Complex architectures of
self-assembling spherical active
colloids
by
Edo Vreeker
September 14, 2020
Student Number Supervisors/examiners 11879815 Prof. Dr. Peter Bolhuis Prof. Dr. Peter Schall
the complex collective behaviour of active spherical colloids, focusing on the self-assembly into finite architectures rather than dynamic phase behaviour such as particle jamming or flocking.
The majority of studies presented here discuss the formation of linear struc-tures such as chains, tubes or networks of branches. Research into chains of active particles that show flagella- or cilia-like oscillations are also reported. Furthermore, experimental studies are discussed that present non-linear struc-tures such as rings and chiral spinners. All the assemblies have been ascribed to arise due to the intrinsic properties of the active particles. Systems often show various phases of collective behaviour at specific parameters and this interplay of variables will also be discussed.
The ability to control the assembly pathway of active colloids has multiple benefits in various research areas. Potential applications are the formation of novel metamaterials or new approaches to transport molecular cargo via e.g. colloidal microtubes. A fundamental understanding of how physical properties of assemblies are shaped by its building blocks can for example also help to explain how organic molecules obtain chirality.
Contents
1 Introduction 3
2 Theoretical studies on active particles forming complex
archi-tectures 5
2.1 Aggegration of active particles into chains and string-like structures 5 2.2 Chains of active particles showing cilia- and flagella- like motion 8 3 Experimental studies on the emergence of
complex structures formed by active particles 17 3.1 Experimental assays on the formation of chains and string-like
structures . . . 17 3.2 Active particle chains showing cilia- and flagella-like behaviour . 22 3.3 Non-linear assemblies of active particles . . . 25
4 Conclusions 29
Appendix A 31
between 1-1000 nm are dispersed throughout a dense medium [1]. Such colloids are subjected to a high degree of inter-particle interactions, which makes it com-plex to investigate these suspensions systematically. To analyse the relevant physics and chemistry that govern the behaviour in colloidal suspensions, it is useful to categorise colloids based on their intrinsic properties. When assessing particles at the colloidal size scale, one way is to distinguish between passive and active particles. Systems of passive colloidal particles in fluid media are abundant in our daily life (with examples like milk, muddy water, or blood), but can also be found in applications in pharmaceutical and cosmetic industries. The particle dynamics in these dispersions can be described by Brownian mo-tion and have been thoroughly studied both theoretically via simulamo-tions and experimentally by using e.g. dynamic light scattering [2, 3, 4].
Active particles differ from passive particles, as the dynamics through the medium is not solely governed by Brownian motion. These colloids also have the ability to induce a locomotion by harnessing energy from its direct environ-ment [5]. Active colloidal systems can both be biological (e.g. motile eukaryotic cells [6]) and artificial (e.g. self-propelling microswimmers [7]). The generation of motion is fundamentally different between biological and artificial systems. Whereas in biology helical flagella are a popular tool to create propulsion, arti-ficial self-propelling particles rely on the environment to deliver energy via e.g. chemical fuel, heat or an external electromagnetic field [8].
Systems containing active particles can show intriguing out-of-equilibrium collective behaviour, such as coherently moving groups or dynamic phase sepa-ration [9, 10]. There is a strong interest to explore the collective behaviour of active particles in systems where the inter-particle interactions are anisotropic. Experimentally, this has been enabled by the synthesis of amphiphilic or dipolar active particles, with Janus particles (JPs) being a prime example [11]. JPs have a divided surface, in which each half has distinctive chemical and physical prop-erties. In this way, JPs can for example be designed to become amphiphilic, by using hydrophilic and/or hydrophobic patches on the surface [12]. Assemblies of such particles have been observed to obtain non-trivial, complex structures such as active chains or colloidal rotors.
Although there is a strong interest in the controlled assembly of active par-ticles to find novel complex structures, this ability has been hindered due to the difficulty to control the directionality of the particles. Research has elucidated how to control the motion of these active particles through its medium. It is
external fields. Systems of active colloids which are driven by chemical fuel have been widely studied, but will not lie within the scope of this review. The knowledge that is acquired by studying the architecture of the formed structures could have potential value in the development of new biomimicking complexes, metamaterials or novel drug delivery methods.
densities of φ = 0.12. The system was set at a dipolar coupling strength λ = 10, that indicates a regime in which dipolar interactions are strong. Particles are coloured based on their dipole orientations, which are indicated by the coloured ring in the inset.
2
Theoretical studies on active particles forming
complex architectures
Theoretical studies have shown that the collective behaviour of active colloidal particles is a complex phenomenon. Often, systems can present multiple phases with distinctive collective behaviour, depending on e.g. the velocity of the active particles or the strength of the interaction potentials. The next sections will present theoretical studies on systems where active particles can occupy states of aggregates with defined geometries.
2.1
Aggegration of active particles into chains and
string-like structures
Various studies have focused on systems where active particles can obtain a state in which they collectively form chains or string-like structures, driven by
short-range steric repulsion (sr) into account: Usr(rij) = 4rep σ rij 12 − σ rij 6 , if rij < rc 0, else. (1)
in which rij represents the distance between the particle centres of i and j, rep
the repulsive strength in KBT , σ the diameter of the particles and rc a
cut-off radius which was set at 21/6σ. Furthermore, the long-range dipole-dipole interactions were regarded via the following equation:
Udd(rij, µi, µj) = µ1µ2 r3 ij − 3(µirij)(µjrij) r5 ij (2) where µi and µj represent the dipole moments of particle i and j [14]. Each
particle was set to have a propulsion, with its direction parallel to its dipole moment. The emerging collective behaviour was observed at different particle density and motility values. At low motility and density the dipolar particles preferred to form of chains (see Figure 1, adapted from [14]).
The occurrence of chain formation is not an uncommon observation among systems of active particles with anisotropic interactions. H. Kogler et al. re-ported how a binary system of active dipolar particles with anisotropic inter-actions were able to form chains that would eventually line up into lanes. The anisotropic interactions were observed to play a stabilising factor in the forma-tion of lanes due to an angle-dependent interacforma-tion potential [15].
