Visualization of asymmetric wetting ridges on soft solids with X-ray microscopy

One of the most questionable issues in wetting is the force balance that includes the vertical component of liquid surface tension. On soft solids, the vertical component leads to a microscopic protrusion of the contact line, that is, a ‘wetting ridge’. The wetting principle determining the tip geometry of the ridge is at the heart of the issues over the past half century. Here we reveal a universal wetting principle from the ridge tips directly visualized with high spatio-temporal resolution of X-ray microscopy. We find that the cusp of the ridge is bent with an asymmetric tip, whose geometry is invariant during ridge growth or by surface softness. This singular asymmetry is deduced by linking the macroscopic and microscopic contact angles to Young and Neuman laws, respectively. Our finding shows that this dual-scale approach would be contributable to a general framework in elastowetting, and give hints to issues in cell-substrate interaction and elasto-capillary problems.

Supplementary Figure 2 | Surface profiles calculated from previous models. The models by (a, b) Shanahan et al. 10,11,20 , (c, d) Limat 12 , and (e, f) Style et al. 8 were calculated for (a, c, e) water and (b, d, f) EG 40% systems. Here, for asymmetric ridges in the water system, x‫׳‬ (the position of the contact line) for the calculated profiles were adjusted to the center of the ridges: x‫׳‬ ≈ 2.87 μm (the center of Exp. E = 3 kPa) for the profiles of E = 3, 5 kPa in (a), 5, 10 kPa in (c), and 4~5 kPa in (e); x‫׳‬ ≈ 2.36 μm (the center of Exp. E = 16 kPa) for the others. (a, b) For Shanahan's model, we used d = 200 μm, the distance from the contact line to unstrained part of the surface. The cut-off length ε was estimated for each elastic modulus E according to Eq. (5) in Ref. 10. We note that this model is invalid in the vicinity of the contact line, |x -x‫|׳‬ < ε (dashed parts). (c, d) For Limat's model, we used the average values of the reported surface energies 50,51 , γ S = (γ SV + γ SL )/2 (solid lines) or measured surface stresses, Υ S = (Υ SV + Υ SL )/2 (dashed lines), and the macroscopic length scale Δ = 200 μm. The angles between LS and SV interfaces are over 180°. All forces are given in mN m -1 .

Supplementary Table 1 | Contact angles measured from x-ray images of wetting ridges
on various soft substrates. The macroscopic () and microscopic ( S ,  V and  L ) contact angles (°) were measured using Image-Pro Plus 6.0 software. For each elasticity condition, the average (Ave) and standard deviation (SD) were calculated.
Supplementary Table 2 | Comparison of the slope for each interface at the tip calculated from the Limat's model 12 to our experimental data. θ SV (θ SL ) is the slope of the SV (SL) interface at a ridge-tip. In the calculation of a symmetric case (γ SV = γ SL ), we used the average value of surface energies, γ S = (γ SV + γ SL )/2. θ S is the microscopic angle of solid (see Fig 1e). Here, the angle difference Δθ SX = θ SX (model) -θ SX (exp.).

Supplementary Note 1 | Equilibrium at the triple point
Surface energy. The validity of Neuman law has been discussed in the immediate proximity of the contact line (w ≲ 2ε, i.e. the inelastic zone 10 or w ≲ t where t is the thickness of the liquid-vapor interface 9 ). In our systems (γ W(or EG 40%) > γ PDMS + γ W(or EG 40%)-PDMS ), the Neuman triangle condition is violated in terms of surface energies. Thus, we tried to check here other possible effects that have been proposed, as follows.
Laplace pressure. If we imagine a small drop with the Laplace pressure ΔP L = 2 LV /r, the solid surface near the contact line should undergo a typical rotation of order Δθ = ΔP L /E. In the limit of a contact angle, ΔP L ≈  LV θ/r, which leads to a typical variation of local angles Δθ/θ ≈ l e /r, where l e is the elasto-capillary length. With r ~ 1 mm and l e ~ 10 μm, as our systems, the variation Δθ/θ ~ 10 -2 is completely negligible. Although the angles of our systems are not small, the order of Δθ/θ will not be changed. In addition to this scaling argument, we can also consider the ΔP L term (= (2(1-ν 2 ) LV sinθ/πE)ΔP L ln(r/ε)) in Shanahan's model 11 . In our systems (E ~ 10 3 Pa,  LV ~ 10 -2 N m -1 , r ~ 1 mm, ε ~ 10 -6 m, and ν ~ 1/2, where ν is the Poisson's ratio of the elastic material), the ΔP L term is estimated as ≲ 10 -3 N m -1 (see Table 1), which is small enough to be ignored compared with surface energies 11 .
Disjoining pressure and direct elastic stress. White 15 suggested that the disjoining pressure and the direct elastic force in the three phase region play the role of a line tension. In his model, the disjoining pressure contribution (τ Π ~ h 0 γ LV ) and the elastic force contribution (τ E ~ γ LV y 0 sinθ Y ) affect the deviation of the apparent macroscopic contact angle (θ) from the Young angle (θ Y ) by cosθ = cosθ Y -(τ E + τ Π )/(γ LV r C ) + O((h 0 /r) 2 ) where h 0 is the vertical range of the disjoining pressure, r C is the macroscopically apparent contact radius (r C ≈ rsinθ), and y 0 is the vertical displacement of the substrate at the microscopic triple point (≈ u z (0)). The disjoining pressure contribution estimated in our systems (h 0 ≈ 0.2 nm for the van der Waals type disjoining pressure and h 0 /r ~ 10 -6 ) is very small as ~ 10 -11 N. The direct elastic contribution estimated τ E ~ 10 -7 N is larger than τ Π , but the deviation therefrom is very small as Δθ = θθ Y ~ 0.6 °, which is in the range of experimental errors. These results indicate that the disjoining pressure contribution or the direct elastic contribution is ignorable in our analysis. In fact, the apparent contact angles measured in our systems (Supplementary Table 1) correspond to Young angles, i.e. θ ≈ θ Y , regardless of E. This model also suggested that the microscopic angle of liquid be "0" in the region where z << h 0 , which is, however, beyond our scope of resolution.
Liquid on solid force. We tested the normal force transmission model and the vectorial force transmission model by Snoeijer and Andreotti 13,14 . We calculated the liquid on solid force LS , which is the basis of the two models, for water or EG 40% (see Supplementary Table 3). In both models, the force balance at tips fails because the angle between LS and the solidvapor interface is much larger than 180° in both water (~247°) and EG 40% (~217°) (see Supplementary Fig. 3). In fact, the tangential component LS (~0.031γ LV and ~0.074γ LV in water and EG 40%, respectively) is negligibly small compared with the normal component LS (~1.050γ LV and ~1.004γ LV in water and EG 40%, respectively), i.e. LS ≈ LS (see Supplementary Table 3). The inapplicability of the two models to our experimental data is presumably due to large asymmetry and/or large strain at the tips.