Hydrodynamic Interactions in Colloidal and Biological Systems Dissertation zur Erlangung des akademischen Grades Doktor der Naturwissenschaften (Dr. rer. nat.) an der Universit¨at Konstanz, Mathematisch-Naturwissenschaftliche Sektion, Fachbereich Physik, vorgelegt von Michael Reichert Referenten: PD Dr. Holger Stark Prof. em. Dr. Rudolf Klein Tag der mundlichen¨ Prufung:¨ 14. Juli 2006 Quand tu veux construire un bateau, ne commence pas par rassembler du bois, couper des planches et distribuer du travail, mais r´eveille au sein des hommes le d´esir de la mer grande et large. Antoine de Saint-Exup´ery Contents 1 General introduction 1 1.1 Colloidsasmodelsystems . 1 1.2 Recent studies on hydrodynamic interactions . 2 1.3 Outlineofthisthesis. 4 A Hydrodynamic interactions Theoretical concepts and simulation methods 2 Stokesian dynamics 9 2.1 Hydrodynamicequations . 10 2.1.1 Continuity equation and incompressibility . 10 2.1.2 The Navier-Stokes equation . 10 2.1.3 The Reynolds number . 12 2.1.4 TheStokesequations . 12 2.2 Translation and rotation of a single sphere . 13 2.2.1 Translation .......................... 14 2.2.2 Rotation ........................... 14 2.3 Hydrodynamicinteractions . 15 2.3.1 Theoretical description and definitions . 15 2.3.2 On the propagation of hydrodynamic interactions . 17 2.4 Mobility matrix for configurations of beads . 18 2.4.1 TheOseentensor....................... 18 2.4.2 Method of reflections: the Rotne-Prager approximation . 19 2.4.3 Methodofinducedforces . 22 2.4.4 Lubricationeffects . 24 2.4.5 Mobilities of a two-sphere system . 26 2.4.6 Mobility matrix for rigid clusters . 29 2.5 Stokesian-dynamics simulations . 30 2.5.1 Equationofmotion . 30 2.5.2 Numerical integration schemes . 31 3 Brownian motion 33 3.1 TheLangevinequation . 34 3.1.1 From Newton’s second law to the Langevin equation . 34 3.1.2 Fluctuation-dissipation theorem . 35 3.1.3 Brownian displacements and Wiener processes . 36 ii Contents 3.1.4 Evolution equation of Langevin dynamics . 39 3.1.5 Deterministic vs. diffusive motion . 40 3.2 TheSmoluchowskiequation . 41 3.3 Brownian-dynamics simulations . 43 3.3.1 Numerical integration schemes . 43 3.3.2 The Cholesky decomposition . 45 3.3.3 Generation of Gaussian random numbers . 46 B Two-point microrheology Hydrodynamic coupling of rotating beads in optical traps 4 Optical tweezers and microrheology 51 4.1 Opticaltweezers ........................... 51 4.1.1 Trapping of dielectric particles in a laser spot . 51 4.1.2 Rotation of birefringent particles in polarized traps.... 53 4.2 Concepts of microrheology . 54 4.2.1 One-point microrheology . 54 4.2.2 Two-point microrheology . 54 4.3 Experiments measuring hydrodynamic interactions . 55 5 Theory of coupled motions of two trapped beads 57 5.1 Tweezer setup and dynamics of trapped particles . 57 5.1.1 Forces and translational motion . 57 5.1.2 Torques and rotational motion . 58 5.2 Relaxational eigenmodes of two trapped spheres . 60 5.2.1 Longitudinalmodes . 61 5.2.2 Transversalmodes . 62 5.3 Correlated Brownian motion of two trapped spheres . 65 5.3.1 Longitudinalfluctuations . 67 5.3.2 Transversalfluctuations . 69 6 Brownian-dynamics simulations and experiments 75 6.1 Brownian-dynamics simulations . 75 6.1.1 Trap forces and torques . 75 6.1.2 Correlation functions . 76 6.2 Experiments ............................. 77 7 Conclusions I 79 Contents iii C Drafting of colloids The benefits of hydrodynamic interactions 8 Colloidal particles orbiting in circular optical traps 83 8.1 Opticalvortices............................ 83 8.2 Circulating optical tweezer . 85 9 Collective drafting dynamics of circling particles 87 9.1 Modelsystem............................. 87 9.2 Stabilityofregularclusters . 88 9.2.1 Dynamics of regular clusters . 89 9.2.2 Eigenmodesofperturbedclusters . 90 9.3 Nonlinear dynamics and periodic limit cycle . 93 9.3.1 Dynamic transition to periodic limit cycle . 93 9.3.2 Enhanced particle motion due to drafting effects . 94 9.3.3 Harmonic analysis of the limit cycle . 95 9.3.4 Clustering in weak radial traps . 97 10 Dynamics in a sawtooth potential supported by drafting 99 10.1Modelsystem............................. 99 10.1.1 Sawtooth-modulated circular trap . 99 10.1.2 Electrostatic interaction . 102 10.1.3 Brownian-dynamics simulations . 102 10.2 Experimental realization and validation of the model . ...... 104 10.3 Single-particledynamics . 105 10.4 Cooperative motion of drafting and thermal jumps . 107 10.4.1 The “caterpillar” mode . 107 10.4.2 Enhanced cluster dynamics . 109 10.4.3 On the role of hydrodynamic interactions . 111 10.4.4 Influence of sawtooth asymmetry and temperature . 115 11 Conclusions II 117 D Swimming of microorganisms Synchronization of rotating helical flagella 12 Locomotion of microorganisms 121 12.1 Principles of swimming at low Reynolds number . 