In another theoretical study, performed by Mallory et al., the self-assembly of active Janus particles in a 2-D system was performed via particle inter-actions that were induced by attractive patches covering the particles’ surfaces [16]. The total interaction Uij between two particles i and j was given by:
Uij(rij, θi, θj) = Urep(rij) + Uatt(rij)φ(θi)φ(θj) (3)
in which φ(θi) and φ(θj) represent the angular interaction that was defined as the
angle between the patch unit vector niand the inter-particle vector rji= rj− ri
[16]. The repulsive term Urep(rij) was described by:
Urep(rij) = 4rep σ rij 12 rij ≤ 1.5σ 0 rij > 1.5σ (4)
in which repwas set to 1 KBT . Furthermore, the attractive interactions were
described by: Uatt(rij) = 4 σ/2 rs+ σ/2 × 21/6 12 − σ rs+ σ/2 × 21/6 6 rij ≤ 1.5σ 0 rij > 1.5σ (5)
Figure 2: Typical snapshots of simulations performed by Mallory et al. [16]. Active particles with attractive patches (covering 50% of the surface) were simu-lated at a relative particle velocity vpof 10. The motion was either aligned with
the attractive patch (WP, black and yellow particles) or against the patch (AP, black and blue particles). The results show two structurally different aggregates formed by the two separate systems. Two values of (dimensionless) binding en-ergies were assessed: 12 (upper panels) and 20 (lower panels). Results show that an increased leads to more branched aggregates. Branch points are ener-getically less favourable due to a smaller contact area between the patches, and therefore only observed in regimes of higher binding energies.
in which represents the binding energy in kBT and rs= |rij− σ| and stands
for the distance between the particles’ surfaces. Active JPs covered for 50 % with these attractive patches were studied. To investigate the influence of di-rectionality, systems were explored in which the particles’ self-propelling motion was set to be either aligned with the patch (WP) or against the patch (AP). The results showed that directionality causes structurally different aggregates between particles in the WP and AP regime. This is clearly illustrated in Figure 2 (adapted from [16]), where aggregates are presented from particles with a low velocity. It was found that the binding energy (in units of KBT ) seemed to
have a strong influence on the structural properties of the aggregates (upper panels = 12, lower panels = 20). The systems in the lower panel show aggre-gates with a substantially higher degree of branching. Branches were only able
prised of two terms in which the attractive term was angular dependent, similar to equation (3). However, in this case, the repulsive term was a short-range soft repulsion given by:
Urep(rs) = s σs rs 2 − σs h0 2 1 − 2(rs− h0) h0 , 0 < rs≤ h0 0, otherwise, (6)
where s represents a repulsive energy term at the particle surface, σsthe
sep-aration length between the particles and h0 the distance where the potential
is shifted to make sure the potential is continuous at rs = h0, with a value
that is dependent on the location of the minimum of the attractive potential. Consecutively, the long-range attractive potential was given by a Lennard-Jones potential Uatt(rij): Uatt(rij) = 4 σ rij 12 − ( σ rij 6 (21/6σ) < r ij < rc − rij, < (21/6σ) 0, otherwise. (7)
The systems were studied while varying ξ, a dimensionless parameter that was defined as the ratio between the self-propelling forces and the attractive inter-particle forces. At ξ = 0.1 WP inter-particles were observed to form chains of finite particles. The particles within the chain had a specific configuration, in which the attractive patch was facing inwards and the repellent patch outwards.
Yan et al. explored 3-D systems consisting of active Janus particles, where the surface of the JPs was divided into two hemispheres carrying opposing elec-tric charges [18]. Different phase behaviour was observed when the magnitudes of the electric charges were varied. It was found that when the opposing charges obtained an equal magnitude, the active particles would connect in chains in a similar fashion as the 2-D systems studied above.
2.2
Chains of active particles showing cilia- and
flagella-like motion
While the aggregation of individual active particles into chains is an interesting observation on its own, multiple theoretical studies have taken it a step further and explored the structural properties of the resulting active assemblies.
One exciting phenomenon is that chains of active particles are found to behave like biologically relevant structures in the form of cilia and flagella. These flexible whip-like structures can be found in both prokaryotic and eukaryotic systems and are for example used to propel cells and bacteria at low Reynolds numbers [19].
In order to study active colloidal chains, Gonzalez et al. introduced a system consisting of a binary mixture of non-polar colloids [20]. A theoretical model was introduced that was inspired by experimental systems in which the motion
of colloids is driven by interactions caused by self-diffusiophoresis and steric repulsions. Particles contained a homogeneous surface, disabling self-propelling motion. However, in this binary mixture with particles of types A and B, motion was accomplished due to phoretic interactions between a pair of colloids. Following the far-field approximation, particle B would obtain a motion due to the activity of particle A according to:
−→ VB= −µB∇C|B ∼ αAµB R2−−→r AB D|−−→rAB|3 (8) where µB is a variable inspired by experimental active colloids that is described
as the mobility of particle B and tells how the particle responds to a concen-tration gradient, ∇C|B the diffusion coefficient of particle B, αA a variable of
particle A that is inspired by experimental active colloids and describes the pro-duction or consumption of chemicals at the colloid surface, R equals the particle radius, −−→rAB= −r→B− −r→Aand D is a variable that is inspired by the effective
diffu-sivity of solutes [21]. In turn, the motion obtained by particle A due to activity of particle B is given by:
−→ VA= −µA∇C|A∼ −αBµA R2−−→r AB D|−−→rAB|3 (9) where µA, ∇C|Aand αB have the same physical meaning as their counterparts
in equation (8) [21].
The different particle properties between particle A and B cause a break in the action-reaction symmetry between the interactions of the pair [21]. This causes the pair to move in a velocity that is proportional to (αAµB− αBµA)
and the particles to have an attracting or repelling interation in a velocity proportional to (αAµB+ αBµA).
