# Self-assembly of active amphiphilic Janus particles

S. A. Mallory1, F.Alarcon2, A. Cacciuto1∗, C. Valeriani2<EMAIL_ADDRESS><EMAIL_ADDRESS>1Department of Chemistry, Columbia University
3000 Broadway, New York, NY 10027
2Departamento de Fisica Aplicada I, Facultad de Ciencias Fisica, Universidad
Complutense de Madrid, 28040 Madrid, Spain

###### Abstract

In this article, we study the phenomenology of a two dimensional dilute
suspension of active amphiphilic Janus particles. We analyze how the
morphology of the aggregates emerging from their self-assembly depends on the
strength and the direction of the active forces. We systematically explore and
contrast the phenomenologies resulting from particles with a range of
attractive patch coverages. Finally, we illustrate how the geometry of the
colloids and the directionality of their interactions can be used to control
the physical properties of the assembled active aggregates and suggest
possible strategies to exploit self-propulsion as a tunable driving force for
self-assembly.

††preprint: APS/123-QED

##  I Introduction

Active matter continues to be one of the most exciting new fields in
statistical mechanics and materials engineering. Whether one focuses on the
chaotic dynamics of human crowds or the peculiar swarming of swimming
bacteria, a common feature of all active systems is their ability to convert
environmental or internal energy into systematic movement. The inherent self-
propulsion or activity at the “single particle level” in these systems gives
rise to a remarkable range of different collective behaviors. For a number of
examples and a thorough discussion on the topic, we direct the reader to a
number of comprehensive reviews (see for instance Bechinger  _et al._ (2016);
Wang  _et al._ (2015); Toner  _et al._ (2005); Cates and Tailleur (2015);
Ramaswamy (2010); Klapp (2016); Marchetti  _et al._ (2016); Bialké  _et al._
(2015); Zöttl and Stark (2016); Dey and Sen (2017); Wang  _et al._ (2013);
Menzel (2015); Marchetti  _et al._ (2013); J. Ebbens and R. Howse (2010);
Chaté  _et al._ (2008); Romanczuk  _et al._ (2012); J. Ebbens and R. Howse
(2010) and the references therein). In this work, we are predominately
concerned with colloidal active systems, and developing new methods to use
activity as a tool to optimize colloidal self-assembly. Synthetic
microswimmers are expected to fill an important void within the material
science community as they possess the rather unique ability of being able to
manipulate, sense, and transport material at the micro-scale. The potential
applications are far reaching and include: targeted drug delivery Din  _et
al._ (2016), information storage and computation Woodhouse and Dunkel (2017),
and the clean-up and neutralization of environmental pollutants Ebbens (2016).
These applications are all built around the unique self-driven nature of the
microswimmers and in many ways, active colloids look to serve as the perfect
building blocks for the self-assembly of the next generation of functional,
environment-sensing microstructures able to perform specific tasks in an
autonomous and targeted manner.

Through the groundbreaking work of synthetic chemists and material scientists,
there is now a continually growing library of synthetic microswimmers and
active colloids. These active particles are the synthetic analogs of swimming
bacteria. However, a major benefit of these synthetic variants over swimming
bacteria is that both the inter-particle interactions and the swimming
velocity can be systematically controlled. For details about the growing
variety of synthetic microswimmers and their associated propulsion mechanisms,
we direct the readers to several recent reviews Wang  _et al._ (2015, 2013);
Bechinger  _et al._ (2016); J. Ebbens and R. Howse (2010). The majority of the
published work on colloidal active matter focuses on suspensions of active
particles which interact via an isotropic potential (whether it be attractive
Mognetti  _et al._ (2013); Redner  _et al._ (2013); Prymidis  _et al._ (2015);
Schwarz-Linek  _et al._ (2012), or purely repulsive Stenhammar  _et al._
(2015, 2014); Tailleur and Cates (2008); Theurkauff  _et al._ (2012); Speck
(2016); Speck  _et al._ (2014); Buttinoni  _et al._ (2013)). Only recently,
has the field branched out to explore the interplay between activity and
anisotropic pairwise interactions. We are only now developing a systematic
understanding of how active forces can be exploited in conjunction with
anisotropic pair interactions to design macroscopic assemblies with desired
structural properties (see Mallory and Cacciuto (2016); Di Leonardo (2016);
Zhang  _et al._ (2016); Singh  _et al._ (2017); Johnson  _et al._ (2017); Gao
_et al._ (2017); Pu  _et al._ (2017) and references therein).

