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Capillary origami uses surface tension to fold an initially flat elastic plate around a liquid droplet. The fold trades a reduction in surface energy for an increase in bending energy; the initial outline helps select the resulting enclosure.^[ref]

In the 2007 experiments of Charlotte Py and collaborators, 40–80 micrometre polydimethylsiloxane sheets rested on a water-repellent support. A water droplet reached the sheet's corners; time-lapse images followed its evaporation. Squares changed from four-corner bending to a tubular shape; triangles formed a tetrahedron. Smaller or thicker sheets could bend and then reopen without enclosing the droplet.^[ref]

The elastocapillary length makes the thickness dependence quantitative. For fixed Young's modulus, Poisson's ratio and surface tension, the bending stiffness obeys $B\propto h^3$ and ${\ell }_{\mathrm{ec}}\propto h^{3/2}$. Halving thickness therefore reduces ${\ell }_{\mathrm{ec}}$ by $2^{3/2}$, about $2.83$. At the same lateral size $L$, the sheet elastocapillary energy ratio $\eta =\gamma L^2/B$ increases eightfold. This calculation explains why making the sheet thinner can enable wrapping even without changing its outline.

The sheet elastocapillary energy ratio compares available energy scales; it does not determine a unique final shape. The contact angle, initial liquid volume and outline also matter. A cut chosen to permit several panels to meet is a mechanical design choice, while the surface tension supplies the driving force. Predicting the complete folding path requires more than the scalar energy comparison. The same group reported cross-shaped templates forming a cube and flower-shaped templates forming nearly spherical enclosures.^[ref]

An elastic sheet is a body whose thickness $h$ is small relative to its lateral dimensions and which responds elastically to deformation. Its geometry separates in-plane stretching from out-of-plane bending. A thin elastic plate is a useful model when the assumptions of linear elasticity and small through-thickness strain apply.

The different thickness dependences explain much of a sheet's behaviour. For uniform uniaxial loading with free lateral contraction, Hooke's law gives $\sigma =E\epsilon$. Multiplying by thickness gives the tensile force per unit width $N=Eh\epsilon$: the uniaxial sheet stretching stiffness is $Eh$. For a homogeneous, isotropic thin elastic plate, its bending stiffness iswhere $E$ is Young's modulus and $\nu$ is Poisson's ratio.^[ref]

For an original scaling example, halve $h$ while keeping material and lateral dimensions fixed. The uniaxial tensile stiffness halves, but the bending stiffness falls to one eighth. Thin sheets can therefore become much easier to bend without becoming equally easy to stretch. In simple bending to radius $R$, a layer a distance $z$ from the middle surface has strain approximately $z/R$; choosing $h=0.1\mathrm{mm}$ and $R=10\mathrm{mm}$ gives outer-layer strain about $0.005$, or $0.5\%$.

The 2007 capillary origami study made this competition observable by placing a water droplet on cut polydimethylsiloxane sheets and following their folding during evaporation. Whether a sheet enclosed the drop depended on size, thickness and outline.^[ref] The sheet elastocapillary energy ratio compares the resulting capillary and bending energy scales.

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.