# Microscopic Dynamics and Topology of Polymer Rings Immersed in a Host Matrix of Longer Linear Polymers: Results from a Detailed Molecular Dynamics Simulation Study and Comparison with Experimental Data

^{1}

^{2}

^{*}

## Abstract

**:**

_{e}of linear PEO at the same temperature but then they become practically time-independent approaching plateau values. The plateau values are different for different wavevectors; they also depend on the length of the host linear chains. Our results are supported by a geometric analysis of topological interactions, which reveals significant threading of all ring molecules by the linear chains. In most cases, each ring is simultaneously threaded by several linear chains. As a result, its dynamics at times longer than a few τ

_{e}should be completely dictated by the release of the topological restrictions imposed by these threadings (interpenetrations). Our topological analysis did not indicate any effect of the few ring probes on the statistical properties of the network of primitive paths of the host linear chains.

## 1. Introduction

_{e}of linear chains at the same temperature.

## 2. Molecular Model and Simulated Systems

_{2}–O–(CH

_{2}–CH

_{2}–O)

_{N}–CH

_{2}– while the linear PEO chains by the formula CH

_{3}–O–(CH

_{2}–CH

_{2}–O)

_{N}–CH

_{3}. Guided by the experimental work of Goossen et al. [21], the ring molecules were taken to consist of N = 456 monomers per chain corresponding to a molecular weight of 20,064 g/mole (so, for brevity, we will denote them as PEO-20k rings), while for the linear chains we considered three different chain sizes, N = 41, 228 and 456, corresponding to molecular weights equal to 1806 g/mole (the L-02k matrix), 10,034 g/mole (the L-10k matrix) and 20,066 g/mole (the L-20k matrix). The weight fraction of rings in all three different blends was kept small, equal to $\mathsf{\varphi}$ = 0.1, see Table 1. Also shown in Table 1 are the numbers of ring and linear chain molecules considered for each system in the simulation cell.

^{5}atoms) to minimize finite system size effects. The equations of motion were integrated using a single time step equal to 2 fs.

## 3. Results and Discussion

#### 3.1. Conformational Properties

**u**(s) as a function of contour length s along the chain from one of its two end points and fitting the resulting curve with an exponential function. The packing length $\tilde{l}$, on the other hand, was computed through $\tilde{l}=\frac{M}{\rho {N}_{\mathrm{Av}}\langle {R}_{\mathrm{g}}^{2}\rangle}$ where M is the molecular weight, ρ is the density, N

_{Av}the Avogadro number and $\langle {R}_{\mathrm{g}}^{2}\rangle $ the mean-square radius-of-gyration. The simulation results for ${l}_{\mathrm{p}}$ and $\tilde{l}$ are shown in Table 3. The packing length of ring molecules in their own melt is significantly larger (by almost three times) than that of linear chains, a direct manifestation of the more compact structure of rings. We also note that the packing length of the guest PEO 20k ring molecules attains its smallest value in the L-02k blend and its largest value in the L-20k blend. In addition, according to Table 3: (a) the persistence length of linear PEO chains is independent of their molecular length (at least for the chain lengths studied here); (b) the persistence length of linear PEO chains is slightly larger than that of ring molecules; and (c) the persistence length of both ring molecules and linear chains is the same (within the statistical uncertainty of the simulation results) in all systems studied here, either pure melts or blends.

**R**of linear chains by the diameter vector

**R**

_{d}of rings). Recently, Ge et al. [24] have proposed a scaling model of self-similar conformations for the size of nonconcatenated entangled ring polymers. Accounting for topological constraints through such a model forces these ring polymers into compact conformations with fractal dimension d

_{f}= 3 (also called fractal loopy globules).

#### 3.2. Normal Mode Analysis

_{B}the Boltzmann constant, T the temperature, and b

^{2}the average mean-square-distance between adjacent beads at equilibrium. The longest relaxation (Rouse) time for a ring molecule is therefore

_{p}are well above those corresponding to the pure PEO-20k ring melt. This demonstrates that dynamics of rings in these entangled blends is significantly slower than in their own melt at all length scales. Indeed, for the L-10k and L-20k blends, the slope is more or less around 2, but all points are shifted upwards.

