Dynamics of capillary coalescence and breakup: Quasi-two-dimensional nematic and isotropic droplets

We observed the formation of small satellite droplets from the bridge at droplet coalescence. Investigations were made using a Hele-Shaw cell in the two-phase region at the nematic-isotropic phase transition. In previous works on coalescence it was considered that before the start of coalescence there exists a bridge between the outer fluid connecting regions on the two sides of the droplets (outer bridge). After the start of coalescence, a bridge connecting the two droplets appears (droplet bridge) and the outer bridge is broken. We have shown that there are coalescence processes where after the start of coalescence both the droplet bridge and the outer bridge can exist. This cardinally changes the coalescence process. During the first coalescence stage the size of the outer bridge decreases and the size of the droplet bridge increases. During the second stage the outer bridge becomes unstable which leads to pinch-off, formation of pointed end domains, secondary instability, splitting of pointed end domains, and formation of satellite droplets. We found the linear dependence of the minimum bridge radius on time near bridge breakup. This behavior confirms the capillary viscous regime of bridge breakup. Our work connects two areas of fluid dynamics: coalescence and breakup with formation of satellite droplets.

Figure 1

Schematic illustration of the conventional model of quasi-2D droplet coalescence in an outer fluid. Droplets before coalescence with the outer bridge (a), (b) and during coalescence with the droplet bridge (c), (d). The outer and droplet bridges are perpendicular to each other.

Figure 2

Dynamics of transformation of the shape and interior structure of nematic ($N$) droplets in surrounding isotropic liquid. Line [yellow stripe between the droplets (b)], elongated (c), and circular domains (d)–(f) are formed at coalescence. Thickness of the cell $h=5\mu \mathrm{m}$. Times after start of coalescence are 0.04 s (b), 0.38 s (c), 2.06 s (d), 4.26 s (e), and 8.76 s (f). The orientation of polarizer, analyzer, and nematic n director is shown.

Figure 3

The width of droplet bridge $W$($t$) versus time for nematic droplets with radius $R\approx 12.3\mu \mathrm{m}$ (diamonds), $R\approx 8.9\mu \mathrm{m}$ (circles), $R\approx 7.2\mu \mathrm{m}$ (triangles). $h=5\mu \mathrm{m}$. The time of coalescence increases with increasing droplet size. The inset shows a log-log plot for $R\approx 7.2\mu \mathrm{m}$.

Figure 4

(a) Log-log dependence of $W\left(t\right)/2R$ on $t/{\tau }_S$ for droplets from $R=7.2\mu \mathrm{m}$ to $R=18.1\mu \mathrm{m}$. Data were fitted by the dependence $W\left(t\right)/R={(t/{\tau }_S)}^{1/4}$. (b) Scaling times ${\tau }_S$ in the middle stage of coalescence (circles) and relaxation times ${\tau }_R$ for the droplet relaxation to circular shape (crosses) versus the initial droplet radius. The solid line is the power law $\sim R^3$. $h=5\mu \mathrm{m}$.

Figure 5

The intermediate region (meniscus) between nematic ($N$) and isotropic liquid ($L$) in a Hele-Shaw cell. (b) Cross cut in the middle of the cell. Meniscus shape [dots in (b)] determined from the position of interference fringes. The curve in (b) is a semicircle. The photograph was taken with a green filter ($\lambda =545\mathrm{nm}$). Thickness of the cell $h=44\mu \mathrm{m}$.

Figure 6

Front view (a)–(c). Bridge region before coalescence of two isotropic droplets in nematic environment (a). Side view of the bridge region (a1)–(c1), cross cut in the middle of the cell. Dots show the shape of the bridge determined from the positions of interference fringes. After start of coalescence the transverse size of the outer bridge decreases (a1)–(c1), the droplet bridge appears and increases (b1), (c1). Time after frame (a) is 2.16 s (b) and 2.6 s (c). $R$ is about 100 μm. The photographs were taken with a green filter. $h=20\mu \mathrm{m}$.

Figure 7

Bridge region when the width of the bridge is modulated (a), pinch-off (b), central and satellite droplets (c). The photographs were taken with crossed polarizers. $h=20\mu \mathrm{m}$. Times after frame (c) in Fig. 6 are 0.03 s (a), 0.07 s (b), and 0.18 s (c).

Figure 8

Instability and breakup of the outer bridge. Modulation of the bridge width appears and increases (b), (c), followed by pinch-offs with formation of an elongated droplet (d). The elongated droplet breaks and two satellite droplets form (e). The photographs were taken in white light. Time after frame (a) is 0.03 s (b), 0.06 s (c), 0.11 s (d), 0.23 s (e). $h=20\mu \mathrm{m}$.

Figure 9

Dependence of the minimum bridge diameter $2r_{\min }$ on time $t_0-t$, $t_0$ is the moment of breakup. The dashed curve is the power law dependence $r_{\min }\sim {(t_1-t)}^{0.2}$, the solid line is the linear dependence. $h=20\mu \mathrm{m}$.