Micron-sized double emulsions and nematic shells generated via tip streaming

Here we show that the so-called Confined Selective Withdrawal technique [Evangelio, Campo-Cortés, and Gordillo, J. Fluid Mech. 804, 550 (2016)] can be used to produce compound drops whose radii and thicknesses can be easily decreased down to length scales of the order of $5\mu \text{m}$ and even smaller. This method is based on the generation of a highly stretched thin coaxial jet of two immiscible fluids that flow inside a third bulk coflowing fluid under creeping flow conditions. The subsequent capillary breakup of such coaxial jet produces, in a single step, compound droplets at rates that can be easily modulated to values of the order of 10 KHz. Using algebraic equations deduced using slender body theory, we report that the production frequency, the diameter, and the shell thickness follow well-defined scaling laws, enabling us to generate, from millimeter-sized injection tubes, and in a highly controlled way, uniformly sized double emulsions with the smallest diameters reported to date. We illustrate our results by producing micron-sized nematic shells with radii below $5\mu \text{m}$ and show that the radius of curvature of the surface and the shell thickness can be fixed independently.

Figure 1

Sketch of the experimental setup for the production of nematic liquid crystal shells. The coaxial injector is composed of an inner capillary tube with an inner diameter of $d_i=0.4$ mm and length of 12.5 cm. The middle fluid flows through a second tube, coaxial with the innermost one, of diameter $d_o=1.1$ mm and length 5 cm. The square glass capillary tube possesses a length $L_T\simeq 4\times {10}^{-2}$ m and a width $L={10}^{-3}\mathrm{m}$. A square section extraction tube is employed to avoid optical distortions. The coaxial alignment between the injection and extraction tubes is ensured by checking the front and side views of the setup provided by two cameras placed perpendicular to each other. The pressure chamber is designed to resist pressure differences $P_0-P_a$ in the range of 0–5 bars (0–500 KPa). The sketch shows one of the two possible setups explored, in which fluids are injected by imposing manually different gas pressures over the three immiscible liquids stored in different reservoirs.

Figure 2

The liquid velocity at the extraction tube, $U$, depends linearly on the pressure difference $P_o-P_a$, as expected from the fact that ${\text{Re}}_o<1$ [see Eq. (1)]. The blue, red, and green circles represent, respectively a fixed $Q_i$ at 0.1 ml/h, 0.5 ml/h, and 1 ml/h, while the black dots represent the prediction given in Eq. (1).

Figure 3

The regime of periodic generation of micron-sized compound drops with diameters much smaller than the diameter of the injection tube is characterized by the conditions ${\text{Re}}_o\lesssim O\left(1\right), \text{Ca}>{\text{Ca}}^{*}$, and $q\ll 1$; see Eqs. (2, 3, 4).

Figure 4

The diameters of the produced shells, $D_d$, decrease upon increasing the velocity of the fluids at the extraction tube, $U$. $U$ is controlled through the pressure difference $P_o-P_a$, which depends linearly on $U$; see Fig. 2. The inner and middle flow rates, $Q_i$ and $Q_m$, are controlled using syringe pumps. (a)–(c) $Q_i=1\mathrm{ml}/\mathrm{h}, Q_m=0.5\mathrm{ml}/\mathrm{h}, {\lambda }_i={\mu }_i/{\mu }_o=7.7\times {10}^{-4}$. (a) $U=0.035\mathrm{m}/\mathrm{s}, q=(Q_i+Q_m)/\left(U{L}^2\right)=0.012, {\text{Re}}_o=0.044, \text{Ca}=1.35$. (b) $U=0.08\mathrm{m}/\mathrm{s}, q=0.005, {\text{Re}}_o=0.1, \text{Ca}=3.07$. (c) $U=0.15\mathrm{m}/\mathrm{s}, q=0.002, {\text{Re}}_o=0.19, \text{Ca}=5.78$. (d) $Q_i=0.05\mathrm{ml}/\mathrm{h}, Q_m=0.025\mathrm{ml}/\mathrm{h}, {\lambda }_i=7.7\times {10}^{-4}, U=0.29\mathrm{m}/\mathrm{s}, q=2.8\times {10}^{-4}, {\text{Re}}_o=0.36, \text{Ca}=11.15$.