The inspiration and motivation for this work was derived from the recent
experimental synthesis of active colloids with directed anisotropic
interactions. By altering the surface chemistries of an active colloid, it is
possible to introduce a number of different directed anisotropic interactions.
In a recent publication, Yan et al. Yan  _et al._ (2016) presented a general
strategy, that exploits electrostatic imbalance between the two hemispheres of
an active dipolar Janus particle, to reconfigure their assembly into a number
of different collective states. Several research groups Kaiser  _et al._
(2015); Guzmán-Lastra  _et al._ (2016) have also begun to probe the clustering
behavior of a small number of dipolar Janus particles with a specific focus on
analyzing how the structural properties of the self-assembled clusters are
affected by the introduction of self-propelling forces. An alternative
approach to introducing directed anisotropic interactions is to pattern the
hydrophobic region on the surface of the active colloid Gao  _et al._ (2013).
Unlike dipolar active Janus particles that consists of two hemispheres of
opposite charge, amphiphilic active patchy particles (APPs) consist of a polar
and a hydrophobic region. A recent experimental realization of an active
amphiphilic patchy particle consists of silica microspheres passivated with a
hydrophobic ligand (octadecyltrichlorosilane, OTS), whose hemisphere is then
covered by a Pt cap. These active APPs experience a short ranged attraction
between the hydrophobic domains, while the interaction between the Pt capped
hemispheres is purely repulsive. In essence, the Pt cap restricts the inter-
particle attraction to the hydrophobic region which gives rise to well defined
assemblies. By tuning the size of the hydrophobic domain and that of the Pt
cap, it is possible to control the angular range of the interaction, and thus
the number of colloids that can simultaneously stick to one another.

In this paper, we report the self-assembly dynamics of a two dimensional
dilute suspension of active APPs and study the structural and morphological
properties of the aggregates that spontaneously form in the presence of active
forces. We consider active APPs with different sizes of the hydrophobic
regions spanning from 25%, 50% and 75% of the total area of the particle, and
for two different propelling directions: one that is in the same direction as
the hydrophobic patch and one that is in the opposite direction of the
hydrophobic patch. Our results unveil a nontrivial morphological dependence of
the self-assembled structures on the active forces and hydrophobic coverage.

##  II Model

Figure 1: (a) Illustration of the three patch coverages considered in this
study. The hydrophobic patch is shown in black with percentage coverage $C$
corresponding to, from left to right, $C=0.25$, $C=0.50$, and $C=0.75$,
respectively. The large arrow indicates the propulsion direction. The top row
corresponds to patchy colloids propelled in the direction of the patch $(WP)$,
whereas the bottom row against the patch $(AP)$. (b) Potential $U_{ij}$ given
as a function of the separation distance $r_{ij}$. The potential for the two
primary binding energies $\varepsilon=12$ and $\varepsilon=20$ are shown, as
well as the purely repulsive component of the potential. (c) Potential
$U_{ij}$ given as a function of the angle $\theta_{i}$ for the various patch
coverages considered in this study.

In this manuscript, we consider three different attractive patch coverages,
$C$, defined as the ratio between the area of the hydrophobic patch and the
total area of the particle. In Figure 1, we illustrate from left to right the
different attractive patch coverages: $C=0.25$, $C=0.50$, and $C=0.75$. The
large arrow in the figure indicates the direction of self-propulsion with
respect to the attractive patch. The top row corresponds to APPs that are
propelled in the same direction as the patch,
$\boldsymbol{\eta_{i}}=\boldsymbol{n_{i}}$, or with the patch, which we
abbreviate as $(WP)$. The bottom row corresponds to APPs propelled against the
attractive patch, $\boldsymbol{\eta_{i}}=-\boldsymbol{n_{i}}$, which is
abbreviated as $(AP)$. In Fig 1, the black portion of the particles correspond
to the attractive patches, and we use yellow and blue to differentiate between
the $(WP)$ and $(AP)$ cases, respectively. For each patch coverage, we will
first study the behavior of the passive counterpart at the same density,
represented as green (repulsive patch) / black (attractive patch) colloids.

We simulate a two dimensional dilute suspension of $N=2000$ amphiphilic patchy
particles (APPs) at an area fraction of $\phi=0.1$. Each APP is modeled as a
disk with diameter $\sigma$, and undergoes Brownian dynamics at a constant
temperature $T$ according to the coupled translational and rotational
equations:

$\dot{\boldsymbol{r}}(t)=\frac{1}{\gamma}\boldsymbol{F}(\\{\boldsymbol{r}_{ij},\boldsymbol{\eta}_{i},\boldsymbol{\eta}_{j}\\})+\upsilon_{p}\,\boldsymbol{\eta}(t)+\sqrt{2D}\,\boldsymbol{\xi}(t)$
(1)

$\dot{\boldsymbol{\eta}}(t)=\frac{1}{\gamma_{r}}\boldsymbol{T}(\\{\boldsymbol{r}_{ij},\boldsymbol{\eta}_{i},\boldsymbol{\eta}_{j}\\})+\sqrt{2D_{r}}\,\boldsymbol{\xi}(t)\times\boldsymbol{\eta}(t)$
(2)

