor for
particle dynamics.
Dumbbell
top view
Trumbbell
top view
1s
Self-aligned states:
R2
R
Fig. 6.2: Particle architectures considered in this work. A dumbbell comprises
hydrodynamically interacting discs, with R = R 1 /R 2 = 1.5. The disc centers
are connected by a stiff Hookean spring with equilibrium length -= s/R 2 = 3.5.
A trumbbell has two "tails" separated by angle / = 500. The two tail discs are
connected by a third stiff spring (not shown) so that this angle remains fixed.
Through hydrodynamic self-interaction, both the dumbbell and trumbbell align
under flow so that the head disc is upstream.
This single particle picture provides a starting point for consideration of how flowing crystals
might self-organize in multiple particle systems. In Fig. 6.1(a), we consider a single aligned and
focused particle. The particle translates in the flow direction without any rotation or
lateral motion,
and is part of an infinite lattice of real and image particles. If one or more of the image particles
is exchanged for a real particle, and the associated image channels are made real, the resulting
configuration should also steadily translate with no relative particle motion. Each real particle
in the "triplet" at the right of Fig. 6.1(a) experiences the same flow fields at its head and tail
discs as the "singlet" at left, and therefore must have the same motion as the singlet. The triplet
configuration is a "fixed point" in phase space for the dynamics of the system, and, in view of the
infinite lattice, can be regarded as a one-dimensional flowing crystal.
These considerations extend to two-dimensional crystals. Fig. 6.1(b) shows two "columns" of a
"doublet crystal" that has translational symmetry in the streamwise direction. This configuration
is also a fixed point. Consider a particle in the column at left. The lateral components of its head
and tail flow fields are still zero. The contributions of the right column to the lateral components
vanish by the combination of the translational symmetry of the column and the mirror symmetry
of the dipolar form.
We have argued that one and two-dimensional flowing crystals are dynamical fixed points, i.e.
6.3. Theory and numerical method
113
have no relative particle motion. However, we have not examined their stability. For instance, a
lattice might be linearly unstable, subject to a clumping instability resembling that found for a row
of sedimenting spheres [42]. Even if a lattice is linearly stable, its basin of attraction might not be
significant if the dynamics of the system are multistable [111].
In this work, we develop a numerical technique to study suspensions and small clusters of flowdriven q2D particles. This technique can accommodate any particle shape that can be constructed
from discs and springs. Applying this technique to initially disordered dumbbells, we find that while
small clusters can assemble with significant yield into one-dimensional lattices, large suspensions
fail to crystallize. We trace this failure to the formation of tightly bound pairs, or "defects," which
tend to pack into aggregates. To eliminate these defects, we rationally design a "trumbbell" shape
for which chaining of particles in the streamwise direction is favored by the influence of particle
shape on the flow field. Through a multistage self-organization process, the trumbbells can form
perfect doublet crystals.
6.3
Theory and numerical method
In polymer dynamics, a common and widely successful approach is to represent a polymer as a
coarse-grained chain of beads connected by springs. These bead-spring models can incorporate
many different physical effects, including hydrodynamic interactions in confined geometries [21].
In this section, we develop model equations suited to disc-spring representations of q2D particles.
The equations include dipolar hydrodynamic interactions through a q2D mobility tensor, derived
below, and can accommodate conservative interaction potentials. We discuss our method for numerical integration of the deterministic model equations. Finally, we present the model particle
architectures and conservative interaction potentials which will be used in this work.
6.3.1
Motion of a single disc
Let us revisit the simple two-dimensional problem of Chapter 3. A single disc with radius R and
velocity UP is subject to a uniform external flow U. The fluid obeys the two-dimensional Brinkman
equation:
-VP2D + pHV 2 u
-
(12p/H)u = 0,
(6.1)
where P2D is the two-dimensional pressure field, u is the fluid velocity field, and H has dimensions
of length. The quantity p is a three-dimensional dynamic viscosity. (The units of dynamic viscosity
depend on dimensionality, and a two-dimensional dynamic viscosity could be defined as p2D =H-)
The fluid obeys no-slip and no-penetration conditions on the solid boundary. Solution of Eq. 6.1
determines two items of interest: i.) the velocity field u, including the disturbance created by the
disc, and ii.) the force fq2D on the disc from the fluid, obtained from (i.) by integrating the fluid
stress tensor over the particle surface.
Eq. 6.1 is motivated as a model of fluid dynamics in a thin gap between two parallel plates. We
make the Hele-Shaw approximation that the fluid has a Poiseuille velocity profile in the direction
normal to the plates. H is the gap height, and u and U are depth-averaged quantities. With only
the first two terms, Eq. 6.1 would be the Stokes equation. The third term represents the friction
exerted by the plates on the fluid. Notably, this term breaks Galilean invariance. The force fq2D
114
6.3. Theory and numerical method
is exerted by the fluid flowing around the disc. The disc also has thin gaps separating it from the
plates [Fig. 6.1(a)], but forces from these lubricating layers are not included in fq2DIn Appendix A, we calculate the force fq2D as
fq2D =((U - UP) +
irR2 (12p/H)U,
(6.2)
where C is a drag coefficient
E2
EK1 (E)
4wrpH - +
(4
Ko(E)
(6.3)
,
KO and K 1 are modified Bessel functions, and E = vlihR/H. Since Galilean invariance is broken,
fq2D does not only depend on the difference between fluid and disc velocities, but also directly on
the fluid velocity. The second term of Eq. 6.2 is due to the external pressure needed to drive the
fluid through the gap.
Now we consider the complete set of forces on a disc. We assume the zero Reynolds number
limit, so that all forces must balance. We write
2
((U - UP)+rR
(12pt/H)U -
7rR 2 (2p/h)UP +
fn.h. = 0.
(6.4)
We have assumed shear flow in the two thin lubricating gaps, each of height h, that separate a
disc from the confining plates, resulting in the frictional third term. f .A. represents any nonhydrodynamic forces on the disc. We define friction coefficients -y,
7rR 2 (12/i/H) and 'p a
2
7rR (2p/h), and rearrange Eq. 6.4 as
('(U - UP) - (7Yp - Yc)Up + f..h. = 0
(6.5)
+ -y . Eq. 6.5 exposes some essential physics
where we have defined a new drag coefficient (' of discs confined to q2D. Even supposing fn.h. = 0, a disc will translate more slowly than the local
fluid velocity if it is subject to stronger friction than the fluid. Secondly, Eq. 6.5 cleanly separates
the hydrodynamic forces into an effective drag, proportional to the difference of fluid and particle
velocities, and an effective friction, proportional to particle velocity.
We rearrange again to obtain
Up = U - (7Yp - 7Yc)Up/('+ fn.h./(',
(6.6)
or, if we include the friction and non-hydrodynamic forces together in a quantity f,
UP = U + f/(.
(6.7)
The disc moves at the fluid velocity plus a linear superposition of perturbations from friction
and other forces. A force on the disc is related to a velocity perturbation by the single particle
mobility 1/(. In the following subsections, we will seek to generalize this single particle quantity
to a mobility tensor for systems of multiple interacting discs.
Before turning to the many-disc problem, we consider item (i.), the flow field u. Leaving the
details of the solution to Appendix A, we note a few salient features. Far away from the disc, the
velocity field is a potential flow:
U = U + V~d.p..
(6.8)
6.3. Theory and numerical method
115
The dipole potential is
Od.p.
=
B(U -UP).r
2
(6.9)
'9
where the position r is evaluated in relation to the disc center. Importantly, the strength of the
flow disturbance is proportional to the difference between fluid and particle velocities. Substituting
Eq. 6.7, we obtain
(6.10)
Br2 f
= -B.r.
The coefficient B, given in Appendix B, is proportional to the disc area: B ~ R 2
6.3.2
Systems of multiple discs
We seek a generalization of Eq. 6.7 for a system of N discs:
V=VO+M-F.
(6.11)
Here, V is a vector of 2N disc velocity components; Vo is a vector of 2N external velocity components, evaluated at each disc; F is a vector of 2N force components on the discs, including friction;
and M is the 2N by 2N mobility tensor. Crucially, this tensor includes disc-disc hydrodynamic
interactions in off-diagonal components, and, as will be shown, can encode the effect of confining
side walls. It will be derived in detail in the next subsection. The location of disc i is given by
ri = (xi, yi) in a reference frame fixed to the channel walls.
We separate the friction on the particles from the non-hydrodynamic forces:
V = VO - M
V + M - F.h.,
(6.12)
where we define a friction tensor
'ij = 6ij(7Yp,i - 7Yi).
(6.13)
Eq. 6.12 can be rearranged to isolate V:
V = (1 + M - F)-(Vo + M -Fa.h.).
(6.14)
Although M is constructed from pairwise interactions, the inversion of (1 + M . r) recovers manybody contributions to particle dynamics. Furthermore, particles interact hydrodynamically even
when Fa.h. = 0, i.e. when they are driven only by external flow.
6.3.3
Mobility tensor
We write M as
M" = 6ij ag/(i + G"
.
(6.15)
The mobility tensor relates a force on disc j in the 3 direction to a contribution to the velocity of
disc i in the a direction. The first term is simply the single particle mobility obtained previously.
The second term contains disc-disc hydrodynamic interactions.
3
in unbounded q2D, i.e.
