Collective dynamics of dipolar and multipolar colloids: from passive to active systems

1.6 MB·

by  Sabine H. L. Klapp

Collective dynamics of dipolar and multipolar colloids: from passive to active systems
Sabine H. L. Klapp∗
Institut fu¨r Theoretische Physik, Sekr. EW 7–1, Technische Universita¨t Berlin,
Hardenbergstrasse 36, D-10623 Berlin, Germany
(Dated: February 2, 2016)
This article reviews recent research on the collective dynamical behavior of colloids with dipolar
or multipolar interactions. Indeed, whereas equilibrium structures and static self-assembly of such
systemsarenowratherwellunderstood,thepastyearshaveseenanexplosionofinterestinunder-
standing dynamicals aspects, from the relaxation dynamics of strongly correlated dipolar networks
over systems driven by time-dependent, electric or magnetic fields, to pattern formation and dy-
namical control of active, self-propelled systems. Unraveling the underlying mechanisms is crucial
for a deeper understanding of self-assembly in and out of equilibrium and the use of such particles
6
asfunctionaldevices. Atthesametime,thecomplexdynamicsofdipolarcolloidsposeschallenging
1
0 physicalquestionsandputsforwardtheirroleasmodelsystemsfornonlinearbehaviorincondensed
2 matter physics. Here we attempt to give an overview of these developments, with an emphasis on
theoretical and simulation studies.
n
a
J I. INTRODUCTION AND SCOPE OF THIS nonlineardynamicssuchassynchronization(andrelated

0 ARTICLE structural) transitions [2]. Third, we discuss the collec-
3
tive (translational) transport properties of dipolar sys-

Colloids with anisotropic, directional interactions play tems in alternating fields [9] and in ”active” systems
]
t nowadaysamajorroleinthefieldofself-assemblyofcol- where the particles are driven by an an internal energy
f
o loidal matter, in microfluidics, the design of functional source [5]. Indeed, active dipoles are an exemplary topic
s devices such as robots and sensors, but also as theoreti- where current research on active-particle systems and
.
t callyandexperimentallyaccessiblemodelsystemsincon- thatonpassivecomplexsystemsmeet,andwhereastim-
a
m densed matter physics. A paradigm example are dipolar ulatinginterplaycanbeforeseen. Infact,therearemany
colloids whose interactions are governed by permanent research themes which are ”hot topics” in both areas,
-
d or field-induced, magnetic or electric dipole moments, as such as the interplay of clustering/aggregation and equi-
n well as particles with more complex multipolar interac- libriumphaseseparation[10,11],aswellasthecontrolof
o tions. (single-particle and collective) motion by external, mag-
c While earlier research has rather focused on under- netic or electric fields [4]. Thus, a comprehensive discus-
[
standing the (often unusual) equilibrium phase behavior sion highlighting the interface of these fields of research

1 and static self-assembly of such systems, the last years is timely.
v have seen substantial progress in understanding dynam- There are a number of topics which are related to
7 ical properties [1], from the single-particle response over the overall theme of this article but are not covered or
0
thecollectivedynamicsofstronglycorrelated,drivensys- touched only briefly here. Examples are the equilibrium
1
tems [2, 3] towards the dynamics of active dipolar sys- structuresofdipolarcolloidsinthegroundstateandatfi-
0
0 tems [4, 5]. The purpose of the present review is to give nitetemperatures(see,e.g. [12]),thebehaviorofelectro-
. anoverviewofrecentdevelopmentsinthisemergingfield and magnetorheological systems in static fields [13], the
2
from a theoretical point of view, and to outline perspec- dynamicsundershearflow[14],thebehaviorofmagnetic
0
tives for future research. The focus lies on systems com- elastomers [15, 16], and the dynamics and growth of the
6
1 posed of spherical particles with dipolar or multipolar closely related patchy particle systems [17, 18].
: interactions since these have been studied in most de-
v
tail so far. However, a trend towards shape-anisotropic
i
X dipolar systems is already foreseeable [6, 7].
II. MODELS
r We start by discussing dynamical properties close to
a
equilibrium, such as the relaxation dynamics and gela-
This section gives an overview of models typically dis-
tion of self-assembled structures. These properties are
cussed in the context of dipolar colloidal systems. We
highly relevant, e.g., for the resulting materials’ elastic
focus here on monodisperse systems of spherical par-
response and conductivity [8]. A second topic is the
ticles. The directional interaction between particles i
non-equilibrium behavior of dipolar systems generated
and j, Uaniso(ij), results either from permanent mag-
by a time-dependent, rotating external field. Indeed,
netic (ferromagnetic) moments or charge distributions,
time-dependent fields have recently shown to induce not
respectively, or it is induced by an external field. In
only unusual (quasi-static) structures, but also complex
addition, the spheres interact via a short-range poten-
tial Usr(ij), such as the purely respulsive hard-sphere
and soft-sphere potentials, or the Lennard-Jones poten-
∗ [email protected] tial which includes attractive (van-der-Waals) interac-
2
gated [28–30] in a quasi-2D set-up (realized by two con-
fining glass plates), where they are dissolved in water
and exposed to an in-plane AC electric field. The lat-
ter creates an induced dipole moment in both parts of
the metallodielectric particles. Interestingly, not only
the magnitude of these dipoles but also their direction
(and thus, the character of the resulting interaction) de-
pends on the frequency (f) of the AC field, the physical
reason being the frequency dependence of the polariz-
ability of the two particle domains. In particular, while
thegoldpatchisstronglypolarizedalongthefieldatany
f, the polarizability of the dielectric part (together with
its counterionic atmosphere) switches its sign from pos-
itive, Fig. 1(e), to negative, Fig. 1(d), at a critical fre-
quency f . Corresponding models have been suggested
c
in[31,32]. Theresultinganisotropicinteractionbetween
FIG. 1. Sketches of model colloids with dipolar (a)-(c) or two spheres is then the sum of the dipole-dipole inter-
multipolar (d)-(e) character. Specifically: (a) Dipolar hard actions, Eq. (1), between each moment (polarization ef-
or soft sphere with permanent point dipole moment in its fectshavesofarnotbeeninvestigated). Finally,Fig.1(f)
center, (b) sphere with off-centered point dipole shifted lat- sketchesaparticlewithcrossed,extendeddipolesinduced
erally (dipole moment is perpendicular to the radius vector), byacombinationoftwoexternalfields[6,33]. Evenmore
(c) sphere with two charges representing an extended (per-
complex charge distributions have been suggested in the
manent or induced) dipole. (d) Janus-like sphere with two
context of ”inverse patchy colloids” [34, 35].
induced,off-centeredpointdipolesorientedinoppositedirec-
tion,(e)Janus-likespherewithtwoinduced,paralleldipoles,
(d) sphere with two crossed, electric and magnetic, dipoles
induced by bidirectional fields.