Activity or self-propulsion is introduced through a directional propelling
velocity of constant magnitude $\upsilon_{p}$ and is directed along a
predefined orientation vector $\boldsymbol{\eta}$. $\boldsymbol{\eta}$ is
centered at the origin of each particle and is restricted to rotate in the 2D
plane of the system. The translational diffusion coefficient $D$ is related to
the temperature $T$ and the translational friction $\gamma$ via the Stokes-
Einstein relation $D=k_{\rm B}T/\gamma$. We make the typical assumption that
the rotational diffusion coefficient $D_{r}=k_{\rm B}T/\gamma_{r}$ satisfies
the relation $D_{r}=(3D)/\sigma^{2}$. Thus, the Péclet number can be defined
as $Pe=\sigma v_{p}/D$. The solvent induced Gaussian white-noise terms for
both the translational and rotational motion are characterized by
$\langle\boldsymbol{\xi}(t)\rangle=0$ and
$\langle\xi_{i}(t)\xi_{j}(t^{\prime})\rangle=\delta_{ij}\delta(t-t^{\prime})$.
The conservative inter-particle forces and torques acting on each colloid are
given by
$\boldsymbol{F}(\\{\boldsymbol{r}_{ij},\boldsymbol{\eta}_{i},\boldsymbol{\eta}_{j}\\})$
and
$\boldsymbol{T}(\\{\boldsymbol{r}_{ij},\boldsymbol{\eta}_{i},\boldsymbol{\eta}_{j}\\})$
respectively, and can be computed as

$\boldsymbol{F}(\\{\boldsymbol{r}_{ij},\boldsymbol{\eta}_{i},\boldsymbol{\eta}_{j}\\})=\sum_{j}\boldsymbol{F}_{ij}=-\sum_{j}\boldsymbol{\nabla}_{\boldsymbol{r}_{ij}}U_{ij}\left(r_{ij},\theta_{i},\theta_{j}\right)$
(3)

and

$\boldsymbol{T}(\\{\boldsymbol{r}_{ij},\boldsymbol{\eta}_{i},\boldsymbol{\eta}_{j}\\})=\sum_{j}\boldsymbol{T}_{ij}=\sum_{j}(\boldsymbol{\eta}_{i}\times\dfrac{\partial
U_{ij}\left(r_{ij},\theta_{i},\theta_{j}\right)}{\partial\boldsymbol{\eta}_{i}})$
(4)

where $U_{ij}$ is the interaction potential between colloids. The anisotropic
pair potential is given by

$U_{ij}\left(r_{ij},\theta_{i},\theta_{j}\right)=U_{rep}(r_{ij})+U_{att}(r_{ij})\phi(\theta_{i})\phi(\theta_{j})$
(5)

where

$U_{rep}(r_{ij})=\begin{dcases}4\varepsilon_{rep}\left(\frac{\sigma}{r_{ij}}\right)^{12}&r_{ij}\leq
1.5\sigma\\\ 0&r_{ij}>1.5\sigma\end{dcases}$ (6)

and

$U_{att}(r_{ij})=\begin{dcases}4\varepsilon\left[\left(\frac{\sigma/2}{r_{s}+\sigma/2\times
2^{1/6}}\right)^{12}-\left(\frac{\sigma/2}{r_{s}+\sigma/2\times
2^{1/6}}\right)^{6}\right]&r_{ij}\leq 1.5\sigma\\\
0&r_{ij}>1.5\sigma\end{dcases}$ (7)

Here $\varepsilon$ is the binding energy, $r_{ij}$ is the distance between the
centers of particles $i$ and $j$, while $r_{s}=|r_{ij}-\sigma|$ is the
distance between the particles surfaces. Here, we choose a value of
$\varepsilon_{rep}=1$ for the repulsive component of the potential. The
potential $U_{ij}$ as a function of the separation distance $r_{ij}$ is given
in Figure 1(b) for different pair configurations.

The angular interaction $\phi(\theta_{i})$ is given by

$\phi(\theta_{i})=\begin{dcases}1&\theta_{i}\leq\theta_{max}\\\
\cos^{2}\left(\frac{\pi(\theta_{i}-\theta_{max})}{2\theta_{tail}}\right)&\theta_{max}\leq\theta_{i}\leq\theta_{max}+\theta_{tail}\\\
0&\mbox{otherwise}\end{dcases}$ (8)

