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Topological growth control allows to produce a narrow distribution of outgrown colloidal rods with defined and adjustable length. We use an external magnetic field to assemble paramagnetic colloidal spheres into colloidal rods of a chosen length. The rods reside above ametamorphic hexagonalmagnetic pattern. The periodic repetition of specific loops of the orientation of an applied external field renders paramagnetic colloidal particles and their assemblies into active bipeds that walk on the pattern. The metamorphic patterns allow the robust and controlled polymerization of single colloids to bipeds of a desired length. The colloids are exposed to this fixed external control loop that causes multiple simultaneous responses: Small bipeds and single colloidal particles interpret the external magnetic loop as an order to walk toward the active zone, where they assemble and polymerize. Outgrown bipeds interpret the same loop as an order to walk away from the active zone. The topological transition occurs solely for the growing biped and nothing is changed in the environment nor in the magnetic control loop. As in many biological systems the decision of a biped that reached its outgrown length to walk away from the reaction site is made internally, not externally.

We measure the trajectories of macroscopic magnetic particles pulled against gravity between twisted alternating magnetic square patterns in a superposed homogeneous magnetic field normal to both patterns. The two patterns are built from a set of magentic cubes having a distribution of magnetization. The magnetic potential between the patterns is a sum of three contributions: two being periodic on two lattices with different magnitude and orientation, and the third random contribution arising from the distribution of magnetization of the cubes. As one varies the twist angle between the two patterns each time the twist angle coincides with a magic twist angle one of the two periodic lattices becomes a sublattice of the other lattice. Simulations of particles moving through patterns with a precise cube magnetization produce pronounced mobility peaks near magic twist angles that are associated with flat channels. Weak random fluctuations of the cube magnetization in the experiment and the simulations cause enhanced random disorder of the potential and reduce the mobility by scattering particles into the interior of the twisted Wigner Seitz cells. The mobility undergoes an Anderson transition from magic to generic behavior as the magnetization disorder increases beyond half of a percent of the cube magnetization.

We simulate the transport of colloidal particles driven by a static and homogeneous drift force, and subject to the optical potential created by two lattices of optical tweezers. The lattices of optical tweezers are parallel to each other, shifted, and rotated by a twist angle. Due to a negative interference between the potential of the two lattices, flat channels appear in the total optical potential. At specific twist angles, known as magic-angles, the flat channels percolate the entire system and the colloidal particles can then be transported using a weak external drift force. We characterize the transport in both square and hexagonal lattices of twisted optical tweezers

We simulate the dynamics of paramagnetic colloidal particles that are placed above a magnetic hexagonal pattern and exposed to an external field periodically changing its direction along a control loop. The conformation of three colloidal particles above one unit cell adiabatically responds with half the frequency of the external field creating a time crystal at arbitrary low frequency. The adiabatic time crystal occurs because of the non-trivial topology of the stationary manifold. When coupling colloidal particles in different unit cells, many body effects cause the formation of topologically isolated time crystals and dynamical phase transitions between different adiabatic reversible and non-adiabatic irreversible space-time-crystallographic arrangements., Comment: submission to SciPostPhys. 5 videos in the ancillary directory

We report on the motion of a spinning sleeping top on an inclined plane. Below a critical inclination angle the sleeping tops are force free. The trajectory of a sleeping top on weakly inclined planes in the adiabatic limit is invariant of the angular frequency of the top and thus invariant under a rescaling of the time, however not invariant under time reversal. The stationary trajectory of the sleeping top is characterized by its angle to the in plane horizontal direction. At larger inclinations of the plane the stationary motion of the top becomes unstable and the top accelerates downhill. The behavior points towards a complex law of dry friction of the contact point between the top tip and the material of the inclined plane that depends on a slip parameter. We propose a phenomenological law of dry friction that can explain the relaxation of the top into the sleeping position, the geometric behavior of the top trajectories, and the instability of the stationary motion at larger inclination angles.

