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

Using an exact Green function method, we calculate analytically the substrate deformations near straight contact lines on a soft, linearly elastic incompressible solid, having a uniform surface tension gamma(s). This generalized Flamant-Cerruti problem of a single contact line is regularized by introducing a finite width 2a for the contact line. We then explore the dependence of the substrate deformations upon the softness ratio l(s)/a, where l(s)=gamma(s)/(2 mu) is the elastocapillary length built upon gamma(s) and on the elastic shear modulus mu. We discuss the force transmission problem from the liquid surface tension to the bulk and surface of the solid and show that the Neuman condition of surface tension balance at the contact line is only satisfied in the asymptotic limit a/l(s) -> 0, the Young condition holding in the opposite limit. We then address the problem of two parallel contact lines separated from a distance 2R, and we recover analytically the 'double transition' upon the ratios l(s)/a and R/l(s) identified recently by Lubbers et al. (2014 J. Fluid Mech. 747, R1. (doi:10.1017/jfm.2014.152)), when one increases the substrate deformability. We also establish a simple analytic law ruling the contact angle selection upon R/l(s) in the limit a/l(s) << 1, that is the most common situation encountered in problems of wetting on soft materials.