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Artificial Chemotaxis of Self-Phoretic Active Colloids:
CollectiveBehaviorPublished as part of the Accounts of Chemical
Research special issue “Fundamental Aspects of Self-PoweredNano-
and Micromotors”.
Holger Stark*
Technische Universitaẗ Berlin, Institute of Theoretical
Physics, Hardenbergstrasse 36, D-10623 Berlin, Germany
CONSPECTUS: Microorganisms use chemotaxis, regulated by internal
complexchemical pathways, to swim along chemical gradients to find
better living conditions.Artificial microswimmers can mimic such a
strategy by a pure physical process calleddiffusiophoresis, where
they drift and orient along the gradient in a chemical density
field.Similarly, for other forms of taxis in nature such as photo-
or thermotaxis the phoreticcounterpart exists.In this Account, we
concentrate on the chemotaxis of self-phoretic active colloids.
Theyare driven by self-electro- and diffusiophoresis at the
particle surface and thereby acquire aswimming speed. During this
process, they also produce nonuniform chemical fields intheir
surroundings through which they interact with other colloids by
translational androtational diffusiophoresis. In combination with
active motion, this gives rise to effectivephoretic attraction and
repulsion and thereby to diverse emergent collective behavior.
Aparticular appealing example is dynamic clustering in dilute
suspensions first reported by agroup from Lyon. A subtle balance of
attraction and repulsion causes very dynamicclusters, which form
and resolve again. This is in stark contrast to the relatively
static clusters of motility-induced phaseseparation at larger
densities.To treat chemotaxis in active colloids confined to a
plane, we formulate two Langevin equations for position and
orientation,which include translational and rotational
diffusiophoretic drift velocities. The colloids are chemical sinks
and develop theirlong-range chemical profiles instantaneously. For
dense packings, we include screening of the chemical fields.We
present a state diagram in the two diffusiophoretic parameters
governing translational, as well as rotational, drift and,thereby,
explore the full range of phoretic attraction and repulsion. The
identified states range from a gaslike phase overdynamic clustering
states 1 and 2, which we distinguish through their cluster size
distributions, to different types of collapsedstates. The latter
include a full chemotactic collapse for translational phoretic
attraction. Turning it into an effective repulsion,with increasing
strength first the collapsed cluster starts to fluctuate at the
rim, then oscillates, and ultimately becomes a staticcollapsed
cloud. We also present a state diagram without screening. Finally,
we summarize how the famous Keller−Segel modelderives from our
Langevin equations through a multipole expansion of the full
one-particle distribution function in position andorientation. The
Keller−Segel model gives a continuum equation for treating
chemotaxis of microorganisms on the level of theirspatial
density.Our theory is extensible to mixtures of active and passive
particles and allows to include a dipolar correction to the
chemicalfield resulting from the dipolar symmetry of Janus
colloids.
1. INTRODUCTION
In nature microorganisms have developed the ability to
sensetheir environment to direct their motion. An
importantbehavioral response is called taxis. Microorganisms can
senseand move along the gradient of an external stimulus or
fieldand thereby are able to find better living conditions.1 The
mostprominent example is chemotaxis, where the gradient is formedby
the density of a chemical species,2 and many micro-organisms employ
this strategy.2−6 Other forms of taxis foundfor living organisms
are gravitaxis,7 rheotaxis,8 magnetotaxis9
phototaxis,10 or thermotaxis.11 To implement a taxis
strategy,microorganisms change their swimming mode in response
tofield gradients. For example, bacteria modify their tumble
rate
using chemical sensors and an internal biochemical
signalingpathway12−15 or light sensing algae alter the breast
stroke oftheir flagella, which is triggered by light
receptors.10
The rapidly evolving field of artificially designed
micro-swimmers16,17 is also driven by the idea of mimicking
differenttaxis strategies, now based on pure physical principles,
tocontrol their swimming paths and explore a wide field
ofintriguing applications. Indeed, in colloidal systems
phoreticmotion induced by field gradients (phoresis) is
wellestablished.18 Combined with activity, colloids exhibit
novel
Received: June 3, 2018Published: October 16, 2018
Article
pubs.acs.org/accountsCite This: Acc. Chem. Res. 2018, 51,
2681−2688
© 2018 American Chemical Society 2681 DOI:
10.1021/acs.accounts.8b00259Acc. Chem. Res. 2018, 51, 2681−2688
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complex behavior while they sense temperature
gradients,19−22
gravitational fields,23 perform phototaxis,24 or show
diffusio-phoresis,25−29 where they drift and orient along
chemicalgradients.In this Account, we concentrate on how chemical
fields
influence the collective motion of self-phoretic
activecolloids,28,30−33 which produce nonuniform chemical fieldsby
consuming chemical reactants and generating products.Prominently,
they show dynamic clustering25−27,29,34 orstimulate the assembly of
passive colloids.35 In suspensionsof active AgCl particles a
colloidal collapse and clusteroscillations were observed.36,37
Continuum theories reveal awealth of pattern formation.28,38,39 A
detailed diffusiophoreticand hydrodynamic modeling reproduces the
motion of self-phoretic colloids near boundaries with step-like
topography.40
On the molecular level enzymes are established as
nanomotors,which undergo chemotaxis.41−44 Hydrodynamic modeling
ofchemical nanomotors either of Janus type or sphere dimersshow
different kinds of clustering45−47 or orientational orderwhen
pinned to a substrate.48 Finally, also emulsion dropletsexhibit
chemotaxis.49
This Account summarizes our work on the chemotaxis
ofself-phoretic active colloids.27,29 It was very much inspired
byexperiments of ref 25. Gold particles half covered withplatinum,
which catalyzes the reaction of H2O2 into waterand oxygen,
self-propel through a combination of self-diffusio-and
electrophoresis. They create nonuniform chemical fields intheir
surroundings with chemical gradients, along whichneighboring
particles drift (translational diffusiophoresis) orreorient
(rotational diffusiophoresis). In combination with self-propulsion
this gives rise to an effective phoretic attraction andrepulsion. A
subtle balance of these effective interactions thencauses dynamic
clustering in dilute suspensions, where clustersform and resolve
again.25,26 This is in stark contrast to therelatively static
clusters of motility-induced phase separation atlarger
densities.50,51
In the following we summarize our particle-based Langevinmodel
for addressing collective motion of self-phoretic activecolloids.