where $\theta_{i}$ is defined as the angle between the patch unit vector
$\boldsymbol{n}_{i}$ and the inter-particle vector
$\boldsymbol{r}_{ji}=\boldsymbol{r}_{j}-\boldsymbol{r}_{i}$, hence
$\cos(\theta_{i})=\boldsymbol{r}_{ji}\cdot\boldsymbol{n}_{i}$ (similarly
$\theta_{j}$ is the angle between patch vector $\boldsymbol{n}_{j}$ and inter-
particle vector $\boldsymbol{r}_{ij}=-\boldsymbol{r}_{ji}$). The corresponding
expression for $\phi(\theta_{j})$ is also given by Eq. 8. The angular
potential $\phi$ is a smooth step function that modulates the angular
dependence of the potential and is equal to 1 within the region
$\theta_{i}<\theta_{max}$ and decays to zero following the expression above.
The percentage coverage $C=\\{0.25,0.5,0.75\\}$ is related with
$\theta_{max}=\\{45,90,135\\}$ degrees, respectively. The particular value of
$\theta_{tail}=25$ degrees has been selected to generate a sufficiently smooth
potential at the Janus interface. The potential $U_{ij}$ is shown in Fig. 1c
as a function of the angle $\theta_{i}$ for two particles at contact while the
orientation of particle $j$ is held fixed. This potential was adapted from the
model introduced by Miller & Cacciuto Miller and Cacciuto (2009), and is
appropriate to describe the behavior of patchy particles at high salt
concentration. In fact, the nature of the interaction between two Janus
particles is very much dependent on the degree of electrostatic screening of
the polar hemisphere of the particle. As observed in experimentsHong  _et al._
(2008), when unscreened, the polar side of the particles would provide a
significant angular dependence on the orientation of the particles, and even
prevent their aggregationHong  _et al._ (2008). In our simulations $\sigma$
and $k_{\rm B}T$ are used as the basic units of length and energy, while
$\tau=\sigma^{2}/D$ is our unit of time. All simulations were run for a
minimum of $10^{7}$ steps with time step $\Delta t=10^{-5}\tau$.

##  III Results

###  III.1 Dilute suspension of active APPs with patch coverage $(C=0.25)$

Figure 2: (a) Snapshot of the stable trimer structure for passive suspension
of APPs with patch coverage $C=0.25$. (b) Typical configuration observed for
binding energy $\varepsilon=20$.

We begin by considering the bulk properties of a low density suspension of
passive APPs with a small attractive patch. For all patch coverages, the
passive system serves as a reference to the equilibrium behavior. In the
passive system at $C=0.25$, the maximum number of attractive interactions or
bonds a given APP can form is limited to two, resulting in clusters of three
APPs at most. In this configuration the APPs arrange into equilateral
triangles with the attractive patch oriented toward the barycenter of the
triangle as shown in Fig. 2(a). A snapshot of a typical configuration at large
binding energy ($\varepsilon=20$) is given in Figure 2(b). As one would
expect, the binding energy determines the clustering behavior of these
suspensions, and the shape of the largest possible stable cluster is
determined by the geometric constraints imposed by the shape of the attractive
patch: a trimer in this small patch case ($C=0.25$).

To study how these particles self-assemble, we compute the average degree of
aggregation $\langle\Theta\rangle$ of the system defined as in Ref.Mognetti
_et al._ (2013); Prymidis  _et al._ (2015):

$\langle\Theta\rangle=1-\frac{\langle N_{c}\rangle}{N}=1-\frac{1}{\langle
S\rangle}$ (9)

where $\langle N_{c}\rangle$ is the average number of clusters in the system
and $\langle S\rangle$ is the average number of APPs in a cluster. For a
system, that has aggregated into a single cluster $\langle\Theta\rangle=1$,
and conversely a system composed of single particle clusters will give a value
$\langle\Theta\rangle=0$. In Figure 3(a), we show how the average degree of
aggregation $\langle\Theta\rangle$ depends on the binding energy
$\varepsilon$. For small binding energies the system mainly consists of
monomers and dimers, while for large $\varepsilon$ it almost exclusively
consists of trimers. For the largest binding energy, we find that
$\langle\Theta\rangle\simeq 0.66$, which is consistent with the theoretical
maximum value of $\langle\Theta\rangle=2/3$ occurring when all aggregates are
trimers.

Figure 3: (a) Average degree of aggregation $\langle\Theta\rangle$ as a
function of $\varepsilon$ for passive suspensions with $C=0.25$. (b) Average
degree of aggregation $\langle\Theta\rangle$ as a function of $\upsilon_{p}$
for active suspensions with $C=0.25$. Two different biniding energies are
considered: $\varepsilon=12$ (light green) and $\varepsilon=20$ (magenta).
Solid lines correspond to the (WP) case, while the (AP) case is indicated by
the dotted lines.

