Hydrodynamic Interactions in Colloidal and Biological Systems

Abstract

Colloids are widely considered as model systems to elucidate fundamental processes in atomic systems. However, there is one feature truly specific to colloidal suspensions that distinguishes them fundamentally from atomic systems: hydrodynamic interactions, which can lead to fascinating collective behavior.In this thesis, we present analytical work and simulation results for several micron-scale colloidal and biological systems where the dynamics is predominantly governed by hydrodynamic interactions.The first part deals with hydrodynamic interactions in two-point microrheology, a method to explore the viscoelastic behavior of soft materials (such as biological tissue) on the micron scale. We consider the overdamped motion of two birefringent colloidal beads immersed in a Newtonian fluid. The particles are assumed to be trapped by optical tweezers with respect to both their position and orientation. On the basis of a Langevin description of this system, we analyze the thermal fluctuations and obtain a rich spectrum of correlation functions. In particular, we focus on the rotational degrees of freedom and how they couple to translation, thus extending recent investigations restricted to translational correlations. An important feature of our system is the self-coupling of translation and rotation of one particle mediated by the neighboring particle. It thus shows a characteristic time delay that is clearly visible in the appropriate self-correlation function. Finally, we compare our analytical results with correlation functions determined both from Brownian-dynamics simulations that we performed and from available experimental data.In the second part, we study the dynamics of spherical particles circling in a ring-shaped harmonic trap. Hydrodynamic interactions completely determine their characteristic collective behavior. At first, the particles are driven by constant forces. A linear stability analysis for regular clusters of circling particles is performed, and we illustrate the periodic limit cycle to which the system converges. We clarify that drafting of particle doublets is essential to interpret this limit cycle. When we apply a spatially periodic sawtooth potential along the circular trap, in addition to the constant force, we find a novel caterpillar-like motional sequence that is dominated by the long-range hydrodynamic interactions and that promotes the surmounting of potential barriers by the particles. Our numerical findings are in good agreement with experiments. Such collective effects in sawtooth potentials may also be relevant in thermal ratchets that are commonly used to describe, e.g., biological motors.The issue of the third part is locomotion of microorganisms. Many types of bacteria use several rotating helical flagella to swim. Typically, the flagellar filaments form bundles, which means that their rotations must be synchronized. The central question of our study is whether hydrodynamic interactions are capable of such a synchronization. In a first approach, we consider two stiff helices that are modeled by rigidly connected beads, neglecting any elastic deformations. They are driven by constant and equal torques, and they are fixed in space by anchoring their ends in harmonic traps. For finite anchoring strength, we do indeed observe a synchronization of the helix rotations. However, the speed of phase synchronization decreases with increasing trap stiffness, and in the limit of infinite trap stiffness, the helices do not synchronize. This leads to the conclusion that some kind of flexibility is essential. Thus, as a second step, we refine our model and consider elastic deformations of the helices within the nontrivial theory of helical elastic rods. Again, we observe that the rotations of the two helices are synchronized. In particular, the additional flexibility of the helices further increases the synchronization speed. Besides the phase locking of the helices, we furthermore observe the "onset" of flagellar bundling.