Active Particles in Complex and Crowded Environments

8.5 MB·

by  Clemens Bechinger

Active Particles in Complex and Crowded Environments
Clemens Bechinger
(1) 2. Physikalisches Institut,
Universit¨at Stuttgart,
Pfaffenwaldring 57,
70569 Stuttgart,
Germany
(2) Max-Planck-Institut fu¨r Intelligente Systeme,
Heisenbergstraße 3,
70569 Stuttgart,
Germany
6
1 Roberto Di Leonardo
0
(3) Dipartimento di Fisica,
2
Universit`a “Sapienza”, I-00185, Roma,

c Italy
e (4) NANOTEC-CNR Institute of Nanotechnology,
D
Soft and Living Matter Laboratory, I-00185 Roma,
Italy
9
2
Hartmut L¨owen

] (5) Institut fu¨r Theoretische Physik II: Weiche Materie,
t
f Heinrich-Heine-Universit¨at Du¨sseldorf,
o D-40225 Du¨sseldorf,
s Germany
.
t
a Charles Reichhardt
m
(6) Theoretical Division,
-
d Los Alamos National Laboratory, Los Alamos,
n New Mexico 87545 USA
o
c Giorgio Volpe
[
(7) Department of Chemistry,

2 University College London,
v 20 Gordon Street,
1 London WC1H 0AJ,
8 United Kingdom
0
0 Giovanni Volpe∗
0
(8) Department of Physics,
.
2 University of Gothenburg,
0 SE-41296 Gothenburg,
6 Sweden
1 (9) Soft Matter Lab,
: Department of Physics,
v
i and UNAM — National Nanotechnology Research Center,
X Bilkent University, Ankara 06800,
r Turkey
a
(Dated: December 30, 2016)
DifferentlyfrompassiveBrownianparticles,activeparticles,alsoknownasself-propelled
Brownianparticlesormicroswimmersandnanoswimmers,arecapableoftakingupen-
ergy from their environment and converting it into directed motion. Because of this
constant flow of energy, their behavior can only be explained and understood within
the framework of nonequilibrium physics. In the biological realm, many cells perform
directed motion, for example, as a way to browse for nutrients or to avoid toxins. In-
spired by these motile microorganisms, researchers have been developing artificial par-
ticles that feature similar swimming behaviors based on different mechanisms; these
manmade micro- and nanomachines hold a great potential as autonomous agents for
healthcare, sustainability, and security applications. With a focus on the basic physi-
2
calfeaturesoftheinteractionsofself-propelledBrownianparticleswithacrowdedand
complex environment, this comprehensive review will put the reader at the very fore-
frontofthefield,providingaguidedtourthroughitsbasicprinciples,thedevelopment
of artificial self-propelling micro- and nanoparticles, and their application to the study
ofnonequilibriumphenomena,aswellastheopenchallengesthatthefieldiscurrently
facing.
