Coalescence of Islands in Freely-Suspended Smectic Films

The coalescence of two fluid objects and the converse process, the breakup of fluid drops, are simple, beautiful examples of singular physical phenomena involving the divergence of a physical quantity. Early work in this field was carried out by J. J. Thomson, who studied the evolution of ink droplets as they merged with water Thomson and Newall (1885). More recent topics of investigation include the dynamics of drop pinch-off Shi et al. (1994); Doshi et al. (2003) and the expansion of the fluid bridge connecting two coalescing drops Aarts et al. (2005); Case and Nagel (2008); Sprittles and Shikhmurzaev (2012); Xia et al. (2019); Klopp et al. (2020). Hopper’s hydrodynamic modeling of the coalescence dynamics of two infinite, fluid cylinders Hopper (1984, 1993) and further works by Eggers, Lister and Stone on the crossover between viscous and inertial regimes during coalescence Eggers et al. (1999) have stimulated several experimental and theoretical studies of both three-dimensional ($3$D) Menchaca-Rocha et al. (2001); Burton and Taborek (2007); Paulsen et al. (2011); Hack et al. (2020); Anthony et al. (2020) and quasi-two-dimensional ($2$D) fluids Mann et al. (1995); Delabre and Cazabat (2010); Nguyen (2011); Petrov et al. (2012); Shuravin et al. (2019). Fluid membranes represent near-ideal model systems for studying hydrodynamics in reduced dimensions, described theoretically by Saffman and Delbrück Saffman and Delbrück (1975); Saffman (1976). Usually such membranes have finite thickness and are immersed in another fluid (or are confined to a fluid-fluid interface) and so are only quasi-two-dimensional. Nevertheless, they can display $2$D character when the Saffman length, the ratio between the $2$D and $3$D viscosities, is sufficiently large, as we have demonstrated experimentally for inclusions in smectic films Nguyen et al. (2010); Eremin et al. (2011); Qi et al. (2014); Kuriabova et al. (2016); Qi et al. (2016). Saffman and Delbrück’s pioneering treatment of hydrodynamics in $2$D has been extended recently to describe $2$D hydrodynamic interactions between embedded particles Oppenheimer and Diamant (2011); Di Leonardo et al. (2011) and the shape relaxation of fluid lipid domains Camley and Brown (2010).

Fluid smectic liquid crystal (LC) films freely suspended in air are near-ideal realizations of homogeneous $2$D fluids that can be used as model $2$D hydrodynamic and structural systems. We describe experiments using high-speed video microscopy to observe the coalescence of pairs of flat, disk-shaped fluid smectic islands embedded in thinner background films of the same LC material, in events driven by the line tension along the island boundaries. In these investigations, pairs of islands with similar radii and the same thickness merge to form single islands of the same total area. The coalescence dynamics were characterized by measuring the width of the bridge connecting the merging islands as a function of time.

Analysis of such events is based on treatment of the islands as incompressible, viscous discs of fixed thickness much smaller than their radius, embedded in the background film and in Saffman-Delbrück contact with the surrounding air. A quantitative hydrodynamic description of the dynamics of coalescence in such a system is available only for later times in the relaxation, where a model by Mann et al. Mann et al. (1995) predicts that the merged island shape becomes elliptical and relaxes exponentially to a circular shape in a time that depends on the island boundary line tension, the LC and air shear viscosities, and the island and background film thicknesses. Measurements of the relaxation times in the smectic A phase of the liquid crystal $8$CB, where these materials properties are known, enables testing and verification of the Mann model using no fitting parameters, confirming that accounting for dissipation in the island, air, and the background film is necessary and sufficient to describe quantitatively the relaxation at long times.

The shape relaxation of a thin, quasi-$2$D fluid domain of equilibrium radius $R$ is predicted in Mann’s theory Mann et al. (1995) to occur in a characteristic time that depends on its reduced radius $\varepsilon=R/\ell_{S}$, where, in the case of a smectic film that is in contact with the air both above and below, the Saffman length is given by $\ell_{S}=\eta_{2D}/(2\eta^{\prime})$. When $\varepsilon\ll 1$, most of the dissipation occurs in the film and the relaxation has two-dimensional character Mann et al. (1995), while when $\varepsilon\gg 1$, the dominant dissipation is in the surrounding air and the dynamics are three-dimensional. In previous experiments on the shape fluctuations of fluid domains in Langmuir films, typically $\varepsilon={\cal O}(100)$ Stone and McConnell (1995). However, in our freely-suspended smectic films, $\varepsilon$ can be tuned in the critical range from $0.1$ to $1$ by varying the film thickness in order to control $\ell_{S}$ Nguyen et al. (2010). As we shall see below, in this regime small deviations from the equilibrium domain shape decay at a rate dictated by dissipation in both $2$D and $3$D, the overall relaxation time constant $\tau_{r}$ being the sum of the characteristic times associated with these individual modes, i.e., $\tau_{r}=\tau_{2D}+\tau_{3D}$ Mann et al. (1995).

