  A polymer chain containing $N$ monomers confined in a finite cylindrical tube
of diameter $D$ grafted at a distance $L$ from the open end of the tube may
undergo a rather abrupt transition, where part of the chain escapes from the
tube to form a "crown-like" coil outside of the tube. When this problem is
studied by Monte Carlo simulation of self-avoiding walks on the simple cubic
lattice applying a cylindrical confinement and using the standard
pruned-enriched Rosenbluth method (PERM), one obtains spurious results,
however: with increasing chain length the transition gets weaker and weaker,
due to insufficient sampling of the "escaped" states, as a detailed analysis
shows. In order to solve this problem, a new variant of a biased sequential
sampling algorithm with re-sampling is proposed, force-biased PERM: the
difficulty of sampling both phases in the region of the first order transition
with the correct weights is treated by applying a force at the free end pulling
it out of the tube. Different strengths of this force need to be used and
reweighting techniques are applied. Using rather long chains (up to N=18000)
and wide tubes (up to D=29 lattice spacings), the free energy of the chain, its
end-to-end distance, the number of "imprisoned" monomers can be estimated, as
well as the order parameter and its distribution. It is suggested that this new
algorithm should be useful for other problems involving state changes of
polymers, where the different states belong to rather disjunct "valleys" in the
phase space of the system.
