**4. Dynamic clustering and chemotactic collapse in a system of active colloids 49**

**4.6. Collapse Regimes**

For sufficiently large ζ_{tr} or ζ_{rot} all N particles in the system collapse and accumulate
in one single cluster (see Fig. 4.3, bottom right). To quantify the crystalline hexagonal
order in such a cluster, we introduce the global 6-fold bond orientational parameter

q_{6} :=

1 N

N

k=1

q^{(k)}_{6}

∈[0,1] with q_{6}^{(k)}:= 1
6

j∈N_{6}^{(k)}

e^{i6α}^{kj}. (4.11)

Here, N_{6}^{(k)} is the set of six nearest neighbors of particle k and α_{kj} is the angle between
the vector connecting particle k to j and some prescribed axis [102]. The local bond

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ì ì

ì ì

ò ò ò ò ò ò ò ò ò ò ò ò

10 12 14 16 18 20

5 10 15 20 25 30 35

Pe
*N**c*

Ζ0

0.09 0.25 0.47 1.13

Figure 4.9.: Mean cluster size N_{c} versus Pe for different lines in the full parameter space
defined via a parametrization with x ∈ [0,1]. We vary Pe as in experiments of Ref.

[14], Pe = 9.5 + 11.5x, and choose ζtr = 4.8 + 16.6x and ζrot = −0.16−ζ0x, where the
parameterζ_{0} defines the different graphs. The transition between clustering states 1 and
2 roughly occurs at the intersection of the two straight lines.

parameter q^{(k)}_{6} becomes one if all six nearest neighbors form a regular hexagon around
the central colloid and the global order parameter becomes one in a hexagonal lattice.

We now use the temporal behavior of q_{6} to identify different regimes in the collapsed
state, as indicated in the state diagram of Fig. 4.2. In Fig. 4.10(a) we plot q_{6} versus
time for several ζ_{tr}. We set ζ_{rot} = 4.5 to a sufficiently large value to guarantee the
collapse of the system for all chosen ζ_{tr}. For positive ζ_{tr} we find the order-parameter
value q_{6} ≈0.89 nearly constant in time. In this situation all particles are packed in one
crystalline cluster and q_{6} is only smaller than 1 since colloids at the rim of the cluster
are not surrounded by six particles on a hexagon. Already for ζ_{tr} = 0 small fluctuations
in the order parameter are visible. The cluster is no longer static. It even can become
more circular in shape and q_{6} assumes values above 0.89. The fluctuations increase for
ζ_{tr} <0, i.e., when the particles effectively repel each other due to translational phoretic
motion. Single particles or even clusters occasionally leave the main large cluster and
rejoin it after a while. For decreasing ζ_{tr} the fluctuations in q_{6} strongly increase. For
example, for ζ_{tr} = −6.4 the main cluster’s integrity is occasionally disrupted, which is
reflected by a q_{6} significantly varying in time.

Upon further decreasing ζ_{tr} one enters a regime, where these fluctuations transform
into surprisingly regular oscillations, which marks the oscillation regime in the state
dia-gram of Fig. 4.2. The origin of these oscillations in q_{6} is the following cyclic process: one
large cluster evolves dynamically (large q_{6}), it resolves and particles disperse (small q_{6}),
they rejoin to form the cluster again, and so on. More precisely, the cluster oscillates
between a crystalline structure and a cloud of confined colloids. In the crystalline cluster
the diffusiophoretic interaction is strongly screened and the particles are not perfectly

Figure 4.10.: (a) Time evolution of the bond orientational parameter q_{6} for different ζ_{tr}.
Further parameters are Pe = 19, ζ_{rot} = 4.5, and σ = 0.05. (b) Standard deviation ∆q_{6} of
the fluctuating q_{6} plotted versus ζ_{tr} for different ζ_{rot}. The curve at ζ_{rot} = 4.5 quantifies
the fluctuations in the graphs of (a).

oriented towards the cluster center. Thermal fluctuations locally disturb the hexagonal packing such that the screening is weakened. Consequently, the repulsion due to trans-lational diffusiophoresis becomes more long-range. This destabilizes the cluster, which appears more like a cloud [a situation similar to Fig. 4.14(b)]. Now, the long-range dif-fusiophoretic interaction orients all particles back to the cloud center and the compact cluster forms again. Therefore, essentially the pulsating takes its origin in two different time-scales with which the particles respond to a sudden change of∇c. The translational repulsion acts immediately, whereas the orientational attraction acts only after a typical reorientation time.