At early times, it is instructive to apply Hopper’s model describing the coalescence of two fluid cylinders in a hypothetical inviscid fluid (summarized in Appendix A) to the island coalescence case, under the assumption that the effects of flow of the surrounding air and of the background film may be ignored. While we find that the observed functional form of the bridge width dynamics is similar to the Hopper model, in agreement with the results of Shuravin et al.  Shuravin et al. (2019), our experiments yield measured characteristic growth rates that are only about half the Hopper prediction, contradicting Shuravin’s claim of overall agreement with Hopper’s model. This indicates that the dissipative effects of both the surrounding air and the background film must also be included at short times.

Smectic liquid crystals can be drawn into stable films freely-suspended in air. Because of the smectic layering, such films are of uniform thickness and are homogeneous in their materials properties such as viscosity Young et al. (1978). Smectic films have low vapor-pressure and exhibit Newtonian viscosity, making them useful for hydrodynamics experiments in $2$D. Islands can appear spontaneously in the films or can be created intentionally using airjets Pattanaporkratana et al. (2004). These islands are bounded by edge-dislocation loops, sketched in Fig. 1, with an energy per unit length that may be expressed as a line tension $\lambda\propto\Delta N=N_{i}-N_{b}$, where $N_{i}$ and $N_{b}$ are respectively the number of molecular layers in the island and background films Géminard et al. (1997). This line tension minimizes the island perimeter and acts as the driving force in island coalescence. The dynamics are limited by dissipation arising from the bulk viscosity of air $\eta^{\prime}$ and the $2$D viscosity of the film $\eta_{2D}=\eta h$, where $\eta$ is the $3$D viscosity of the liquid crystal and $h=Nd$ is the island or background film thickness, with $d$ being the thickness of a single smectic layer. The absence of a substrate in these experiments allows us to obtain high-contrast images of the islands using reflection-mode microscopy. The film and island layer numbers were determined by optical reflectivity Rosenblatt and Amer (1980); Pankratz et al. (2000).

The liquid crystal used in these investigations is $8$CB ($4^{\prime}$-n-octyl-$4$-cyanobiphenyl, Sigma-Aldrich), which at room temperature ($22^{\circ}$C) is in the smectic A phase, with an in-plane viscosity $\eta=0.052\;\mathrm{Pa\cdot s}$ Schneider (2006) and a layer thickness $d=3.17\;\mathrm{nm}$ Davidov et al. (1979). The surrounding air is assumed to have a viscosity $\eta^{\prime}=1.827\cdot 10^{-5}\;\mathrm{Pa\cdot s}$ Weast (1973). We drew freely-suspended films with $N_{b}$ in the range $2$–$8$ smectic layers across a $3\;\mathrm{mm}$-diameter, horizontal opening in a glass cover slip and generated islands with radii in the range $4$–$100\;\mathrm{\mu m}$ and thickness $N_{i}\lesssim 30$ layers. The large difference in thickness $\Delta N$ between the islands and the background film gives good optical contrast and helps stabilize the island size for up to an hour against draining, a slow process in $8$CB films in which material is transported from the film into the meniscus at the film boundary. The line tension of an elementary layer step $\lambda_{1}=\lambda/\Delta N$ was found previously for $8$CB to be $10.5\pm 0.6\;\mathrm{pN}$ by Zou et al. Zou et al. (2007), $7\pm 1\;\mathrm{pN}$ by Géminard et al. Géminard et al. (1997), and $10\pm 1\;\mathrm{pN}$ in our measurements of the boundary deformation of sessile islands in a vertical film pulled by gravity to the meniscus at the bottom of the film.

The films were enclosed in a sealed chamber in order to minimize disturbances from the surrounding air. Pairs of islands were selected and then brought close together using optical tweezers, eliminating having to wait for the islands to approach each other by random diffusion. In $8$CB films, even when two islands touch, they often do not coalesce immediately but once they began to merge, we recorded the shape evolution of the merged island at rates of up to $67{,}065$ frames/sec (fps) using a high-speed digital video camera (Phantom v12.1, Vision Research) with $512\times 512$ pixels, providing a spatial resolution of about $0.5\;\mathrm{\mu m}$.

A typical coalescence event (two $27$-layer islands of approximately the same size merging in a $4$-layer film, captured at $21{,}005\;\mathrm{fps}$) is shown in Fig. 2. Soon after the islands connect (at $t=0$), a narrow, fluid bridge forms between them. The bridge then expands and the initially sharp, concave cusps on either side of the bridge move steadily apart and become rounded. Once the bridge spans the width of the merged island, its ends become convex (at $t\approx 5\;\mathrm{ms}$), following which the line tension slowly forces the merged island to change from approximately elliptical to circular. During the entire coalescence process, the two islands remain along a fixed symmetry axis passing through their centers. The bridge expands normal to this axis and has a well-defined boundary, allowing us to measure the bridge width accurately to within $\pm 1\;\mathrm{\mu m}$ for $t\gtrsim 0.5\;\mathrm{ms}$. We have analyzed the island coalescence in order to determine the bridge width vs. time first by measuring directly the neck width of thresholded coalescence images, and second by analyzing time-series images (kymographs) created by making intensity scans parallel to the bridge (see Appendix B for a more detailed explanation of this procedure). These analysis methods produce similar results. At early times, leading up to coalescence, the island boundaries near the point of contact are very close together and essentially tangent to each other. Since the island boundaries are extremely narrow, with widths that are below the optical diffraction limit, they cannot be resolved individually in the microscope, giving an apparent bridge even before coalescence begins. The initiation of coalescence of the two islands (where $w=0$) was therefore taken to be the instant at which the perceived bridge width began to grow.

