Hydrodynamic interactions in freely suspended liquid crystal films

Hydrodynamic interactions play an important role in biological processes in cellular membranes, a large separation of length scales often allowing such membranes to be treated as continuous, two-dimensional (2D) fluids. We study experimentally and theoretically the hydrodynamic interaction of pairs of inclusions in two-dimensional, fluid smectic liquid crystal films suspended in air. Such smectic membranes are ideal systems for performing controlled experiments as they are mechanically stable, of highly uniform structure, and have well-defined, variable thickness, enabling experimental investigation of the crossover from 2D to 3D hydrodynamics. Our theoretical model generalizes the Levine-MacKintosh theory of point-force response functions and uses a boundary-element approach to calculate the mobility matrix for inclusions of finite extent. We describe in detail the theoretical and computational approach previously outlined in Z. Qi et al., Phys. Rev. Lett. 113, 128304 (2014) and extend the method to study the mutual mobilities of inclusions with asymmetric shapes. The model predicts well the observed mutual mobilities of pairs of circular inclusions in films and the self-mobility of a circular inclusion in the vicinity of a linear boundary.

Figure 1

Schematic illustration of (a) a smectic A film suspended across an aperture and (b) the cross section of a circular island, eight molecular layers thick, embedded in a three-layer film.

Figure 2

Drawing of a circular smectic island in the vicinity of a linear boundary (top view). In the far-field limit, the flow field due to the island may be approximated as the flow field due to a single force applied at the center of the island. In the BEM computations, we consider continuous distributions of forces along both the inclusion's circumference and along the linear boundary.

Figure 3

Calculated $x$ component of the fluid velocity field $v_x$ along the $x$ axis (dashed line) and along the $y$ axis (solid line) due to a single circular inclusion modeled as a rigid ring of radius (a) $a=0.1{\ell }_S$ and (b) $a=10{\ell }_S$ moving with velocity $\mathbf{V}$ along the $x$ axis in a 2D unbounded membrane. Note that the computed velocity field inside the rigid ring is nearly uniform, validating our simplified modeling of a disklike inclusion using point forces only on the inclusion boundary. For distances $r<{\ell }_S$ from the center of the ring and comparable to the size of the inclusion, $v_x$ falls off (a) logarithmically along both $x$ and $y$ in the case of small inclusions, (b) as $1/r$ along $x$ and as $1/r^2$ along $y$ when the inclusions are large.

Figure 4

Calculated mobility of a circular inclusion (scaled by the HPW mobility ${\mu }_0$) moving (a) parallel and (b) perpendicular to a linear film boundary as a function of the distance $d$ from the inclusion center to the boundary (scaled by the inclusion radius $a$). The colored (gray) curves are the results of BEM computations, with the black dashed curves showing the far-field approximations.

Figure 5

Measured and predicted translational mobilities ${\mu }_{\parallel }$ and ${\mu }_{\perp }$ of islands of different sizes as a function of the dimensionless distance $d/{\ell }_S$ to a linear boundary.

Figure 6

Island and silicone oil droplet pairs in thin 8CB films viewed in reflection. (a) Islands with radii $a_1$ and $a_2$ and center-to-center separation $s$ subject to the forces $F_1$ and $F_2$. (b) Schematic cross section of a five-layer island in a two-layer film (the vertical dimension is greatly exaggerated for clarity). (c) Silicone oil droplets embedded in a six-layer film. (d) Cross section of a typical silicone oil inclusion (drawn with expanded vertical scale), measured using optical interference in monochromatic light [37].

Figure 7

Pairs of membrane inclusions moving in response to applied forces. The arrows represent simultaneously the forces applied to the inclusions (which all have the same magnitude) and the resultant velocities of the inclusion centers. The fluid at infinity is assumed to be at rest. Determination of the viscous drag force found by applying Eq. (17) to each case yields the following mobility combinations: (a) $M_{11}^{rr}+M_{12}^{rr}$, (b) $M_{11}^{rr}-M_{12}^{rr}$, (c) $M_{11}^{\theta \theta }+M_{12}^{\theta \theta }$, (d) $M_{11}^{\theta \theta }-M_{12}^{\theta \theta }$.

Figure 8

Calculated self-mobilities scaled by the mobility of an isolated inclusion in an unbounded domain for inclusions moving (a) along the line connecting their centers (b) perpendicular to the line connecting their centers as a function of the nondimensionalized center-to-center distance $s/{\ell }_S$ between the inclusions for various inclusion radii scaled by the Saffman length.

Figure 9

Measured and calculated mutual mobilities (a) $M_{12}^{rr}$ (radial) and (b) $M_{12}^{\theta \theta }$ (tangential) as a function of dimensionless separation $s/{\ell }_S$ for smectic islands of various sizes in the regime $a>{\ell }_S$ [16].

Figure 10

Two discrete clusters, each comprising, in this example, four small blobs arranged on the vertices of a square oriented so that there is reflection symmetry in $x$ and $y$. The mutual and self-mobilities are computed by allowing the clusters to move along the $x$ and $y$ directions with velocity $V$, as in Fig. 7.

Figure 11

Relative deviation of the approximate radial and angular mutual mobilities given by the leading term in the multipole expansion from the extended BEM calculations. The relative deviations (a) $\mathrm{\Delta }{M}_{12}^{rr}$ (b) $\mathrm{\Delta }{M}_{12}^{\theta \theta }$ are plotted vs the dimensionless distance $s/{\ell }_S$ between the cluster centers. The radii of the small blobs forming the clusters were set to $\varepsilon =0.01{\ell }_S$. The vertices of the clusters lie on circles of radius $a/{\ell }_S=10$ at angular positions ${\varphi }_j=2\pi (j-1)/N$, with $j=1,...,N$, where $N$ is the number of blobs in each cluster and the angles ${\varphi }_j$ are measured from the $x$ axis.

Figure 12

Radial mutual mobility of two elliptical inclusions with different aspect ratios plotted against distance $s$ between their centers scaled by the Saffman length. The black dashed curve is the LM response function ${\alpha }_{\parallel }$, describing well the case of two circular inclusions of radius ${\ell }_S$.