Phototaxis of synthetic microswimmers in optical landscapes

Abstract

Many microorganisms, with phytoplankton and zooplankton as prominent examples, display phototactic behaviour, that is, the ability to perform directed motion within a light gradient. Here we experimentally demonstrate that sensing of light gradients can also be achieved in a system of synthetic photo-activated microparticles being exposed to an inhomogeneous laser field. We observe a strong orientational response of the particles because of diffusiophoretic torques, which in combination with an intensity-dependent particle motility eventually leads to phototaxis. Since the aligning torques saturate at high gradients, a strongly rectified particle motion is found even in periodic asymmetric intensity landscapes. Our results are in excellent agreement with numerical simulations of a minimal model and should similarly apply to other particle propulsion mechanisms. Because light fields can be easily adjusted in space and time, this also allows to extend our approach to dynamical environments.

Figure 1. Particle motion with mere position-dependent propulsion velocity.

(a) Time evolution of the probability distribution P(x) of self-propelled particles with variable self-propulsion velocity for x≥−a as obtained from Brownian dynamics simulations (Methods). Although P(x) becomes asymmetric at long times, no systematic drift of the particles to regions of higher motility is observed. (b) Time evolution of (N⁺−N⁻)/(N⁺+N⁻), where N⁺ and N⁻ are the numbers of particles at x>0 and x<0, respectively. Obviously, a mere position-dependent motility does not induce a macroscopic particle current.

Figure 2. Light-activated self-propulsion mechanism.

(a) Sketch of the sample cell with a capped particle suspended in a binary critical mixture of water–2,6-lutidine. Illumination with light leads to heating of the cap and thus to local demixing resulting in active motion. (b) Experimental set-up for creation of periodic illumination landscapes by a scanned line focus of a laser beam (Methods). (c) Propulsion velocity v_p versus illumination intensity I. Symbols (with error bars representing the s.d.) correspond to homogeneous illumination of the sample cell, while coloured lines were obtained in the presence of light profiles with different gradients (|∇I|=2 × 10⁻³ μW μm⁻³ (pink), 0.027 μW μm⁻³ (green), 0.037 μW μm⁻³ (red), 0.115 μW μm⁻³ (blue) and 0.156 μW μm⁻³ (grey)). The agreement between the data for different intensity gradients demonstrates that v_p is only determined by the local intensity incident on the particle.

Figure 3. Phototactic particle motion in constant light gradients.

(a) Trajectory of an active particle in a gradient |∇I|=0.042 μW μm⁻³. (b) Time evolution of the angle θ for |∇I|=0.02 μW μm⁻³ and I=0.55 μW μm⁻². The data are averaged over 10 runs. Upper inset: snapshot of a particle and its trajectory (solid curve) during reorientation. Lower inset: sketch of an active colloid in a non-uniform light field with gradient ∇I. The slip velocity (green arrows) becomes axially asymmetric, which results in an angular velocity ω (ref. 41). (c) Plot of the maximum angular velocity ω_max (left axis) and the corresponding reorientation time τ_ω (right axis, see Methods for details) as a function of the gradient |∇I| for different initial local intensities, that is, velocities (I=0.94 μW μm⁻² (v_p=12 μm s⁻¹), squares; I=0.69 μW μm⁻² (v_p=7 μm s⁻¹), circles; I=0.55 μW μm⁻² (v_p=5 μm s⁻¹), triangles; I=0.44 μW μm⁻² (v_p=3 μm s⁻¹), inverted triangles; I=0.35 μW μm⁻² (v_p=1.5 μm s⁻¹), diamonds). The error bars represent the s.d., and the solid curves show the theoretical fits (Methods).

Figure 4. Rectification mechanism.

