Coalescence of holes in two-dimensional free-standing smectic films

We investigate in free-standing smectic films coalescence of holes (circular regions with thickness smaller than the surrounding film). This process can be considered as a two-dimensional analog of coalescence of bubbles in a three-dimensional fluid. A high speed video camera was used to study the evolution of domains at different stages of coalescence. Special attention was given to investigations of the dependence of the size of the bridge between two holes at the initial stage of coalescence, which was considered in numerous theoretical works and bears information on the coalescence mechanism. It is established that the scaling law is applicable for the description of the transformation of bridges for holes of different radius $R$. We found that in the regime corresponding to the experimental situation the length of the bridge $H$ increases with the scaling law $H/R={\left(t/{\tau }_R\right)}^{1/2}$. The characteristic time ${\tau }_R$ determined from the scaling law is larger than the theoretical time, which can be connected with dissipation of energy both in the film and inside the holes.

Figure 1

Free-standing smectic film with holes (dark circular regions) whose thickness is smaller than the thickness of the film. The arrow indicates a smectic island with thickness greater than the thickness of the film. The horizontal size of the image is 450 μm.

Figure 2

Coalescence of two holes with thickness eight smectic layers in a film with thickness ten layers. Photos were taken 2.6 ms (b), 8.6 ms (c), 28.6 ms (d), 48.6 ms (e), and 198.6 ms (f) after start of coalescence. The horizontal size of the images is 159 μm.

Figure 3

The half length of the bridge $H\left(t\right)$ versus time for domains with final radius $R_f=193\mu \mathrm{m}$ (■) and $R_f=107\mu \mathrm{m}$ ($•$). Characteristic time of coalescence increases with increasing hole size.

Figure 4

Comparison of temporal dependence of the experimental data $H\left(t\right)/R$ with Hopper's law (solid curve). The radii of coalescing holes are $R_1=57.4\mu \mathrm{m}, R_2=47.5\mu \mathrm{m}$ (▼); $R_1=77.5\mu \mathrm{m}, R_2=73.5\mu \mathrm{m}\left(•\right)$; $R_1=108.7\mu \mathrm{m}, R_2=98.6\mu \mathrm{m}$ (▲); $R_1=148.4\mu \mathrm{m}, R_2=123.2\mu \mathrm{m}$ (■). Data (■, $•$) correspond to the data shown in Fig. 3.

Figure 5

(a) Experimental data on the half length of the bridge $H\left(t\right)$ scaled by the hole radius $R$. Time $t$ was scaled by ${\tau }_R$. The solid curve is the dependence $H\left(t\right)/R={\left(t/{\tau }_R\right)}^{1/2}$. (b) Log-log dependence of $H\left(t\right)/R$ on $t/{\tau }_R$. The solid line is the power-law dependence with exponent 1/2. Data were fitted by the dependence $H\left(t\right)/R={\left(t/{\tau }_R\right)}^{1/2}$ in the range $H/R<0.7$. Symbols in Fig. 5 correspond to the same experimental data as in Fig. 4.

Figure 6

Dependence of the characteristic time ${\tau }_R$ on $R$. The dashed line shows ${\tau }_R$ calculated from material parameters in accordance with the simple scaling ${\tau }_v=\eta R/\gamma$.