Contact line depinning from sharp edges

With the aim of finding mathematical criteria for contact line depinning from sharp corners, we have studied the equilibrium and stability of a semi-infinite planar liquid layer pinned at the vertex of a wedge. Equilibrium is compatible with a fan of apparent contact angles ${\theta }_0$ bracketed by the equilibrium contact angles of both flanks of the wedge, so the contact line could remain pinned if ${\theta }_0$ is within this fan. However, the equilibrium becomes unstable at critical depinning advancing ${\theta }_0^a$ and receding ${\theta }_0^r$ contact angles, which are found as subcritical saddle-node bifurcations. Equilibrium is possible (stable) within the interval ${\theta }_0^r<{\theta }_0<{\theta }_0^a$, but the contact line depins from the vertex beyond these critical angles if they occur within the equilibrium fan.

Figure 1

Geometry of the liquid layer interface (pink line) pinned at the vertex of a wedge of inner angle $\varphi$. The plotted interface is the equilibrium shape corresponding to an angle ${\theta }_0={240}^{\circ }$, or equivalently, to a Bond number $\text{Bo}=\rho g{h}_{\infty }^2/\sigma =3$, i.e., a thickness upstream $h_{\infty }=\sqrt{3}{\ell }_c$ with ${\ell }_c^2=\sigma /\left(\rho g\right)$ the capillary length. As shown below, this is an equilibrium but unstable configuration.

Figure 2

The continuous blue and dashed red lines represent the equilibrium (thickness) Bond number $\text{Bo}=\rho g{h}_{\infty }^2/\sigma$ as a function of the apparent angle ${\theta }_0$ at the edge, as given by the expression (2). The blue line represents the stable equilibrium solutions branch, whereas the red dashed lines are unreachable equilibrium solutions because depinning occurs before. The inserts show the shape of the interfaces for some representative values of ${\theta }_0$ on the stable branch.

Figure 3

Interface shapes for different values of the effective thickness $\mathrm{\Delta }{h}_{\infty }=h_{\infty }-h_0$ of the liquid layer. The thick vertical blue arrow and the thin curved green one indicate the direction of increasing values of $\mathrm{\Delta }{h}_{\infty }$ and ${\theta }_0$, respectively. ${\theta }_0$ range from negative (the two lowest darker blue interfaces) to positive (the highest light blue one) values, with ${\theta }_0=0$ corresponding to a straight horizontal interface at the wedge's vertex height. The arrows also indicate the evolution as the liquid layer is forced to advance against the vertex (green curve starting at $b$ in the ${\theta }_0$-Bo plane of Fig. 4), whereas for a layer receding (purple curve starting at $a$ in Fig. 4) back from the vertex the evolution is reversed.

Figure 4

The complete bifurcation diagram including positive as well as negative apparent contact angles. The purple arrow, starting at $a$, represents the evolution path for a receding interface, eventually depinning at ${\theta }_0=-{180}^{\circ }$. The green one, starting at $b$, corresponds to the evolution of an advancing liquid front with an initial negative effective thickness $\mathrm{\Delta }{h}_{\infty }$, eventually depinning at ${\theta }_0={180}^{\circ }$. The solution branches in dash-dot lines are inaccessible because they are unstable and depinning occurs before, at ${\theta }_0=\pm {180}^{\circ }$.

Figure 5

The solid blue and dash-dotted red lines respectively represent the stable and unstable equilibrium solutions in the plane $\text{Bo}-\mathcal{P}$, with the contact angle as a parameter along them. This is the Maddocks' preferred plane [53] to analyze the stability of the solutions of the problem defined by (10). The fold ($d\text{Bo}/d{\theta }_0=0$) at $|{\theta }_0|={180}^{\circ }$ ($\text{Bo}=4$) represents the loss of stability of the blue stable branch which corresponds to $|{\theta }_0|<{180}^{\circ }$.

Figure 6

At the left, sketch of a ${90}^{\circ }$ wedge supporting a pinned interface with a pinning angle ${\theta }_0$. The interface depins when $\alpha ={\theta }_0-{90}^{\circ }$ reaches the equilibrium contact angle on the wedge's vertical wall. In the case of the semi-infinite liquid layers analyzed here for which the limit of stability is ${\theta }_0={180}^{\circ }$, liquids wetting the vertical wall, i.e., with equilibrium angles smaller than ${90}^{\circ }$, depin before becoming unstable. However, nonwetting liquids, which have contact angles larger than ${90}^{\circ }$, only depin once the stability limit ${\theta }_0={180}^{\circ }$ is reached. It can be noticed that the stability limit is independent of the geometry, whereas the critical value of ${\theta }_0$ to reach the equilibrium contact angle depends on the orientation of the wedge's angle, in this case ${90}^{\circ }$. At the right, an interface of a nonwetting liquid that remains pinned beyond ${\theta }_0={180}^{\circ }$ and is about to touch the next wedge's corner. The orange interfaces represent the configuration a moment later after the blue interface has touched the corner. This is the basic mechanism underlying rolling drops over superhydrophobic substrates as shown by Schellenberger et al. [8]. Clearly, the maximum pinning angle increases with the separation $\mathcal{L}$ between corners, as indeed found experimentally [8].