In formulating the model, we wanted to concentrateon the essential
features. Exploring the full range of phoreticattraction and
repulsion, we illustrate the collective dynamicsof self-phoretic
colloids ranging from a gas-like state, overdynamic clustering 1
and 2, to different types of collapsedstates including a full
chemotactic collapse. The latter can berationalized by the
Keller−Segel equation,52,53 which derivesfrom our Langevin
model.
2. MODELTo explore the collective dynamics of self-phoretic
colloidsinduced by diffusiophoretic or chemotactic interactions, we
setup a simplified model system. We consider a colloidalmonolayer
close to a bounding plate with fluid in the infinitehalf-space
above it. The active colloids are Janus particlespartially covered
by a catalyst, which catalyzes a chemicalreaction in the
surrounding fluid. In general, through acombined process of
self-electro- and diffusiophoresis, thecolloids move with a
velocity v0 along the unit vector e, whichgives a direction fixed
in the particle. In the following, weassume that v0 and e are not
changed by the presence of othernearby particles. Instead of taking
into account all reactantsand products of the chemical reaction, we
simply consider theself-phoretic colloid as a chemical sink. It
consumes a chemicaland thereby produces a nonuniform chemical field
with
concentration c, in which nearby colloids perform
translationaland rotational diffusiophoretic motion. Hydrodynamic
inter-actions, which have a reduced range close to a no-slip
surface,are neglected against these effective interactions. The
influenceof hydrodynamic flow has been studied in different
systemsand is reviewed in ref 17.2.1. Translational and Rotational
Diffusiophoresis
The molecules or solutes of the chemical field interact with
thesurface of a colloid. This results in a body force on the
fluid,which influences fluid pressure. Ultimately a gradient in
theconcentration c along the particle surface causes a
pressuregradient, which then drives a slip velocity vs = ζ∇c along
thecolloidal surface, where the slip velocity coefficient ζ
dependson the surface interaction potential between the chemical
andthe colloidal surface. Averaging vs over the particle
surfacegives the translational diffusiophoretic velocity18
ζ ζ ∇= [⟨ ⟩ − ⟨ ⊗ − ⟩] cv 1 n n 1(3 )/2D (1)
where ∇c is evaluated at the particle center, n is the
localsurface normal, and ⟨...⟩ means average over the
particlesurface. For particles with uniform surface properties but
alsofor half-coated Janus colloids the quadrupolar term in eq
1vanishes and we obtain the diffusiophoretic drift velocity,which
we use in the following
ζ ∇= − cvD tr (2)
Here, ζtr ≔ −⟨ζ⟩ is the translational
diffusiophoreticparameter.The slip velocity field also causes a
diffusiophoretic
rotational velocity18
ω ζ ζ∇ ∇= ⟨ ⟩ × = − ×a
c cn e9
4 iD rot (3)
where the rotational diffusiophoretic parameter ζrot is
definedby (9/4a)⟨ζn⟩ = −ζrote. Spherical particles with a
uniformsurface (constant ζ) do not rotate. However, for
half-coatedJanus particles ωD is nonzero, as long as the solutes
interactdifferently with the two sides, and symmetry dictates
thatswimming direction e is parallel to ⟨ζn⟩.Since the slip
velocity coefficient ζ can be controlled by
choosing appropriate materials for the Janus colloids and
theircaps, but also by the geometry and number of the capscovering
the colloidal surface,28 we expect the phoreticparameters ζrot and
ζtr to be tunable to either positive ornegative values. This opens
the possibility to induce andexplore a variety of collective
colloidal dynamics. For example,since each active particle acts as
a chemical sink in our setting,neighboring particles will drift
toward the sink for positivetranslational diffusiophoretic
parameter ζtr in eq 2, whichcorresponds to an effective attraction,
while ζtr < 0 gives rise toan effective repulsion. Similarly, a
positive rotational parameterζrot in eq 3 rotates the swimming
direction of an active colloidtoward a neighboring chemical sink
and the colloid movestoward the sink. Hence, rotational phoresis
also acts like anattractive colloidal interaction while it becomes
repulsive forζrot < 0.2.2. Langevin Dynamics and Chemical
Field
At the micron scale inertia is negligible and position ri
andorientation ei of the ith Janus colloid obey overdampedLangevin
equations. Adding up the deterministic translationalvelocities from
self-propulsion and diffusiophoretic drift (v0ei +
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vD) and using the kinematic relation for the orientation
vector,ei̇ = ωD × ei, we obtain
ξζ ∇̇ = − +v cr e r( )i i i i0 tr (4)
μζ ∇̇ = − − ⊗ + ×ce 1 e e r e( ) ( )i i i i i irot (5)
The additional vectors represent translational (ξi)
androtational (μi) white noise of thermal origin with zero meanand
respective time correlation functions ⟨ξi(t) ⊗ ξi(t′)⟩ =2Dtr1δ(t −
t′) and ⟨μi(t)⊗μi(t′)⟩ = 2Drot1δ(t − t′), where Dtrand Drot are the
translational and rotational diffusioncoefficients, respectively.
