Topological constraint in ring polymers under theta conditions studied by Monte Carlo simulation.

We studied equilibrium conformations of trivial-, 3(1)-, and 5(1)-knotted ring polymers together with a linear counterpart over the wide range of segment numbers, N, from 32 up through 2048 using a Monte Carlo simulation to obtain the dependence of the radius of gyration of these simulated polymer chains, R(g), on the number of segments, N. The polymer chains treated in this study are composed of beads and bonds placed on a face-centered-cubic lattice respecting the excluded volume. The Flory's critical exponent, ν, for a linear polymer is 1/2 at the θ-temperature, where the excluded volume is screened by the attractive force generated among polymer segments. The trajectories of linear polymers at the θ-condition were confirmed to be described by the Gaussian chain, while the ν values for trivial-, 3(1)-, and 5(1)-knots at the θ-temperature of a linear polymer are larger than that for a linear chain. This ν value increase is due to the constraint of preserving ring topology because the polymer chains dealt with in this study cannot cross themselves even though they are at the θ-condition. The expansion parameter, β, where N-dependence vanishes by the definition, for trivial-, 3(1)-, and 5(1)-knotted ring polymers is obtained at the condition of ν = 1/2. It has been found that β decreases with increasing the degree of the topological constraint in the order of trivial (0.526), 3(1) (0.422), and 5(1) knot (0.354). Since the reference β value for a random knot is 0.5, the trivial ring polymer is swollen at ν = 1/2 and the other knotted ring polymers are squeezed.