Common theoretical and computational methods
tions. Some important representatives of such parti-
cles are shown in Fig. 1. The ”paradigm” model for
Many of the theoretical studies on passive dipolar sys-
ferromagnetic particles (permanent moment) are hard
temsinvolveparticle-resolvedcomputersimulationssuch
spheres with embedded point dipoles µ and µ (the
i j asMolecularDynamics(MD),thatis,solutionofthecou-
so-called ”dipolar hard sphere” (DHS) model) in their
pledNewtonianequationsofmotion;LangevinDynamics
center, Fig. 1(a), where the anisotropic part of the inter-
(LD),whichisbasedonunderdampedstochasticdifferen-
action is given by the usual dipole-dipole potential,
tial equations with friction and white noise, and Brown-
µ ·µ 3(r ·µ )(r ·µ ) ianDynamics(BD)basedonoverdampedLangevinequa-
UD(r ,µ ,µ )= i j − ij i ij j , (1)
ij i j r3 r5 tions. Combinations of MD and the Navier-Stokes equa-
ij ij
tion to incorporate hydrodynamic flows have been used
with r being the connecting vector and r = |r |. as well [36]. Besides particle-resolved computations, the
ij ij ij
Figure 1(b) shows a variant characterized by an off- dynamicsofdipole-coupledcolloidshasbeeninvestigated
centered, laterally shifted permanent point dipole; this by a variety of effective single-particle theories (to which
(and related off-centered) model(s) have been recently we will refer in the text) and field methods such as (dy-
introduced[19–22]todescribethebehaviorofchemically namical) density functional theory. The latter is based
heterogeneoussphericalparticlessuchasmagneticJanus on a generalized diffusion equation, where particle inter-
spheres composed of two different magnetic materials actions are incorporated adiabatically via a free energy
(see,e.g.,[23,24]). Insomecasesthepointdipoleapprox- density functional derived from equilibrium (static) den-
imationhasfoundtobeinappropriateorcomputationally sity functional theory (see, e.g., [37]). Similar to the
inefficient, thus, models with spatially separated charges passice case, currenttheoretical studiesof activeparticle
(seeFig.1(c))interactingeitherwithtrueCoulombicin- systems heavily involve particle-based simulations such
teractions [8, 25, 26] or exponentially screened, Yukawa- as BD, but also methods incorporating hydrodynamics
like potentials [27] are used as well. at low Reynolds numbers where the Navior-Stokes equa-
The bottom row of Fig. 1 shows models of parti- tion reduces to the Stokes equation. Common repre-
cleswith”multipolar”interactions,involvingeithermore sentativesaremulti-particlecollisiondynamics(see,e.g.,
than one (point) dipole moment, or more than two spa- [38,39])andStokesianDynamics[40]. Atthesametime,
tially separated charges. The models in Fig. 1(d) and thesesystemsarestudiedonthebasisofkinetic(Fokker-
(e)havebeeninspiredbymetallodielectricJanusspheres Planck-type) equations [41], by dynamical density func-
such as polystyrene colloids with gold patches. Exper- tional theory [42] as well as via continuum approaches
imentally, such particles have been extensively investi- [43].
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III. AGGREGATION AND RELAXATION MSD as tα with α < 1. At the same time, one observes
DYNAMICS a marked slowing down of the intermediate scattering
(cid:68) (cid:69)
function F (q,t) = N−1(cid:80)N exp[iq·r (t)−r (0)] ,
s j=1 j j
In this section we summarize recent theoretical work
yielding a finite non-ergodicity parameter F (q,t → ∞)
s
on the dynamics of dipolar and multipolar colloids close
[8, 25, 26]. Overall, the behavior is thus similar to that
to thermodynamic equilbrium, that is, in the absence of
observed in short-ranged attractive systems forming re-
a driving field or an intrinsic propulsion mechanism. In
versiblegels[17]. Moreover,thedumbbellsystemsshows
particular, we discuss aspects of kinetic aggregation and
interesting nonlinear response to an external magnetic
dynamical slowing-down.
field [8].
We further note that some aspects of gel-like dynam-
ics, particularly sub-diffusion and dynamically heteroge-
A. Simple dipolar systems: self-diffusion and neous behavior, has also been observed in a MD simula-
gelation tion study of strongly coupled dipolar soft spheres in a
homogeneous external field [46]. The latter ”helps” the
particles in forming extended structures, that is, chains
Dipolarparticlesareprototypesofself-assemblingsys-
along the field, which finally percolate in the field direc-
tems: even for the simplest systems, that is, spheres
tion. Interestingly,thesesystemsadditionallydisplayev-
with centered point dipoles (see Fig. 1(a)), the resulting
idence of super-diffusion (characterized by a MSD ∝ tα
interaction given in Eq. (1) is characterized by strong
with α > 1) at larger times, consistent with dynamic
anisotropy favoring head-to-tail ordering. Equilibrium
lightscatteringexperiments[47]ofsuchstronglycoupled
aspects of the resulting cluster- and chain formation in
systems. On the other hand, slow dynamical relaxation
model systems with dipolar (or multipolar) interactions
related to static percolation has also been predicted to
have been studied extensively by computer simulations,
occurinaLangevindynamicsstudyofacobaltnanopar-
ground state calculations, and association theories (see
ticle suspension [48], as well as in Discontinuos Molecu-
e.g.,[12,44,45]andreferencestherein)andarenowadays
lar Dynamics simulations of a 2D system of spheres with
quitewellunderstood. Thisconcernsboth,systemsinex-
short-ranged dipole-like interactions [27].
