Line tension and morphology of a sessile droplet on a spherical substrate

The effects of line tension on the morphology of a sessile droplet placed on top of a convex spherical substrate are studied. The morphology of the droplet is determined from the global minimum of the Helmholtz free energy. The contact angle between the droplet and the spherical substrate is expressed by the generalized Young's formula. When the line tension is positive and large, the contact angle jumps discontinuously to ${180}^{\circ }$, the circular contact line shrinks towards the top of the substrate, and the droplet detaches from the substrate, forming a spherical droplet if the substrate is hydrophobic (i.e., the Young's contact angle is large). This finding is consistent with that predicted by Widom [J. Phys. Chem. 99, 2803 (1995)]; the line tension induces a drying transition on a flat substrate. On the other hand, the contact angle jumps to $0^{\circ }$, the circular contact line shrinks towards the bottom of the substrate, and the droplet spreads over the substrate to form a wrapped spherical droplet if the substrate is hydrophilic (i.e., the Young's contact angle is small). Therefore, not only the drying transition of a cap-shaped to a detached spherical droplet but also the wetting transition of a cap-shaped to a wrapped spherical droplet could occur on a spherical substrate as the surface area of the substrate is finite. When the line tension is negative and its magnitude increases, the contact line asymptotically approaches the equator from either above or below. The droplet with a contact line that coincides with the equator is an isolated, singular solution of the first variational problem. In this instance, the contact line is pinned and cannot move as far as the line tension is smaller than the critical magnitude, where the wetting transition occurs.

Figure 1

Liquid droplet on a spherical substrate. The centers of the droplet with radius $r$ and that of the spherical substrate with radius $R$ are separated by a distance $C$. The contact angle is denoted by $\theta$. Note that the three-phase contact line passes through the equator when $\varphi ={90}^{\circ }$.

Figure 2

Mechanical-force balance among the three surface tensions ${\sigma }_{\mathrm{lv}}, {\sigma }_{\mathrm{sl}}$, and ${\sigma }_{\mathrm{sv}}$ and the tension ${\sigma }_{\tau }$ from the line tension $\tau$ of the droplet on a convex, spherical substrate. The line-tension contribution ${\sigma }_{\tau }$ vanishes when $\varphi ={90}^{\circ }$ or $\theta ={\theta }_{\mathrm{c}}$, as given by Eq. (15).

Figure 3

(a) Spherical droplet sitting on top of a spherical substrate. The contact angle is $\theta ={180}^{\circ }$. (b) A spherical droplet wrapped around a spherical substrate. The contact angle is $\theta =0^{\circ }$. (c) A spherical droplet with a boiled-egg-like structure; it has the same free energy as that in part (b). Therefore, this structure is also one of the structures which has the same free energy as that for $\theta =0^{\circ }$.

Figure 4

Wetting-drying boundary ${\theta }_{\mathrm{Yw}}$ of detached and wrapped spherical droplets, defined by Eq. (24) as a function of the radius (volume) ${\rho }_{180}$ of the droplet. Below ${\theta }_{\mathrm{Yw}}$, a wrapped droplet is more stable than a detached droplet. Therefore, if the substrate is hydrophilic (${\theta }_{\mathrm{Y}}<{\theta }_{\mathrm{Yw}}<{90}^{\circ }$), the complete wetting of the wrapped droplet is favorable.

Figure 5

Size parameter $\rho \left(\theta \right)$ as a function of the contact angle $\theta$ for ${\rho }_{180}=0.7,1.5,\mathrm{and}2.0$. The scaled radius $\rho$ of the droplet increases as the contact angle $\theta$ decreases from ${\rho }_{180}$ for $\theta ={180}^{\circ }$.

Figure 6

Characteristic contact angle ${\theta }_{\mathrm{c}}$ as a function of the size parameter ${\rho }_{180}$ (solid curve). We also show the contact angle $\theta ={cos}^{-1}\left(1/{\rho }_{180}\right)$ for the fixed radius ${\rho }_{180}$ (dashed curve), which is characteristic of the nucleation problem [12, 22]. The contact line can cross the equator as the size parameter $\rho \left(\theta \right)$ changes as a function of $\theta$ (see Fig. 4) when the droplet volume is fixed by ${\rho }_{180}^3$. If the scaled radius of the droplet is fixed, the droplet cannot reach the equator if ${\rho }_{180}<1$, and no characteristic contact angle ${\theta }_{\mathrm{c}}$ can be found as a solution of Eq. (15). We also show the wetting-drying boundary (dash-dotted curve) depicted in Fig. 4, which is always larger than ${\theta }_{\mathrm{c}}$.

Figure 7

Morphological phase diagram of the droplet on a spherical substrate when ${\rho }_{180}=0.7$. There are four regions separated by two solid curves and two vertical lines. These four regions correspond to a wrapped spherical droplet (complete wetting, $\theta =0^{\circ }$), a detached spherical droplet (complete drying, $\theta ={180}^{\circ }$), a cap-shaped droplet ($\theta >{\theta }_{\mathrm{c}}$) whose contact line is on the upper hemisphere of the substrate, and the one ($\theta <{\theta }_{\mathrm{c}}$) whose contact line is on the lower hemisphere.

