Geometry and wetting of capillary folding

Capillary forces are involved in a variety of natural phenomena, ranging from droplet breakup to the physics of clouds. The forces from surface tension can also be exploited in industrial applications provided the length scales involved are small enough. Recent experimental investigations showed how to take advantage of capillarity to fold planar structures into three-dimensional configurations by selectively melting polymeric hinges joining otherwise rigid shapes. In this paper we use theoretical calculations to quantify the role of geometry and fluid wetting on the final folded state. Considering folding in two and three dimensions, studying both hydrophilic and hydrophobic situations with possible contact-angle hysteresis, and addressing the shapes to be folded to be successively infinite, finite, curved, kinked, and elastic, we are able to derive an overview of the geometrical parameter space available for capillary folding.

Figure 1

Experimental self-assembly of a 500-$\mu$m metallic dodecahedron using surface-tension-induced folding from molten hinges. Image reproduced from Ref. [14] with permission (Copyright 2009 IOP Publishing).

Figure 2

Idealized folding in two dimensions. A fluid droplet is located at the intersection of two freely hinged walls, opened by a semiangle $\alpha$ (the folding angle). The equilibrium contact angle of the droplet on the solid is denoted by ${\theta }_E$ and we assume that there is no contact-angle hysteresis (that assumption is relaxed in Sec. 3). The length of the wetted region is denoted by $R$ and the radius of curvature of the droplet by $r$.

Figure 3

Total surface energy $E$ including solid-solid, solid-liquid, and liquid-gas surfaces (arbitrary units) as a function of the folding angle $\alpha$ (radians) at constant volume for (a) the hydrophilic case (${\theta }_E=\pi /6$), in which the minimum energy is obtained for complete folding, and (b) the hydrophobic case (${\theta }_E=3\pi /4$), in which the stable folding angle is given by $\alpha ={\theta }_E-\pi /2$, which is $\pi /4$ here.

Figure 4

(a) Hydrophobic folding with ${\alpha }_{\mathrm{eq}}=-\pi /2+{\theta }_E$ (energy minimum); the expansion of the droplet and the hinge intersect at the same location. (b) Hydrophilic case with ${\alpha }_{\mathrm{eq}}=\pi /2+{\theta }_E$ (energy maximum); here also, the droplet and the two walls intersect at the same point.

Figure 5

Equilibrium folding angle ${\alpha }_{\mathrm{eq}}$ as a function of the contact angle ${\theta }_E$. In the hydrophilic case ${\theta }_E<\pi /2$, the system folds completely. In contrast, in the hydrophobic case ${\theta }_E>\pi /2$, the folding angle is finite and follows the relationship ${\alpha }_{\mathrm{eq}}=-\pi /2+{\theta }_E$.

Figure 6

Folding angle ${\alpha }_{\mathrm{eq}}$ (radians) as a function of the droplet volume $V$ (arbitrary units) for finite-size walls. Before the critical volume, the droplet does not reach the edges of the walls and the opening angle remains constant (dotted lines). The cases depicted here correspond to (a) ${\theta }_E=2\pi /3$, (b) ${\theta }_E=3\pi /4$, and (c) ${\theta }_E=5\pi /6$. After the critical volume, the contact line is pinned and the opening angle increases monotonically (solid line). The critical volume depends on the value of contact angle (or, alternatively, on the critical folding angle, as shown here).

Figure 7

The left shows shapes chosen for concave and convex walls and the right shows that for a given $\alpha$ and a known wall shape, we can calculate the corresponding angles $\nu$ and $\mu \left(\nu \right)$. Mechanical equilibrium requires that the expansion of the arc of the circle of the droplet intersects the hinge of the walls, which is equivalent to the relationship $\alpha +\nu =-\pi /2+{\theta }_E+\mu \left(\nu \right)$.

