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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]

Charlotte Py has investigated elasto-capillarity, including how surface tension deforms an elastic sheet. She is the first author of the 2007 study introducing capillary origami: spontaneous wrapping of a liquid droplet by a flexible sheet.^[ref]

The experiment used thin polydimethylsiloxane sheets cut into different shapes. As water evaporated, surface tension progressively curved the sheet around the diminishing droplet. Square and triangular cuts followed different folding sequences, while sufficiently small or stiff sheets reopened rather than completely enclosing the liquid. This connected an accessible observation with a competition between surface energy and elastic bending.^[ref]

A complementary loop experiment measured the elastocapillary length: a wetting meniscus held a long strip in a loop whose shape could be compared with an elastic calculation. The resulting length scale helped organize the threshold for enclosure across sheet thicknesses and geometries.^[ref]

A droplet is a bounded amount of liquid surrounded by another phase, such as a gas or an immiscible liquid. Its interface carries surface energy. For a free droplet at fixed volume with uniform positive surface tension, reducing surface area lowers this contribution to the free energy; when other effects are negligible, the equilibrium shape is a sphere. This is the physical setting behind the isoperimetric inequality.

For a spherical droplet of radius $R$ and constant surface tension $\gamma$, write $F_s=4\pi \gamma R^2$ and $V=4\pi R^3/3$. Quasistatic expansion against the pressure difference givesThis is the spherical Young–Laplace equation: the pressure inside exceeds the outside pressure. An illustrative $\gamma =0.072\mathrm{N}{\mathrm{m}}^{-1}$ gives $\Delta p=144\mathrm{Pa}$ at $R=1\mathrm{mm}$, and twice that pressure difference at half the radius.

Real surface tension depends on material and temperature; the IAPWS release, for example, tabulates measured water values and supplies an interpolation formula.^[ref] Gravity, contact with a solid and motion can deform a droplet, so the spherical calculation states a regime rather than defining every possible droplet shape.

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.

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.