  We investigate the scaling of the interfacial adsorption of the
two-dimensional Blume-Capel model using Monte Carlo simulations. In particular,
we study the finite-size scaling behavior of the interfacial adsorption of the
pure model at both its first- and second-order transition regimes, as well as
at the vicinity of the tricritical point. Our analysis benefits from the
currently existing quite accurate estimates of the relevant (tri)critical-point
locations. In all studied cases, the numerical results verify to a level of
high accuracy the expected scenarios derived from analytic free-energy scaling
arguments. We also investigate the size dependence of the interfacial
adsorption under the presence of quenched bond randomness at the originally
first-order transition regime (disorder-induced continuous transition) and the
relevant self-averaging properties of the system. For this ex-first-order
regime, where strong transient effects are shown to be present, our findings
support the scenario of a non-divergent scaling, similar to that found in the
original second-order transition regime of the pure model.
