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

Abstract

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 γ s . This generalized Flamant–Cerruti problem of a single contact line is regularized by introducing a finite width 2 a for the contact line. We then explore the dependence of the substrate deformations upon the softness ratio l s / a , where l s = γ s /(2 μ ) is the elastocapillary length built upon γ s and on the elastic shear modulus μ . 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 2 R , 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.