Surface energy-driven crumpling transition in a thin sheet under compression

Abstract

In our common experience, crumpling a sheet requires external compressive force and leads to a random network of folds. However, thin sheets have been theoretically predicted to spontaneously transition from a flat to a crumpled state driven by thermal fluctuations, a phenomenon that has been elusive in experiments. We report the first observation of a similar crumpling transition driven instead by surface energy. Using a sensitive experimental protocol, when we gently compress a thin polymer sheet weakly adhered to a hydrogel substrate it transitions to a self-crumpling state at a well defined critical compression independent of system details. The transition is marked by the percolation of a fold network, and a power law increase in fold density. Most remarkably, the crumpled state shows a tunable order of folds establishing the phenomenon's potential as a simple and scalable technique to do origami with extremely thin sheets.

Figures (5)

• Figure 1: Capillary crumpling transition: (A) A schematic representation of the interplay between deformation and surface energies during crumpling of a thin sheet. (B) If the deformation energy $U_\epsilon$ is less than the surface energy $U_\gamma$, we get a self-crumpling phase for $\tilde{W}<\tilde{W_c}$. (C) The experimental scheme employed for controllably compressing a thin solid sheet. (D) A typical top-view image of the sheet (yellow dashed circle) on the hydrogel substrate (white dashed circle) used to measure $\tilde{W}(t)$ and $\tilde{R}(t)$. (E) Experimental data showing $\tilde{W}(\tilde{R})$. Initially ($\tilde{R}\sim\tilde{W}\sim1$), the sheet compresses slowly: $\tilde{W}=1-a\log(1+b(1-\tilde{R}))$, with fitting parameters $a$ and $b$ (red line). After reaching a threshold point ($\tilde{R}_c, \tilde{W}_c$), the sheet begins to compress much more rapidly. (F) The sheet’s response $\chi=\mathrm{d}\tilde{W}/\mathrm{d}\tilde{R}$ increases rapidly near $\tilde{W}=\tilde{W_c}$. Inset: The quantity $-\mathrm{d}\chi/\mathrm{d}\tilde{R}$ shows a peak at the transition.
• Figure 2: Transition point: (A) $\tilde{R}_c$ and (B) $\tilde{W}_c$ plotted as a function of $W_0/R_0$. $W_0$ is the initial radius of flat sheets, and $R_0$ the initial radius of the spherical and the cylindrical substrates. All data on flat substrate is plotted at $W_0/R_0=0$. The legends describe the various substrate and sheet types on which experiments have been performed. Legends where substrate material is not explicitly mentioned correspond to hydrogel-I. See Supplementary Text for more details.
• Figure 3: Self-crumpling after transition: (A) Variation of $\chi_\mathrm{s}$, the slope of $\tilde{W}(\tilde{R})$ curve after transition, with $W_0/R_0$. A value of $\chi_s \geq 1$ implies that after the transition the sheet compresses faster than the substrate. The symbols have the same meaning as described in the legend in Fig. 2. The relaxation of $\tilde{W}$ once the experiment is stopped at time $t_\mathrm{w}$ (B) before, and (C) after the transition (large circles). The observed variation in $\tilde{R}$ is also plotted for comparison (small dots) in the respective graphs. The relative change in the fold density $f$ observed during these two time periods are plotted in (D) and (E), respectively. While on stopping the experiment before the transition, the fold density decreases and the sheet relaxes towards an open state, it keeps compressing further on its own with increasing fold density after the transition.
• Figure 4: Growth of folds near the transition: (A)-(D) Fluorescence images of the sheet ($W_0 =2.3$ mm, $h =300$ nm) at various times across the transition. Near the transition point (panel C) the folds grow rapidly and a system spanning fold network emerges. (E) The variation of $\ell$ with $\tilde{W}$. Near the transition, $\ell$ increases rapidly and attains the value $1$, at $\tilde{W}=\tilde{W}_\ell$, reflecting the percolation phenomena. Inset: A Comparison of $\tilde{W}_\ell$ and $\tilde{W}_c$ shows that their values lie close to each other. (F) Variation of $f$ with $\tilde{W}$. Inset: A log-log plot of $f-f_c$ versus $\tilde{W}_c-\tilde{W}$ after the transition. The dashed line is a power-law fit $f-f_c=A(\tilde{W}_c-\tilde{W})^\alpha$ giving an $\alpha=0.81$. (G)-(J) Optical microscope images of the folds and their evolution across the transition, confirm the emergence of fold percolation at the transition.
• Figure 5: A technique for ultra-thin sheet origami: Fluorescence images of thin flat sheets taken after the crumpling transition ($h\sim 300$ nm) showing the different patterns of folds obtained on (A) spherical substrate (B) cylindrical substrate (C) flat substrate with an array of linear grooves (D) flat substrate with a square grid of grooves. (E) Image of a flat sheet crumpled on a spherical substrate after being released from the substrate inside water showing that the sheet retains its spherical shape. Image of a crumpled sheet inside water (F) just before, and (G) after it is adsorbed onto the water-air interface. (H) Image of the same sheet as in (G) taken under a microscope. Scale bars in all images are $0.5$ mm long. Images (F), (G), and (H) are from the same experiment.