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 <Rg(2)> 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.