# Difference between revisions of "How wet paper curls"

Entry by Zin Lin, AP 225.

## Keywords

Capillary wetting, molecular diffusion

## Reference

[1] E. Reyssat and L. Mahadevan, How wet paper curls, EPL, 93 (2011) 54001

## Summary

When a thin strip of tracing paper is gently placed on water surface, the paper curls and rolls up as water penetrates into the mass of the paper (see Fig. 1).

Figure 1: curling of wet paper on water surface. Figure taken from [1]
The effect is due to differential swellings in the paper before it is homogeneously wetted throughout – an effect much similar to the behavior of a bimetallic thermostat when heated. As such, the authors attempted to explain the time-dependent curvature $~\kappa$ of the paper strip by using the classical theory of a bimetallic thermostat while wetting mechanism was assumed to be capillary imbibition which usually gives rise to two distinct dry and wet layers at any particular time. However, the authors found discrepancies between the experiment and theoretical predictions, and concluded that a simple bilayer model based on capillary action is inadequate. Rather, the authors posited that the wetting mechanism is due to molecular diffusion and the wetted paper is not a simple bilayer of dry and wet parts but a gradient of wettedness along the paper's thickness at any particular time. The authors tested this assumption by introducing surfactant into the water. If the wetting mechanism were due to capillary action, curvature of the paper must be different in the presence of surfactant than in its absence (see Fig. 2a).
Figure 2: (a) curvature of the curling paper strip as a function of time with or without surfactant (b) experimental results and theoretical fit of the curvature. Figure taken from [1]
But the test showed that the curvature doesn't change with surfactant, indicating molecular diffusion to be the main actor at play. Writing down the diffusive equation for the water content $~\phi (z,t)$ along the paper thickness $~z$
$\displaystyle \frac{\partial \phi}{\partial t} = \frac{\partial}{\partial z} \Big( D \frac{\partial \phi}{\partial z} \Big),$ the authors calculated the time-dependent water content and curvature, finding good agreement between the molecular theory predictions and experimental results at long time (see Fig. 2b). The authors also noted that the concavity of the curvature at shorter times might be due to significant swelling of the paper strip along its thickness plus nonlinearity effects that arise when the diffusivity constant $~D$ itself is a function of the water content $~\phi$. Finally, the authors examine the radius $~R_0$ of the cylinder which forms as the paper curls up and rolls. Surprisingly, this radius is a constant in time, which the authors accounted for as a result of competition between elastic bending forces that help curling and capillary forces that prevent the paper from lifting above the water surface.

## Discussion

The paper discusses in exquisite details about how a wet strip of paper curls. The authors managed to find out that the conventional wisdom of capillary wetting through porous media doesn't apply here. Much as the curling behavior of the paper strip resembles that of a heated bimetallic strip, a simple bilayer explanation no longer works. Rather it is a more sophisticated and smoother process at work - molecular diffusion. Instead of two discrete layers, a gradient of multi-layers is created. The key here is the fiber density in the paper that can switch on or off the rapid capillary wetting in the absence of which the slower molecular diffusion takes over.