Thermocapillary-actuated contact-line motion of immiscible binary fluids over substrates with patterned wettability in narrow confinement

We investigate thermocapillary-driven contact-line dynamics of two immiscible fluids in a narrow fluidic confinement comprising substrates with patterned wettability variations. Our study, based on phase field formalism, demonstrates that the velocity of the contact line is a strong function of the combined consequences of the applied thermal gradient and the substrate wetting characteristics. Finally, we evaluate different energy transfer rates and show that the dissipation due to fluid slip over the solid surface plays a dominating role in transferring energy into the contact-line motion. Our analysis, in effect, provides an elegant way of controlling the capillary filling rate in a narrow fluidic confinement by tailoring the applied temperature gradient and the substrate wettability in tandem.

Figure 1

Schematic depicting the solution domain along with the physical dimensions. The solution domain is the symmetric upper half of the channel and the origin is placed at the center of the left end of the channel. Fluids A and B initially reside in the left and right side of the channel, respectively. The channel walls are chemically patched so as to provide a designed wettability. The fluid temperatures at the inlet and outlet of the channel are maintained at $T_H$ and $T_C$, respectively. Initial temperatures of both the fluids are $T_r$. An external temperature gradient [$\nabla T=\left(T_H-T_C\right)/L$] acts on the system, which, in conjunction with the substrate wettability, dictates the contact-line dynamics.

Figure 2

Schematic of the solution domain showing the governing transport equations and the appropriate boundary conditions. The channel wall is chemically patched, characterized by predefined contact angle ${\theta }_s$. There is an externally applied temperature gradient, which actuates the fluid motion into the channel. The velocities ${\overline{v}}_{\parallel }$ and ${\overline{v}}_{\perp }$ are the components along the parallel and perpendicular to the surface.

Figure 3

Model benchmarking: (a) Contact-line velocity shown as a function of distance along the capillary. (b) Temperature distribution along the length of the channel. The simulations results using the present model are represented by the solid lines and the square markers are used to represent the results reported in Refs. [13] and [40], respectively. The results of the present numerical framework show fairly accurate match with the reported numerical and experimental results. Grid independence study: Contact-line velocity as a function of the location along the channel for (c) three different values of grid sizes and (d) three different time step values. For (c) and (d) the different parameters considered are patch width $P_w=1.5,\left\{T_H,T_r,T_C\right\}=\left\{32,16,0\right\},\mathrm{Re}=0.01,\mathrm{Ca}=0.01,\mathrm{Ma}=0.01,\mathrm{Pe}=0.02$, and $B=-0.04$.

Figure 4

Contact-line velocity $v_{cl}$ vs $x$: (a) for constant surface wettability ${\theta }_s=70.6^{\circ },77.6^{\circ }$ and $\left\{T_H,T_r,T_C\right\}=\left\{32,16,0\right\}$. For three different patch widths $P_w=0.75$, 1.5, and 3: (b) $\left\{T_H,T_r,T_C\right\}=\left\{32,16,0\right\}$ and (c) $\left\{T_H,T_r,T_C\right\}=\left\{20,10,4\right\}$. The other parameters are $\mathrm{Re}=0.01,\mathrm{Ca}=0.01,\mathrm{Ma}=0.01,\mathrm{Pe}=0.02$, and $B=-0.04$ in case (b) and $-0.02$ in case (c). For the constant wettability case, the contact-line velocity increases initially, reaches maximum, and finally becomes zero. The contact-line velocity shows oscillation as it moves over the chemically patterned surface, solely attributable to the (de)pinning mechanism of the contact line. The velocity of the contact line decreases gradually as the interface moves along the channel. For the smaller patch width $P_w=0.75$ and 1.5, the interface traverses the patterned section completely. For the larger patch width $P_w=3.0$, retreating of contact-line velocity following the reverse movement of the interface initiates as the interface comes over the unfavorable patch (B-like patch) before it crosses the entire patterned section of the channel.