CONTENTS D. Sortingofmicroswimmers 41
1. Staticpatterns 41
I. Introduction 2 2. Chiralparticleseparation 42
II. Non-interactingActiveParticlesinHomogenous VI. TowardstheNanoscale 43
Environments 5
A. Brownianmotionvs. activeBrownianmotion 5 VII. OutlookandFutureDirections 44
B. Phenomenologicalmodels 6
1. ChiralactiveBrownianmotion 7 VIII. Acknowledgments 46
2. Modelsforactiveparticlereorientation 8
3. Non-sphericalactiveparticles 9 References 46
4. Modelingactivemotionwithexternalforcesand
torques 10
5. Numericalconsiderations 11
C. Effectivediffusioncoefficientandeffective I. INTRODUCTION
temperature 11
D. Biologicalmicroswimmers 12
Active matter systems are able to take energy from
E. Artificialmicroswimmers 12
their environment and drive themselves far from equilib-
1. Propulsionbylocalenergyconversion 13
2. Propulsionbyexternalfields 14 rium(Ramaswamy,2010). Thankstothisproperty,they
3. SynthesisofJanusparticles 14 featureaseriesofnovelbehaviorsthatarenotattainable
bymatteratthermalequilibrium,including,forexample,
III. Hydrodynamics 15
A. Microhydrodynamicsofself-propulsion 15 swarming and the emergence of other collective proper-
B. Particle–particlehydrodynamicinteractions 17 ties (Schweitzer, 2007). Their study provides great hope
C. Hydrodynamiccouplingtowalls 19 to uncover new physics and, simultaneously, to lead to
D. Non-Newtonianmedia 20
the development of novel strategies for designing smart
IV. InteractingParticles 22 devices and materials. In recent years, a significant and
A. Classificationofparticleinteractions 22 growing effort has been devoted to advancing this field
1. Aligninginteractions,Vicsekmodel,and
and to explore its applications in a diverse set of dis-
swarming 22
ciplines such as statistical physics (Ramaswamy, 2010),
B. Collectivebehaviorsofactiveparticles 23
1. Clusteringandlivingcrystals 23 biology (Viswanathan et al., 2011), robotics (Brambilla
2. Self-jammingandactivemicrorheology 24 et al., 2013), social transport (Helbing, 2001), soft mat-
3. Activeturbulence 25
ter (Marchetti et al., 2013), and biomedicine (Wang and
C. Mixturesofactiveandpassiveparticles 25
1. Activedoping 25 Gao, 2012).
2. Phaseseparationandturbulentbehavior 26 An important example of active matter is consti-
3. Activebaths 26 tuted by natural and artificial objects capable of self-
4. Directedmotionandgears 27
propulsion. Self-propelled particles were originally
5. Activedepletion 29
6. Flexiblepassiveparticlesandpolymers 31 studied to model swarm behavior of animals at the
macroscale. Reynolds (1987) introduced a ‘Boids model’
V. ComplexEnvironments 32
tosimulatetheaggregatemotionofflocksofbirds,herds
A. Interactionwithawall 32
B. Activeparticlesinaconfinedgeometry 33 of land animals, and schools of fish within computer
1. Non-Boltzmannpositiondistributionsforactive graphics applications; Vicsek et al. (1995) then intro-
particles 33 duced his namesake model as a special case. In the Vic-
2. Activematterforcesandequationofstate 35
sek model, a swarm is modeled by a collection of self-
3. Collectivebehaviorsinconfinedgeometries 35
C. Interactionwithobstacles 35 propellingparticlesthat move withaconstantspeedbut
1. Captureandconcentrationofactiveparticles 35 tend to align with the average direction of motion of the
2. Ratcheteffectsanddirectedmotion 37 particlesintheirlocalneighborhood(Chat´eetal.,2008a;
3. Motionrectificationinamicrochannel 38
Czir´okandVicsek,2000). Swarmingsystemsgiveriseto
4. Extendedlandscapesofobstacles 38
5. Subdiffusionandtrappingofmicroswimmers 40 emergentbehaviors,whichoccuratmanydifferentscales;
furthermore, some of these behaviors are turning out to
be robust and universal, e.g. they are independent of
the type of animals constituting the swarm (Buhl et al.,
∗ [e-mail] 2006). It has in fact become a challenge for theoretical
3
100000
Biological Swimmers
a
10000
h
spermatozoa
E.coli
) 1000 Artificial Swimmers
−1 m
s T.majus
m
µ b: Janus rods
(
d
e k
spe 100 q p g, j: Janus spheres
s spermatozoa
r
o
f
l E.coli f: chiral particles
n i
10
g,j b
c
l: vesicles
d e
1
0.1 1 10 100 1000 10000
size(µm)
FIG. 1 (Color online) Self-propelled Brownian particles are biological or manmade objects capable of taking up energy from
their environment and converting it into directed motion; they are micro- and nanoscopic in size and have propulsion speeds
(typically) up to a fraction of a millimeter per second. The letters correspond to the artificial microswimmers in Table I. The
insets show examples of biological and artificial swimmers. For the artificial swimmers four main recurrent geometries can be
identified so far: Janus rods, Janus spheres, chiral particles, and vesicles.
physics to find minimal statistical models that capture Brownian particles are provided in Fig. 1 and Table I.
these features (Bertin et al., 2009; Li et al., 2008; Toner While the motion of passive Brownian particles is
et al., 2005). driven by equilibrium thermal fluctuations due to ran-
Self-propelled Brownian particles, in particular, have dom collisions with the surrounding fluid molecules
come under the spotlight of the physical and biophys- (Babiˇc et al., 2005), self-propelled Brownian particles
ical research communities. These active particles are exhibit an interplay between random fluctuations and
biological or manmade microscopic and nanoscopic ob- active swimming that drives them into a far-from-
jects that can propel themselves by taking up energy equilibrium state (Erdmann et al., 2000; H¨anggi and
from their environment and converting it into directed Marchesoni, 2009; Hauser and Schimansky-Geier, 2015;
motion (Ebbens and Howse, 2010). On the one hand, Schweitzer, 2007). Thus, their behavior can only be ex-
self-propulsion is a common feature in microorganisms plainedandunderstoodwithintheframeworkofnonequi-
(Cates, 2012; Lauga and Powers, 2009; Poon, 2013) and librium physics (Cates, 2012), for which they provide
allows for a more efficient exploration of the environ- ideal model systems.
ment when looking for nutrients or running away from From a more applied perspective, active particles pro-
toxic substances (Viswanathan et al., 2011); a paradig- vide great hope to address some challenges that our
maticexampleistheswimmingbehaviorofbacteriasuch society is currently facing — in particular, personal-
as Escherichia coli (Berg, 2004). On the other hand, ized healthcare, environmental sustainability, and secu-
tremendousprogresshasrecentlybeenmadetowardsthe rity (Abdelmohsen et al., 2014; Ebbens, 2016; Gao and
fabrication of artificial micro- and nanoswimmers that Wang,2014;Nelsonet al.,2010;Patraet al.,2013;Wang
canself-propelbasedondifferentpropulsionmechanisms; andGao,2012). Thesepotentialapplicationscanbebuilt
some characteristic examples of artificial self-propelled around the core functionalities of self-propelled Brown-
4
TABLEI Examplesofexperimentallyrealizedartificialmicroswimmersandrelativepropulsionmechanisms. Thelettersinthe
first column correspond to the examples plotted in Fig. 1.
