Singular Nature of the Elastocapillary Ridge

The mechanical and chemical properties of soft solids are crucial to many applications in biology and surface science. Recent studies use wetting by liquid drops to probe the surface mechanics of reticulated polymer networks, leading to controversial interpretations. This controversy relates to the long-standing paradox of Young’s law for the liquid contact angle, which invokes only a horizontal force balance. Recent work shows that, for very soft materials, the solid’s surface tension plays a key role for the vertical force balance, involving a singular ridgelike deformation exactly at the point where the droplet pulls on the network. A hotly debated question is whether unexpected measurements on this singular deformation can be attributed to nonlinear bulk elasticity or whether these provide evidence for an intricate surface elasticity, known as the Shuttleworth effect. Here, we theoretically reveal the nature of the elastocapillary singularity on a hyperelastic substrate with various constitutive relations for the interfacial energy. First, we finely resolve the vicinity of the singularity using goal-adaptive finite-element simulations. This simulation confirms that bulk elasticity cannot affect the force balance at the contact line. Subsequently, we derive exact solutions of nonlinear elasticity that describe the singularity analytically. These solutions are in perfect agreement with numerics and show that both the angles and stretch at the contact line, as previously measured experimentally, consistently point to a strong Shuttleworth effect. Finally, using Noether’s theorem, we reveal the quantitative link between Young’s law, hysteresis, and the nature of the elastocapillary singularity. Our contribution closes the issue of the missing normal force at the contact line and opens up the development of modern techniques in polymer surface science.

Popular Summary

In interfacial science, one infers the surface properties of a solid by measuring the contact angle between a liquid droplet and the substrate. Recently, researchers have used this technique of “wetting” to measure the surface tension of polymeric solids, which are exceedingly soft and squishy. However, the interpretation of these results has remained controversial.

Here, we address the issues underlying this controversy and show that these experiments point to intricate surface elasticity in polymeric solids. Until now, interpretation of these “soft wetting” experiments has been hindered by two issues. First, the droplet pulls on the soft substrate, forming a ridge along the droplet’s edge—one wonders if such a deformation would change the angles from which the surface properties are inferred. Second, it is debated whether or not the solid surface tension is modified by the amount of deformation, signifying the presence or absence of surface elasticity. This inherent coupling between surface and bulk properties of soft solids makes the explanation of experimental observations notoriously difficult.

We resolve these pertinent issues by analyzing the wetting ridge due to capillary forces within the droplet with unprecedented detail. Our numerical simulations in combination with theoretical modeling establish that bulk elasticity cannot affect the selection of contact angles and that existing experimental results are, in fact, a consequence of surface elasticity.

Moving beyond experiments, we reveal the connection between the singular nature of the wetting ridge and contact angle hysteresis in soft substrates, which opens up new avenues for designer polymeric surfaces.

Figure 1

Symmetric wetting ridges under large deformation induced by a localized contact line force, with and without the Shuttleworth effect. (a) Typical numerical solution using a finite-element method, where successive magnifications show the adaptive resolution of the elastocapillary ridge. The example is a case without the Shuttleworth effect, with equal liquid and solid surface energies $\gamma$ (giving a solid angle ${\theta }_S=120°$). The scales are expressed in the corresponding elastocapillary length $\gamma /G$. (b) The solid angle ${\theta }_S$ versus the ratio of liquid-vapor surface tension ${\gamma }_{\mathrm{LV}}$ and solid surface tension ${\mathrm{\Upsilon }}_S$. Symbols are numerical results with the Shuttleworth effect (open symbols, ${\mathrm{\Upsilon }}_S$ measured at the contact line) and without the Shuttleworth effect (closed symbols). We vary both ${\gamma }_{\mathrm{LV}}$ (circles) and the amount of prestretch of the substrate from ${\lambda }_{\infty }=1$ to 2 (squares). The solid line corresponds to Neumann’s law (15), with ${\mathrm{\Upsilon }}_S$ based on its value at the contact line.

Figure 2

Geometry of the elastocapillary ridge upon stretching the substrate. (a) Solid angle ${\theta }_S$ as a function of the stretch at the contact line ${\lambda }_{\mathrm{cl}}$. Open circles correspond to FEM in the presence of a strong Shuttleworth effect ($c_0=c_1=1$, with ${\gamma }_0={\gamma }_{\mathrm{LV}}$). The solid line is the analytical prediction by Neumann’s law (15). The closed circles (several measurements superimposed) correspond to FEM without a Shuttleworth effect ($c_0=c_1=0$, with ${\gamma }_0={\gamma }_{\mathrm{LV}}$). (b) Relation between the stretch at the contact line ${\lambda }_{\mathrm{cl}}$ and the globally imposed stretch ${\lambda }_{\infty }$. In the presence of a strong Shuttleworth effect, the two stretches takes on very similar values (the red dashed line, ${\lambda }_{\mathrm{cl}}={\lambda }_{\infty }$, is a guide to the eye). Without a Shuttleworth effect, ${\lambda }_{\mathrm{cl}}=\sqrt{\pi /{\theta }_S}$ takes on a constant value (dash-dotted line).