With an understanding of the equilibrium behavior, we now turn our attention
to the effect of self-propulsion. Specifically, we consider the two cases
where APPs are either propelled in the direction of the attractive patch
$(WP)$ or in the opposite direction $(AP)$. For these active suspensions, we
only consider two different values of the binding energy, $\varepsilon=12$ and
$\varepsilon=20$, and a range of activities up to $\upsilon_{p}=50$. Fig. 3(b)
summarizes the results of this study for $C=0.25$. Interestingly, the
clustering behavior of active APPs is highly sensitive to the direction of
self-propulsion relative to the attractive patch. At the smaller binding
energy $\varepsilon=12$, thermal fluctuations in the passive case are able to
easily break the bond between two APPs. As a result, $\langle\Theta\rangle$ in
the passive system is quite small and takes a value of
$\langle\Theta\rangle=0.3$. As we introduce self-propulsion in the $(WP)$
direction, we observe an increase in $\langle\Theta\rangle$ to values above
$\langle\Theta\rangle=0.50$. Here activity stabilizes the formation of dimers
and trimers structures. Whereas the opposite occurs when the propulsion is in
the $(AP)$ direction, where we observe $\langle\Theta\rangle$ rapidly dropping
to nearly zero where even small clusters quickly break into unpaired monomers.
This behavior is even more pronounced at higher binding energies. At
$\varepsilon=20$, the system mainly consists of trimers for the $(WP)$
direction. Interestingly, since in this configuration, the net sum of the
active forces is zero, these trimers are essentially inert, resulting in a low
density fluid where each trimer undergoes Brownian diffusion. Once more, in
the $(AP)$ direction we observe $\langle\Theta\rangle$ dropping to nearly zero
(even though slower than in the $\varepsilon=12$ case).

This behavior is particularly interesting as it further reinforces the notion
that activity in the $(WP)$ direction can be used to strengthen particle-
particle interactions and significantly improves self-assembly of compact
structures Mallory  _et al._ (2017). It is worth pointing out, however, that
the small lowering in $\langle\Theta\rangle$ for the largest activities at
$\varepsilon=12$ is due to either collisions of a trimer with fast moving
unpaired APPs, or the formation of unstable configurations where the vectorial
sum of the propelling axes in the trimer deviates sufficiently from zero to
generate internal torques able to break the trimer. These destructive events
are very rare for $\varepsilon=20$ as most APPs occupy a stable configuration
within a trimer, and thus we observe no significant decline with increasing
$\upsilon_{p}$ (at least within the range of velocities considered in this
study). These trimer destroying events occur more frequently at lower binding
energies, as there are more unpaired APPs and the interaction energy is not as
effective at maintaining fairly strict angular orientations within the trimer.

Although, it is tempting to think of the role of activity in this system as
having the simple effect of shifting the strength of the inter-particle
interactions as $\varepsilon_{\rm eff}=\varepsilon\pm\alpha\upsilon_{p}$, with
the plus or minus corresponding to the $(WP)$ and $(AP)$ directions
respectively, and where $\alpha$ is a phenomenological constant, the reality
is more complicated. In fact, while on the one hand, self-propulsion can
either strengthen or weakens the interaction energy of the particles in a
cluster, on the other hand self-propulsion increases the translational
velocity of the unpaired particles in the bulk (that one could think of as an
effective ”temperature”), which makes a straightforward mapping to an
effective equilibrium system rather nontrivial.

###  III.2 Dilute suspension of active APPs with patch coverage $(C=0.50)$

Figure 4: (a) Average degree of aggregation $\langle\Theta\rangle$ as a
function of $\varepsilon$ for passive suspensions with $C=0.50$. (b) Average
degree of aggregation $\langle\Theta\rangle$ as a function of $\upsilon_{p}$
for active suspensions with $C=0.50$. Two different binding energies are
considered: $\varepsilon=12$ (light green) and $\varepsilon=20$ (magenta).
Solid lines correspond to the (WP) case, while the (AP) case is indicated by
the dotted lines.

We now consider the behavior of APPs with patch coverage $C=0.50$. For this
patch coverage, the maximum number of bonding interactions for a given APP is
limited to four. For this reason, unlike the $C=0.25$ case, there isn’t a
simple compact structure determining the maximum cluster size for a given set
of system parameters (i.e. density, binding energy, temperature, etc). In the
passive system, the most stable local configuration is achieved when each APP
is located at the vertices of a parallelogram with the attractive patches
oriented inward toward one another. This results in the most abundant
macrostructures being two-particle thick, rigid chain-like clusters, whose
length is controlled by the binding energy $\varepsilon$ (see top panels in
Figures LABEL:fig:5 and LABEL:fig:6). For small values of $\varepsilon$ (top
panel of Fig. LABEL:fig:5), clustering is limited to the dynamic assembly and
disassembly of chain like clusters. In this regime, predominately linear chain
structures are observed and the typical number of APPs in a single cluster is
on the order of 20. As expected, the length of these chains increases with
increasing binding energy. However, as the binding energy increases to large
values, branches and kinks develop along the chain backbone (top panel of Fig.
LABEL:fig:6). For the largest binding energies considered, we observe the
formation of large branched networks spanning the size of the system. This
qualitative description of the behavior of the passive system can be
represented quantitatively with average degree of aggregation
$\langle\Theta\rangle$ shown in Fig 4(a).