Microswimmer Propulsion mechanism Medium Dimensions Max. Speed
a PDMS platelets coated with Pt Bubbles generated in a H O H O aqueous 1cm 2cms−1
2 2 2 2
(Ismagilov et al., 2002) aqueous solution by asymmetric meniscus
patterns of Pt
b Rod-shaped particles consisting of Catalysis of oxygen at the Pt end of Near a boundary 2µm 10µms−1
Au and Pt segments (Paxton et al., the rod in H O acqueous (length),
2 2
2004) solution 370nm
(width)
c Linear chains of DNA-linked External actuation of the flexible Aqueous solution 30µm 6µms−1
magnetic colloidal particles attached artificial flagella by oscillating
to red blood cells (Dreyfus et al., magnetic fields
2005)
d Janus spherical particles with a Self-diffusiophoresis catalyzed by a H O aqueous 1.6µm 3µms−1
2 2
catalytic Pt patch (Howse et al., chemical reaction on the Pt surface solution
2007)
e DNA-linked anisotropic doublets Rotation induced by a rotating Near a boundary 3µm 3.2µms−1
composed of paramagnetic colloidal magnetic field in aqueous
particles (Tierno et al., 2008) solution
f Chiral colloidal propellers (Ghosh External actuation by a magnetic Aqueous solution 2µm 40µms−1
and Fischer, 2009) field (length),
250nm
(width)
g Janus particles half-coated with Au Self-thermophoresis due to local Aqueous solution 1µm 10µms−1
(Jiang et al., 2010) heating at the Au cap
h Catalytic microjets (Sanchez et al., H O catalysis on the internal H O aqueous 50µm 10mms−1
2 2 2 2
2011) surface of the microjet solution (length),
1µm (width)
i Water droplets containing bromine Marangoni flow induced by a Oil phase 80µm 15µms−1
(Thutupalli et al., 2011) self-sustained bromination gradient containing a
along the drop surface surfactant
j Janus particles with light-absorbing Local demixing of a critical mixture Critical mixture 0.1 to 10µm 10µms−1
patches (Buttinoni et al., 2012; due to heating associated to (e.g.
Ku¨mmel et al., 2013; Volpe et al., localized absorption of light water-2,6-lutidine)
2011)
k Rod-shaped particles consisting of Self-acoustophoresis in a ultrasonic Aqueous solution 1 to 3µm 200µms−1
Au and Pt (or Au and Ru) standing wave (length),
segments (Wang et al., 2012) 300nm
(width)
l Pt-loaded stomatocytes (Wilson Bubbles generated in a H O H O acqueous 0.5µm 23µms−1
2 2 2 2
et al., 2012) aqueous solution by entrapped Pt solution
nanoparticles
m Colloidal rollers made of PMMA Spontaneous charge symmetry conducting fluid 5µm 1mms−1
beads (Bricard et al., 2013) breaking resulting in a net (hexadecane
electrostatic torque solution)
n Polymeric spheres encapsulating Self-phoretic motion near a Near a boundary 1.5µm 15µms−1
most of an antiferromagnetic boundary due to the decomposition in H O acqueous
2 2
hematite cube (Palacci et al., 2013) of H O by the hematite cube when solution
2 2
illuminated by ultraviolet light
o Water droplets (Izri et al., 2014) Water solubilization by the reverse Oil phase with 60µm 50µms−1
micellar solution surfactants above
the critical
micellar solution
p Janus microspheres with Mg core, Bubble thrust generated from the Aqueous solution 20µm 110µms−1
Au nanoparticles, and TiO shell Mg-water reaction
2
layer (Li et al., 2014a)
q Hollow mesoporous silica Janus Catalysis powered by Pt or by three Aqueous solution 50 to 500nm 100µms−1
particles (Ma et al., 2015a,b) different enzymes (catalase, urease,
and glucose oxidase)
r Janus particles half-coated with Cr AC electric field Aqueous solution 3µm 60µms−1
(Nishiguchi and Sano, 2015)
s Enzyme-loaded polymeric vesicles Glucose catalysis powered by Aqueous solution 0.1µm 80µms−1
(Joseph et al., 2016) catalase and glucose oxidase
5
ian particles, i.e. transport, sensing, and manipulation. Brownian motion (Section II.A) and serve as a starting
Infact, thesemicro-andnanomachinesholdthepromise pointtodiscussthebasicmathematicalmodelsforactive
ofperformingkeytasksinanautonomous, targeted, and motion (Section II.B). We will then introduce the con-
selective way. The possibility of designing, using, and cepts of effective diffusion coefficient and effective tem-
controllingmicro-andnanoswimmersinrealisticsettings perature for self-propelled Brownian particles, as well
of operation is tantalizing as a way to localize, pick-up, as their limitations, i.e. differences between systems at
and deliver nanoscopic cargoes in several applications equilibrium at a higher temperature and systems out
— from the targeted delivery of drugs, biomarkers, or of equilibrium (Section II.C). We will then briefly re-
contrastagentsinhealthcareapplications(Abdelmohsen viewbiologicalmicroswimmers(SectionII.D).Finally,we
et al., 2014; Nelson et al., 2010; Patra et al., 2013; Wang will conclude with an overview of experimental achieve-
and Gao, 2012) to the autonomous depollution of water mentsconnectedtotherealizationofartificialmicro-and
and soils contaminated because of bad waste manage- nanoswimmersincludingadiscussionoftheprincipalex-
ment, climate changes, or chemical terroristic attacks in perimentalapproachesthathavebeenproposedsofarto
sustainability and security applications (Gao and Wang, build active particles (Section II.E).
