## Power spectrum

Because the entire image shakes, no feature can be used as a reference point to follow the movement. To quantify the oscillation, the real image was formed directly on a 2D position-sensitive detector, which returned the (x, y) coordinate of the optical center-of-mass (CM weighted by the local light intensity *I(x,y;t)*) at time *t*.

$${\overrightarrow{r}}_{OCM}\left(t\right)= \frac{\sum _{x,y}I\left(x,y;t\right)\overrightarrow{r}\left(x,y\right)}{\sum _{x,y}I\left(x,y;t\right)}$$

1

This analog signal was FFT in real time to yield the power spectrum (Fig. 1b). The large peak around 4 Hz is due to the microscope body and is always present regardless of the sample. SO appears as a single sharp peak at approximately 20 Hz. There were no other peaks over the frequency range up to 3 kHz, the frequency bandwidth of the detector.

### Kuramoto Model

Synchronization is treated theoretically in a framework of limit-cycle oscillations11. In particular, the Kuramoto model has been studied intensively since it can be solved analytically14,15. The interaction governing the time development of the j-th oscillator phase is given by the sine of the phase differences with others.

where \({\omega }_{j}\) is the natural frequency, \(K\) is the mean-field interaction strength, and \(N\) is the number of interacting units. By introducing an order parameter \(R\), expressing the phase coherence, by

and initial phase distribution, it can be shown (for \(N\to \infty\)) that a threshold strength \({K}_{c}\) exists to synchronize, and that \(R\) increases as

$$R= \sqrt{\frac{K-{K}_{c}}{K}}$$

4

as shown in Fig. 2a.

### Surfactant Concentration Dependence

Because van der Waals (VDW) attractions always act on every CNT16, CNTs aggregate rapidly in the solution. To reduce this attraction, a non-ionic surfactant is added. The surfactant molecules are adsorbed spontaneously on the CNT surface. When the surfactant concentration exceeds a critical surface aggregation concentration, the CNT surface is sufficiently covered by the adsorbed molecules so that neighboring CNTs cannot come close to each other. This steric effect keeps the VDW attraction small, and the dispersion becomes stable. In other words, at a concentration below the critical concentration, some parts of the CNT surface are free of the surfactant layer. At the exposed area, neighboring CNTs can approach close enough that the VDW attraction becomes large. Also, it is known that the dye (rhodamine 6G) molecules are adsorbed on the free CNT surfaces17,18. Moreover, they tend to aggregate in the solution at high concentrations. Then, neighboring CNTs can be bridged by the adsorbed dye molecules at the exposed area. In this study, the dye concentration is high (250 mmole/L), so the dye-bridging effect may be significant. Both the VDW interaction and the dye-bridging effect take place in the surfactant-free area. Thus, lowering the surfactant concentration effectively increases the attractive interaction between CNTs and the number of interacting CNTs.

Figure 2b shows the surfactant concentration dependence of SO. The synchronized frequency remains the same over this range. At high concentrations, the VDW attraction is negligible, and the dispersion is stable. No SO is detected. As the concentration decreases, SO suddenly appears at around 1.0 wt% and grows stronger. The dispersion becomes unstable around 0.5–0.6 wt% and forms large aggregates below 0.4 wt%.

The existence of a threshold concentration agrees with the Kuramoto model. Because the phase coherence *R* and the interaction strength *K* are the parameters in the purely mathematical model, their exact relationships to the peak intensity and the surfactant concentration are not known. We think that the agreement between the experimental data and Eq. (4) in Fig. 2b is better-than-expected. Nevertheless, all repeated experiments have yielded concave down, increasing curves, indicating that the Kuramoto model can be applicable to the CNT SO by making a suitable transformation of *K*. It also demonstrates that the peak intensity can be used as a measure of synchronization.

### External Vibrations

Investigating the effects of external vibration is important for understanding the origin of SO and its response. A single-frequency, mechanical vibration was applied to the solution (Fig.3a). When the external frequency was varied in the neighborhood of the synchronized frequency (Fig. 3b, black line), SO was not affected at all (Fig.3b, red line). Also, SO did not resonate when the external frequency matched, exhibiting its immune nature to external stimuli. We repeated the experiment over frequencies ranging from 3 to 50 Hz and obtained the same result. We also confirmed that applying the external vibration to a stable dispersion (not oscillating) did not induce SO. Thus, SO is not caused by external vibration.

### Power-law Noise

Although SO was observed quite reproducibly, it was absent in some samples despite our consistent experimental procedure. We have noted that, when the sample does not synchronize, the baseline of a power spectrum becomes linear (Fig. 4b), whereas when it synchronizes, the baseline always exhibits \(\frac{1}{{f}^{n}}\) dependence (Fig. 4a. In this case, n = 20). The power-law spectrum is known to be caused by complex heterogeneous networks in solution19. We conjecture that SO and the power-law noise share the same origin that is related to the thermal motion of heterogeneous CNT networks.