Cargo capture and transport by colloidal swarms

Abstract

Controlling active colloidal particle swarms could enable useful microscopic functions in emerging applications at the interface of nanotechnology and robotics. Here, we present a computational study of controlling self-propelled colloidal particle propulsion speeds to cooperatively capture and transport cargo particles, which otherwise produce random dispersions. By sensing swarm and cargo coordinates, each particle’s speed is actuated according to a control policy based on multiagent assignment and path planning strategies that navigate stochastic particle trajectories to targets around cargo. Colloidal swarms are shown to dynamically cage cargo at their center via inward radial forces while simultaneously translating via directional forces. Speed, power, and efficiency of swarm tasks display emergent coupled dependences on swarm size and pair interactions and approach asymptotic limits indicating near-optimal performance. This scheme exploits unique interactions and stochastic dynamics in colloidal swarms to capture and transport microscopic cargo in a robust, stable, error-tolerant, and dynamic manner.

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APPLIED SCIENCES AND ENGINEERING
Cargo capture and transport by colloidal swarms
Yuguang Yang and Michael A. Bevan*
Controlling active colloidal particle swarms could enable useful microscopic functions in emerging applications at
the interface of nanotechnology and robotics. Here, we present a computational study of controlling self-propelled
colloidal particle propulsion speeds to cooperatively capture and transport cargo particles, which otherwise produce
random dispersions. By sensing swarm and cargo coordinates, each particle’s speed is actuated according to a
control policy based on multiagent assignment and path planning strategies that navigate stochastic particle
trajectories to targets around cargo. Colloidal swarms are shown to dynamically cage cargo at their center via inward
radial forces while simultaneously translating via directional forces. Speed, power, and efficiency of swarm tasks
display emergent coupled dependences on swarm size and pair interactions and approach asymptotic limits indicating
near-optimal performance. This scheme exploits unique interactions and stochastic dynamics in colloidal swarms
to capture and transport microscopic cargo in a robust, stable, error-tolerant, and dynamic manner.
INTRODUCTION
The relatively recent ability to synthesize self-propelled colloidal
particles has generated considerable interest in understanding and
manipulating these particles based on their similarities to micro-
organisms (1,2) and their potential use in microscopic devices (3,4).
Collections of these particles are often referred to as active matter
and are scientifically interesting because of their nonequilibrium nature
and unique emergent behaviors and properties, including their
effective thermodynamics (5–7), collective motions and stresses (8–10),
and connections to living systems (11). From a technological stand-
point, self-propelled particles have been proposed for use in nano/
microrobotics (4,12), biomedical devices (13), environmental re-
mediation (14), sensing (15), separations (3), assembly (16), and
micromachines (17,18). Many approaches to harnessing self-propelled
particles are based on single particles, although cases involving
assembly and micromachines begin to consider how self-propelled
particle ensembles can be exploited to perform useful functions.
Without control, self-propelled particle random walks (19) often
produce by default randomly fluctuating fluid structures (20), effective
phase separation (5), or small clusters (16) but do not spontaneously
produce useful work characteristic of machines.
One potential route for exploiting self-propelled particles in
applications is to formally control their individual and collective
trajectories to perform useful microscopic functions. Examples of
feedback-controlled self-propelled particles include navigation in free-
space (21) and mazes (22). These examples involve noninteracting
particles, which simplifies control schemes but also limits the com-
plexity of emergent behaviors and functions. External fields have also
been used for feedback control of interacting particles in stochastic
self-assembly (23) of perfect crystals (24) as well as deterministic cargo
transport within clusters (18), but without the ability to address in-
dividual particles. We recently learned of a study, while finalizing
this manuscript (25), where feedback was used to specify individually
addressable self-propelled particle speeds (26); however, the objective
was mimicking group formation in living systems. We are unaware
of any precedent for feedback control of interacting and independently
addressable self-propelled colloidal particles to perform precision
microscopic functions.
Desirable features of animal and robot swarms provide motivation
to exploit useful properties of interacting and independently address-
able self-propelled colloidal particles to perform useful functions,
although with consideration of additional unique aspects such as
stochastic motion and pair potentials. For example, ants collaborate
to carry cargo (e.g., food) beyond any individual’s capabilities through
both instantaneous local observations (distributed control) and
additional nonlocal information based on memory, communication,
social structure, etc. (effective central control) (27). Such an approach
is necessary to efficiently perform coordinated work in a collabora-
tive manner, which has been mimicked in robot swarms (28). In a
contrasting example on microscopic length scales, bacterial swarm
motion is generally not sufficiently organized to perform complex
tasks in a stable efficient manner. Although a bacterial swarm has
been shown in a simple demonstration to deliver colloidal cargo (29),
this example illustrates intelligent human intervention to coordinate
bacterial motion on a pattern, which is effectively introducing central
control. The common feature in these examples is the performance
of tasks by autonomous agents interacting via nontrivial rules, which
is difficult to program into colloidal interactions. Although light-
activated self-propelled particles have been assembled into unsteady
(16) and steady-state (30) structures via controlled illumination, the
particles in these examples are not independently addressable and
have not performed any obvious measurable work.
Here, we aim to understand whether individually addressable
self-propelled colloidal particles can be controlled collectively as a
swarm to perform useful machine-like functions. In particular, we
report computer simulations of a scheme to enable feedback control
of an ensemble of self-propelled colloidal particles to operate as
a reconfigurable swarm capable of carrying out tasks including
capturing and transporting microscopic Brownian cargo. Capture
is achieved via caging of cargo within the microstructure of a self-
propelled colloidal swarm, which has repulsive interactions with
the cargo, so cargo remains unattached to swarm particles. This
strategy is adaptive and flexible as it avoids covalent or noncovalent
anchoring of cargo to self-propelled particles (3,4). Feedback-controlled
swarm reconfigurability provides the ability to adapt and respond
to stochastic disturbances via corrective actions. Our approach is
based on self-propelled colloids with individually actuatable speeds
mediated by an underlying array, which could be realized by a
number of externally but locally actuated transport mechanisms
(Fig.1 and movie S1).
Chemical and Biomolecular Engineering, Johns Hopkins University, Baltimore, MD
21218, USA.
*Corresponding author. Email: mabevan@jhu.edu
Copyright © 2020
The Authors, some
rights reserved;
exclusive licensee
American Association
for the Advancement
of Science. No claim to
original U.S. Government
Works. Distributed
under a Creative
Commons Attribution
NonCommercial
License 4.0 (CC BY-NC).
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There are a number of challenges with this scheme from a control
perspective based on the unique behaviors of colloids. Because
each particle’s propulsion direction is determined by uncontrolled
Brownian rotation, not all degrees of freedom can be controlled
deterministically (i.e., the swarm is underactuated and stochastic).
An essential aspect to enable swarm functions is the determination
of a control policy to achieve the necessary coordinated motion. In
short, swarm capture and transport of cargo require multiagent
stochastic control to direct particles to cage cargo and simultaneously
generate directional forces in a cooperative manner. In addition to
executing these functions over minimum times and distances, we also
consider the optimization of energy efficiency based on task power
output relative to propulsion power input, which enables the use of
scarce resources. By mimicking intelligent swarms (e.g., animals and
robots), this approach demonstrates centralized multiagent stochastic
control over self-propelled colloidal swarms to perform cargo capture
and transport in a robust, stable, and efficient manner.
