= Sheet elastocapillary energy ratio
{title2=$\eta=\gamma L^2/B$}
The \b[sheet elastocapillary energy ratio] compares capillary and bending <energy> scales for an <elastic sheet>:
$$
\boxed{\eta=\frac{\gamma L^2}{B}=\left(\frac{L}{\ell_{\rm ec}}\right)^2,\qquad
\ell_{\rm ec}=\sqrt{\frac{B}{\gamma}}.}
$$
Here $L$ is a lateral size, $\gamma$ is <surface tension>, $B$ is plate <bending stiffness> and $\ell_{\rm ec}$ 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>.<Capillary Origami: Spontaneous Wrapping of a Droplet with an Elastic Sheet by Charlotte Py and others>[p.156103-2, bending-energy density and Equation1]{ref}
For an original calculation, take $B=2\times10^{-8}\,\mathrm{N\,m}$, $\gamma=0.05\,\mathrm{N/m}$ 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.
\Image[/elastocapillary-scaling-paths.png]
{title=Two size-scaling paths for the <sheet elastocapillary energy ratio>}
{id=sheet-energy-ratio-scaling-paths}
{description=Reuses the original normalized scaling plot: fixed thickness gives $s^2$, while geometric similarity gives $s^{-1}$. The curves are dimensional estimates, not measured wrapping thresholds.}
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.
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