ternal fields and zero-field systems involving permanent
dipoles. We note, however, that establishing the precise
relationbetweenclusteringandtheequilibriumphase di-
agram, particularly the existence of a first-order vapor- B. Multipolar particles: Networks and aggregation
liquid transition predicted by perturbation theories, has
been a challenge for decades even for the paradigmatic Advances in material chemistry have enabled fabrica-
DHS model [44]. tionofnovelparticlesystemswheretheanisotropicinter-
Thestrongtendencyofdipolarspherestoself-assemble particle interactions can not be approximated by those
into clusters and chains prompts the question whether between dipoles, but have rather multipolar character.
simple dipolar particles can form gels, similar to what is In many cases, these interactions are induced by exter-
found in systems of ”patchy” particles [17]. Patchy par- nal electric or magnetic fields (for recent reviews from
ticles interact via short-range attractive potentials be- the experimental side, see [6, 49]).
tween a finite number of interaction sites on each par-
ticle. This leads to long-lived physical bonds forming
eventually a system-spanning network. The latter is a
Multipolar particles in uni-directional fields
prerequiste for a dynamically arrested state with bro-
ken ergodicity. In conventional, DHS-like, models parti-
Aprimeexamplearepatchymetallodielectricparticles
cles arrange in clusters and chains which, however, ap-
suchaspolystyreneparticleswithgoldpatches. Inauni-
pearnotsufficientlyconnectedtoformgels. Therefore,a
directionalfield,theseparticles(forsketchesseeFig.1(d)
number of recent MD simulation studies [8, 25, 26] have
and (e)) can form a variety of novel structures, such as
considered dumbbell-like particles made up of two in-
zig-zag chains along the field (also reported for magnetic
terpretating soft spheres carrying oppositie charges ±q
Janusspheres[24])and,importantly,stringsandclusters
at their centres, separated by a fixed distance d. The
perpendicular to the field [30].
resulting dipole moment is µ = qd (see Fig. 1(c) for
The latter phenomenon, which occurs at high fre-
a sketch of corresponding spherical particle). The ex-
quencies f > f of the applied AC field, was stud-
tended dipole enhances string formation and facilitates c
ied theoretically in [31] by means of MC and MD sim-
branching of the chains, leading to true threedimen-
ulations. The model consists of a binary mixture of
sional (3D) networks. Dynamical signatures of this
spheres with two laterally shifted, oppositely oriented,
process are, e.g., the emergence of sub-diffusive ranges
dipole moments (see Fig. 1(d)), and spheres with only
in the mean-squared displacement (MSD) ∆r2(t) =
(cid:68) (cid:69) one dipole moment (representing ”defect” particles in
N−1(cid:80)Ni=1(rj(t)−rj(0))2 , where t is the time. Sub- realsystems). Alldipolevectorsareorientationallyfixed
diffusion is characterized by a time-dependence of the due to their induced character. The simulations demon-
4
tem’s elastic response and its transport properties.
Completely different behavior of metallodielectric
spheres occurs at certain frequencies below the critical
one: Here, the particles do not only interact in a differ-
entmanner;theyalsodisplayself-propulsionindirections
perpendiculartothefield[29]. Thecollectivedynamicsof
the resulting ”dipolar swimmers” is discussed in Sec. V.
Multipolar particles in bi-directional fields
Another recent topic is the aggregation behavior
of polarizable colloidal spheres under the influence of
bi-directional fields. An exemplary system are la-
tex spheres with embedded paramagnetic nanoparticles;
these spheres are responsive to both, electric and mag-
FIG. 2. Formation of crosslinked 2D networks in systems netic fields. Using fields in an orthogonal set-up the par-
of metallodielectric particles (mixed with simple dielectric ticleshavebeenshowntobuild2D,percolatednetworks,
particles) in an uniaxial electric field [31]. (a) Optical mi- whose structure appears to depend on time and whose
croscopy images of the experimental system of 5 µm parti- topological (connectivity) properties can be tuned inde-
cles, (b) MC simulation snapshot of a corresponding model pendentlybythetwofields[6]. In[33],BrownianDynam-
(black: multipolar particle, see Fig. 1(d), grey: simple dipo- ics simulations were carried out on the basis of a simpli-
lar particle, Fig. 1(a)) at a temperature below the transver-
fied, yet generic model suitable for (spherical) particles
salpercolationthreshold,(c)and(d)groundstatestructures
in crossed fields. The model contains four charges with
of small clusters illustrating energetically preferred configu-
variable separation δ (see Fig. 1(f)), which interact via
rations, (e) transversal percolation probability as function of
reducedtemperatureT∗ fordifferentsystemsizes,revealinga screened(thus,short-ranged)Yukawapotentials. Atvery
transversal percolation transition. The longitudinal percola- lowtemperaturesthemodelpredictsnon-equilibriumag-
tiontransitionoccursatsomewhatlargerT∗. (f)Bondcorre- gregation into large clusters with square- or hexagonal-
lationfunctionand(g)mean-squareddisplacement(transver- likelocalorder. Theoverallbehavior(includingtheclus-
saldirection)asfunctionsoftime(t∗). In(g),thelowestcurve ter’s fractal dimension and behavior of bond time corre-
pertains to a temperature below transversal percolation. lations) is similar to diffusion-limited aggregation. Upon
increasing the temperature the clusters ”melt” and the
system transforms into a fluid phase, consistent with re-
sultsofasimplemean-fielddensityfunctionaltheory[33].
strate formation of percolated, gel-like structures both
Importantly, dynamical clustering is a ”hot topic” not
parallel (longitudinal) and perpendicular (transversal)
only in the area of passive colloids discussed so far, but
to the field, consistent with optical microscopy experi-
also in systems of active, self-propelled colloids (see also
ments on metallodielectric particles [30] (see Fig. 2(a)-
Sec. VB). Experiments [51] and simulations [52, 53] in-
(b)). The transformation between these structures takes
volving (chemically driven) colloidal swimmers report
place via two clear, longitudinal and transverse perco-
formation of a novel cluster phase (resulting from a per-
lation transitions, with the novel 2D-crosslinked struc-
manent dynamical particle merging and separation) at
ture being stable for a broad range of concentrations
intermediatedensities. Contrarytopassivesystem, clus-
and coupling strengths [31]. Moreover, the networks
ters can already occur in purely repulsive active systems
are characterized by strongly hindered translational dy-
due to a self-trapping mechanism [54]. An intriguing
namics: The MSD shows a plateau in transversal direc-
questionforboth,passive[33]andactivecolloids[10,11],
tion after the second percolation transition. Further,
concernstherelationbetweenclusteringandequilibrium
orientational correlations reveal an extremly long bond
phaseseparation. Herewemayexpectthatthetwofields
lifetime, suggesting a strong persistence of the network
can strongly benefit from one another, especially when
(see Fig. 2(f)-(g)). Similar dynamic behavior has been
active colloids with directional interactions are consid-
very recently reported in MD many-particle simulations
ered [32, 113].
