Draping Films: A Wrinkle to Fold Transition

A polymer film draping over a point of contact will wrinkle due to the strain imposed by the underlying substrate. The wrinkle wavelength is dictated by a balance of material properties and geometry; most directly the thickness of the draping film. At a critical strain, the stress in the film will localize, causing hundreds of wrinkles to collapse into several discrete folds. In this Letter, we examine the deformation of an axisymmetric sheet and quantify the force required to generate a fold. We observe that the energy of formation for a single fold scales nearly linearly with the film thickness. The onset of folding, in terms of a critical force or displacement, scales as the thickness to the four-ninth power, which we predict from the energy balance of the system. The folds increase the tension in the remainder of the film causing the radial stress to increase, thereby decreasing the wavelength of the remaining wrinkles.

Figure 1

(a) Wrinkles on a thin film floating on water lifted by a spherical probe (inset: image via optical profilometry). (b) As the vertical displacement of the probe increases the wrinkles localize into sharp folds. (c) Schematic of the experimental setup, where (i) depicts the axisymmetric thin film floating on water; (ii) the formation of wrinkles as the probe displaces the film vertically, and (iii) the critical displacement at which wrinkles collapse into folds.

Figure 2

A plot of the number of wrinkles, $N$, versus film thickness, $h^{-1}$, normalized by material and geometric properties $C_{\delta }{E}^{-1/4}$. In our experiments, $C_{\delta }=0.005{\mathrm{Pa}}^{1/4}\mathrm{m}$ with $a=0.5\mathrm{mm}$, $\delta =1\mathrm{mm}$, $L=17.5\mathrm{mm}$, and $\rho g=9.8\mathrm{kPa}/\mathrm{m}$. Images of wrinkles were obtained by optical microscopy. The dotted line is the predicted scaling of unity from Eq. (2).

Figure 3

Number of wrinkles, $N$, versus displacement, $\delta$, along with images of wrinkles obtained through optical microscopy for a 91 nm thick polystyrene film.

Figure 4

Optical profilometry images with corresponding profile plots show the change in wrinkle amplitude. These images show the film with a constant wrinkle wavelength (i), with a growth in wrinkle amplitude (ii), and finally with a further increase in amplitude and decrease in number of wrinkles before folding (iii).

Figure 5

(a) Force, $P$, vs displacement, $\delta$, for a 375 nm thick PS film. A fold develops at ${\delta }_c$ at which point the load decreases. Lowering the film causes it to unfold at ${\delta }_u\approx 0.5\mathrm{mm}$ which causes the load to rapidly increase. (b) Hystersis energy of single fold formation, $U_{\mathrm{hys}}$, vs film thickness, $h$ with a linear scaling from Eq. (5). (c) Plot of the critical force and critical displacement for the wrinkle-to-fold transition, $P_c$ and ${\delta }_c$ vs film thickness, $h$. The dashed line represents a scaling of $h^{4/9}$ from Eq. (3). (d) Radial stress after folding, ${\sigma }_{rr}^f/{\sigma }_{rr}^w$ vs film thickness, $h$ with an empirical trend line of $h^{-1}$.