Figure 8

Equilibrium folding angles ${\alpha }_{\mathrm{eq}}$ for concave (top), flat (middle), and convex (bottom) boundaries, for a contact angle $\theta =3\pi /4$, as a function of the droplet volume $V$. The volume is given in normalized units corresponding to the two curves $y=x+x^2$ and the reciprocal function given by $y=\left(-1+\sqrt{1+4x}\right)/2$. For concave walls the folding angle $\alpha$ increases with the volume $V$, while it decreases in the convex case. Note that the two curves are not symmetric with each other.

Figure 9

If the kink angle $\varphi$ is larger than $\pi -{\theta }_E$, the contact line remains pinned at the kink and the system behaves similarly to the finite-folding case studied in Sec. 2b.

Figure 10

Equilibrium folding angle ${\alpha }_{\mathrm{eq}}$ as a function of the droplet volume $V$ when there is a kink along the walls (${\theta }_E=5\pi /6$). As the droplet volume increases, three regimes appear: (i) The folding angle ${\alpha }_{\mathrm{eq}}$ is constant and the configuration similar to the infinite-size wall. (ii) When the interface reaches the kink, the contact line remains pinned, as in the finite-size folding problem, and the folding angle increases. (iii) When the angle of contact with the second part of the wall reaches ${\theta }_E$, the triple line moves again and ${\alpha }_{\mathrm{eq}}$ decreases, possibly with a sudden jump (hysteresis).

Figure 11

In the situation with two droplets from the same liquid, the stable equilibrium configuration has one droplet satisfying the single-droplet equilibrium ${\alpha }_{\mathrm{eq}}={\theta }_E-\pi /2$, while the other one merely touches the hinged point.

Figure 12

Variation of the folding angle with the volume ratio if the additional droplet (droplet 2) is more hydrophilic than the original droplet (droplet 1).

Figure 13

Folding with two hydrophobic droplets: equilibrium folding angle as a function of the ratio of volumes. Droplet 1 has a contact angle ${\theta }_1={160}^{\circ }$ and we consider three different contact angles for droplet 2 (145${}^{\circ }$, 150${}^{\circ }$, and 155${}^{\circ }$). Without droplet 2, the equilibrium folding angle would be ${\alpha }_{\mathrm{eq}}={\theta }_1-{90}^{\circ }={70}^{\circ }$. The most hydrophilic droplet pulls the system closer towards its own equilibrium folding angle. When the folding angle reaches $\pi /2$ (open configuration), the instability reverses the folding into the situation of Fig. 12 (right) with droplet 2 inside and 1 outside.

Figure 14

Dewetting from the hinge leads to no change in surface energy with a change in the droplet position provided there is no contact-angle hysteresis. Configurations (a), (b), and (c) have the same surface energy, although the folding angles are different. In cases (a) and (b) the folding angle satisfies $\alpha <-\pi /2+{\theta }_E$, while (c) corresponds to the limit case where $\alpha ={\alpha }_{\mathrm{eq}}=-\pi /2+{\theta }_E$.

Figure 15

Total surface energy $E$ (arbitrary units) as a function of the folding angle $\alpha$ for ${\theta }_E=3\pi /4$, in two dimensions. When we allow dewetting to occur, the same minimum of surface energy is obtained for the range of folding angles $0\le \alpha \le -\pi /2+{\theta }_E$.

Figure 16

Surface energy $E$ as a function of the contact angle $\theta$ for a fixed location of the contact line. Since the volume remains constant, the folding angle $\alpha$ decreases as $\theta$ increases. The surface energy is minimal at the angle ${\theta }_m$. If ${\theta }_r\le {\theta }_m\le {\theta }_a$, $\theta$ and $\alpha$ will vary until $\theta$ reaches ${\theta }_m$. If ${\theta }_m\le {\theta }_r$, then $\theta$ will decrease (and $\alpha$ will increase) until it reaches ${\theta }_r$ and the contact line will recede. If ${\theta }_a\le {\theta }_m$, $\theta$ will increase until it reaches ${\theta }_a$ and the contact line will advance.