The effects of self-propulsion are not as simple to unravel as in the $C=0.25$
case, where self-propulsion can either stabilizes or destabilizes a cluster
depending on the direction of self-propulsion relative to the attractive
patch. In Figure 4(b), we show the average degree of aggregation
$\langle\Theta\rangle$ as a function of activity for both binding energies
considered. In both cases, $\langle\Theta\rangle$ decreases as $\upsilon_{p}$
is increased, and in both cases, the $(AP)$ direction experiences a
significantly faster decay of $\langle\Theta\rangle$. Thus, one would naively
concluded that the main effect of activity is that of reducing the average
size of the aggregates in solution until a fluid state is recovered for very
large $\upsilon_{p}$. This line of thought also suggests that larger active
forces are necessary in the $(WP)$ direction to break the clusters with
respect to the $(AP)$ direction. However, this is not the entire story: a
closer look at the evolution of the different configurations that develop in
the two propelling directions reveals a complex structural behavior. Figures
LABEL:fig:5 and LABEL:fig:6 present steady-state snapshots of typical
structure that develop as activity $\upsilon_{p}$ is increased for
$\varepsilon=12$ and $\varepsilon=20$, respectively.

A binding energy of $\varepsilon=12$ is somewhat ideal for the passive system,
as the APPs self-assemble into long linear aggregates with only a sporadic
formation of kinks along the chain (Fig. LABEL:fig:5) . No branching of the
chains is observed at these binding energies. When activity is introduced in
the $(AP)$ direction (right column of Figure LABEL:fig:5), the length of the
structures decreases with increasing $\upsilon_{p}$, and the formation of
kinks becomes less probable. The system eventually devolves into a hot fluid
for even larger values of $\upsilon_{p}$. When activity is introduced in the
$(WP)$ direction (left column of Figure LABEL:fig:5), patchy particles feel a
stronger attraction when in the condensed phase. This leads to the more
frequent formation of kinks along the chain, but simultaneously internal
torques develop at the location of the kinks that act as pivots around which
the chain can fold. A comparison of the structures for $\upsilon_{p}=25$ in
the $(WP)$ and $(AP)$ directions is quite revealing (lowest panels in Fig.
LABEL:fig:5). In the $(WP)$ direction most of the particles are involved in
the formation of small, rigid and compact clusters, while in the $(AP)$
direction most particles are in a fluid phase. This is consistent with the
data shown in Fig. 4(b).

A much more dramatic illustration of this effect can be observed for
$\varepsilon=20$. Here, large branched clusters develop in the passive case
(Fig. LABEL:fig:6, top panel). As activity is introduced either in the $(AP)$
(right column) or in the $(WP)$ direction (left column), we observe an overall
collapse of the structures. The collapse or folding points in these structure
are those locations where kinks or branching exist. The kink portions of the
structure act as a fulcrum, and the two linear segments surrounding this
fulcrum generate a force imbalance leading to the compaction and folding of
the structure. However, while these structures, regardless of the direction of
the active force, tend to become more compact for small values of
$\upsilon_{p}$, at intermediates values of $\upsilon_{p}$, we observe
diverging structural behaviors. In both $(WP)$ and $(AP)$ cases the large
branched aggregates systematically break down into smaller chuncks. This is
because kinks and branching points are the structural weak points of these
assemblies and thus more susceptible to be unraveled by the active forces.
However, when considering the $(WP)$ direction, we observe small compact
structures where multiple linear pieces merge together into activity-
stabilized open-mesh clusters, whereas in the $(AP)$ direction, we observe a
suspension of linear chains. This phenomenon is most evident as soon as the
active forces become comparable in strength to the dispersion forces
$\upsilon_{p}\simeq 20$.

Figure 5: Average radius of gyration $\langle R_{g}\rangle$ as a function of
cluster size $S$ for binding energy $\varepsilon=20$. The $WP$ case shows a
compaction of the clusters with increasing $\upsilon_{p}$, while the $AP$ case
exhibits reentrant behavior where clusters undergo a collapse before an
eventual expansion of $R_{g}$ as $\upsilon_{p}$ increases.

This diverging structural behavior for the two propelling directions can be
more quantitatively expressed by measuring the average radius of gyration of
the clusters formed by particle in the AP and WP directions. This analysis is
shown in Figs. 5, where the average radius of gyration $\langle R_{g}\rangle$
is given as a function of clusters size $S$ for $\varepsilon=20$ at different
propelling velocities $\upsilon_{p}$. Figure. 5(a) refers to the data for the
$(WP)$ direction and illustrates how $R_{g}$ systematically decreases when
going from the passive case, $\upsilon_{p}=0$ (blue curve), to the largest
activity considered, $\upsilon_{p}=50$ (red lowest curve). The $(AP)$
direction is given in figure. 5(b) and shown how $\langle R_{g}\rangle$
initially slightly decreases when compared to the passive reference system,
but eventually moves up to larger values for sufficiently large activities
(already for $\upsilon_{p}=20$). Unfortunately, it is very hard to obtain
these data with sufficient statistics and for large enough clusters to extract
a reliable size exponent. Nevertheless the two trends are quite clear.
Structurally speaking, apart from the decreasing overall size of the clusters
with the increase of $\upsilon_{p}$, one can think of the aggregates formed by
the $(WP)$ particles as polymers in a bad solvent with a linear persistence
length consisting of about 10 particles, and of the aggregates formed by the
$(AP)$ particles, at least beyond the re-entrant point, as semiflexible
polymers. Lastly, it is also worth noting that the activation of the APPs in
the (AP) direction has a practical application as a method to wash away
unwanted kinks and branches from a configuration of passive particles
assembling with interactions that are too sticky.