#### 3.3. Dynamic Structure Factor

**q**, ${R}_{\mathrm{nm}}\left(t\right)$ denotes the magnitude of the displacement vector ${\mathbf{R}}_{\mathrm{nm}}\left(t\right)={R}_{\mathrm{n}}\left(t\right)-{R}_{\mathrm{m}}\left(0\right)$ between chain segments n and m, and N is the number of atomistic units along the chain. In Figure 6a–c, the MD predictions for $\frac{S(q,t)}{S(q,0)}$, for several scattering vector magnitudes q (= 0.05, 0.08, 0.1, 0.13, and 0.2 Å

^{−1}) are the solid and dashed lines, while the open symbols are the experimentally measured NSE spectra [21] for the L-02k and L-20k systems. According to Figure 6:

- (a)
- In the short L-02k blend (Figure 6a1), the computed $\frac{S(q,t)}{S(q,0)}$ curves for all q’s decay monotonically and smoothly over short and long time scales (that extend up to 600ns). The rate of decay is steeper at short times and deceases as the time increases, but in a very smooth way. As a result, only quantitative differences are observed from the corresponding $\frac{S(q,t)}{S(q,0)}$ spectra computed for the neat ring 20k PEO melt (Figure 6a2).
- (b)
- In the L-10k blend, a totally different picture emerges. Compared to the pure 20k ring or the behavior of rings in the unentangled L-02k blend, the simulation results here indicate an initial rapid decay at short times but then a rather asymptotic and time-independent behavior which leads to plateau values for $\frac{S(q,t)}{S(q,0)}$ that depend on the momentum transfer q. The time scale of the fast initial decay depends on the wavenumber q but overall is seen to be between 30 and 100 ns, i.e., on the order of the entanglement time τ
_{e}for entangled linear PEO melts at 413K, see Section 3.4 and Section 3.6 below. A closer inspection reveals that the initial fast decay is even steeper than the one recorded in the corresponding pure ring PEO-20k melt, implying more freedom for motion. According to Goossen et al. [21], at these short times, rings in the L-10k blend enjoy free 3-d Rouse motion in the tubes formed by the surrounding (moderately entangled, number of entanglements Z ≅ 5) L-10k linear chains. At later times, strong topological interactions (hindrance effects [21]) set in, which cause $\frac{S(q,t)}{S(q,0)}$ to cross over to time-independent plateau values exhibiting no sign of any further decay. - (c)
- In the longer L-20k blend, the dynamic structure factor exhibits the same qualitative behavior as in the L-10k blend. Again, we observe the rapid initial decay at very short times (on the order of 30 to 100 ns depending on the wavenumber q), followed by the time-independent behavior towards plateau values that are practically the same with those observed in the L-10k blend.
- (d)
- The qualitative agreement between predicted and experimentally measured $\frac{S(q,t)}{S(q,0)}$ spectra in the L-02k and L-20k blends is excellent. The quantitative agreement, on the other hand, is fairly good but improves for larger q values (larger than approximately 0.10 Å
^{−1}). For lower q values the simulation results systematically overpredict the measured NSE data. We believe that any quantitative differences between simulated and experimentally measured spectra should be attributed to the united-atom nature of the force-field employed in the MD simulations.

#### 3.4. Mean Square Displacement of Atomistic Segments

#### 3.5. Mean-Square-Displacement of Chains Centers-of-Mass

^{1}), which allows us to compute the corresponding diffusion coefficient D of 20k PEO rings in this blend. The result is 0.50 ± 0.04 Å

^{2}/ns, in excellent agreement with the experimentally measured value of 0.49 Å

^{2}/ns by Goossen et al. [21]. Linear chains, on the other hand, in the three blends exhibit dynamics that is identical to that in their own neat melt.

_{e}is accelerated compared to the pure melt, whereas in linear PEO matrices with molecular weight M well above M

_{e}it is significantly slowed down.