All the results presented in thisreview are generated by a
two-dimensional version of eqs 4 and5, since the Janus colloids
move in a monolayer. An effectivehard-core repulsion between the
colloids is implemented.Whenever they overlap during the
simulations, we separatethem along the line connecting their
centers to the point ofcontact. Finally, hydrodynamic flow fields
are not includedhere. Close to a bounding plane they are less
long-ranged andwe also assume that chemical interactions
dominate.According to our model assumption the colloids consume
a
chemical with rate k, which typically diffuses much faster onthe
micron scale than the Janus colloids swim. Therefore, whenthe
colloids move, they carry around with them a staticconcentration
field c, which obeys the Poisson equation [for adiscussion of this
approximation and its impact, see refs 38 and39]
∑ δ∇= − −=
D c k r r0 ( )i
N
ic2
1 (6)
where we approximate the Janus colloids as point-like
chemicalsinks. We neglect here higher-order contributions that
wouldbreak the radial symmetry of the concentration field because
ofthe dipolar character of the Janus colloid. The solution is
givenby
∑π
= −| − |=
c hckh
Dr
r r( )
41
i
N
i2D 0
c 1 (7)
Here, c0 is the concentration field far away from the
colloids,and we have multiplied with the thickness of the
colloidalmonolayer h = 2a to obtain a two-dimensional density in
theplane of the monolayer. Note that according to our model
thechemical field still diffuses in an infinite three-dimensional
half-space. A no-flux boundary condition at the bounding
surfacedoes not change the principal 1/r dependence in eq 7.If the
chemotactic attraction between the active colloids is
sufficiently strong, they form compact clusters, where
thechemical substance cannot diffuse freely between the colloids.We
roughly take this into account by implementing a screenedchemical
field, whenever a colloid is surrounded by six closelypacked
neighbors with distances below rs = 2a(1 + ϵ). Then,we replace the
algebraic decay 1/r = 1/|r − ri| in eq 7 byexp[−(r − ξ)/ξ]/r, where
we introduce the screening length ξ≔ rs. We set ϵ = 0.3 but have
checked that varying ϵ by 50%does not change the results. [In the
Supporting Information ofref 26, the authors also mention a cutoff
of the particleattraction at distances larger than three particle
diameters. So,in particular, dynamic clustering should also be
visible forlarger screening lengths.]
2.3. Essential Parameters
Rescaling the dimensions of all quantities in the equations
ofmotion can be used to reduce the number of systemparameters and
thereby reveals the essential parameters. Onepossibility is to set
the rescaled diffusion constants, whichdetermine the noise
strengths in the Langevin equations 4 and5, to one. This is
achieved when rescaling time by tr = 1/(2Drot) and length by = =l D
D a/ 2.33r tr rot . The thermaldiffusion coefficients in a bulk
fluid would give
=D D a/ 1.15tr rot , while the experiments of ref 25 measurelr =
1.79a for colloids moving close to a bottom wall. Thehigher value
used for all reported results in this Account doesnot change the
qualitative behavior of our system.Ultimately, one obtains four
essential system parameters: the
Peclet number = v D DPe /(2 )0 tr rot , the rescaled
translational
diffusiophoretic parameter ζ π ζ→kh D D D/(8 ) /tr c rot tr3
tr, therescaled rotational diffusiophoretic parameter
ζrotkh/(8πDcDtr)→ ζrot, and the area fraction σ defined as the
ratio of projectedarea of all Janus colloids to the area of the
simulation box. Notethat the factor kh/(4πDc) from eq 7 is already
subsumed intothe rescaled diffusiophoretic parameters, when using
∇c2D(ri)in the Langevin eqs 4 and 5.2.4. Numerical
Implementation
The Langevin eqs 4 and 5 are solved by a typical Euler schemein
a two-dimensional square simulation box. Always 800particles are
used and different area fractions σ are realized byadjusting the
size of the simulation box. To implement thehard-core interactions
mentioned earlier, a sufficiently smalltime step has to be chosen,
which makes them the numericallymost expensive part of the
simulations. Therefore, a neighborlist is implemented such that the
search for overlappingparticles could be restricted to eight
immediate neighbors, atmost. Furthermore, due to the small
translational steps of thecolloids it is sufficient to update the
chemical concentrationfield every 50th time step, which further
saves computationaltime.Since only bulk properties are of interest
here, boundary
conditions are implemented, which keep the colloids awayfrom the
boundary of the simulation box. Thus, wheneverhitting the boundary,
the particles are reflected into a randomlychosen direction.
3. RESULTS AND DISCUSSION
3.1. Overview: State Diagram
Figure1 shows a typical state diagram at low area fractions
ofthe colloids for the two phoretic parameters ζtr and ζrot. It
isroughly divided by a diagonal line, which separates
collapsedstates, where all colloids form one cluster with
differentproperties, from states with varying cluster size. In
particular,for ζtr and ζrot both positive meaning that
translational androtational phoretic motion act like an effective
attractionbetween the colloids, a sharp transition from a gaslike
to acollapsed state occurs, where all particles are packed into
onesingle static cluster (see snapshot in Figure 2, bottom
right).This behavior is reminiscent of the chemotactic collapse
inbacterial systems.52,53 Making both phoretic
interactionsrepulsive (ζtr < 0 and ζrot < 0), the gaslike
state is realized,where small transient clusters are possible.From
here two directions are possible. On the one hand,
inreasing ζtr to sufficiently large ζtr > 0, so that
translational
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diffusiophoresis acts attractive, dynamic clustering occurs
intwo different states 1 and 2. Motile clusters form that
stronglyfluctuate in shape and size and may ultimately dissolve
again(see snapshots in Figure 2, top right and bottom left).
