Ay. Grosberg et al., FLORY-TYPE THEORY OF A KNOTTED RING POLYMER, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics, 54(6), 1996, pp. 6618-6622
A mean field theory of the effect of knots on the statistical mechanic
s of ring polymers is presented. We introduce a topological invariant
which is related to the primitive path in the ''polymer in the lattice
of obstacles'' model and use it to estimate the entropic contribution
to the free energy of a nonphantom ring polymer. The theory predicts
that the volume of the maximally knotted ring polymer is independent o
f solvent quality and that the presence of knots suppresses both the s
welling of the ring in a good solvent and its collapse in a poor solve
nt. The probability distribution of the degree of knotting is estimate
d and it is shown that the most probable degree of knotting upon rando
m closure of the chain grows dramatically with chain compression. The
theory also predicts some unexpected phenomena such as ''knot segregat
ion'' in a swollen polymer ring, when the bulk of the ring expels all
the entanglements and swells freely, with all the knots concentrated i
n a relatively small and compact part of the polymer.