# Shear Rheology of Unentangled and Marginally Entangled Ring Polymer Melts from Large-Scale Nonequilibrium Molecular Dynamics Simulations

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{0.4}followed by the terminal relaxation zone. The latter is faster than in linear counterparts and obeys the expected exponential decay with time [2], which cannot be fully captured by theoretical models available today. This terminal relaxation appears to be characterized by a slow mode linked either with linear impurities that are not accounted for in theoretical models [1,3] or, as suggested by atomistic simulations, by ring–ring threading events [4,5]. Although these observations are supported by experiments [6,7] and simulations with coarse-grained models [8,9], further studies are definitely needed to fully elucidate and quantify the microscopic mechanisms responsible for the slow terminal relaxation exhibited by polymer rings.

_{e}, the power law dependence of η

_{0}on M

_{w}for rings becomes stronger. For example, the viscosity data of Doi et al. [6] suggest that for entangled ring melts $b=2.4\pm 0.1$. This is in contrast to the prediction of the lattice animal model according to which $b=3/2$ [1] or to the recent fractal loopy globule model for which $b=1.33$ [11]. On the other hand, several recent molecular dynamics (MD) simulations based either on coarse-grained [8] or atomistic [12] models report values for the exponent b that lie in the region of 1.4–1.7. Further studies are thus needed to also elucidate this issue.

^{+}measurements of Yan et al. [13], rings exhibit similar qualitative behavior with linear chains. In particular, the η

^{+}data collapse at low times to define what we call the linear viscoelastic envelope, followed by the characteristic overshoot maxima before steady-state values are reached. However, the maxima for the rings are smaller than for the linears. Moreover, the well-known undershoot succeeding the stress overshoot at strong shear rates typically observed for entangled linear melts is not observed for rings. Overall, and despite recent advances in purification techniques, the fundamental understanding of the linear and non-linear shear rheology of rings remains still a challenging issue [1,13].

_{w}< M

_{e}) up to fully entangled (${M}_{\mathrm{w}}\ge {10M}_{\mathrm{e}}$), which is currently missing. As a first step to this direction, we report here results from a detailed atomistic NEMD simulation study addressing well-characterized poly(ethylene oxide) (PEO) melts in this crossover regime. We chose to work with PEO because: (a) there exist in the literature several experimental data sets for its dynamics and rheology with which we can directly compare simulation findings and validate the accuracy of the obtained results, and (b) recent equilibrium MD simulations with the force field of Fischer et al. [20,21] yielded results for its equilibrium conformation and dynamics in excellent agreement with state-of-the-art experimental measurements [19]. The NEMD simulation results reported here have been obtained with the same force field and, as we will see, this will be reflected both in the quality of the simulation predictions and in their comparison with available experimental data. To compare with the corresponding rheology of linear melts, we also carried out NEMD simulations for the corresponding linear PEO melts. In all cases, the NEMD simulations were executed with very large simulation cells (containing in some cases up to two million interacting units) to ensure complete absence of system size effects and to add to the accuracy and reliability of the predicted rheological and conformational behavior.

## 2. Systems Studied and Simulation Details

_{2}−O−(CH

_{2}−CH

_{2}−O)

_{N}−CH

_{2}− for rings and CH

_{3}−O−(CH

_{2}−CH

_{2}−O)

_{N}−CH

_{3}for linear chains, where N denotes the number of monomers per molecule (or, equivalently, the degree of polymerization). For both types of melts, we have considered three different chain sizes characterized by N = 29, 40 and 120. The entanglement molecular weight M

_{e}of linear PEO is M

_{e}= 2020 g/mol, which corresponds to N

_{e}= 46. We understand then, that the linear systems with N = 29 and N = 40 are unentangled while the system with N = 120 is marginally entangled characterized by approximately Z = 2.5 entanglements per chain. The corresponding molecular weights (they differ slightly between ring and linear chains) are 1322, 1846 and 5326 g/mol. In the following, we will refer to the three systems as PEO-1k, PEO-2k and PEO-5k, respectively. The ring melts will be further denoted as R-1k, R-2k and R-5k, while the linear ones as L-1k, L-2k and L-5k.