#### 3.6. Topological Analysis

_{t}of the host linear chains in the three blends. The tube diameter is a very difficult quantity to calculate and, in the literature, is typically taken to be equal to the corresponding Kuhn step length of the PP. However, Stephanou et al. [63] have proposed a method that leads directly to the calculation of the tube diameter d

_{t}by monitoring the mean-square displacement (msd) of the innermost chain segments $\varphi (t)=\langle {\left({r}_{\mathrm{n}}(t)-{r}_{\mathrm{n}}(0)\right)}^{2}\rangle $ vs. time t and observing the 1st break signaling the onset of tube constraints on segmental dynamics. The tube diameter d

_{t}is then estimated as ${d}_{\mathrm{t}}=2\sqrt{\varphi \left({t}^{*}\right)}$ where $\varphi ({t}^{*})$ is the value of $\varphi (t)$ at the time $t={t}^{*}$ where the slope of $\varphi (t)$ starts to change as the msd leaves the initial t

^{1/2}regime to enter the next t

^{1/4}regime. The method was applied by Stephanou et al. [63] to melts of entangled linear polyethylene and cis-1,4-polybutadiene melts with remarkable success. It was also found that the usual approach to take the tube radius equal to the Kuhn step length of the PP overestimates d

_{t}by 15%–30%.

_{t}with the Stephanou et al. [63] method is shown in Figure 13; it refers to the pure 20k linear PEO melt. We see that as the linear chain segments “feel” the tube constraints, a break in their displacement appears; the resulting value of the tube diameter then is d

_{t}= 40 ± 5 Å. Given that the segmental msd’s of 20k linear chains are identical in the L-20k blend and in their own melt (see Figure 7), we understand that the same value of d

_{t}should characterize the L-20k blend as well. For the 10k linear chains, the corresponding prediction for the tube diameter (both in the L-10k blend and in their melt) is d

_{t}= 33 ± 3 Å. Our estimate (d

_{t}= 40 ± 5 Å) for the tube diameter in the L-20k blend is in excellent agreement with the value reported by Goossen et al. [21] (d

_{t}= 42 ± 1 Å) through a description of tube constraints on their measured $\frac{S(q,t)}{S(q,0)}$ at times beyond τ

_{e}with a Gaussian distribution. Equally excellent is the agreement between MD data and NSE measurements as far as the order of magnitude of the entanglement time τ

_{e}(~20 ns) is concerned.

## 4. Conclusions

_{e}of linear PEO at the same temperature) but then become practically time-independent approaching constant asymptotic values that are different for different wave-vectors and depend strongly on the molecular weight of the host linear chains.

_{e}should be completely dictated by ring–linear disentanglement events, which explains the asymptotic values exhibited by the dynamics structure factor spectra at times longer than approximately a few τ

_{e}. In striking contrast, our topological analysis did not indicate any appreciable effect of the few rings on the average size or the statistics of the network of primitive paths of the host linear chains. A direct calculation of the tube diameter gave a value which was seen to coincide with the value extracted indirectly from the measured $\frac{S(q,t)}{S(q,0)}$ spectra [21] by describing tube constraints with a Gaussian distribution of obstacles. MD simulation predictions and NSE measurements agree also on the value of the entanglement time τ

_{e}for linear PEO (~20 ns at 413 K).

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Dependence of: (

**a**) $\langle {R}_{\mathrm{g},\mathrm{R}}^{2}\rangle $, (

**b**) $\langle {R}_{\mathrm{g},\mathrm{L}}^{2}\rangle $, (

**c**) $\langle {R}_{\mathrm{d}}^{2}\rangle $ and (

**d**) $\langle {R}_{\mathrm{ee}}^{2}\rangle $ on the chain length N

_{L}(L stands for linear) of the host linear chains. Results are also shown for the same quantities in the corresponding pure melts.

**Figure 2.**MD simulation predictions (symbols) for the probability distribution function of: (

**a**) the magnitude of the end-to-end distance vector for the pure linear PEO-02k and PEO-10k melts; and (

**b**) the magnitude of the diameter vector for the ring PEO-20k molecules in their own melt and in the L-20k blend. For comparison (lines), we also show the distributions according to the analytical expression, Equation (1), using the values of $\langle {R}^{2}\rangle $ and $\langle {R}_{\mathrm{d}}^{2}\rangle $ computed from the MD simulations.

**Figure 3.**The squared amplitudes of the Rouse normal modes $\langle {\mathbf{X}}_{\mathrm{p}}{(0)}^{2}\rangle $ for the ring molecules in the three blends and in a pure ring PEO-20k melt as a function of $\raisebox{1ex}{$N$}\!\left/ \!\raisebox{-1ex}{${p}^{2}$}\right.$ in a log–log plot. The dashed line has been drawn with a slope of 1 as a guide for the eye and corresponds to the Rouse scaling.

