Emergence of oscillatory states of self-propelled colloids under optical confinement

TL;DR

The study addresses how self-propelled colloids behave under optical confinement and identifies a novel oscillatory trapping regime arising from the interplay of self-propulsion and light-induced torques. It combines experiments on carbon-coated Janus beads and rod-shaped variants with a minimal overdamped stochastic model that includes a center-seeking torque, successfully reproducing orientation dynamics and four-regime translational motion. The key contribution is the demonstration that confinement can induce self-sustained oscillations whose frequency grows with propulsion speed, plus a framework explaining how anisotropy-driven torques enable back-and-forth motion within a light field. These findings advance understanding of non-equilibrium dynamics in confined active matter and have potential implications for designing light-driven micro-engines and controlled transport in complex fluids.

Abstract

We investigate experimentally the single-particle motion in water of silica colloidal beads half-coated with carbon under the action of a converging laser beam. The beads are self-propelled in this medium by means of self-thermophoresis resulting from local heating as a result of light absorption by their carbon cap. Within a certain laser power range, we find that these particles exhibit a quasi-two-dimensional active motion near a solid surface with stochastic rotational reversals when propelling themselves away from the region of maximum intensity, which leads to a stable trapping with oscillatory-like behavior inside the illuminated region. The orientation autocorrelation function of this type of confined active motion displays damped oscillations whose characteristic frequency increases with increasing propulsion speed, thus resulting in four regimes of translational motion depending on the observation time scale: thermal diffusion, ballistic motion, oscillatory behavior, and confinement. Our experimental findings are well described by a minimal phenomenological model that includes the nonlinear effect of a torque that reorients the particle toward the center of the optical confinement, which in combination with rotational diffusion gives rise to the observed orientational changes that allow their oscillatory trapping inside the light field. We also show that a similar active trapping mechanism emerges in the case of Janus colloidal rods, even though the periodicity is hindered by their three-dimensional rotation in the laser beam.

Figures (5)