###  III.3 Dilute suspension of active APPs with patch coverage $(C=0.75)$

Figure 6: (a) Average degree of aggregation $\langle\Theta\rangle$ for passive
suspensions of patchy particles with patch coverage $C=0.75$ as a function of
the binding energy $\varepsilon$. (b) Average degree of aggregation
$\langle\Theta\rangle$ as a function of $\upsilon_{p}$ for active suspensions
with $C=0.75$. Two different binding energies are considered: $\varepsilon=12$
(light green) and $\varepsilon=20$ (magenta). Solid lines correspond to the
(WP) case, while the (AP) case is indicated by the dotted lines.

We conclude this study by analyzing the bulk properties of a low density
suspension of APPs with patch coverage $C=0.75$. At this large patch coverage
the geometry of the interaction is wide enough that each APP can accommodate
more than 4 other APPs. This leads to the formation of more isotropic
aggregates. In Fig. 6(a), we show $\langle\Theta\rangle$ as a function of the
binding energy $\varepsilon$ for the passive system. The system fully
aggregates into a single cluster for any $\varepsilon$ greater than about
$10$, and for smaller values of $\varepsilon$, we observe small isotropic
clusters with overall hexagonal symmetry. At $\varepsilon=12$, we report the
formation thick crystalline branched structures. See top panels of Fig. 7 for
a typical conformations of the passive suspension with $C=0.75$ and
$\varepsilon=12$. Figure 6(b) shows the average degree of aggregation
$\langle\Theta\rangle$ as a function of activity for binding energies
$\varepsilon=12$ and $\varepsilon=20$. Notably, when compared to the two
smaller patch geometries considered in this study the difference in
$\langle\Theta\rangle$ between the $AP$ and $WP$ directions is far less
dramatic. For both binding energies, $\langle\Theta\rangle$ decrease
monotonically with increasing self-propelling velocity $\upsilon_{p}$.

Figure 7: Typical simulation snapshot for active APPs with patch coverage
$C=0.75$ and $\varepsilon=12$. The passive reference system $\upsilon_{p}=0$
is shown at the top of the figure. The $(WP)$ case is reported in the left
column whereas the $(AP)$ case is in the right column. Each snapshot is
accompanied by an enlarged snapshot highlighting the local cluster geometry.

One of the most compelling features of the clusters formed both at
$\varepsilon=12$ and at $\varepsilon=20$ is the change in cluster shape as
soon as activity is introduced. Unlike the typical branched structures formed
by the passive system, clusters formed by active APPs immediately collapse
into truly two dimensional objects with a size exponent equal to $1/2$, which
means that system form isotropic clusters, size exponent of $1/2$ comes from
the calculation of the radius of gyration as a function of the cluster size.
This is due in large part to the increased degree of rotational freedom of an
APP allowing for more efficient folding of the cluster. Figures 7 (lower
panels) and 8 (lower panels) show the typical aggregates formed for binding
energy $\varepsilon=12$ and $\varepsilon=20$, respectively. Apart from the
slightly faster decay in $\langle\Theta\rangle$ for the $(AP)$ direction with
respect to the $(WP)$ direction, there aren’t significant structural
differences across the range of active velocities considered in this study.

Figure 8: Typical simulation snapshot for active APPs with patch coverage
$C=0.75$ and $\varepsilon=20$. The passive reference system $\upsilon_{p}=0$
is shown at the top of the figure. The $(WP)$ case is shown in the left column
while the $(AP)$ case is shown in the right column. Each system snapshot is
accompanied by an enlarged snapshot highlighting the local cluster geometry.

What is, however, significant is the dynamical nature of the clusters
themselves. In fact, the clusters that form are active in their own right, and
they ”move, break apart and form again” over the life-time of the simulations.
The behavior here is reminiscent of the behavior observed in experiments with
attractive colloidal swimmers Palacci  _et al._ (2013). The main difference
between our system and that with active spherical particles interacting via an
isotropic attractive potential is that in our case, the axis of propulsion of
the particles in the condensed phase is essentially frozen, and thus patchy
particles in the cluster do not rotate significantly. This is in stark
contrast with the isotropic attractive case where active particles undergo
rotational Brownian motion. For APPs, what drives these clusters unusual
dynamics is their orientational and translational frustration. In fact, far
from being hexagonal, these clusters are frozen into rather disordered
configurations with frequent gaps within the structure and with the presence
of several tetramers with square symmetry organized in such a way that the
hydrophilic, rather than the hydrophobic sides of the particles point toward
each other. The large torques and forces that develop within these clusters
can therefore easily destabilize them as soon as they grow to a sufficiently
large size, or, as we often observe, as soon as a small cluster in solution
binds to it forcing orientational re-arrangements of the particles that can
destabilize their already precarious balance of forces. Finally, it is worth
mentioning that while living crystals observed for $C=1.00$ are typically in
dynamic equilibrium with a fluid of unpaired particles, at low velocities, we
observe fluids of active clusters, with no unpaired APPs, where clusters
either break upon interacting with other clusters, or after large and slow
internal cluster re-arrangements.

