  By Molecular Dynamics simulation of a coarse-grained bead-spring type model
for a cylindrical molecular brush with a backbone chain of $N_b$ effective
monomers to which with grafting density $\sigma$ side chains with $N$ effective
monomers are tethered, several characteristic length scales are studied for
variable solvent quality. Side chain lengths are in the range $5 \le N \le 40$,
backbone chain lengths are in the range $50 \le N_b \le 200$, and we perform a
comparison to results for the bond fluctuation model on the simple cubic
lattice (for which much longer chains are accessible, $N_b \le 1027$, and which
corresponds to an athermal, very good, solvent). We obtain linear dimensions of
side chains and the backbone chain and discuss their $N$-dependence in terms of
power laws and the associated effective exponents. We show that even at the
Theta point the side chains are considerably stretched, their linear dimension
depending on the solvent quality only weakly. Effective persistence lengths are
extracted both from the orientational correlations and from the backbone
end-to-end distance; it is shown that different measures of the persistence
length (which would all agree for Gaussian chains) are not mutually consistent
with each other, and depend distinctly both on $N_b$ and the solvent quality. A
brief discussion of pertinent experiments is given.