As a simple initial demonstration, we consider how to obtain G"ag
neglecting the effect of side walls. Clearly, G" = 0, since a disc is not subject to its own flow
116
6.3. Theory and numerical method
disturbance. To obtain G" for i 7 j, consider the quantity G'j - (fj,/Q). This quantity expresses
the fluid velocity disturbance at disc i created by the force fj on disc j, as can be seen in Eq. 6.11.
Since the velocity disturbance is dipolar, we compare this quantity against Eq. 6.10 to obtain
G'3= (1- 5 ig)V (
,
(6.16)
where rij is the vector from the center of disc j to the center of disc i. The detailed form of G"'
is given in Appendix B.
Throughout this work, we use G 2 , for a channel geometry, obtained by the method of images
[61]. A disc generates an infinite series of image discs across the two channel side walls. The disturbance field created by this set of discs satisfies the no-penetration boundary condition at the two
side walls. We sum over the series of discs, obtaining a "dressed" or effective disc-disc interaction.
This interaction is screened in the streamwise direction, with a screening length proportional to
the channel width W. Furthermore, G Jp 7 0, as a disc will experience a flow disturbance created
by its own images. We give G2(43' for a channel geometry in detail in Appendix B.
6.3.4
Particle architecture and conservative forces
We consider two particle architectures in this work (Fig. 6.2). A dumbbell comprises two hydrodynamically interacting discs with centers connected by a stiff spring. A trumbbell has two tail discs,
each connected to the head by a stiff spring. The tail discs have the same radius, and the two tails
have identical length. The angle between the two tails is 0. We connect the two tail discs by a
third stiff spring in order to maintain this angle. (Alternatively, a three-body angle potential could
be accommodated in our mobility framework.) For both architectures, the radius of a head disc
is R 1 and the radius of a tail disc is R 2 . The tail length is determined by the equilibrium spring
distance s.
The spring force is Hookean. If discs i and j are connected, the force on i due to j is
F'-, = -kspr(rij- s)
spr
rij
(6.17)
All discs, whether within the same or different particles, interact via repulsive excluded volume
forces [112]:
Fe-K(rii-(Ri+RJ)) rF"EV- = FO
EV
1 - e-r(rij-(Ri+R)) rj
(6.18)
(
Discs are also repelled from the channel side walls. Via the length scale K- 1 , the excluded volume
interaction can be tuned to resemble a hard disc interaction (K-1 -+ 0) or a screened electrostatic
interaction in q2D (r-1 ~ H) [113].
Both dumbbells and trumbbells align under flow via hydrodynamic self-interaction so that the
head disc is upstream. It should be noted that we have neglected the rotation of individual discs.
For instance, if a rigid dumbbell comprising two linked discs rotates, the two discs should rotate
with the same angular velocity. (To see this, consider the motion of a point marked on a disc edge.)
Disc rotations do not contribute to far-field hydrodynamics, since they require no displacement
of fluid mass. However, the rotational resistance of a disc does contribute to the overall torque
6.4. Results
117
balance on a particle. Previously, we have shown that inclusion of disc rotations has a moderate
quantitative effect on particle dynamics, but does not change the qualitative behavior sustained by
hydrodynamics (Appendix C).
6.3.5
Numerical integration scheme
In order to integrate the equations of motion, we modify an adaptive time-stepping scheme from
[114]. At each timestep, we calculate M and the spring and excluded volume forces from the disc
positions, and obtain V from Eq. 6.12. We calculate Atmax as the largest timestep that can be
taken in an Euler step without disc/disc or disc/wall overlap. We choose At as gAtmx, where
g = 0.2. We then advance the simulation over a timestep At via the midpoint method.
6.4
Results
We consider two model problems. In the first model problem, we consider the dynamics of small
clusters of particles in a channel. The particles are initially seeded with random initial angle and
position in an finite area with length 1_ and width equal to the channel width W. Angles and
positions are chosen with uniform probability. The simulation domain is unbounded in the flow
direction.
In the second model problem, we consider large particle suspensions. Periodic boundary conditions are imposed in the flow direction. When particles cross a periodic boundary, they are
mapped to the other side of the simulation domain. Moreover, particles experience disturbance
flows created by periodic images. Since the hydrodynamic interaction decays exponentially in the
flow direction, we consider only two image cells on each side of the real domain. The simulation
domain has length L, in the flow direction, and particles are seeded with random initial angle and
position in the domain.
If the particles are effectively rigid, the dimensionless parameters governing the model problems
are purely geometric. Particle architecture is characterized by head/tail asymmetry f - R1/R2,
bond length § = s/R 2 , dimensionless lubricating gap height h - h/H, and trumbbell angle @. The
channel has dimensionless width W = W/R 2 and height H
H/IR 2 . We fix R = 1.5, 9 = 3.5,
= 500, h = 0.08, and H = 1.6. W varies as indicated in the text. Additionally, suspensions
N/(L2W), where N is the
are characterized by the two-dimensional particle number density qo
number of particles in the simulation domain.
'Ps = 50, where G is
To approximate a rigid constraint, we use a stiff spring constant Ik
evaluated for the head disc. The dimensionless constant ks, characterizes the ability of a spring
to resist stretch or compression by viscous hydrodynamic forces. We found negligible quantitative
difference when we compared the dynamics of a single dumbbell with this spring constant and
previous results for a single dumbbell with an explicit rigid constraint [92]. For the excluded
= 50.
volume potential, we use R -= rR2 = 10 and F 0 -=
We take the external flow to be Uo = Uo. Particles move in the xy plane, with channel side
walls bounding the y direction. Owing to the linearity of Stokes flow, the only effect of Uo is to
set a timescale R 2 /Uo for particle dynamics. The particles' trajectories in space do not depend
on the magnitude of U0 . Therefore, when showing results, we parameterize trajectories by average
particle position in the streamwise direction, x, = 1 E xi, instead of by time.
118
6.4.1
6.4. Results
Small cluster of dumbbells
The dynamics of a dumbbell pair provide an obvious starting point for our investigation. A doublet
is the simplest self-organized structure predicted by the geometric framework of Fig. 6.1. Moreover,
pairwise interactions are likely to play a significant role in determination of the behavior of large
suspensions. In Fig. 6.3, we show results from a pair seeded in a section with length l = 2W
of a channel with width W = 30. One hundred random initial configurations were simulated for
'T = 5.0 x 104.
Pair behaviors can be divided into four classes, with simulation snapshots of each shown in
Fig. 6.3(c). As predicted by the geometric construction of Fig. 6.1(c), pairs can self-organize
into doublets. The two particles of a doublet are positioned at y/W = 1/4 and y/W = 3/4.
In Fig. 6.3(b), we show the final positions of dumbbell head discs in the transverse direction.
Approximately one fifth of the trajectories form doublets. Given that particles were seeded over a
large area and with any possible angle, it is clear that the doublet basin of attraction in the system's
five-dimensional phase space is substantial. We also obtain singlets, in which both particles align
and focus to the channel centerline. We previously obtained this behavior for a single dumbbell
[92], and it is most likely to occur in the two particle system when the particles are initially widely
separated. There is no steady x separation for two singlets; they weakly repel.
However, we also obtain two behaviors that were not predicted by symmetry considerations.
These "defects" are undesirable from the standpoint of crystal self-organization. In the stable
defect, two particles adopt a staggered formation near a side wall, remaining in a steady transverse
position. In the unstable defect, particles adopt a head-to-tail configuration. The particles translate
back and forth across a section of the channel width before eventually breaking up to form a doublet.
The time-dependent behavior of the unstable defect stands out in Fig. 6.3(a), which shows the
y position of head discs as a function of downstream position x. A majority of trajectories are
clearly included in the exponential envelope that focuses to the centerline; these are singlets. The
doublets and stable defects can also be seen quite easily.
6.4.2
Dumbbell suspension
For the large volume of parameter space tested, dumbbell suspensions fail to form the two-dimensional
crystals predicted in Fig. 6.1(c). Instead, the dominant behavior is the formation of large aggregates of dumbbells. Nevertheless, dumbbell suspensions qualitatively and quantitatively show
tantalizing signs of intermittent self-organization. Hindrance of this self-organization can be traced
qualitatively and quantitatively to two-body defect formation.
Snapshots from a representative simulation are shown in Fig. 6.4. In (a), particles are initially
randomly seeded throughout the channel. In (b), the particles have formed large aggregates. The
configuration of neighboring pairs in the aggregates strongly resembles the unstable and stable
defects formed by isolated pairs. For instance, particles 9 and 20 have a head-to-tail configuration,
like the unstable defect, while particles 7 and 17 are staggered, like the stable defect. The aggregates
are dynamic, and continuously gain and lose particles. For instance, in (c), several particles have
been from the aggregates in the channel center, and move upstream. On the other hand, emerging
doublet order can be glimpsed downstream of the aggregates. In (d), four particles have successfully
organized into a doublet crystal configuration. Particle velocity depends on local density and
microstructure. The doublet configuration moves more slowly than the disordered particles, and
hence moves upstream relative to the rest of the suspension. In (e), particles 19 and 10 have been
6.4. Results
119
(a)
- W
0.8
0.6
0.4
0.2
0
0
200
600
400
800
1000
XC/W
(b)
(C)
stable defect
toulet
unstable defect
two singlets
0.80.6* 0.4 -0
0.2-0
0.2
0.4
0.6
Y head,final /W
0.8
1
Fig. 6.3: (a) Trajectories of an isolated pair of dumbbells from one hundred random initial conditions. Blue and green curves are obtained from each run as the
y positions of the two head discs plotted against the pair center of mass x. A
majority of dumbbells focus to the centerline; these trajectories have a characteristic exponential envelope. A substantial number focus to the doublet crystal
positions y/W = 1/4 and y/W = 3/4. Other pairs form stable defects that are
attracted to positions near the side walls, or unstable but long-lived oscillatory
defects that eventually break up to form doublet crystals. The channel width is
W = 30. (b) Histogramof the final head disc positions of the trajectoriesin (a).
