  We study the effect of polydispersity on the structure of polymer brushes by
analytical theory, a numerical self-consistent field approach, and Monte Carlo
simulations. The polydispersity is represented by the Schulz-Zimm chain-length
distribution. We specifically focus on three different polydispersities
representing sharp, moderate and extremely wide chain length distributions and
derive explicit analytical expressions for the chain end distributions in these
brushes. The results are in very good agreement with numerical data obtained
with self-consistent field calculations and Monte Carlo simulations. With
increasing polydispersity, the brush density profile changes from convex to
concave, and for given average chain length $N_n$ and grafting density
$\sigma$, the brush height $H$ is found to scale as
$(H/H_{\mathrm{mono}}-1)\propto(N_w/N_n-1)^{1/2}$ over a wide range of
polydispersity indices $N_w/N_n$ (here $H_{\mathrm{mono}}$ is the height of the
corresponding monodisperse brush. Chain end fluctuations are found to be
strongly suppressed already at very small polydispersity. Based on this
observation, we introduce the concept of the brush as a near-critical system
with two parameters (scaling variables), $(N_{n}\sigma^{2/3})^{-1}$ and
$(N_w/N_n-1)^{1/2}$, controlling the distance from the critical point. This
approach provides a good description of the simulation data. Finally we study
the hydrodynamic penetration length $l_\mathrm{p}$ for brush-coated surfaces in
flow. We find that it generally increases with polydispersity. The scaling
behavior crosses over from $l_\mathrm{p}\sim N_\mathrm{n}^{1/2}\sigma^{-1/6}$
for monodisperse and weakly polydisperse brushes to $l_\mathrm{p}\sim
N_{n}^{2/3}$ for strongly polydisperse brushes.