Figure 8

Free-energy landscape $f$ given by Eq. (4) as a function of the contact angle $\theta$ for ${\rho }_{180}=0.7$ when ${\theta }_{\mathrm{Y}}={60}^{\circ }>{\theta }_{\mathrm{Yw}}$. The local minimum at $\theta =0^{\circ }$ corresponds to the wrapped spherical droplet (complete wetting) and that at $\theta ={180}^{\circ }$ corresponds to the detached spherical droplet (complete drying). The local minimum indicated by short arrows corresponds to the stable and the metastable cap-shaped droplets whose contact angle ${\theta }_{\mathrm{e}}$ is determined from Eq. (11). The arrows on the stability limit ${\tilde{\tau }}_{\mathrm{st}}$ indicate the contact angle ${\theta }_{\mathrm{e}}$ of the unstable cap-shaped droplet determined from Eq. (11). The stable structure with the minimum free energy is a spherical droplet with $\theta ={180}^{\circ }$ when $\tilde{\tau }>{\tilde{\tau }}_{\mathrm{dry}}\simeq 0.322$. The metastable cap-shaped droplet becomes unstable when $\tilde{\tau }>{\tilde{\tau }}_{\mathrm{st}}\simeq 0.458$. On the other hand, the cap-shaped droplet is stable and the equilibrium contact angle ${\theta }_{\mathrm{e}}$ approaches the characteristic contact angle ${\theta }_{\mathrm{c}}\simeq 18.0^{\circ }$ when the line tension is large negative.

Figure 9

Free-energy landscape $f$ as a function of the contact angle $\theta$ for ${\rho }_{180}=0.7$ when ${\theta }_{\mathrm{Y}}={\theta }_{\mathrm{Yw}}\simeq 43.3^{\circ }$. In this special case of the hydrophilic-hydrophobic boundary, both the drying transition and the wetting transition can occur. When $\tilde{\tau }={\tilde{\tau }}_{\mathrm{dry}}={\tilde{\tau }}_{\mathrm{wet}}\simeq 0.410$, the landscape shows three minimums of equal depth at $\theta =0^{\circ }$, at $\theta ={180}^{\circ }$, and at ${\theta }_{\mathrm{e}}=71.5^{\circ }$, where the cap-shaped droplet may transform into either the detached spherical droplet or the wrapped spherical droplet.

Figure 10

Free-energy landscape $f$ as a function of the contact angle $\theta$ for ${\rho }_{180}=0.7$ when ${\theta }_{\mathrm{Y}}={30}^{\circ }<{\theta }_{\mathrm{Yw}}\simeq 43.3^{\circ }$. In this case, the substrate is hydrophilic. Therefore, the wetting transition occurs when $\tilde{\tau }={\tilde{\tau }}_{\mathrm{wet}}\simeq 0.143$. However, a pseudodrying transition from the metastable cap-shaped droplet to the metastable detached spherical droplet occurs when $\tilde{\tau }={\tilde{\tau }}_{\mathrm{dry}}\simeq 0.470$.

Figure 11

Free-energy landscape $f$ as a function of the contact angle $\theta$ for ${\rho }_{180}=0.7$ when ${\theta }_{\mathrm{Y}}={10}^{\circ }<{\theta }_{\mathrm{c}}\simeq 18.0^{\circ }$. In this case, not only the cap-shaped droplet with a contact line on the lower hemisphere but also the one with a contact line on the upper hemisphere can exist. There are three stability limits ${\tilde{\tau }}_{\mathrm{st}}$. At the first stability limit $\tilde{\tau }={\tilde{\tau }}_{\mathrm{st}}\simeq 0.012$, the cap-shaped droplet with a contact line on the lower hemisphere ($\theta <{\theta }_{\mathrm{c}}\simeq 18.0^{\circ }$) becomes unstable. At the next stability limit $\tilde{\tau }={\tilde{\tau }}_{\mathrm{st}}\simeq 0.387$, the metastable cap-shaped droplet with a contact line on the upper hemisphere ($\theta >{\theta }_{\mathrm{c}}\simeq 18.0^{\circ }$) reappears. This cap-shaped droplet with a contact line on the upper hemisphere may transform into the metastable detached sphere at ${\tilde{\tau }}_{\mathrm{dry}}\simeq 0.529$. Finally, this metastable cap-shaped droplet with a contact line on the upper hemisphere becomes unstable at the third stability limit ${\tilde{\tau }}_{\mathrm{st}}\simeq 0.740$.

Figure 12

Morphological phase diagram of the droplet on a spherical substrate when ${\rho }_{180}=2.0$, which also shows four regions.

Figure 13

Free-energy landscape $f$ as a function of the contact angle $\theta$ for ${\rho }_{180}=2.0$ when ${\theta }_{\mathrm{Y}}={\theta }_{\mathrm{c}}\simeq 60.8^{\circ }$. In this case, the contact angle is fixed at $\theta ={\theta }_{\mathrm{c}}$, which means that the contact line of the cap-shaped droplet cannot move and it is fixed at the equator of the spherical substrate.