# Difference between revisions of "Smooth Cascade of Wrinkles at the Edge of a Floating Elastic Film"

Entry by Helen Wu, AP225 Fall 2010

Figure 1. (a) Image of the wrinkled PS sheet, (b) geometry of the system, (c) wavelength of wrinkles as a function of thickness, which is fit with $q_0=(\frac{\rho g}{B})^{1/4}$ .

## Reference

J. Huang, B. Davidovitch, C. D. Santangelo, T. P. Russell, N. Menon, Physical Review Letters, 105, 038302 (2010).

## Overview

Figure 2. (a) Wave number as a function of distance from the edge, which is almost independent of thickness. (b) Wave number vs. distance scaled with capillary length. (c) same as (a), scaled with distance.
Figure 4. (a) Image of cascade. (b) Histogram of separation between crests (scaled with $q_0d/(2\pi)$ at x distances from the edge.

At the edge of a pattern, the symmetry usually breaks due to the boundary's tendency to be flat. The authors of this paper studied wrinkling using this phenomenon by looking at a thin sheet that has a pattern due to elastic instability and proposed a mechanism by which the pattern and the flat boundary can both exist.

When a thin rectangular sheet floating on a liquid surface is compressed from two sides along the same axis, it forms the pattern in Figure 1 - large wrinkles of wavelength $\lambda \ll$ the width of the sheet and much smaller wrinkles near the boundary. The wave amplitude is expected to decrease to minimize the surface energy of the interface and the wave number should increase to maintain inextensibility. This model has a point where the cost of bending offsets the gain in surface energy. The paper explores parameters that affect the wave cascade, the amplification of wavelength, and the length of the cascade.

## Results and discussion

The authors propose that 2 principles determine the pattern we see:

1. a thin sheet is basically inextensible, so the wavelength and amplitude are proportional
2. the wavelength is a compromise between the bending energy (favors long wavelengths) and gravitational energy (favors small ones).

They demonstrate that the scaling of the wrinkles in the bulk goes as $q_0=(\frac{\rho g}{B})^{1/4}$ where $q_0$ is the wave number, B is the bending modulus, $\rho$ is the fluid density. (see Figure 1c for the graph)

Figure 2 shows that the increase in wave number at the edge happens at approximately the same distance for all film thicknesses (with systematic deviations). This penetration distance is around $1.8 \pm 0.2mm$. They used energy calculations, estimating the cost of a wave number at the edge as well as the effect of breaking symmetry in the wave pattern. They found that persistence length $l_p \approx$ between $l_c$, the capillary length, and $1/q_e$ at various wave numbers between the edge and the large bulk wrinkles.

A current model for such elastic cascades was proposed by Pomeau and Rica, which explained the smaller wrinkles near the edge using a branching hierarchy. However, the system observed in the paper behaved differently than this would predict, particularly that the smooth amplitude reduction reflects a finite number of Fourier components being mixed as you approach the edge. The Pomeau-Rica case would predict sharp ridges and folds rather than a smooth cascade. Figure 4 shows the observed cascade and also a histogram of the separation between wave crests. If the branching theory were correct, there would be peaks at the lower end of the separation axis for each distance from the edge as the two waves become one.

Gravity, bending, and capillarity forces are all important in this system but they all scale differently. The authors suggest that they can be used to tune the system independently and allow for further studies.

## Experimental Setup

Polystyrene (PS) sheets of dimensions 3x2cm with thicknesses between 50-400nm were prepared by spin coating onto glass substrates, then transferred to a dish of DI water. The sheet floated because of the hydrophobic nature of PS.