Approximately 20% of dumbbell pairsform doublet crystals. (c) Pair behaviors
obtained in the simulations of (a). The unstable defects translate back and forth
across a section of the channel width before breaking up. Singlets weakly repel
each other in the flow direction; there is no steady separationin x.
120
6.4. Results
10
20
19g1
go
8
5
5
18"
12
7
1
(b)
17
16
se-l
18
4'f5
00
2
(C)
13
e
0 .. 7.13
10mk
5
9
0
14
8'.,OP
16
1
163
2" 7%
_20_646%S
(d)
12
1319
8018
15 19
14
11
3
288
15e
f)
4%ee1
6
2000
8
19
10
*v.
1
3
16
Fig. 6.4: Frames from a representative simulation of a dumbbell suspension.
The channel width is W = 20 and the simulation contains N = 20 dumbbells.
There are periodic boundary conditions in the flow direction, and the simulation
box has length lx/W = 7.5. Particles are labeled by number. (a) The randomly
seeded initial condition. (b) A large number of particles have aggregated.
The
configuration of neighboring particles in the aggregate strongly resembles
the
defect states obtained in Fig. 6.3. For instance, particles 9 and 20 appear
to be
an unstable defect, and particles 7 and 17 resemble a stable defect. (c) Particles
2, 9, 6, 5 have escaped the central aggregate. Relatively separated from
each
other and the rest of the suspension, they behave like isolated single particles
and
migrate towards the channel centerline. Downstream of the central aggregate,
emerging spatial order can be discerned in particles 11, 14, 12, 16, 8, 18, and
3.
(d) Particles 14, 11, 12, and 16 have formed a doublet crystal, which is moving
upstream towards the aggregate. (e) Particles 11 and 16 have left the crystal,
and particles 19 and 10 have been recruited to it. The crystal is approaching
a
stable defect (particles 13 and 1) and an unstable defect (particles 4 and 15).
Further downstream, particles 3 and 18 have formed a doublet crystal. (f) One
doublet crystal has broken up from encountering defects, while another (particles
3 and 18) approaches an aggregate.
6.4. Results
121
recruited to the doublet crystal, but particles 11 and 16 have left it. The doublet crystal approaches
unstable and stable defects. In (f), the crystal has been disrupted by its encounter with the defects.
These observations prompt the following ideas: In dumbbell suspensions, intermittent selforganization of flowing crystals is hindered by the formation of disordered aggregates. In a typical
crystal/aggregate encounter, the crystal is disrupted and its particles are recruited to the aggregate.
Aggregation is driven by formation of defects, which is a two-body effect.
These ideas are substantiated by quantitative study of thirty-two dumbbell suspension simulations, each carried out for a time T = 5.0 x 104 and with the same parameters as in Fig. 6.4. In
Fig. 6.5, we present the statistics of these simulations. The pair distribution function g(AX, Ay)
expresses the probability of finding two head discs separated by Ax in the external flow direction
and Ay in the transverse direction. It is normalized by the suspension density #o. The central
dark area is sterically forbidden. There is clearly a ring of strongly enhanced probability around
the excluded volume region. Particles pair as closely as excluded volume allows. Although there
are faint signs of a second ring, aggregation is largely a two-body effect.
The pair distribution function in Fig. 6.5(a) is strongest for small Ay partly due to the fact
that the channel is bounded in y. For large Ay, there are few head disc positions in y for which Ay
is not outside the channel boundaries; even if the particles were uniformly distributed, observation
of small Ay is more likely than observation of large Ay. Moreover, particles are not uniformly
distributed across the channel, as shown in Fig. 6.5(d). We correct for these effects in Fig.
6.5(b). The correction is motivated by the following considerations: the probability of separation
Ar = (Ax, Ay) can be expressed as p(Ar) = E p(r + Arjr)p(r), where the sum is taken over
head disc positions r. On the right hand side, a single body probability multiples a conditional
probability. In the absence of any two-body correlation, this expression reduces to the product of
single body probabilities E p(r + Ar)p(r). Hence, this expression provides a normalization factor
that distinguishes two-body from single body effects. The probability p(r) is simply given by the
particle density q(y) in Fig. 6.5(d). The resulting corrected distribution function in Fig. 6.10(b)
preserves the key features of 6.5(a). However, the peaks that correspond to formation of doublets
and triplets become more clearly visible, as indicated by arrows. Doublets occur for Ax = 0 and
Ay = ±W/2, while triplets occur for Ax = 0 and Ay = ±W/3 or Ay = ±2W/3.
Triplets can be isolated with a conditional pair distribution function g(AX, Y2IY1 = W/6). This
function expresses the probability of finding a head disc at (x + AX, y2), given that there is a head
disc at (x, y1 = W/6). In Fig. 6.5(c), there are clearly strong peaks at triplet positions Ax = 0,
Y2 = W/2 and AX = 0, Y2 = 5W/6.
While we have presented our findings for one particular set of parameters, we obtain similar
results upon varying W and #0.
6.4.3
Engineering hydrodynamic interactions via particle shape
In dumbbell suspensions, crystal self-organization is frustrated by formation of defects. In a defect,
particles pair side-by-side in either a staggered or head-to-tail configuration. We wish to design
a particle architecture that disfavors defect formation while preserving the essential features promoting crystal self-organization. For instance, defect formation would be disfavored for rod-like
particles with a particular anisotropic interaction: repulsion in the direction of the short axis and
attraction along the long axis. Such particles would preferentially chain in the flow direction. If
the particles are fore-aft asymmetric, they would still order laterally as doublets and triplets.
122
6.4. Results
(a)
-
(b)
-1/2
15
0
10
5
1/2
-1/2
6
0
5
4
3
1/2
2
downvt reamiii
-2/3 -1/3
(c)
0
1/3
2/3
-2/3
-1/3
0
1/3
2/3
(d)
Ay/W
-1/2
x
253
20
2.5
15 0
2
10
5
1/2
-
,----.
e- 1.5
0.50
1/6
1/2
y2/W
5/6
0
1/8
1/4
3/8
1/2
y/W
Fig. 6.5: Statistics of thirty-two different runs of a dumbbell suspension with the
same parameters as in Fig. 6.4. Simulations are run for time T = 5.0 x 10 4 .
(a) The pair correlationfunction g(/Ax, Ay) showing the probability with which
two head discs are separated by (Ax, Ay). The function is normalized by the
probability function for a suspension with uniform density 0. The sterically
excluded area is indicated by a dashed line. There is a bright ring around this
region, indicating a short-range attraction responsible for defect formation and
aggregation. (b) A correlation function corrected for the variation of particle
density across channel width. With this correction, peaks in pair separation at
(Ax = ±W/3, Ay = 0), (Ax = ±W/2, Ay = 0), and (Ax = ±2W/3, Ay =
0), indicated by white arrows, become more clearly visible. These peaks are
due to transientformation of doublet and triplet crystals. (c) The conditional
correlationfunction g(Ax, y2 IY1 = W/6). This function expresses the probability
of finding a head disc at position (xI + Ax, y2), given that a second head disc
is at y1 = W/6. There are triplet crystal peaks at (Ax = 0, y2 = W/2) and
(Ax = 0, y2 = 5W/6), in addition to defects surrounding the excluded volume
region. The function is normalized by particle density profile in (d). (d) The
variation of particle density across the channel. Particlesare depleted from the
channel center and enriched along the side walls, as the frames in Fig.
6.4
suggest.
6.4. Results
123
(a)
0.05
(b)
0-
-0.10
(C)
0.os
(d)
-- trumell
3
---dumbbell
0
2
-0.05
-7
x =7.4
1
-. 1s
05
10
15
20
Ax
Fig. 6.6: (a) Disturbance flow created by a single isolated "trumbell" in unbounded q2D, calculated numerically. Streamlines are shown in black. The
total flow disturbance is due to the superposition of dipole singularities: yellow
arrows show dipoles from friction on the discs, and white arrows show dipoles
from internal spring forces. Notably, the streamlines are fore-aft asymmetric,
bent in the downstream direction. The color field indicates the x component of
the disturbance velocity. To focus attention on the far-field disturbance, we do
not show the area immediately around the particle. (b) The disturbance flow
field can be regarded as the sum of multipole components. The lowest order
contribution is a point dipole. The quadrupolar correction bends the streamlines downstream. (c) In contrast, the disturbance streamlines for a dumbbell
are bent upstream. Accordingly, its quadrupolar correction has opposite sign
as the trumbbell quadrupole. (d) Effective potentials for two dumbbells (dashed
red line) and two trumbells (solid black line) aligned in the flow direction in
unbounded q2D. The quadrupolarcontribution to the interaction of two dumbbells is repulsive. For two trumbbells, the quadrupolarcomponent is attractive.
Higher order, shorter range multipole components are repulsive, stabilizing the
trumbbells against collision. As a result, two trumbbells have an equilibrium
separation A.eq = 7.4.