RESULTS AND DISCUSSION
Controlling colloidal swarms
A particle-scale dynamic model of self-propelled colloidal swarm
particles and the cargo particle is implemented to capture the dominant
contributions of Brownian rotation and translation and colloidal
interactions commonly observed in colloidal feedback control ex-
periments (see Methods and fig. S1) (23,24,31). Although more
rigorous and complex models of hydrodynamic interactions [e.g.,
translation-rotation coupling (32), concentration and configuration
dependence (33)] and propulsion mechanisms (34) could introduce
quantitative differences, the conceptual problem and general multi-
agent control algorithms presented in the following are not expected
to change. Practically, for an equation of motion with different or
additional terms, only the details of planning paths of swarm particles
to surround cargo would need to be updated in the general approach
demonstrated in this work.
Feedback control of self-propelled particle swarms requires the
following elements (Fig.1A): (i) sensing the system state via particle
tracking, i.e., the self-propelled particle positions and orientations
and cargo position; (ii) calculating self-propulsion speeds of each
particle based on the current state and desired future state (based on
a control objective); and (iii) actuating each particle’s speed by con-
sidering light-activated self-propelled colloids on a light array [e.g.,
a liquid-crystal display (35) screen surface]. Actuation could also be
achieved on spatially addressable electrodes or other arrays that
mediate locally actuated transport mechanisms (36–39), which could
modify some terms of the dynamic model, but the control problem
would be conceptually similar.
The nontrivial element in this control scheme is the determination
of the policy to close the loop between sensing the system state and
actuating each particle’s speed to achieve the specified control
objectives. To achieve both cargo capture and transport, at each control
update time, tC, the control policy must assign swarm particles to
track target locations around cargo and plan paths to navigate particles
toward multiple dynamic targets (Fig.1B). Throughout the process
of self-propelled particles surrounding and capturing the cargo within
a cage, a lattice of target sites is dynamically constructed to track the
Fig. 1. Cargo capture and transport using multiagent feedback-controlled colloidal swarm. (A) Feedback control system consists of multiple self-propelled Janus
particles and a cargo particle on a light or electrode array. At each control update period, imaging senses the system state including swarm particle positions and orien-
tations and cargo position. On the basis of the system state and a given control objective, the policy determines each particle’s propulsion speed, which is actuated for
each particle by a light array (or other arrays that mediate locally actuated transport mechanisms). (B) Control policy determines each swarm particle propulsion speed
by (i) first assigning swarm particles to track target locations around cargo (represented by dashed lines) and (ii) then path planning. (C) Scheme for controlling swarm
particles to navigate to target lattice to cage cargo (Algorithm 1), which is unmodified for transient and steady-state capture processes. (D) Following cargo capture,
cargo transport is realized by maintaining steady-state capture while simultaneously identifying a subset of swarm particles oriented within a threshold of the desired
direction to produce swarm and cargo translation without swarm breakup (Algorithm 2).
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Brownian cargo particle (Fig.1C). In the reference frame of the cargo
particle, the lattice target sites are fixed. Each swarm particle is
assigned to a unique target to minimize the sum of distances between
all particles and their targets, which can be expressed by a time-
dependent assignment function, g*, given by
g * = arg min
g∈G
∑
i∈
I S
‖
r S,i (t ) − r T,g(i∣t) (t )
‖
(1)
where rS and rT are swarm particle and target position vectors in-
dexed by the set IS = [1, 2, …, N], g(i|t) denotes the assignment for
particle i such that rT,g(i|t) is the target location particle i is assigned
to track at time step t, and G is the set of all possible assignments
(see Methods). The solution of Eq. 1 is obtained using the Hungarian
algorithm for combinatorial optimization (40). The assignment
decision captured by g* can be interpreted as maximizing the co-
ordination and cooperation of the self-propelled particles making
up the swarm. Each swarm particle is assigned to track target posi-
tions around the cargo to minimize the total traveled distance of all
particles instead of its own traveled distance (e.g., self-propelled
particles do not necessarily track their nearest target; see fig. S2).
The assignment solution also naturally avoids crossing paths that
might result in swarm particle collisions (fig. S2).
After assigning self-propelled particles to targets around cargo,
the propulsion speed for each particle is determined as part of path
planning. Given particle assignment to targets, g*, the self-propulsion
speed for each particle, v*
S,i, can be formulated as
( v
S,1
* , v
S,2
* , … , v
S,n
* ) = arg min
0≤ v S,i ≤ v max ,i∈ I S ∑
i∈ I S 〈‖ r S,i (t +  t C )
− r T,g*(i∣t) (t +  t C ) ‖〉
(2)
where the left hand side is the vector of speeds for each particle and
the right hand side indicates the minimization of the average dis-
tance between swarm particles, rS,i(t + tC), and their assigned targets,
rT,g*(i|t)(t + tC), in the future at the next control update period given
a maximum allowed propulsion speed, vmax. Because the probability
distribution of future swarm particle and target positions depends
on particle interactions and stochastic dynamics, the solution of Eq. 2
is not trivial [e.g., deterministic solutions remain the subject of deep
learning methods (41)].
We make several simplifying assumptions to yield a solution to
Eq. 2, which can be practically tested on the basis of the performance
of the resulting control policy. By considering relatively short control
update times, we assume the following: (i) Particle positions are
primarily determined by propulsion rather than interactions with
nearby particles, which allows independent optimization of each
particle’s speed; (ii) rotational diffusion has limited time to reorient
particles, which allows consideration of current orientations (instead
of future probabilistic orientations); and (iii) target locations are
relatively unchanged on the basis of their reference frame connected
to the relatively slow cargo translational diffusion, which allows
consideration of current target positions (instead of future probabilistic
positions). Note that because feedback control generally corrects
errors, these assumptions do not have to be strictly satisfied. In any
case, by considering these assumptions, an approximate solution of
Eq. 2 can be written as (see Methods and fig. S3)
v
S,i
* =
{
min( d i /  t C , v max ) , d i > 0
0, d i ≤ 0 (3)
where di = (rT,g*(i|t) − rS,i)· ni is the projected distance of the swarm
particle-target vector onto the swarm particle orientation vector.
The policy in Eq. 3 optimizes each particle’s speed to minimize the
expected distance between each self-propelled particle and its assigned
target position at each control update time. When targets are in front
of self-propelled particles, propulsion is adjusted in proportion to
the projected distance (up to a maximum). If targets are behind self-
propelled particles (di ≤ 0), then no propulsion is applied to avoid
increasing distance to targets. The assignment and path planning
strategies together avoid lattice vacancies and dynamic arrest since
cooperative motion is automatically programmed into this policy.
The policy in Eq. 3 and its performance is similar to our previous
findings (22) in the limit of single particles navigating to single targets
(fig. S3), but the current policy is amenable to continuous propulsion
and real-time computation with multiple particles and targets (com-
pared with the infinite horizon Markov decision process framework
used in our prior work).