of ”inverse patchy colloids” characterized by heteroge-
neouslychargedsurfaces[50]: Thesesystemsdonotonly
self-assemble in a variety of structures depending on the
charge distribution [35]; they also display a drastic de- IV. COLLECTIVE BEHAVIOR IN ROTATING
FIELDS
crease of the diffusion coefficient with decreasing tem-
perature or increasing density, indicating again the exis-
tenceofagel-likestate[50]. Fromanapplicationalpoint We now turn to systems driven out of equilibrium by
of view, the gelation is of prime importance for the sys- means of rotating, magnetic or electric, external fields
5
(see [55] and [7], respectively, for recent reviews covering the interparticle vector r and the direction perpendic-
ij
the experimental and material science perspective). ular to the plane of the field (i.e., the z-direction). The
Earlierexperimentalandtheoreticalstudiesinthearea time-averagedpotentialcorrespondstoaninverteddipo-
of rotating fields typically focus on the field-induced dy- lar (ID) potential, which is attractive if the angle Θ
ij
namics of an isolated colloid such as a magnetic rod [56– satisfies cos2Θ < 1/3, i.e. if the particles i and j are
ij
60],amagneticchain[61]orfilament[62],oranoptically approximatelyinthesameplanewithrespecttothefield.
excitablenanorod[63]inaviscousmedium. Understand- Conversely, if the angle Θ satisfies 1/3 < cos2Θ , the
ij ij
ing the resulting single-particle rotational dynamics is particles repel each other. This combination of in-plane
important for the advancement of actuators [58], sensors attraction and repulsion along the rotation axis [66, 68]
[56], molecular switches, particles in optical traps [63], explains why layers are a favorable configuration.
and in the more general context of microfluidics [59]. From a theoretical perspective, the above time-
However,evenmoreintriguingisthecollective,dynam- averaged potential allows for an ”effective equilibrium
ical self-assembly of colloids exposed to rotating fields. description” of the rotating-field-driven dipolar fluid.
Indeed,inmaterialscience,rotatingfieldshavebeenreal- It should be noted, however, that the resulting over-
ized as a powerful tool to control self-assembly processes all (quasi-equilibrium) phase behavior additionaly de-
into functional materials [24, 64]. In fact, even relatively pends on the remaining parts of the effective pair in-
simple dipolar systems can form novel structures, which teraction. UsingMCsimulations,SmallenburgandDijk-
do not exist in equilibrium. stra [69] have shown that layered-fluid phase do appear
for charged spheres with inverted dipolar interactions,
Eq. (4). On the other hand, the phase diagram of col-
A. Polarizable spheres in rotating fields loidalhardsphereswithinverteddipolarinteractionsdis-
plays a gas-liquid transition, a hexagonal ABC stacked
A classical example in this context, first discussed by crystalphase,astretchedhexagonal-close-packedcrystal,
Martinetal. [65]are3Dsystemsofparamagnetic(orpo- but no layered structures.
larizable) spherical particles in magnetic (electric) fields Anotherintriguingexampleoftheuseofrotatingfields
rotating in a plane (biaxial field). Specifically, let us to create a specific time-averaged (quasi-equilibrium) in-
assume a field B(t) rotating with frequency ω in the teractionhasbeenproposedbyOstermanet al. [70]who
0
x-y-plane, considered superparamagnetic spheres in a precessing
magneticfield. Here,astaticandarotatingfieldarecom-
√
B(t)=B0(excosω0t+eysinω0t). (2) bined in a ”magic” opening angle (θ = arctan(1/ 3 ≈
54.70), such that the spheres effectively feel an isotropic
Since the particles are polarizable and spherical, they pair attraction similar to the van der Waals force be-
aquire an induced dipole moment parallel to the field at tween atoms. However, additional polarization interac-
all times, that is tions, and thus, many-body interactions, lead to the for-
mation of crosslinked network, which coarsen and even-
µ (t)=µ (t)=µ(e cosω t+e sinω t) (3)
i j x 0 y 0 tually form ”self-healing” membranes. These investiga-
tions have been recently extended towards 2D systems
with µ ∝ B . Note that this simple dipole-field relation
0 [71]: depending on the opening angle θ, a broad range of
does not hold for the induced dipole moment of non-
quasi-equilibrium structures from hexagonal crystals to
spherical paramagnetic particles characterized by differ-
froth-like patterns is observed both in experiments, and
ent polarizabilities along and perpendicular to the sym-
in parallel numerical (MC) simulations of a model sys-
metry axes; here, the direction of µ (t) can deviate from
i tem involving a mixture of patchy and non-patchy par-
that of the field (see, e.g. [60]).
ticles. This study [71] underlines the role of many-body
For sufficiently large field strength B , both experi-
0 effects for structure formation in complex paramagnetic
ments and computer simulations [65–69] reveal the for-
systems.
mation of layers in the field plane, i.e. a spatial symme-
try breaking induced by the rotating field. The phys-
ical mechanism becomes clear by considering the time-
averaged dipolar potential [66] defined as B. Permanent dipoles in rotating fields: Layering
and synchronization
(cid:90) t0+τ
UID(rij)=τ−1 UD(rij,µi(t),µj(t))dt As discussed above, many phenomena in rotating-field
t0 driven systems of induced dipoles can be explained from
=−µ2(1−3cos2Θij). (4) an equilibrium perspective involving the free energy and
2r3 resulting phase behavior in the time-averaged field. Less
ij
is known about the corresponding behavior of particles
Inthisequation,UDisthedipole-dipolepotentialEq.(1) with permanent dipole moments, such as the (ferromag-
between the induced dipoles given in Eq. (3), τ =2π/ω netic)particlesofaferrofluid. Here,theindividualorien-
0
is the oscillation period, and Θ is the angle between tationscanbedifferentfromtheoneoftherotatingfield
ij
6
duetothermaleffectsandeffectsfromthesolvent. Thus,
theessentialprerequisiteforbuildingatime-averagedpo-
tential, that is, full synchronization with the field (see
Eq. (3)), can break down.