**Figure 4.**Log-linear plots of the normalized time autocorrelation functions of several Rouse modes for the PEO-20k ring molecules in: (

**a**) the L-02k blend, (

**b**) the L-10k blend, (

**c**) the L-20k blend, and (

**d**) the pure ring PEO-20k melt.

**Figure 5.**The scaling of the characteristic times ${\mathit{\tau}}_{\mathrm{p}}$ describing the relaxation of the Rouse normal modes p with chain length N. The dashed line indicates the Rouse model, namely, ${\mathit{\tau}}_{\mathrm{p}}~{\left(\raisebox{1ex}{$N$}\!\left/ \!\raisebox{-1ex}{$p$}\right.\right)}^{2}$, and has been drawn as a guide for the eye.

**Figure 6.**Computed (continuous or dashed lines) and experimentally measured (diamonds) $\frac{S(q,t)}{S(q,0)}$ plots for several wavenumbers q, for: (

**a**) ring molecules in the L-02k blend together with data from Reference [21]; (

_{1}**a**) ring molecules in the L-02k blend and in the pure PEO-20k ring melt (dashed lines); (

_{2}**b**) ring molecules in the L-10k blend; and (

**c**) ring molecules in the L-20k blend together with data from Reference [21].

**Figure 7.**Log–log plot of the segmental mean-square displacement of ring PEO-20k molecules with time t in the three blends and in their own melt.

**Figure 8.**Log–log plot of the mean-square displacement of atomistic segments of linear chains with time t in: (

**a**) the L-02k blend and the pure linear PEO-02k melt, (

**b**) the L-10k blend and the pure linear PEO-10k melt, and (

**c**) the L-20k blend and the pure linear PEO-20k.

**Figure 9.**Log–log plot of the mean-square displacement of the centers-of-mass of ring molecules (

**a**) and linear chains (

**b**) with time t for all simulated systems.

**Figure 11.**(

**a**) Percentage of linear chains that are involved in threading events with ring molecules in the three blends; and (

**b**) percentage of linear chains involved in multiple threading events.

**Figure 12.**Examples of multiple threading. The snapshots have been taken from the combined geometric/topological analysis of: (

**a**) the L-02k blend, and (

**b**) the L-20k blend. (

**a**) The blue ring molecule is simultaneously threaded by fifteen linear chains. (

**b**) The blue ring chain is simultaneously threaded by ten linear chains (here, for simplicity, only the PP of the red linear chain is shown in full; for all other linear chains, only the part of their PP whose strands are involved in the threading is shown).

**Figure 13.**Calculation of the tube diameter d

_{t}based on the segmental mean-square displacement of the innermost chain segments $\varphi (t)=\langle {\left({r}_{\mathrm{n}}(t)-{r}_{\mathrm{n}}(0)\right)}^{2}\rangle $ versus time t. The tube diameter is estimated as ${d}_{\mathrm{t}}=2\sqrt{\varphi \left({t}^{*}\right)}$ where $\varphi ({t}^{*})$ is computed at time $t={t}^{*}$ where the first break is observed as segments leave the initial t

^{1/2}regime to enter the next t

^{1/4}regime.

**Table 1.**Simulated poly(ethylene oxide) (PEO) blend systems and number of ring and linear PEO molecules in each one of them.

System | Host matrix | Number of ring PEO-20k molecules | Number of linear PEO chains | Volume fraction of ring molecules |
---|---|---|---|---|

1 | L-02k | 8 | 720 | 0.1 |

2 | L-10k | 8 | 144 | 0.1 |

3 | L-20k | 8 | 72 | 0.1 |

**Table 2.**MD predictions for $\langle {R}_{\mathrm{g},\mathrm{R}}^{2}\rangle $, $\langle {R}_{\mathrm{g},\mathrm{L}}^{2}\rangle $ , $\langle {R}_{\mathrm{d}}^{2}\rangle $ and $\langle {R}_{\mathrm{ee}}^{2}\rangle $ in all simulated PEO systems (pure and blends).