Using the assignment and planning strategies in Eqs. 1 and 3, we
designed an algorithm to capture Brownian cargo by navigating
self-propelled particles to sites surrounding cargo (see Methods,
Algorithm 1). To study different swarm sizes, lattice sites around
cargo consist of one to five complete hexagonal non–close-packed
coordination shells (6, 18, 36, 60, and 90 particles; fig. S3). We chose
lattice spacing as the pair potential minimum separation to minimize
swarm potential energy (fig. S1), which prevents swarm particles from
propelling against pair repulsion. A control update time, tC = 0.1 s,
was empirically optimized and satisfies the assumptions underlying
Eq. 3(<1/DR, <a2/DT). By measuring the system state at each control
update time, new target lattice coordinates are constructed, and tar-
get assignments and propulsion speeds are determined using Eqs. 1
and 3. Practically, updating assignments based on updated system
states is critical to avoid swarm particles blocking each other from
reaching targets and is essential to enabling the simple navigation
policy in Eq. 3. In contrast, if particle-target assignments were fixed,
then a much more sophisticated and computationally expensive
navigation policy would be required for each swarm particle to
dynamically navigate a reconfigurable maze of its self-propelled
neighbors (22).
The cargo transport algorithm (see Methods, Algorithm 2) has
many of the same elements as the capture algorithm with several
new aspects. Cargo transport requires the swarm to continue caging
cargo but also assigns a subset of swarm particles to use propulsion
to simultaneously generate a net directional force (Fig.1D). Particles
are assigned to the transport task based on the following: (i) proximity
to their targets, which indicates particles that can afford to generate
propulsion in another direction for a control update period; (ii) their
coordination number, where a greater number indicates that they
are likely to remain on the lattice to maintain cargo capture; and
(iii) their orientation relative to the desired transport direction, where
the projected component of their velocity determines their contri-
bution available for transport. Particles selected for transport are set
to maximum propulsion for one control update period to produce
translation. Particles not selected for transport continue to have target
assignments and speeds optimized via the capture algorithm.
As previewed in Introduction, it can be noted that the centralized
multiagent control scheme developed in this work is more similar
to animal and robot swarms than to microorganism swarms. Central-
ized control based on global information is necessary to gain sufficient
control authority and effective coordination to overcome challenges
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due to underactuation (no control over rotation) and stochastic
positions and orientations (Brownian motion). Coordination is achieved
in the central control design by (i) specifying a target hexagonal lattice
that is energetically favorable and minimizes energy usage once targets
are reached and (ii) assigning targets and path planning in a manner
that minimizes travel distances, crossing paths, and target competition.
In the following, this degree of coordination via central control of
multiple agents is shown to be an effective strategy that established
a benchmark for robust, stable, and near-optimal control based on
task performances approaching asymptotic theoretical limits.
Swarm cargo capture and transport demonstration
We now demonstrate cargo capture and transport for transient and
steady-state processes using the feedback control algorithms for
colloidal swarms of varying sizes and pair interactions. For swarms
starting from initial configurations encircling cargo with a radius of
~300a, the average transient time to capture cargo takes ~300 to 450 s,
which increases slightly for larger swarm sizes presumably due to
cooperative motion (fig. S3). For different combinations of swarm
size and pair attractions, representative images (fig. S3) and movies
(movies S2 to S5) depict transient cargo capture processes starting
from a radius of ~20a and steady-state captured configurations.
To better illustrate the transient cargo capture and its evolution
into steady-state cargo transport, we track a number of quantitative
metrics for the N = 90 swarms and three levels of pair attractions
(mechanism are most pronounced in this largest swarm studied)
(Fig.2). Representative configurations and trajectories of all swarm
particles and the cargo are plotted for the course of an entire capture
and transport process for a ~1300-s total time period (Fig.2,AandB,
movies S7 to S9 for N = 90, and movie S10 for other cases). Trajectories
show pair attraction improves the process effectiveness including
maintaining cargo capture and transporting cargo over longer
distances.
In all cases, self-propelled particles rapidly approach targets from
a starting radius of ~20a in ~50 s based on particle distances from
targets, dS (Fig.2C). The cargo is captured in a similar or slightly
Fig. 2. Trajectories in demonstration of cargo capture and transport by 90 particle swarms. (A) Representative configurations for 90 particle swarms with 0kT attraction
at t = 0 s (I), 30 s (II), 240 s (III), and 870 s (IV). (B) Cargo (black) and swarm particle (inset: spectrum time scale) trajectories (movies S7 to S9) for 90 particles with pair
attraction of 0kT (left), 3.2kT (middle), and 5.3kT (right) during capture (<300 s) and transport (>300 s) using feedback control (Algorithms 1 and 2). Cargo starts at origin,
and swarm is in initial circular configuration ~20a from cargo. (C) All swarm particle distances to their assigned targets, dS, (gray lines) and their mean value (black line).
(D) Cargo distance relative to the swarm center of mass, dC. (E) Instantaneous power (kT per second) based on propulsion speeds for the entire swarm (red), swarm particles
involved in capture (green), and swarm particles involved in transport (blue), where WS = WC + WT (red is on top of the green curve during initial capture process).
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shorter time than the swarm particles reaching their targets, as can
be seen from small fluctuations of the cargo position about the
swarm centroid, dC (Fig.2D). The degree to which self-propelled
particles maintain target lattice positions at steady state, after the
transient capture process but before translation is commenced, is
enhanced for increased attraction with dS ≈ 2a, 1a, and 0.2a for
pair potential energy minima of UM = 0kT, 3.2kT, and 5.3kT. The
degree of cargo capture at steady state is also correlated with particles
maintaining target positions and, thus, an effective cage, which is
apparent from steady-state values of dC ≈ 0.8a, 0.2a, and 0.1a for
increasing pair attraction.
After cargo capture, transport is initiated and quantified by the
directed translational motion of the cargo along the transport direction
of x axis (Fig.2B). The ability of the feedback control algorithms to
maintain target positions and cargo capture while generating direc-
tional translational forces is enhanced by particle pair attraction.
Two-dimensional swarm and cargo trajectories (Fig.2B) visually
show a correlation between swarm attraction and transport effec-
tiveness. No pair attraction produces the shortest directed translation
(xC ≈ 200a), the greatest stochastic orthogonal motion (yC ≈ 40a),
and the greatest fluctuations of swarm and cargo particles about targets
(〈dS〉 ≈ 4a, 〈dC〉 ≈ 2a). These results are consistent with the tendency
for uncontrolled active particle ensembles to display large density
fluctuations (2,20,42). In contrast, swarms with 5.3kT pair attrac-
tion yield the greatest translated distance (xC ≈ 700a) with minimal
diffusion in the orthogonal direction (yC ≈ 5a) while maintaining
close adherence to targets (〈dS〉 ≈ a, 〈dC〉 ≈ a). Feedback-controlled
swarms with more pair attraction appear to better maintain cargo
capture and simultaneously transport steadier, faster, and further
(~50 to 200%).
Cargo capture and transport effectiveness versus pair attraction
(Fig.2) illustrate how feedback control together with swarm poten-
tial energy resists the natural tendency for the swarm to disperse via
Brownian motion (for no propulsion and full propulsion). On the
basis of this consideration, we quantify how self-propulsion energy
is injected into the swarm at each control update time [in contrast
to a uniform energy addition that simply raises the effective tempera-
ture (7)]. The instantaneous input power to the swarm at each time
step is a sum of the power to each self-propelled particle given by
W S (t ) = ∑
i=1
N
W S,i (t ) , W S,i (t ) = 6 av
S,i
2 (4)
where the instantaneous power of each particle is the product of its
propulsion velocity, vS,ini, and propulsion force, 6avS,ini (Stokes
drag times velocity is force balancing propulsion force at low Reynold’s
number). Because the assignment of particles to capture and transport
processes is defined in our algorithms, it is also possible to track the
energy input to each subpopulation of the swarm performing either
capture (WC) or transport (WT) where WS(t) = WC(t) + WT(t).
