Engineering living worms and active crystals with colloids propelled by attractive time-delayed feedback

Using computer simulations, we study the dynamics of colloidal particles with time-delayed feedback interactions. In particular, here, we consider "feedback-pullers," i.e., colloidal particles that are pulled away from their current position toward an attractive ring centered around their past position. For a single particle, small rings lead to reduced diffusive motion, while large rings render activity to the particle. For multiple particles, the particles not only feel their own attractive ring but are also attracted by the rings around all other particles. As expected, for ring sizes larger than the particle diameter, the feedback leads to crystallites whose lattice constant is set by the feedback ring radius. However, here, we demonstrate that for long delays (compared to the Brownian time), the colloidal particles start to oscillate around their lattice positions, with the crystallites ultimately collapsing to a close-packed lattice whose lattice constant corresponds to the particle diameter. This effect is caused by the time delay between the particle misplacement within the lattice and the corresponding change in the feedback force. Furthermore, we show that apart from the expected hexagonal crystallites, the time delay may result in the formation of uncommon new states in the case that the ring size is chosen slightly smaller than the particle diameter. Here, particles self-assemble into and move collaboratively as "living worms" or as active square-lattice crystallites.