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

4.6. Collapse Regimes

For sufficiently large ζtr or ζrot 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

q6 :=

 1 N

N

k=1

q(k)6

∈[0,1] with q6(k):= 1 6

j∈N6(k)

ei6αkj. (4.11)

Here, N6(k) is the set of six nearest neighbors of particle k and αkj is the angle between the vector connecting particle k to j and some prescribed axis [102]. The local bond

æ æ æ æ æ

æ æ

à à à à à à

à à

à

ì ì ì ì ì ì ì ì

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 Nc

Ζ0

0.09 0.25 0.47 1.13

Figure 4.9.: Mean cluster size Nc 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 parameterζ0 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 q6 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 q6 versus time for several ζtr. We set ζrot = 4.5 to a sufficiently large value to guarantee the collapse of the system for all chosen ζtr. For positive ζtr we find the order-parameter value q6 ≈0.89 nearly constant in time. In this situation all particles are packed in one crystalline cluster and q6 is only smaller than 1 since colloids at the rim of the cluster are not surrounded by six particles on a hexagon. Already for ζtr = 0 small fluctuations in the order parameter are visible. The cluster is no longer static. It even can become more circular in shape and q6 assumes values above 0.89. The fluctuations increase for ζtr <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 ζtr the fluctuations in q6 strongly increase. For example, for ζtr = −6.4 the main cluster’s integrity is occasionally disrupted, which is reflected by a q6 significantly varying in time.

Upon further decreasing ζtr 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 q6 is the following cyclic process: one large cluster evolves dynamically (large q6), it resolves and particles disperse (small q6), 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 q6 for different ζtr. Further parameters are Pe = 19, ζrot = 4.5, and σ = 0.05. (b) Standard deviation ∆q6 of the fluctuating q6 plotted versus ζtr for different ζrot. The curve at ζrot = 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 negativeζtrno 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 q6 ≈ 0.35, which is close to the value q6 = 1/3 for systems with homogeneous particle distribution.

To quantify the fluctuations of the bond orientational order parameter q6, 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 ζrot fluctuations continuously increase with decreasing ζtr 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 ζrot = 2.2 and 3.0 the sharp drop with decreasingζtr 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

[q6(ti+jτ)− ⟨q6⟩][q6(ti)− ⟨q6⟩]

(∆q6)2 . (4.12)

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

Q6(ω) =

k

j=1

C(jτ) exp(−iωjτ), (4.13)

which, according to Wiener-Khinchine’s theorem, is equal to the power spectrum of q6. The results for different ζtr are plotted in Fig. 4.11(a). We fit the spectrum with a non-normalized Gaussian function and detect its maximum at the position ωmax. In the fluctuating-cluster regime no peak in the power spectrum can be distinguished, rather more, the curve for ζtr =−3.2 decreases monotonically. By contrast, in the oscillating-cluster regime a clear maximum at non-zero frequency ωmax exists. We identify the oscillation state in the state diagram if ωmax > 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 ζtr < 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 ωmax versus ζtr. From above (fluctuating cluster) and from below (collapsed cloud) a sharp increase ofωmaxindicates the onset of the oscillation regime. The curves for ωmax display a plateau-like maximum with a value essentially independent of ζrot. For example, the curves in Fig. 4.11(b) belong to Pe = 19 and we find ωmax≈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 ωmax 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, ζrot and ζtr was rationalized in Sec. 4.5.3.