For very negativeζ_{tr}no oscillations occur any more. The translational repulsion forces
are strong and lead to separation of particles, as shown in the snapshot of Fig. 4.14(b).

In this collapsed cloud the hexagonal bond order nearly vanishes leading to q_{6} ≈ 0.35,
which is close to the value q_{6} = 1/3 for systems with homogeneous particle distribution.

To quantify the fluctuations of the bond orientational order parameter q_{6}, we plot its
standard deviation ∆q6 := [⟨(q6− ⟨q6⟩)^{2}⟩]^{1/2} in the full range of ζtr in Fig. 4.10(b). For
large values of ζ_{rot} fluctuations continuously increase with decreasing ζ_{tr} and then when
entering the collapsed-cloud regime a sharp decrease occurs. However, the fluctuations
do not indicate the transition to the oscillatory regime. The dependence onζtr is smooth
at the transition between fluctuating and oscillating clusters. Note that for ζ_{rot} = 2.2
and 3.0 the sharp drop with decreasingζ_{tr} indicates the transition into the gaslike state.

To identify the oscillating regime we determine the power spectrum of the bond ori-entational parameter. For this purpose, we first define the time-autocorrelation function

C(jτ) = 1 n

n

i=1

[q_{6}(t_{i}+jτ)− ⟨q_{6}⟩][q_{6}(t_{i})− ⟨q_{6}⟩]

(∆q_{6})^{2} . (4.12)

Here, {t1, . . . , tn} is a set of equally spaced time points from the stationary state withn
typically around 10000, τ =t_{i}−ti−1, andj ranges from 1 to 1000. We perform a discrete
Fourier transform,

Q_{6}(ω) =

k

j=1

C(jτ) exp(−iωjτ), (4.13)

which, according to Wiener-Khinchine’s theorem, is equal to the power spectrum of q_{6}.
The results for different ζ_{tr} are plotted in Fig. 4.11(a). We fit the spectrum with a
non-normalized Gaussian function and detect its maximum at the position ω_{max}. In the
fluctuating-cluster regime no peak in the power spectrum can be distinguished, rather
more, the curve for ζ_{tr} =−3.2 decreases monotonically. By contrast, in the
oscillating-cluster regime a clear maximum at non-zero frequency ω_{max} exists. We identify the
oscillation state in the state diagram if ω_{max} > 0.01. This value is slightly larger than
zero, in order to being able to clearly identify a maximum. In Ref. [40] the authors
formulated continuum equations for diffusiophoretically coupled active colloids. In the
case where the diffusiophoretic translational velocity acts repulsively, i.e., for ζ_{tr} < 0,
they predict an instability with the onset of spontaneous oscillations. We have shown
that oscillations persist in steady state in a certain region in the state diagram.

As it turns out, activity, rather than phoresis, determines the frequency of the pulsating
cluster. In Fig. 4.11(b) we plot ω_{max} versus ζ_{tr}. From above (fluctuating cluster) and
from below (collapsed cloud) a sharp increase ofω_{max}indicates the onset of the oscillation
regime. The curves for ω_{max} display a plateau-like maximum with a value essentially
independent of ζ_{rot}. For example, the curves in Fig. 4.11(b) belong to Pe = 19 and we
find ω_{max}≈0.012±0.002 for the maximum value. Indeed, the oscillation frequencies are
strongly determined by the activity of the particles. In the inset of Fig. 4.11(b) we plot
the maximum value of ω_{max} versus Pe. Beyond the regime where thermal fluctuations
dominate, which is set by a defined threshold value Pe≈20, the characteristic frequency
exhibits a nearly linear increase in Pe. So, the oscillations become faster if the active
colloids are faster.

We note, that upon increasing Pe the phoretic strengths need to be adjusted as well to
reach the oscillation state. This common scaling of the parameters Pe, ζ_{rot} and ζ_{tr} was
rationalized in Sec. 4.5.3.