Detailed control over the motion of colloidal particles is relevant in many applications in colloidal science such as lab-on-a-chip devices. Here, we use an external magnetic field to assemble paramagnetic colloidal spheres into colloidal rods of several lengths. The rods reside above a square magnetic pattern and are transported via modulation of the direction of the external magnetic field. The rods behave like bipeds walking above the pattern. Depending on their length, the bipeds perform topologically distinct classes of protected walks above the pattern. We demonstrate that it is possible to design parallel polydirectional modulation loops of the external field that command up to six classes of bipeds to walk on distinct predesigned paths. We use such parallel polydirectional loops to induce the collision of reactant bipeds, their polymerization addition reaction to larger bipeds, the separation of product bipeds from the educts, the sorting of different product bipeds, and also the parallel writing of a word consisting of several different letters.

Single and double paramagnetic colloidal particles are placed above a magnetic square pattern and are driven with an external magnetic field precessing around a high symmetry direction of the pattern. The external magnetic field and that of the pattern confine the colloids into lanes parallel to a lattice vector of the pattern. The precession of the external field causes traveling minima of the magnetic potential along the direction of the lanes. At sufficiently high frequencies of modulation only the doublets respond to the external field and move in direction of the traveling minima along the lanes, while the single colloids cannot follow and remain static. We show how the doublets can induce a coordinated motion of the single colloids building colloidal trains made of a chain of several single colloids transported by doublets.

The transport on top of a periodic two-dimensional hexagonal magnetic pattern of (i) a single macroscopic steel sphere, (ii) a doublet of wax/magnetite composite spheres, and (iii) an immiscible mixture of ferrofluid droplets with a perfluorinated liquid is studied experimentally and analyzed theoretically. The transport of all these magnetic objects is achieved by moving an external permanent magnet on a closed modulation loop around the two-dimensional magnetic pattern. The transport of one and also that of two objects per unit cell is topologically protected and characterized by discrete displacements of the particles as we continuously scan through a family of modulation loops. The direction and the type of transport is characterized by the winding numbers of the modulation loops around special objects in control space, which is the space of possible directions of the external magnetic field. The number of winding numbers necessary for characterizing the topological transport increases with the number of particles per unit cell. The topological character of the transport is destroyed when transporting a large collection of particles per unit cell like it is the case for a macroscopic assembly of magnetic nano particles in a ferrofluid droplet for which the transport is geometrical and no longer topological. To characterize the change in the transport from topological to geometrical, we perform computer simulations of the transport of an increasing number of particles per unit cell., Comment: 9 page, 5 figures

Edge currents of paramagnetic colloidal particles propagate at the edge between two topologically equivalent magnetic lattices of different lattice constant when the system is driven with periodic modulation loops of an external magnetic field. The number of topologically protected particle edge transport modes is not determined by a bulk-boundary correspondence. Instead, we find a rich variety of edge transport modes that depend on the symmetry of both the edge and the modulation loop. The edge transport can be ratchet-like or adiabatic, and time or non-time reversal symmetric., Comment: 16 pages, 7 figures

Driven or active suspensions can display fascinating collective behavior, where coherent motions or structures arise on a scale much larger than that of the constituent particles. Here, we report experiments and numerical simulations revealing that red blood cells (RBCs) assemble into regular patterns in a confined shear flow. The order is of pure hydrodynamic and inertialess origin, and emerges from a subtle interplay between (i) hydrodynamic repulsion by the bounding walls which drives deformable cells towards the channel mid-plane and (ii) intercellular hydrodynamic interactions which can be attractive or repulsive depending on cell-cell separation. Various crystal-like structures arise depending on RBC concentration and confinement. Hardened RBCs in experiments and rigid particles in simulations remain disordered under the same conditions where deformable RBCs form regular patterns, highlighting the intimate link between particle deformability and the emergence of order. The difference in structuring ability of healthy (deformable) and diseased (stiff) RBCs creates a flow signature potentially exploitable for diagnosis of blood pathologies.

Topological insulators insulate in the bulk but exhibit robust conducting edge states protected by the topology of the bulk material. Here, we design a colloidal topological insulator and demonstrate experimentally the occurrence of edge states in a classical particle system. Magnetic colloidal particles travel along the edge of two distinct magnetic lattices. We drive the colloids with a uniform external magnetic field that performs a topologically non-trivial modulation loop. The loop induces closed orbits in the bulk of the magnetic lattices. At the edge, where both lattices merge, the colloids perform skipping orbits trajectories and hence edge-transport. We also observe paramagnetic and diamagnetic colloids moving in opposite directions along the edge between two inverted patterns; the analogue of a quantum spin Hall effect in topological insulators. We present a new, robust, and versatile way of transporting colloidal particles, enabling new pathways towards lab on a chip applications.