Interesting collective behaviour was observed in regimes where αA, µA> 0
and αB, µB< 0 [21, 20]. In these regimes, similar particles repelled each other
and particles of different types attracted each other. This allowed the formation of stable active chains comprising of an alternating order of particles A and B in a particle ratio of 1:2. A selection of chains with different values of k, in which k indicates the number of particles according to 3(k+1), is presented in Figure 3 (adapted from [20]).
Figure 3: Assemblies of active particles into chains, as studied by [20]. The studied systems comprised of a binary mixture of particles A and B. Each par-ticle was labelled with parameters α and µ, inspired by the surface activity (α) and particle mobility (µ) of experimental active colloids. When parameters were set such that αA, µA > 0 and βB, µB < 0, A-B inter-particle interactions
became attractive while A-A and B-B became repellent. This enabled the for-mation of stable chains where particles A and B assembled in alternating order (A in magenta, B in cyan). Chains were labelled with k, in which k describes the total amount of particles 3(k + 1). The chain at k = 2 illustrates how the first two particles of the chain have lowered mobility (black dots), while other particles enjoy soft degrees of freedom (black arrows).
To create cilia-like motions, the first two particles at one of the chain’s ends were tethered in the x,y-plane. A phase diagram could then be obtained presenting distinct regimes of collective behaviour, governed by α and µ (see Figure 4, adapted from [20]). Four different phases could be observed for all values of k: I) An unstable phase where the chain would disintegrate. II) A phase where the linear configuration of the chain would be stable. III) A phase where the chains could perform stable oscillations. IV) A phase where the chains would collapse due to the oscillations. Springs attached to initially neighbouring colloids expanded the stability of oscillations to a larger regime by restricting the maximum distance between particles, ultimately diminishing phase IV. This helped to create stable oscillating chains ranging from k = 1 to k = 9, where stable configurations could only be realised due to springs for k = 7 and 9. The results can be observed in Figure 5 (adapted from [20]). The pattern of the chain, while oscillating, was observed to make a travelling wave. The back and forth motion was non-reciprocal, which is a vital property that would allow these oscillating chains to make a net movement at low Reynolds number (known as
Figure 4: Phase diagram depicting four distinguishing phases obtained by chains of aligned active colloids, as studied by [20]. Phase I: Unstable phase where the chain disintegrates. Phase II: Phase where the initial linear configu-ration remains stables. Phase III: Chain performs stable osillations. Phase IV: Chain collapses due to the heavy oscillations.
the scallop theorem [22]).
Gonzalez et al. also looked at the possibility to create flagella-like active chains in 2-dimensional systems. In order to achieve this, the first two particles at one of the chain’s ends were given a lower mobility to simulate the head of a microorganism, that would then be propelled by flagella. By allowing the head group to move, rather than tethering it, the chain makes a net movement. The obtained oscillations showed a similar profile as seen with the cilia-like motion (see Figure 6, adapted from [20]). Similarly to the cilia, there was no net velocity observed if the chains were not oscillating. Simulations were performed at different values of αA, being 1.6 and 1.7 respectively, while µAwas set at 0.5,
αB and µB at -1.0 and k at 2. The latter value of αAwas above a critical level
at which a second oscillation at the end of the tail occurred, which affected the amplitude. The second oscillation affected the trajectory of the head, leading to a larger displacement in the same time interval for the cargo compared to the chain at αA= 1.6.
Figure 5: Cilia-like oscillations made by chains of active colloids, studied by Gonzalez et al. [20]. Each chain is built up of 3(k + 1) particles, with the appropriate k value labelled underneath each aggregate. The oscillations were observed to be travelling waves, with no reciprocal movements. Different colours indicate different points of time within one period. Chains travelled in a se-quence of: green, cyan, blue, magenta, red, yellow and back to green. At k = 7, the end of the chain is indicated by a black dot to indicate the non-reciprocal motion. At k = 9 a secondary oscillation is observed at the right part of the chain (sample rate is increased to visualisation). All chains were simulated at μAof 0.5 (with μBremaining constant at -1), except the k = 9 chain, which was
Figure 6: Flagella-like oscillations made by chains of active colloids, studied by Gonzalez et al. [20]. Flagella-like motion was obtained by lowering the mobility of the first two particles in the chain, simulating cargo. The oscillations were observed to be similar as the cilia-like oscillations in Figure 3. As the head group was allowed to move, the chain made a net movement while oscillating. Two situations were assessed, one at αA = 1.6 and one at αA = 1.7. Other
parameters were set at µA = 0.5, αB = µB = -1.0, and k = 2. The latter
value of αAwas above a critical level, inducing a second oscillation at the end of
the tail. This second oscillation would affect the amplitude of the oscillations. The trajectory of the head group was affected by the occurrence of the second oscillation, creating a larger displacement for the cargo compared to the αA =
Chelakkot et al. used simulations and theory to observe tightly attached, active, polar colloids in 2-dimensional systems [23]. Chains comprised of N par-ticles with diameter σ and coordinates ri(i = 1, ..., N ). Each particle interacted
with its linked neighbour via a harmonic potential Ul at an equilibrium linker
length b according to: Ul= κl 2kBT N −1 X i=1 |ri+1− ri| − b 2 (10) where κlrepresents the linker stiffness, with a value chosen to be large enough
such that the actual distance between particle N and N +1 is close to b [23]. Ad-ditionally, a potential was added that described the resistance to chain bending according to a three-body-bending potential:
Ub= κb 2kBT N −1 X i=2 bi+1− bi 2 , (11)
where bi = (ri−1− ri)/|ri− ri−1| is the unit bond vector that is oriented
anti-parallel to the local tangent and κb is the bending rigidity [23]. The polar
direction ˆpi of each particle was biased to direct along bi by using an angular
harmonic potential: Ua= κa 2kBT ˆ pi− bi 2 (12) where κa represents the angular stiffness [23]. In the limit of κa → ∞ the
propulsion forces of the active particles are exactly aligned with the local tangent of the chain. In this limit, the bond between two particles in the chain is strongly coupled and parallel to the direction of the self-propulsion of the particles. Lower values of κa translate to a lower coupling, leading to a fluctuation in polarity
along the chain [23].