124
6.4. Results
The far-field disturbance created by a disc is dipolar. There are no other long-range terms; all
other contributions to the flow disturbance are exponentially screened. The complete disturbance
field created by a particle composed of linked discs is a superposition of dipolar fields. Since the
various dipoles are located at different points on the particle, the complete particle field is not
strictly dipolar. However, the complete field can be expressed as a multipole expansion which is
always dipolar at leading order. The universality of the dipolar term owes to the fact that all
particles, regardless of shape, obstruct and redirect the external flow.
The quadrupole is the first subleading term, decaying as 1/r 3 . Unlike the dipole, it captures
the effects of shape anisotropy. While the dipolar term expresses the total fluid mass displaced
by the particle, the quadrupolar term captures how much of the mass displacement owes to the
front of the particle and how much to the back. Consider Fig. 6.6(a). We have plotted the
disturbance flow field created by a "trumbbell" particle comprising three discs linked by springs.
The complete disturbance flow field is a superposition of nine dipolar fields: three from friction
on the discs (yellow arrows), and six from the spring forces (white arrows). The disturbance flow
field is approximately dipolar, but streamlines are bent towards the back of the particle. The two
rear discs are responsible for more mass displacement than the front disc. In Fig. 6.6(b), we show
how a superposition of a dipolar term and a quadrupolar term can produce bent streamlines. The
dipole streamlines are fore-aft symmetric. When we add a quadrupole field of appropriate sign,
we enhance mass transport towards the rear of the particle and decrease mass transport from the
front, bending the streamlines towards the rear.
Quadrupolar interactions bear a crucial difference from dipolar interactions: quadrupolar interactions can drive relative motion of two identical particles, as shown by Janssen et al. [60] Consider
two particles separated in the external flow direction (Ax $ 0, Ay = 0), with particle A upstream
of particle B. Considering only the dipolar term, particle A drives particle B upstream, and particle
B drives A upstream with equal strength. Via dipolar interactions, the two particles are slower
than they would be individually, but do not move relative to each other. However, if we add a
quadrupolar interaction with the same sign as in Fig. 6.6(b), particle A drives B upstream, and
particle B drives A downstream. The particles are hydrodynamically attracted. A quadrupolar
interaction with opposite sign would drive the particles apart.
We are now in a position to understand why two dumbbell singlets repel each other, as observed
during study of dumbbell pair dynamics. The disturbance flow field of a single dumbbell is shown
in Fig. 6.6(c). The streamlines are bent forward, since the head disc displaces more mass than the
tail disc. Accordingly, the quadrupolar field created by a dumbbell is repulsive.
Although two trumbbells are attracted through quadrupolar interactions, repulsion by higher
order and shorter range multipole terms stabilizes the particles against collision. As a result,
there is an equilibrium pairing distance for trumbbells, as previously found for deformable droplets
[60]. This equilibrium is most clearly demonstrated with an effective potential. We define Ueff
by dAx/dt = -dUeff/dx. For the trumbbell architecture studied, a pair has a potential well with
5
minimum at Az1
eq = 7.4, as shown in Fig. 6.6(d). Dumbbells clearly repel with no equilibrium
pairing distance.
Significantly, quadrupolar interactions disfavor the side-by-side configurations characteristic of
defects. As can be seen in Fig. 6.6(b), quadrupole streamlines issue from the side of a trumbbell
and terminate at the front and rear. Two side-by-side trumbbells will rotate around each other until
Ay = 0. If the sign of the quadrupole is negated, as with the dumbbells, quadrupolar interactions
favor side-by-side pairing.
6.4. Results
6.4.4
125
Trumbbell suspensions
General observations. A suspension of trumbbell particles can self-organize into two-dimensional
flowing crystals. Crystal formation occurs through multiple stages. These can be summarized as:
i.) self-alignment of individual particles with the flow; ii.) local self-organization, both laterally,
through doublet and triplet formation, and in the flow direction, through formation of strings;
iii.) formation of channel length-spanning lanes with propagating lattice defects; and iv.) defect
annihilation and relaxation to an unstrained lattice. Crucially, quadrupolar interactions drive
the formation of strings, which preferentially align in the flow direction, precursing lanes. For
dumbbells, hydrodynamic interactions drive organization in the lateral direction, but provide no
mechanism driving organization in the flow direction. For trumbbells, hydrodynamic interactions
drive organization in both directions.
The simulation snapshots in Fig. 6.7 illustrate the various stages of self-organization. In (a),
particles are initially placed with random positions and angles. In (b), nearly all particles have
individually aligned with the flow. Sections of the suspension have locally self-organized: some
particle pairs have formed doublets spanning the channel width, while other particles have formed
strings in the flow direction via quadrupolar HI. In (c), the particles have completely partitioned
into two separate lanes located near the doublet crystal positions. However, the lanes are not spread
evenly across the channel, and many particles are not unambiguously matched with a partner in a
neighboring lane. This mismatch produces two characteristic lattice defect structures. In a triangle
formation, two particles seem to share a partner; for instance, particles 4 and 10 share particle 1.
For other particles, such as 14 and 9, the partner position is vacant. In (d), the lanes have spread
across the channel, but vacancies and triangle formations persist. In (e), the suspension resembles
a strained crystal. Vacancies have annihilated by pairing across the two lanes. The crystal is on the
threshold of relaxation to an unstrained lattice: particle 2 is about to capture particle 16, allowing
6 and 19 to pair. Finally, in (f), the suspension has relaxed to an apparently perfect lattice.
Quantitatively, in Fig. 6.8(a) we show six different order parameters as a function of average
downstream position x,. The values of x, for frames (b) through (f) of Fig. 6.7 are indicated on
the plot. We briefly describe the parameters here, leaving their details to Appendix D. 4D describes
the average orientation of particles with the external flow. T4, @4,,, and 96 are bond orientational
order parameters, expressing whether neighboring particles generally have a square, rectangular,
or hexagonal structure. These parameters capture the geometric arrangement of particles, but not
whether they are in the correct spatial positions. The parameter XFT quantifies translational order:
how evenly the particles are spread in the streamwise direction, and how close the row separation
is to W/2. Finally, XFT,V isolates the particles' partitioning into lanes separated by W/2 in the y
direction.
At the beginning of the simulation, particles individually align with the flow through hydrodynamic self-interaction, and <D rapidly evolves to unity. All five qI parameters exhibit a crossover at
frame (e), confirming that it marks the beginning of rapid relaxation to an unstrained crystalline
structure. As the appearance of Fig. 6.7(f) suggests, the lattice is rectangular, and 94,r evolves to
unity. On the other hand, %Ftjumps quickly to the vicinity of unity, but approaches it only slowly.
In fact, although Fig. 6.7(f) appears to be a perfect lattice, there are very small imperfections in
the particle positions in the flow direction. In Fig. 6.8(b), we show particle positions in the flow
direction as function of x,. A small amplitude, low frequency density wave propagates in x, but
particles are otherwise evenly spaced. The value of xc for frame (f) of Fig. 6.7 is indicated on the
126
6.4. Results
(a)
e.g
61
INSV
(b): ec se
VA
4c
S
1
(d)e
4___
SC
1
@18
10C
SC S12
Sce
@C5
S
(f)
41
13
14
et
ac
(4
Sw
S
C5
S
10
(CQ
of-
C9
g3
SC12
6C
17
SC_4
ea s ec
c
acc
cc
SC11 6
414
S15
____
g11
6C
70
20
sc
sc
13SC
74
80C
20SC
4
S10 S5
SC3
S14
SC 19
g16
SC1 1
S13
4C7
SC 1
S18 S9
SC12
SC15
SC 6
SC 2
SC1 7
SC 8
SC 2 0
Fig. 6.7: An initially disorderedsuspension of trumbbells can self-organize into
a two-dimensional crystal. There are N = 20 particles in a simulation box
with W = 20 and Lx/W = 7.5. (a) Trumbbells are initially placed with random positions and angles. (b) The particles have aligned with the flow. Groups
of particles have locally self-organized, either laterally, as doublets, or in the
streamwise direction, as strings held together by the quadrupolar interaction.
(c) The particles have entirely partitioned into two separate lanes located near
the doublet crystal positions y = W/4 and y = 3W/4. In this case, the two
lanes have an equal number of particles. However, not every particle has found
a partner. For some particles the partner position is vacant (e.g. particle 9.)
For others, the partner is shared (e.g. particles 4, 10, 1) in a triangular configuration. (d) The particles have spread more evenly across the channel, but
vacancies and triangleformations remain. (e) The suspension now resembles a
strained crystal, and is on the threshold of relaxation to an unstrained lattice.
Particle 2 will capture particle 16, allowing 6 and 19 to partner. (f) The particles have settled into an apparently "perfect" crystal, and have approximately
the same neighbors as in frame (c)
6.4. Results
127
bc
d
e
(b)
f
0.8
0.8
CL
100.2 1
0
f
500
1000
xc/W
1500
-
02
2000
0
2000
4000
6000
xc/W
Fig. 6.8: . (a) Evolution of the orderparameterswith center of mass position x,
for the trajectory shown in Fig. 6.7. Dashed lines indicate the values of x, for
the frames shown in that figure. (b) Long time evolution of the particlepositions
in the streamwise direction. Aside from a small amplitude, low frequency density
wave, particles positions in x are approximately evenly spaced and steady.
plot.