The transport of a steel sphere on top of two dimensional periodic magnetic patterns is studied experimentally. Transport of the sphere is achieved by moving an external permanent magnet on a closed loop around the two dimensional crystal. The transport is topological i.e. the steel sphere is transported by a primitive unit vector of the lattice when the external magnet loop winds around specific directions. We experimentally determine the set of directions the loops must enclose for nontrivial transport of the steel sphere into various directions.

The topologically protected transport of colloidal particles on top of magnetic patterns of all possible single lattice constant two dimensional magnetic point group symmetries is studied experimentally, theoretically, and with numerical simulations. We examine the transport of colloidal particles in response to modulation loops of the external field. We classify the modulation loops into topologically distinct classes causing different transport. We show that the lattice symmetry has a profound influence on the transport modes, the accessibility of transport networks, and the individual addressability of paramagnetic versus diamagnetic colloidal particles. We show how the transport of colloidal particles above a two fold symmetric stripe pattern changes from universal adiabatic transport at large elevations via a topologically protected ratchet motion at intermediate elevations toward a non-transport regime at low elevations. Transport above four fold symmetric patterns is closely related to the transport above two fold symmetric patterns. There exists a family of three fold symmetric patterns that vary as a function of the phase of the pattern. We show how this family can be divided into two topologically distinct classes supporting different transport modes and being protected by proper and improper six fold symmetries. Both classes support individual control over the transport of paramagnetic and diamagnetic particles. We discuss the topological transition when moving the phase from one class of pattern to the other class. The similarities and the differences in the lattice symmetry protected transport of classical over-damped colloidal particles versus the topologically protected transport in quantum mechanical systems are emphasized

We theoretically study the motion of magnetic colloidal particles above a magnetic pattern and compare the predictions with Brownian dynamics simulations. The pattern consists of alternating square domains of positive and negative magnetization. The colloidal motion is driven by periodic modulation loops of an external magnetic field. There exist loops that induce topologically protected colloidal transport between two different unit cells of the pattern. The transport is very robust against internal and external perturbations. Theory and simulations are in perfect agreement.

Topological protection allows robust transport of localized phenomena such as quantum information, solitons, and dislocations. The transport can be either dissipative or non-dissipative. Here, we experimentally demonstrate and theoretically explain the topologically protected dissipative motion of colloidal particles above a periodic hexagonal magnetic pattern. By driving the system with periodic modulation loops of an external and spatially homogeneous magnetic field, we achieve total control over the motion of diamagnetic and paramagnetic colloids. We can transport simultaneously and independently each type of colloid along any of the six crystallographic directions of the pattern via adiabatic or deterministic ratchet motion. Both types of motion are topologically protected. As an application, we implement an automatic topologically protected quality control of a chemical reaction between functionalized colloids. Our results are relevant to other systems with the same symmetry.

We report a technique to realize reconfigurable colloidal crystals by using the controlled motion of particle defects above an externally modulated magnetic substrate. The transport of particles is induced by applying a uniform rotating magnetic field to a ferrite garnet film characterized by a periodic lattice of magnetic bubbles. For filling factor larger than one colloid per bubble domain, the particle current arises from propagating defects where particles synchronously exchange their position when passing from one occupied domain to the next. The amplitude of an applied alternating magnetic field can be used to displace the excess particles via a swapping mechanism, or to mobilize the entire colloidal system at a predefined speed.

We study the collective transport of paramagnetic colloids driven above a magnetic bubble lattice by an external rotating magnetic field. We measure a direct ratchet current which rises in integer and fractional steps with the field amplitude. The stepwise increase is caused by excluded volume interactions between the particles, which form composite clusters above the bubbles with mobile and immobile occupation sites. Transient energy minima located at the interstitials between the bubbles cause the colloids to hop from one composite cluster to the next with synchronous and period doubled modes of transport. The colloidal current may be polarized to make selective use of type up or type down interstitials., Comment: 4 pages, 4 figures