##  IV Discussion and conclusions

Figure 9: Summary of the Average degree of aggregation $\langle\Theta\rangle$
for all patch coverages considered in this study. The left hand side plots
show the results for the $WP$ case, while the plots on the right hand side
show the results for the $AP$ case. As a reference, these plots also include
the data for fully isotropic attractive particles corresponding to $C=1$.

In this paper, we analyzed how active forces can be used to stabilize or
destabilize aggregates formed by active amphiphilic patchy particles (APPs).
We study the phase behavior of dilute suspensions of these patchy particles
for three different hydrophobic patch coverages: $C=0.25$, $C=0.50$, and
$C=0.75$. We also considered explicitly the role of the direction of the
active forces with respect to the placement of the attractive patch, i.e. with
the patch ($WP$) and against the patch ($AP$) for a range of different
swimming velocities $\upsilon_{p}$ and hydrophobic attractions $\varepsilon$.
Figure 9 nicely summarizes all of our results concerning how the average
degree of aggregation $\langle\Theta\rangle$ depends on the swimming velocity
$\upsilon_{p}$ for two different values of $\varepsilon$. The left column
shows results for the $(WP)$ direction while the right column corresponds to
the $(AP)$ direction. For completeness we also added to these plots the data
corresponding to the fully hydrophobic particles (isotropic attraction) for
which $C=1.00$.

Our study illustrates how the geometry of the interactions between the
particles plays a pivotal role in how active forces affect the stability of
self-assembled aggregates. Unlike the very special case of spherical active
particles interacting with an isotropic potential, where the rotational
degrees of freedom of each particle aren’t lost when in the condensed phase,
using active APPs with a tunable size of the hydrophobic region enables us to
systematically control the parameters responsible for their rotational
dynamics. Interestingly, we find that under strong angular confinement $-$
condition that leads to the formation of small finite size clusters $-$ it is
possible to increase the particle’s aggregation yield by aligning the active
forces in the direction of the patch $(WP)$. We have also shown how, when only
half of the particle is covered by a hydrophobic patch, it is possible to
break apart and either compactify or wash away branches from large branched
networks of particles. In doing so we highlighted the significant role played
by the direction of the propelling forces for particle with 50% hydrophobic
coverage. Finally, we discussed the remarkable behavior of the isotropic
active clusters that self-assemble when 75% of the particle area is covered by
a hydrophobic patch, and point out how frustration of the frozen active forces
within the clusters leads to their dynamic, ”living” behavior. One can think
of these aggregates as being in between those formed by active attractive
rods, and those formed by isotropic active Lennard Jones particles. In fact,
in clusters formed by rods (or dumbbells), particles are unable to rotate even
when a second cluster merges with it. This leads to the formation of ever
growing macroscopic aggregates that have a net rotational speed that depends
on the details of the particle’s relative orientations, and that decreases
with the cluster size Schwarz-Linek  _et al._ (2012). In clusters formed by
spherical particles, as mentioned above, the colloids are completely free to
rotate within the clusters, and for a range of active forces and for
sufficiently low densities, this leads to the formation of ”living” clusters
which do not seem to grow beyond a certain size Palacci  _et al._ (2013).
Active APPs with 75% coverage form clusters where the directions of the active
forces of the particles is mostly frozen, leading to rotating clusters with a
net rotational speed, but in this case the propelling forces can undergo
significant re-arrangements when interacting with another cluster, leading to
cluster fractures, large global re-arrangements and eventual disassembly.

Hydrodynamic interactions may play an important role on the phase behavior and
structural properties of the assemblies formed by these particles. Extensive
work on the subject is currently underway and it will be published elsewhere.

## Acknowledgements

We thank Clarion Tung and Joseph Harder for insightful discussions and helpful
comments. C.V. acknowledges financial support by a Ramon y Cajal tenure track,
the Spanish Ministry of Education (MINECO, FIS2016-78847-P) and UCM-Santander
(PR26/16-10B-3). A.C. acknowledges financial supported from the National
Science Foundation under Grant No. DMR-1408259. SAM acknowledges financial
support from the National Science Foundation Graduate Research Fellowship.
This work used the Extreme Science and Engineering Discovery Environment
(XSEDE), which is supported by National Science Foundation grant number
ACI-1053575.

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