The dynamics of the chain was governed by the evolution of both the position ri and orientation ˆpi of each particle (i = 1,...,N ), which was simulated with
over-damped Brownian dynamics: ˙ri= 1 KBT D Fli+ Fbi+ FExi + 1 KBT Dfpˆp + √ 2Dξi (13) and ˙ˆp = 1 KBT Drτi× ˆp + p 2Drξri. (14) In these equations, Fl
i= ∂Ui/∂rirepresents the axial force along the linker, Fbi =
−∂Ub/∂rithe bending force, fpp the propulsion force and τˆ i = −ˆp × ∂Ua/∂ ˆpi
the torque that is caused by the angular potential [23]. D and Dr are the
translational and rotational diffusion coefficients that satisfy the Stokes-Einstein relation Dr= 3D/σ2. ξ and ξr were described as the zero mean, unit variance
was described as a pairwise excluded-volume repulsive force, given by the Weeks-Chandler-Anderson (WCA) potential:
Uex(rij) = 4 σ rij 12 − σ rij 6 + , if rij< 21/6 0, elsewhere (15)
that arises between any pair of particles that come into proximity, not just the nearest-neighbours one bond further in the chain [23].
To study flagellar dynamics with these chains of active particles, one chain end was clamped into a vertical direction. The initial configuration of the chain was straight, with the internal forces of each particle directed towards the clamped end. Two chain parameters were varied to observe the collective behaviour at different chain lengths. The first parameter was the dimension-less κa, that would describe whether active particles could perfectly align their
propulsion force with the local tangent (κa→ ∞) or whether noise would affect
this. The other parameter was the propulsion force which was oriented in an anti-parallel direction to the local tangent vector. This resulted in the tip of the chain to be deflected by a small distance h ( l, the length of the active chain) in the transverse direction. Ultimately, the initially straight chain configuration was found to become unstable when the propulsion force was set above a specific threshold value. This threshold value, regarded as the critical propulsion force, was found to be approximately equal to:
fc∼ C1
κ l3
(16) in which C1 is a constant depending on κa (that needs to be determined from
simulations close to the critical point) and κ the chain’s bending stiffness. When the propulsion force would surpass this critical value, the chain would buckle. However, in a viscous fluid, rather than buckling the chain would oscillate. To study the consequent behaviour of these unstable chains, two different bound-ary conditions were explored. Chains of short lengths (80 particles) and weak angular noise (κa 1) showed regular beatings that were reminiscent of flagella
(see Figure 7, adapted from [23]). In this regime of κa, these chains were found
to oscillate at a frequency that was only dependent on the propulsion force. At high regimes of angular noise, the motion of the chain would decorrelate leading to erratic oscillations.
Figure 7: Chain of oscillating active colloids, studied by Chelakkot et al. [23]. To create flagella-like beatings, the first colloid in the chain (at arclength s = 0) is clamped in a vertical direction. Initially linear chains were observed to show a beating pattern after the particles’ propulsion forces exceeded a threshold value. Short chains (in this case, l = 80 particles) were observed to show flagella-like oscillations at low values of angular stiffness (κa = 20). The propulsion force
was observed to affect the frequence of the oscillations. The evolution of the chain during an oscillation is presented by numbers 1-5 and colours (from red to green), showing the chain’s configuration at different stages as the chain propagates.
To create a self-propelling force in active colloidal systems, a variety of spherical particles can be used. JPs have for example been decorated with various metals that enable the conversion of different chemical fuels into motion through a cat-alytic site, such as for example is reported with Pd-decorated JPs [24]. However, active particles fueled by chemicals do not lie in the scope of this review. Instead, particles will be discussed that are able to convert external electromagnetic fields into kinetic energy. JPs that are able to respond to external electromagnetic fields are capable to create anisotropic inter-particle interactions. Additionally, external fields can affect the direction of the particles’ motion. Harnessing these properties can ultimately lead to the creation of complex assemblies that show a variety in architectures. To achieve motion in particles without a chemical fuel, one popular approach is to prepare JPs that respond asymmetrically to external electromagnetic fields. For example, Gangwel et al. have prepared JPs made of one metallic hemisphere and one dielectric hemisphere [25]. It was reported that these particles were capable to create a self-propelling motion when the system was subjected to an external low frequency alternating current (AC) electric field. The motion was ascribed to induced-charge electrophoresis (ICEP). This phenomenon describes how motion arises due to an asymmetric response within the metallodielectric JP to the applied homogeneous A.C. electric field, in which the JPs move in a direction perpendicular to the applied field. The process is described more thoroughly in section 3.2.
3.1
Experimental assays on the formation of chains and
string-like structures
One of the directions that experimental research has focused on, is the assembly of active particles into structures as chains and strings. Kokot et al. looked at dispersions of magnetic colloids that were able to be energised by a uni-axial alternating magnetic field [26]. The particles, nickel colloids of approximately 90 μm in size, were immersed in a system in which they were confined to the air-liquid interface due to surface tension. The alternating magnetic field was able to apply a torque on the colloids, that would dissipate around the surface of the particles into the environment, creating a hydrodynamic flow. Ultimately, two factors created an interaction between the particles: magnetic dipole-dipole interactions and hydrodynamic forces. The interplay between these forces could
Figure 8: Micrographs of self-assembling magnetic colloids, as studied by Kokot et al. [26]. Various quasi-stable states could be deserved, depending on the frequency and amplitude of the magnetic field. a At a frequency of 180 Hz and magnetic field of 4 mT, colloids were observed to form long wires par-allel to the magnetic field HAC. The aggregates were observed to assemble and
disassemble reversibly. Scale bar is 2 mm. b When an out-of-equilibrium state was created by applying a small static (D.C.) magnetic field perpendicular to the interface, parallel dynamic wires were observed. The in-plane magnetic field had a frequency of 200 Hz with a magnetic field of 5.4 mT. The additional D.C. magnetic field perpendicular to the system was set at 2 mT.
the dynamic particles were observed to self-assemble into long wires parallel to the applied magnetic field (see Figure 8a, obtained from [26]). The formed aggregates were observed to assemble and disassemble reversibly. Moreover, by creating an out-of-equilibrium state using an additional small static (D.C.) mag-netic field perpendicular to the interface, parallel arrays of dynamic wires could be obtained (see Figure 8b, obtained from [26]). The in-plane A.C. magnetic field was set at 5.4 mT with a frequency of 200 Hz, while the perpendicular D.C. magnetic field was set at 2 mT.