For the N = 90 swarms at different pair attraction levels, trajectories
of total swarm power, WS, as well as capture and transport sub-
population powers, WC and WT, are shown during capture and
transport processes (Fig.2E). For each case, power is initially high
as all particles are assigned to the capture process, and many have
maximum propulsion values (Fig.2A,I); the initial power is ~50%
of the maximum power for all particles with vmax. This initially
high-power input decays markedly over ~50 s as particles start
reaching assigned targets (Fig.2A,II) and continue performing
minor adjustments to retain captured cargo at steady state at >100 s
(Fig.2A,III). The steady-state capture power, WC, plateaus at
decreasing values with increasing pair attraction. This reflects the role
of decreased swarm potential energy via pair attraction competing
against entropically driven swarm dispersion, so less power is required
via feedback-controlled propulsion.
After cargo transport is initiated at 300 s, WC and WT quickly
reach time-averaged plateaus with different means, fluctuations, and
relative levels for each pair attraction. During cargo transport, less
power is required to maintain cargo capture as pair attraction in-
creases, which allows more resources to be directed toward transport.
At the highest pair attraction, transport power exceeds that for
cargo capture; this results from swarm pair attraction maintaining a
crystalline cage around cargo (movie S9), and a resulting substantial
drop in the power requirement for the feedback-controlled capture
processes. These results indicate an opportunity to manage swarm
resources as part of optimizing its functionality subject to different
objectives (e.g., fastest transport, greatest distance, efficiency). In
the following sections, we first analyze swarm configurations and
forces to provide mechanistic insights into swarm functions and
then return to a comprehensive analysis of energy management as a
design consideration.
Swarm structures and forces: Cargo capture
With successful demonstration of controlled colloidal swarm capture
and transport of cargo, we now examine emergent microstructures
and directional forces generated during steady-state processes. Continu-
ing with the N = 90 swarms as a benchmark, the measured steady-
state time-averaged two-dimensional density profile and inward
radial force components are reported versus swarm pair attraction
(Fig.3A). In the case of no attraction (movie S2), the steady-state
swarm density profile has no obvious angular dependence, and the
radial dependence indicates a crystalline cage around the core cargo
that is surrounded by a nonuniform fluid structure decays exponen-
tially with a long tail to ~20a. As swarm pair attraction increases to
3.2kT and 5.3kT (movies S3 and S4), the proportion of particles in
the crystalline core increases, and the fluid diminishes. At the highest
pair attraction, the swarm fluid periphery all but vanishes, with a
few particles infrequently escaping via stochastic disturbances, as
a sort of gas phase, but feedback control always returns particles
to the swarm. Swarm pair attraction together with feedback control
mediates different swarm configurations caging cargo.
Because pair attraction <6kT is insufficient to crystallize passive
colloids in two dimensions (e.g., movie S6) (43), crystalline cages
around cargo are clearly nonequilibrium structures formed via feed-
back control. Because entropy disperses swarms with either no pro-
pulsion or uncontrolled full propulsion, the feedback scheme must
generate in a controlled manner a net inward radial force that over-
comes dispersion. The inward radial force toward the central cargo,
as well as swarm center of mass, due to self-propulsion can be ob-
tained from the radial component of each particle’s instantaneous
velocity vector as Fa = 6avSn. For comparison, we also compute
each particle’s instantaneous force component in the tangential di-
rection (i.e., orthogonal to the radial direction). Results show that
feedback control produces a net positive inward radial component
due to self-propulsion (Fig.3A). Tangential forces are ~10× smaller and
randomly oriented, which shows that the control algorithm generates
no other undesirable net directional stresses. Microstructures and forces
for smaller swarms are qualitatively similar to the N = 90 swarms (e.g.,
see fig. S3 for varying swarm sizes, and figs. S5 and S6 for N = 60 swarms).
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Swarm radial forces decrease as pair attraction increases, which
can be understood based on swarm potential energies and concen-
tration profiles. We can show there exists a balance between controlled
self-propelled particle inward pressure and the two-dimensional swarm
osmotic pressure. The swarm osmotic pressure is determined by
sedimentation equilibrium profiles (44), which agree with the hard
disk equation of state (45) in the absence of attraction or propulsion
(see the Supplementary Materials and fig. S4). As higher pair attrac-
tions decrease swarm osmotic pressure, the net inward pressure due
to controlled self-propelled swarm particles also correspondingly
decreases. Although the control policy is designed to steer inde-
pendently addressable particles to targets around cargo, this analysis
shows from another viewpoint that, collectively, a delicate swarm-
scale force balance emerges. Furthermore, these results demonstrate
a consistent interpretation of the swarm concentration, microstructure,
and force profiles including the presence of crystalline cores co-
existing with peripheral nonuniform fluids.
To characterize steady-state cargo capture performance versus
swarm size and pair attraction, we report metrics for cargo location
relative to the swarm center (Fig.3B), cargo diffusivity relative to its
free space value (Fig.3C), and swarm particle locations relative to
targets (Fig.3D). All metrics are obtained from mean squared dis-
placement (MSD) analyses (fig. S7). Cargo localization relative to
the swarm center and swarm particle localization relative to target
sites are both finite in all cases, indicating successful capture. Cargo
localization less than the particle radius is achieved for greater swarm
sizes and pair attraction due to strong confinement via a locally
crystalline core (as a sort of freezing criterion) (46). Relative cargo
diffusivities greater than unity indicate apparent cargo heating (7)
via buffeting by swarm particles for small swarms and pair attrac-
tion. In contrast, vanishing diffusivities for large swarms and pair
attraction again indicate crystallization in the swarm core. Swarm
particle localization on targets indicates the degree of control fidelity,
which is also aided by increased pair attraction and swarm size,
which both favor crystallization on target sites [as a sort of critical
nucleus size (47)]. Briefly, increasing swarm size and pair attraction
together with controlled inward propulsion forces overcome entropy
to maintain a dense crystalline cage around cargo with a coexisting
lower density fluid periphery.
To demonstrate that this feedback control scheme without mod-
ification is robust and stable in the presence of disturbances (besides
Brownian motion), we measured cargo and swarm particle MSDs
relative to targets for steady-state cargo capture processes in the
presence of added noise. For all swarm sizes and pair potentials
reported in Fig.3, Gaussian uncertainty with a relative SD of 5% was
simultaneously added to particle positions and orientations, swarm
particle speeds, and swarm particle attraction. Slight increase in
steady-state MSD plateau values relative to the noise free cases (fig. S7)
shows that the control algorithm is robust in the presence of disturbances
to sensors, actuators, and system state data. Mechanisms to ensure
robust stable control include (i) a controller design for stochastic
Brownian motion already suited to correct for uncertainties, (ii) actua tion
AB
C
D
Fig. 3. Swarm structure and forces during steady-state capture. (A) Column 1: Two-dimensional density profile of 90 particle swarms at steady-state surrounding
Brownian cargo under feedback control for pair attractions of 0kT (top), 3.2kT (middle), and 5.3kT (bottom) (coordinate system relative to swarm center of mass). Column
2: Magnitude of inward radial force component (units are N/a2) due to swarm particle self-propulsion. Swarm size and pair attraction dependence of (B) cargo mean
displacement relative to swarm center of mass, (C) cargo diffusivity relative to free-space value, and (D) swarm mean displacement from targets.