The behavior of a 3D system of permanent, strongly
coupled dipoles in a planar rotating field has been in-
vestigated in Ref. [72] on the basis of LD simulations
and density functional theory. In this study, a full
non-equilibrium state diagram as function of the driv-
ingfrequencyandthefieldstrengthwasmappedout,see
Fig. 3(b). At small frequencies and sufficiently large B ,
0
the system is in a synchronized state, where the individ-
ual particles follow the field with (on average) constant
phase difference. This is indicated by a single-peaked
distribution of phase differences, f(Φ), where Φ is the
i
in-plane angle between dipole vector i and the external
field, see Fig. 3(c). Thus, the overall magnetization is
FIG. 3. Structure formation in simulated systems of per-
M(t) ≈ M (e cos(ω t+φ )+e sin(ω t+φ )) where
0 x 0 0 y 0 0 manent dipoles in rotating (planar) fields [22, 72, 73]. (a)
φ is the average phase difference. The synchronization
0 Layerformationina3Dsystemofcentereddipoles,Fig.1(a).
is accompanied by 3D layer formation, similar to sys- (b)Correspondingnon-equilibriumstatediagraminthefield
tems of induced dipoles in rotating fields. At very low strength (B∗)–frequency (ω∗) plane. The dashed line in-
0 0
frequencies, the translational structure within these lay- dicates the results from the effective single-particle theory,
ers is characterized by small chains with head-tail order- Eq. (5). (c) and (d): Distribution of phase differences in the
ing (similar to what is seen in equilibrium dipolar sys- layered(c)andunlayered(d)region,showingthatlayeringis
tems). At somewhat larger frequencies, the local struc- intimatelyrelatedtosynchronizationoftheparticles[72]. (e)
Clusterformationinthesynchronizedregimeofa2Dsystem
ture rather resembles that in a isotropically interact-
of permanent (centered) dipoles in a rotating field [73]. (f)
ing 2D system which tend to form hexagonal lattices.
Doublelayerformationina3Dsystemofoff-centereddipoles
However,forpermanentdipolesthelayeringbreaksdown
(see Fig. 1(b)) [22] in the synchronized regime.
even at small ω when the field strength B becomes too
0 0
small. AsshowninRef.[72]thetransformationofanun-
layered(smallB )intoalayered/synchronized(largeB )
0 0
state can be described as a quasi-equilibrium transition for the phase angle Φ,
induced by the competition between the time-averaged,
inverted dipolar interaction (which favors layering) and
thelossoftranslationalentropyrelatedtotheonsetof1D dΦ = ω0 −sinΦ, (5)
translational order. This has been demonstrated on the dτ ω
c
basis of an effective (equilibrium) free energy functional.
Completelydifferentbehaviorisfoundathighfrequen- where ω = µB /γ and τ = ω t (see dashed line in
c 0 c
cies and field strengths. Under these conditions, the Fig. 3(b)). We note that this type of equation occurs in
picture of synchronously rotating dipoles breaks down. variouscontextsrelatedtosynchronizingsystems[75,76];
Instead, one observes a mixture of rotating and counter- there it is often referred to as Adler equation [77]. For
rotatingorrestingparticles,asananalysisoff(Φ)reveals 0 ≤ ω < ω Eq. (5) has two fixed points (φ˙ = 0, i.e.,
0 c
(see Fig. 3(d)). Similar behavior has been observed in constant phase difference), with the stable one given by
systems of rotating ellipsoids [63] and theoretically ana- φ = arcsin(ω /ω ). Here, the torque due to friction
0 c
lyzedinRef.[74]bydynamicaldensityfunctionaltheory. equals the torque due to the field. At ω = ω , i.e. at
0 c
φ = π/2, the two solutions form a saddle-node bifurca-
The desynchronization induces, at the same time, a
tion and there are no fixed points for ω >ω . At these
breakdown of the translational, layered structure. De- 0 c
highfrequencies,themaximaltorquethatcanbeexerted
spite this complex many-particle behavior, the high-field
bythefieldisinsufficienttobalancethefrictionaltorque.
boundary between layered and non-layered states can be
welldescribedwithinaneffectivesingle-particleapproach The main simulation result of Ref. [72], namely that
for the rotational motion of a permanent dipole in a the rotational dynamics of the individual particles (syn-
viscous medium. This dipole experiences, first, a field- chronous versus asynchronous motion) determines the
inducedtorquegivenbyτ =µ×B=µB sinΦ(where structure of a many-body system of permanent dipoles,
B 0
Φ is again the phase lag), and second, a viscous (dissipa- has been recently confirmed by experiments involving
tive) torque τ =γdΦ/dt. For large friction constants magnetic Janus spheres with both, ferromagnetic and
diss
γ, the motion can be considered as overdamped, leading paramagnetic response [24, 78]. We come back to these
to a first-order (in time), nonlinear equation of motion studies in Sec. IVD.
7
C. 2D systems in rotating fields: Clustering plex. In fact, already spheres with off-centered perma-
nent dipoles, Fig. 1(b), yield different structures such
as colloidal double-layers [22] with phase-ordered time-
Further interesting phenomena occur when the parti-
dependent orientations, see Fig. 3(f).
clesaretranslationallyconfinedtotheplaneoftherotat-
ing field. Corresponding experiments with micron-sized
(paramagnetic) particles [79] and magnetic Janus parti-
cles[24]haverevealedtheformationofclusterstructures,
whoseinternalstructureisgovernedbyhexagonalorder-
ing and whose size varies with time. Such clustering and
coarsening has also been observed in Langevin and MC
An experimental system attracting much attention [3]
computer simulations [73] of a model system involving
inthiscontextaremagnetic”Janus”spheresinvestigated
permanent dipoles (see Fig. 3(e)). There, the clustering
byYanetal. [2,24,78]. TheseJanusspheresinvolvetwo
is associated to synchronous rotational motion (similar
different magnetic materials, Ag and Ni, leading to both
to the layering in 3D). The simulations [73] further sug-
para- and ferromagnetic properties (for other types, see
gest that the clustering can be related to an underlying
e.g. [7]). Whenplacedinaprecessingmagneticfieldwith
condensation phase transition of the (quasi-equilibrium)
an opening angle θ, the particles investigated by Yan et
system defined by the time-averaged dipolar potential.