Thestrongly dynamic clusters form due to a delicate balance of
theattractive translational phoresis (holding the clusters
together)and the fact that active colloids turn and swim away from
thecluster (effective repulsion). This behavior is
decisivelydifferent from the motility-induced phase separation of
hard-core active particles.50,51 On the other hand, keeping ζtr
< 0(effective repulsion) but making ζrot sufficiently large so
thatactive colloids orient and therefore swim toward each
other(effective attraction), different types of collapsed states
occur.With decreasing ζtr first the collapsed cluster fluctuates at
the
rim due to particles leaving and rejoining it. Then the
clusterdevelops nearly regular oscillations and, finally, for
stronglynegative ζtr a static collapsed cloud appears, where
colloids donot touch and their mutual distance increases toward the
rimof the cloud.It is possible to derive the famous Keller−Segel
equation
from the Langevin eqs 4 and 5 (see section 3.5 and ref 27). Asin
bacterial systems it here predicts a chemotactic collapse ofthe
active colloids at 8πσ(ζtr + ζrotPe)/(1 + 2Pe
2) = b, where bis a positive constant. In the ζrot−ζtr state
diagram, the linearseparation line between the gaslike and
collapsed state has anegative slope and it is shifted along the
vertical to positivevalues. Thus, it gives the right trend for the
state diagram ofFigure 1. The Keller−Segel equation is a mean-field
equation,it only contains the density of the colloids. Therefore,
it cannotdescribe subtle states such as dynamic clustering 1 and 2.
Forexample, if ζtr is sufficiently large so that the phoretic
driftvelocity vD exceeds the swim speed v0, the active colloids
willalways stay attached to a cluster, regardless how negative
ζrotbecomes. Hence, the separation line in Figure 1 deviates
fromthe simple straight line.Finally, the state diagram of Figure 1
does not change
qualitatively with Pe and σ, as long as Pe is well above one
andσ sufficiently low so that phase separation does not occur.In
the following two sections we characterize the dynamic
clustering states and collapsed states in detail, report on
thestate diagram, when screening of the chemical field in
densecolloid clusters is switched off, and comment on how to
derivethe Keller−Segel equation from our Langevin model.3.2. Gas
Phase and Dynamic Clustering
3.2.1. Signature of Dynamic Clustering. Dynamicclustering states
are distinguished from the gaslike state bythe occurrence of larger
cluster and a pronounced increase incluster size between states 1
and 2. This is already pictured inFigure 1, where the dynamic
clustering state 1 shows anincrease of the mean cluster size Nc in
a small region. To bettercharacterize these states and justify, why
there are twoclustering states, Figure 3 shows the cluster size
distributionfor increasing ζtr for fixed rotational phoretic
parameter ζrot =−0.38. The curves until ζtr = 15.4 are well fitted
by
= −β−P n c n n n( ) exp( / )0 0 (8)
where the cutoff size n0 is a measure how large the clusters
canbecome. While for pure steric interaction (ζtr = 0, blue
curve)
Figure 1. Full state diagram ζtr versus ζrot at Pe = 19 and
surfacefraction σ = 0.05. The mean cluster size Nc for the gaslike
anddynamic-clustering state 1 are color-coded. A full discussion
isprovided in the main text. Reprinted with permission from ref
29.Copyright 2015 European Physical Journal.
Figure 2. Snapshots of colloid configurations for increasing
ζtrans atζrot = −0.38 and Pe = 19. Top left: Gas-like state. Top
right: Dynamicclustering 1. Bottom left: Dynamic clustering 2.
Bottom right:Collapsed state. Nc is the mean cluster size.
Reprinted with permissionfrom ref 27. Copyright 2014 American
Physical Society.
Figure 3. Cluster size distributions P(n) for increasing
translationalphoretic parameter ζtr at Pe = 19 and ζrot = −0.38.
The transitionbetween dynamic clustering states 1 and 2 occurs
between the red andgreen curves. Inset: mean cluster size Nc versus
ζtr. The transition isindicated. Reprinted with permission from ref
27. Copyright 2014American Physical Society.
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and small ζtr (gray curve) an exponential decay is
predominant,the distributions for larger ζtr (orange and red
curves) followfirst the power law before they are cut off by the
exponential.The exponent β = 2.1 ± 0.1 gradually decreases for
morenegative ζrot and the fit is robust against increasing
particlenumber.27 So there is a smooth transition from the gaslike
tothe dynamic clustering state 1.Further increasing ζtr, the
distribution P(n) develops an
inflection point (green curves), which marks the onset of
thedynamic clustering state 2. At the transition the mean
clustersize strongly increases as the inset indicates. Now, the sum
oftwo power-law-exponential curves have to be used for fittingthe
distributions,
= − + −β β− −P n c n n n c n n n( ) exp( / ) exp( / )1 1 2 21 2
(9)with β1 = 2.1 ± 0.2 and β2 ≈ 1.5. Both dynamic clusteringstates
are observed for all negative ζrot. However, turning offscreening
of the chemical field within clusters, dynamicclustering is less
pronounced and the clustering state 2 doesnot occur at all.27
Finally, all the exponents βi decrease forlarger area fraction σ
and the sharp increase of Nc at thetransition from state 1 to 2
vanishes (see Figure 4). In
literature similar cluster-size distributions including
thetransition indicated by the occurrence of an inflection
pointwere observed in experiments on gliding bacteria, when
thebacterial density was varied.54 While this system shows
purehard-core interactions causing nematic alignment, we
insteadvary the strength of the diffusiophoretic coupling.3.2.2.