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of swarm particles in proportion to distance and alignment with
targets and with a maximum speed to avoid overshooting and input
energy spikes, and (iii) pair attraction to maintain swarm structure
integrity. Practically, the level of disturbance investigated in fig. S7
is beyond anything encountered in prior feedback-controlled colloidal
assembly experiments (23,24,31), but implementation of these control
approaches in complex media (e.g., tissue, soil) could encounter more
severe disturbances.
Swarm structure and forces: Cargo transport
In comparison to steady-state cargo capture, unique changes arise
in swarm structures and forces structure during steady-state cargo
transport. Cargo transport by the N = 90 swarms for different pair
attraction levels produces anisotropic concentration profiles and
directional force polarization in the transport direction (Figs.2A,IV,
and 4A and movies S7 to S10). The concentration profile during
transport is compressed on the swarm’s leading edge and expanded
on the trailing side. Distortion to the swarm concentration profile
and structure decreases with increasing pair attraction. A low-
concentration longtail is observed in all cases corresponding to
detachment of individual particles from the swarm, which then
undergo free navigation to rejoin the swarm (22).
The swarm force distribution in the translation direction, based
on the projection of self-propelled velocity vectors (Fig.4A and see
Methods), has a bipolar structure that is most evident in the cases
with less pair attraction. These plots indicate the swarm’s trailing
side has particles with positive forces, indicating they are directed
toward the cargo and in the transport direction; in contrast, the
swarm’s leading side has many particles with predominantly negative
forces, indicating they are also directed toward the cargo for caging
purposes and pointed in the opposite direction to the transport. The
force profile in the direction of transport results from a compromise
of simultaneously translating the cargo and maintaining the caging
structure, which can be aided by increasing pair attraction. After
subtracting directional forces, the remaining component of the
radial forces during steady-state transport is positive, indicating
a net inward pressure on the cargo similar to steady-state capture
(Fig.3). In all cases, the inward self-propelled force via feedback
control is essential to avoid the tendency for swarm dispersion
during transport (movie S11).
Within the bipolar force distribution in Fig.4A, as expected, there
is a net force in the positive transport direction that produces trans-
lation of the swarm (i.e., the red lobe is greater than the blue lobe).
As swarm pair attraction is increased, the proportion of propulsion
vectors with greater magnitudes in the positive direction increases,
which allows the transport speed to be increased. Because pair
attraction maintains cargo capture more easily with less assignment
of particles to the capture process, more self-propulsion resources
can be assigned to generate forces in the transport direction. This
finding is consistent with the analyzed energy distribution between
capture and transport functions in Fig.2 but now can be understood
in terms of particle-scale directional forces that simultaneously maintain
capture while generating translational forces.
As in the capture process, steady-state transport performance versus
swarm size and pair attraction is characterized by cargo localization
(Fig.4B), cargo diffusivities (Fig. 4C), and swarm localization
(Fig.4D) (from MSD analyses in fig. S8). Similar to steady-state
capture, during steady-state transport, trends indicate increased swarm
size and pair attraction both produce better cargo and swarm particle
localization and decreased diffusivities. However, during transport,
cargo and swarm particles show less localization at targets compared
with the capture process for the same swarm size and interactions.
This finding is perhaps consistent with expectations since swarm
particles must split resources between two functions and maintain
condensed caging structures with lower fidelity. Results in fig. S8
for addition of extra Gaussian noise to simulations of steady-state
cargo transport (same noise introduced in capture process in fig. S7)
again show the control scheme is robust in the presence of simulta-
neous disturbances to sensors, actuators, and system state variables.
In addition to mechanisms already noted for stabilizing control,
limiting the number of particles that contribute directional forces
for transport enables the majority of swarm particles to maintain
cargo capture and the swarm structural integrity.
Swarm power management
We now investigate how swarm parameters determine energy
management and efficiencies to optimize functions based on different
objectives (e.g., speed and distance) and constraints (e.g., limited
resources). We first discuss mechanisms using rendered configura-
tions to illustrate the intricate interplay of pair attraction, swarm
size, and swarm particle assignments to capture and transport
processes (Fig.5A). During transport, maintaining cargo capture via
the caging structure integrity is inherently a priority over transport
(Algorithm 2); this is coded into criteria for whether particles are
eligible to produce transport based on their orientation, positions
relative to targets, and coordination number. For smaller swarms
and weaker particle attraction, a substantial portion of propulsion
energy is necessarily directed toward capture since condensed struc-
tures enclosing cargo disassemble more easily. As swarm size and
pair attraction increase, crystalline cages around cargo require sig-
nificantly less feedback-controlled propulsion input, thus freeing
up resources for a greater proportion of propulsion to be directed
toward transport.
Because transport speed and energy consumption are inherently
linked and because maximizing speed is an interesting design criterion
in its own right, we first analyze how speed depends on swarm
parameters. We summarize cargo transport speeds versus swarm size
and pair attraction (Fig.5B and movies S7 to S10). Transport speeds
are negligible for small swarms (N < 60) with low attraction (<3kT)
due to their inability to simultaneously maintain cargo capture and
produce propulsion for transport (<4 particles contribute to transport;
Fig.5A). While all swarm size and pair attraction combinations
result in successful cargo capture, the locus of points indicating finite
transport speeds in Fig.5B indicates a threshold for successful cargo
transport. Low attraction swarms (<3kT) can have relatively large
speed fluctuations (Fig.5B), consistent with large swarm particle
position fluctuations (Fig.4A and fig. S6). At attraction ~15kT, the
large fluctuations in the N = 6 swarms occur as the result of struc-
tures that cannot easily reconfigure to maximize transport speed.
As more swarm particles contribute to transport, faster transport
speeds are obtained for larger swarms and more pair attraction, up
to a limit (Fig.5,AandB). For each swarm size, a marked speed
increase occurs upon swarm crystallization around cargo, which
occurs at different attraction levels for each swarm size. This appears
as a sort of critical crystal size similar to heterogeneous nucleation,
with the critical difference that feedback control is involved in
determining nonequilibrium steady-state microstructures. Swarm
crystallization aids cargo capture and requires less feedback control,
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which allows more propulsion to be committed to transport. Maxi-
mum transport speeds with increasing swarm size and pair attrac-
tion approach a theoretical limit (see the Supplementary Materials),
based on assuming all swarm particles have propulsion assigned to
transport, as given by
v
c
∞ = v max N [2(N + 1 ) ] −1 (5)
which is approached in the limit of high pair attraction, and the
1/2 factor accounts for particles being correctly oriented for trans-
port ~1/6 of the time based on Brownian rotation. On the basis of
these findings, swarm size and potential energy can be chosen to
design the cargo transport speed.
Instead of maximizing transport speed, power use in the presence
of limited resources could be another design criterion. Our results
can be analyzed to find how swarm parameters influence energy
consumption including total energy, distribution between functions,
and transport efficiency. Similar to the analysis of instantaneous power
for the N = 90 swarms (Fig.2), we summarize steady-state power
during capture and transport processes versus swarm size and attrac-
tion (Fig.5C and fig. S9). Larger swarms require more energy for
both capture and transport. For example, for cargo capture in the
absence of attraction, more power is required to make more self-
propelled particles maintaining steady-state targets (Fig.5C). In
this limiting case, power per particle is ~15kT/s for all swarm sizes,
which is close to one-sixth of constant full-propulsion power per
particle (i.e., 6avmax2/6 = 116.4/6 = 19.42kT/s; particles oriented
within ±30° of targets use near full propulsion or one-sixth of the
time for Brownian rotation). This shows that in the absence of
attraction, particles consume energy near the maximum rate to
maintain targets.
