Front tracking simulation of droplet displacement on solid surfaces by soluble surfactant-driven flows

We provide comprehensive numerical insights into the displacement of droplets subject to soluble surfactant-driven flows. The effects of soluble surfactants on the dynamics of moving contact lines are introduced by the surfactant-dependent generalized Navier boundary condition. We show that surfactant transport significantly influences the displacement patterns of droplets on solid surfaces, affecting both the equilibrium state of the sliding motion and the critical conditions for detachment. In particular, a linear increase in the displacement velocity of a droplet with the dimensionless adsorption depth $K$ is observed. This rate of increase is more pronounced at higher elasticity numbers, as evidenced by a more significant increase in the advancing contact angle. The critical condition for droplet detachment depends on the surfactant's ability to swiftly adsorb from the bulk and replenish at interfaces, which is improved as the Biot number Bi or $K$ increases. Adsorption is enhanced by an increase in Bi, resulting in a decrease in the required time $t_d$ for droplet detachment. However, this enhancement effect becomes nonmonotonic at high Bi values. In contrast, consistently increasing the bulk Peclet number decreases $t_d$, eventually approaching the convection limit where the Marangoni-induced drag force ceases to increase. In addition, surfactant transfer near the moving contact line at a moderate Damköhler number restricts the motion of the advancing contact lines, promoting droplet detachment. For all detachment scenarios, we find that detachment necessitates a critical effective capillary number, and an increase in this number results in an exponential decline in $t_d$.

Figure 1

(a) Schematic illustration of a liquid droplet adhering to a solid surface in a parabolic flow of surfactant solution of concentration $C$. The surface surfactant concentrations $\mathrm{\Gamma }$ due to the adsorption are shown in (b) and (c), respectively.

Figure 2

The diffusion and adsorption test: Bulk surfactant concentration profiles at different times caused by the adsorption in a quiescent fluid with an initial uniform concentration $C=C_{\infty }$. The desorption term in the bulk surfactant concentration equation is switched off and the source term in Eq. (19) is reduced to $S_{\mathrm{\Gamma }}=k_a{C}_s$. The inset shows the corresponding surface surfactant concentration at different times. Our results are compared with the analytical results of Muradoglu and Tryggvason [40], i.e., Eqs. (24) and (25).

Figure 3

Bulk convection test. A clean droplet is placed on a solid surface and then suddenly subjected to shear flow with a surfactant solution. The diffusion and source terms in the evolution equation, i.e., Eq. (18), are switched off. (a) The scatter represents the bulk surfactant concentrations $C$ at the center line of the plane $z=0$ along the $x$ axis at $t=0$, 1, and 2. The dashed lines represent the indicator function, i.e., 1−$I$, which implicitly indicates the location of the interface. (b) The bulk surfactant concentration at the $z=0$ plane. The white line is the contour $I=0.5$ showing the contact line.

Figure 4

Distribution of bulk surfactant concentration and the snapshots of the droplet displacement on a solid surface when subjected to a shear flow with a surfactant solution for (b) ${\mu }_{\infty }/{\mu }_0=1$, (c) ${\mu }_{\infty }/{\mu }_0=2$, and (d) ${\mu }_{\infty }/{\mu }_0=4$ at $t=4$, 8, and 12. For each ${\mu }_{\infty }/{\mu }_0$, the droplet shape and the bulk concentration $C$ at $t=0$ are shown in (a). The evolutions of velocity $U_d$ and wetted area $A_w$ enclosed by the contact line are plotted in (e) and (f), respectively. Here, Ca, $K$, Da, Bi, ${\mathrm{Pe}}_b$, ${\mathrm{Pe}}_s$, and ${\beta }_s$ are 0.16, 0.1, 0.01, 10, 10, 10, and 0.2, respectively.

Figure 5

(a) Snapshots of droplets in surfactant solution with different viscosities, i.e., ${\mu }_{\infty }/{\mu }_0=1$, 2, and 4, for $\mathrm{Ca}=0.08$ and $K=0.1$ at $t=0$, 10, 20, and 25. The dashed profiles are steady-state shapes for droplets in surfactant solution with ${\mu }_{\infty }/{\mu }_0=1$ and 2, respectively, whereas in the case of ${\mu }_{\infty }/{\mu }_0=4$ the droplet finally detaches from the solid surface after the initial sliding. (b) The corresponding snapshots and distributions of the surface surfactant concentration Γ at $t=10$. (c) The steady-state wetting area $A_w$ and contact-line velocity $U_{\mathrm{cl}}$ at different ${\mu }_{\infty }/{\mu }_0$ for $\mathrm{Ca}=0.04$. Other parameter values in (a)–(c) are $\mathrm{Da}=0.01$, $\mathrm{Bi}=10$, ${\mathrm{Pe}}_s=10$, ${\mathrm{Pe}}_b=10$, and ${\beta }_s=0.2$.

Figure 6

(a) The displacement velocity $U_d$ and the deformation $D$ of the droplets are plotted as a function of the dimensionless adsorption depth $K$ for different elasticity numbers, i.e., ${\beta }_s=0.2$ and 0.5. Color contours are plotted in (b) to show the distribution of surfactant concentration on the droplet surface. Here, Da, Bi, ${\mathrm{Pe}}_b$, ${\mathrm{Pe}}_s$, and Ca are 0.01, 10, 10, 10, and 0.04, respectively.

