Solid Drops: Large Capillary Deformations of Immersed Elastic Rods

Under the effect of surface tension, a blob of liquid adopts a spherical shape when immersed in another fluid. We demonstrate experimentally that soft, centimeter-size elastic solids can exhibit a similar behavior: when immersed into a liquid, a gel having a low elastic modulus undergoes large, reversible deformations. We analyze three fundamental types of deformations of a slender elastic solid driven by surface stress, depending on the shape of its cross section: a circular elastic cylinder shortens in the longitudinal direction and stretches transversally; the sharp edges of a square based prism get rounded off as its cross sections tend to become circular; and a slender, triangular based prism bends. These experimental results are compared to analysis and nonlinear simulations of neo-Hookean solids deformed by surface tension and are found to be in good agreement with each other.

Figure 1

Shortening of an initially circular cylinder made of agar gel when immersed in toluene. The axial stretch $\lambda \le 1$ of a circular elastic cylinder is plotted for different values for the shear modulus $\mu$ and of the initial radius ${\rho }_0$ (in the range 0.45 to 2 mm), as a function of the dimensionless surface energy $\overline{\gamma }$. It is compared to the analytical formula in Eq. (2) (solid curve). Inset: Experimental pictures for $\mu =200\mathrm{Pa}$ and ${\rho }_0=0.8\mathrm{mm}$. The agar cylinder is formed inside a translucent mold (top); after dissolution of the mold, it lays on a hydrophobic grid and shortens as the result of surface stress (bottom). To aid visualization, the boundaries of the agar cylinder are highlighted using a dashed brown overlay.

Figure 2

Rounding of the edges of an initially square based prism made of polyacrylamid gel, immersed in silicone oil. (a) The amount of rounding is measured based on the surface area $\Delta \mathcal{A}$ defined in the text (yellow region). (b) Experimental shape of the cross section (solid red curve) for a shear modulus $\mu =35\mathrm{Pa}$, surface stress $\gamma =42.6\mathrm{mN}/\mathrm{m}$, and initial edge length $a_0=5\mathrm{mm}$, and comparison to the simulation ($\overline{\gamma }=0.26$, light blue dots). (c) The dimensionless measure of rounding $\Delta \mathcal{A}/a^2$ in the experiments is plotted as a function of the dimensionless surface stress $\overline{\gamma }$ for various initial cross-sectional widths $a_0$ and shear moduli $\mu$, and compared to simulations (solid line).

Figure 3

Bending of a triangular prism by uniform surface stress. (a) Initial geometry. Note that the centroid $H$ of the triangular cross section is distinct from the centroid $G$ of its boundary, called the Spieker center. As a result, the uniform surface stress induces a nonzero bending moment. (b) Dimensionless curvature $\overline{\kappa }$ of a triangular prism with apex angle $\theta =20°$ and aspect ratio $L/h=5.55$, as a function of the dimensionless surface stress $\overline{\gamma }$: nonlinear 3D finite element simulation for a neo-Hookean incompressible material and comparison to the linear beam theory from Eq. (3) for $L\gg h$ and $\overline{\gamma }\ll 1$. (c) 3D shapes of the deformed prism for selected values of $\overline{\gamma }$. The color code shows the pressure contribution $p$ to the stress enforcing incompressibility. (d) Experimental evidence of bending of agar gels immersed in silicone oil ($h=4.5\mathrm{mm}$, $L=2.5\mathrm{cm}$, $\theta =20°$, $\overline{\gamma }=0.0172$) and comparison to the corresponding finite-element simulation labeled (B).