Borge ten Hagen

Circular motion of asymmetric self-propelling particles.

Micron-sized self-propelled (active) particles can be considered as model systems for characterizing more complex biological organisms like swimming bacteria or motile cells. We produce asymmetric microswimmers by soft lithography and study their circular motion on a substrate and near channel boundaries. Our experimental observations are in full agreement with a theory of Brownian dynamics for asymmetric self-propelled particles, which couples their translational and orientational motion.

Gravitaxis of asymmetric self-propelled colloidal particles

Many motile microorganisms adjust their swimming motion relative to the gravitational field and thus counteract sedimentation to the ground. This gravitactic behaviour is often the result of an inhomogeneous mass distribution, which aligns the microorganism similar to a buoy. However, it has been suggested that gravitaxis can also result from a geometric fore-rear asymmetry, typical for many self-propelling organisms. Despite several attempts, no conclusive evidence for such an asymmetry-induced gravitactic motion exists. Here, we study the motion of asymmetric self-propelled colloidal particles which have a homogeneous mass density and a well-defined shape. In experiments and by theoretical modelling, we demonstrate that a shape anisotropy alone is sufficient to induce gravitactic motion with either preferential upward or downward swimming. In addition, also trochoid-like trajectories transversal to the direction of gravity are observed.

Phototaxis of synthetic microswimmers in optical landscapes

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.

Brownian motion and the hydrodynamic friction tensor for colloidal particles of complex shape

We synthesize colloidal particles with various anisotropic shapes and track their orientationally resolved Brownian trajectories using confocal microscopy. An analysis of appropriate short-time correlation functions provides direct access to the hydrodynamic friction tensor of the particles revealing nontrivial couplings between the translational and rotational degrees of freedom. The results are consistent with calculations of the hydrodynamic friction tensor in the low-Reynolds-number regime for the experimentally determined particle shapes.

How does a flexible chain of active particles swell

We study the swelling of a flexible linear chain composed of active particles by analytical theory and computer simulation. Three different situations are considered: a free chain, a chain confined to an external harmonic trap, and a chain dragged at one end. First, we consider an ideal chain with harmonic springs and no excluded volume between the monomers. The Rouse model of polymers is generalized to the case of self-propelled monomers and solved analytically. The swelling, as characterized by the spatial extension of the chain, scales with the monomer number defining a Flory exponent ν which is ν = 1/2, 0, 1 in the three different situations. As a result, we find that activity does not change the Flory exponent but affects the prefactor of the scaling law. This can be quantitatively understood by mapping the system onto an equilibrium chain with a higher effective temperature such that the chain swells under an increase of the self-propulsion strength. We then use computer simulations to study the effect of self-avoidance on active polymer swelling. In the three different situations, the Flory exponent is now ν = 3/4, 1/4, 1 and again unchanged under self-propulsion. However, the chain extension behaves non-monotonic in the self-propulsion strength.

Can the self-propulsion of anisotropic microswimmers be described by using forces and torques?

The self-propulsion of artificial and biological microswimmers (or active colloidal particles) has often been modelled by using a force and a torque entering into the overdamped equations for the Brownian motion of passive particles. This seemingly contradicts the fact that a swimmer is force-free and torque-free, i.e. that the net force and torque on the particle vanish. Using different models for mechanical and diffusiophoretic self-propulsion, we demonstrate here that the equations of motion of microswimmers can be mapped onto those of passive particles with the shape-dependent grand resistance matrix and formally external effective forces and torques. This is consistent with experimental findings on the circular motion of artificial asymmetric microswimmers driven by self-diffusiophoresis. The concept of effective self-propulsion forces and torques significantly facilitates the understanding of the swimming paths, e.g. for a microswimmer under gravity. However, this concept has its limitations when the self-propulsion mechanism of a swimmer is disturbed either by another particle in its close vicinity or by interactions with obstacles, such as a wall.