2014).
The field of active matter is now confronted with var-
ious open challenges that will keep researchers busy for A. Brownian motion vs. active Brownian motion
decades to come. First, there is a need to understand
In order to start acquiring some basic understanding
how living and inanimate active matter systems develop
of the differences between passive and active Brownian
socialand(possibly)tunablecollectivebehaviorsthatare
motion, a good (and pedagogic) approach is to compare
not attainable by their counterparts at thermal equilib-
two-dimensional trajectories of single spherical passive
rium. Then, there is a need to understand the dynamics
and active particles of equal (hydrodynamic) radius R
ofactiveparticlesinreal-lifeenvironments(e.g. inliving
in a homogenous environment, i.e. where no physical
tissues and porous soils), where randomness, patchiness,
barriers or other particles are present and where there is
and crowding can either limit or enhance how biological
a homogeneous and constant distribution of the energy
and artificial microswimmers perform a given task, such
source for the active particle.
as finding nutrients or delivering a nanoscopic cargo. Fi-
The motion of a passive Brownian particle is purely
nally,thereisstillastrongneedtoeffectivelyscaledown
diffusive with translational diffusion coefficient
tothenanoscaleourcurrentunderstandingofactivemat-
ter systems. k T
D = B , (1)
With this review, we provide a guided tour through T 6πηR
the basic principles of self-propulsion at the micro- and
nanoscale, the development of artificial self-propelling wherekB istheBoltzmannconstant,T theabsolutetem-
micro- and nanoparticles, and their application to the perature, and η the fluid viscosity. The particle also
study of far-from-equilibrium phenomena, as well as undergoes rotational diffusion with a characteristic time
through the open challenges that the field is now facing. scale τR given by the inverse of the particle’s rotational
diffusion coefficient
k T
II. NON-INTERACTING ACTIVE PARTICLES IN DR =τR−1 = 8πBηR3 . (2)
HOMOGENOUS ENVIRONMENTS
Fromtheaboveformulas,itisclearthat,whilethetrans-
Before proceeding to analyze the behavior of active
lational diffusion of a particle scales with its linear di-
particles in crowded and complex environments, we will
mension, its rotational diffusion scales with its volume.
set the stage by considering the simpler (and more fun-
For example, for a particle with R ≈ 1µm in water,
damental) case of individual active particles in homo- D ≈ 0.2µm2s−1 and D ≈ 0.17rad2s−1 (τ ≈ 6s),
T R R
geneous environments, i.e. without obstacles or other
while, for a particle 10 times smaller (R ≈ 100nm),
particles. We will first introduce a simple model of an D ≈ 2µm2s−1 is one order of magnitude larger but
T
active Brownian particle,1 which will permit us to un- D ≈ 170rad2s−1 is three orders of magnitude larger
R
derstandthemaindifferencesbetweenpassiveandactive
(τ ≈6ms).
R
In a homogeneous environment, the translational and
rotational motions are independent from each other.
1 Weremarkthattheterm“activeBrownianparticle”hasmainly
beenusedintheliteraturetodenotethespecific,simplifiedmodel
of active matter described in this section, which consists of re-
pulsive spherical particles that are driven by a constant force tothisspecificmodelanditsstraightforwardgeneralizations(see
whose direction rotates by thermal diffusion. In the following, Section II.B.1), while we will use the terms “active particle” or
we will use the term “active Brownian particle” when we refer “self-propelledparticle”whenwerefertomoregeneralsystems.
6
FIG. 2 (Color online) Active Brownian particles in two dimensions. (a) An active Brownian particle in water (R = 1µm,
η = 0.001Pas) placed at position [x,y] is characterized by an orientation ϕ along which it propels itself with speed v while
undergoing Brownian motion in both position and orientation. The resulting trajectories are shown for different velocities (b)
v = 0µms−1 (Brownian particle), (c) v = 1µms−1, (d) v = 2µms−1, and (e) v = 3µms−1; with increasing values of v, the
active particles move over longer distances before their direction of motion is randomized; four different 10-s trajectories are
shown for each value of velocity.
Therefore, the stochastic equations of motion for a pas- i.e. (cid:104)x(t)(cid:105)=(cid:104)y(t)(cid:105)≡0, where (cid:104)...(cid:105) represents the ensem-
sive Brownian particle in a two-dimensional space are bleaverage. Foranactiveparticleinstead,theaverageis
a straight line along the x-direction (determined by the
 (cid:112)
 x˙ = 2DTξx prescribed initial orientation),
(cid:112)
y˙ = 2DTξy (3) v (cid:20) (cid:18) t (cid:19)(cid:21)
ϕ˙ = (cid:112)2D ξ (cid:104)x(t)(cid:105)= DR [1−exp(−DRt)]=vτR 1−exp −τR ,
R ϕ
(5)
where [x,y] is the particle position, ϕ is its orientation while(cid:104)y(t)(cid:105)≡0becauseofsymmetry. Thisimpliesthat,
(Fig.2a),andξ ,ξ ,andξ representindependentwhite on average, an active Brownian particle will move along
x y ϕ
noisestochasticprocesseswithzeromeanandcorrelation the direction of its initial orientation for a finite persis-
δ(t). Interestingly, the equations for each degree of free- tence length
dom (i.e. x, y, and ϕ) are decoupled. Inertial effects v
L= =vτ , (6)
are neglected because microscopic particles are typically D R
R
in a low-Reynolds-number regime (Purcell, 1977). Some
examplesofthecorrespondingtrajectoriesareillustrated before its direction is randomized.