As pair attraction is increased, energy consumption for steady-
state capture decreases (Fig.5C). This results from swarm attraction
maintaining targets with less propulsion (~1kT/s per particle at
highest attraction investigated). The lowest capture power is obtained
for the smallest swarm size at the highest and lowest attraction levels.
At intermediate attraction levels, slightly larger swarms have the
lowest power, presumably as the result of the increased size aiding
formation of condensed swarm states around cargo. In contrast
to steady-state capture, steady-state transport consumes energy in
proportion to swarm size but is mostly independent of attraction
(fig. S9). The independence from attraction arises from the fact that
during transport, all swarm particles experience self-propulsion
the whole time. However, the proportion of propulsion assigned to
maintaining capture and producing transport changes continuously
as pair attraction changes.
In addition to the qualitative mechanistic interpretation already
provided in Fig.5A, the proportion of self-propelled particle power
dedicated to transport relative to the total swarm power is quanti-
fied by the ratio, WT/WS = WT/(WC + WT), as a function of swarm
size and pair attraction (Fig.5D). Trends in percent swarm power
directed toward transport mimic the maximum velocity results
AB
C
D
Fig. 4. Swarm structure and forces during steady-state transport. (A) Column 1: Two-dimensional density profile of 90 particle swarms at steady-state surrounding
Brownian cargo under feedback control for pair attractions of 0kT (top), 3kT (middle), and 5kT (bottom) (coordinate system relative to swarm center of mass). Magnitudes
of force densities (units are N/a2) in (column 2) transport direction and (column 3) inward radial directions due to swarm particle self-propulsion. Swarm size and pair at-
traction dependence of (B) cargo mean displacement relative to swarm center of mass, (C) cargo diffusivity relative to free-space value, and (D) swarm mean displacement
relative to target sites.
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(Fig.5B), which occurs for similar reasoning (i.e., velocity is directly
related to transport power). At low pair attraction, little to no trans-
port occurs since nearly all resources are assigned to maintaining
cargo capture. As pair attraction increases, a sharp increase in
transport power occurs since pair attraction assists cargo caging.
These trends are most obvious in the N = 90 swarms, where only
15% of swarm power is assigned to transport with no pair attraction
but jumps to 80% at ~5kT attraction and asymptotically approaches
~100% with >20kT attraction. One consideration is that the capture
algorithm in combination with limited propulsion rates cannot
overcome swarm particle aggregation for >8kT pair attraction;
however, this is overcome easily by assembling swarm particles
around cargo at lower pair attraction and then increasing attraction
after swarm particles are on lattice target positions. These findings
show finite resources must be assigned to capture power for all
practical conditions, since activating transport alone will disperse
the swarms at all attraction levels investigated (movie S11).
One way to quantify the effectiveness of swarm functions is to
define an efficiency based on work performed relative to input
energy [e.g., uncontrolled active colloids (48)]. However, in the con-
text of a colloidal swarm acting as a transport device, efficiency may
also be considered on the basis of distance covered divided by energy
input (e.g., miles per gallon for cars). Transport efficiency can be
defined for colloidal swarms based on average cargo transport speed
and average swarm power during transport as
 = 〈 v C 〉 / 〈 W S 〉 (6)
Using this definition, for each swarm size, higher pair attraction
leads to monotonically increasing higher energy efficiency (Fig.5E).
The dependence on pair attraction emerges for similar reasons as
the vC and WT dependence on attraction (Fig.5,BandD); more
power is available for transport as pair attraction assists the mainte-
nance of cargo capture.
To understand how transport efficiency depends on swarm size,
we first consider the high pair attraction limit where small swarms
are most efficient. This can be understood based on the fact that the
max velocity at high attractions is weakly dependent on swarm size
(Eq. 5 and Fig.5B), whereas swarm transport power at high attraction
is nearly proportional to swarm size (fig. S9). These observations
can be captured by an asymptotic analysis of the high pair attraction
limit of Eq. 6 to give (see the Supplementary Materials)
 ∞ =
[
2  2  av max (N + 1 )
]
−1 (7)
which displays good agreement with measured values (Fig.5E),
particularly for larger swarms where nearly all energy is assigned to
transport. This dependence arises through net drag increasing faster
than net velocity in larger swarms (reminiscent of small economy
cars versus large fast cars).
In contrast to the high attraction limit, larger swarms are found
to be most efficient as swarm pair attraction vanishes. This arises
from capture being more efficient in large swarms in the absence of
attraction; this allows a greater proportion of power to be directed
toward transport functions, which ultimately also makes transport
more efficient in large swarms in the absence of attraction. A con-
tinuous transition between low and high attraction limits leads to a
uniquely efficient swarm size for each pair attraction level. In summary,
(i) increasing pair attraction for a given swarm size increases efficiency,
(ii) small swarms are most efficient at high attraction, (iii) large swarm s
Fig. 5. Steady-state swarm speed, power, and efficiency versus swarm size and attraction. (A) Scheme to illustrate the interplay of self-propulsion assignments, pair
attraction, and swarm size during transport. (B) Cargo transport mean speed and its SD as fraction of max single particle speed, vmax. Dashed line is theoretical maximum
relative speed of ~(2)−1 (Eq. 5). Gray triangles show locus of points for threshold combinations of swarm size and attraction to generate finite transport speeds. (C) Swarm
capture power (where WS = WC, WT = 0) mean and its SD. Gray triangles show locus of points indicating lowest power at each level of attraction, which progressively
changes from the largest swarm at no attraction to the smallest swarm at the highest attraction. (D) Ratio of transport power to total swarm power during cargo transport
while simultaneously maintaining cargo capture (where WS = WC + WT). (E) Energy efficiency (fuel efficiency) during cargo transport. Gray triangles show locus of points
indicating highest efficiency at each level of attraction. Inverted black triangles are theoretical maximum efficiency from Eq. 7 for each system size.
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are most efficient at low attraction, and (iv) intermediate swarms
sizes are most efficient at intermediate attraction levels. Swarm design
for cargo transport energy efficiency has a nontrivial solution.
Analysis of swarm functions and parameters illustrates an oppor-
tunity for quantitative optimization based on design constraints such
as swarm size, energy budget, feasible pair attraction, minimum
transport speeds, etc. Intermediate-sized swarms of N = 36 to 60
and ~5kT of pair attraction are perhaps most practical in terms of
energy efficiency and avoiding conditions that cause swarm aggre-
gation. Smaller swarms might more easily navigate in confined
spaces to transport cargo through porous media, whereas larger
swarms produce more stable caging of cargo via multiple coordination
shells. Ultimately, we have demonstrated a scheme for capturing
and transporting cargo based on individually addressable colloidal
particles on an array and show how physics associated with the
colloidal domain such as Brownian motion and particle interac-
tions provide unique challenges and opportunities compared with
macroscopic insects or robot swarms.
