Elasto-capillarity

Elastocapillarity is the coupling between surface tension and elastic deformation. Changing an interface can reduce surface energy while bending a compliant solid increases bending energy. Thin structures can deform strongly when the available capillary energy competes with their elastic resistance.

For an isotropic thin elastic plate, its bending stiffness iswhere $E$ is Young's modulus, $h$ the thickness and $\nu$ Poisson's ratio. A sheet bent over a size $L$ has curvature of order $1/L$. Its bending energy scales as $B(1/L^2)L^2\sim B$, while the capillary energy scale is $\gamma L^2$, where $\gamma$ is surface tension. Balancing these scales gives the elastocapillary lengthHere $B$ has units of energy and $\gamma$ energy per area, so the ratio has units of length squared. At fixed material properties, ${\ell }_{\mathrm{ec}}\propto h^{3/2}$; doubling thickness multiplies this length by $2\sqrt{2}$. Geometry and contact angle enter the more detailed wrapping criterion.^[ref]

The 2007 capillary origami experiments of Charlotte Py and collaborators provide a direct demonstration. They cut polydimethylsiloxane membranes, 40–80 micrometres thick, into planar shapes, placed them on a water-repellent support, added a water droplet and recorded the deformation during evaporation. Suitable triangular sheets folded around the shrinking drop into a tetrahedral enclosure. Smaller or stiffer sheets failed to close completely. The observations connected a controlled material thickness and initial shape to the competition between surface energy and bending energy.^[ref]