Figure 3

Elastic stress along the free surface near the ridge singularity (symmetric ridges). (a) Pressure $p$ versus distance to the contact line $X$, scaled as indicated on the axes. Data correspond to the situation without a Shuttleworth ($c_{0,1}=0$) with different ${\theta }_S$ obtained by varying the ratio ${\gamma }_{\mathrm{LV}}/{\gamma }_0$. (b) Pressure $p$ versus distance to the contact line $x$, scaled as indicated on the axes. Data correspond to the situation with a Shuttleworth ($c_{0,1}=1$) with different amounts of prestretch ${\lambda }_{\infty }$. (c) Shear stress ${\sigma }_{nt}$ versus distance to the contact line $x$, scaled as indicated on the axes. Data correspond to the same cases as in (b).

Figure 4

Marangoni stress ${\sigma }_{nt}/G$ at the contact line, as a function of the stretch at the contact line ${\lambda }_{\mathrm{cl}}$. The Marangoni stress can be positive or negative depending on whether the stretch at the tip is larger or smaller than ${\lambda }_{\infty }$. Open circles are obtained by FEM, the solid line from similarity solutions described in Sec. 4 in combination with Neumann’s law. The grids represent geometry of the ridge and deformation within it for negative (A) and positive (B) Marangoni stresses, as obtained from the similarity solutions. The gray lines denote the undeformed grid, and the arrows indicate the direction of the Marangoni stress.

Figure 5

Similarity solutions for symmetric corners obtained from Eqs. (24) and (29), all with ${\theta }_S=120°$. (a) Without the Shuttleworth effect, the shear stress vanishes at the interface, and one recovers the Singh-Pipkin solution [49]. (b),(c) The Shuttleworth effect induces Marangoni stresses, giving positive (b) or negative (c) elastic shear stress at the interface, the direction indicated by the arrows.

Figure 6

Determining the liquid contact angle ${\theta }_L$ upon global displacement of the solution. (a) Lagrangian point of view: On the domain of material coordinates, the shift is achieved by a change of material point $\delta {\mathbf{X}}_{\mathrm{cl}}=\delta U\mathbf{T}$, where the contact line applies. Without pinning, the displacement is energetically neutral, while, in the presence of a pinning defect, an energy $-\mathrm{\Gamma }\delta U$ is dissipated at the contact line. (b) Eulerian point of view: The displacement $\delta U$ leads to a variation of the entire solution as given in Eq. (34). At a large distance from the contact line, the change of the surface energies reads $({\gamma }_{\mathrm{SL}}-{\gamma }_{\mathrm{SV}}){\lambda }_{\infty }\delta U$.

Figure 7

(a) The strength of the surface defect, quantified by the discontinuity of chemical potential $[\mu {]}_{-}^{+}$, plotted versus the liquid contact angle ${\theta }_L$. Solid lines are from the theoretical prediction (39), open symbols from FEM with the Shuttleworth effect ($c_0=c_1=1$). Blue data: Symmetric surface energies ${\gamma }_{0,\mathrm{SV}}={\gamma }_{0,\mathrm{SL}}={\gamma }_{\mathrm{LV}}$, so that the equilibrium angle ${\theta }_Y=90°$. Red data: Asymmetric surface energies ${\gamma }_{0,\mathrm{SV}}=4/5{\gamma }_{\mathrm{LV}}$ and ${\gamma }_{0,\mathrm{SL}}=6/5{\gamma }_{\mathrm{LV}}$, so that based on ${\lambda }_{\infty }=1$ we find $\cos {\theta }_Y=-2/5$. The numerics confirm that $[\mu {]}_{-}^{+}$ provides the pinning force; when pulling at ${\theta }_L={\theta }_Y$, there is no pinning and $[\mu {]}_{-}^{+}=0$. (b) The grid plot represents the asymmetric ridge for symmetric surface energies, resulting from a contact angle ${\theta }_L>{\theta }_Y$. This ridge corresponds to the data point marked by an arrow in the main panel.

Figure 8

Typical example of the approach of the Neumann angle: the interface angle $\phi$ with respect to the horizontal as a function of the scaled distance to the contact line $rG/\gamma$. Light gray symbols correspond to FEM without a Shuttleworth effect ($c_0=c_1=0$, with ${\gamma }_0={\gamma }_{\mathrm{LV}}$); large symbols represent a typical range accessible to experiments. The red dashed line gives the asymptotics (a1), while the solid black line (a2) gives an improved fit at the scale $r\sim \gamma /G$. The latter is fitted over the range typically accessible in experiment (large symbols, appearing as a thick gray line).