Figure 17

Case ${\theta }_m<{\theta }_r$. The contact angle decreases, while the opening angle increases, and $\theta$ reaches ${\theta }_r$, after which the contact line moves until $\alpha =-\pi /2+{\theta }_r$ is reached. The initial folding angle was ${\alpha }_i=\pi /18$ (10${}^{\circ }$), the initial contact angle ${\theta }_i=3\pi /4$ (135${}^{\circ }$), and ${\theta }_E=3\pi /4$ (135${}^{\circ }$). Note that the chosen values for ${\theta }_a$ and ${\theta }_r$ are arbitrary and the shape of the energy profile shown here is generic.

Figure 18

The left shows the evolution from a folded configuration (small value of $\alpha$). Surface energy $E$ is plotted as a function of the folding angle $\alpha$. The value of $\alpha$ increases with $\theta ={\theta }_r$ fixed until the configuration with a local minimum of energy is reached (solid line). If the system goes backward and the hinge closes, $\theta$ is no longer constant but the contact line is pinned and the surface energy goes back up, but along a different path (dashed lines). The right shows hysteresis for the contact angle as a function of the folding angle if we open and close successively the system (${\theta }_r=\pi /2$, ${\theta }_a=3\pi /4$).

Figure 19

The left shows the intersection between a sphere and a planar surface that leads to a constant contact angle along the intersection line. The right shows two planar surfaces intersecting a sphere and forming a corner that satisfy the contact-angle condition along both contact lines. Removing the caps leads to the actual droplet configuration.

Figure 20

Total surface energy $E$ as a function of the folding angle $\alpha$, in three dimensions, for ${\theta }_E=3\pi /4$. For $\alpha \le -\pi /2+{\theta }_E$, the energy does not depend on $\alpha$, as in two dimensions.

Figure 21

In three dimensions, and similarly to two dimensions, the energy increases with $\alpha$ in the hydrophilic case (computations with surface evolver). The system is driven towards $\alpha =0$.

Figure 22

In the hydrophilic case, for small values of (and changes in) $\alpha$ we assume that the droplet remains of a similar shape (see the text for notation).

Figure 23

Example of a final three-dimensional folded configuration visualized with surface evolver, assuming a contact angle ${\theta }_E=\pi /2$. The size of the square supports is $1\times 1$ and the volume of the droplet 0.5. The edges deform the otherwise spherical shape.

Figure 24

Folding angle ${\alpha }_{\mathrm{eq}}$ vs droplet volume $V$ for finite-size three-dimensional folding, obtained numerically using surface evolver. The behavior is similar to the two-dimensional calculations (Fig. 6). For ${\theta }_E=3\pi /4$, ${\alpha }_{\mathrm{eq}}=\pi /4$ and is independent of volume until the droplet reaches the edges of the wall. After that stage, the folding angle increases with the droplet volume. For ${\theta }_E=\pi /2$, the edges are reached for all volumes and the folding volume always increases with the droplet volume. In both cases, ${\alpha }_{\mathrm{eq}}$ slowly asymptotically approaches $\pi /2$ as $V\to \infty$.

Figure 25

Elastic sheet bent by a droplet in two dimensions. On the left the contact angle is ${\theta }_E$ and the folding angle $\alpha$. The droplet wets a total length $L$ along the rod. The right shows the specific notation used to enforce force and moment balance along the flexible sheet (see the text for details).

Figure 26

Folding angle $\alpha$ as a function of the nondimensional droplet curvature $\overline{r}$. For small $\overline{r}$ (large stiffness), $\alpha$ is close to $\pi /2$ and there is no folding. In the opposite small-stiffness limit of large $\overline{r}$, $\alpha$ tends to $\pi /2-{\theta }_E$.

Figure 27

Folding angle $\alpha$ as a function of the dimensionless droplet volume $\overline{V}=V\gamma /B$ for three values of the equilibrium contact angle ($\pi /4,\pi /2$, and $3\pi /4$). The dashed lines represent cases where $\overline{r}<0$.