In another study performed by Yan et al., systems were studied comprising of magnetic Janus particles [27]. These particles were prepared by covering one hemisphere of the silica bead with a thin nickel coat. The JPs were suspended in deionised water and subjected to a spatially homogeneous, precessing magnetic field at a fixed frequency of 20 Hz and a field strength of 5 mT (see Figure 9a, adapted from [27]). The particles had an anisotropic response to the magnetic field, with the response towards the Janus director (red arrow in Figure 9a) being smaller than towards the perpendicular direction. This precessing mag-netic field created a torque on the JPs that induced a motion reminiscent to a gyroscope’s nutation. When two of these dynamic particles would come into proximity, they were found to undergo an attractive interaction until an electro-static repulsion would come into play at 200 nm distance. At this distance, one particle was capable of applying a torque on the neighbouring particle, which would ultimately lead to the two particles being in an antiphase steady state.
Figure 9: a Overview of the precessing magnetic field acting on the JP system studied by Yan et al. [27]. The precessing magnetic field H is set at an angle θ relative to the z-axis. The magnetic JP responds as follows: the Janus director (indicated by the red arrow) rotates around axis z at an angle φ. Additionally, the JP oscillates perpendicularly to the Janus director, with an angle α. b Mi-crographs and models presenting dynamic microtubes formed by active particles (scale bars represent 3 μm). JPs were prepared consisting of silica beads with one hemisphere covered in a nickel coat. When immersed in deionised water and subjected to a homogeneous precessing magnetic field with a frequency of 200 Hz and a field strength of 5 mT, these particles were found to obtain a torque that would result in a motion similar to a gyroscope’s nutation. Suspensions of these JPs were found to create a variety of detect-free microtubes, the diameter dependent on the angle θ between the magnetic field and the z-axis. The tubes were observed to roll on the surface of the substrate. JPs within the aggregate maintained their constant nutation-like motion. The particles were found to synchronise, keeping the metallic hemispheres inwards.
When this steady state was achieved, the particles were found to rotate at a slower pace around a common central axis, with their spatial positions in a head-to-tail direction. Suspensions of these particles were assessed as function of the angle θ between the magnetic field and the z-axis. At values of low θ, a variety of defect-free microtubes were observed, in which θ influenced the width of the microtubes (see Figure 9b). These tubes were found to be achiral and consisted of stacks of polygons formed by the active particles. Once the particles had aggregated into the tubes, the constructs were observed to roll along the surface of the substrate. The JPs within the microtubes remained in a constant
Figure 10: A study by Snezhko et al. investigated the formation of magnetic snakes by nickel particles at the liquid-air interface [28]. Particles sized 90 μm were subjected to an alternating magnetic field perpendicular to the quasi-2D system. The particles were driven into self-assembly due to surface deforma-tions of the fluid and the collective response to the alternating magnetic field. Particles would form ferromagnetic segments that would line-up in an antifer-romagnetic configuration to create a snake-like assembly. At a magnetic field strength of 10 mT, the snake was found to initiate a self-propelling motion when the frequency of the magnetic field was increased to 100 ± 30 Hz and above. The snake would propel in a straight line until it was perturbed by its environment. The velocity could be controlled by the frequency of the external magnetic field: an increasing frequency would lead to an increasing velocity.
perpendicular to the quasi-2D system (see Figure 10, adapted from [28]). The self-assembly was reported to be driven by surface deformations of the fluid and the collective response of the particles to the alternating magnetic field. More information about the assembly pathway can be found in [29]. The first step in the formation of snakes after activation of the spherical particles was an initial assembly into linear chains. Within these chains, particles would align in a ferromagnetic orientation. These chains would then further assemble into a snake, where each chain would form a separate segment of the snake (see the smaller grey constructs within the snake in Figure 10). The separate segments within the snake were ordered in an anti-ferromagnetic configuration.
These snakes showed self-propelling motion when the magnetic field strength was set at 10 mT and the frequency was increased to 100 ± 30 Hz and above, as is demonstrated in Figure 10 where the snake has shown to move in a time frame of 1.40 sec. A snake was found to swim into a straight line until it would collide with the wall of the experimental chamber, or other snakes. The velocity of the self-propelling snakes was found to be controlled by the frequency of the external magnetic field, with an increase in frequency causing an increase in velocity.
systems by using magnetic fields as external energy sources. Especially the study of Kokot et al. and Snezhko et al. are good comparisons, as both used nickel particles of the same size (∼ 90 μm) at the liquid-air interface - only different system parameters were set [26, 28]. Both results reveal the complex interplay of the frequency and magnitude of an A.C. magnetic field on the activation of particles. It would be interesting to explore if both set-ups are able to achieve both states (the dynamic wires and mobile snakes). Combining both phenomena could perhaps lead to interesting novel applications. A mobile snake that could transport itself through the system to eventually form a dynamic wire at a point of interest could perhaps function as a local mobile conductor. A dynamic wire that is capable to break up and form a mobile snake could in turn form an on-demand transporter in systems with narrow dimensions. The application of the swimming snake to transport cargo has already been presented by Snezhko et al., as it was observed that such snakes are capable of pushing a glass bead towards the direction of motion [28].