al. [2]bothrotateanddisplayverticaloscillationssimilar
Indeed, in the 2D set up considered, the time-averaged
to nutations of a gyroscope, with the corresponding fre-
dipolar potential (see Eq. (4] for the 3D case) is purely
quenciesbeingtunablebyθ. Thepresenceofverticalos-
attractive and long-ranged,
cillationsimpliesanadditionaldegreeoffreedom(”phase
variable”) not present for chemically homogeneous par-
µ2
UID(r )=− . (6) ticles. Thus, the particles can not only synchronize with
2D ij 2ri3j thefield,theyalsocansynchronizewitheachother(phase
locking), similar to networks of coupled oscillators (pen-
yielding a first-order vapor-liquid transition in the 2D dulums, fire flies ...) [76]. As shown in [2] by a combi-
system. Inside the corresponding coexistence curve, the nation of experiment, theory and computer simulations,
system forms domains of low and high density with the synchronization (which arises due to the dipolar in-
the domain size growing in time. Moreover, for suffi- teractions at small distances) leads to the formation of
ciently large coupling strengths the translational struc- different types of phase-locked dipolar dimers and even-
ture within the clusters is hexagonal. The thermody- tually tubular structures [2]. These structures disappear
namicconditionsmatchthoseforrotatingfields,support- when the field conditions are changed such that the par-
ingtherelationbetweenrotating-field-inducedclustering ticles desynchronize. From the theory side, synchroniza-
and equilibrium phase separation [73]. tion transitions of coupled oscillators are often described
Importantly, these results remain qualitatively un- within the Kuramoto model [83]. As shown in Ref. [2],
changed when hydrodynamic interactions (HI) in a far- it is indeed posible to describe the collective behavior of
field approximation are taken into account [80]. The theJanussphereswithintheKuramotolanguage. Inpar-
corresponding expressions for dipolar fluids, involv- ticular, one can set up an effective single-particle equa-
ing translation-translation coupling (Rotne-Prager ten- tion resembling the Adler equation given in Eq. (5). In
sor [81]), translation-rotation and rotation-rotation cou- addition, the authors have performed MD simulations of
pling are given in [82]. As shown in Ref. [80], the main LJsphereswithoff-centeddipoles(supplementedbyself-
effects of HI are a) an acceleration of the cluster for- consistent field calculations to taken into account polar-
mation process and b) a rotation of the clusters due to izability) supporting the experimentally observed struc-
hydrodynamic translation-rotation coupling. The latter ture formation. In summary, this study forges a link be-
effect has also been observed in 2D experiments of mag- tween synchronization and self-assembly, which will cer-
netic Janus colloids [78]. Physically, the cluster rotation tainly stimulate further investigations at the interface of
can be explained by the presence of solvent-induced, un- dynamic soft matter and nonlinear dynamics [3]. For
balanced shear forces at the edges of the cluster. instance, from a more abstract perspective the suspen-
sion of magnetic Janus particles can be viewed as a net-
work of oscillators with non-local coupling. Such net-
D. Janus-like magnetic colloids: works have been predicted [84] to spontaneously form
Self-synchronization and novel structures ”Chimera”stateswhichcontainspatialdomainsexhibit-
ing synchronized dynamics and others that have desyn-
So far, our discussion of rotating-field induced struc- chronized dynamics. Experimentally, such states have
ture formation in systems of permanent dipoles was fo- been observed only recently e.g. in systems of coupled
cused on ”simple”, spherical particles with central, per- chemical (Belousov-Zhabotinsky) oscillators [85]. It is
manent dipole moments. Novel structures besides lay- an intriguing question whether suspensions of complex
ers (3D) or hexagonal clusters (2D) can form when magnetic particles are also capable of forming Chimera
the particle’s response to the field becomes more com- states or other complex nonlinear behavior.
8
V. COLLECTIVE TRANSPORT AND PATTERN So far, most studies in this area have been under-
FORMATION taken for single colloids [90, 92] or systems with negli-
gible interactions. However, recent experiments suggest
In the preceding section we have concentrated on the that interaction effects are important for the transport
collective rotational motion and associated structure for- ofmagneticchainsinmagneticratchets[95]andtheself-
mation of dipolar colloids. Another emerging area is assembly of particles into patterns on magnetic lattices
thedirectedtranslationalmotion(transport)anditscon- [96]. Only very recently [97], first theoretical steps have
trollability by external fields. Here we focus on two as- been undertaken to investigate ratchet-driven transport
pects: transport by ratchet-like external potentials and of interacting ensembles of paramagnetic colloids. In
transport by self-propulsion. A closely related topic, not Ref. [97], a deterministic, analytically tractable ratchet
covered in the present article, is the recently discovered modelwasdevelopedtodescribetheimpactofattractive
pattern formation of magnetic particles at fluctuating or repulsive, field-induced interactions in the framework
liquid-air and liquid-liquid interfaces (for a review, see of a time-averaged field. Depending on the sign of the
[55]): Here the interplay of alternating magnetic fields, effective interaction one observes the formation of sta-
(ferro-)magnetic interactions and hydrodynamic inter- ble doublets or oscillating pairs of particles, which move
actions induced by local oscillations of the liquid sur- withconstantspeed. Anexperimentalapplication[98]of
face yields a number of fascinating structures such as these cooperative transport phenomena is the assembly
”snakes”, ”asters” [4] and swimmers [86]. These struc- and transport of straight magnetic chains and the use of
tures have been extensively investigated by experiments, magnetic ratchets for microsieving of non-magnetic par-
aphenomenologicalapproach(involvingamplitudeequa- ticles.
tions forparametric waves) and MD simulations coupled
Another interesting effect of particle interactions has
to thin-film hydrodynamics [36, 55].
