  What are the fundamental laws for the adsorption of charged polymers onto
oppositely charged surfaces, for convex, planar, and concave geometries? This
question is at the heart of surface coating applications, various complex
formation phenomena, as well as in the context of cellular and viral
biophysics. It has been a long-standing challenge in theoretical polymer
physics; for realistic systems the quantitative understanding is however often
achievable only by computer simulations. In this study, we present the findings
of such extensive Monte-Carlo in silico experiments for polymer-surface
adsorption in confined domains. We study the inverted critical adsorption of
finite-length polyelectrolytes in three fundamental geometries: planar slit,
cylindrical pore, and spherical cavity. The scaling relations extracted from
simulations for the critical surface charge density $\sigma_c$-defining the
adsorption-desorption transition-are in excellent agreement with our analytical
calculations based on the ground-state analysis of the Edwards equation. In
particular, we confirm the magnitude and scaling of $\sigma_c$ for the concave
interfaces versus the Debye screening length $1/\kappa$ and the extent of
confinement $a$ for these three interfaces for small $\kappa a$ values. For
large $\kappa a$ the critical adsorption condition approaches the planar limit.
The transition between the two regimes takes place when the radius of surface
curvature or half of the slit thickness $a$ is of the order of $1/\kappa$. We
also rationalize how $\sigma_c(\kappa)$ gets modified for semi-flexible versus
flexible chains under external confinement. We examine the implications of the
chain length onto critical adsorption-the effect often hard to tackle
theoretically-putting an emphasis on polymers inside attractive spherical
cavities.