Figure 5

Temperature gradient acting across the contact line as a function of $x$ with variation of (a) patch width for $\left\{T_H,T_r,T_C\right\}=\left\{32,16,0\right\}$ and (b) thermal actuation strength for $P_w=3.0$. It is important to note that as the interface progress, the temperature gradient acting across the contact line, which is the primary driving force, decreases monotonically. When the magnitude of the temperature gradient decreases below a threshold, the actuating force is not strong enough to drive the interface along and we observe interface pinning.

Figure 6

Time sequence of the interface profiles for patch width $P_w=1.5$ and 3.0 for (a) $\left\{T_H,T_r,T_C\right\}=\left\{32,16,0\right\}$ and (b) $\left\{T_H,T_r,T_C\right\}=\left\{20,10,4\right\}$. The crowding zone of the interface profiles as seen is essentially because of the pinning mechanism of the interface over unfavorable patches [B-like patches; shown by green color (light patch)]. With the progression of the interface along the channel, the net driving force acting over the interface decreases and the crowding zone of the interface profiles becomes relatively denser as indicated by I, II. A deep crowding zone of the interface profiles for $P_w=3.0$ is due to the reverse motion of the interface following the retreating of the contact-line velocity. The reverse motion of the interface starts at the time when the interface comes over the unfavorable patch [B-like patches; shown by green color (light patch)] and the net driving force reduces substantially. For a relatively lower applied temperature gradient along the channel, the deep crowding zone appears earlier in the process owing to smaller magnitude of net driving force acting over the interface.

Figure 7

Contact-line velocity $v_{cl}$ vs $x$ for (a) $\left\{T_H,T_r,T_C\right\}=\left\{32,16,0\right\}$ for patch width $P_w=1.5$ for different values of contrast in wettability $r=0.7578,$ 0.6453, 0.48, 0.40, 0.3373. The other parameters are $\mathrm{Re}=0.01,\mathrm{Ca}=0.01,\mathrm{Ma}=0.01,\mathrm{Pe}=0.02$, and $B=-0.04$. The contact-line velocity shows oscillation as it moves over the chemically patterned surface. For the smallest contrast in wettability $r=0.3373$, the amplitude of oscillations becomes higher and the amplitude of oscillations in contact-line velocity keeps on decreasing with increasing value of $r$. For all the values of $r$ considered, the interface traverses the patterned section completely, albeit the retreating of the contact-line velocity follows the reverse movement of the interface initiates earlier as $r$ decreases (see inset B).

Figure 8

(a) Time $\left(t\right)$ vs capillary filling distance $\left(x\right)$, for different patch widths: (a) $\left\{T_H,T_r,T_C\right\}=\left\{32,16,0\right\}$ and $B=-0.04$; (b) $\left\{T_H,T_r,T_C\right\}=\left\{20,10,4\right\}$ and $B=-0.02$. The other parameters are $\mathrm{Re}=0.01,\mathrm{Ca}=0.01,\mathrm{Ma}=0.01$, and $\mathrm{Pe}=0.02$. All the parameters and boundary conditions considered here correspond to those taken in Fig. 4. For a given applied temperature gradient and other different parameters, the filing time decreases with the decrease in patch width. For the larger patch width given by $P_w=3.0$, the filing time shows oscillation following the phenomenon of the retreating of the contact line, and the displacing phase fluid cannot traverse the entire patterned section of the channel.

Figure 9

Evolution of different energy transfer rates as a function of time: (a) $\left\{T_H,T_r,T_C\right\}=\left\{32,16,0\right\}$ and (b) $\left\{T_H,T_r,T_C\right\}=\left\{20,10,4\right\}$. The different parameters are $\mathrm{Re}=0.01,\mathrm{Ma}=0.01,\mathrm{Ca}=0.01$, and $\mathrm{Pe}=0.02$. All the parameters and boundary conditions considered here correspond to those as taken in obtaining Fig. 4. Energy dissipation due to fluid slip at the wall is the highest among the three different rates of energy dissipations. All the energy dissipation rates are seen to decrease over the hydrophobic patches and become zero, while the contact line crosses the patterned section and moves towards the hydrophobic patch. The oscillation of the rate of the kinetic energy dissipation leads to a negative value in its variation.