been reported for systems of paramagnetic (iron-doped
Similar to the other areas discussed in this review,
polystyrene)particlesexposedtoanopticalpotentialen-
investigations of (collective) dipolar transport are moti-
ergy landscape composed of a (symmmetric) potential,
vated,ontheonehand,byfundamentalquestionssuchas
a biasing force and an oscillating field [9]. In this situ-
understanding Brownian motion in complex geometries,
ation, already a single colloid can display complex dy-
exploring the emerging dynamical structures in interact-
namics characterized by dynamical mode locking, where
ing systems, and, most recently, the interplay between
the average particle velocity increases step-wise (rather
self-propulsion and dipolar forces. On the other hand,
than monotonically) with the external drive. This syn-
the topic of dipolar transport receives increasing inter-
chronizationphenomenonissignificantlyenhancedwhen
est from the application side due to the potential use
instead of a single colloid a flexible magnetic chain, held
of magnetic colloids e.g. in biomedical applications (as
together by magnetic interactions, is exposed to the po-
drug delivery cargo), as micropropellers in lab-on-a-chip
tential. The chain then exhibits a ”breathing mode”
devices, or as active microrheological probes.
which leads to caterpillar-like motion and to stabiliza-
tionofthesynchronizedstate. Fromthetheoreticalside,
theobservedeffectscanbenicelydescribedintheframe-
A. Collective transport in ratchet potentials
work of the Frenkel-Kontorova-Model [99] of solid-state
physics, a generic model for nonlinear transport and ex-
Ratchet potentials are characterized by combinations citations in complex geometries.
of a static, asymmetric periodic potential and an al-
ternating field. In such a potential landscape, non- We finally mention recent theoretical work on the in-
equilibrium fluctuations can induce net particle trans- terplay of ratchet potentials and phase separation in 2D
portintheabsenceofabiasingdeterministicforce. This magneticcolloidalmixtures. In[100],abinarymixtureof
genericeffectarisesinmanyareasofphysicsandbiology ferromagnetic particles (modeled by spheres with classi-
[87]; and it has been studied in a large variety of optical cal Heisenberg spins) and non-magnetic spheres subject
[88], magnetic [57, 89–91] and biological systems (e.g., to a 1D rocking ratchet potential was investigated by
molecular motors, molecular sieves). meansofdynamicaldensityfunctionaltheory. Intheab-
Static, magnetic periodic potentials can be created, sence of the external potential the system undergoes a
e.g., by using ferrite garnet films [92] (where ferromag- first-order fluid-fluid demixing transition (driven by the
netic domains with opposite magnetization direction are magnetic interactions), resulting in spinodal decomposi-
aligned in stripe-like fashion) or by using a periodic ar- tionincertainparameterranges[101]. Theinterplaybe-
rangement of micromagnets on “lab-on-a-chip” devices tweenthisintrinsic,thermodynamicinstabilityandtime-
[91, 93, 94]. An additional oscillating field is introduced dependent external forces then leads to a novel dynami-
by combining the static potential with a rotating mag- cal instability where stripes against the symmetry of the
netic field. The latter yields a periodic increase (de- external potential form. Moreover, the structural transi-
crease) of the size of domains with parallel (antiparal- tion associated to stripe formation suppresses the usual,
lel)magnetization,whicheventuallyenablestransportof ratchet-driven transport of particles along the longitudi-
(para-)magnetic colloidal particles. nal direction [100].
9
B. Pattern formation in systems of dipole-coupledassembliesonaliquid-airinterfacedueto
(self-)propelled, active dipolar particles apulsatingmagneticfield. Thelocomotionisinducedby
periodicdeformationsoftheparticlearrangement(rather
The collective behavior of active, self-propelled parti- thanbycapillarywaves[86]). Finally,wementionexper-
cles or ”swimmers” has received an explosion of interest iments on quasi-2D suspensions of dielectric patchy par-
during the last years (for recent reviews from the the- ticles in vertical electric fields [112]. These particles per-
ory perspective, see Refs. [41, 43, 102, 103]). From the form random, in-plane swimming motion while interact-
experimental side, many types of artifical swimmers in- ing via repulsive dipolar interactions. The experiments
volving colloidal particles (contrary to biological swim- [112] reveal mesoscopic turbulence similar to what has
mers such as bacteria) have been proposed, with differ- been seen in bacterial solutions.
ent propulsion mechanism, typically based on catalytic From the theoretical side, Kaiser et al. [113] have in-
or phoretic effects. One prominent model of such swim- vestigated cluster formation of active dipolar particles
mers, particularly suitable for spherical objects such as with permanent (magnetic) dipoles by means of numer-
those considered in this review, is the so-called squirmer ical solution of overdamped, deterministic (noise-free)
[104, 105], that is, spheres with a pre-described tangen- equations of motion. Specifically, the active dipoles are
tial velocity profile on the surface which initiates the represented by dipolar soft spheres with a propulsion
swimming. The parameters of this profile then deter- force in the direction of the dipole moment. The self-
mine whether the squirmer is a pusher or puller. Orig- propulsion is shown to create complex cluster dynam-
inally introduced to mimic the synchronized beating of ics, involving a variety of states with different (non-
cilia[104,105],themodelisnowconsideredasgenericfor equilibrium) internal structures and different modes of
spherical microswimmers including, e.g., active droplets motion of the cluster’s center of mass.
and diffusiophoretic particles [106]. Particle-based sim-
ulations of squirmer suspensions at different concentra-
tions have demonstrated that, as a result of hydrody- Lane formation
namicinteractionsbetweenthesquirmers[38],thesesys-
temdisplayveryrichdynamicalbehaviorincludinglong- Aspecialclassofdipolarcolloidalswimmersisrealized
range polar order [40] and a phase separation between a by metallodiectric (gold-patched polystyrene) spheres in
gas-like and a cluster phase [39]. a 2D set-up with an in-plane AC (electric) field. At
certain frequencies below the critical one (for the high-
frequency regime, see Sec. IIIB), experiments [29] re-
Swimmers with dipolar interactions port spontaneous motion of each particle in one of the
two directions orthogonal to the field, specifically, away
A significant number of artificial swimmers involves from the particles gold patch. The underlying mecha-
magnetic materials and thus, exhibits magnetic dipolar nism foots on the difference in strength of the (parallel)
properties. Recent examples of magnetic swimmers are induceddipolemomentinthemetallicanddielectricpart
sphericalJanusparticleswithamagneticcap, whichcan ofthesphere. Thisasymmetryyieldsanasymmetricflow
be driven catalytically [107] or by thermophoresis [108]. of solvent charges induced by the (in-plane) AC electric
Moreover, magnetic particles with anisotropic shapes field (induced-charge electrophoresis).
