Capillary Pressure and Contact Line Force on a Soft Solid

The surface free energy, or surface tension, of a liquid interface gives rise to a pressure jump when the interface is curved. Here we show that a similar capillary pressure arises at the interface of soft solids. We present experimental evidence that immersion of a thin elastomeric wire into a liquid induces a substantial elastic compression due to the solid capillary pressure at the bottom. We quantitatively determine the effective surface tension from the elastic displacement field and find a value comparable to the liquid-vapor surface tension. Most importantly, these results also reveal the way the liquid pulls on the solid close to the contact line: the capillary force is not oriented along the liquid-air interface, nor perpendicularly to the solid surface, as previously hypothesized, but towards the interior of the liquid.

Figure 1

Top: Scenarios for distribution of capillary forces on an elastic wire partially immersed in a liquid [10, 11, 14, 15, 16, 17, 18, 19, 20, 21]. The liquid-on-solid force near the contact line (black arrows) could be oriented (a) normal to the wire, (b) along the liquid-vapor interface, (c) pointing into the liquid phase. Recovering the total thermodynamic force (per unit contact line), $F_{\mathrm{ext}}={\gamma }_{LV}\cos \theta$, requires a liquid-induced capillary pressure on the bottom of the wire (gray arrows) with an effective surface tension: (a) $\Gamma =-{\gamma }_{LV}\cos \theta$, (b) $\Gamma =0$, (c) $\Gamma ={\gamma }_{LV}$. Bottom: Displacement field $u(z)$ inside the wire due to immersion. Depending on the spatial distribution of capillary forces, these result into (a) a stretching, (b) no deformation, or (c) a compression of the immersed part of the wire.

Figure 2

Schematic representation of the experiment: (a) side view and (b) top view. The vertical axis is denoted $z$, with $z=0$ at the liquid free surface. The deformation is characterized by a vertical displacement field $u(z)$, determined from markers in the wire [by convention, $u(0)=0$]. The horizontal axes are noted $x$ and $y$, with $x=0$ and $y=0$ at the intersection of the camera optical axes. One camera is used to accurately calibrate the scale of the other. (c),(d) Images of the wire with markers used to determine the displacement $u$.

Figure 3

Local displacement $u(z)$ of the elastic wire compared to the situation when it is hanging from its own weight. Each point corresponds to the displacement of one polystyrene bead. Measurements above the contact line (○) and below (▵) are performed by raising the liquid level. Solid dark gray (red) lines are the best fit by Eqs. (3, 4). The calibration error bars are shown by the colored regions. The light gray (orange) line shows the (compressive) contribution of hydrostatic pressure.

Figure 4

(a) Schematic showing the origin of the tangential force component exerted by the entire liquid (blue wedge) on the solid near to the contact line, due to attraction of solid by liquid elements (gray arrow). (b) Sketch of the polymers at the free surface of the elastomer, as they are pulled towards the interior of the liquid.