Figure 5

Bright-field images (upper row) and cross-polarized images (lower row) of nematic shells produced with the CSW technique. The images show that the shell diameter, $D_d$, decreases with the flow-rate ratio $q$, defined in Eq. (4). (a) $q=4.8\times {10}^{-3}, q_i=3.47\times {10}^{-3}, \text{Ca}=3.07, {\text{Re}}_o=0.1, D_d=112\mu \mathrm{m}, h=8\mu \mathrm{m}$. (b) $q=1.8\times {10}^{-3}, q_i=9.26\times {10}^{-4}, \text{Ca}=5.76, {\text{Re}}_o=0.19, D_d=96\mu \mathrm{m}, h=7\mu \mathrm{m}$. (c) $q=3.4\times {10}^{-4}, q_i=1.85\times {10}^{-4}, \text{Ca}=5.76, {\text{Re}}_o=0.19, D_d=46\mu \mathrm{m}, h=6\mu \mathrm{m}$. (d) $q=2.4\times {10}^{-4}, q_i=4.79\times {10}^{-5}, \text{Ca}=11.15, {\text{Re}}_o=0.36, D_d=20\mu \mathrm{m}, h=2\mu \mathrm{m}$. (e) $q=1.25\times {10}^{-4}, q_i=3.47\times {10}^{-5}, \text{Ca}=15.3, {\text{Re}}_o=0.504, D_d=14.5\mu \mathrm{m}, h=2.5\mu \mathrm{m}$. (f) $q=1.04\times {10}^{-4}, q_i=2.7\times {10}^{-5}, \text{Ca}=15.3, {\text{Re}}_o=0.5, D_d=10.5\mu \mathrm{m}, h=1.75\mu \mathrm{m}$. (g) $q=8.68\times {10}^{-5}, q_i=4.34\times {10}^{-5}, \text{Ca}=12.3, {\text{Re}}_o=0.4, D_d=8.5\mu \mathrm{m}, h=1.2\mu \mathrm{m}$. Here $h=0.5\left(D_d-D_i\right)$ indicates the thickness of the nematic shell and $q_i=Q_i/\left(U{L}^2\right)$.

Figure 6

The diameters of the produced shells, $D_d$, decrease when reducing either $Q_i$ or $Q_m$ while keeping the rest of the parameters constant. (a)–(c) The effect of varying $Q_i$ for fixed values of $U=0.08\text{m}{\text{s}}^{-1}$ and $Q_m\simeq 0.25\mathrm{ml}/\mathrm{h}$ ($\text{Ca}=3.07, {\text{Re}}_o=0.1$): (a) $Q_i=1\mathrm{ml}/\mathrm{h}$, (b) $Q_i=0.1\mathrm{ml}/\mathrm{h}$ and (c) $Q_i=0.01\mathrm{ml}/\mathrm{h}$. (d)–(f) The effect of varying $Q_m$ for fixed values of $U=0.08\text{m}{\text{s}}^{-1}$ and $Q_i\simeq 0.1\mathrm{ml}/\mathrm{h}$ ($\text{Ca}=3.07, {\text{Re}}_o=0.1$): (d) $Q_m=0.9\mathrm{ml}/\mathrm{h}$, (e) $Q_m=0.5\mathrm{ml}/\mathrm{h}$ and (f) $Q_m=0.4\mathrm{ml}/\mathrm{h}$. The scale bar indicates $200\mu \mathrm{m}$.

