Experimental realization of an incompressible Newtonian fluid in two dimensions

The Brownian diffusion of micron-scale inclusions in freely suspended smectic-$A$ liquid crystal films a few nanometers thick and several millimeters in diameter depends strongly on the air surrounding the film. Near atmospheric pressure, the three-dimensionally coupled film-gas system is well described by Hughes-Pailthorpe-White hydrodynamic theory but at lower pressure ($p\lesssim 70$ torr), the diffusion coefficient increases substantially, tending in high vacuum toward the two-dimensional limit where it is determined by film size. In the absence of air, the films are found to be a nearly ideal physical realization of a two-dimensional, incompressible Newtonian fluid.

Figure 1

Experimental apparatus for observing inclusions in smectic liquid crystal films at low pressure. A resistive filament coated with silicone oil is briefly heated with an electric current to generate an oil vapor, part of which then condenses as droplets on the film. The buffer chamber shields the partially evacuated film chamber from sudden changes in pressure.

Figure 2

Oil droplets on smectic films. (a) Oil droplets on a three-layer, freely suspended liquid crystal film viewed in reflection. (b) Interference fringes in a large oil droplet. (c) Cartoon of an oil droplet of radius $a$ near the center of a film of radius $R$. (d) Cartoon of the cross section of an oil droplet deposited on a thin smectic film. The relative thickness of the film is greatly exaggerated here.

Figure 3

Mean-squared displacement of an oil droplet ($a=8\mu \mathrm{m}$) near the center of a film ($R=2$ mm, $N=3$ layers) as a function of time for different surrounding air pressures (symbols). The solid lines are fits whose slopes give the corresponding diffusion coefficients. The black, dashed line is the theoretical limit corresponding to two-dimensional diffusion confined by the film boundary.

Figure 4

Effect of surrounding air pressure and film size on droplet diffusion. (a) Diffusion coefficient of a single droplet ($a=8\mu \mathrm{m}$) near the center of a film ($R=2\mathrm{mm}, N=3$ layers) as a function of surrounding air pressure (symbols). The green curve corresponds to SD-HPW theory. The black curve shows the diffusion predicted by a model assuming free air molecules impinging on the film. The horizontal dashed red line shows the $2\mathrm{D}$ confinement limit predicted by Saffman for vanishingly small air viscosity and no-slip boundaries. The dashed black line shows the mean free path of the air molecules (right axis) scaled by the film diameter ($[{\lambda }_{\mathrm{reduced}}=\lambda /\left(2R\right)]$ as a function of pressure. The background shading indicates three distinct behavioral regimes corresponding to different air pressure ranges: free molecules, slip, and continuity. (b) Reduced mobility $m=4\pi \eta h\mu$ of single oil droplets in a film in high vacuum as a function of reduced film radius. The model curve is Saffman's prediction for a $2\mathrm{D}$ fluid with no-slip boundaries.

Figure 5

Reduced mobility of oil-drop inclusions diffusing parallel (red) and perpendicular (blue) to a straight film boundary in high vacuum. $a$ is the radius of the inclusion and $d$ the distance from the boundary. The model curves are analytical predictions from Eq. (5). The two mobilities show distinct behavior, as predicted by theory, increasing logarithmically with distance from the boundary as expected in $2\mathrm{D}$. In these experiments, the oil-drop radii were in the range $3<a<20\mu \mathrm{m}$.

Figure 6

Hydrodynamic boundaries of freely suspended liquid crystal films and oil droplets. The fluid thickness and areal viscosity both increase linearly at the boundaries of the SmA film with the outer meniscus (a) and with oil droplets (c). Reflected light microscope image of a silicone oil droplet in a thin, freely suspended liquid crystal film near the meniscus at a straight boundary. The fringes are from optical interference caused by a steady increase in thickness [(a) and (c), right]. The effective viscosity in both the meniscus and the oil drop increases by several orders of magnitude within a short distance of the film boundary [(a) and (c), left]. The boundaries are indicated here by dashed red lines. Note that the viscosity and thickness are plotted on logarithmic and linear scales respectively.

Figure 7

Film velocity profile near a boundary. The transverse velocity $v$ in the film (blue region, $x<0$) falls off linearly on approaching an oil droplet or the film meniscus, penetrating a distance $l_{\mathrm{slip}}$ into the thicker region (yellow wedge, $x>0$). The velocity normal to the boundary is identically zero. The smectic film and meniscus-droplet have viscosities $\eta$ and ${\eta }_D$ respectively.