Figure 7

Successive snapshots of a droplet for the capillary number above the critical value for different Biot numbers, i.e., (a) $\mathrm{Bi}=0.01$, (b) $\mathrm{Bi}=0.1$, (c) $\mathrm{Bi}=1$, and (d) $\mathrm{Bi}=10$, at $t=4$, 8, and 12. (e) Time evolution of the average surfactant concentration $M_s/A$, the average contact-line velocity $U_{\mathrm{cl}}$, the maximum ${\mathrm{\Gamma }}_{max}$, and the minimum ${\mathrm{\Gamma }}_{\min }$ concentration on the droplet surface. The scatter in (e) shows the moment of detachment and its corresponding critical value. Here, $K=1.0$, $\mathrm{Da}=0.1$, ${\mathrm{Pe}}_s=10$, ${\mathrm{Pe}}_b=10$, $\mathrm{Ca}=0.1$, and ${\beta }_s=0.5$.

Figure 8

(g) The critical time $t_d$ for droplet detachment as a function of the Peclet number ${\mathrm{Pe}}_b$. The corresponding snapshots of the droplet and the distribution of the surfactant concentration in the displacing fluid for (a) ${\mathrm{Pe}}_b=0.01$, (b) ${\mathrm{Pe}}_b=1$, (c) ${\mathrm{Pe}}_b=10$, (d) ${\mathrm{Pe}}_b=100$, (e) ${\mathrm{Pe}}_b=1000$, and (f) ${\mathrm{Pe}}_b=10000$ at $t=4$, 8, and 11. (h) Surface concentration distribution in the plane of symmetry at $t=4$. Here, ϕ is the angle between the direction vector of the surface point relative to the wetted center and the $x$ direction. Other parameter values are $K=1.0$, $\mathrm{Da}=0.1$, $\mathrm{Bi}=1$, ${\mathrm{Pe}}_s=10$, $\mathrm{Ca}=0.1$, and ${\beta }_s=0.5$.

Figure 9

Distribution of surfactant concentration both in the displacing fluid (left side) and on the droplet surface (right side) for (a) $\mathrm{Da}=0.01$, (b) $\mathrm{Da}=0.05$, (c) $\mathrm{Da}=0.1$, and (d) $\mathrm{Da}=0.5$. (e) Distributions of the surface concentration and the velocity at the contact line for different Damköhler numbers. The positive and negative signs of $u_{\mathrm{cl}}$ mean that the contact lines are advancing and receding, respectively. Other parameters are fixed at $K=1.0$, $\mathrm{Bi}=1$, ${\mathrm{Pe}}_s=10$, ${\mathrm{Pe}}_b=10$, $\mathrm{Ca}=0.1$, and ${\beta }_s=0.5$.

Figure 10

Effect of parameters that are relevant or specific to surfactant transport on the conditions for droplet sliding and detachment. The fixed parameters are ${\mathrm{Pe}}_c={\mathrm{Pe}}_b=10$. The dotted line shows the boundary corresponding to the onset of droplet detachment.

Figure 11

The detachment time $t_d$ for droplets displaced by surfactant solutions for different critical effective capillary numbers ${\mathrm{Ca}}_{e,c}=\mathrm{Ca}/\left(1+{\beta }_{s}ln\left[1-{\left(M_s/A\right)}_c\right]\right)$, where ${\left(M_s/A\right)}_c$ is the critical value of the average surfactant concentration at the interface at the time of droplet detachment. The fitted curve from our data is represented by the red dashed line. The inset plot shows the upper and lower limits of the relative error between the fitted curve and our numerical data. $t_{d,\mathrm{line}}$ is the value obtained from the fitted curve.

Figure 12

Implementation of the boundary condition for the surface concentration equation at the contact lines. The yellow triangular element represents the front element adjacent to the contact-line element. ${\mathit{n}}_e$ is the unit normal of the triangular element and $\mathrm{\Delta }{l}_i$ and ${\mathit{t}}_i$ are the length and tangential vector of the contact-line element. ${\mathit{n}}_{\mathrm{cl}}$ is the normal vector of the contact-line element. The red arrows represent the plane tangential vectors at the midpoints of the three sides of the triangle. The diffusive flux along the tangential vectors (i.e., the dashed arrow) is forced to zero in the equation for the evolution of the surface surfactant concentration.

Figure 13

Normalization of the distribution function. The circle shows the range of action of the distribution function for the front element indicated by the pentagon. The squares (i.e., $I<0.5$ and $z>0$) indicate the Eulerian grid points that are involved in the distribution of the source term $S_{\mathrm{\Gamma }}$.

Figure 14

Convergence study of the surfactant loss $e\left(M_{sf}\right)$ and wetted area $A_w$ plotted against the dimensionless time $\gamma t$ with different Eulerian grid resolutions at $\mathrm{Ca}=0.1$, ${\mathrm{Pe}}_s=1$, and ${\beta }_s=0.4$. Here $N_R$ is the number of Eulerian grids per unit length $R$. The insets show the color contours on the droplet surface representing the interfacial concentration at $\gamma t=6$.

Figure 15

Convergence study of the surfactant loss $e\left(M_{sf}\right)$ and interfacial concentration plotted against the dimensionless time $\gamma t$ with different Lagrangian grid resolutions at $\mathrm{Ca}=0.1$, ${\mathrm{Pe}}_s=1$, and ${\beta }_s=0.4$. Here $N_{La}$ is the number of Lagrangian grids. The colored contours on the droplet surface represent the interfacial concentration at $\gamma t=10$.