Active microrheology of smectic membranes

Thin fluid membranes embedded in a bulk fluid of different viscosity are of fundamental interest as experimental realizations of quasi-two-dimensional fluids and as models of biological membranes. We have probed the hydrodynamics of thin fluid membranes by active microrheology using small tracer particles to observe the highly anisotropic flow fields generated around a rigid oscillating post inserted into a freely suspended smectic liquid crystal film that is surrounded by air. In general, at distances more than a few Saffman lengths from the meniscus around the post, the measured velocities are larger than the flow computed by modeling a moving disklike inclusion of finite extent by superposing Levine-MacKintosh response functions for pointlike inclusions in a viscous membrane. The observed discrepancy is attributed to additional coupling of the film with the air below the film that is displaced directly by the shaft of the moving post.

Figure 1

Oscillating post assembly for generating flow in smectic films. A thin metal post mounted on top of a small permanent magnet is carefully raised until it penetrates the liquid crystal film above it. The magnet is attached to a leaf spring that allows it to move from side to side. By varying the current in the coil of a nearby electromagnet, the post can be displaced laterally at a controlled rate. The resulting flow induced in the film is monitored using reflected light video microscopy.

Figure 2

Flow generated by a big inclusion around a moving post located far from any boundaries in an ($N=3$)-layer 8CB film. The location of the post is shown in yellow, and the inclusion is shown in green. (a) Reflected microscope image of the inclusion (radius of $a=110\mu \mathrm{m}$), which is moving along $y$. The white dots are ash particles embedded in the film. (b) Experimental flow field obtained by analyzing trajectories of tracer particles. The instantaneous velocity of the post, also measured by velocimetry, is $v_0=143\mu \mathrm{m}/\mathrm{s}$. (c) Corresponding model 2D flow field predicted by generalized LM theory. Although similar in overall appearance, the model agrees only qualitatively with experiment.

Figure 3

Normalized flow velocity profiles measured (a) along and (b) perpendicular to the direction of post motion for the big inclusion in the ($N=3$)-layer 8CB film shown in Fig. 2. The post (radius of $b=6\mu \mathrm{m}$) is shown in yellow, and the smectic film and the inclusion (radius of $a=110\mu \mathrm{m}$ or $8.1{\ell }_S$) embedded in it in green. The black curves are LM model predictions for a disklike inclusion, whereas the blue curves show the flow velocity profiles of air generated by an infinitely long moving post with no-slip boundary conditions. Away from the inclusion boundary, the experimental data lie above the LM curves, suggesting that flow in the film is boosted by air flow generated by the shaft of the post. The dashed curve shows the flow profile obtained by adding 26% of this air flow to the LM prediction. The velocities are scaled by the instantaneous post velocity $v_0$ and distances both by the inclusion radius $a$ and the Saffman length ${\ell }_S$. The velocity measurements were averaged over ten different oscillation cycles with the error bars showing the standard deviation of the mean.

Figure 4

Computed flow around a moving post inserted into a fluid liquid crystal film. The post (yellow) moves along the $y$ direction with velocity $v_0$. The nominal flow field in the film around the inclusion (green) is estimated by integrating LM point response functions for a disklike inclusion, whereas the quasi-2D flow in the air is found by solving the 2D Stokes equations around an infinite cylinder, assuming no-slip conditions at the boundaries defined by the film holder. The air couples to the flow in the membrane to give a slower decay of the film velocity than predicted by LM theory alone.

Figure 5

Flow field generated by a small inclusion (radius of $a=33.5\mu \mathrm{m}$ or $2.58{\ell }_S$) formed around a thin moving post inserted near the middle of an ($N=3$)-layer 8CB film. (a) Measured flow field superimposed on the corresponding microscopic image, showing the locations of the post (yellow) and inclusion (green). Flow velocity profiles were extracted along the directions (b) parallel and (c) transverse to the inclusion motion. The deviation of experimental data from LM theory (black curve) is larger here than for the bigger inclusion shown in Fig. 3. The velocity profile in air around an infinite cylindrical post is shown in blue. The dashed curve shows the flow profile obtained by adding 45% of this air flow to the LM prediction.

Figure 6

Flow field generated by an inclusion of intermediate size (radius of $a=80.5\mu \mathrm{m}$ or $3.58{\ell }_S$) moving parallel to a straight film boundary. (a) Observed flow field superimposed on the reflected microscopic image of an ($N=5$)-layer film. The center of the post is located $232\mu \mathrm{m}$ from the edge of the film and is moving along $y$. (b) Model flow field associated with a disklike inclusion moving under the same conditions as in (a). LM theory predicts that recirculation of the flow field should be observed on both sides of the inclusion but in the experiment this is only clearly visible on the side further from the boundary. The meniscus at the edge of the film is shown as gray, and the glass film holder is shown as blue.

Figure 7

Normalized film flow velocity profiles near the inclusion moving parallel to the film boundary shown in Fig. 6, measured (a) along $y$ and (b) along $x$. The black curves are LM model predictions for a disklike inclusion. Negative velocities indicate flow with a component opposite the post motion, i.e., flow reversal.

Figure 8

Flow field generated by an inclusion of intermediate size (radius of $a=75.5\mu \mathrm{m}$ or $5.59{\ell }_S$) moving perpendicular to a straight film boundary. (a) Observed flow field superimposed on the reflected microscopic image of an ($N=3$)-layer film. The center of the inclusion is located $258\mu \mathrm{m}$ from the film boundary, and it is moving along $y$. (b) Model flow field associated with a disklike inclusion moving under the same conditions as in (a). The proximity to a no-slip boundary causes significant vorticity on both sides of the post. The meniscus at the film edge is shown as gray, and the glass film holder is shown as blue.

Figure 9

Normalized film flow velocity profiles near the inclusion moving perpendicular to the film boundary shown in Fig. 8, measured (a) along $y$ and (b) along $x$. The black curves are LM model predictions for a disklike inclusion. Negative velocities indicate flow with a component opposite the post motion.