Thermal singularity and droplet motion in one-component fluids on solid substrates with thermal gradients

Using a continuum model capable of describing the one-component liquid-gas hydrodynamics down to the contact line scale, we carry out numerical simulation and physical analysis for the droplet motion driven by thermal singularity. For liquid droplets in one-component fluids on heated or cooled substrates, the liquid-gas interface is nearly isothermal. Consequently, a thermal singularity occurs at the contact line and the Marangoni effect due to temperature gradient is suppressed. Through evaporation or condensation in the vicinity of the contact line, the thermal singularity makes the contact angle increase with the increasing substrate temperature. This effect on the contact angle can be used to move the droplets on substrates with thermal gradients. Our numerical results for this kind of droplet motion are explained by a simple fluid dynamical model at the droplet length scale. Since the mechanism for droplet motion is based on the change of contact angle, a separation of length scales is exhibited through a comparison between the droplet motion induced by a wettability gradient and that by a thermal gradient. It is shown that the flow field at the droplet length scale is independent of the statics or dynamics at the contact line scale.

Figure 1

The difference $\Delta \theta ={\theta }_{app}-{\theta }_{cx}$ plotted as a function of the capillary number $\mathrm{Ca}={\eta }_l{v}_l/\gamma$ in Eq. (13). The circles (squares) represent the apparent contact angles measured on the hydrophobic (hydrophilic) substrate. A linear least squares fitting is also shown for each case, with the solid line for the hydrophobic case and the dashed line for the hydrophilic case. (Inset) The coefficient ${\alpha }_{\theta }$ in the linear proportional relation $\Delta \theta ={\alpha }_{\theta }\mathrm{Ca}$ (obtained from the linear least squares fitting) plotted as a function of the initial droplet radius $R_0$. Note that this linear relation holds for both positive and negative $\mathrm{Ca}$ values, which correspond to heated and cooled substrates, respectively.

Figure 2

The migration velocity $V_{\mathrm{mig}}$ plotted as a function of the temperature difference $\Delta T\equiv T_L-T_R$, with $T_L$ and $T_R$ denoting the local bottom substrate temperatures at the left and right contact lines. The data (circles) are obtained for droplets of different sizes on hydrophobic substrates with different temperature gradients. A linear least squares fitting gives $V_{\mathrm{mig}}/V_0=0.44\Delta T/T_c$, represented by the solid line.

Figure 3

The tangential velocity $v_x$, measured on the middle line of the droplet, plotted as a function of $z$ at $t=2000{\tau }_0$. The circles (squares) represent the data obtained for the hydrophobic (hydrophilic) substrate. The initial radius of the droplet is $R_0=30\ell$. The solid (dashed) line is a quadratic least squares fitting for the hydrophobic (hydrophilic) case.

Figure 4

Schematic illustration for a migrating droplet on a substrate with a temperature gradient $d{T}_w/dx<0$. The apex of the droplet divides the liquid-gas interface into two circular arcs with constant curvatures ${\kappa }_L$ (left arc) and ${\kappa }_R$ (right arc). These two curvatures are geometrically determined by the contact angles ${\theta }_L$ and ${\theta }_R$, which depend on the local substrate temperatures $T_L$ and $T_R$, respectively. Here the droplet height $h_0$ is defined along the vertical middle line from $z=0$ to the apex, and $d_c$ is the contact length.

Figure 5

The migration velocity $V_{\mathrm{mig}}$ plotted as a function of $\Delta T/d_c$, for a droplet with initial radius $R_0=30\ell$ on hydrophobic substrates with different thermal gradients. A least squares fitting of $V_{\mathrm{mig}}$ by $V_{\mathrm{mig}}={\alpha }_{ts}{V}_{\mathrm{mig}}^{ts}$ (shown as solid line) gives the fitting coefficient ${\alpha }_{ts}=1.08$, with $V_{\mathrm{mig}}^{ts}$ being the migration velocity given by Eq. (17).

Figure 6

The migration velocity $V_{\mathrm{mig}}$ plotted as a function of $\left(\cos {\theta }_R-\cos {\theta }_L\right)$, for droplet motion driven by wettability gradient (squares) and that driven by thermal gradient (circles).

Figure 7

Density (color) and velocity (arrow) fields for droplets on solid substrates with (a) wettability gradient and (b) thermal gradient.