Statistics of Crumpled Paper

A statistical study of crumpled paper is allowed by a minimal 1D model: a self-avoiding line bent at sharp angles—in which the elastic energy resides—put in a confining potential. Many independent equilibrium configurations are generated numerically and their properties are investigated. At small confinement, the distribution of segment lengths is log-normal in agreement with previous predictions and experiments. At high confinement, the system approaches a jammed state with a critical behavior, whereas the length distribution follows a gamma law in which the parameter is predicted as a function of the number of layers in the system.

Figure 1

(a) A compact equilibrium state of the 1D model. A self-avoiding closed line bent at sharp angles ${\varphi }_i$ with $N$ segments of free lengths $l_i$ is put in a confining quadratic potential of strength $\lambda$. Here the number of vertices is $N=80$, the thickness $h={10}^{-3}$ (in units of total line length) and the confinement $\lambda ={10}^6$. (b) Picture of a cut through a crumpled ball of paper, courtesy E. Couturier.

Figure 2

Gyration radius of the line $R_{\mathrm{g}}$ as a function of the strength of the confining potential $\lambda$. Dotted (dashed) line: average over 60 samples; continuous line: maximum (minimum) value of $R_{\mathrm{g}}$ among all samples. $N=80$ and $h={10}^{-3}$. The differences between maximum and minimum are relatively large; nevertheless, the standard deviation is only of the order of ${510}^{-4}$.

Figure 3

The distribution of segments lengths $P(l/⟨l⟩)$ normalized by the mean length ($N=60$ and $h={10}^{-3}$). (a) Low confinement $\lambda =900$. The line is a best fit to a log-normal distribution $P_{\mathrm{LN}}$ [Eq. (2)] with width $\sigma =1.59\pm 0.03$. (b) High confinement $\lambda ={10}^6$. The two lines are a best fit to a log-normal distribution $P_{\mathrm{LN}}$ [Eq. (2)] and to a gamma distribution $P_{\Gamma }$ [Eq. (3)] with parameter $\alpha =1.64\pm 0.03$.

Figure 4

The parameter $\alpha$ of the Gamma distribution $P_{\Gamma }$ [Eq. (3)] as a function of the number of layers $N_{\mathrm{L}}$ [defined as the mean number of intersections of a half-infinite straight line with the system (see Fig. 1), averaged over all directions of the line and all samples].