Mechanics of large folds in thin interfacial films

A thin film confined to a liquid interface responds to uniaxial compression by wrinkling, and then by folding, that has been solved exactly before self-contact. Here, we address the mechanics of large folds, i.e., folds that absorb a length much larger than the wrinkle wavelength. With scaling arguments and numerical simulations, we show that the antisymmetric fold is energetically favorable and can absorb any excess length at zero pressure. Then, motivated by puzzles arising in the comparison of this simple model to experiments on lipid monolayers or capillary rafts, we discuss how to incorporate film weight, self-adhesion, or energy dissipation.

Figure 1

A thin interfacial film responds to uniaxial confinement first by wrinkling (top) and then by forming a large fold (bottom). The invariance of the system along $\widehat{z}$ allows us to parametrize the shape of the film by a function $r\left(s\right)=\left[x\right(s),y(s\left)\right]$.

Figure 2

Fold energy as a function of the imposed displacement for the symmetric (squares) and antisymmetric (circle) folds. The solid blue line is the exact solution, Eq. (3), valid before self-contact. Symmetric (top) and antisymmetric (bottom) configurations are shown before self-contact (left, exact solutions from Diamant and Witten [22] are shown as thick dashed lines) and after self-contact (right). After self-contact, the size of the fold $\Delta /2$ absorbs the excess length, while bending is localized in highly curved zones of length $l^{\prime }$.

Figure 3

Energy of the antisymmetric fold as a function of the confinement length $\Delta$. Red circles are the results of the numerical simulations, the continuous line is the exact solution, Eq. (3), which is valid before self-contact, i.e., for $\Delta \lesssim 6.6$. Inset: typical configurations showing the different folding steps (numbers indicate the confinement length).

Figure 4

Discretized model used for the simulations. The shaded discs represent the repulsion interaction, the range of which is given by the rest length $l_0$ between two neighbors to prevent self-crossing between two parts of the rod.