Paradox of Contact Angle Selection on Stretched Soft Solids

The interfacial mechanics of soft elastic networks plays a central role in biological and technological contexts. Yet, effects of solid capillarity have remained controversial, primarily due to the strain-dependent surface energy. Here we derive the equations that govern the selection of contact angles of liquid drops on elastic surfaces from variational principles. It is found that the substrate’s elasticity imposes a nontrivial condition that relates pinning, hysteresis, and contact line mobility to the so-called Shuttleworth effect. We experimentally validate our theory for droplets on a silicone gel, revealing an enhanced contact line mobility when stretching the substrate.

Figure 1

Contact angles on a soft solid: The liquid angle ${\theta }_L$ of the drop (main panel) and the solid angle ${\theta }_S$ of the wetting ridge (zoom). In the presented theory, the three interfaces ($i=1$, 2, 3) are parametrized using curvilinear coordinates $s$, related to the reference coordinate $S$ by the strain ${\epsilon }_i(S)$, while the local angle of the interface is ${\theta }_i(\mathrm{S})$. The interfaces meet at the contact line position ${\overrightarrow{x}}_{\text{cl}}$. The effect of surface tensions ${\mathrm{\Upsilon }}_{\mathrm{SV}}$ and ${\mathrm{\Upsilon }}_{\mathrm{SL}}$ is illustrated by a force balance on the circular zone near the contact line on the right [Eqs. (7), (8)].

Figure 2

Equilibrium requires that the energy is stationary with respect to any possible mode of contact line displacement, as sketched in panels (a) and (b). (a) Displacement mode at fixed material point ($\delta R=0$), where the molecules from each phase joining at the contact line are moved together, gives rise to the Neumann condition (5). (b) Displacement mode with variable material point ($\delta R\ne 0$) gives rise to a no-pinning condition (6). Color indicates whether material points prior to displacement belonged to the wet part (green) or dry part (red) of the solid. At equilibrium, both Eqs. (5) and (6) must be satisfied.

Figure 3

Pinning and enhanced contact line mobility. (a) Liquid-vapor macroscopic contact angle ${\theta }_L$ with respect to the undeformed solid surface as a function of the contact line velocity $v$ (liquid: fluorosilicone oil). Closed and open symbols: unstrained and strained (${\epsilon }_{\infty }=0.93$) samples respectively. (b) Contact angle rotations ${\theta }_L-{\theta }_{a/r}$ vary like $v^n$, with $n$ obtained from loss modulus measurement as expected from (10) ($n=0.55$ for ${\epsilon }_{\infty }=0$ and $n=0.50$ for ${\epsilon }_{\infty }=0.93$). (c) Liquid contact angles ${\theta }_{a/r}$ in the limit $v\to 0^{+}/0^{-}$, as a function of the applied strain ${\epsilon }_{\infty }$. (d) Contact line friction factor $\alpha$ defined by Eq. (10) as a function of strain ${\epsilon }_{\infty }$.