Contact Angles on a Soft Solid: From Young’s Law to Neumann’s Law

The contact angle that a liquid drop makes on a soft substrate does not obey the classical Young’s relation, since the solid is deformed elastically by the action of the capillary forces. The finite elasticity of the solid also renders the contact angles differently from those predicted by Neumann’s law, which applies when the drop is floating on another liquid. Here, we derive an elastocapillary model for contact angles on a soft solid by coupling a mean-field model for the molecular interactions to elasticity. We demonstrate that the limit of a vanishing elastic modulus yields Neumann’s law or a variation thereof, depending on the force transmission in the solid surface layer. The change in contact angle from the rigid limit to the soft limit appears when the length scale defined by the ratio of surface tension to elastic modulus $\gamma /E$ reaches the range of molecular interactions.

Figure 1

Geometry near the three-phase contact line obtained by coupling elasticity to a DFT model. Contact angles continuously vary from Young’s law to Neumann’s law by reducing the stiffness of the solid. (a) Rigid solid, $\gamma /(Ea)\ll 1$. The surface is undeformed, and the liquid contact angle follows Young’s law down to molecular scale $a$. (b) Soft solid, $\gamma /(Ea)\gg 1$. Surface elasticity is negligible on the scale of molecular interactions, and the contact angles obey Neumann’s law. The solid is deformed over a distance $\sim \gamma /E$ from the contact line. (c) Very soft solid, $\gamma /(Eh)\sim 1$. The change of the contact angles saturates when $\gamma /E$ becomes comparable to the thickness of the elastic film. The solid angle measured at scale $h$ becomes identical to the microscopic solid angle at scale $a$. (d) Surface elevation profile measured by Jerison et al. [17] induced by a water drop on a silicon gel ($E=3\mathrm{kPa}$). The solid deforms into a “cusp” of solid angle ${\theta }_S=164°$.

Figure 2

Main graph: Relation between ${\theta }_L$ and ${\theta }_S$ predicted by the DFT model. The solid line is the analytic formula (2) for ${\theta }_Y=0.96$. The symbols are the angles obtained numerically for the normal force transmission model (▪) and the vectorial force transmission model (●), as defined in the text. Upper inset: Definition of ${\theta }_L$ and ${\theta }_S$. Lower inset: Forces acting on a corner of liquid (bright region, light blue). Black: Force exerted by the solid. Red: Repulsive liquid-liquid force induced by the presence of the solid. White: Attractive force exerted by the liquid due to the missing half-domain of liquid.

Figure 3

Forces acting on the corner of the solid near the contact line [indicated by the bright (light orange) region near the contact line]. (a) Normal force transmission model. Black: Force exerted by the liquid. Red: Force exerted by the solid due to pressure build induced by the liquid. White: Force exerted by the solid due to the missing half-domain of solid. (b) Vectorial force transmission model. The difference with respect to (a) is the absence of surface stress (red).

Figure 4

Transition of the contact angles ${\theta }_L$ (white) and ${\theta }_S$ (gray) upon increasing the “softness” parameter $\gamma /(Ea)$ for ${\theta }_Y=0.96$ and ${\gamma }_{SV}=\gamma$. The symbols correspond to DFT numerical solutions for $h/a=1000$ for the normal transmission force (□) and for the vectorial transmission force (○).