_{x}of the simulation cell exceeding 170 nm in all cases (see Table 1). For example, for the longest ring melt addressed here (R-5k), the mean radius of gyration ${R}_{\mathrm{g}}^{\mathrm{eq}}\left(={\langle {R}_{\mathrm{g},\mathrm{eq}}^{2}\rangle}^{\frac{1}{2}}\right)$ at equilibrium is ${R}_{\mathrm{g}}^{\mathrm{eq}}$ = 18 ± 1 Å while the magnitude of the maximum diameter vector (${R}_{\mathrm{d}}^{\mathrm{max}}$) corresponding to a fully extended ring conformation is ${R}_{\mathrm{d}}^{\mathrm{max}}$ ≈ 220 Å. This ${R}_{\mathrm{d}}^{\mathrm{max}}$ is estimated under the approximation that it is equal to one half the fully extended end-to-end length of the linear analogue, ${R}_{\mathrm{d}}^{\mathrm{max}}=0.5{R}_{\mathrm{ee}}^{\mathrm{max}}$. The dimensions of the corresponding NEMD simulation cell were chosen to be equal to (207 nm) × (11 nm) × (11 nm) in the x, y and z directions, respectively. This indicates that the simulation cell was ~9.4 times larger than ${R}_{\mathrm{d}}^{\mathrm{max}}$ in the direction of flow and ~6.0 times larger than ${R}_{\mathrm{g}}^{\mathrm{eq}}$ in the other two directions, implying that finite system size effects are kept to a minimum.

**p**

_{ia},

**q**

_{ia}and

**F**

_{ia}are the momentum, position and force vectors of atom a in molecule i, of mass m

_{ia}. In the above equations, n denotes the total number of atoms, T the absolute temperature, k

_{B}the Boltzmann constant, and D the space dimensionality (in our case, D = 3). Also, ζ and p

_{ζ}are the coordinate- and momentum-like variables of the Nosé–Hoover thermostat, ${Q=Dnk}_{B}{T\tau}^{2}$ denotes the mass parameter of the thermostat, and $\nabla \mathbf{u}$ represents the velocity gradient tensor. According to Equations (1)–(4), flow in our work is imposed at the level of the microscopic equations of motion through the relevant velocity gradient tensor. In the case of shear flow, $\nabla \mathbf{u}$ has the form:

## 3. Results

#### 3.1. Rheological Properties

_{w}ring PEO melts simulated, we observe the typical characteristic behavior of the shear viscosity with shear rate already known for linear polymer melts: (a) At low shear rates (${Wi}_{\mathrm{C}}\le 1$), the viscosity is practically constant defining what we know as the Newtonian plateau. (b) At higher shear rates, the viscosity starts decreasing, exhibiting what we know from the corresponding behavior of linear polymers as shear thinning. (c) At even higher shear rates, we enter the highly nonlinear regime where the viscosity drops rapidly with applied shear rate. It is interesting that for the PEO-1k system, ring and linear melts in the Newtonian regime exhibit similar viscosity values. On the other hand, for the PEO-2k and PEO-5k systems, the viscosity of the linear melt in the Newtonian regime is larger than that of the corresponding ring, which is in full agreement with experimental measurements [1,2,6,13,33] and previous simulation studies [8,12,14]. Overall, it appears that differences in the viscosity between ring and linear PEO melts in the Newtonian regime become more important as the M

_{w}increases.

