  Scaling predictions and results from self-consistent field calculations for
bottle-brush polymers with a rigid backbone and flexible side chains under good
solvent conditions are summarized and their validity and applicability is
assessed by a comparison with Monte Carlo simulations of a simple lattice
model. It is shown that under typical conditions, as they are also present in
experiments, only a rather weak stretching of the side chains is realized, and
then the scaling predictions based on the extension of the Daoud-Cotton blob
picture are not applicable.
  Also two-component bottle brush polymers are considered, where two types
(A,B) of side chains are grafted, assuming that monomers of different kind
repel each other. In this case, variable solvent quality is allowed for, such
that for poor solvent conditions rather dense cylinder-like structures result.
Theories predict ``Janus Cylinder''-type phase separation along the backbone in
this case. The Monte Carlo simulations, using the pruned-enriched Rosenbluth
method (PERM) then are restricted to rather short side chain length.
Nevertheless, evidence is obtained that the phase separation between an A-rich
part of the cylindrical molecule and a B-rich part can only occur locally. The
correlation length of this microphase separation can be controlled by the
solvent quality. This lack of a phase transition is interpreted by an analogy
with models for ferromagnets in one space dimension.