Figure 7

(a)–(c) Bright-field images of 5CB shells produced when $U=0.15\mathrm{m}{\text{s}}^{-1}$ is kept constant and $Q_m$ is varied. These images illustrate that the method employed here enables the production of double emulsions with practically the same outer diameter $D_d$ but different shell thicknesses: (a) $Q_m=0.12\mathrm{ml}/\mathrm{h}, q=3.82\times {10}^{-4}, q_i=1.85\times {10}^{-4}, D_d=43\mu \mathrm{m}, h=6.5\mu \mathrm{m}$. (b) $Q_m=0.1\mathrm{ml}/\mathrm{h} q=3.47\times {10}^{-4}, q_i=1.85\times {10}^{-4}, D_d=46\mu \mathrm{m}, h=5.5\mu \mathrm{m}$. (c) $Q_m=0.08\mathrm{ml}/\mathrm{h} q=3.33\times {10}^{-4}, q_i=1.85\times {10}^{-4}, D_d=46\mu \mathrm{m}, h=4\mu \mathrm{m}$. Panels (d)–(f) show that double emulsions with virtually the same thickness but different outer diameters can be produced using the CSW technique: (d) $Q_i=0.1\mathrm{ml}/\mathrm{h} Q_m=0.07\mathrm{ml}/\mathrm{h}, U=0.165\mathrm{m}{s}^{-1}, q=2.86\times {10}^{-4}, q_i=1.68\times {10}^{-4}, D_d=42\mu \mathrm{m}, h=1.5\mu \mathrm{m}$. (e) $Q_i=0.08\mathrm{ml}/\mathrm{h}, Q_m=0.03\mathrm{ml}/\mathrm{h}, U=0.19\mathrm{m}{\text{s}}^{-1}, q=1.625\times {10}^{-4}, q_i=1.18\times {10}^{-4}, D_d=32\mu \mathrm{m}, h=1.5\mu \mathrm{m}$ (f) $Q_i=0.08\mathrm{ml}/\mathrm{h}, Q_m=0.08\mathrm{ml}/\mathrm{h}, U=0.4\mathrm{m}{\text{s}}^{-1}, q=1.04\times {10}^{-4}, q_i=5.55\times {10}^{-5}, D_d=16\mu \mathrm{m}, h=2\mu \mathrm{m}$. $q_i=Q_i/\left(U{L}^2\right)$.

Figure 8

(a) Left: schematic of a glass capillary device that combines coflow and flow focusing. Right: schematic of a nematic liquid crystal shell, with an inner droplet of Diameter $D_i$ that is contained inside a bigger liquid crystalline droplet of Diameter $D_o$. (b) Production of liquid crystal shells in a glass capillary device. (c) Optical micrograph showing liquid crystal shells, with diameters of the order of the diameter of the constriction through which the three liquids flow.

Figure 9

(a) The experimentally measured frequencies depend linearly on $U/\left(L{r}_{\infty }\right)$, as predicted by Eq. (14). (b)–(c) The experimentally measured inner and outer diameters of the double emulsion, $D_i$ and $D_d$, respectively, are compared with the predictions given by Eq. (15), finding an excellent agreement between observations and theory. The red box indicates those experiments for which the inner and intermediate flow rates have not been set using syringe pumps, but varying the pressure differences $P_i-P_o$ and $P_m-P_o$, which depend linearly on $Q_i$ and $Q_m$, respectively.

Figure 10

Schematic representation of a shell with (a) hybrid boundary conditions and (c) planar boundary conditions. The dashed lines represent the director field across the shell. The arrows point out the presence of topological defects. In panel (a) the topological defects are $+1$ surface defects or boojums, inducing a $2\pi$ rotation of the director, as represented in the inset, which shows a top view of the defect. In panel (c) the defects are $+1/2$ disclination lines, inducing a $\pi$ rotation of the director. Note that the other two $+1/2$ disclination lines needed to fulfill the topological constrains in panel (c) are in a different plane and, thus, not represented. Panels (b) and (d) are cross-polarized images showing the top view of two shells with hybrid and planar boundary conditions respectively. Fluorinated oil is the inner fluid in panel (b), whereas water with $1\%$ wt of polyvinyl alcohol (PVA) is the inner fluid in panel (d).

Figure 11

Top view of shells with planar boundary conditions and diameters below $10\mu \text{m}$: (a) bright field and (b) cross-polarization. The two black brushes (indicated by yellow arrows) emerging from the defects in cross-polarization seem to indicate a $+1/2$ topological charge. The small size of the shells produced and the visualization technique employed prevent us from asserting, with absolutely no doubt, that the defects correspond to $+1/2$ disclination lines.