In the same article in which Yan et al. presented the theoretical studies on chains of JPs, an experimental part was discussed in which 2D-systems contain-ing metal-dielectric JPs were studied where the particles were subjected to a perpendicular A.C. electric field (see Figure 11a, adapted from [18]). JPs of size 3 μm were prepared by coating one hemisphere of a silica bead with titanium and consecutively coating it with a thin layer of SiO2to create an outer surface
that is chemically homogeneous. The particles were immersed in deionised wa-ter and settled on the bottom of the experimental chamber. By subjecting this quasi-2D system to an A.C. electric field from a perpendicular direction, parti-cles would orient their equators vertically as the hemispheres became polarised. However, due to the different properties of the hemispheres, both are polarised differently. This difference had two consequences: 1) ionic flows along the two different surfaces were imbalanced in magnitude, leading to self-propulsion in a direction perpendicular to the electrical field (this motion is known as induced-charge electrophoresis). 2) the two hemispheres obtained two different dipole moments, leading to imbalanced electrostatics along the particle. As the dipoles were directed perpendicularly to the plane, isotropic dipole-dipole interactions were realised that were compared to charge-charge interactions. The dielectric response was reported to be frequency-dependent, meaning that the imbalance of the dipole moments between the two hemispheres and the magnitude and di-rection of the swimming force could be controlled by the electric-field frequency f . The direction of the motion was found to be reversed if f was increased. When f was set in the range of MHz, the dipole-dipole interactions were dom-inating as the oscillating field would move too fast for the ions to follow. The
Figure 11: a Illustration of the experimental set-up containing metal-dielectric JPs as studied by Yan et al. [18]. Particles reside at the bottom of the exper-imental chamber in deionised water. The system was then sandwiched by two electrodes to create an A.C. electric field E in the z-direction. This created a motion in the particles within the x,y-plane. b Active particles forming chains due to dipolar interactions (scale bar represents 5 μm). Silica beads were cov-ered with one hemisphere by titanium and consecutively by a thin SiO2layer to
create a chemically homogeneous surface. When immersed in deionised water, an A.C. electric field was created perpendicularly to the system. Due to the inherently different properties of the two hemispheres, particles were observed to obtain a motion and dipolar interactions between different hemispheres were realised. It was found that the direction and magnitude of the swimming force and the imbalance of the dipolar moment between the two hemispheres could all be controlled with the frequency f of the electric field. When f was set into values of MHz, the dipolar interactions dominated and particles were found to form chains. Chains were dynamic, with particles aligning in a head-to-tail configuration. The aggregates had a tendency to form linear chains. However, perturbations could occur due to the environment, for example when chains would collide with other chains.
3.2
Active particle chains showing cilia- and flagella-like
behaviour
Nishiguchi et al. studied self-propelling JPs that were fuelled by an A.C. elec-tric field in a quasi-2D system [30]. The JPs were prepared by coating one hemisphere of silica beads by a thin titanium film and consecutively coating the whole surface again with a silica layer to ensure homogeneity. The particles had a size of 3 μm and were subjected to the A.C. electric field from a direction perpendicular to the quasi-2D system (in a similar set-up as in Figure 11a). The JPs were immersed in a NaCl solution (0.1-1.0 mM) and sedimented on the bottom of the experimental chamber.
Two-dimensional motion of the particles was observed when the electric field was turned on. Turning on the electric field creates surface charges that gather counter ions from the bulk solution, creating an electric double layer. The
counter ions in this layer are driven by the electric field, creating an electro-osmotic flow around the particle. This flow is asymmetric due to the different physical properties of the hemisphere, creating self-propulsion perpendicular to the electric field (known as induced-charge electrophoresis - ICEP).
In this study, the influence of frequency and salt concentration are investi-gated on the ICEP of JPs. Results showed that the direction of the particle’s motion was dependent on both the frequency (f ) and salt concentration. At low frequencies (10-50 kHz), particles were found to follow the direction of the silica hemisphere (ICEP). However, at 70 kHz and a salt concentration of 0.1 mM, particles were observed to revert their motion towards their titanium hemisphere (rICEP). This behaviour was observed for every salt concentration, with higher concentrations requiring higher frequencies (200-1000 kHz) to show similar behaviour.
Chain formation was observed in the rICEP regime at f ∼ 1 MHz. Addi-tionally, this was also observed in the ICEP regime for high salt concentrations of ∼ 1 mM. In the ICEP regime, it was observed that chains could randomly collide with aggregates at the bottom electrode. These aggregates would then be pushed forward, creating an oscillating chain carrying cargo at the head. In the rICEP regime, oscillations of chains was observed when head particles happened to be tethered to the electrode.
The events enabled the researchers to study flagellar beatings in both the ICEP and rICEP regime at high frequencies (see Figure 10 A, adapted from [30]). Oscillations were caused by rear particles pushing the front particles, creating accumulating internal stress across the chain. This would disturb the stability of the chain, which could lead to beating behaviour comparable to flagella. Chains of multiple lengths (N = 5, 10, 14) were investigated at voltage V values between 12-20 Vpp (peak to peak voltage) at f = 1 MHz and 0.1 mM
NaCl. Voltage was observed to only affect the time scale of the chain dynamics. Various experimental data of the oscillating chains was presented, such as the trajectories of the particles and the mean square displacement (see Figure 10 B, data representative for a chain of 14 particles, figure adapted from [30]).
Finally, also a theoretical model to estimate the effects of hydrodynamics was discussed. A flow pattern was obtained that was similar to the profile of eukaryotic flagella, however it was emphasised that this was still a rough estimation.