Dynamic Clustering for Varying Pećlet Num-
ber. In Figure 5, we plot the state diagram ζtr versus
Pećletnumber Pe. Dynamic clustering becomes more pronounced
forlarger ζtr and then also occurs at larger Pe, which is
necessaryto establish the delicate balance between translational
phoreticattraction (ζtr > 0) and effective repulsion (ζrot <
0), where theactive colloids swim away from the clusters. However,
forconstant ζtr large clusters disappear with increasing Pe. This
isin contrast to the experiments with diffusiophoretic
coupling,which showed a linear scaling of the mean cluster size
with Pe:Nc ∼ Pe, when increasing fuel concentration.25To
rationalize the experimental scaling prediction, we note
that increasing the fuel concentration c0 means higher
reactionrate k on the colloidal surface and therefore more
self-activitybut also larger phoretic forces, which is encoded in
the reduced
phoretic parameters ζtr, ζrot ∝ k introduced in section
2.3.Assuming Michaelis−Menten kinetics for the reaction rate inthe
linear regime well before saturation, k ∼ c0, we not onlyfind Pe ∝
c055 but also ζtr ∝ ζrot ∝ c0. Therefore, varying c0defines a
straight line in the ζtr−ζrot−Pe parameter space. Inthis
three-dimensional space the dynamic clustering states 1and 2 are
separated by a plane. We choose different lines,which always hit
the transition plane and plot in Figure 6 the
mean cluster size Nc versus Pe along the lines. The blue
andpurple curves show the strong increase of Nc when theclustering
state 2 is entered, since the respective lines arenearly normal to
the transition plane. Tilting the lines more,the increase of Nc
becomes smoother. In particular, the greengraph shows an almost
linear increase of Nc in the range Pe =10−20. Thus, Figure 6
demonstrates that the relation of clustersize and swimming speed
might take different forms dependingon the relation between fuel
concentration c0, activity Pe, andphoretic strengths ζtr and
ζrot.
3.3. Clustering States
To classify the different collapsed states in Figure 1 and
todescribe the hexagonal order in the N-particle cluster,
weintroduce the global 6-fold bond orientational parameter
Figure 4. Mean cluster size Nc plotted against ζtr for different
arealfractions σ. The dashed line indicates the transition between
dynamicclustering states 1 and 2. The rotational diffusiophoretic
parameter isζrot = −0.38. Reprinted with permission from ref 29.
Copyright 2015European Physical Journal.
Figure 5. State diagram ζtr versus Pe at ζrot = −0.38 and
surfacefraction σ = 0.05. The mean cluster size Nc for the gaslike
anddynamic-clustering state 1 are color-coded. Reprinted with
permissionfrom ref 27. Copyright 2014 American Physical
Society.
Figure 6. Mean cluster size Nc versus Pe for different lines in
theζtr−ζrot−Pe parameter space. The lines are defined via a
para-metrization with x ∈ [0, 1], where Pe varies as in the
experiments of,25Pe = 9.5 + 11.5x, ζtr = 4.8 + 16.6x, and ζrot =
−0.16 − ζ0x. Theparameter ζ0 defines the different graphs. The
transition betweenclustering states 1 and 2 roughly occurs at the
intersection of the fittedtwo straight lines. Reprinted with
permission from ref 27. Copyright2014 American Physical
Society.
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∑
∑
≔ ∈ [ ]
≔ α
=
∈
qN
q
q e
10, 1
with16
k
Nk
k
j
i
61
6( )
6( ) 6
k
kj
6( ) (10)
The second sum goes over the set k6( ) of six nearest
neighbors
of particle k and αkj is the angle between the line
connectingparticle k to j and some prescribed axis.17 The local
bondparameter q6
(k) becomes one if all six nearest neighbors form aregular
hexagon around the central colloid and the global orderparameter is
one in a hexagonal lattice.The temporal evolution of q6 is plotted
in Figure 7 for
several ζtr at constant ζrot = 4.5. At positive ζtr the
order
parameter is nearly constant in time indicating a
staticcrystalline cluster, while q6 < 1 results from the
colloids atthe rim of the cluster, which do not have six neighbors
on ahexagon. At ζtr = 0 and especially for negative ζtr,
whereparticles effectively repel each other due to
translationalphoretic motion, the order parameter shows
increasinglystrong fluctuations. The cluster fluctuates close to
the rim,where particles leave and rejoin it frequently (ζtr = −6.4
inFigure 7). A video of the fluctuating collapsed state is
attachedto ref 29. Further decreasing ζtr the order parameter
developsnearly regular oscillations, where the cluster oscillates
betweendensely packed and a cloud of confined colloids (ζtr = −12.8
inFigure 7). When the dense packing develops, the diffusiopho-retic
interaction becomes strongly screened and the particleslose their
orientations toward the cluster center. Repulsion dueto
translational phoresis takes over and the cluster expandsuntil the
particles are oriented back to the center. This initiatesthe
collapse of the cloud and the cycle starts again. Thepulsating
cluster is visualized in a video attached to ref 29. Thepower
spectrum of q6 shows a broad peak at a nonzerofrequency and
confirms the regular oscillations.29 Finally,further decreasing
ζtr, the oscillations abruptly stop and a staticcollapsed cloud
forms, where the strong effective repulsionprevents direct contact
between the particles (ζtr = −16.0 inFigure 7). In this state the
hexagonal bond order is small sinceq6 ≈ 0.35 is close to the value
q6 = 1/3 of uniformly distributedparticles.
3.4. State Diagram without Screening
Some of the states in the full state diagram of Figure
1discussed so far depend on the screening of the chemical
field,which is implemented when the colloids are densely packed.To
demonstrate its influence, we show in Figure 8 the state
diagram in the absence of any screening for the sameparameters.
A comparison reveals that without screeningdynamic clustering 1 is
less pronounced producing on averagesmaller clusters and the
dynamic clustering state 2 disappearscompletely. The rich
phenomenology of the collapsed states atnegative ζtr is replaced by
a single core−corona state, where adensely packed core is
surrounded by a cloud of colloids (seeFigure 9). The extension of
the core decreases, when ζtrbecomes more negative.
