Topological Linking Drives Anomalous Thickening of Ring Polymers in Weak Extensional Flows

Molecular dynamics simulations confirm recent extensional flow experiments showing ring polymer melts exhibit strong extension-rate thickening of the viscosity at Weissenberg numbers $\mathrm{Wi}\ll 1$. Thickening coincides with the extreme elongation of a minority population of rings that grows with Wi. The large susceptibility of some rings to extend is due to a flow-driven formation of topological links that connect multiple rings into supramolecular chains. Links form spontaneously with a longer delay at lower Wi and are pulled tight and stabilized by the flow. Once linked, these composite objects experience larger drag forces than individual rings, driving their strong elongation. The fraction of linked rings depends nonmonotonically on Wi, increasing to a maximum when $\mathrm{Wi}\sim 1$ before rapidly decreasing when the strain rate approaches $1/{\tau }_e$.

Figure 1

(a) Elongated $N=400$ rings from the terminal state of the $\mathrm{Wi}=0.17$ flow. (b) Chains from (a) after PPA, revealing tight links connecting rings into supramolecules. (c) Close-up of a tight-link framed in (a) and (b) before and after PPA. (d) Diagram of the formation of a link and (e) the final structure once it is pulled taut.

Figure 2

Start-up extensional viscosity ${\eta }_{\mathrm{ex}}/{\eta }_{\mathrm{ex}}^N$ vs $t/{\tau }_{\mathrm{ring}}$ for a $N=400$ ring melt (upper data) and PS experiments of Huang et al. [7] (lower data) at the indicated Wi. Dashed and dot-dashed lines show linear envelopes for experiments and simulations, respectively. Both simulations and experiments are for $Z\approx 14$ and experiments are shifted down by a factor of 100 for clarity. Experiments terminate at lower Hencky strains ($ϵ\sim 5$) then simulations ($>12$). Crosses on simulated curves indicate $ϵ=5$.

Figure 3

Steady-state ${\eta }_{\mathrm{ex}}$ vs $\dot{ϵ}$ for all three ring molecular weights compared to linear melt data with $N_{\ell }\approx N/2$. $N_{\ell }$ are chosen so elongated rings and linears have similar dimensions at full extension. Unfilled symbols are points that did not reach steady state by $ϵ=12$.

Figure 4

Steady-state $P(R)$ for $N=400$ rings (solid) and $N=200$ linears (dashed) at three Wi. Relative stress contributions $P(R){\mathcal{L}}^{-1}(h)h$ for rings (dotted) are scaled to fit on the same axes.

Figure 5

(a) Fraction of topologically linked rings ${\varphi }_{\mathrm{top}}$ and fraction of the entropic stress ${\sigma }_{\mathrm{top}}/{\sigma }_{\text{ent}}$ they contribute vs Wi for $N=400$ rings. (b) Steady-state stress ${\sigma }_{\mathrm{ex}}$ vs Wi compared to ${\sigma }_{\text{ent}}$ from Eq. (1) for an $N=400$ ring melt.