Contact-angle-hysteresis effects on a drop sitting on an incline plane

We study the contact-angle hysteresis and morphology changes of a liquid drop sitting on a solid substrate inclined with respect to the horizontal at an angle $\alpha$. This one is always smaller than the critical angle, ${\alpha }_{\text{crit}}$, above which the drop would start to slide down. The hysteresis cycle is performed for positive and negative $\alpha$'s ($\left|\alpha \right|<{\alpha }_{\text{crit}}$), and a complete study of the changes in contact angles, free surface, and footprint shape is carried out. The drop shape is analyzed in terms of a solution of the equilibrium pressure equation within the long-wave model (lubrication approximation). We obtain a truncated analytical solution describing the static drop shapes that is successfully compared with experimental data. This solution is of practical interest since it allows for a complete description of all the drop features, such as its footprint shape or contact angle distribution around the drop periphery, starting from a very small set of relatively easy to measure drop parameters.

Figure 1

Sketch of the drop cross section along the symmetry vertical plane on an incline.

Figure 2

Scheme of the contact-angle-hysteresis cycle. Point O corresponds to the initial situation where the drop volume is $V$ and $\mathrm{\Delta }r=0$. The line OA shows the path to enter the cycle ABCD. Both $V$ and $\theta$ decrease (increase) from A to C (C to A). Note that there is a pinned contact line ($\mathrm{\Delta }r=\text{const}$) along AB and CD, while there is depinning along BC and DA [see Fig. 2(b) in 24].

Figure 3

Inclination angle, $\alpha$, as a function of the number of steps for ${\alpha }_{\text{max}}={25}^{\circ }$. The complete cycle is $0^{\circ }\to {25}^{\circ }\to 0^{\circ }\to -{25}^{\circ }\to 0^{\circ }\to {25}^{\circ }$. Each step corresponds to $\mathrm{\Delta }\alpha =2.5^{\circ }$. The numbers and letters indicate the branches and their extreme points, respectively.

Figure 4

Images as obtained from the goniometer of the side (top line) and top (bottom line) views of the drop for three inclination angles, $\alpha$. (a) $\alpha =0^{\circ }$, (b) $\alpha =12.5^{\circ }$, and (c) $\alpha ={25}^{\circ }$. The green lines are the contours extracted from the image analysis. The bar in (c) gives the scale of all pictures.

Figure 5

Maximum plane inclination ${\alpha }_{\text{max}}={25}^{\circ }$ and initial contact angle ${\theta }_0={55}^{\circ }$: (a) Contact line displacements $\mathrm{\Delta }{x}_l$ and $\mathrm{\Delta }{x}_r$ (with respect to their positions at $\alpha =0^{\circ }$) vs $\alpha$. (b) Contact angles ${\theta }_l$ and ${\theta }_r$ vs $\alpha$. The horizontal dotted (dot-dashed) lines in (b) correspond to ${\theta }_{\text{rcd}}={46}^{\circ }$ and ${\theta }_{\text{adv}}={52}^{\circ }$ (${\theta }_{\text{min}}={40}^{\circ }$ and ${\theta }_{\text{max}}={55}^{\circ }$). The drop volume is $V=0.657a^3=2.17{\text{mm}}^3$.

Figure 6

Contact angles, ${\theta }_l$ and ${\theta }_r$, vs contact line displacements, $\mathrm{\Delta }{x}_l$ and $\mathrm{\Delta }{x}_r$, for ${\alpha }_{\text{max}}={25}^{\circ }$ and ${\theta }_0={55}^{\circ }$ for the drop in Fig. 5. The horizontal dotted (dot-dashed) lines correspond to ${\theta }_{\text{rcd}}={46}^{\circ }$ and ${\theta }_{\text{adv}}={52}^{\circ }$ (${\theta }_{\text{min}}={40}^{\circ }$ and ${\theta }_{\text{max}}={55}^{\circ }$).

Figure 7

Contact angles, ${\theta }_l$ and ${\theta }_r$, vs contact line displacements, $\mathrm{\Delta }{x}_l$ and $\mathrm{\Delta }{x}_r$, for (a) ${\alpha }_{\text{max}}={25}^{\circ }$ and ${\theta }_0={48}^{\circ }$ ($V=0.474a^3=1.57{\text{mm}}^3$). (b) ${\alpha }_{\text{max}}={15}^{\circ }$ and ${\theta }_0={55}^{\circ }$ ($V=0.519a^3=1.72{\text{mm}}^3$). The horizontal segment in (b) corresponds to the error bar for $\mathrm{\Delta }x$.

Figure 8

Branch 1: (a) Thickness profiles at $\alpha =0^{\circ }$ (solid lines), $12.5^{\circ }$ (dashed lines), and ${25}^{\circ }$ (dot-dashed line). (b) Corresponding footprints shifted so that the leftmost points are coincident. The drop volume is $V=0.876a^3=2.9{\text{mm}}^3$.

Figure 9

Thickness profiles for $\alpha =12.5^{\circ }$ at branch 1 (dashed line) and branch 2 (dot-dashed lines). The solid line corresponds to point O at $\alpha =0$ and it is shown for comparison. (b) Corresponding footprints shifted so that the leftmost points are coincident.

Figure 10

(a) Thickness profiles for $\alpha =0^{\circ }$ at different extreme points of the cycle: Solid line at start (point O), dashed line at the end of branch 2 (point B), and dot-dashed line at the end of branch 4 (point D). (b) Thickness profiles in (a) shifted toward left and right, respectively, until both maxima are at $x=0$.

Figure 11

Comparison between the experimental footprint (symbols) and those predicted by the truncated approximation, Eq. (8) (dashed line), when using only the positions of the three points indicated by the green dots (and the experimental value ${\theta }_r$) for (a) $\alpha =0^{\circ }$, (b) $\alpha =12.5^{\circ }$, and (c) $\alpha ={25}^{\circ }$.

Figure 12

Comparison between the predictions of $h_{\text{trunc}}$ and the experimental data for (a) the contact angle at the left, (b) the maximum drop thickness, and (c) the drop volume.

Figure 13

Azimuthal distribution of contact angle, $\theta$, as given by the truncated solution for $\alpha =12.5^{\circ }$ (red solid line, $\text{Bo}=1.01$) and $\alpha ={25}^{\circ }$ (blue dashed line, $\text{Bo}=2.15$). The symbols at $\phi =0^{\circ }$ and ${180}^{\circ }$ stand for the experimental data, namely ${\theta }_l$ and ${\theta }_r$ for $\alpha =12.5^{\circ }$ (red circles) and $\alpha ={25}^{\circ }$ (blue squares).

Figure 14

Contact angles as functions of Bond number, Bo, along branch 1: (a) ${\theta }_r/{\theta }_{\text{max}}$ (diamonds) and the relative deviation of ${\theta }_l$: $\mathrm{\Theta }=({\theta }_l-{\theta }_{\text{min}})/({\theta }_{\text{max}}-{\theta }_{\text{min}})$ (squares). The thick solid line is the best quadratic fit of the squares. (b) Experimental data and best quadratic fit for the ratio of both contact angles ${\theta }_l/{\theta }_r$. The dashed line corresponds to [25].

Figure 15

Experimental data plotted in terms of the left- and right-hand sides of Eq. (A1) compared with the identity line.