Elastocapillary length (${\ell }_{\mathrm{ec}}=\sqrt{B/\gamma }$)

For a thin elastic plate interacting with a liquid interface, the elastocapillary length isHere $B$ is the plate's bending stiffness, $\gamma$ the surface tension, $E$ Young's modulus, $h$ the thickness and $\nu$ Poisson's ratio. In this plate convention $B$ has units $\mathrm{N}\mathrm{m}$, while $\gamma$ has units $N/m$; their ratio has units of length squared. The quantity is different from the gravity-dependent capillary length.

To see the balance, bend a portion of sheet of lateral size $L$ through an angle of order one. Its curvature is of order $1/L$, so its bending energy scales as $B(1/L{)}^2{L}^2\sim B$. The competing surface energy scale is $\gamma L^2$. Their dimensionless quantity is the sheet elastocapillary energy ratioLarge $\eta$ means that the capillary energy scale is large relative to the bending energy scale. This dimensional analysis does not make $L={\ell }_{\mathrm{ec}}$ a universal threshold for complete closure in capillary origami.^[ref]

For chosen illustrative values $B=10^{-9}\mathrm{N}\mathrm{m}$ and $\gamma =0.05N/m$, the elastocapillary length is $\sqrt{2\times 10^{-8}}\mathrm{m}\simeq 141\mu \mathrm{m}$. A lateral size of $0.1\mathrm{mm}$ gives $\eta =0.5$, whereas $1\mathrm{mm}$ gives $\eta =50$. These are calculations with assumed parameters, not measurements of a particular material.

Two ways of changing size give opposite trends. Keep Young's modulus, Poisson's ratio and surface tension fixed, and let $s=L/L_0$ be a lateral scale factor. At fixed thickness, $B$ stays fixed and $\eta /{\eta }_0=s^2$. If all sheet dimensions shrink in the same proportion, then $h/h_0=s$, $B/B_0=s^3$ and $\eta /{\eta }_0=s^{-1}$. Halving lateral size alone decreases the sheet elastocapillary energy ratio fourfold; halving lateral size and thickness together doubles it. The phrase "smaller sheets" therefore needs a specified thickness rule.

The length can also be calibrated directly: the 2007 study fitted an elastic-loop model to a long polydimethylsiloxane strip held by an ethanol meniscus. For their complete-wetting configuration, loop width was $0.89{\ell }_{\mathrm{ec}}$. The coefficient belongs to that model and geometry.^[ref]

The sheet elastocapillary energy ratio compares capillary and bending energy scales for an elastic sheet:Here $L$ is a lateral size, $\gamma$ is surface tension, $B$ is plate bending stiffness and ${\ell }_{\mathrm{ec}}$ is the elastocapillary length. The ratio is a dimensionless quantity: $\gamma L^2$ and $B$ both have units of energy.

The estimate assumes bending through an angle of order one over size $L$. Then curvature scales as $1/L$ and integrating the bending energy density over area $L^2$ gives a scale $B$. A comparable change in interfacial area gives a surface energy scale $\gamma L^2$. The same balance underlies the capillary origami study's definition of elastocapillary length.^[ref]

For an original calculation, take $B=2\times 10^{-8}\mathrm{N}\mathrm{m}$, $\gamma =0.05N/m$ and $L=1\mathrm{mm}$. Then $\eta =2.5$. Doubling $L$ alone raises it to 10. Halving thickness at fixed $L$, Young's modulus and Poisson's ratio divides $B$ by eight and raises $\eta$ to 20.

Changing all dimensions gives another trend. Write $s=L/L_0$ and keep material properties and surface tension fixed. At fixed thickness, $\eta /{\eta }_0=s^2$; at fixed aspect ratio $h/L$, $B/B_0=s^3$ and $\eta /{\eta }_0=s^{-1}$. Thus halving $L$ alone gives one quarter of the initial ratio, while halving both $L$ and $h$ doubles it.

A larger ratio makes capillary energy more competitive with bending, but does not establish complete closure. Actual wetting, contact angle, droplet volume, contact-line pinning and accessible deformation paths also matter. The ratio compares scales rather than computing the full change in energy along a folding path.