Motion of lattice defects. We have not yet examined the specific mechanism by which lattice
defects propagate through the lattice and eventually annihilate. In the snapshots of Fig. 6.7, we
identified triangle formations and vacancies as characteristic defect structures. On closer examination, both appear as transient structures in a three-body mechanism of defect propagation. In
Fig. 6.9(a), we show a single particle (particle 3) in a doublet position y = 3W/4 approaching a
doublet pair (particles 1 and 2.) The single particle has a "vacant" partner position. The doublet
moves more quickly than particle 3, owing to the dipolar hydrodynamic interactions between the
pair. If we consider the form of the dipole in Fig. 6.1(a), it is clear that two particles oriented
perpendicular to the flow will experience "collective drag reduction," i.e. will speed up relative to
a single particle. Therefore, the doublet and particle 3 collide, forming a transient triangle formation. Particle 3 displaces particle 2. Particles 1 and 3 pair and move downstream as a doublet, and
particle 2 is left upstream. In Fig. 6.9(b), the same mechanism is clearly responsible for vacancy
propagation in a two-dimensional lattice. Notably, the two lanes slide past each other through the
partner swap that occurs in defect propagation. In the earlier results of Fig. 6.7, the particles
have the same neighbors in frames (c) and (f). Despite defect propagation, there is no net sliding
because there is an equal number of vacancies in the two lanes.
Lattice defect annihilation and crystal relaxation.
Owing to differences in local microstructure, lattice defects can move at different speeds. Two
lattice defects in different lanes can annihilate each other on close approach. Fig. 6.10 shows a
doublet crystal with a lattice defect in each lane. In frame (a), the vacancy
in the bottom lane,
associated with particle 12, is close to the vacancy in the top lane, associated with particle 14. Via
the mechanism of defect propagation just discussed, two triangle formations are induced in frame
128
6.4. Results
(a)
(b)
2
3
1
ecec
1
2
3
2
22
3
17
1
2
22
eclec
ec
e
1
24
26
atec
17
1
1
2
22
17
2
t
15
f
10 5
24
26~
15 10
24
5
26
e
ec
15
10
5
Fig. 6.9: Vacancy defects propagate by a simple mechanism. (a) A cluster of
three isolated particles. Particles 1 and 2 are initially placed in a doublet crystal configuration, and particle 3 is placed next to particle 2. Due to dipolar HI,
a doublet crystal has a greater downstream velocity than an isolated particle,
since each particle in the crystal increases the local fluid velocity of its partner ("transverse anti-drag.") The crystal collides with the slower particle 3.
Particle 3 slows down particle 2 and speeds up particle 1, forming a triangular
configuration. Particle 1 swaps partners, leaving particle 2 upstream. (b) The
same mechanism occurs in a two-dimensional crystal. As a result of defect motion, the two rows of the crystal slide past each other: particle 24 has exchanges
particle 15 for particle 10, and particle 22 is about to exchange particle 2 for
particle 15.
6.4. Results
129
6
(a)
10
..
5
28
16
9
11
22
25
18
26
:t
8
17
7
21
.:3.3 .:.
(C)
15
.
19
3
14
18
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3ete
27
2
13
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69
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t
3
e: ~ ~
1
e0 49ta
3
18
.
.
(g)
6
10
e
: e at
:
5
20
21
28
26
18
7
9
11
15
17
4
29
13
4
is
17
29
~s
e0
27
12
:
14
2
27
12
* S:iSC
19
3
22
14
25
18
2
e
3: .
8
19
3
13
17
4
29
:.3. .3.3:.3
.
19 3
25
18
~te ~
2
3.3.
e.
.3.3
.e .a.s..
eteie~~
3:.3
13
t 64t
.e 6Cs t
14
25
:.3e
e: .:
...
27
12
30
29
.3.3.3.3.3.3.3.3
s:
s: s:e:.3.3
~ te~
19
.0 :
14
2
23
.3
24
e Sie at 0% g : SCet
25
(d)
A
1
:..
.0
.3s.3
19
.
29
20
19 3
14
27
25
18 2
12
(e)
4
SC
: 4. 4. .4 . 4. .: Se.
.
12
Se
.
69690C
e 0: . Se a: e
e
or
(b)
2
0: 0..
.0
0 e.
St
27
12
.3
14
27
e
13
4
17
29
t
e
13
4
.3
13
:
.
t
e
17
29
.3
17
1
.3
S
24
23
:
30
Fig. 6.10: Two vacancy defects in parallel lanes can annihilate each other on
close approach. If an equal number of particles partition to two lanes, defect
annihilationprecedes relaxation to a defect-free crystal. (a) Particles 27 and 14
are associatedwith vacancies. (b) Particles 27 and 14 form triangularconfigurations alongside particles 14 and 2. (c) Particle 12 has successfully partnered
with 27, pushing particle 2 upstream. Particle 14 has not fully partnered with
particle 18. (d) Instead of partnering with particle 18, particle 14 is captured
by particle 2. (e) Particle 3, instead of partnering with 25, is likewise attracted
to a downstream particle, particle 18. (f) A shear wave propagates down the
lattice. (g) The shear wave is dissipated, and the crystal relaxes to equilibrium.
130
6.4. Results
(b). If propagation were to proceed as usual, particle 12 would pair with 27, leaving the bottom
lane vacancy with particle 2, and particle 14 would pair with 18, leaving the top lane vacancy
with particle 3. In (c), particles 12 and 27 have successfully paired, but particles 14 and 18 have
not yet done so. This allows particle 2, which just received the bottom row vacancy, to capture
particle 14 in frame (d); particle 14 "rebounds" downstream. Subsequently, particles 18 and 3,
which were originally paired in frame (a), begin to pair again in frame (e); i.e. particle 3 rebounds.
The annihilation of the two lattice defects creates a shear wave that propagates down the lattice
in frame (f). This wave dissipates, and the lattice relaxes to an apparently perfect crystal in (g).
Crystallization with permanent lattice defects. We have considered the self-organization of essentially perfect doublet crystals. All lattice defects in the top and bottom lanes eventually pair
and annihilate. However, crystals can also self-organize with two types of permanent lattice defect.
If an unequal number of particles partition between the two lanes, then it is not possible for all
vacancies to pair and annihilate, and some permanently flow through the lattice. Secondly, a stray
particle can flow between the two lanes of a doublet crystal on the channel centerline. Both types
of lattice defect are shown in Fig. 6.11. Particle 7, the "centerline inclusion," strains the lattice
as it moves downstream, forming transient triplet structures with particles in the top and bottom
lanes. Due to the transverse anti-drag effect, the particles in a triplet flow faster than the rest of
the suspension. The temporary partners of particle 7 are continuously exchanged through collisions with downstream particles. Former partners, left upstream, return to the doublet transverse
positions. Through this mechanism, the two lanes of the doublet lattice flow around the centerline
inclusion.
Wide channels. For larger channel widths W, self-organizing crystals with more than two lanes
(e.g. triplet or quadruplet crystals) are possible in principle. We have not observed the formation
of perfect triplet crystals. However, for W = 30, particles generally organize into three lanes that
are approximately located at the triplet transverse positions. Lanes have always been observed
to have unequal numbers of particles. In Fig. 6.12, an initially random configuration of N = 45
particles in a wide channel (W = 30, LX/W = 6) evolves into a configuration of three lanes through
stages of self-alignment and local self-organization. Notably, local self-organization includes such
events as transient formation of quadruplets, as shown in frame (b), and the organization of long
strings of particles held together by quadrupolar HI, appearing in frames (b), (c), and (d). When
finally organized, as in frame (e), the lanes have density heterogenities in the flow direction that
propagate through the lanes.
As with dumbbell suspensions, we can quantitatively examine self-organization in wide channels
with correlation functions. The head disc pair correlation functions g(Ax, Ay) of Fig. 6.13(a) and
(b) were calculated for the entire trajectory of Fig. 6.12. The function in (b) was corrected for the
variation of particle density across channel width. Notably, the central dark spot extends beyond
the sterically excluded region. Hydrodynamic interactions are repulsive at short range, preventing
particles from aggregating. There are clear peaks in the triplet positions (Ax = 0, Ay = ±W/3)
and (Ax = 0, Ay = ±2W/3), as well as (Ax = ±W/2,Ay = ±W/3). These peaks are indicative
of local crystalline order. In Fig. 6.12(e), particles adopt local crystalline configurations where the
densities of neighboring lanes match. Finally, the streaks in Fig. 6.13(a) and (b) extending over all
Ax/W are due to both the relative motion of neighboring lanes and the variation of density within
a lane. In Fig. 6.13(c) and (d), we have limited calculation of g(Ax, Ay) to the beginning of the
simulation through frame (c) of Fig. 6.12. The streaks and crystal peaks have not yet appeared,
but quadrupolar HI drives pairing in the flow direction, with a peak at (Ax = t6.5R2 , Ay = 0).
6.4. Results
131
16
(a).
6
2
1
15
64c. ac
12
18
9
ac6 ac4ac
4
ac
14
.
74
6C
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16
5
6
(b) e
4
19
646
44
11
is
20
13
2
1
'Coe
64
7 S
IC 4
(C)
6
444
5
6
19 13
2
1
11
15
20
12
sc
C
4C cc
OC
4C
I4
6:
.4e
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2
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6
4
a:
.4a
13
11
15
1
17
8
9
4
14
64 64
64
C64
3
16
7O
(d)
64
S6464
10
18
ac
10
18
6
17
9
4C
8
4
14
4
64C
64 .