CONCLUSIONS AND OUTLOOK
A multiagent stochastic control algorithm was developed to enable
a swarm of individually addressable self-propelled colloidal particles
to perform nontrivial machine-like tasks of capturing and trans-
porting microscopic cargo. By sensing cargo and swarm particle
positions, assignment and path planning strategies enable feedback
control over swarms by simply specifying each particle’s propulsion
speed (whereas direction is stochastic via Brownian rotation). The
control policy enables swarm functions in a minimally complex
model that cannot be reduced to a simpler heuristic, and without
control, swarms randomly disperse. This scheme exploits unique
interactions and stochastic dynamics in colloidal swarms to capture
and transport microscopic cargo in a robust, stable, error-tolerant,
and dynamic manner that is unconventional compared with macro-
scopic swarms.
Swarm size, interactions, entropy, and control together influence
the ability of swarms to simultaneously capture and transport cargo.
The control policy navigates independently addressable swarm particles
to targets around cargo, which collectively results in a number of
nontrivial emergent swarm behaviors including controlled condensa-
tion around cargo, generation of net caging and translational forces,
and the ability to tune task efficacy, efficiency, and speed. Forces
within swarms show feedback control generates a net inward radial
force that balances swarm osmotic pressure, which determines
average swarm radial microstructures. Microstructures are polarized
as some swarm propulsion resources are redirected toward cargo
transport via translational forces. The extent of swarm capabilities
for cargo localization, transport speed, power management, and
fuel efficiency displays nontrivial dependencies on swarm size and
interactions. Careful design of swarm characteristics is shown to
optimize functions subject to application constraints and enables
robust stable control in the presence of substantial disturbances to
sensors, actuators, and system state variables. The approach of cargo
transport speeds and efficiency toward expected theoretical asymptotic
limits suggests the central controller scheme developed in this work
provides a benchmark for near-optimal performance.
Our scheme for controlling colloidal swarms provides a general
framework to incorporate additional physics, different control objec-
tives, and strategies to address more severe disturbances to control
variables. The algorithm as reported could be implemented in a
number of currently available experimental systems involving light- or
field-mediated local transport mechanisms, but continued advances
in particle synthesis, externally triggered transport mechanisms, and
imaging could enable implementation of related approaches in other
materials and applications. Depending on system details, our algo-
rithms could be modified to include, for example, different hydro-
dynamic or propulsion models or different particle shapes that
influence local cargo corralling. In addition, the central control scheme
in this work can be adapted to include, for example, distributed control
elements suitable for larger swarms with more sophisticated capa-
bilities (e.g., memory and communication), filtering to deal with noisy
sensor data, and data-driven approaches (e.g., reinforcement learn-
ing) to achieve simultaneous dynamic modeling and path planning.
Increasingly ambitious control objectives for colloidal swarms could
include navigating cargo through mazes (fig. S10), herding many
cargo particles, assembling passive colloids, or performing as micro-
scopic machines (fig. S10). Ultimately, the control and analysis
framework for colloidal swarms provide a basis to develop robotic
colloidal systems that could enable unique technologies in nano-,
micro-, bio-, and environmental systems and applications.
METHODS
Brownian dynamic simulations
An equation of motion for the positions, rS, and orientations, , of
N self-propelled swarm particles and a single cargo particle is given by
r S,i (t + t ) = r S,i (t ) + D t
─
kT ⋅ F i t +  r
S,i
B + v S,i (t ) n i
 i (t + t ) =  i (t ) +  
i
B
r C (t + t ) = r C (t ) + D t
─
kT ⋅ F C t +  r
C,i
B
(8)
where Dt and Dr are the translational and rotational diffusivities for
spheres, vS,i is propulsion speed (i.e., the light modulated control
input), ni = (cos(i), sin(i)) is the particle orientation, kT is thermal
energy, ∆t is the time step, i is the particle index, and superscript B
indicates Brownian displacements (with zero mean and variances of
2Dtt and 2Drt). Forces due to particle pair interactions, F, are
computed from the gradients of scalar potentials as
F i = − (∇ u C,i ( r ij ) + ∑
j≠i, j∈ I M ∇ u i,j ( r ij ) )
F C = − ∑
j∈ I S ∇ u C,j ( r C,j ) (9)
where the swarm pairwise interactions are modeled as the superpo-
sition of electrostatic repulsion and depletion attractions as (43)
u i,j ( r ij ) = Bexp
[
− ( r ij − 2a )
]
+  V ex ( r ij ) (10)
where B contains material property constants, −1 is Debye length, a
is particle radius, rij = ||rS,i − rS,j|| is particle pair separation, Vex is
excluded volume between spheres (43), given by
V ex ( r ij ) = (4 / 3 )  (a + L) 3
[
1 − 3
─
4
(
r ij
─
a + L
)
+ 1
─
16
(
r ij
─
a + L
)
3
]
(11)
and  is osmotic pressure difference between bulk and excluded
volume region, which is adjusted to tune net pair potentials (fig. S1).
For simplicity, the swarm-cargo pair potential is given by an electro-
static repulsion of the same form as the swarm pair potential.
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Simulations were performed using an integration time step of
0.1 ms and using other parameters listed in Table1. The maximum
propulsion speed, vmax, is based on typical experiments (3,4). The
control update time of 0.1 s is based on prior feedback control
experiments (22,24), although it is validated in practice as well.
Multiagent assignment and control policy
An assignment function, g(·|t), was used to represent the target
assignment decision at time, t. For example, let IS = [1, 2, 3] be set of
three swarm particles, and an assignment function g(· | t = 0) at time
0, denoted by g0, be set as g0(1) = 2, g0(2) = 3, and g0(3) = 1 to represent
assigning swarm particles 1, 2, and 3 to targets 2, 3, and 1, respec-
tively. Mathematically, the function g(·|t) is a permutation from the
set of swarm particles, IS, to the set of targets, IS (targets have the
same index set). After obtaining the assignment, g*, using Eq. 1 as
described in the Results and Discussion, the self-propulsion speed for
each swarm particle, vi*, can be formulated using Eq. 2. Solving Eq. 2
is difficult because the probability of future swarm and target po-
sitions rS,i(t + tC) and rT,i(t + tC) are complicated because of parti-
cle interactions and stochastic dynamics. By assuming on short time
scales that swarm particle positions are primarily determined by
propulsion and not influenced by interactions with nearby parti-
cles, the optimization of Eq. 2 can be decomposed into the following
optimization problem for each swarm particle, g*(i|t) ∈ IS, as
v
S,i
* = arg min
0≤ v
S,i
* ≤ v max 〈 ‖ r S,i (t + Δ t C ) − r T,g*(i∣t) (t + Δ t C ) ‖ 2 〉 (12)
where the future position of swarm particle i, rS(t + tC) under
velocity, vS,ini, is given by integration of Eq. 8 from t to t + tC as
r S,i (t +  t C ) = r S,i (t ) +  r
i
B + ∫0
 t C ( v S,i n i ) dt (13)
and on short time scales, tC < < 1/Dr, the probability densities of
rS,i(t + tC) and rT,g*(i|t)(t + tC) are concentrated around their mean
values, and an approximate solution to Eq. 12 is given as
v
S,i
* = arg min
0≤ v
S,i
* ≤ v max ‖〈 r S,i (t + Δ t C ) 〉 − 〈 r T,g*(i∣t) (t + Δ t C ) 〉‖ 2 (14)
where 〈rS,i(t + tC)〉 ≈ rS,i(t) + vS,initC for translation with stochastic
orientations (49) and 〈rT,g*(i|t)(t + tC)〉 = rT,g*(i|t)(t) since the target
is undergoing driftless Brownian translation. With the mean posi-
tion formulated, the solution to Eq. 14 is given by Eq. 3.
