Capillary Deformations of Bendable Films

We address the partial wetting of liquid drops on ultrathin solid sheets resting on a deformable foundation. Considering the membrane limit of sheets that can relax compression through wrinkling at negligible energetic cost, we revisit the classical theory for the contact of liquid drops on solids. Our calculations and experiments show that the liquid-solid-vapor contact angle is modified from the Young angle, even though the elastic bulk modulus ($E$) of the sheet is so large that the ratio between the surface tension $\gamma$ and $E$ is of molecular size. This finding indicates a new elastocapillary phenomenon that stems from the high bendability of very thin elastic sheets rather than from material softness. We also show that the size of the wrinkle pattern that emerges in the sheet is fully predictable, thus resolving a puzzle in modeling “drop-on-a-floating-sheet” experiments and enabling a quantitative, calibration-free use of this setup for the metrology of ultrathin films.

Figure 1

(a), (b) Views of the wrinkle pattern formed by placing a water drop on a floating ultrathin polystyrene sheet ($t=72\mathrm{nm}$). The radius of the contact line is $R_i$. Radial wrinkles appear in (a) an annular zone $R_i<r<L_O$ and (b) a narrow annulus $L_I<r<R_i$ beneath the drop. Panel (b) is obtained from confocal slices of a fluorescent sheet [14]. (c) A schematic cross section depicts the forces acting at the contact line (wrinkles are azimuthal undulations around this radial profile).

Figure 2

(a), (b) Computed values of $\alpha$, $\tau$ for ${10}^{-5}<\tilde{\gamma }<{10}^{-1}$. Here ${\vartheta }_Y=\pi /2$ and ${\gamma }^{\prime }/\gamma$ is 2 (red line; dark gray), 1 (purple line; medium gray), 0.5 (blue line; light gray). For $\tilde{\gamma }\ll 1$ the various plots collapse upon rescaling $\tau \to \tau {\gamma }^{\prime }/\gamma$. For sufficiently large $\tilde{\gamma }$, both $\alpha$ and $\tau$ are below their respective threshold values; hence, the state is unwrinkled and described by the axisymmetric solution. As $\tilde{\gamma }$ decreases, $\alpha (\tilde{\gamma })$ and $\tau (\tilde{\gamma })$ exceed their critical values (dashed lines: ${\alpha }_c\approx 5.16$; ${\tau }_c=2$) and our calculations describe the wrinkled state of the highly bendable sheet ($ϵ\to 0$) using the FT theory. (c) A log-log plot of the angle $\varphi$ (see Fig. 1). Dashed line (guide to the eye) has a slope $1/3$. Data points: solid circles are taken from confocal fluorescence microscopy measurements of the sheet’s profile under the drop. Both bath and drop are water. The open circles are from a different, but comparable configuration (where the bath is glycerol [14]). Error bars are comparable to the symbol size. (d) The ratio $({\vartheta }_Y-\vartheta )/\varphi \to 1/2$ as $\tilde{\gamma }\to 0$.

Figure 3

The stresses ${\sigma }_{rr}$ [light gray (blue) line] and ${\sigma }_{\theta \theta }$ [dark gray (red) line] beneath the drop [($I$), left] and outside the contact line [($O$), right] for representative post-threshold values of the confinements: $\alpha =100$, $\tau =5$. Solid curves are the compression-free (FT) limit [15, 19]. Dashed curves are the axisymmetric (NT) limit.

Figure 4

(a), (b) Comparison between observed and predicted wrinkle radius $L_O$ (normalized by $R_i$). Predictions shown for the FT (solid lines) and NT (dashed lines) theories. In (a), $\gamma ={\gamma }^{\prime }=72\mathrm{dyn}/\mathrm{cm}$ and $\tilde{\gamma }$ varies (data obtained by varying $t$ from 31 to 233 nm [7]). The dotted line has a slope $-1/3$. In (b), $t=50\mathrm{nm}$ [dark gray (red) line], $t=152\mathrm{nm}$ [light gray (blue) line], and ${\gamma }^{\prime }/\gamma$ is varied using a concentration of surfactant (pefluorododecanoic acid) in the aqueous bath. Vertical error bars result from taking the standard deviation of several wrinkles. The measurement of the wrinkle length is described in 18. (c) The measured profile beneath the drop, obtained by confocal microscopy [14] and the predicted profile (blue curve).