Figure 1

Free energy profiles as a function of ${\phi }_R$ for the melt with short rings $N/g^{*}={\left(0.4\right)}^2$ (left panel) and long rings $N/g^{*}=10$ (right panel).Reuse & Permissions

Figure 2

Master curve of the ring size $R/{\xi }^{*}$ as a function of the chain length $N/g^{*}$ (double logarithmic scale) obtained by numerically minimizing the free energy Eq. (5).Reuse & Permissions

Figure 3

Topological mesh size ${\xi }_{♮}/{\xi }^{*}$ as a function of the chain length $N/g^{*}$. The left panel is a semilog plot showing a logarithmic increase toward ${\xi }_{♮}\to {\xi }^{*}$, while the right panel is a double logarithmic plot showing an effective power law ${\xi }_{♮}\sim N^{1/5}$ in the intermediate range of the chain length ($N_1<N<N_2$).Reuse & Permissions

Figure 4

Dependence of the coordination number $N_R$ on the chain length $N$. The solid line is the theoretical prediction, while points are results from numerical simulations (read off from Fig. 8 in Ref. [11] and Fig. 13 in Ref. [12]). Circle: Monte Carlo simulation by Vettorel et al. [$N_e\simeq 175,N_R\left(N_2\right)\simeq 15$] [11]. Square: Molecular dynamic simulation by Halverson et al. [$N_e\simeq 28,N_R\left(N_2\right)\simeq 17$] [12].Reuse & Permissions

Figure 5

Schematics illustrating the spatial structures in ring polymer melts. Top: almost ideal ring regime ($N<N_1$). Second line: onset of the topological constraints ($N=N_1$). Third line: crossover regime ($N_1<N<N_2$). Bottom: onset of the compact statistics ($N=N_2$). The right column shows a reference ring (shaded) and its neighbors, which are represented by soft spheres. During the crossover, the coordination number $N_R$, thus, the degree of the overlap measured by ${\phi }_R$, increases with $N$ as shown in the right column. The middle column represents the global structure focusing on how the size and the correlation of topologically active clusters grow with $N$, where the density of the background color reflects the value of ${\phi }_R$. The left column shows the internal structure (meshes represented by dotted circles) of individual rings, and the finer intermesh structure with (a part of) the reference ring (thick line) and a part of other neighboring ring (thin line) penetrating the reference ring region.Reuse & Permissions

Figure 6

Dependence of intrinsic topological length scale $N_1$ on the stiffness parameter $p$. Curves are drawn according to Eq. (35), where numerical coefficients are appropriately adjusted so that the two curves coincide at $p=p^{*}\simeq 2$. Also plotted is the entanglement length $N_e$ in linear chain melts determined by primitive path analysis (points and error bars: from Table I in Ref. [14]).Reuse & Permissions