20
12
10
18
17
9
8
14
4
.:4N.:.4.4.4.464
3
a
7.4
0
(e) :
5
6
C
19
2
13 11
1 15
20
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is
9
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S
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cc
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eer
5
19
13
11
at
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OC
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17
3
8
14
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.
S
8
3
Fig. 6.11: Suspensions can crystallize with two types of permanent defect. If
an unequal number of particles is partitioned between the crystal lanes, then
vacancies have no means to heal. Secondly, a stray particle on the centerline
can propagate freely through a doublet crystal. Frames (a) through (e) show
both types of defect. A vacancy switches from particle 4 to particle 9 through
the mechanism discussed in Fig. 6.9, moving upstream. Particle 7 strains the
lattice as it moves down the centerline. Particles flow around it, returning to
their previous y positions upstream of the defect. Due to transverse anti-drag,
the inclusion has a higher velocity than doublet pairs.
132
6.4. Results
(a)A
(b)IwwC
fe 4e
.4
4.
SC
C
4
fe
(C)
4C
4cc
*4C
4 4
4Crq
4'*
a44C4 e0
. 4e
ac 4C
ce
4
ac
C
G
CZ
y4e 4Ce QC.C
Gcac dtc 4C
4
V
C
e fe
se
4
4
SWi
4C
4
'4
(d
(d )
4444
4 A.~44
44
4GO
4C .c4 .cSC. accC;4C
4
sc 6Cat
c
4 se
4 4s dces ec 4 c
ed c
.4
(e)
(e) 4c
44444
c acsc s
s
.
ac
.e
a4
44 4
4C
ac acecseac mese
4 4 4C
4 4C 46se 4C
44
4C 4
4C 44 se 4C
C
4 C
4C 4C
4 C 44
Fig. 6.12: Self-organization of three lanes in a wide channel. There are N = 45
particles in a simulation box with W = 30 and Lx/W = 6. (a) Particles are
seeded with a random initial configuration. (b) Particles have aligned with the
flow and organized into strings and, at right, a quadruplet. (c) At left, particles
have sorted themselves into three lanes. Strings of particles are joining these
lanes. (d) Lanes now extend over most of the simulation box. A long string
of particles flows around an "inclusion" that is not in a lane. (e) Particles
are now entirely within the three lanes. The lanes contain different numbers
of particles, and vary in density in the streamwise direction. These density
variations propagate through the lanes.
6.4. Results
133
(a)
(b)
-1/2
-1/2
303
S0
2
1/2
-2/3 -1/3
1
0
1/3
2/3
2
1/2
0
-2/3 -1/3
AY/W
1
0
1/3
2/3
0
Ay/W
(c)
4
4
-1/2
(d)
4
-1/2
3
S
0
2
1/2
-2/3 -1/3
11
0
AY/W
1/3
2/3
0
02
1/2
-2/3 -1/3
1
0
1/3
2/3
0
Ay/W
Fig. 6.13: (a) Head disc pair correlation function g(/Ax, Ay) calculated over
the entire trajectory of Fig. 6.12 and normalized by the probability function
for uniform density 0 . The dashed white line indicates the sterically excluded
area. Particles are depleted from close contact not only by
steric interactions,
but also by hydrodynamic interactions. The five streaks are due to particle
laning. (b) The correlationfunction of (a) corrected for variation of particle
density across the channel width. Each lane has localized peaks, indicatinglocal
crystalline order. In Fig. 6.12(e), particles and their neighbors in other lanes
adopt crystalline order where the the local lane densities match. (c) Correlation
function normalized by 0o and calculatedfrom the beginning of the simulation
through frame (c) of Fig. 6.12. Particles have not yet partitioned
into lanes,
but particle pairing by quadrupolarHI leads to peaks at (LAx = ±6.5, Ay = 0).
(d) The correlationfunction of (c), corrected for variation of particle density
across the channel width.
6.5.
134
6.5
Conclusions
Conclusions
We have shown that a flow-driven suspension of microparticles can self-organize into flowing crystals
under suitable conditions of particle shape and geometric confinement. Shape and confinement
modify hydrodynamic interactions between particles. They can be tailored so that hydrodynamic
interactions drive organization in multiple spatial directions and over multiple time and length
scales.
In our system, organization into a two-dimensional crystal occurs through several stages in which
one, two, and multiple-body effects are successively important. Particles first align with the flow
via hydrodynamic self-interaction. This self-alignment generically occurs for asymmetric particles
in quasi-two-dimensional confinement [92]. In the next stage, particles spatially order in both the
streamwise and transverse directions via two distinct two-body mechanisms. While the transverse
mechanism is generic for asymmetric q2D particles, we engineered the streamwise mechanism by
breaking the coaxial geometry of the model particle. Particles form lanes with lattice defects that
propagate and eventually annihilate via coordinated motions of multiple particles. Finally, the
lattice collectively relaxes to an unstrained and nearly perfect crystal.
Our study is to first to demonstrate that flowing lattices can be stabilized purely by viscous
hydrodynamic interactions. It provides a starting point for further exploration of the collective
dynamics of complex particles in q2D confinement. Our theoretical and numerical framework can
accommodate suspension polydispersity, particle deformability, other interaction potentials, and
more complex particle morphologies. For instance, a trumbbell with unequal size tail discs is
a model chiral particle. The dynamics of DNA driven by flow in slit-like confinement [115] or
flow-driven confined fibers [116] or microsprings [117] could be studied with a bead-spring chain
representation. Furthermore, we have exploited the effects of only the first two terms in the multipole expansion of a particle's disturbance field. Hydrodynamic interactions could be engineered in
finer detail by tuning higher order terms via particle shape. Shape and multipole expansions can
be systematically related via conformal mapping techniques [118]. The statistics and kinetics of
dumbbell aggregation could be studied in further detail, especially in comparison with the dynamic
clustering that occurs in q2D droplet suspensions [59, 119].
Our findings could be tested experimentally with Continuous Flow Lithography (CFL) [94]. In
this technique, hydrogel particles with two-dimensional extruded shape are "optically stamped"
with UV light in a flowing stream of photopolymerizable prepolymer solution. Particle shape is
dictated by the choice of photomask. Since particles are fabricated in situ, this technique allows
precise control over the initial positions and orientations of one or more quasi-two-dimensional
particles. In a recent work, we used CFL to study the dynamics of a single asymmetric dumbbell,
obtaining qualitative and semi-quantitative agreement between theory and experiment [92]. We
note that the model problems considered here are not ideal for experimental study. For instance, the
suspension in Fig. 6.7 settled into a flowing lattice only after flowing approximately one thousand
channel widths downstream. For a typical channel width of 300 pm, the required channel length
would be 30 cm, which is impractical. Fortunately, the control over initial configuration afforded
by CFL, guided by insight provided by numerics, should allow study of initial configurations that
self-organize over realistic microchannel lengths. Furthermore, the completely disordered initial
condition of Fig. 6.7 does not represent typical device operation conditions. Ordinarily, particles
in solution are continuously injected into a channel through an inlet port. A possible experiment
would be to synthesize particles with CFL, collect them at the channel outlet, and then inject them
6.5.
Conclusions
135
into a flow-through device. We previously performed such an experiment with dilute solutions
in order to study single particle dynamics [92]. Our numerical scheme could be modified so that
particles continuously enter and exit the flow domain, modeling typical device operation conditions.
CHAPTER 7
Summary and Outlook
The four studies of this thesis constitute a natural progression. Beginning with the simplest shape discs - and the simplest problem in collective dynamics - the two-body problem - we subsequently
considered larger collections of particles and more complicated shapes. This progression culminated
with the achievement, via theoretically informed particle design, of suspensions that self-organize
into flowing crystals.
Our first study was in Chapter 3, where we considered two hydrodynamically coupled discs in
a quasi-two-dimensional channel. We developed model equations for disc dynamics and a scheme
for their numerical integration. The hydrodynamic interaction tensor, which we obtained by the
method of images, is the core quantity in these equations, and was used throughout this thesis.
Crucially, it includes the effect on disc-disc interactions by confining side walls, mathematically
described as a "dressed" interaction. Side walls change pair dynamics from a rather trivial motion
- translation with no relative motion - to a nonlinear oscillatory bound state. We also demonstrated
that this bound state can be manipulated via patterning of confining boundaries.
In the subsequent study of Chapter 4, we considered the interaction of multiple discs. We provided symmetry principles for the a priori construction of flowing crystalline states. We developed
a Lattice Boltzmann code complementary to the model equations, and used both to investigate
the crystals' collective modes, finding excellent agreement between the two methods. The Lattice
Boltzmann method recovers q2D hydrodynamics "from the ground up," validating our theoretical
approach. The crystalline states generalize the two-body bound state to more complex configura-
138
6.5. Conclusions
tions and motions. We also widened our study of the dynamical landscape, discovering "sticky" or
metastable states that include exquisitely coordinated cyclical motions with large particle excursions.
In Chapter 5, we turned to consideration of particle shape. We found that a single fore-aft
shape asymmetric particle (an asymmetric "dumbbell") can self-steer: reliably align with the flow
and focus to the channel centerline. We isolated three viscous hydrodynamic mechanisms that
together produce this self-steering, and which are generic to asymmetric particles in q2D channels.