Figure 12: Experimental results of active particle chains showing flagella-like beatings, observed by Nishiguchi et al. [30]. JPs were prepared by coating one hemisphere of silica beads with titanium, followed by a complete coverage of a thin layer of silica to make the particle homogeneous. The JPs were activated by an A.C. electric field. A Upper series of micrographs: In the ICEP regime, chains that collided with aggregates at the bottom electrode were observed to show flagella-like beatings. The aggregate became cargo, which was pushed by the chain. In this case, a chain of 14 particles is presented at a peak to peak volt-age of Vpp= 20 V, f = 170 kHz, NaCl = 1.0 mM. Voltage would only affect the
time scale of the dynamics. Lower series of micrographs: when the head group of chains was tethered to the bottom electrode, flagella-like oscillations were also observed in the rICEP regime. Again, a chain of 14 particles is presented at Vpp = 20 V, f = 1 MHz, NaCl = 1.0 mM. B Experimental data obtained
from 14-particle chains in the rICEP regime. Left panel: Representative data of the trajectories of all the particles in the chain over a period of 7.5 s. Right panel: representative data of the mean square displacement versus time, for all the particles. The oscillations are also noticeable in these results, with all the particles returning to MSD = 0 after a period of 7.5 s.
3.3
Non-linear assemblies of active particles
To obtain a good overview of the available stable states that assemblies of active particles can obtain, Long et al. have offered a useful insight with their combined theoretical and experimental study [31]. Machine learning was used to create an overview of the available assembly landscape for active particles at given system parameters. The landscapes show how experimental parameters affect the eventual assembly pathway of the active particles.
The researchers used JPs made from silica beads where for each particle one of the two hemispheres was coated with a thin layer of titanium and sil-ica, respectively. Mixtures were prepared consisting of these JPs and so-called linker particles, untreated silica beads, in a 10:1 ratio. These mixtures were con-secutively subjected to an A.C. electric field, creating an asymmetrical dipole moment between the hemispheres (set-up was similar to Figure 11a). At high field frequencies, motions were observed towards the titanium hemisphere (rI-CEP).
The strength of the approach to use machine learning for elucidating assem-bly mechanisms was that knowing the underlying physics was unnecessary to obtain knowledge about the assembly roadmaps, empirical observations on the range of formed aggregates sufficed. The power lied in reducing the vast compu-tational costs of observing N particles (in a 3N -dimensional phase space). The phase space can be substantially decreased by making use of the fact that self-assembly needs cooperative inter-particle interactions. By coupling the so-called ’building block degrees of freedom’, a separation of time scales could be made. The long time evolution of the system could then be observed individually and this is governed by a small number of collective modes. By extracting these collective modes, a low-dimensional subspace was obtained which included the important dynamical features.
Machine learning offered the researchers the opportunity to obtain the im-portant collective modes together with assembly pathways and accessible ag-gregates. This technique was combined with experimental particle tracking to come up with the assembly pathways and final aggregates. A total of 537 ex-periments were performed, revealing 739246 clusters (including free monomers) that belonged to a total of 338 distinct aggregates.
Histograms covering the aggregates formed in the long term evolution of the system were presented in two-dimensional diffusion maps. These maps identify a low-dimensional subspace within the vast 3N-dimensional phase space, where the important assembly dynamics reside (described by the top two collective modes−→
ψ2,
− →
Figure 13: Effective free energy profiles revealing available architectures for active particle assemblies, as function of system parameters [31]. Research was performed by Long et al., in which active metallodielectric JPs exposed to an electric field were experimentally tracked. By using machine learning, two di-mensional maps could be created showing the relevant aggregates at different regimes based on data gathered from 537 experiments. Data presented here belong to the top two collective modes describing the subspace of configura-tions showing the long term aggregates (−→ψ2,
− →
ψ3 ). Each column represents a
different regime of electric field strength E: low (167-333 V cm-1), intermediate (417-583 V cm-1) and high (667 - 833 V cm-1). Furthermore, each row belongs
to a different regime in frequency f of the A.C. electric field: low (50-300 kHz), intermediate (400-800 kHz) and high (900 kHz - 3 MHz). In each system a con-centration of 0.01 mM NaCl (low regime) was present. The colour scale indicates the effective free energy value (β ˆF ) in a specific tile in the histogram. Results show that larger values of electric field strength and/or frequencies create more stable states in which larger aggregates are present. Large active aggregates are observed to form multiple stable states, such as long strings, networks of branches and ring structures.
An interesting example of non-linear active aggregation is the formation of stable active chiral spirals. Zhang et al. investigated this phenomenon in a system comprising of active JPs mixed with silica beads while being subjected to a vertical A.C. electric field in a similar set-up as presented earlier in Figure 11a [32]. The JPs were prepared by coating one hemisphere of silica particles with a 20 nm layer of titanium followed by 20 nm of silica. The electric field was set at a frequency of 300 kHz and an electric field strength of 40 V mm-1(rICEP
regime). Particles were immersed in a solution of 0.1 mM NaCl, in which they sedimented to the bottom to create a quasi-2D system.
By applying an electric field, dipole moments were induced in both types of particles: for silica beads this dipole was located in the centre of the particle, while for the JPs each hemisphere had its own dipole moment. Attractive interactions were then established between the dipole in the metal and in the silica, such that JPs favoured to form head-to-tail configurations. Furthermore, attractive interactions also occurred between the metal hemispheres and silica beads.
Aggregates were observed to form rotating spirals, with a silica bead particle as a central point connected to 3 arms (for 3 μm particles) or 4 arms (for 4 μm particles), as presented in Figure 14 (adapted from [32]). The angle between two neighbouring JPs was found to be π/4, instead of the expected 0 that would be the case when solely dipolar interactions are present. It was suggested that the shift was probably caused by hydrodynamic interactions, but no quantitative feedback was provided.
Assemblies were found to form in two mechanisms. In the first one, a random cluster would be formed that would start to rotate and structurally adapt until a spiral was formed. The alternative formation was a more controllable growth, in which arms were observed to form on a smaller spiral. Formed spirals were found to be chiral, with opposite handedness and rotation direction and no preference in what direction it would rotate. The arms of the spiral were found to be flexible, able to fluctuate by external perturbations, while spinning around the centre with an equal distance between the arms.
