How wet paper curls

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Entry by Zin Lin, AP 225.

General Information

Authors: E. Reyssat and L. Mahadevan

Publication: How wet paper curls, EPL, 93 (2011) 54001

Keywords: Capillary wetting, molecular diffusion

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 curling of a bimetallic thermostat due to differential expansion upon being heated. As such, the authors attempted to explain the time-dependent curvature <math>~\kappa</math> of the paper strip by using the classical theory of a bimetallic thermostat. Such a model seems plausible if the underlying wetting mechanism is 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 predictions of the bilayer model; they concluded that the fiber density in the paper strip is too high and the pore connectivity is too low for any operative capillary action. Therefore, a simple bilayer model based on capillary wetting is found to be 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 hypothesis 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 <math>~\phi (z,t)</math> along the paper thickness <math>~z</math>
<math>\displaystyle \frac{\partial \phi}{\partial t} = \frac{\partial}{\partial z} \Big( D \frac{\partial \phi}{\partial z} \Big),

</math> the authors calculated the time-dependent water content and paper 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 <math>~D</math> itself is a function of the water content <math>~\phi</math>. Finally, the authors examine the radius <math>~R_0</math> 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 steady state equilibrated out 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 a large fiber density in the paper that has switched off the rapid capillary wetting. In other words, in the absence of rapid capillary action, wetting is brought about by the slower molecular diffusion.

Reference

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