We carried out experiments with Continuous Flow Lithography (CFL), finding qualitative and
semi-quantitative agreement with our theoretical predictions. Obtaining statistics from hundreds
of particle trajectories, we provided a convincing experimental demonstration of self-steering for
device applications. Theoretically, we answered the question posed at the beginning of this work:
reversibility is compatible with self-steering. In the language of dynamical systems, reversibility
is not violated if an asymptotic attractor accompanies an asymptotic repeller in phase space.
Moreover, to our knowledge, this study provides the first demonstration that rigid particles can
focus to the centerline in channel flow. We explained how this self-steering is ultimately made
possible by the approximately fixed upstream orientation of the q2D dipolar flow singularity.
Finally, in Chapter 6, we considered small clusters and large suspensions of particles with
complex shape. Motivated by the mobility formalism of polymer dynamics, we developed a more
general numerical and theoretical framework that can recover the collective dynamics of many
"disc-spring" particles that interact both hydrodynamically and via conservative potentials. As
a starting point, we successfully demonstrated the simplest example of crystal self-organization formation of a "doublet" by two particles - and obtained statistics for the two particle system.
These statistics demonstrated that the doublet has a substantial basin of attraction in phase space
- but also that particles can pair as "defects," which are undesirable. When we considered large
suspensions of dumbbells, we found that defect pairing drove formation of large aggregates, hindering self-organization. To tame this aggregation, we rationally redesigned particle shape to promote
chaining if particles in the flow direction via two-body interactions. To achieve this chaining,
we noted that the disturbance flow created by a particle can be described by a multipole expansion. Since the higher order terms are sensitive to shape, we engineered "trumbbell" particles that
have attractive quadrupolar interactions. As result, we could achieve self-organization of large,
two-dimensional flowing crystals. We revealed how crystal self-organization occurs through a multistage process. In successive stages, one, two, several, and finally many-body interactions become
important. This study was the first to demonstrate that flowing lattices can be stabilized purely
by viscous hydrodynamic interactions.
There are many interesting directions opened by this work. Most obviously, the CFL experiments - which were highly successful in Chapter 5 - could be extended to the other studies. In
this connection, the results of Chapter 6 are most promising. The marginally stable configurations
of Chapters 3 and 4 would be sensitive to perturbations and channel defects. In contrast, in Chapter 6 we considered how particles can be designed to robustly self-organize. As a starting point,
we suggest experimental investigation of the self-organization of two particles. For instance, two
trumbbells can either organize in the flow direction, forming a "chain," or in the lateral direction,
forming a doublet. The effective potentials for two particles separated in the flow direction can be
measured and compared to numerical results. The angle and asymmetry of a trumbbell could be
systematically varied.
We have hardly begun to exploit the theoretical and numerical framework of Chapter 6. For
6.5.
Conclusions
139
instance, this framework easily handles particle deformability. It can be used to study the motion
of DNA driven by flow in "slit-like" confinement. In this regime, the height of a confining slit is less
than the DNA molecule's radius of gyration in free solution. The DNA molecule therefore adopts a
q2D conformation. Our framework can acconunodate rigid particles with more complex shape, or
hetereogeneous suspensions that have particles of different shape. Other self-organizing collective
behaviors might be possible in such systems. Furthermore, it can be used to study self-propelled
q2D particles. Our work provides a starting point for further consideration of shape effects. We
exploited only the first two terms in the multipole expansion of a particle's disturbance field.
Shape and multipole expansions could be systematically and quantitatively related via conformal
mapping techniques. Our general strategy to achieve self-organization can find application even
outside q2D suspensions. We designed particles that align via self-interaction (a one-body effect),
and subsequently order in both spatial directions via two or several-body interactions. Every degree
of freedom was engineered for self-organization, and the self-alignment of the first stage simplified
particle design for the second.
With so many promising opportunities for continued work, the outlook for quasi-two-dimensional
hydrodynamics is bright. With all modesty, we hope and believe that the studies presented here
will endure to provide insight and inspiration for future discovery.
Appendix A
Single disc flow field
This Appendix consolidates material originally presented in the four studies of Chapters 3 through
6. Specifically, we consider the following fluid dynamical problem: A single disc with radius R and
velocity UP is subject to a uniform external flow with velocity U. We wish to obtain the force on
the particle and the flow field u. The fluid obeys the two-dimensional Brinkman equation
-VP2D + p2D V2 U - pHa2 u = 0,
(A.1)
where P2D is the two-dimensional pressure field, P2D = pH is the two-dimensional dynamic viscosity, the bulk dynamic viscosity is p, and the H is the height of the confining slit. The two confining
plates exert friction on the fluid, which is modeled by the last term of the equation.
If u is the fluid velocity in the channel midplane, as in Chapters 3 and 4, then a 2 = 8/H 2 . If u
is the depth-averaged velocity, as in Chapters 5 and 6, then a 2 = 12/H 2 . In both cases, we assume
that the flow profile is parabolic in the direction normal to the plates.
Since this problem is two-dimensional, can be solved with a stream function approach. In
cylindrical coordinates, the stream function T is defined by
1 OF
r-
09 q
Ur,
UO.
--
(A.2)
Eq. A.1 becomes
V4 q - a2V2q
=
0.
(A.3)
142
6.5.
Conclusions
By the linearity of the Brinkman equation, the complete problem can be split into two subproblems:
(a) the disc is stationary and subject to an external flow U, and (b) the disc is moving with velocity
UP through a quiescent fluid. We will discuss the solution to (b). The solution to (a) follows by
example.
Without loss of generality, we take UP = UP.
The solution for Ub, where the subscript
designates the subproblem, is subject to the following boundary conditions. Far away from the
disc, the flow field must vanish, so that Ub,r r-+o = 0 and Ub,Olr-+oo = 0. On the edge of the disc,
the no-slip and no-penetration conditions hold, and we have Urlb,r=R = UP cos(9) and Ub,Olr=R =
-UP sin(9). These boundary conditions suggest a solution of the form XI(r, 0) = f(r) sin(9). The
boundary conditions reduce to conditions on f(r): f(R) = RUP and ;IrR
= Up. Substituting
f(r)sin(9) into Eq. A.3 and solving the equation, we obtain
f(r) = cr + E + c 3 Ki(ar).
r
(A.4)
Imposing the boundary conditions, we find the integration constants to be cl = 0,
C2= UP 1+
Ki(aR) R 2,
RKo(aR'
(A.5)
and
C3
-2U
AKo(aR)
(A.6)
Eq. A.4 has a revealing interpretation. The second, long-range term is the dipolar flow disturbance.
The third term decays exponentially with screening length - H. This term represents the viscous
boundary layer in the vicinity of the disc. The boundary layer is associated with the Laplacian
term in Eq. A.1, which constitutes a singular perturbation to the equation. This problem is an
interesting example in which a boundary layer can be obtained via exact solution.
The (complete) flow field u determines the stress tensor a-. Integration of the stress vector
a- f
t over the edge of the disc gives the force per unit length F/. Since the particle height is
approximately the same as the slit, we take l ;: H. We note that this approximation is more
realistic when u is the depth-averaged velocity, i.e. when a 2 = 12/H 2 , than when a2 = 8/H 2 . On
the other hand, we do not seek a high degree of accuracy from our model. We assumed the flow
is everywhere two-dimensional, whereas it is likely to have three-dimensional components in the
vicinity of the disc. Secondly, we have assumed the disc has sharp edges, whereas a disc fabricated
with the Continuous Flow Lithography method of Chapter 5 will have rounded edges.
We therefore obtain
F = ((U - UP) + 7rR 2 aHU,
(A.7)
where
( 47rpH
E2
- +
cK1 (c)
,
(A.8)
(4
Ko(E) /
and E _ aR. Notably, the second term in A.7 breaks Galilean invariance, and is due to the external
pressure needed to drive flow through a slit. The first term in Eq. A.8 is also due to pressure, while
the second term is due to viscous tangential stress. If the tangential stress is neglected, Eq. A.7
reduces to the expression force obtained in [120] for the force on a "pancake" droplet anchored to
remain stationary under flow.
Appendix B
Hydrodynamic interaction tensor
The hydrodynamic interaction tensor for unbounded q2D can be obtained directly from the disturbance field (u - U) of Appendix A. It is non-zero only for i 7 j:
2K 1 (aR)
(1 + \RiKo(aRj))
BZ
X
2
G
(Xi - Xj),
Y
(yi - yA)
= Bj(X 2 _ y 2)/r!-, G O = 2BjXY/r!-
G
-=
G
,
G(23) = -G
.
To obtain the HI tensor in a channel geometry, we must include the effect of confining side walls.
The no-penetration condition on the side walls can be imposed by the method of images. A disc
and its images split into two sets. The "far" set includes the real disc, as well as periodic images
displaced from the real disc in the y direction with periodicity 2W. Each disc in the "far" set has
the same velocity as the real disc. The "near" set is seeded from the original disc's mirror image
across the closest side wall and includes periodic copies of this image. The "near" set includes the
image nearest to the real disc. The y component of velocities in this set are negated relative to the
original disc, while the x component remains unchanged. Summing over images, the self-interaction
144
(i =
6.5.
j)
is determined as:
Rit
2W
-)
Ci 1 +2K1(Al
aRjKo(aR))
a3-
Gnear
xfar
(G)
Gyfar =
G()
=
G")ear=
2
aB,far
3
xx,far
yyfar
= ()
0,
0,
-C, sin (,ry/W),
0.