Swarm cargo capture algorithm
The cargo capture algorithm (Algorithm 1) was repeated after every
control update time interval, tC. Initially, the target sites, rT, were
constructed on the basis of the current cargo position. Target sites
were positioned as hexagonal lattices with a minimum spacing cor-
responding to a minimum pair potential energy. Target assignments
and self-propulsion speeds were obtained via Eqs. 1 and 3 based on
the current system state (positions and orientations of swarm particles
and target sites). The system state was updated after a control update
time of tC = 0.1 s.
Algorithm 1.
Cargo capture algorithm
1 Loop with frequency tC:
2 Reconstruct the target sites around the cargo.
3 Calculate target assignments and self-propulsion speeds for
swarm particles from Eqs. 1 to 3.
4 Update the position of swarm particles and cargo within tC.
5 End Loop
Swarm cargo transport algorithm
The cargo transport algorithm (Algorithm 2) was repeated after
every control update time interval, tC. Initially, the target sites, rT,
were constructed on the basis of the current cargo position. Target
sites were positioned as hexagonal lattices with a minimum spacing
corresponding to a minimum pair potential energy. A subset of swarm
particles was selected by satisfying the following: (i) an orientation
condition (|i −  | < c, i.e., swarm particle i is oriented within a
tolerance of the transport direction), (ii) a position condition (||rS,i −
rT,g*(i)|| < a, i.e., swarm particle i is positioned within a tolerance
of its assigned target), and (iii) a coordination number condition
[coordination number (i) > nC, i.e., swarm particle i has sufficient
neighbors within a tolerance to promote swarm structural integrity].
The selected subset of swarm particles was assigned maximum self-
propulsion to generate swarm transport. Remaining swarm particles
were assigned targets and speeds according to Algorithm 1. The sys-
tem state was updated after a control update time of tC = 0.1 s.
Parameters used in the cargo transport algorithm include c = 30°,
nC = 3 for N = 90, nC = 4 for N = 60, nC = 5 for N = 36 and 18, and
nC = 2 for N = 6. For pair attraction >8kT for all swarm sizes, nC = 0
to investigate the asymptotic speeds and efficiencies when all swarm
particles are engaged in transporting.
Algorithm 2.
Cargo transport in  direction algorithm
1 Loop with frequency tC:
2 Construct the target sites around the cargo
3 For each swarm particle i:
4 If |i −  | < c and ||rS,i − rT,g*(i|t)|| < a and neighbor(i) > nC
5 Swarm particle i is labeled as transporter.
Table 1. Brownian dynamic simulation parameters.
Parameter Equation Value Parameter Equation Value
a (nm)* 8 1000 Dt (m2/s)†8 2.145 × 10−13
B (a/kT)‡10 2.29 Dr (rad2/s)§8 0.161
−1 (nm)‖10 50 N¶8 6, 18, 36, 60, 90
L#10 200 vmax (m/s)** 8 5 × 10−6
 (10−8kT/nm3)†† 10 0, 3.8, 5.8, 8.8 tC (s)‡‡ 1 and 2 0.1
*Particle radii. †Translational diffusivity. ‡Electrostatic prefactor. §Rotational diffusivity. ‖Debye length. ¶Swarm size.
#Depletant radius. **Maximum propulsion speed. ††Osmotic pressure. ‡‡Control update time.
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6 Set the self-propulsion speed for swarm particle i to be
maximum speed.
7 Else
8 Swarm particle i is labeled as capturer.
9 Calculate self-propulsion speed for swarm particle i from
Eqs. 1 to 3.
10 End For
11 Update the position of swarm particles and cargo within tC.
12 End Loop
SUPPLEMENTARY MATERIALS
Supplementary material for this article is available at http://advances.sciencemag.org/cgi/
content/full/6/4/eaay7679/DC1
Section S1. Nomenclature
Section S2. Supplemental Methods and Results
Fig. S1. Swarm particle pair potentials with different amounts of attraction.
Fig. S2. Examples illustrating the optimal assignment solution.
Fig. S3. Optimal control policy for single particles–single targets, multiple particles–multiple
targets, and cargo capture.
Fig. S4. Osmotic pressure from sedimentation equilibrium and swarm configurations.
Fig. S5. Steady-state density and force distribution during the capture process for N = 60.
Fig. S6. Steady-state density and force distribution during the transport process for N = 60.
Fig. S7. MSD analysis for cargo and swarm particles during steady-state capture.
Fig. S8. MSD analysis for cargo and swarm particles during steady-state transport.
Fig. S9. Steady-state energy consumption rate for swarm particles during cargo transport.
Fig. S10. Example extensions of colloidal swarm functions.
Movie S1. High-quality three-dimensional rendering of cargo capture by N = 60 swarms for
UM = 5.3kT.
Movie S2. Cargo capture by N = 90 swarms for UM = 0kT.
Movie S3. Cargo capture by N = 90 swarms for UM = 3.2kT.
Movie S4. Cargo capture by N = 90 swarms for UM = 5.3kT.
Movie S5. Cargo capture by N = 6 (0kT), 18 (5.3kT), 36 (5.3kT), and 60 (3.2kT) swarms.
Movie S6. Crystalline swarm melting without capture control for N = 18, 36, 60, and 90 and
UM = 5.3kT.
Movie S7. Cargo transport by N = 90 swarms for UM = 0kT.
Movie S8. Cargo transport by N = 90 swarms for UM = 3.2kT.
Movie S9. Cargo transport by N = 90 swarms for UM = 5.3kT.
Movie S10. Cargo capture and transport by N = 36 (3.2kT), 36 (5.3kT), 60 (3.2kT), and 60 (5.3kT)
swarms.
Movie S11. Cargo transport without capture control for N = 18 (8.7kT), 36 (5.3kT), 60 (5.3kT),
and 90 (5.3kT) swarms.
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Acknowledgments
Funding: We acknowledge financial support from the NSF Cyber Enabled Discovery and
Innovation grant (CMMI-1124648). Author contributions: Y.Y. performed and analyzed the
computer experiments. Y.Y. and M.A.B. designed the computer experiments and their analysis,
interpreted the data, and wrote the manuscript. Competing interests: The authors declare
that they have no competing interests. Data and materials availability: All data needed to
evaluate the conclusions in the paper are present in the paper and/or the Supplementary
Materials. Additional data related to this paper may be requested from the authors.
Submitted 17 July 2019
Accepted 20 November 2019
Published 24 January 2020
10.1126/sciadv.aay7679
Citation: Y. Yang, M. A. Bevan, Cargo capture and transport by colloidal swarms. Sci. Adv. 6,
eaay7679 (2020).
on January 25, 2020http://advances.sciencemag.org/Downloaded from

Cargo capture and transport by colloidal swarms
Yuguang Yang and Michael A. Bevan
DOI: 10.1126/sciadv.aay7679
(4), eaay7679.6Sci Adv
ARTICLE TOOLS http://advances.sciencemag.org/content/6/4/eaay7679
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SUPPLEMENTARY http://advances.sciencemag.org/content/suppl/2020/01/17/6.4.eaay7679.DC1
REFERENCES http://advances.sciencemag.org/content/6/4/eaay7679#BIBL
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