• Figure 1: (a) Sketch of a spherical Janus bead (radius $a = 2.32$ µm) self-propelling in water with velocity v due to the action of a laser beam (wavelength $\lambda = 532$ nm), which propagates in the $+z$ direction and is tightly focused inside the sample cell at a distance of $h = 7$ µm from its lower inner surface, thus illuminating a circular area of radius $w \approx 10$ µm. The coordinate system $xyz$ used to characterize the particle motion is also outlined. (b) Snapshot of a silica bead half-coated with a 100 nm thick carbon layer (dark side). The blue spot represents the coordinates ${\bf{r}}= (x,y)$ of its center of mass, while the red arrow depicts the projection onto the $xy$ plane of its orientation $\mathbf{n}$, which defines the angle $\theta$. (c) Example of a trajectory segment of a Janus bead moving actively in the converging laser beam at $P = 0.5$ mW. The color map portrays the time elapsed over 2 minutes, $0 \le t \le 120$ s. The arrows indicate the instantaneous particle orientation every 0.5 seconds. (d) Dependence of the characteristic propulsion speed of the Janus particles on the applied laser power. The dashed line is a linear fit of the experimental data. (e) Translational diffusion coefficient of the Janus particles at short time scales as a function of the laser power. The horizontal dashed line represents the value estimated in the bulk by means of the Stokes-Einstein relation.
• Figure 2: [(a)-(c)] Stochastic time evolution over 300 s of the radial position of a Janus particle relative to the beam axis (a) and its orientation angle (b) under the action of the converging laser beam at $P = 0.5$ mW. An expanded view of both coordinates [$r(t)$ in green and $\theta(t)$ in red] during 30 s is shown in (c). [(d)-(f)] Stochastic motion over 300 s of the radial position of the same Janus particle (d) and its orientation angle (e) subject to the converging beam at $P = 1.0$ mW. An enlarged view of both coordinates [$r(t)$ in green and $\theta(t)$ in red] during 30 s is shown in (f). The shaded areas in (c) and (f) depict sudden orientational changes of $|\delta \theta| \approx \pi$ rad occurring when the particle reaches comparatively large radial distances. (g) Stochastic evolution during 150 s of the radial distance of an uncoated silica bead in the laser beam at $P=0.5$ mW. [(h)-(i)] Free motion over 150 s of the radial position of a Janus bead (h) and its orientation angle (i) at $P = 0$.
• Figure 3: (a) Experimental orientation autocorrelation function of the Janus beads defined by equation (\ref{['eq:aacf']}) for three representative propulsion speeds achieved in the converging laser beam: $v_0= 0.840$ µm s$^{-1}$ (dotted line), 1.238 µm s$^{-1}$ (dotted-dashed line), 2.334 µm s$^{-1}$ (solid line). The dashed line represents the exponentially decaying behavior with relaxation time $\tau_r \approx 70$ s of the OACF experimentally measured for a Janus bead in the absence of the laser beam ($v_0=0$). (b) Experimental mean-square displacement of the Janus particle position for the same propulsion speeds shown in figure \ref{['fig:3']}(a) represented with the same line style. Thin solid and dotted lines depict diffusive and ballistic behavior, respectively. (c) Dependence of the characteristic oscillation frequency of active Janus particles under the optical confinement on their propulsion speed determined from the corresponding experimental OACF curve ($\Box$). Each data point results from the analysis of a single trajectory of an independent particle moving at a fixed laser power. The solid line is a numerical curve calculated using the stochastic model described by equation (\ref{['eq:xy']}) and (\ref{['eq:theta']}). [(d)-(e)] Numerical results of the simulation of the stochastic dynamics described by the model of equation (\ref{['eq:xy']}) and (\ref{['eq:theta']}): (d) orientation autocorrelation function, and (e) positional mean-square displacement. The values of the propulsion speed used in the simulation are the same as those determined in the experiments represented in figures \ref{['fig:3']}(a) and \ref{['fig:3']}(b), and are also depicted with the same line style. (f) Colormap representing the magnitude $M$ of the torque (normalized by $\kappa d w$) relative to the direction $+z$ that is included in the rotational part of the model, see equation (\ref{['eq:theta']}), as a function of the radial position $r$ of the particle (normalized by $w$) and the angle $\varphi$ between the particle orientation, ${\bf{n}}$, and minus its position, $-{\bf{r}}$. The arrows indicate the sense of rotation of ${\bf{n}}$ in the $xy$ plane induced by the torque depending on $\varphi$: counterclockwise ($0\le \varphi < \pi$), and clockwise ($\pi < \varphi \le 2\pi$).
• Figure 4: (a) Picture of a colloidal rod of length $L = 4\,\mu$m and diameter $D = 2\,\mu$m half-coated with a $100$ nm tick carbon cap (dark side on the left). The coordinates $(x,y)$ and $\theta$ used to characterize its motion in the converging laser beam are also depicted. (b) Example of a trajectory segment of 3 minutes of a Janus rod self-propelling in the beam at $P = 1$ mW. The orientation of the projection of its main axis onto the $xy$ plane defined by the azimuthal angle $\theta$ is traced as bars every 0.5 s. The colormap depicts the time elapsed during the interval $0 \le t \le 180$ s. [(c)-(d)] Stochastic time evolution over $0\le t \le 1200$ s in the presence of the laser beam at $P = 1$ mW of: (c) the radial position $r(t)$ of the same Janus rod; (d) its azimuthal angle $\theta(t)$. (e) An expanded view of the coordinates $r(t)$ (green) and $\theta(r)$ (red) plotted in (c) and (d), respectively, over the last 240 s.
• Figure A1: (a) Experimental velocity autocorrelation function of a Janus bead recorded at $f_s = 100$ Hz freely moving at $P = 0$ ($\circ$) and in the converging laser beam at $P = 0.5$ mW ($\Box$). Inset: expanded view of the main figure that highlights the extrapolation $(v_0^{(est)})^2$ to $\tau = 0$ of the oscillatory part of the VACF to estimate $v_0$ and $D_T$. (b) Numerical velocity autocorrelation function obtained from the simulation of equations (\ref{['eq:xy']}) and (\ref{['eq:theta']}) for a particle freely diffusing with $v_0^{(sim)} = 0$ ($\circ$) and actively moving under confinement at $v_0^{(sim)} = 0.840\,\mu$m s$^{-1}$ ($\Box$). Inset: expanded view of the main figure that highlights the extrapolation $(v_0^{(est)})^2$ to $\tau = 0$ of the oscillatory part of the VACF. (c) Comparison between the values $v_0^{(est)}$ of the characteristic propulsion speed estimated through the VACF and those used in the simulations of the model, $v_0^{(sim)}$ (solid line). The identity function (dotted line) depicts perfect agreement.