We review some recent theoretical studies on the mean square radius of gyration for ring polymers in dilute solution at a temperature close to the θ temperature of the corresponding linear polymers. Near the θ temperature, the topological effect becomes enhanced so that the mean square radius of gyration is much larger than that of phantom ring chains. We call the phenomenon topological swelling. For ring polymers in good solvents near the θ temperature we give a phenomenological explanation for the topological swelling. For ring polymers in the θ solvent we suggest that the scaling behavior of ring polymers shows novel criticality. Here, we recall that the topological constraint plays a central role in the theoretical random models of real ring polymers.

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In this article, initiator-free polymerizations of cyclic disulfides are described. Cyclic disulfides such as 1,2-dithiane (DT) and 1,4-dihydro-2,3-benzodithine (XDS), and lipoic acid (LPA) can be polymerized easily at their melting temperatures to give high molecular weight polymers that were characterized by GPC. From the analysis of NMR and ESI-MS spectra, the polymers obtained from the polymerization of cyclic disulfides were found to have a cyclic structure. Moreover, thermal and mechanical properties, and the decomposition behaviors of the polymers demonstrate that polycatenane structures are present. From polymerization of cyclic disulfides in the presence of cyclic poly(ethylene oxide) and crown ether, a polycatenane consisting of two different cyclic polymer was obtained. The polymers showed a shape-memory effect. Furthermore, a gel was obtained by the polymerization of bifunctional monomers.

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Recent progress in topological polymer chemistry has been outlined. First, we focus on recent developments in an “electrostatic self-assembly and covalent fixation (ESA-CF)” process in conjunction with effective linking/cleaving chemistry including a metathesis process and an alkyne-azide click reaction. A variety of novel cyclic polymers having specific functional groups and unprecedented multicyclic macromolecular topologies have been realized by combining intriguing synthetic protocols. By making use of these topological polymers, unprecedented opportunities have now been realized to provide new insights on fundamental polymer properties either in solution or bulk, in static or dynamic states, or in self-assemblies. Moreover, unusual properties and functions for polymer materials have now been revealed based on their cyclic topologies, i.e., topology effects, unattainable either by linear or branched counterparts.

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Imine-bridged pseudorotaxanes can be prepared by forming imine bonds between the diformyl hydrindacene axle and diamino-substituted macrocycles. Such pseudorotaxanes can be used as intermediates for a series of interlocked molecules with multiple rotaxane substructures, since they do not dethread under neutral or basic reaction conditions. Such imine-bridged oligorotaxanes were prepared by connecting the imine-bridged pseudorotaxane using a triethylene glycol spacer followed by introducing end-caps under basic conditions. The resulting oligorotaxanes with different subunit numbers were separated by GPC.
   The imine-bridged oligorotaxanes are interconvertible with the hydrogen-bonded oligorotaxanes under acidic hydrolyzing conditions by using trifluoroacetic acid. Upon heating the hydrolyzed solution of the oligorotaxane, the imine-bridged oligorotaxane was regenerated since the bridged forms are the thermodynamically favored species.

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The second virial coefficient A₂ (A₂(θ_l)) for ring polymers in dilute solution at the theta temperature of the corresponding linear polymers, θ_l, is investigated. It is interesting that the ring polymers at θ_l show a positive value of A₂(θ_l), although the excluded volume of the polymer segments apparently disappears at θ_l. This is a consequence of the topological interaction due to a constraint that the topological state of each of the ring polymers is to be conserved. The value of A₂(θ_l) can be directly derived from the linking probability P_link, which is defined by the probability that two ring polymers are mutually entangled so that they make a nontrivial link type. Recently, we have numerically evaluated P_link precisely for random polygons (RPs) with several different values of the step number N. We have found a good approximate formula for P_link as a function of N which should be valid for arbitrary values of N. Consequently, we obtained a theoretical curve of A₂(θ_l) as a function of N through this formula. Rather recently, Takano et. al. obtained extremely pure ring polystyrenes by using liquid chromatography at the critical condition (LCCC) and measured the dependence of A₂(θ_l) on their molecular weight M_w. In this paper, we compare the theoretical curve of A₂(θ_l) vs. N calculated by P_link with the experimental data of A₂(θ_l) vs. M_w. Here we assume that M_w should be proportional to N. The qualitative behavior of the theoretical curve is consistent with the experimental data; quantitatively, however, the theoretical values are slightly larger than the experimental values. This result suggests that we should take into account some other effects also in the theory such as the three-body interaction (A₃), as well as the topological interaction.

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