in Fig. 2b. The relative importance of directed motion versus dif-
Foraself-propelledparticle withvelocityvinstead,the fusion for an active Brownian particle can be character-
directionofmotionisitselfsubjecttorotationaldiffusion, ized by its P´eclet number
which leads to a coupling between rotation and transla-
v
tion. The corresponding stochastic differential equations Pe∝ √ , (7)
D D
are T R
 (cid:112) where the proportionality sign is used because the lit-
 x˙ = vcosϕ+ 2DTξx erature is inconsistent about the value of the numerical
y˙ = vsinϕ+(cid:112)2D ξ (4) prefactor. If Pe is small, diffusion is important, while, if
T y
ϕ˙ = (cid:112)2D ξ PeTishelamrgoed,edlirfeocrteadctmivoetiBonropwrneivaanilsm. otion described by
R ϕ
Eqs. (4) can be straightforwardly generalized to the case
Some examples of trajectories for various v are shown
of an active particle moving in three dimensions. In this
in Figs. 2c, 2d, and 2e: as v increases, we obtain active
case, the particle position is described by three Carte-
trajectories that are characterized by directed motion at
sian coordinates, i.e. [x,y,z], and its orientation by the
short time scales; however, over long time scales the ori-
polar and azimuthal angles, i.e. [ϑ,ϕ], which perform
entation and direction of motion of the particle are ran-
a Brownian motion on the unit sphere (Carlsson et al.,
domized by its rotational diffusion (Howse et al., 2007).
2010).
To emphasize the difference between Brownian mo-
tion and active Brownian motion, it is instructive to
consider the average particle trajectory given the ini- B. Phenomenological models
tial position and orientation fixed at time t = 0, i.e.
x(0) = y(0) = 0 and ϕ(0) = 0. In the case of passive In this section, we will extend the simple model in-
Brownian motion, this average vanishes by symmetry, troduced in Section II.A to describe the motion of more
7
complex (and realistic) active Brownian particles. First, chirality (or helicity) to the path, the sign of which de-
we will introduce models that account for chiral active termineswhetherthemotionisclockwise(dextrogyre)or
Brownianmotion(SectionII.B.1). Wewillthenconsider anti-clockwise (levogyre). The result is a motion along
more general models of active Brownian motion where circulartrajectoriesintwodimensions(circleswimming)
reorientation occurs due to mechanisms other than ro- andalonghelicaltrajectoriesinthreedimensions(helical
tational diffusion (Section II.B.2) and where the active swimming).
particles are non-spherical (Section II.B.3). Finally, we Theoccurrenceofmicroorganismsswimmingincircles
will discuss the use of external forces and torques when was pointed out more than a century ago by Jennings
modeling active Brownian motion (Section II.B.4) and (1901) and, since then, has been observed in many dif-
wewillprovidesomeconsiderationsaboutnumerics(Sec- ferent situations, in particular close to a substrate for
tion II.B.5). bacteria (Berg and Turner, 1990; DiLuzio et al., 2005;
Before proceeding further, we remark that in this sec- Hill et al., 2007; Lauga et al., 2006; Schmidt et al.,
tion the microscopic swimming mechanism is completely 2008; Shenoy et al., 2007) and spermatozoa (Friedrich
ignored; inparticularhydrodynamicinteractionsaredis- and Ju¨licher, 2008; Riedel et al., 2005; Woolley, 2003).
regarded and only the observable effects of net motion Likewise, helical swimming in three dimensions has been
are considered. While the models introduced here are observed for various bacteria and sperm cells (Brokaw,
phenomenological, they are very effective to describe the 1958,1959;Corkidiet al.,2008;Crenshaw,1996;Fenchel
motion of microswimmers in homogenous environments. and Blackburn, 1999; J´ekely et al., 2008; Jennings, 1901;
Wecancastthispointintermsofthedifferencebetween McHenry and Strother, 2003; Thar and Fenchel, 2001).
“microswimmers” and “active particles”. Microswim- Figures 3a and 3b show examples of E. coli cells swim-
mers are force-free and torque-free objects capable of ming in circular trajectories near a glass surface and at
self-propulsion in a (typically) viscous environment and, a liquid–air interface, respectively. Examples of non-
importantly, exhibit an explicit hydrodynamic coupling
with the embedding solvent via flow fields generated by
the swimming strokes they perform. Instead, active par- (a) (b)
ticlesrepresentamuchsimplerconceptconsistingofself-
propelled particles in an inert solvent, which only pro- 20µm
videshydrodynamicfrictionandastochasticmomentum
transfer. While the observable behavior of the two is the
same in a homogenous environment and in the absence
of interactions between particles, hydrodynamic interac-
tions may play a major role in the presence of obstacles
or other microswimmers. The simpler model of active
particles delivers, however, good results in terms of the
particle’sbehaviorandismoreintuitive. Infact,theself-
propulsion of an active Brownian particle is implicitly
(c) (d)
modeledbyusinganeffectiveforcefixedintheparticle’s
5µm 5µm
body frame. For this reason, in this review we typically
consider active particles, while we will discuss hydrody-
namic interactions in Section III (see also Golestanian
et al. (2010), Marchetti et al. (2013) and Elgeti et al.
