Surfactant reorientation under shear: dynamic surface tension and droplet deformation

Abstract

We study the deformation of a surfactant-covered droplet under shear flow using a phase-field model that explicitly accounts for both the surfactant concentration and its polarization, representing the average molecular orientation. We first consider a flat interface and show that an imposed tangential shear causes the surfactant polarization to tilt away from the interface normal. This reorientation reduces the ability of surfactants to lower the interfacial free energy, leading to an increase in the effective surface tension and demonstrating that surface tension can be dynamically modified by shear. We then examine droplet deformation under shear in both weakly and strongly confined geometries. In the weak-confinement regime, numerical results recover the linear Taylor scaling at small capillary numbers, while at larger capillary numbers they are accurately described by a modified Maffettone-Minale phenomenological model. The presence of surfactants enhances deformation through a reduction in the effective surface tension. In the strong-confinement regime, wall effects further increase droplet deformation, with results qualitatively captured by including the Shapira-Haber correction. Overall, our findings show that surfactant reorientation under flow provides a microscopic mechanism for shear-dependent surface tension and has significant implications for droplet deformation in confined multiphase flows.

Figures (7)

• Figure 1: Snapshot of a simulation showing a flat interface located at $y=L_y/2=32$ under an imposed tangential shear flow in the $x$-direction. Black arrows denote the surfactant polarization (average orientation) $\mathbf{p}(\mathbf{r})$, while blue arrows represent the fluid velocity field $\mathbf{u}(\mathbf{r})$ in the steady state. The color map shows the value of the binary fluid order parameter $\phi(\mathbf{r})$, ranging from yellow ($\phi>0$) to green ($\phi<0$). The tilting angle $\theta$ is defined to be the angle between the polarization and the interface normal. In equilibrium ($\dot{\gamma}=0$), $\mathbf{p}$ will be perpendicular to the interface ($\theta=0$). Parameters used: $L_x=L_y=64$, $\dot{\gamma}=0.5$, $c_0=0.244$, $\varepsilon=1$ and $M=\Gamma_t=\beta=B=\eta=1$.
• Figure 2: Tilting angle $\theta$ of the polarization field $\mathbf{p}$ relative to the interface normal under tangential shear flow as a function of the shear rate $\dot{\gamma}$. The orange points show the simulation results, while the blue curve represents the analytic solution obtained from perturbation theory. Parameters used: $c_0=0.244$, $\varepsilon=1$ and $M=\Gamma_t=\beta=B=\eta=1$.
• Figure 3: Effective surface tension $\sigma$ as a function of the average surfactant concentration $c_0$ for different shear rates $\dot{\gamma}$. As in the equilibrium case, the addition of surfactant lowers the effective surface tension, whereas an imposed tangential shear counteracts this reduction. Parameters used: $\varepsilon=1$ and $M=\Gamma_t=\beta=B=\eta=1$.
• Figure 4: Snapshot of a deformed droplet under an imposed shear flow at steady state. The lines represent the streamlines of the fluid velocity field $\mathbf{u}(\mathbf{r})$, with the color scale indicating its magnitude $|\mathbf{u}|$. The black arrows indicate the polarization field $\mathbf{p}(\mathbf{r})$, or the average orientation of the surfactants. Initial droplet radius is $R=10$ and channel width is $L_y=32$. Other parameters used: $L_x=64$, $\dot{\gamma}=0.01$, $c_0=0.244$, $\varepsilon=1$ and $M=\Gamma_t=\beta=B=\eta=1$.
• Figure 5: Plots of deformation parameter $D$, over time, $t$, for two different capillary numbers $\text{Ca}=0.22$ and $0.53$ (colored purple and pink respectively). We also plot the pure unbounded case (PU) against three other cases: surfactant unbounded (SU), pure bounded (PB) and surfactant bounded (SB). See Table \ref{['tab:cases']} for the parameter values.