such as magnetic filaments [62], nanowires, spheres with The collective behaviour of these driven particles was
helical tails, clusters of DNA-linked paramagnetic col- investigated theoretically in [32] based on the asymmtric
loids [109, 110] and magnetic ”asters” [86] can be set particle model sketched in Fig. 1(e). Specifically, the
into motion via different magnetic fields, see [7] for a study reports results from BD simulations of a binary
summary of recent experimental work. In fact, the ex- mixtureofparticleswiththeirsmallerdipolemomentei-
ternal control or ”guidance” of swimmers by magnetic ther in the right or in the left part. The self-propulsion
fields is an important topic from an applicational point is modelled by a constant force whose direction depends
of view. Clearly, a main effect of the external field is on the particle type, see Fig. 4. The resulting equations
the suppression of rotational diffusion. In this context, of motion are given by γr˙i = (cid:80)Nj=1∇U(ij)+fsd,i +ζi
arecentcombinedexperimentalandsimulationstudy[5] where U(ij) is the full pair interaction, γ is the friction
has reported interesting collective effects of magnetically constant, ζ represent white noise, and fs is the con-
i d,i
sensitiveswimmers. Theswimmers(sphereswithalight- stant driving force. At zero drive (f = 0), the par-
d
activatedhematitecube)interactviaphoreticattraction, ticles assemble into staggered chains, which have been
leading to spontaneous aggregate into ”living”, crystal- previously observed experimentally [28]. Switching on
like clusters. The motion of these clusters can be di- the driving force, the system displays a transition to-
rected by an external magnetic field. Thus, the overall wards a laned state (Fig. 4(c)). Indeed, lane formation
non-equilibrium self-assembly behavior of this correlated is a protoype of a non-equilibrium self-organization pro-
system is tunable by a combination of light and mag- cess,whereanoriginallyhomogenousmixtureofparticles
netic fields. Another recent experimental study [111] re- (such as charged colloids) moving in opposite directions
portstheonsetofcooperativeswimmingofferromagnetic segregates into macroscopic lanes composed of different
10
tures, aggregation and phase transitions), from nonlin-
ear dynamics of synchronizing networks, and from the
physics of pattern formation in driven or autonomous,
dissipative systems, to name just a few. At the same
time, the field attracts substantial attention from ma-
terial science in the context of developing novel materi-
als with ”programmable” response to mechanical stress,
shear, magnetic and thermal fields [6], as well as from
microfluidics [7]. It is the interplay between these sci-
entific communities which has led to the recent burst of
attention in this area.
One obvious trend for future (theoretical and experi-
mental) research is a shift towards the collective dynam-
ics of anisotropic dipolar or multipolar particles, e.g.,
dumbbells, ellipsoids, or disks. Compared to spheres,
shape anisotropy can induce an enhanced sensitivity to
externalfields(regardingparticularlyrotationalmotion),
FIG. 4. Lane formation in binary systems of dipolar mi- but also additional degrees of freedom (”phase vari-
croswimmers moving into opposite directions in response to ables”) in oscillating fields, enabling new types of syn-
a propulsion force fd (results from a BD simulation study chronization and associated structure formation. More-
[32]). (a) Assembly into staggered chains at zero force, con- over, in the area of active systems, already non-polar
sistentwithexperimentalresults(b)[28]. (c)Snapshotofthe
shape-anisotropic particles display complex swimming
laned state. (d) Laning state diagram at fixed particle den-
modes not seen for spheres [114]. It would be very in-
sity, showing the occurence of lanes (measured by the order
teresting to explore the swimming behavior of shape-
parameter Φ) at sufficiently large driving forces and certain
rangesofthedipolarcouplingstrengthµ∗. Verticalline: cou- anisotropic, dipolar swimmers which moreover offer ad-
plingstrengthsrelatedtoaspinodalinstability(condensation ditional ways of external guidance. Another promising
transition) in the equilibrium system (f = 0). Blue-dotted direction concerns the control of dynamical structures in
d
line: Estimate for the break-down of laning based on a bal- suchsystems: Hereonemayenvisiontheselectionaspe-
ance between the attractive force stemming from multipolar cificdynamicalstructure(ormodeofmotion)fromapool
interactions and the driving force [32]). ofcompeting”candidates”byadaptingtheexternalfield
based on information from the systems. Such feedback
control[115]strategies(whicharewell-establishedinthe
species. As shown in [100], anisotropic dipolar interac- fieldofnonlinearscience[116])arealreadyusedforguid-
tions induce several novel features within the (otherwise ing(phoretic)swimmers[117]andtomanipulatedynam-
well-studied) laning transition. In particular, laning oc- ical structures [118] and dynamical assembly of simple
cursonlyinawindowofinteractionstrengths, Fig.4(d), colloids [119]. Applications to the collective dynamics of
which is closely related to phase separation process (as dipolarandmultipolarcolloidsthusseemverypromising.
confirmed by density functional arguments [100]). This
is yet a further example demonstrating the intimate re-
lationship between complex out-of-equilibrium collective VII. ACKNOWLEDGEMENT
behavior and equilibrium properties.
I would like to acknowledge collaborations and stimu-
lating discussions with Sebastian J¨ager, Heiko Schmidle,
Florian Kogler, Arzu B. Yener, Ken Lichtner, Carol K.
VI. CONCLUSIONS
HallandOrlinD.Velev. Thisworkwassupportedbythe
Deutsche Forschungsgemeinschaft (DFG) through the
Inthepresentarticlewehavediscussedrecentadvance- International Research Training Group ”Self-assembled
mentsinunderstandingthecollectivedynamicsofdipolar soft matter nanostructures at interfaces” IRTG 1524,
and multipolar colloids in and out of equilibrium. Theo- theResearchTrainingGroup”Non-equilibriumcollective
retical progress in this emerging, very active field of re- phenomena in condensed matter and biological systems”
search involves a broad variety of concepts from equi- RTG1558,andtheCollaborativeResearchCenter”Con-
librium statistical physics (targeting ground state struc- trol of self-organizing nonlinear systems” SFB 910.
[1] J. Dobnikar, A. Snezhko, and A. Yethiraj, Emergent 9, 3693 (2013).
colloidaldynamicsinelectromagneticfields,SoftMatter