(2015) for extensive reviews on the role of hydrodynam-
Aucoating 10µm Aucoating 10µm
ics in active matter systems). We will provide a more
detailed theoretical justification of why active particles
FIG. 3 (Color online) Biological and artificial chiral active
are a good model for microswimmers in Section II.B.4.
Brownian motion. (a) Phase-contrast video microscopy im-
ages showing E. coli cells swimming in circular trajectories
nearaglasssurface. Superpositionof8-svideoimages. From
1. Chiral active Brownian motion Laugaetal.(2006). (b)Circulartrajectoriesarealsoobserved
for E. coli bacteria swimming over liquid–air interfaces but
thedirectionisreversed. FromDiLeonardoet al.(2011). (c-
Swimmingalongastraightline—correspondingtothe
d) Trajectories of a (c) dextrogyre and (d) levogyre artificial
linearly directed Brownian motion considered until now
microswimmer driven by self-diffusiophoresis: in each plot,
— is the exception rather than the rule. In fact, ideal the red bullet correspond to the initial particle position and
straightswimmingonlyoccursiftheleft–rightsymmetry the two blue squares to its position after 1 and 2 minutes.
relativetotheinternalpropulsiondirectionisnotbroken; Theinsetsshowmicroscopeimagesoftwodifferentswimmers
evensmalldeviationsfromthissymmetrydestabilizeany with the Au coating (not visible in the bright-field image)
indicated by an arrow. From Ku¨mmel et al. (2013).
straight motion and make it chiral. One can assign a
8
(a) v (b) (c) (d)
5µm
FIG.4 (Coloronline)ChiralactiveBrownianmotionintwodimensions. (a)Atwo-dimensionalchiralactiveBrownianparticle
hasadeterministicangularvelocityωthat,iftheparticle’sspeedv>0,entailsarotationaroundaneffectiveexternalaxis. (b-
d)Sampletrajectoriesofdextrogyre(red/darkgray)andlevogyre(yellow/lightgray)activechiralparticleswithv=30µms−1,
ω=10rads−1, and different radii (R=1000, 500 and 250nm for (b), (c) and (d), respectively); as the particle size decreases,
the trajectories become less deterministic because the rotational diffusion, responsible for the reorientation of the particle
direction, scales according to R−3 (Eq. (2)).
living but active particles moving in circles are spherical and L¨owen, 2008; Volpe et al., 2014a)
camphors at an air–water interface (Nakata et al., 1997)
 (cid:112)
a(Kndu¨mchmirealle(tLa-ls.h,2a0p1e3d)).cFoilnloaildlya,ltsrwaijmecmtoerrisesoonfadesfuobrmstarabtlee  x˙ = vcosϕ+ 2DTξx
(cid:112)
y˙ = vsinϕ+ 2D ξ (8)
active particles (Ohta and Ohkuma, 2009) and even of T y
completely blinded and ear-plugged pedestrians (Obata ϕ˙ = ω+(cid:112)2D ξ
R ϕ
et al., 2005) can possess significant circular characteris-
tics. Some examples of trajectories are shown in Figs. 4b, 4c,
and 4d for particles of decreasing radius. As the particle
Theoriginofchiralmotioncanbemanifold. Inpartic- size decreases, the trajectories become less determinis-
ular,itcanbeduetoananisotropyintheparticleshape, tic because the rotational diffusion, responsible for the
which leads to a translation–rotation coupling in the hy- reorientation of the particle direction, scales according
drodynamic sense (Kraft et al., 2013), or an anisotropy to R−3 (Eq. (2)). The model given by Eqs. (8) can be
inthepropulsionmechanism;Ku¨mmeletal.(2013)stud- straightforwardly extended to the helicoidal motion of a
ied experimentally an example where both mechanisms three-dimensional chiral active particle following an ap-
are simultaneously present (Figs. 3c and 3d). Further- proach along the lines of the discussion at the end of
more, even a cluster of non-chiral swimmers, which stick Section II.A.
togetherbydirectforces(Redneretal.,2013a),byhydro- Itisinterestingtoconsiderhowthenoise-averagedtra-
dynamics, or just by the activity itself (Buttinoni et al., jectory given in Eq. (5) changes in the presence of chiral
2013; Palacci et al., 2013), will in general lead to situa- motion. In this case, the noise-averaged trajectory has
tions of total non-vanishing torque on the cluster center the shape of a logarithmic spiral, i.e. a spira mirabilis
(Kaiser et al., 2015b), thus leading to circling clusters (van Teeffelen and L¨owen, 2008), which in polar coordi-
(Schwarz-Linek et al., 2012). Finally, the particle rota- nates is written as
tion can be induced by external fields; a standard ex-
ρ∝exp[−D (ϕ−ϕ(0))/ω] , (9)
ample is a magnetic field perpendicular to the plane of R
motionexertingatorqueontheparticles(C¯ebers,2011).
where ρ is the radial coordinate and ϕ is the azimuthal
We remark that, even though the emergence of circular
coordinate. In three dimensions, the noise-averaged tra-
motion can be often attributed to hydrodynamic effects,
jectoryisaconcho-spiral(WittkowskiandL¨owen,2012),
in this section we will focus on a phenomenological de-
which is the generalization of the logarithmic spiral.
scriptionandleavetheproperhydrodynamicdescription
Stochatic helical swimming was recently investigated in
to Section III.C.