The boundary of the merged island can be fitted to a Fourier expansion in the first three even powers of $\cos\theta$, using

Even at the fastest observed bridge expansion velocity ($v\sim 25\;\mathrm{mm/s}$) and largest relevant length scale ($\ell\sim 150\;\mathrm{\mu m}$) in these experiments, the Reynolds number is small ($\mathrm{Re}=\rho v\ell/\eta={\cal O}(0.1)$, where $\rho$ is the density of the liquid crystal). The hydrodynamics in the plane of the film are thus expected to be Stokes-like. The evolution of the bridge width $w$ during the coalescence of a typical pair of islands is plotted in Fig. 3(a), together with Hopper’s prediction for the coalescence of two infinite, viscous cylinders of isotropic fluid surrounded by inviscid fluid Hopper (1984) taking the effective surface tension as $\gamma=\lambda_{1}/d=3.15\;\mathrm{mN/m}$, where $\lambda_{1}=10\;\mathrm{pN}$, and using the viscosity of $8$CB. The coalescence of several pairs of islands of the same thickness and similar size vs. time scaled by the Hopper relaxation constant $\tau_{H}=N_{i}\eta dR_{f}/(\Delta N\lambda_{1})$ is shown in Fig. 3(b), with the raw data plotted in the Inset, and on a log-log scale in Fig. 3(c). In all of the experiments, the growth at early times is observed to be about a factor of two slower than Hopper’s model, in accord with the observations of Shuravin et al. Shuravin et al. (2019), suggesting that there are sources of dissipation in addition to those due to material flow in the island interior arising from processes not present during the coalescence of two isolated fluid disks, such as shear flow in the surrounding film and in the air, that need to be considered.

We have observed directly the coalescence of disc-shaped, smectic fluid islands in thin, quasi-two dimensional smectic films in a regime governed mainly by two-dimensional hydrodynamics. Coalescence of the islands is driven by the line tension along the island boundary, at a rate limited principally by viscous drag associated with shear flow in the island and the background liquid crystal film. The merged island evolves rapidly from the narrow-waisted shape formed immediately after coalescence to a rounded dumbbell, then becomes elliptical and finally relaxes exponentially to being circular. Application of Mann’s model for the relaxation of fluid domains in Langmuir films, which includes dissipation in both $2$D and $3$D, shows that at long times in the island coalescence process, dissipation occurs principally in two dimensions, in the thin fluid smectic membrane rather than in the surrounding air.

This work was supported by NASA Grants NAG-NNX07AE48G and NNX-13AQ81G, by the Soft Materials Research Center under NSF MRSEC Grants DMR 0820579 and DMR-1420736, and by the German Space Management DLR Grant OASIS-Co (50WM1744). K.H. was also supported by German Science Foundation (DFG) Grants HA8467-1/1 and HA8467/2-1. We thank R. Stannarius for fruitful discussions.

An exact, analytical solution for the evolution of the shape of two viscous, infinite cylinders of equal size coalescing in an idealized inviscid fluid to form a single cylinder of radius $R_{f}$ has been derived by Hopper Hopper (1984, 1990). Coalescence is driven in this model by the surface tension $\gamma$ and the dynamics are limited by the viscosity $\eta$. The flow is assumed to be strictly planar.

The evolution of the island shape at any stage of coalescence can be described Hopper (1984) by a pair of parametric equations

Finally, we note that the definition of the complete elliptic integral of the first kind in Hopper’s $1990$ paper Hopper (1990) appears to be incorrect, being inconsistent with the rest of that paper and with Hopper’s $1984$ publication Hopper (1984): the upper integration bound should presumably be $\pi/2$.

Selected images from a typical coalescence experiment are shown in Fig. 5. Before the islands merge (which starts at around frame $40$ in this sequence), the two islands are very close, with a separation that is smaller than the optical resolution of the microscope: the thin background film cannot be discerned between the islands and the interfacial region is somewhat brighter than the background film. The initiation of coalescence ($t=0$) was taken to be the time at which the center of the bridge became as bright as the two islands. In order to determine the coalescence bridge width at early times, we constructed time series images (kymographs) from intensity scans across the bridge region carried out on the entire sequence of video frames. The kymograph obtained for the coalescence event depicted in Fig. 5 is shown in Fig. 6. The positions of the outer edges of the bridge were identified at early times by making horizontal scans through the kymographs and using dynamic thresholding to determine the edge locations, while at later times during the coalescence process, we analyzed vertical scans. Direct measurements of thresholded versions of the original island images carried out for comparison gave similar results.