  We study the free energy of the worm-like-chain model, in the
constant-extension ensemble, as a function of the stiffness for finite chains
of length L. We find that the polymer properties obtained in this ensemble are
"qualitatively" different from those obtained using constant-force ensembles.
In particular we find that as we change the stiffness parameter, the polymer
makes a transition from the flexible to the rigid phase and there is an
intermediate regime of parameter values where the free energy has three minima
and both phases are stable. This leads to interesting features in the
force-extension curves.