0, G()
yxfar
=
2
G()near
G-near
=
yx,near
xyj,nea
For i
Conclusions
# j,
Y+
7gr(xi - Xj)/2W,
X-
-
,r(yi t yj)/2W
2
2
2
2
- cosh X= - cos y- cosh 2XGxxfa
2
Gx,far = C4
(cosh X- - cos y-)
2
y-
X- sinY- sinh X
20 cosY-cosh
2
2
2
G(--)
xy,far
(cosh X-
Gjyx,far =G
-xy,far,
G
C
xx,near
=
y-)
Ga yy,far = -G xx,far
2 cos Y+ cosh 2 X- - cosh 2 X - - COS2 y+
(cosh 2 X- - cos 2 y+)2
-2
cos Y+ cosh X- sin Y+ sinh X(cosh 2 X cos 2 y+)2
-G(j)
G
xy,near,
yx,near
- cos
2
xy,near
G
- cos
2
-
G(j)
yy,near
xx,near
G a3--- G(af,near +G() 4a3,far
For fixed Y+ and Y-, the two body interaction decays exponentially with screening length W/7r
or W/2,r as |X1
- -* oc.
For some analytical derivations, we find it convenient to use the unbounded q2D HI tensor in
cylindrical coordinates (r,#):
Bi=
1+
2K1 (aRj)
aRiKo(aRi)
G
G
B/4lr?,
-,n
=0,
G
R?
R
= -Bjlr?.
G~j) =0.
The subscripts of Gi now refer to directions ? and 0. f is a unit vector in direction of the vector
from the center of disc j to the center of disc i. q5 is defined as the angle between ? and the 2
is orthogonal to F and in the direction of increasing #.
direction. The unit vector
Appendix C
Effect of disc rotations
In Chapter 5, we model the motion of individual discs in a rotating dumbbell as strictly translational. However, each individual disc should also rotate with the same angular velocity as the
dumbbell. This can be seen by following the trajectory of a single point labeled on a disc edge
during dumbbell rotation.
The rotation of individual discs does not contribute to far-field hydrodynamics, since it does not
involve the displacement of fluid mass. However, it does contribute to the overall torque balance on
the dumbbell. In neglecting it, we underestimated the torque on a rotating dumbbell, and therefore
underestimated the timescale for self-alignment. In this section, we quantitatively consider the error
entailed by this omission via both theory and numerics.
Taking an approach similar to Appendix A, we consider an isolated disc with radius Ri rotating
with angular velocity w in unbounded q2D. We obtain two contributions to the disc torque r:
T =
) W,
Cri +
-
(C.1)
where the first term arises from the fluid around the disc [72], and the second from friction from
the confining plates above and below the disc. The coefficient (r,i is given by
=,,i472r
R
(I + aRiK 2 (aRi))
z(
2K1 (aR,)
(C.2)
146
6.5.
where a 2 = 12/H
2
Conclusions
, 7p = 2p/h, and p2D = pH.
Now we consider a dumbbell rotating with angular velocity w. The total torque rtot is given by
translational and rotational contributions from the individual discs. For a dumbbell rotating with
angular velocity w, the torque from disc rotations is
T
rot =
YPR4+(r,2+ 1yp7,R
Cri+
-
(C.3)
W.
We can estimate the translational contributions as follows. The two discs translate with velocity
ws/2. The translational drag coefficent for disc 1, including both wall friction and drag from the
surrounding q2D fluid, is ((1 + -yp7rR2), so that the force on the disc is
F 1 = ((1 + -yp7rR')ws/2,
(C.4)
neglecting hydrodynamic interactions. To include HI, we note that disc 2 produces a velocity
disturbance at the location of disc 1 in the direction of translational motion, effectively reducing
the rotational drag coefficient. Including HI, the force is
F 1 = ((1 + 7prR2)(1 - B 2/s 2 )ws/2.
(C.5)
Therefore, the translational torque is
Ttrans =
-
[(Ci + yp7rR2)(1 - B 2 /s 2) + ((2 + yp7R2)(1 - Bi/s 2 )]
ws 2/4.
(C.6)
+ Trot)/w and
R, H, and h, the complete rota-
Now we can compare dumbbell rotational drag coefficients Cr,db-complete = -(Ttrans
(r,db-trans
= -Ttrans/W.
For typical values of the parameters
s,
tional drag coefficient is typically ~ 1.5 times the rotational drag coefficient calculated with only
translational contributions. The timescales for self-alignment should be linearly proportional to
the rotational drag coefficient, so that the timescales should be increased by the same factor. This
increase can partially account for the discrepancy between theoretical and experimental timescales
in Figure 2 of the main text.
We briefly discuss a numerical model augmented to include disc rotations, and compare its
results to the analytical predictions just given. The system of equations for disc velocities is
expanded to include two disc torque balances and two constraints on disc rotations, which can be
expressed as:
ri
-
(r'i + 1 prR) wi = 0,
(r21
Wdb
X
(U
- U8)) -
Ir2l1|
(C.7)
)
Here, Ti are torques of constraint on the discs that set the disc angular velocities wi equal to the
dumbbell angular velocity Wdb. In our framework, these constraint torques must be distributed to
the discs as forces in order to affect dumbbell motion. In contrast with disc translation, disc rotation
cannot contribute a net force to the entire dumbbell. Therefore, we apply equal and opposite forces
F = (Ti + 72)/s to the two discs.
We apply the augmented model to the self-alignment of an isolated dumbbell with initial angle
6.5. Conclusions
147
6 = 900, comparing the time evolution of dumbbell angle with and without the inclusion of disc
torques. We use the same parameters h, H, 9 as in Fig. 5.15, and vary R. As expected, disc
torques slow down dumbbell self-alignment. We rescale time by the rotational drag coefficients
calculated above. For each value of R, the rescaled curves with and without the inclusion of disc
torques collapse quite well, with some visible difference for high R. In the inset, we plot the ratio
of
Cr,db-complete and
r,db-trans-
We conclude that disc rotations must be included in a quantitatively accurate model. Their
consideration partially resolves the discrepancy between theoretical and experimental timescales
in Fig. 5.6. However, for the semiquantitative accuracy desired in this work, omission of disc
rotations from our minimalistic simplifies our analysis and restricts our focus to the hydrodynamics
underlying self-alignment and focusing.
90
75
-
60 _
S=
=
-=
-=
a:).
R =1.3, with disc torque
R = 1.5, with disc torque
R = 2.0, with disc torque
= 2.5, with disc torque
45-
1.3,
1.5,
2.0,
2.5,
no disc torque
no disc torque
no disc torque
no disc torque
30-15-
0S
0
I
I
0.001
I
I
0.002
0.003
t/
0.004
0.005
,db
Fig. C.1: Evolution of dumbbell angle with time with and without disc torques.
We rescale time with our theoretically estimated values of the dumbbell rotational drag coefficient. For each R, the curves obtained with and without disc
torques collapse, since the self-alignment timescale is proportionalto the dumbbell rotational drag coefficient. The ratio of rotational coefficients with and
without disc torques is shown in Supplementary Figure C.2 for each value of R.
The parameters h = 0.06, H = 1.6, and 9 = 3.3 are the same as in Fig. 5.6.
148
6.5.
Conclusions
I
1.8 -
01.6
2 1.4
1.
1.2
1.2
1.4
1.6
1.8
,
2
2.2
2.4
2.6
R
Fig. C.2: Comparison of dumbbell rotational drag coefficients. We show the
ratio of theoretically estimated dumbbell rotational drag coefficients with and
without contributions from disc rotations as a function of R. The parameters
h = 0.06, H = 1.6, and s = 3.3 are the same as in Fig. 5.6. In Fig. C.1, these
theoretically estimated coefficients were shown to collapse data from numerical
simulations. Torque from the discs does not wholly account for the quantitative
discrepancy between theoretical and experimental self-alignment timescales in
Fig. 5.6.
Appendix D
Order parameters
The order parameters used in Chapter 6 are defined as follows. <D measures the average alignment
of particles with the flow:
<(X)
=I
n
cos(0j).
(D.1)
The angle Oi is defined by the flow direction and the vector between particle i's head disc and the
midpoint between its two tail discs.
The bond order parameters XF4, W'4,r, T6 capture whether the particles are in a square, rectangular, or hexagonal lattice, respectively. For each frame of a simulation, we find each particle's
neighbors via a Voronoi tesselation, taking periodic boundaries into account. We obtain T 4 and
T 6 as
T4
1 1:
nENnbr
(D.2)
and
fl
Nnbr
(D.3)
where the inner sum is taken over all Nnbr neighbors j of particle i. The angle Oij is defined by the
flow direction 1 and the vector between the head discs of i and j.
6.5. Conclusions
150
The bond order parameter qI4 will be less than one for a rectangular lattice, since the set of
next nearest lattice neighbors will not generally have angles Oij that are an integer multiple of 7r/4.
The inner sum in the rectangular bond order parameter is limited to the closest three particles:
'F4,,
e,8"
:1
=
n
.
(D.4)
3
We consider three particles instead of four because the doublet lattice has only two lanes of real
particles.
We define %FTas
IT
eI(mlkzxi+mikxyj)
n.
(D.5)
where kx = 27r/Lx and k. = 27r/W. For twenty particles partitioned into two rows, ml = 10 and
m2 = 2. Unlike the bond order parameters, TT is sensitive to imperfections in the particles' x
positions.
Finally, we define
TT,y
-
el"&9
(D.6)
which quantifies the partitioning of particles into lanes separated by the lattice length given by the
geometric construction of Fig. 6.1.
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