Interfacial behavior of surfactant-covered double emulsion in extensional flow

We analyze the interface-interface interactions of a surfactant-covered double emulsion using the lattice Boltzmann method and study the interaction of the inner and outer interfaces and the local surfactant distribution under a uniaxial extensional flow. First, the capillary effects are analyzed. Upon surfactant application, the outer droplet deformation increases and the inner droplet deformation decreases. The concentrated surfactants on the outer interface increase deformation, and the inner droplet is affected by the inner flow. At a fixed Péclet number (Pe), the surfactant concentration at the outer interface increases with an increase in capillary number (Ca); however, such a tendency is difficult to identify at the inner interface. Next, the Pe effects are analyzed. With an increase in Pe, the deformation of the inner droplet decreases significantly. The local distribution of the surfactant considerably affects the double emulsion stabilization, which is analyzed in terms of internal flow. The interfacial tension gradient induced by the surfactant generates vortices internally, which is verified by applying the surfactant to each interface independently. The radius ratio affects droplet deformation and surfactant transport. The compression of the inner flow region increases the viscous force and decreases the interface velocity. Therefore, with an increase in radius ratio, the deformation increases, and the surfactant transport becomes slow.

Figure 1

Schematic summarizing the objectives of this study.

Figure 2

Nondimensional time evolution of the deformation of clean outer and inner droplets. Droplets maintain a steady shape below ${\mathrm{Ca}}_{\mathrm{cr}}$. The deformation direction of the inner droplet changes over time, and we divide this into two regions. $\kappa =0.5,\mathrm{Ca}=0.09$.

Figure 3

The streamlines in the region of (a) the inertia of extension and (b) the vortex generation region.

Figure 4

Inner and outer deformation indices of clean double emulsion in uniaxial extensional flow. The simulation results are compared with those of Stone and Leal. Blue circles and triangles represent ${\mathrm{DI}}_{\mathrm{out}}$ and $|{\mathrm{DI}}_{\mathrm{in}}|$ of the presented model, respectively. The upper and lower solid lines represent the ${\mathrm{DI}}_{\mathrm{out}}$ and $|{\mathrm{DI}}_{\mathrm{in}}|$, respectively, determined by Stone and Leal [26], and the red dashed lines represent the analytical models. The change in the morphology of the double emulsion with Ca is shown on the right. $\kappa =0.5$.

Figure 5

Change in DI with Pe. (a) Absolute DI. (b) Relative DI based on $\mathrm{Pe}=50$. $\mathrm{Ca}=0.06$. Compared with the outer droplet, the inner droplet is greatly affected by Pe.

Figure 6

(a) Phase field and XY plane of double emulsion. (b) Velocity field of surfactant-covered outer droplet interface. (c) Surfactant distribution in (b).

Figure 7

Schematic of the effect of surfactant distribution on double emulsion deformation. (a) Stagnation points are generated by an external velocity field, where the surfactant transport is slow. (b) Mechanism of vortex generation in the XY plane upon surfactant application. The vortex is generated by the interfacial tension gradient, and the inner droplet resists deformation in the minor axis direction.

Figure 8

$|{\mathrm{DI}}_{\mathrm{in}}|$ with or without surfactant applied on each interface. I: Surfactant applied only to the inner interface. II: Clean interface. III: Surfactant applied to both the interfaces. IV: Surfactant applied only to the outer interface. When surfactant is applied to the outer interface, the deformation of the inner droplet decreases due to the vortex. When surfactant is applied to the inner interface, the deformation of the inner droplet increases due to the concentrated surfactants.

Figure 9

(a) Inner and outer deformation indices of surfactant-covered double emulsion. Compared with clean droplets, the deformation of the outer droplets increased, while that of the inner droplets decreased. (b) Surfactant contour on outer droplet interface. Surfactants transported to the tips increased the deformation of the outer droplet. $\kappa =0.5,\mathrm{Pe}=100.$

Figure 10

(a) Difference in dimensionless surfactant concentration of outer droplet and inner droplet with Ca. (b) The surfactant contour at both the droplet interfaces. Although Pe is fixed ($\mathrm{Pe}=100$), surfactant transport accelerated at the outer droplet interface with increasing Ca.

Figure 11

(a) Deformation indices of inner and outer droplets and (b) difference in dimensionless surfactant concentration of outer droplet as functions of radius ratio. $\mathrm{Pe}=100,\mathrm{Ca}=0.06.$