Colonial Choanoflagellates (Kirkegaard et al., 2016).
For a two-dimensional chiral active Brownian particle
(Fig. 4a), in addition to the random diffusion and the 2. Models for active particle reorientation
internal self-propulsion modeledby Eqs.(4), the particle
orientation ϕ also rotates with angular velocity ω, where The simple models presented so far, and in particu-
thesignofω determinesthechiralityofthemotion. The lar the one discussed in Section II.A, consider an ac-
resulting set of equations describing this motion in two tive particle whose velocity is constant in modulus and
dimensions is (Mijalkov and Volpe, 2013; van Teeffelen whose orientation undergoes free diffusion. This type of
9
FIG.5 (Coloronline)SampletrajectoriesofactiveBrownianparticlescorrespondingtodifferentmechanismsgeneratingactive
motion: (a)rotationaldiffusiondynamics;(b)run-and-tumbledynamics;and(c)Gaussiannoisedynamics. Thedotscorrespond
to the particle position sampled every 5s.
dynamics — to which we shall refer as rotational diffu- tions of motion of the active particle contain no stochas-
sion dynamics (Fig. 5a) — is often encountered in the tic terms and the particle keeps on moving ballistically
case of self-propelling Janus colloids (Buttinoni et al., along straight lines until it interacts with some obstacles
2012; Howse et al., 2007; Palacci et al., 2013). There or other particles. Such a limit is reached, e.g., for suffi-
are, however, other processes that generate active Brow- ciently large active colloids or for active colloids moving
nianmotion;herewewillconsider,inparticular,therun- through an extremely viscous fluid.
and-tumble dynamics and the Gaussian noise dynamics
(Koumakis et al., 2014). More general models include
also velocity- and space-dependent friction (Babel et al., 3. Non-spherical active particles
2014; Romanczuk et al., 2012; Taktikos et al., 2011). It
The models presented until now — in particular,
hasalsobeenrecentlyspeculatedthatfinite-timecorrela-
Eqs. (4) and (8) — are valid for spherical active par-
tions in the orientational dynamics can affect the swim-
ticles. However, while most active particles considered
mer’s diffusivity (Ghosh et al., 2015).
in experiments and simulations are spherically or axi-
The run-and-tumble dynamics (Fig. 5b) were intro-
ally symmetric, many bacteria and motile microorgan-
duced to describe the motion of E. coli bacteria (Berg,
isms considerably deviate from such ideal shapes, and
2004; Berg and Turner, 1979; Schnitzer et al., 1990).
this strongly alters their swimming properties.
They consist of a random walk that alternates linear
Inordertounderstandhowwecanderivetheequations
straight runs at constant speed with Poisson-distributed
of motion for non-spherical active Brownian particles, it
reorientation events called tumbles. Even though their
is useful to rewrite in a vectorial form the model pre-
microscopic (short-time) dynamics are different, their
sented in Section II.A for the simpler case of a spherical
long-time diffusion properties are equivalent to those
active particle:
of the rotational diffusion dynamics described in Sec-
tion II.A (Cates and Tailleur, 2013; Solon et al., 2015a;
γr˙ =Feˆ+ξ , (10)
Tailleur and Cates, 2008).
In the Gaussian noise dynamics (Fig. 5c), the active where γ = 6πηR denotes the particle’s Stokes friction
particle velocity (along each direction) fluctuates as an coefficient(forasphereofradiusRwithstickyboundary
Ornstein–Uhlenbeck process (Uhlenbeck and Ornstein, conditionsattheparticlesurface),risitspositionvector,
1930). Thisis,forexample,agoodmodelforthemotion F is an effective force acting on the particle, eˆ is its ori-
of colloidal particles in a bacterial bath, where multiple entation unit vector, and ξ is a random vector with zero
interactions with the motile bacteria tend to gradually meanandcorrelation2k TγIδ(t),whereIistheidentity
B
change the direction and amplitude of the particle’s ve- matrix in the appropriate number of dimensions. If the
locity, atleastaslongastheconcentrationisnotsohigh particle’s orientation does not change, i.e. eˆ(t) ≡ eˆ(0)
to give rise to collective phenomena (Wu and Libchaber, (e.g. being fixed by an external aligning magnetic field),
2000). the particle swims with a self-propulsion speed v =F/γ
Finally,wecanalsoconsidertheinterestinglimitofthe alongitsorientationeˆanditstrajectoryistriviallygiven
rotational diffusion dynamics when the rotational diffu- by r(t) = r(0)+vteˆ(0). If the particle orientation can
sioniszero, orsimilarlyintherun-and-tumbledynamics insteadchange,e.g. ifeˆissubjecttorotationaldiffusion,
when the run time is infinite. In this case, the equa- the particle will perform active Brownian motion.
