Strain-Dependent Solid Surface Stress and the Stiffness of Soft Contacts

Surface stresses have recently emerged as a key player in the mechanics of highly compliant solids. The classic theories of contact mechanics describe adhesion with a compliant substrate as a competition between surface energies driving deformation to establish contact and bulk elasticity resisting this. However, it has recently been shown that surface stresses provide an additional restoring force that can compete with and even dominate over elasticity in highly compliant materials, especially when length scales are small compared to the ratio of the surface stress to the elastic modulus, $\mathrm{\Upsilon }/E$. Here, we investigate experimentally the contribution of surface stresses to the total force of adhesion. We find that the elastic and capillary contributions to the adhesive force are of similar magnitude and that both are required to account for measured adhesive forces between rigid silica spheres and compliant, silicone gels. Notably, the strain dependence of the solid surface stress contributes to the stiffness of soft solid contacts at leading order.

Popular Summary

Sticky notes, adhesive bandages, product labels, and packing tape are all examples of everyday objects that rely on soft adhesives. Despite their ubiquitous application, much remains unknown about the physics of soft contact. Meanwhile, it is often assumed that classic theories developed for much stiffer materials can be used to interpret experiments with soft materials. We present experiments that demonstrate that soft solids stick differently than their stiffer counterparts.

Our experiment looks at the pull-off of small glass spheres from silicone gel substrates. We stick the spheres (ranging in diameter from about $16\mu \mathrm{m}$ to $\text{64\hspace{0.17em}\hspace{0.17em}}\mu \mathrm{m}$) to the silicone gel and then slowly pull them back off. By measuring force and imaging the shape of the contact zone at the same time, we find that the surface of the gel contributes significantly to its stiffness. The role of the surface also becomes more pronounced as we pull more.

These findings change the way we think about soft contacts. In the future, a complete theory of soft adhesion will need to account for both elastic mechanics and strain-dependent solid surface stresses.

Figure 1

Structure and force of soft adhesive contacts. (a) Schematic of the experiment. A force $F$ is required to hold the sphere rigidly at each displacement $D$. (b–e) Raw images of adhesive contact between a $10.0\text{−}\mu \mathrm{m}$-radius sphere and an initially flat, compliant silicone substrate with a Young modulus of $E=5.6\mathrm{kPa}$. (See Ref. [39] for movies of initial contact and quasistatic pull.) (f) Examples of force-displacement measurements for a range of sphere sizes from initial contact through detachment. Sphere radii are labeled. The shaded region indicates the part of the stable contact regime that is fit to measure the initial contact stiffness. (g,h) Initial contact force $F_0$ and initial contact stiffness $k_0$ plotted vs sphere radius $R$, respectively. Elastic contact mechanics predictions [17] are overlaid as blue dot-dashed lines. Simple capillary predictions with a fixed value of surface stress, $\mathrm{\Upsilon }\approx 0.02\mathrm{N}/\mathrm{m}$ [11], are plotted as red dashed lines.

Figure 2

Geometry of soft adhesive contact. (a) Mapped profiles from the same images shown in Figs. 1 (overlapping gray points). Predictions from elastic theory [17] are overlaid on the left (blue lines). Constant total curvature fits are overlaid on the right (red lines). (b) Size of the domain of constant curvature $\mathrm{\Delta }S$ vs sphere displacement $D$ for three example experiments with spheres of radius $10.0\mu \mathrm{m}$ (black circles), $18.8\mu \mathrm{m}$ (blue triangles), and $32.0\mu \mathrm{m}$ (red squares). (c) Total curvature $-\kappa$ vs $D$ for the same three examples as in (b). (d) Plot of fit initial length scales vs sphere radius $R$: $\mathrm{\Delta }{S}_0$ (blue triangles, mean $\pm \mathrm{std}\mathrm{dev}=5.5\pm 0.6\mu \mathrm{m}$) and $-1/{\kappa }_0$ (red squares, mean $\pm \mathrm{std}\mathrm{dev}=3.5\pm 0.4\mu \mathrm{m}$). (e) Log-log plot of fit exponential length scales $L_{\mathrm{\Delta }S}$ (blue triangles) and $L_{\kappa }$ (red squares) vs $R$. Lines of slope 1 (dot-dashed line) and slope $1/2$ (dashed line) are shown as guides to the eye.

Figure 3

Predicting adhesive forces. (a,b) Initial force $F_0$ and initial contact stiffness $k_0$, respectively, vs sphere radius $R$. Plotted are measured data (black symbols), predictions of elastic theory (blue dot-dashed lines) [17], and capillary force predictions accounting for a strain-dependent surface stress (red dashed lines). Solid gray lines show the sum of the capillary and elastic predictions. (c) Measured force vs sphere displacement for an example experiment using an $18.9\text{−}\mu \mathrm{m}$-radius sphere (black points). Theoretical predictions are overlaid as follows: elastic theory (blue dot-dashed line), capillary prediction with constant or strain-dependent surface stress (red dot-dashed line and red dashed line, respectively), and total estimated force with constant or strain-dependent surface stress (gray dot-dashed line and solid gray line, respectively).