Topological and threading effects in polydisperse ring polymer solutions

Figures & data

Figure 1. (a) Probability of finding an unknotted configuration in the sample of simulated rings. The black solid line indicates the best fit $p_{\mathrm{unknot}}(N)=a\cdot {\mathrm{e}}^{-N/N_0}$ for $N\ge 500$ ($a\simeq 1.0007$). The error bars indicate the standard error over two independent simulation runs. (b) Probability of finding a specific knot configuration in the sample of simulated rings for few N.

Figure 2. (a) Scaling of the ring's mean square radius of gyration with the polymerisation degree N. The black line is the fit $R_{\mathrm{g}}^2\sim N^{2\nu }$ of the MC results (open blue circles), yielding $\nu =0.588\pm 0.001$ for $N\gtrsim {10}^3$. Right inset: normalised distributions of $X={\hat{R}}_{\mathrm{g}}^2/R_{\mathrm{g}}^2$ for different ring lengths N. Left inset: snapshot of a ring with $N={10}^4$. (b) Scaling of the mean area of the ring's minimal surface with N. The black line is the fit $⟨A⟩\sim N^x$ of the simulation results (open blue circles), yielding $x=1.25\pm 0.01$ for $N\gtrsim 200$. Right inset: normalised distributions of $X=A/⟨A⟩$ for different ring lengths N. Left inset: snapshot of a minimal surface of a ring with $N=6400$.

Figure 3. Effective interactions between monodisperse ring polymers at infinite dilution. (a) The effective potential $V_{\mathrm{eff}}(r)$ between a pair of rings of the same length N for different N. (b) The steric part of the effective potential $V_{\mathrm{ste}}(r)$ that does not account for the non-concatenation condition. (c) The topological part of the effective potential $V_{\mathrm{top}}(r)=V_{\mathrm{eff}}(r)-V_{\mathrm{ste}}(r)$. (d) The comparison between $V_{\mathrm{eff}}(r)$, $V_{\mathrm{ste}}(r)$, and $V_{\mathrm{top}}(r)$ for $N=3200$.

Figure 4. Effective interactions between polydisperse ring polymers at infinite dilution. (a) The effective potential $V_{\mathrm{eff}}(r)$ between a pair of rings of length $N_1$ and $N_2$. The same length ratios $N_2/N_1$ are shown with the same color. (b) The steric part of the effective potential $V_{\mathrm{ste}}(r)$ that does not account for the non-concatenation condition. (c) The topological part of the effective potential $V_{\mathrm{top}}(r)=V_{\mathrm{eff}}(r)-V_{\mathrm{ste}}(r)$. (d) The amplitude of the effective potential, $V_{\mathrm{eff}}(r=0)$ shown for different $N_1$ as a function of $N_2/N_1$. The dashed gray line indicates the apparent scaling $V_{\mathrm{e}\mathrm{f}\mathrm{f}}(r=0)\sim (N_2/N_1{)}^{-0.89\pm 0.01}$. The inset highlights the fact that the behaviour of the zero-separation value of the effective potential as a function of $z\equiv N_1/N_2<1$ is bracketed between the asymptotic behaviours $\sim z$ and $\sim z^{0.76}$, see the text.

Figure 5. The ratio between the topological and effective potential $V_{\mathrm{top}}(r)/V_{\mathrm{eff}}(r)$ as a function of the centre-of-mass separation between two rings r for different length ratios $N_2/N_1$.

Figure 6. Threading properties as a function of ring centre-of-mass separation r and the length ratio $N_2/N_1$. Probability of finding a threading conformation as a function of r for (a) monodisperse and (b) polydisperse ring pairs. $p_{ij}$ is the probability that the ring i threads the ring j placed at a given centre-of-mass distance r. $⟨Q_{ij}⟩$, which quantifies the depth of threading, computed on the threading ring pairs ($Q_{ij}\ne 0$, solid lines) and all ring pairs (dashed lines) as a function of r for (c) monodisperse and (d) polydisperse ring pairs. (e) Distribution of $Q_{12}=Q_{21}$ (including cases with $Q_{ij}=0$) at different fixed distances between the rings' centres of mass for the case with $N_1=200$ and $N_2=200$. Distribution of (f) $Q_{12}$ and (g) $Q_{21}$ (including cases with $Q_{ij}=0$) at different fixed distances between the rings' centres of mass for the case with $N_1=100$ and $N_2=200$.

Figure 7. Threading contributions to the effective potentials in the case of (a) $N_1=100$, $N_2=100$, (b) $N_1=200$, $N_2=200$, (c) $N_1=100$, $N_2=200$. Note that ij-threading implies that ring i threads ring j.