Brownian dynamics of elongated particles in a quasi-two-dimensional isotropic liquid

We demonstrate experimentally that the long-range hydrodynamic interactions in an incompressible quasi-two-dimensional isotropic fluid result in an anisotropic viscous drag acting on elongated particles. The anisotropy of the drag is increasing with increasing ratio of the particle length to the hydrodynamic scale given by the Saffman-Delbrück length. The microrheology data for translational and rotational drags collected over three orders of magnitude of the effective particle length demonstrate the validity of the current theoretical approaches to the hydrodynamics in restricted geometry. The results also demonstrate crossovers between the hydrodynamical regimes determined by the characteristic length scales.

Figure 1

Experimental setup for the study of Brownian motion in a freely suspended liquid crystal film. The film is drawn on an airtight sealed cylindrical container. Before deposition of the particles, we bend the smectic films lightly downward by an depression in the container, to avoid immediate drift of the particle into the meniscus. The pressure is then slowly equilibrated so that the experiment is performed with a flat film.

Figure 2

(a) Polarizing microscopy images of a rod embedded in the freely suspended film, observed with a full wave plate. The images are taken at time instances of 3, 120, and 300 s after the deposition. The bright birefringent area of the meniscus indicates a complex inner structure where the smectic layers are deformed. Its boundary schematically marked by a dash line. The arrows on the right show the orientations of the polarizer P, analyzer A, and the wave plate $\lambda$. (b) Time dependence of the meniscus width for the rod in (a). (c) A fragment of a typical trajectory of a rod and (d) an example of the mean squared displacement MSD (drift-free) with a linear fit (solid line), in the inset, there is a distribution of displacements measured between positions on the trajectory separated by the time interval of 1/60 s.

Figure 3

(a) Parallel and perpendicular drag coefficients for thin rods as a function of their dimensionless length $\lambda =L/L_s$. The solid lines are theoretical predictions by the Levine model [24] for a rod aspect ratio $\rho =2$ and for infinitely thin rods $\left(\rho =\infty \right)$, (LM [24]). The dash-dotted line corresponds to the Petrov-Schwille equation with an argument $\lambda =L/L_s$ [13]. (b) Ratio of the perpendicular component of the viscous drag to the parallel component, ${\zeta }_{\perp }/{\zeta }_{\parallel }$ as a function of $\lambda$. The solid line is the theoretical prediction for thin rods from Ref. [24]. The experimental data points are separated in two groups of rods with different aspect ratios $\rho =L/W$: $2<\rho <7$ (open symbols) and $7<\rho <35$ (filled symbols). Due to only a small dispersion of the rod widths, the data points corresponding to small $\rho$ occur at small $\lambda$. The dependence of the mean drag on the aspect ratio is shown in the inset of (b).

Figure 4

Dependence of the rotational drag coefficient ${\zeta }_R$ on the specific length $\lambda$. The experimental data are compared with the prediction of the Levine theory [24] (solid line) for infinitely thin rods.