122 12.2 Flagellarpropulsion . 123 iv Contents 12.2.1 Ciliarybeating . 124 12.2.2 Phenomenology of propulsion by rotating helical flagella . 125 12.3 Hydrodynamic interactions in flagellar systems . 126 13 Synchronized rotation of rigid helical filaments 129 13.1Modelsystem............................. 129 13.2 Symmetryconsiderations . 133 13.3 Synchronizationdynamics . 134 13.3.1 Phase synchronization . 134 13.3.2 On the role of “flexibility” . 137 13.3.3 Robustness against torque differences . 139 13.3.4 “Extrapolation” to real flagella . 140 14 Synchronization of flexible flagella 143 14.1Modelsystem............................. 143 14.1.1 The Frenet-Serret equations . 143 14.1.2 Helical worm-like chain model . 144 14.1.3 Discrete bead-spring model of a flexible helix . 146 14.1.4 Equilibrium configuration . 149 14.1.5 Elastic forces and torques . 150 14.1.6 Motorforces ......................... 152 14.1.7 Equationsofmotion . 153 14.2 Synchronizationdynamics . 156 14.2.1 Dynamics and deformations of rotating helices . 156 14.2.2 Phase synchronization . 157 14.2.3 Dependence on flexibility . 159 14.2.4 Flexible vs. rigid helices . 159 15 Conclusions III 163 Bibliography 165 List of publications 179 Zusammenfassung 181 Danksagung 185 1 General introduction Colloids are widely considered as model systems to elucidate fundamental pro- cesses in atomic systems. However, there is one feature truly specific to colloidal suspensions which distinguishes them fundamentally from atomic systems: hydro- dynamic interactions. In this introductory chapter, we first give a rough overview of the physics of colloids and their applications as model systems. As a motivation of this work, we then discuss representative examples of colloidal and biological systems where hydrodynamic interactions lead to fascinating collective behavior. Finally, an outline of this thesis is given in Sect. 1.3. 1.1 Colloids as model systems Colloids are mesoscopic particles that are suspended in a continuous medium and whose size ranges from 10 nm to 1 µm, such as fog (water droplets in air) or milk (fat globules in water). In this regime, the suspended particles are much larger than the solvent molecules, but small enough to exhibit vivid thermal (Brownian) motion, so they rapidly sample the accessible configuration space [60, 80]. Suspensions of more or less compact particles in liquids form the basis of a wide variety of systems of scientific and technological importance, including inks, paints, detergents, and biological cells. Since the characteristic length scales are close to (or larger than) the wave- length of visible light, many optical techniques such as confocal microscopy [87, 242], total-internal-reflection microscopy [100, 210], and light scattering [56, 139] can be employed to study colloidal suspensions. Real-space trajectories of the suspended particles can be monitored by means of video microscopy [48, 105], which allows for direct comparison with analytical and numerical results and leads to a close and fruitful interplay between theory, computer simulation, and experiment. Colloidal suspensions have many features in common with atomic systems; in particular, with increasing volume fraction of colloidal particles, fluid, crystalline, and glassy phases are observed [197, 200]. In some sense, colloidal suspensions can be regarded as “analog computers” which allow real-time “calculations” in complex situations such as crystal nucleation and growth [87], particle dynamics near the glass transition [242], interactions of dislocations [69], and many more. The striking advantage of using colloids as model systems is the fact that their interactions can be tailored as desired [60, 80]. For instance, the screened 2 1 General introduction electrostatic repulsion of like-charged particles in suspension is an adjustable short-range interaction, while the dipole-dipole interaction of superparamagnetic particles yields long-range forces. Moreover, the magnetic dipole interactions can be tuned from repulsive to attractive (depending on the orientation of the magnetic field that induces the magnetic dipole moments) [82]. Colloidal par- ticles can be controlled and manipulated rather easily by external fields [156], such as electric and magnetic fields or optical traps, so-called optical “tweez- ers”. Therefore, colloids are often referred to as versatile model systems to study fundamental processes in atomic systems and to address novel concepts in the context of statistical physics, such as entropic forces [210], light-induced phase transitions [243], or two-dimensional melting scenarios [68]. However, besides the potential forces described above, which are sort of analo- gous to the interactions