Dynamics of viscous droplet coalescence in the confined geometry of optical cells

The dynamics of quasi-two-dimensional coalescence of isotropic droplets in nematic liquid crystal environment was studied. Investigations were made in confined geometry of a Hele-Shaw optical cell with different transverse droplet sizes. The existence of three distinct dynamic regimes was found for coalescence, namely, short-, middle-, and long-time regimes. The fast dynamics of bridge transformation was visualized. At short time the dynamics of droplet transformation is similar to the transformation of free (three-dimensional) droplets. At later stages, two regimes of the coalescence at different timescales are determined by Poiseuille flow. Experimental data are discussed on the basis of existing theories.

Figure 1

A snapshot sequence of transformation of the droplet shape at coalescence. Isotropic droplets are located in a homeotropic nematic sample. Thickness of the cell is 19 µm. Droplet radii R are 21.8 and 22.5 µm. Times after the start of coalescence are 0.016 s (b), 0.05 s (c), 0.34 s (d), 3 s (e), and 16 s (f).

Figure 2

A schematic of two equal droplets at the early stage of their coalescence (left) and bridge region between the droplets (right). R, W, and $r$ are droplet radius, bridge width, and curvature radius at the edge of the bridge, respectively.

Figure 3

The width of droplet bridge W versus time for isotropic droplet coalescence (a). Log-log (b) presentation of the experimental dependence of W on time. (c) Log-time representation of the shape factor $D=(L/W-1)$ at droplet relaxation to circular form. The radii of initial droplets are $R_1\approx 22.5\mu \mathrm{m}$, $R_2\approx 14.6\mu \mathrm{m}$, and $R_3\approx 13.4\mu \mathrm{m}$. Thickness of the cell is $h=5.5\mu \mathrm{m}$.

Figure 4

Linear (a) and log-log (b) representation of the experimental dependence of W on time in a thick cell with $h=19\mu \mathrm{m}$. The radii of initial droplets R are about 27.4, 16.7, and 11.3 µm.

Figure 5

The time evolution of the bridge width W for droplets with approximately the same radius but different cell thickness. Blue open circles: $R\approx 22.2\mu \mathrm{m}$, $h=19\mu \mathrm{m}$; red closed squares: $R\approx 22.5\mu \mathrm{m}$, $h=5.5\mu \mathrm{m}$. (a),(b) Dependence of W on time in linear and log-log representation; (c) log-time representation of the shape factor $D=(L/W-1)$. The solid lines in (c) are exponential functions.

Figure 6

The time dependence of bridge width W (a),(b) and shape factor $D=(L/W-1)$ in (c) depend on droplet radius R and cell thickness $h$ but do not depend on heating rate ($0.05{}^{\circ }\mathrm{C}/min$, solid symbols and $0.2{}^{\circ }\mathrm{C}/min$, open symbols). Red closed squares, green open circles: $h=5.5\mu \mathrm{m}$. Orange closed circles, green open triangles: $h=19\mu \mathrm{m}$.

Figure 7

Scaled experimental data demonstrating three different regimes. (a) The experimental dependence of scaled bridge width $W/2R$ on scaled time $(t-t_0)/\tau$ for droplets of different sizes that demonstrates linear dependence at the initial stage of coalescence. Thickness of the cell $h=19\mu \mathrm{m}$. (b) The dependence of $W/2\sqrt{Rh}$ on $(t-t_0)/{\tau }_s$. The data were scaled in the middle-time region, $h=5.5\mu \mathrm{m}$. (c) The dependence of scaled shape factor $D/D_0$ on $(t-t_0)/{\tau }_r$ shown for the late stage of coalescence. The solid line is the dependence $ln(D/D_0)=-t/{\tau }_r$. $h=5.5\mu \mathrm{m}$.

Figure 8

(a) Characteristic times for different stages of coalescence and different film thickness. Circles, diamonds, and squares are characteristic times ($\tau$, ${\tau }_s$, and ${\tau }_r$) for early, middle, and final stages of coalescence. Closed symbols: $h=5.5\mu \mathrm{m}$; open symbols: $h=19\mu \mathrm{m}$. Early-stage characteristic times $\tau$ are close in thin and thick samples. The characteristic times for the later stages are given for the thin cell ($h=5.5\mu \mathrm{m}$). The straight lines are guides to the eye; the solid lines have the slope 1, the dashed line corresponds to $\sim R^3$ dependence. (b) Characteristic times at the initial-stage $\tau$ (circles), rescaled middle-stage ${\tau }_s^{*}$, and late-stage ${\tau }_r^{*}$ times rescaled to $R\mu /\gamma$ for the 5.5 µm cell (diamonds and squares).