Elastocapillary deformations on partially-wetting substrates: rival contact-line models

Abstract

A partially-wetting liquid can deform the underlying elastic substrate upon which it rests. This situation requires the development of theoretical models to describe the wetting forces imparted by the drop onto the solid substrate, particularly those at the contact-line. We construct a general solution using a displacement potential function for the elastic deformations within a finite elastic substrate associated with these wetting forces, and compare the results for several different contact-line models. Our work incorporates internal contributions to the surface stress from both liquid/solid Σls and Σsg solid/gas solid surface tensions (surface stress), which results in a non-standard boundary-value problem that we solve using a dual integral equation. We compare our results to relevant experiments and conclude that the generalization of solid surface tension Σls ≠ Σsg is an essential feature in any model of partial-wetting. The comparisons also allow us to systematically eliminate some proposed contact-line models.

Elastocapillarity: Surface Tension and the Mechanics of Soft Solids

“…In other words, Equation (7) reduces to Neumann's vector balance, Eq. (6), which requires the three interfaces to meet with fixed orientations (Figure 5b) [14,16,67,[72][73][74]. Note that this analysis ignores any long-range forces between the interfaces [75,76].…”

Surface tension and the mechanics of liquid inclusions in compliant solids

“…This has been suggested both by theoretical models of nanoscale inclusions [12][13][14], and by recent experiments which have shown that surface tension (isotropic, strain-independent surface stress) can also significantly affect soft solids at micron and even millimetric scales. For example, solid capillarity limits the resolution of lithographic features [15][16][17][18], drives pearling and creasing instabilities [19][20][21][22], causes the Young-Dupré relation to break down for sessile droplets [23][24][25][26][27][28], and leads to a failure of the JohnsonKendall-Roberts theory of adhesion [29][30][31][32][33]. Of particular relevance are our recent experiments embedding droplets in soft solids, where we found that Eshelby's predictions could not describe the response of inclusions below a critical, micron-scale elastocapillary length [34].…”

Contact lines on soft solids with uniform surface tension: analytical solutions and double transition for increasing deformability

“…However, the precise structure of this ridge is still under debate [19], and, for instance, it is only recently that the surface tension of these materials has been included in modelling [4,5,7] in a way that could allow some direct comparisons with the more well-known case of liquid-liquid wetting. The difficulties of reaching a full theory are still numerous: how to build a reasonably simple formalism combining two different substrate surface tensions (for the wet and dry part of the surface [20]), finite deformations, substrate rheology and more generally dynamical effects?…”