Effect of molecular architecture on ring polymer dynamics in semidilute linear polymer solutions

Abstract

Understanding the dynamics of ring polymers is a particularly challenging yet interesting problem in soft materials. Despite recent progress, a complete understanding of the nonequilibrium behavior of ring polymers has not yet been achieved. In this work, we directly observe the flow dynamics of DNA-based rings in semidilute linear polymer solutions using single molecule techniques. Our results reveal strikingly large conformational fluctuations of rings in extensional flow long after the initial transient stretching process has terminated, which is observed even at extremely low concentrations (0.025 c^*) of linear polymers in the background solution. The magnitudes and characteristic timescales of ring conformational fluctuations are determined as functions of flow strength and polymer concentration. Our results suggest that ring conformational fluctuations arise due to transient threading of linear polymers through open ring chains stretching in flow.

Fig. 1

Schematic of the experimental ring-linear system. Fluorescently labeled tracer ring DNA molecules (45 kbp, shown in red) are uniformly dissolved in a background solution of semidilute linear DNA molecules. Dynamics are studied under a equilibrium (no flow) conditions and in b planar extensional flow. The transient molecular extension of ring polymers l_circ is directly observed using SMFM

Fig. 2

Relaxation of single ring polymers in semidilute linear polymer solutions. a Single molecule relaxation trajectories and average relaxation for molecular subpopulations corresponding to (top) single-mode and (bottom) double-mode exponential relaxation trajectories for ring polymers in a background solution of 0.1 c^* linear chains. Black curves with error bars (standard deviation) denote the ensemble averaged relaxation trajectory, and the solid thin curves (color) correspond to single molecule relaxation trajectories. Molecular ensembles consist of n = 18 molecules for single-mode relaxation and n = 13 molecules for double-mode relaxation at 0.1 c^*. b Longest relaxation times (normalized to dilute solution values) for circular (red diamonds) and linear polymers (blue squares) in semidilute unentangled linear solutions as a function of scaled concentration c/c^*. Source data are provided as a Source Data file

Fig. 3

Single molecule trajectories of ring polymers show large conformational fluctuations. Transient fractional extension of ring DNA polymers in semidilute unentangled linear solutions with concentrations of 0.1 c^* at a Wi = 0.9, b Wi = 1.4, and c Wi = 2.3, and background linear chain concentration of 1.0 c^* at d Wi = 1.5, and e)Wi = 3.0. Individual single molecule trajectories are shown in gray lines and ensemble averaged trajectories are shown in a black line. A characteristic individual single molecule trajectory is highlighted in blue line. Molecular ensembles consist of n = 24, n = 25, and n = 33 molecules, respectively for Wi = 0.9, 1.4, and 2.3 at 0.1 c^*, and n = 38 for both Wi = 1.5 and Wi = 3.0 at 1.0 c^*. The dashed line indicates the time at which the step strain input is stopped. f Magnitude of ring polymer conformational fluctuations as a function of Wi, plotted as average fractional fluctuation values, such that average chain extension fluctuations 〈δ〉 are normalized by the contour length of ring polymers L_circ and linear polymers L, respectively. Data for ring polymers in dilute solution, 0.025 c^* linear semidilute solution, 0.1 c^* linear semidilute solution, and 1.0 c^* linear semidilute solution are denoted as dark cyan right triangle, dark red diamond, red up triangle, and magenta down triangle, respectively. Data for 1.0 c^* linear polymers (black circle) are taken from prior work. Error bars represent standard deviation. Source Data are provided as a Source Data file

Fig. 4

Characteristic transient stretching trajectory and single molecule snapshots. Experimental data shows large magnitude extension fluctuations for ring polymers in extensional flow. a Representative single molecule trajectory for a ring polymer in 1.0 c^* semidilute unentangled linear polymer solution at Wi = 3.0. The characteristic timescale between each peak is 2.8 s corresponding to 4.4 strain units. Source data are provided as a Source Data file. b Single molecule snapshots of the ring polymer corresponding to the trajectory in a, where the Roman numerals correspond to individual time points along the trajectory. Scale bar = 2.5 μm

Fig. 5

Probability distribution of ring polymer extension. Experimental data show ring polymers (red bars) in background solutions of semidilute linear polymers at concentrations of 0.1 c^* and 1.0 c^* and accumulated fluid strains of $ϵ$ = 10, $ϵ$ = 15, $ϵ$ = 20, $ϵ$ = 25. Molecular ensembles consist of a n = 24, b n = 25, c n = 38, and d n = 38 molecules. Data for 1.0 c^* linear polymers (gray bars) are from Hsiao et al.. Source data are provided as a Source Data file

Fig. 6

Quantitative analysis of ring polymer conformational fluctuations. a Autocorrelation of fluctuations in ring polymer extension as a function of Wi and background concentration after the initial transient phase. Molecular ensembles consist of n = 200 molecules for dilute linear chains (black solid line); n = 32 and n = 31 for Wi = 0.8 (red solid line) and Wi = 1.5 (green solid line) at 0.025 c^*; n = 24, n = 25, and n = 33 for Wi = 0.9 (blue solid line), Wi = 1.4 (cyan solid line), and Wi = 2.3 (magenta solid line) at 0.1 c^*; n = 38 for Wi = 1.5 (dark yellow solid line) and Wi = 3.0 (navy solid line) at 1.0 c^*. b Characteristic correlation time (black diamond) of ring polymer fluctuations at Wi = 1.5 as a function of background solution concentration. Error bars represent standard deviation. Molecular ensembles consist of n = 31, n = 25, n = 38 molecules, respectively from 0.025 c^* to 1.0 c^*. c Schematic of two linear polymers threading into a ring polymer in planar extensional flow. Source Data are provided as a Source Data file

Fig. 7

Steady and average unsteady fractional extension of linear and ring polymers in extensional flow. Experimental data for ring polymers in semidilute unentangled linear solutions at 0.025 c^*, 0.1 c^*, and 1.0 c^* are shown in dark red hexagon, red up triangle, and magenta down triangle, respectively. Experimental data for dilute ring polymers (dark cyan right triangle) are taken from Li et al., and data for 1.0 c^* linear polymers (blue circle) are taken from Hsiao et al.. Molecular ensembles consist of at least n ≥ 35 molecules at each concentration and error bars represent standard deviation. Source data are provided as a Source Data file