3.5. Relation to Keller−Segel ModelIn the end, we illustrate
here how to derive the Keller-Segelequation starting from our model
for the chemically interactingactive particles. More details are
presented in ref 27. TheLangevin eqs 4 and 5 are equivalent to the
Smoluchowskiequation for the full one-particle distribution
function P(e, r,t), where we neglect direct interactions between
the colloids:
ζ
ζ
∇ ∇ ∇ ∇
∇
∂ = − · + · +
+ ∂ [ ∂ · ] + ∂φ φ φ
P t v Pe P c D P
c P D P
e r
e
( , , ) ( ) ( )
( ) ( )t 0 tr tr
2
rot rot2
(11)
To formulate the Smoluchowski equation, we used e = (sin φ,cos
φ) and rewrote eq 5 to ∂tφi = ζrot∂φ ei·∇c + μi. One thenderives
dynamic equations for the colloidal density P0(r,t) =∫ P(e, r,
t)dφ, as well as the polar [P1(r,t) = ∫ e P dφ ] and
Figure 7. Time evolution of the bond orientational parameter q6
fordifferent ζtr. Further parameters are Pe = 19, ζrot = 4.5, and σ
= 0.05.Reprinted with permission from ref 29. Copyright 2015
EuropeanPhysical Journal.
Figure 8. Full state diagram ζtr versus ζrot at Pe = 19 and σ =
0.05 inthe absence of screening. Reprinted with permission from ref
29.Copyright 2015 European Physical Journal.
Figure 9. Snapshots of the colloidal configuration in the
core−coronastate: (a) ζtr = −6.4 and (b) ζtr = −16. Other
parameters are Pe = 19and ζrot = 4.1. Reprinted with permission
from ref 29. Copyright 2015European Physical Journal.
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the nematic ∫ φ= ⊗ −( )t PP r e e( , ) d12 2ÄÇÅÅÅÅÅÅ
ÉÖÑÑÑÑÑÑ order parame-
ter. The resulting hierarchy of coupled equations is closed
bysetting P3 to zero. Neglecting time derivatives of P1 and P2
ontime scales much larger than the rotational diffusion time 1/Drot
and also higher-order spatial derivatives, one arrivesultimately at
the Keller−Segel equation for the colloidaldensity
ζ ∇ ∇ ∇∂ = +P P c D P( )t 0 eff 0 eff2
0 (12)
with renormalized phoretic coefficient and effective
transla-tional diffusion constant:
ζ ζζ
= + = +v
DD D
vD2
and2eff tr
rot 0
roteff tr
02
rot (13)
From the Keller−Segel equation the condition for thechemotactic
collapse mentioned in section 3.1 can be derived.
4. CONCLUSIONSThis Account reviews our work on self-phoretic
active colloidsinteracting by self-generated chemical gradients and
therebymimicking chemotaxis well-known from the biological
world.The combination of translational and rotational
diffusiopho-retic drift velocities with self-propulsion gives rise
to effectivephoretic attraction and repulsion. Exploration of the
full rangeof the drift velocities reveals that a variety of dynamic
statesoccur that range from a gas-like state, over dynamic
clustering1 and 2, to different types of collapsed states including
a fullchemotactic collapse. The latter can be rationalized by
theKeller−Segel equation, which derives from our Langevinmodel.We
shortly discuss one point. In our state diagram in Figure
1, the dynamic clustering states only appear in a narrow
region,while in the experiments dynamic clustering seems to be
ageneric feature.25,26 One reason could be that
furtherattractive/repulsive forces between the active colloids
arepresent in the experiments, which are not included in ourmodel.
Indeed, in simulations with a Lennard-Jones potentialdynamic
clustering with cluster size distributions similar to theones
discussed in this article are reported.56,57
Possible extensions of the Langevin model presented in
thisarticle are mixtures of active and passive particles.
Anexperimental realization was published recently in ref
35.Furthermore, Janus particles do not just produce a
monopoledisturbance in the chemical environment but also a
dipolarcontribution due to their polar character. We are
currentlyexploring the consequences of such a contribution.
Ultimately,a full solution of the chemical field with appropriate
boundaryconditions at the colloid surface is needed for being able
totreat particles, which are really close to each other.
However,this will also directly influence the swimming speed of
thecolloids. In addition, a full hydrodynamic treatment needs tobe
included, which has been addressed in recent publica-tions.45−47
Finally, the slip velocity coefficient ζ in eq 1determines the two
diffusiophoretic parameters ζrot and ζtr. Itstrongly depends on
material properties. So, as a challenge tothe fabrication of
self-phoretic colloids, one can ask if it ispossible to purposely
tune the phoretic parameters by choosingappropriate materials,
catalysts, cap sizes, and physicalmechanisms for the phoretic
process?All this will help to further develop the idea of how
artificial
microswimmers can mimic biological taxis strategies to
control
their swimming paths and explore a wide field of
intriguingapplications.
■ AUTHOR INFORMATIONCorresponding Author
*E-mail: [email protected]
Holger Stark: 0000-0002-6388-5390Notes
The author declares no competing financial interest.
Biography
Holger Stark received his Ph.D. from University of Stuttgart
andpursued postdoctoral studies at the University of Pennsylvania
inPhiladelphia. After staying as a Heisenberg fellow at the
University ofKonstanz and a group leader at the
Max-Planck-Institute forDynamics and Self-Organization in
Göttingen, he became a Professorof Theoretical Physics at the
Technical University Berlin. His researchinterests lie in the areas
of nonequilibrium statistical physics of softmatter and biological
systems.
■ ACKNOWLEDGMENTSThe author thanks Oliver Pohl and Julian
Stürmer forcollaborating on the topic and acknowledges funding
fromthe DFG within the research training group GRK 1558 and
thepriority program SPP 1726, project number STA 352/11. Theauthor
also thanks Julian Stürmer for preparing the picture forthe
Conspectus.
■ REFERENCES(1) Dusenberg, D. B. Living at Micro Scale; Harvard
University Press:Cambridge, USA, 2009; Chapter 14.(2) Berg, H.;
Brown, D. Chemotaxis in Escherichia coli analysed
bythree-dimensional tracking. Nature 1972, 239, 500.(3) Grimm, A.
C.; Harwood, C. S. Chemotaxis of Pseudomonas spp.to the
polyaromatic hydrocarbon naphthalene. Appl. Environ.Microbiol.
