Resolving the Apparent Line Tension of Sessile Droplets and Understanding its Sign Change at a Critical Wetting Angle

Despite strenuous research efforts for more than one century, identifying the magnitude and sign of the apparent line tension for a liquid-solid-gas system remains an elusive goal. Herein we accurately determine the apparent line tension from the size-dependent contact angle of sessile nanodrops on surfaces with different wetting properties via atomic force microscopy measurements and molecular dynamics simulations. We show that the apparent line tension has a magnitude of ${10}^{-11}–{10}^{-10}\mathrm{J}/\mathrm{m}$, in good agreement with theoretical predictions. Furthermore, while it is positive and favors shorter contact lines for droplets on very lyophilic surfaces, the apparent line tension changes its sign and favors longer contact lines on surfaces with an apparent contact angle higher than a critical value. By analyzing the density and the potential energy of liquid molecules within the sessile droplet, we demonstrate that the sign of the apparent line tension is a thermodynamic property of the liquid-solid-gas system rather than the local effect of intermolecular interactions in the three-phase confluence region.

Figure 1

(a) Three-dimensional AFM height image of ionic liquid nanodrops on a solid surface with ${\theta }_{\text{app}}\approx 85°$. (b) Three-dimensional spherical fitting (color cap) of the AFM data points (black dots) of a nanodrop.

Figure 2

(a) Size-dependent contact angles of ionic liquid nanodrops on five surfaces with different wetting properties. ${\theta }_{\text{adv}}$ and ${\theta }_{\text{rec}}$ of macroscopic droplets on each surface are plotted for comparison. The vertical line indicates $r=100\mathrm{nm}$. (b) $\cos \theta$ versus $1/r$, where dashed lines are linear fits. (c) ${\theta }_{\infty }$ versus ${\theta }_{\text{app}}$ and (d) $\tau$ as a function of ${\theta }_{\text{app}}$ on surfaces of different materials.

Figure 3

(a) Plot of $\cos \theta$ as a function of $1/r$ for water droplets on surfaces with different binding energies obtained in the MD simulations. The dashed lines are linear fits of the data. Insets are selected snapshots of water nanodrops at equilibrium states. (b) Plot of $\tau$ as a function of the macroscopic contact angle ${\theta }_{\infty }$ and binding energy ${ϵ}_{ws}/k_B$.

Figure 4

(a) Variation of the molecular density for water droplets with $N=3150$ on surfaces with a binding energy of ${ϵ}_{ws}/k_B=117.2\mathrm{K}$ and 204.8 K. (b) Sketch of the strength and direction of intermolecular forces on the three-phase confluence region. (c) Plot of the average potential energy of water molecules in the interfacial region of the liquid-gas interface ($p_{\mathrm{LG}}$), close to the liquid-solid interface ($p_{\mathrm{SL}}$), and of the TPCL ($p_{\mathrm{TPCL}}$) as a function of ${\theta }_{\infty }$ for droplets with $N=1210$. (d) Sketch of the direction of the line tension along the contact line and its effects on the topography of sessile droplets on surfaces with ${\theta }_{\text{app}}$ lower (top) and higher (bottom) than ${\theta }_c$.