  We study using Monte Carlo simulations the finite-size scaling behavior of
the interfacial adsorption of the two-dimensional square-lattice $q$-states
Potts model. We consider the pure and random-bond versions of the Potts model
for $q = 3,4,5,8$ and $q = 10$, thus probing the interfacial properties at the
originally continuous, weak, and strong first-order phase transitions. For the
pure systems our results support the early scaling predictions for the size
dependence of the interfacial adsorption at both first- and second-order phase
transitions. For the disordered systems, the interfacial adsorption at the
(disordered induced) continuous transitions is discussed, applying standard
scaling arguments and invoking findings for bulk critical properties. The
self-averaging properties of the interfacial adsorption are also analyzed by
studying the infinite limit-size extrapolation of properly defined
signal-to-noise ratios.