1997, 63, 4111−4115.(4) Homma, M.; Oota, H.; Kojima, S.; Kawagishi,
I.; Imae, Y.Chemotactic responses to an attractant and a repellent
by the polarand lateral flagellar systems of Vibrio alginolyticus.
Microbiology 1996,142, 2777−2783.(5) Amselem, G.; Theves, M.; Bae,
A.; Bodenschatz, E.; Beta, C. Astochastic description of
Dictyostelium chemotaxis. PLoS One 2012,7, e37213.(6) Pohl, O.;
Hintsche, M.; Alirezaeizanjani, Z.; Seyrich, M.; Beta,C.; Stark, H.
Inferring the Chemotactic Strategy of P. putida and E.coli Using
Modified Kramers-Moyal Coefficients. PLoS Comput. Biol.2017, 13,
e1005329.(7) Had̈eri, D.-P.; Rosumi, A.; Schaf̈er, J.; Hemmersbach,
R.Gravitaxis in the flagellate Euglena gracilisis controlled by an
activegravireceptor. J. Plant Physiol. 1995, 146, 474.(8) Marcos,
H. C. F.; Powers, T. R.; Stocker, R.; et al. Bacterialrheotaxis.
Proc. Natl. Acad. Sci. U. S. A. 2012, 109, 4780−4785.(9) Blakemore,
R. Magnetotactic bacteria. Science 1975, 190, 377−379.(10) Garcia,
X.; Rafaï, S.; Peyla, P. Light Control of the Flow ofPhototactic
Microswimmer Suspensions. Phys. Rev. Lett. 2013, 110,138106.(11)
Mori, I.; Ohshima, Y. Neural regulation of thermotaxis
inCaenorhabditis elegans. Nature 1995, 376, 344−348.(12) Alon, U.;
Surette, M. G.; Barkai, N.; Leibler, S. Robustness inbacterial
chemotaxis. Nature 1999, 397, 168−171.
Accounts of Chemical Research Article
DOI: 10.1021/acs.accounts.8b00259Acc. Chem. Res. 2018, 51,
2681−2688
2687
mailto:[email protected]://orcid.org/0000-0002-6388-5390http://dx.doi.org/10.1021/acs.accounts.8b00259
-
(13) Korobkova, E.; Emonet, T.; Vilar, J. M.; Shimizu, T. S.;
Cluzel,P. From molecular noise to behavioural variability in a
singlebacterium. Nature 2004, 428, 574−578.(14) Sourjik, V.
Receptor clustering and signal processing in E. colichemotaxis.
Trends Microbiol. 2004, 12, 569−576.(15) Berg, H. E. coli in
Motion; Springer, New York, 2004.(16) Bechinger, C.; Di Leonardo,
R.; Löwen, H.; Reichhardt, C.;Volpe, G.; Volpe, G. Active
particles in complex and crowdedenvironments. Rev. Mod. Phys. 2016,
88, 045006.(17) Zöttl, A.; Stark, H. Topical Review: Emergent
behavior inactive colloids. J. Phys.: Condens. Matter 2016, 28,
253001.(18) Anderson, J. L. Colloidal Transport by Interfacial
Forces. Annu.Rev. Fluid Mech. 1989, 21, 61−99.(19) Golestanian, R.
Collective Behavior of Thermally ActiveColloids. Phys. Rev. Lett.
2012, 108, 038303.(20) Cohen, J. A.; Golestanian, R. Emergent
cometlike swarming ofoptically driven thermally active colloids.
Phys. Rev. Lett. 2014, 112,068302.(21) Braun, M.; Würger, A.;
Cichos, F. Trapping of single nano-objects in dynamic temperature
fields. Phys. Chem. Chem. Phys. 2014,16, 15207.(22) Bickel, T.;
Zecua, G.; Würger, A. Polarization of active Janusparticles. Phys.
Rev. E 2014, 89, 050303.(23) Ten Hagen, B.; Kümmel, F.;
Wittkowski, R.; Takagi, D.;Löwen, H.; Bechinger, C. Gravitaxis of
asymmetric self-propelledcolloidal particles. Nat. Commun. 2014, 5,
4829.(24) Lozano, C.; ten Hagen, B.; Löwen, H.; Bechinger, C.
Phototaxisof synthetic microswimmers in optical landscapes. Nat.
Commun.2016, 7, 12828.(25) Theurkauff, I.; Cottin-Bizonne, C.;
Palacci, J.; Ybert, C.;Bocquet, L. Dynamic Clustering in Active
Colloidal Suspensions withChemical Signaling. Phys. Rev. Lett.
2012, 108, 268303.(26) Palacci, J.; Sacanna, S.; Steinberg, A. P.;
Pine, D. J.; Chaikin, P.M. Living Crystals of Light-Activated
Colloidal Surfers. Science 2013,339, 936.(27) Pohl, O.; Stark, H.
Dynamic clustering and chemotacticcollapse of self-phoretic active
particles. Phys. Rev. Lett. 2014, 112,238303.(28) Saha, S.;
Golestanian, R.; Ramaswamy, S. Clusters, asters, andcollective
oscillations in chemotactic colloids. Phys. Rev. E 2014,
89,062316.(29) Pohl, O.; Stark, H. Self-phoretic active particles
interacting bydiffusiophoresis: A numerical study of the collapsed
state and dynamicclustering. Eur. Phys. J. E: Soft Matter Biol.
Phys. 2015, 38, 93.(30) Wang, W.; Duan, W.; Ahmed, S.; Sen, A.;
Mallouk, T. E. FromOne to Many: Dynamic Assembly and Collective
Behavior of Self-Propelled Colloidal Motors. Acc. Chem. Res. 2015,
48, 1938−1946.(31) Banigan, E. J.; Marko, J. F. Self-propulsion and
interactions ofcatalytic particles in a chemically active medium.
Phys. Rev. E: Stat.Phys., Plasmas, Fluids, Relat. Interdiscip. Top.
