Chain conformations of ring polymers under theta conditions studied by Monte Carlo simulation.

We studied equilibrium conformations of trivial-, 31-knot, and 51-knot ring polymers with finite chain length at their θ-conditions using a Monte Carlo simulation. The polymer chains treated in this study were composed of beads and bonds on a face-centered-cubic lattice respecting the excluded volume. The Flory's critical exponent ν in Rg ~ N(ν) relationship was obtained from the dependence of the radius of gyration, Rg, on the segment number of polymers, N. In this study, the temperatures at which ν equal 1∕2 are defined as θ-temperatures of several ring molecules. The θ-temperatures for trivial-, 31-knot, and 51-knot ring polymers are lower than that for a linear polymer in N ≤ 4096, where their topologies are fixed by their excluded volumes. The radial distribution functions of the segments in each molecule are obtained at their θ-temperatures. The functions of linear- and trivial-ring polymers have been found to be expressed by those of Gaussian and closed-Gaussian chains, respectively. At the θ-conditions, the excluded volumes of chains and the topological-constraints of trivial-ring polymers can be apparently screened by the attractive force between segments, and the values for trivial ring polymers are larger than the half of those for linear polymers. In the finite N region the topological-constraints of 31- and 51-knot rings are stronger than that of trivial-ring, and trajectories of the knotted ring polymers cannot be described as a closed Gaussian even though they are under θ-conditions.