Depolarized light scattering from prolate anisotropic particles: The influence of the particle shape on the field autocorrelation function

We provide a theoretical analysis for the intermediate scattering function typically measured in depolarized dynamic light scattering experiments. We calculate the field autocorrelation function g1(VH)(Q,t) in dependence on the wave vector Q and the time t explicitly in a vertical-horizontal scattering geometry for differently shaped solids of revolution. The shape of prolate cylinders, spherocylinders, spindles, and double cones with variable aspect ratio is expanded in rotational invariants flm(r). By Fourier transform of these expansion coefficients, a formal multipole expansion of the scattering function is obtained, which is used to calculate the weighting coefficients appearing in the depolarized scattering function. In addition to translational and rotational diffusion, especially the translational-rotational coupling of shape-anisotropic objects is considered. From the short-time behavior of the intermediate scattering function, the first cumulants Γ(Q) are calculated. In a depolarized scattering experiment, they deviate from the simple proportionality to Q(2). The coefficients flm(Q) strongly depend on the geometry and aspect ratio of the particles. The time dependence, in addition, is governed by the translational and rotational diffusion tensors, which are calculated by means of bead models for differently shaped particles in dependence on their aspect ratio. Therefore, our analysis shows how details of the particle shape--beyond their aspect ratio--can be determined by a precise scattering experiment. This is of high relevance in understanding smart materials which involve suspensions of anisotropic colloidal particles.

Dynamics of a deformable active particle under shear flow

The motion of a deformable active particle in linear shear flow is explored theoretically. Based on symmetry considerations, we propose coupled nonlinear dynamical equations for the particle position, velocity, deformation, and rotation. In our model, both, passive rotations induced by the shear flow as well as active spinning motions, are taken into account. Our equations reduce to known models in the two limits of vanishing shear flow and vanishing particle deformability. For varied shear rate and particle propulsion speed, we solve the equations numerically in two spatial dimensions and obtain a manifold of different dynamical modes including active straight motion, periodic motions, motions on undulated cycloids, winding motions, as well as quasi-periodic and chaotic motions induced at high shear rates. The types of motion are distinguished by different characteristics in the real-space trajectories and in the dynamical behavior of the particle orientation and its deformation. Our predictions can be verified in experiments on self-propelled droplets exposed to a linear shear flow.

Helical paths, gravitaxis, and separation phenomena for mass-anisotropic self-propelling colloids : Experiment versus theory

The self-propulsion mechanism of active colloidal particles often generates not only translational but also rotational motion. For particles with an anisotropic mass density under gravity, the motion is usually influenced by a downwards oriented force and an aligning torque. Here we study the trajectories of self-propelled bottom-heavy Janus particles in three spatial dimensions both in experiments and by theory. For a sufficiently large mass anisotropy, the particles typically move along helical trajectories whose axis is oriented either parallel or antiparallel to the direction of gravity (i.e., they show gravitaxis). In contrast, if the mass anisotropy is small and rotational diffusion is dominant, gravitational alignment of the trajectories is not possible. Furthermore, the trajectories depend on the angular self-propulsion velocity of the particles. If this component of the active motion is strong and rotates the direction of translational self-propulsion of the particles, their trajectories have many loops, whereas elongated swimming paths occur if the angular self-propulsion is weak. We show that the observed gravitational alignment mechanism and the dependence of the trajectory shape on the angular self-propulsion can be used to separate active colloidal particles with respect to their mass anisotropy and angular self-propulsion, respectively.

Brownian motion of a circle swimmer in a harmonic trap

We study the dynamics of a Brownian circle swimmer with a time-dependent self-propulsion velocity in an external temporally varying harmonic potential. For several situations, the noise-free swimming paths, the noise-averaged mean trajectories, and the mean-square displacements are calculated analytically or by computer simulation. Based on our results, we discuss optimal swimming strategies in order to explore a maximum spatial range around the trap center. In particular, we find a resonance situation for the maximum escape distance as a function of the various frequencies in the system. Moreover, the influence of the Brownian noise is analyzed by comparing noise-free trajectories at zero temperature with the corresponding noise-averaged trajectories at finite temperature. The latter reveal various complex self-similar spiral or rosette-like patterns. Our predictions can be tested in experiments on artificial and biological microswimmers under dynamical external confinement.