= Elastocapillary length
{title2=$\ell_{\rm ec}=\sqrt{B/\gamma}$}
For a thin <elastic plate> interacting with a <liquid> interface, the \b[elastocapillary length] is
$$
\boxed{\ell_{\rm ec}=\sqrt{\frac{B}{\gamma}}},\qquad
B=\frac{Eh^3}{12(1-\nu^2)}.
$$
Here $B$ 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 $\mathrm{N\,m}$, while $\gamma$ has units $\mathrm{N/m}$; 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 $B(1/L)^2L^2\sim B$. The competing <surface energy> scale is $\gamma L^2$. Their <dimensionless quantity> is the <sheet elastocapillary energy ratio>
$$
\eta=\frac{\gamma L^2}{B}=\left(\frac{L}{\ell_{\rm ec}}\right)^2.
$$
Large $\eta$ means that the capillary <energy> scale is large relative to the bending <energy> scale. This <dimensional analysis> does not make $L=\ell_{\rm ec}$ a universal threshold for complete closure in <capillary origami>.<Capillary Origami: Spontaneous Wrapping of a Droplet with an Elastic Sheet by Charlotte Py and others>[Equation1 and its dimensional argument]{ref}
For chosen illustrative values $B=10^{-9}\,\mathrm{N\,m}$ and $\gamma=0.05\,\mathrm{N/m}$, the elastocapillary length is $\sqrt{2\times10^{-8}}\,\mathrm m\simeq141\,\mathrm{\mu m}$. 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.
\Image[/elastocapillary-scaling-paths.png]
{title=Changing lateral size at fixed thickness versus scaling all sheet dimensions: opposite trends in the capillary-to-bending energy-scale ratio}
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 $0.89\ell_{\rm ec}$. The coefficient belongs to that model and geometry.<Capillary Origami: Spontaneous Wrapping of a Droplet with an Elastic Sheet by Charlotte Py and others>[p.156103-2, Equation2 and loop-calibration paragraph]{ref}
Back to article page
