Why measured contact-line roughness exceeds the elastic-line model prediction
In plain words
A liquid edge creeping over a dirty surface catches on tiny defects and becomes ragged, and the theory of a springy line pinned by randomly placed defects predicts how ragged. Experiments find a rougher edge than the theory, and nobody knows what the theory leaves out.
Precise statement
Model the contact line as an elastic line with long-range elasticity (elastic $\mathrm{energy}\sim \mid k\mid\,\mid h_k\mid^2$ for a deformation $h_k$ of wavenumber $k$) in a random field of defects. Numerical simulations of this model give the roughness exponent at depinning $\zeta = 0.388 \pm 0.002$ (first-order renormalization-group estimate $1/3$), while experiments (liquid helium on cesium, water-glycerol mixtures on disordered surfaces) have reported values near 0.5. Identify what the model omits (nonlinear elasticity at large deformation, thermal activation, crossover or finite-size effects, viscous dissipation, or analysis artefacts) and show that the corrected model reproduces the measured $\zeta$ and depinning exponents.
What would settle it
An experiment on a surface with designed, characterized disorder that measures $\zeta$ over a wide range of scales, matched by simulations of the line model with the same disorder and any added nonlinear elasticity or thermal effects.