Multiple dynamic reactions within the spinning clusters could be observed, such as particle addition, substitutions of active particles with external particles or ring openings and closures caused by interactions between active particles in neighbouring arms. Such reactions are actually able to switch the chirality of the spirals. Additionally, switching of the chirality was observed in spirals with arms longer than 3 particles due to the short persistence length of each arm.
Figure 14: Assembly of active particles into chiral spinners, observed by Zhang et al. [32]. JPs were mixed with silica beads in a 0.1 mM NaCl solution and consecutively subjected to an A.C. electric field perpendicular to the 2-D sam-ple field. The electric field was set at a frequency f of 300 kHz and an electric field strength of 40 V mm-1, creating a state in the rICEP regime. The electric
field would create dipole moments in all the particles. This induced a motion in the JPs, together with an attractive interaction between the metal hemisphere of the JPs and the silica present in the other hemisphere of JPs and the plain beads. The mixture was observed to form chiral spinners following two differ-ent pathways (see (a)). One pathway comprised of active particles assembling at the surface of a silica bead, in which the silica bead would form the centre of the aggregate and JPs would connect in arms (3 arms for systems of 3 μm sized particles, 4 arms for systems of 4 μm). The aggregate would eventually break symmetry, leading to the formation of chiral spinners. Spinners showed to have no directional preferences. The other assembly pathway comprised of a more templated growth, in which JPs would connect to preformed smaller spin-ners. (b) shows representative micrographs of both symmetric and asymmetric spinners that were observed to form.
particles into complex structures have been discussed. These complex structures typically arise due to induced anisotropic interactions between the active par-ticles within the system. Various theoretical studies presented here have shown how active colloids are capable to form long chains and branched networks of particles. In order to create anisotropic interactions in these theoretical studies, colloids can be labelled with an intrinsic dipole moment or attractive patches at the surface that create pair-wise interactions which are dependent on the rel-ative orientations of the particles. Additionally, particles contain a propelling motion which ultimately defines these particles as active. The strength of the interaction potential, the direction of the motion and magnitude of the veloc-ity were seen to be typical parameters that governed the final structure of the assemblies. Other theoretical studies have focused on the collective behaviour once chains of active particles have been formed. What’s most interesting about these studies, is that it is reported that such active chains can show a similar oscillating behaviour as the biologically relevant cilia and flagella. These oscil-lations are able to arise due to the intrinsic active properties of the particles within these assemblies and the immobilisation of one of the chains’ ends. Typ-ical properties that would affect the oscillations were the particles’ ability to obtain a motion and the stiffness of the chains. Oscillations were also observed to be dependent on the length of the chain.
Experimental studies have been presented that show results of collective behaviour of active particles analogues to the theoretical studies. Anisotropic interactions among active spherical colloids are the main cause of the observed assemblies. Such interactions are often realised by harnessing the properties of Janus particles (JPs). These particles are typically prepared by coating one hemisphere of a silica bead with a metal, such as nickel or titanium. JPs are often observed in quasi-2D systems, where the particles are dispersed in a fluid in which they sediment and reside on the bottom of the experimental chamber. To induce the activity within these particles, an external source in the form of an alternating magnetic or electric field is generally used (directed perpendic-ularly to the system). When these external fields are activated, JPs obtain a dipole moment in each hemisphere. Magnetic fields would apply a torque on these particles, ultimately leading to a hydrodynamic flow. JPs subjected to an electric field are usually dispersed in ionic solutions. When an external electric field was applied, motion was created due to the asymmetric flow of ions along the particles’ surface caused by the physically different hemispheres. Both
ex-form of ring-structures. Free energy landscapes are discussed, calculated with machine learning, that show how stable non-linear aggregates can be achieved depending on frequency and electric field strength. Finally, a study is presented in which a mixture of active particles with silica beads is shown to form chiral spinners. Multiple assembly pathways for these spinners were presented. Spin-ners were found to be able to switch the direction of the rotation, caused by inter-particle reactions within the aggregate or when aggregates had sufficiently long arms.
All these studies show how complex collective behaviour of spherical col-loids in systems arise when these particles are energised and contain anisotropic interactions. The presented studies show how the assembly of active particles into complex architectures could trigger the interest of researchers in a variety of fields. Whether this regards the formation of colloidal wires, microtubes or the emergence of chiral active complexes. The ability to control active particles with a switch, by simply switching off the external field, is another asset which is not found in systems such as passive amphiphilic colloids and which further promotes the use of active particle systems.
Phase diagram of dynamic self-assembled
struc-tures studied by Kokot et al.
Figure 15: Phase diagram of self-assembling magnetic colloids studied by Kokot et al. [26]. a. Overview of the system, where the particles reside at the liquid-air interface while being subjected to a uni-axial alternating magnetic field. b. Phase diagram of multiple states obtained depending on the frequency
tion of the external electric field frequency as studied by
Yan et al.
Figure 16: Different obtained states in systems of metal-dielectric JPs, depend-ing on the frequency of the applied perpendicular A.C. electric field, studied by Yan et al. [18]. When the frequency was set at 5 kHz, the system would obtain a gas state, as presented in micrograph a. When the frequency is increased to 30 kHz, ionic screening is reduced and particles reverse the direction of motion. The repulsion of the metallic hemisphere starts to govern the interactions and a pair of particles align after collisions occur. This allows a state in which the particles are swarming (micrograph at b. When the frequency is increased to 1 MHz regime ions are no longer capable to follow the oscillating field. As a re-sult, the material properties govern the interactions. The dipoles created in the metallic and dielectric hemispheres create a strong attraction leading to active chains in a head-to-tail configuration (micrograph c). Micrograph d represents a state of clustering which was observed at 40 kHz when a NaCl concentration of 0.1 mM was added to the system. Scale bars represent 5 μm for micrographs a-c and 30 μm for micrograph d.
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