Mean Square Angular Displacement Research Articles

Single-file dynamics of colloids in circular channels: Time scales, scaling laws and their universality

In colloidal systems, Brownian motion emerges from the massive separation of time and length scales associated to characteristic dynamics of the solute and solvent constituents. This separation of scales produces several temporal regimes in the colloidal dynamics when combined with the effects of the interaction between the particles, confinement conditions, and state variables, such as density and temperature. Some examples are the short- and long-time regimes in two- and three-dimensional open systems and the diffusive and sub-diffusive regimes observed in the single-file dynamics along a straight line. This work studies the way in which a confining geometry induces new time scales. We report on the dynamics of interacting colloidal particles moving along a circle by combining a heuristic theoretical analysis of the involved scales, Brownian Dynamics computer simulations, and video-microscopy experiments with paramagnetic colloids confined to lithographic circular channels subjected to an external magnetic field. The systems display four temporal regimes in this order: one-dimensional free diffusion, single-file sub-diffusion, free-cluster rotational diffusion, and the expected saturation due to the confinement. We also report analytical expressions for the mean-square angular displacement and crossover times obtained from scaling arguments, which accurately reproduce both experiments and simulations. Our generic approach can be used to predict the long-time dynamics of many other confined physical systems.

Trapped active toy robots: theory and experiment

We characterise the diffusion, mean-square speed, mean-square angular displacement, radial and speed probability distributions of toy robots called ‘hexbugs-nano’ that move on a dish antenna (with added surface roughness) simulating a harmonic well. It is observed that a model considering the system’s translational inertia but neglecting its moment of inertia, together with the inclusion of a constant external torque in the orientational motion, suffices to describe the robots’ dynamics. Langevin dynamics simulation are also performed and a good agreement between theory, simulations and experiments is observed.

Coupling and decoupling between translational and rotational dynamics in supercooled monodisperse soft Janus particles.

We perform dynamics simulations to investigate the translational and rotational glassy dynamics in a glass-forming liquid of monodisperse soft Janus particles. We find that, with decreasing temperature, the mean-square angular displacement shows no clear plateau in the caging region, in contrast with the apparent caging behavior of translational motion. By defining a reorientational mean-square angular displacement, the caging behavior of rotational motion can be recognized. On approaching the glass transition (decreasing temperature), the coupling between translational and rotational relaxation increases, while the coupling between translational and rotational diffusion decreases, whereas the coupling between translational and reorientational diffusion increases. The strong decoupling between translational and rotational diffusion is due to the suppressed translational mobility but promoted rotational mobility of soft Janus particles. We think that the low-T SE and SED decoupling is mainly attributed to hopping motion of soft Janus particles, whereas the high-T SE and SED decoupling is mainly attributed to collective cage motion of soft Janus particles. Our results demonstrate that interaction anisotropy has a critical effect on the translational and rotational dynamics of soft Janus particles.

Janus particle-based microprobes: Determination of object orientation

Employing information extracted from their translational dynamics, colloidal particles have served extensively as micro-sized probes in in situ analytical techniques such as particle image velocimetry and microrheology. Janus particles, partially shadowed by metal caps appear directional when imaged with an optical microscope, and hence detectable rotational dynamics are introduced into an otherwise spherically symmetric system. Current tracking methods require high-resolution imaging, are time consuming, or focus on bulk light intensity variations. Here, we introduce two image analysis methods that identify the actual body center of gold-capped Janus particles (AuJPs), as well as distinguish the particle orientation within the space. Method I tracks the optical centroid, determines the particle’s in-plane tilting angle, and locates its actual body center (i.e., the Optical Centroid-Tilting Angle-Geometry, OCTAG procedure). Method: II tracks the diffraction ring center and the moon-phase center revealing in-plane orientation (i.e., the Ring-Phase Dual Optical Centroids, RPDOC procedure). With both methods we are able to successfully identify actual body centers and orientations of AuJPs with a resolution of 0.5 pixels and an angular uncertainty of sub−10°. It is thus possible to probe the local medium response on sub-micrometer length scales. The data analysis is applied to track dilute suspensions of AuJPs undergoing Brownian motion in a viscous glycerol-water mixture of volume ratio 1 or 2. Mean-square displacement (MSD) and angular mean-square displacement (AMSD) extracted from the experimental measurements are related to medium viscosities according to the Stokes-Einstein equation and the Perrin rotational diffusion, respectively. The viscosities calculated based on the diffusion coefficients determined from analysis of the translational and rotational dynamics of the microprobes differ by less than 6%. The 6% deviation observed is attributed to wall and gravitational effects. Results indicate that Janus particles can serve as novel probes in microrheology, and their asymmetric surface properties may potentially be used to investigate particle-medium interactions.

Colloidal rotation near the colloidal glass transition

We compare, using single-particle optical imaging, trajectories of rotation and translation for micron-sized spheres in index-matched colloidal suspensions near their glass transition. Rotational trajectories, while they show intermittent caged behavior associated with supercooled and glassy behavior, explore a sufficiently wider phase space such that in the averaged mean-square angular displacement there appears no plateau regime, but instead sub-Fickian angular diffusion that follows an apparent power law in time. We infer translation and rotation time constants, the former being the time to diffuse a particle diameter and the latter being the time to rotate a full revolution. Correlation between time constants increases with increasing volume fraction, but unlike the case for molecular glasses, the rotation time constant slows more weakly than the translation time.

Self-assembled autonomous runners and tumblers

A class of artificial microswimmers with combined translational and rotational self-propulsion is studied experimentally. The chemically fueled microswimmers are made of doublets of Janus colloidal beads with catalytic patches that are positioned at a fixed angle relative to one another. The mean-square displacement and the mean-square angular displacement of the active doublets are analyzed in the context of a simple Langevin description, using which the physical characteristics of the microswimmers such as the spontaneous translational and rotational velocities are extracted. Our work suggests strategies for designing microswimmers that could follow prescribed cycloidal trajectories.

Higher-order director fluctuations

Standard theories of the influence of a fluctuating director on spin relaxation in nematic liquid crystals are based on the notion that the mean-square angular displacement 〈θ2〉 of the director is small, such that terms of this and higher orders are negligible. One consequence of this ‘small-angle’ approximation is that director fluctuations contribute only to the spectral density J1(ω0) and not to J2(2ω0) or to J0(0). In this paper the theory of director motion is extended to include terms of order 〈θ2〉. Second-order contributions are found for J2(2ω0) and J0(0). No additional contributions to J1(ω0) occur, but it is shown that the order parameter which appears in formulae for all three spectral densities is that for molecular ordering relative to director-fixed axes rather than laboratory–fixed axes. By assuming Gaussian statistics for the fluctuating director, it is shown that the correlation function relevant for J2(2ω0) and J0(0) is just the square of that used to calculate J1(ω0). Fourier-transformation of this new correlation function leads to an explicit formula for director fluctuation contributions to J2(2ω0) which is found to agree semiquantitatively with experimental data.

A note on the angular dispersion of a fluid line element in isotropic turbulence

The mean-square angular displacement of a fluid material line element is expressed as an integral of the corresponding angular velocity in material coordinates, with forms like those in Taylor's (1921) linear displacement analysis. Measurements using a hydrogen-bubble tracer in isotropic turbulence show that the mean-square angular velocity of a line is of the same order of magnitude as the mean-square vorticity, and that its ‘Lagrangian’ integral time scale is of the order of the inverse of the r.m.s. vorticity. The angular velocity of a line element is also formulated in spatial co-ordinates. Finally, the connexion between angular dispersion and the approach toward isotropy is pointed out.