Topological Analysis and Recovery of Entanglements in Polymer Melts

Abstract

The viscous flow of polymer chains in dense melts is dominated by topological constraints whenever the single-chain contour length, N, becomes larger than the characteristic scale N_e, defining comprehensively the macroscopic rheological properties of the highly entangled polymer systems. Even though they are naturally connected to the presence of hard constraints like knots and links within the polymer chains, the difficulty of integrating the rigorous language of mathematical topology with the physics of polymer melts has limited somehow a genuine topological approach to the problem of classifying these constraints and to how they are related to the rheological entanglements. In this work, we tackle this problem by studying the occurrence of knots and links in lattice melts of randomly knotted and randomly concatenated ring polymers with various bending stiffness values. Specifically, by introducing an algorithm that shrinks the chains to their minimal shapes that do not violate topological constraints and by analyzing those in terms of suitable topological invariants, we provide a detailed characterization of the topological properties at the intrachain level (knots) and of links between pairs and triplets of distinct chains. Then, by employing the Z1 algorithm on the minimal conformations to extract the entanglement length N_e, we show that the ratio N/N_e, the number of entanglements per chain, can be remarkably well reconstructed in terms of only two-chain links.

Figure 1

Examples of ring polymer structures with Gauss linking number [GLN (see eq 3)] equal to 0. (a) Two rings intertwined in the Whitehead link 5₁². (b) Three rings clustered into the Borromean conformation 6₂³. Both conformations have been extracted from numerical simulations of ring polymer melts after the minimization procedure described in the text. To name the conformations here and in the rest of the text, we have used the classical nomenclature introduced in Rolfsen’s book (see section 2.3).

Figure 2

P_unknot (left), probability that a ring is unknotted as a function of the number of monomers, N, and for different bending stiffness values, κ_bend. The shrinking algorithm (solid lines) and Topoly (dashed lines) are in perfect agreement. ⟨L_min⟩ (right), average minimal contour length of rings with N = 640 monomers as a function of knot crossing number, K, and for different bending stiffness values, κ_bend. Each error bar corresponds to the standard deviation calculated for the ring population at the respective crossing number K. The data are described well by the simple power-law behavior ∼K^0.81 (dashed line). The generic label “>12” follows from the fact that Topoly is unable to recognize properly knots with >12 crossings.

Figure 3

⟨n_2link(|GLN|)⟩ (left), mean number of two-chain links per ring as a function of absolute Gauss linking number |GLN|. P(K_i²|GLN = 0) (right), fractional population of two-chain links K_i² (termed according to Rolfsen’s convention) having a GLN of zero. Here, as well as in the right panel of Figure 4 and Figure S4, error bars are estimated by assuming the formula for simple binomial statistics for the probability of observing a given link (knot, in Figure S4) type in the total population. Empty and filled circles represent data for alternating and non-alternating links, respectively, while vertical dotted lines separate link classes with the same number of crossings. The displayed link labels correspond to those links appearing with the highest frequency in their class of number of crossings K. The generic label “>9” follows from the fact that Topoly cannot recognize properly links with >9 crossings. In both panels, data refer to rings with N = 640 and different bending stiffness values, κ_bend.

Figure 4

⟨n_3link⟩ (left), mean number of different three-chain structures per ring. P(K_i³|irreducible) (right), fractional population of three-chain links K_i³ (termed according to Rolfsen’s convention) belonging to the poly(2)catenane+1-ring and Brunnian classes (see the text for details). These are “irreducible” with respect to the simpler compositions of two-chain links. As in Figure 3, empty and filled circles represent data for alternating and non-alternating links, respectively, while vertical dotted lines delimit link classes with the same number of crossings. Similarly, the generic label “>9” follows from the fact that Topoly cannot recognize properly links with >9 crossings. In both panels, data refer to rings with N = 640 and different bending stiffness values, κ_bend.

Figure 5

Entanglement length, N_e, as a function of the number of monomers per chain, N, for different bending stiffness values, κ_bend. Solid and dashed lines depict data after including and removing, respectively, self-entanglements (knots) through the Z1 algorithm (technical details in section 2.4). The inset shows the x and y coordinates of data with self-entanglements normalized by the corresponding asymptotic value, N_e(N = 640), of the entanglement length.

Figure 6

Number of entanglements per ring, N/N_e, as a function of the mean linking degree, ⟨LD⟩, computed (see eq 4) by taking into account the contribution from two-chain links solely (left) and after including (right) also the contribution of three-chain links.