Probing the Spatiotemporal Dynamics of Catalytic Janus Particles with Single-Particle Tracking and Differential Dynamic Microscopy

We demonstrate differential dynamic microscopy and particle tracking for the characterization of the spatiotemporal behavior of active Janus colloids in terms of the intermediate scattering function (ISF). We provide an analytical solution for the ISF of the paradigmatic active Brownian particle model and find striking agreement with experimental results from the smallest length scales, where translational diffusion and self-propulsion dominate, up to the largest ones, which probe effective diffusion due to rotational Brownian motion. At intermediate length scales, characteristic oscillations resolve the crossover between directed motion to orientational relaxation and allow us to discriminate active Brownian motion from other reorientation processes, e.g., run-and-tumble motion. A direct comparison to theoretical predictions reliably yields the rotational and translational diffusion coefficients of the particles, the mean and width of their speed distribution, and the temporal evolution of these parameters.

Figure 1

(a) ISFs of a dilute suspension of Janus particles obtained via DDM (symbols and error bars) over the whole time sequence ($1.2\times {10}^5$ images), fitted with the ABP model (colored lines) and the RTP model (dashed, gray lines), both averaged over a Gaussian speed distribution. ${\tau }_{\mathrm{rot}}=D_{\mathrm{rot}}^{-1}$ denotes the rotational diffusion time. Global fitting of the ISFs [36] using the ABP model yields $⟨v⟩=1.23\pm 0.01\mu \mathrm{m}{\mathrm{s}}^{-1}$, ${\sigma }_v=0.26\pm 0.02\mu \mathrm{m}{\mathrm{s}}^{-1}$, $D_{\mathrm{rot}}=0.24\pm 0.01{\mathrm{s}}^{-1}$, and $\mathrm{D}=0.24\pm 0.02\mu {\mathrm{m}}^2{\mathrm{s}}^{-1}$. Separately fitting the RTP model provides the tumbling rate $\lambda =0.20\pm 0.03{\mathrm{s}}^{-1}$, and $⟨v⟩=1.21\pm 0.03\mu \mathrm{m}{\mathrm{s}}^{-1}$, ${\sigma }_v=0.05\pm 0.01\mu \mathrm{m}{\mathrm{s}}^{-1}$, $D=0.26\pm 0.02\mu {\mathrm{m}}^2{\mathrm{s}}^{-1}$. (b)–(c) Temporal variation of the motility parameters obtained from the ABP model fit of the ISFs extracted from DDM and tracking of fifteen 160 s submovies. The black line is $\propto \exp (-t/T)$ with $T=140\min$. (d) ABP-fitted ISFs from DDM (hollow symbols, solid line) and tracking (solid symbols, dashed line) for the submovie at 25 min.

Figure 2

ISFs for a single Janus particle (symbols and error bars, as in Fig. 1). Colored solid lines correspond to the theoretical predictions for ABPs ($v=1.64\pm 0.02\mu \mathrm{m}/\mathrm{s}$, $D=0.16\pm 0.01\mu {\mathrm{m}}^2/\mathrm{s}$, and $D_{\mathrm{rot}}=0.21\pm 0.02{\mathrm{s}}^{-1}$) and gray dashed lines for equivalent RTPs ($v=1.58\pm 0.02\mu \mathrm{m}/\mathrm{s}$, $D=0.17\pm 0.01\mu {\mathrm{m}}^2/\mathrm{s}$, and $\lambda =0.14\pm 0.01{\mathrm{s}}^{-1}$), both corrected for a finite exposure time $T_e$. Colored dashed lines correspond to the uncorrected ABP ISF [Eq. (3)]. Dash-dotted, black lines indicate the corrected large and uncorrected small wave number approximations, $\exp [-k^{2}D(\tau -T_e/3)]J_0(vk\tau )$ and $\exp (-k^2{D}_{\mathrm{eff}}\tau )$, respectively. Note that the ISF does not approach the small wave number approximation closely even at the smallest $k$ displayed, for which $kL=0.43$.