10
We can now generalize these simple considerations for (2012),Bialk´eetal.(2012),Redneretal.(2013b),Reich-
a spherical particle to more complex shapes, as system- hardt and Olson-Reichhardt (2013a), Elgeti and Gomp-
atically discussed in ten Hagen et al. (2015). When the per(2013),MijalkovandVolpe(2013),Filyet al.(2014),
particlehasarigidanisotropicshape,theresultingequa- Costanzo et al. (2014) and Wang et al. (2014).
tions of motion can be written in compact form as These simple considerations for a spherical active par-
ticlecanbegeneralizedtomorecomplexsituations,such
H·V=K+χ, (11)
as to Eqs. (11) for non-spherical active particles. In gen-
where H is the grand resistance matrix or hydrodynamic eral, the following considerations hold to decide whether
a model based on effective forces and torques can be ap-
friction tensor (see also Section III.A) (Fernandes and
plied safely. On the one hand, the effective forces and
delaTorre,2002;HappelandBrenner,1991),V=[v,ω]
torques can be used if we consider a single particle in
isageneralizedvelocitywithvandωtheparticle’strans-
an unbounded fluid whose propulsion speed is a generic
lational and angular velocities, K = [F,T] is a general-
explicit function of time (Babel et al., 2014) or of the
ized force with F and T the effective force and torque
particle’s position (Magiera and Brendel, 2015). On the
actingontheparticle,andχisarandomvectorwithcor-
relation2k T Hδ(t). Equation(11)isbestunderstoodin otherhand,bodyforcesandtorquesarising,e.g.,froman
B
the body frame of the moving particle where H, K, and externalfieldorfrom(non-hydrodynamic)particleinter-
actions can simply be added to the effective forces and
V are constant, but it can also be transformed to the
torques, under the sole assumption that the presence of
laboratory frame (Wittkowski and L¨owen, 2012). In the
the body forces and torques should not affect the self-
deterministic limit (i.e. χ = 0), the particle trajectories
propulsion mechanism itself; a classical counter-example
are straight lines if ω = 0, and circles in two dimen-
tothisassumptionarebimetallicnanorodsdrivenbyelec-
sions (or helices in three dimensions) if ω (cid:54)=0 (Friedrich
trophoresis in an external electric field (Paxton et al.,
and Ju¨licher, 2009; Wittkowski and L¨owen, 2012). In
2006, 2004), as the external electric field perturbs the
the opposite limit when K=0, we recover the case of a
transport of ions through the rod and the screening
freeBrownianparticle,whichhoweverfeaturesnontrivial
around it, and thus significantly affects its propagation
dynamical correlations (Cichocki et al., 2015; Fernandes
(Brown and Poon, 2014).
anddelaTorre,2002;Kraftetal.,2013;MakinoandDoi,
Inordertoavoidapotentialconfusion,weremarkthat
2004).
theuseofeffectiveforcesdoesnotimplythatthesolvent
flow field is modeled correctly; contrarily, the flow field
4. Modeling active motion with external forces and torques is not considered at all. When the propagation is gener-
ated by a non-reciprocal mechanical motion of different
Equation (10) describes the motion of a spherical ac- parts of the swimmer, any internal motion should fulfill
tive particle using an effective “internal” force F = γv Newton’s third law such that the total force acting on
fixed in the particle’s body frame. F is identical to the swimmer is zero at any time. As we will see more in
the force acting on a hypothetical spring whose ends detailinSectionIII,thisimpliesthatthesolventvelocity
are bound to the microswimmers and to the laboratory field u(r) around a swimmer does not decay as a force
(Takatori et al., 2014); hence, it can be directly mea- monopole(i.e. u(r)∝1/r,asiftheparticleweredragged
sured, at least in principle. While this force can be by a constant external force field), but (much faster) as
viewed as a special force field F(r,eˆ) = Feˆ experienced a force dipole (i.e. u(r) ∝ 1/r2). The notion of an ef-
by the particle, it is clearly non-conservative, i.e. it can- fective internal force, therefore, seems to contradict this
not be expressed as a spatial gradient of a scalar poten- generalstatementthatthemotionofaswimmerisforce-
tial. The advantage in modeling self-propulsion by an free. The solution of this apparent contradiction is that
effective driving force is that this force can be straight- the modeling via an effective internal force does not re-
forwardly added to all other existing forces, e.g. body solvethesolventvelocityfieldbutisjustacoarse-grained
forces from real external fields (like gravity or confine- effective description for swimming with a constant speed
ment), forces stemming from the interaction with other along the particle trajectory. Therefore, the concept of
particles, and random forces mimicking the random col- effective forces and torques is of limited utility when the
lisions with the solvent. This keeps the model simple, solventflowfield,whichisgeneratedbytheself-propelled
flexible and transparent. This approach has been fol- particles,hastobetakenintoaccountexplicitly. Thisap-
lowed by many recent works; see, e.g., Chen and Leung plies,forexample,tothefarfieldofthesolventflowthat
(2006), Peruani et al. (2006), Li et al. (2008), van Teef- governs the dynamics of a particle pair (and discrimi-
felen and L¨owen (2008), Mehandia and Prabhu (2008), nates between pullers and pushers (Downton and Stark,
Wensink and L¨owen (2008), ten Hagen et al. (2011), 2009)),tothehydrodynamicinteractionbetweenaparti-
Angelani et al. (2011a), Kaiser et al. (2012), Kaiser cleandanobstacle(Chilukurietal.,2014;Kreuteretal.,
et al. (2013), Yang et al. (2012), Wittkowski and L¨owen 2013; Sipos et al., 2015; Takagi et al., 2014), and to the
(2012), Wensink and L¨owen (2012), McCandlish et al. complicated many-body hydrodynamic interactions in a