2016, 93, 012611.(32) Yan, W.; Brady, J. F. The behavior of active
diffusiophoreticsuspensions: An accelerated Laplacian dynamics
study. J. Chem. Phys.2016, 145, 134902.(33) Moran, J. L.; Posner,
J. D. Phoretic Self-Propulsion. Annu. Rev.Fluid Mech. 2017, 49,
511.(34) Ginot, F.; Theurkauff, I.; Detcheverry, F.; Ybert, C.;
Cottin-Bizonne, C. Aggregation-fragmentation and individual
dynamics ofactive clusters. Nat. Commun. 2018, 9, 696.(35) Singh,
D. P.; Choudhury, U.; Fischer, P.; Mark, A. G. Non-Equilibrium
Assembly of Light-Activated Colloidal Mixtures. Adv.Mater. 2017,
29, 1701328.(36) Ibele, M.; Mallouk, T. E.; Sen, A. Schooling
Behavior of Light-Powered Autonomous Micromotors in Water. Angew.
Chem., Int. Ed.2009, 48, 3308.(37) Ibele, M. E.; Lammert, P. E.;
Crespi, V. H.; Sen, A. Emergent,collective oscillations of
self-mobile particles and patterned surfacesunder redox conditions.
ACS Nano 2010, 4, 4845.
(38) Liebchen, B.; Marenduzzo, D.; Pagonabarraga, I.; Cates, M.
E.Clustering and Pattern Formation in Chemorepulsive Active
Colloids.Phys. Rev. Lett. 2015, 115, 258301.(39) Liebchen, B.;
Marenduzzo, D.; Cates, M. E. PhoreticInteractions Generically
Induce Dynamic Clusters and Wave Patternsin Active Colloids. Phys.
Rev. Lett. 2017, 118, 268001.(40) Simmchen, J.; Katuri, J.; Uspal,
W. E.; Popescu, M. N.;Tasinkevych, M.; Sańchez, S. Topographical
pathways guide chemicalmicroswimmers. Nat. Commun. 2016, 7,
10598.(41) Sengupta, S.; Dey, K. K.; Muddana, H. S.; Tabouillot,
T.; Ibele,M. E.; Butler, P. J.; Sen, A. Enzyme Molecules as
Nanomotors. J. Am.Chem. Soc. 2013, 135, 1406−1414.(42) Dey, K. K.;
Das, S.; Poyton, M. F.; Sengupta, S.; Butler, P. J.;Cremer, P. S.;
Sen, A. Chemotactic Separation of Enzymes. ACS Nano2014, 8,
11941−11949.(43) Duan, W.; Wang, W.; Das, S.; Yadav, V.; Mallouk,
T. E.; Sen, A.Synthetic Nano- and Micromachines in Analytical
Chemistry:Sensing, Migration, Capture, Delivery, and Separation.
Annu. Rev.Anal. Chem. 2015, 8, 311−333.(44) Zhao, X.; Palacci, H.;
Yadav, V.; Spiering, M. M.; Gilson, M. K.;Butler, P. J.; Hess, H.;
Benkovic, S. J.; Sen, A. Substrate-drivenchemotactic assembly in an
enzyme cascade. Nat. Chem. 2017, 10,311−317.(45) Huang, M.-J.;
Schofield, J.; Kapral, R. A microscopic model forchemically-powered
Janus motors. Soft Matter 2016, 12, 5581−5589.(46) Huang, M.-J.;
Schofield, J.; Kapral, R. Chemotactic andhydrodynamic effects on
collective dynamics of self-diffusiophoreticJanus motors. New J.
Phys. 2017, 19, 125003.(47) Colberg, P. H.; Kapral, R. Many-body
dynamics of chemicallypropelled nanomotors. J. Chem. Phys. 2017,
147, 064910.(48) Robertson, B.; Stark, H.; Kapral, R. Collective
orientationaldynamics of pinned chemically-propelled nanorotors.
Chaos 2018, 28,045109.(49) Jin, C.; Krüger, C.; Maass, C. C.
Chemotaxis andautochemotaxis of self-propelling droplet swimmers.
Proc. Natl.Acad. Sci. U. S. A. 2017, 114, 5089−5094.(50) Cates, M.
E.; Tailleur, J. Motility-Induced Phase Separation.Annu. Rev.
Condens. Matter Phys. 2015, 6, 219−244.(51) Bialke,́ J.; Speck, T.;
Löwen, H. Active colloidal suspensions:Clustering and phase
behavior. J. Non-Cryst. Solids 2015, 407, 367−375.(52) Keller, E.
F.; Segel, L. A. J. Theor. Biol. 1970, 26, 399−415.(53) Masoud, H.;
Shelley, M. J. Collective surfing of chemicallyactive particles.
Phys. Rev. Lett. 2014, 112, 128304.(54) Peruani, F.; Starruß, J.;
Jakovljevic, V.; Søgaard-Andersen, L.;Deutsch, A.; Bar̈, M. Phys.
Rev. Lett. 2012, 108, 098102.(55) Moran, J. L.; Wheat, P. M.;
Posner, J. Locomotion ofelectrocatalytic nanomotors due to reaction
induced chargeautoelectrophoresis. Phys. Rev. E 2010, 81,
065302.(56) Mognetti, B. M.; Šaric,́ A.; Angioletti-Uberti, S.;
Cacciuto, A.;Valeriani, C.; Frenkel, D. Living Clusters and
Crystals from Low-Density Suspensions of Active Colloids. Phys.
Rev. Lett. 2013, 111,245702.(57) Alarcoń, F.; Valeriani, C.;
Pagonabarraga, I. Morphology ofclusters of attractive dry and wet
self-propelled spherical particlesuspensions. Soft Matter 2017, 13,
814.
Accounts of Chemical Research Article
DOI: 10.1021/acs.accounts.8b00259Acc. Chem. Res. 2018, 51,
2681−2688
2688
http://dx.doi.org/10.1021/acs.accounts.8b00259