System | Ring molecules | Linear chains | |||
---|---|---|---|---|---|

$\langle {\mathit{R}}_{\mathbf{g},\mathbf{R}}^{\mathbf{2}}\rangle $ (Å^{2}) | $\langle {\mathit{R}}_{\mathbf{d}}^{\mathbf{2}}\rangle $ (Å^{2}) | $\langle {\mathit{R}}_{\mathbf{g},\mathbf{L}}^{\mathbf{2}}\rangle $ (Å^{2}) | $\langle {\mathit{R}}_{\mathbf{ee}}^{\mathbf{2}}\rangle $ (Å^{2}) | $\langle {\mathit{R}}_{\mathbf{ee}}^{\mathbf{2}}\rangle /\langle {\mathit{R}}_{\mathbf{g},\mathbf{L}}^{\mathbf{2}}\rangle $ | |

02k pure linear melt | - | - | 235 ± 55 | 1475 ± 140 | 6.3 ± 0.7 |

10k pure linear melt | - | - | 1410 ± 180 | 8420 ± 310 | 6.0 ± 0.8 |

20k pure linear melt | - | - | 2660 ± 320 | 17,015 ± 930 | 6.4 ± 1.0 |

L-02k blend | 1150 ± 120 | 3040 ± 260 | 240 ± 25 | 1525 ± 105 | 6.3 ± 1.4 |

L-10k blend | 885 ± 105 | 2320 ± 240 | 1380 ± 85 | 8160 ± 220 | 5.9 ± 08 |

L-20k blend | 815 ± 110 | 2055 ± 260 | 2975 ± 185 | 16,925 ± 500 | 5.7 ± 0.8 |

20k pure ring melt | 825 ± 63 | 2250 ± 120 | - | - | - |

**Table 3.**MD predictions for the packing length $\tilde{l}$ and the persistence length ${l}_{\mathrm{p}}$ of all simulated PEO systems (pure and blends).

System | $\tilde{\mathit{l}}$ of ring molecules (Å) | ${\mathit{l}}_{\mathbf{p}}$ of ring molecules (Å) | $\tilde{\mathit{l}}$ of linear chains (Å) | ${\mathit{l}}_{\mathbf{p}}$ of linear chains (Å) |
---|---|---|---|---|

02k pure linear melt | 12 ± 3 | 6.7 ± 0.4 | ||

10k pure linear melt | 12 ± 3 | 6.7 ± 0.4 | ||

20k pure linear melt | 11 ± 3 | 6.7 ± 0.4 | ||

20k pure ring melt | 35 ± 3 | 6.1 ± 0.4 | ||

L-02k blend | 28 ± 3 | 6.1 ± 0.4 | 13 ± 3 | 6.7 ± 0.4 |

L-10k blend | 31 ± 3 | 6.1 ± 0.4 | 12 ± 3 | 6.7 ± 0.4 |

L-20k blend | 38 ± 4 | 6.1 ± 0.4 | 12 ± 3 | 6.7 ± 0.4 |

© 2016 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license ( http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Papadopoulos, G.D.; Tsalikis, D.G.; Mavrantzas, V.G.
Microscopic Dynamics and Topology of Polymer Rings Immersed in a Host Matrix of Longer Linear Polymers: Results from a Detailed Molecular Dynamics Simulation Study and Comparison with Experimental Data. *Polymers* **2016**, *8*, 283.
https://doi.org/10.3390/polym8080283

**AMA Style**

Papadopoulos GD, Tsalikis DG, Mavrantzas VG.
Microscopic Dynamics and Topology of Polymer Rings Immersed in a Host Matrix of Longer Linear Polymers: Results from a Detailed Molecular Dynamics Simulation Study and Comparison with Experimental Data. *Polymers*. 2016; 8(8):283.
https://doi.org/10.3390/polym8080283

**Chicago/Turabian Style**

Papadopoulos, George D., Dimitrios G. Tsalikis, and Vlasis G. Mavrantzas.
2016. "Microscopic Dynamics and Topology of Polymer Rings Immersed in a Host Matrix of Longer Linear Polymers: Results from a Detailed Molecular Dynamics Simulation Study and Comparison with Experimental Data" *Polymers* 8, no. 8: 283.
https://doi.org/10.3390/polym8080283