Self-Similar Conformations and Dynamics in Entangled Melts and Solutions of Nonconcatenated Ring Polymers

Abstract

A scaling model of self-similar conformations and dynamics of nonconcatenated entangled ring polymers is developed. Topological constraints force these ring polymers into compact conformations with fractal dimension d_f = 3 that we call fractal loopy globules (FLGs). This result is based on the conjecture that the overlap parameter of subsections of rings on all length scales is the same and equal to the Kavassalis-Noolandi number O_KN ≈ 10-20. The dynamics of entangled rings is self-similar and proceeds as loops of increasing sizes are rearranged progressively at their respective diffusion times. The topological constraints associated with smaller rearranged loops affect the dynamics of larger loops through increasing the effective friction coefficient but have no influence on the entanglement tubes confining larger loops. As a result, the tube diameter defined as the average spacing between relevant topological constraints increases with time t, leading to "tube dilation". Analysis of the primitive paths in molecular dynamics simulations suggests a complete tube dilation with the tube diameter on the order of the time-dependent characteristic loop size. A characteristic loop at time t is defined as a ring section that has diffused a distance equal to its size during time t. We derive dynamic scaling exponents in terms of fractal dimensions of an entangled ring and the underlying primitive path and a parameter characterizing the extent of tube dilation. The results reproduce the predictions of different dynamic models of a single nonconcatenated entangled ring. We demonstrate that traditional generalization of single-ring models to multi-ring dynamics is not self-consistent and develop a FLG model with self-consistent multi-ring dynamics and complete tube dilation. This selfconsistent FLG model predicts that the longest relaxation time of nonconcatenated entangled ring polymers scales with their degree of polymerization N as τ_relax ~ N^7/3, while the diffusion coefficient of these rings scales as D_3d ~ N^-5/3. For the entangled solutions and melts of rings, we predict power law stress relaxation function G(t) ~ t^-3/7 at t < τ_relax without a rubbery plateau and the corresponding viscosity scaling with the degree of polymerization N as η ~ N^4/3. These theoretical predictions are in good agreement with recent computer simulations and are consistent with experiments of melts of nonconcatenated entangled rings.

Figure 1

(a) Temporary entanglements between linear chains. (b) Permanent self-knot of a ring. (c) Permanent links between rings. (d) Permanent topological constraints between nonconcatenated rings.

Figure 2

(a) Double-folded lattice animal model of a non-concatenated ring in an array of fixed obstacles. Primitive path AB of ring section of g monomers (green) is denoted by the red line. (b) Small fraction of elementary loops (cyan) move along the primitive path AB (red) leading to effective mass transport, while most of elementary loops (magenta) move along the double-folded sections (green), resulting in the change of conformations of these larger loops. The size of an elementary loop is on the order of the spacing a between fixed obstacles, while the size of the primitive path AB with contour length l is r ≈ a(l/a)^1/2 ≈ a(g/N_e)^1/4.

Figure 3

Single-ring stress relaxation modulus G_s (red line) in the DFLA model and the corresponding multi-ring stress relaxation modulus G (blue line) obtained by double reptation. Both G_s and G are normalized by the entanglement plateau modulus G_e and are shown as functions of the time t normalized by the entanglement time τ_e. Both axes are logarithmic.

Figure 4

(a) Schematic sketch of the fractal loopy globule (FLG) conformation of a ring (black line) in a melt of nonconcatenated rings. Regular circles of different colors indicate the length scales where loops of various sizes overlap with similar size neighbors at the same overlap parameter O_KN. Circles associated with other rings are shown as the dimmed background. (b) Fractal structure in the FLG model: root-mean-square size r of ring section with g Kuhn segments on logarithmic scales. (c) Snapshot of a ring in a melt of nonconcatenated rings in molecular dynamics simulations^, and the primitive paths obtained through pulling out loops consisting of less than s monomers.

Figure 5

Average size r of a primitive path segment as a function of the contour length l of the segment along (a) s-PP at indicated s and (b) t-PP at indicated t. Results are obtained based on a molecular dynamics simulation^, of M = 200 rings each with N = 1600 beads. The insets show the primitive path segment contour length l and size r normalized by their respective crossover values l_c and r_c. For both s-PP and t-PP, all normalized data points collapse onto corresponding master curves (orange lines). (c) Primitive path segment sizes r normalized by the crossover values r_c as functions of the primitive path segment contour lengths l normalized by the crossover values l_c for different s-PPs and t-PPs. (d) Crossover segment size r_c for t-dependent primitive paths (red squares) and the characteristic loop size r(g,t) (black triangles) as functions of t. The inset shows the ratio δ/r_c (blue circles) as a function of t, where δ is the root-mean-square deviation between pairs of s-PP and t-PP that have the same crossover segment size r_c.

Figure 6

(a) Stress relaxation modulus G(t) in molecular dynamics simulations^, of nonconcatenated rings with different numbers of monomers per ring (empty symbols). The function in eq 49 is used to simultaneously fit all the simulation data for different N. The least-squares fit results are shown as lines. The best-fit value of the stress relaxation exponent describing the self-similar dynamics of rings in a melt is α = 0.42 ± 0.01, and accordingly the best-fit value of the relaxation time exponent is 1/α = 2.40 ± 0.06. (b) Viscosity η of simulated rings scales with the degree of polymerization N as η ~ N^1.33±0.04.

Figure 7

(a) A characteristic loop of size r(t₁) at time t₁ and (b) a characteristic loop of size r(t₂) at time t₂ > t₁ with the corresponding topological constraints (black circles) and primitive paths (dashed red lines). Chain section within the dashed-line cyan frame in (b) corresponds to the characteristic loop in (a). The topological constraints relevant to the confining tube at t₁ but irrelevant to the one at t₂ are shown as dimmed black circles in (b). The average spacing between topological constraints increases from a(t₁) to a(t₂). In both (a) and (b), only a small fraction of entanglement strands (blue) contribute to the effective diffusion along the primitive path (dashed red line), while most entanglement strands (magenta) do not make contributions. (c) Time dependences of the size r(t) of a characteristic loop (green line) and the average spacing a(t) between topological constraints (red line) for τ_e,0 < t < τ_d. Logarithmic scales.