(a) Measured intensity profile of a one-dimensional asymmetric light field with period length L=33.5 μm. (b) Comparison of experimental (solid line) and numerical (dashed line) intensity profiles. The latter is approximated by segments with constant positive (a) and negative (b) gradients. (c) Particle trajectories for a/b=0.22 and ΔI=1.0 μW μm⁻² (purple), 0.70 μW μm⁻² (cyan) and 0.45 μW μm⁻² (green). Inset: magnification of the framed region demonstrating back and forth motion of a particle. (d,e) Probability distribution functions (PDFs) of τ_r,i (i=a,b) for a/b=0.22 and (d) ΔI=1.0 μW μm⁻² and (e) ΔI=0.70 μW μm⁻², obtained from a segments (red) and b segments (blue), respectively. The vertical dashed lines indicate the corresponding mean values of τ_ω,i taken from Fig. 3c.

Figure 5. Particle current in periodic asymmetric light profiles.

(a) Average velocity versus intensity amplitude ΔI obtained from experiments (symbols) and numerical simulations (solid curves), for a/b=0.22 (squares), 0.28 (circles), 0.33 (triangles), 0.55 (diamonds), 0.75 (inverted triangles) and 1.0 (pentagons). The experimental data points were obtained by averaging the velocity over ∼200 periods each. Error bars correspond to 95% of the confidence interval. (b,c) PDFs of for a/b=0.22 and (b) ΔI=1.0 μW μm⁻² and (c) ΔI=0.22 μW μm⁻², obtained from experiments (symbols) and numerical simulations (filled areas). (d) Average velocity for different particle diameters σ, for a/b=0.22 and ΔI=0.92 μW μm⁻².

Figure 6. Saturation of the aligning torque.

(a) Schematic of an active Janus particle in a non-uniform light field with gradient ∇I. The inhomogeneous illumination leads to different temperatures T₂>T₁ at the two sides of the particle cap, which is approximated by the brighter region in the middle of the particle. The resulting asymmetric slip velocity profile (indicated by green arrows) induces an angular velocity ω. (b) Close-up view of the particle cap in the effectively one-dimensional model. As the local slip velocity v_s increases with higher illumination intensity, the advective coupling between the solvent and the heat flux j_Q through the particle surface A_⊥=dh leads to a higher flux at the left side of the particle (j_Q,2>j_Q,1). (c) Resulting temperature profile T(x) inside the particle cap and (d) corresponding temperature gradient between the two sides of the particle as a function of the intensity gradient according to equation (27). This reduced temperature difference leads to the observed saturation behaviour of the phoretic torque. The values of the various input parameters for the theory correspond to the experimental data or are obtained from literature, respectively (C_c=500 J kg⁻¹ K⁻¹, C_f=4,200 J kg⁻¹ K⁻¹, ρ_c=2 × 10³ kg m⁻³, ρ_f=0.99 × 10³ kg m⁻³, α=1.5 K m² J⁻¹, κ=0.4 W K⁻¹ m⁻¹, λ_s=180 nm, h=20 nm, γ=1.1 × 10⁵ s⁻¹, b=10⁻⁹ m³ W⁻¹ s⁻¹, I₀=1 μW μm⁻², v₀=12 μm s⁻¹).

Figure 7. Particle dynamics without saturation of the aligning torque.

(a) Calculated trajectories of active Janus particles in a periodic asymmetric velocity profile v_p(x) for different initial angles ϕ₀. Different from the experimental situation, here a linear relation between the angular velocity and the gradient of the translational velocity is considered. Depending on the initial orientation, particles either perform an oscillating motion in a valley of the velocity profile (green and magenta trajectories) or they periodically move in positive or negative x direction (orange and cyan trajectories). (Note that the green trajectory is scaled by a factor of 2/5 in the y direction for reasons of presentation.) In the absence of saturation, the probabilities and the mean velocities of particles moving to the left and to the right are the same. Thus, no net particle current occurs. (b) Particle orientation ϕ versus position x for the trajectories shown in a. Black crosses represent the initial positions, and the arrows indicate the evolution with time.