_{0}we fit the simulation data to the Carreau Model [34,35]:

_{0}, along with their standard deviation, are listed in Table 2. Very rough estimates of η

_{0}were also obtained from the longest relaxation times for linears and rings ${\tau}_{1,\mathrm{L}}$ (and ${\tau}_{2,\mathrm{R}}$, respectively) as computed directly from the equilibrium MD simulations using the corresponding Rouse equations, namely ${\eta}_{0,\mathrm{L}}=\left(\frac{{\pi}^{2}\rho RT}{12}\right){\tau}_{1,\mathrm{L}}$ and ${\eta}_{0,\mathrm{R}}=\left(\frac{{\pi}^{2}\rho RT}{6}\right){\tau}_{2,\mathrm{R}}$; they are reported in the fourth and fifth column of Table 2. In the sixth and seventh column of Table 2, we report available experimental data for the zero-shear rate viscosity of PEO [12,19].

_{0}

_{,}

_{L}proved very efficient allowing us to reliably estimate the ratio ${\eta}_{0,\mathrm{L}}/{\eta}_{0,\mathrm{R}}$ of the zero-shear rate viscosity of the linear melt to the corresponding zero shear rate viscosity of the ring. As already mentioned, for the shortest melt examined (PEO-1k), the NEMD prediction for the viscosity ratio between linear and ring melt is ${\eta}_{0,\mathrm{L}}/{\eta}_{0,\mathrm{R}}=1.0\pm 0.1$. This observation agrees nicely with the experimental measurements of Nam et al. [36] for short ring and linear PEO melts at a lower temperature (T = 329 K) than in our NEMD simulations (T = 363 K), where the ratio ${\eta}_{0,\mathrm{L}}/{\eta}_{0,\mathrm{R}}$ was equal to 1 for melts with M

_{w}≈ 1500 g/mol. We remind the reader that according to the Rouse model the ratio ${\eta}_{0,\mathrm{L}}/{\eta}_{0,\mathrm{R}}$ is equal to 2 [10]. The reason for the deviation of the NEMD prediction from the Rouse model should be due to the excess free volume phenomena present in the linear melts due to chain ends, which accelerate chain dynamics and, in turn, cause the viscosity of the linear melt to decrease. Of course, as the chain length increases, these excess free volume phenomena become less and less important, thus we expect the ratio of the two viscosities to come closer to the value of 2.0 predicted by the Rouse theory. Indeed, for the PEO-2k melt, the corresponding ratio is ${\eta}_{0,\mathrm{L}}/{\eta}_{0,\mathrm{R}}=2.1\pm 0.1$, a result which is fully consistent with the Rouse model [10]. On the other hand, by further increasing M

_{w}, entanglements start developing between chains in the linear melt which restrict their dynamics; thus, now, we expect the viscosity of the linear melt to increase faster than the viscosity of the ring. Indeed, according to our NEMD simulations, for PEO-5k (a marginally entangled melt), ${\eta}_{0,\mathrm{L}}/{\eta}_{0,\mathrm{R}}=2.6\pm 0.1$. The fact that a change in the relaxation mechanism takes place as we cross over from PEO-2k to PEO-5k that cannot be accommodated by the Rouse model is also reflected in the unrealistically large value of the ratio ${\eta}_{0,\mathrm{L}}/{\eta}_{0,\mathrm{R}}$ (=8.5 ± 1), see data in the fourth and fifth column in Table 2 for PEO-5k, predicted by naïve application of the Rouse model equations on the basis of the computed chain orientational relaxation times for the ring and linear melt from the equilibrium MD data. Overall, we can say that, according to our NEMD simulations, the ratio ${\eta}_{0,\mathrm{L}}/{\eta}_{0,\mathrm{R}}$ for very short PEO melts (well below M

_{e}) starts from a value close to 1 and increases smoothly with M

_{w}, reflecting the faster increase of the viscosity of the corresponding linear melt as inter-chain entanglements start playing a role. Our NEMD data for η

_{0}are also in very favorable agreement with experimentally measured data from Refs. [19,33].

_{1}is positive and Ψ

_{2}negative (this explains why in Figure 3, we plot −Ψ

_{2}), and that the magnitude of Ψ

_{2}is much smaller than the magnitude of Ψ

_{1}. For weak-to-moderate shear rates (${Wi}_{\mathrm{C}}<100$), ring melts exhibit smaller Ψ

_{1}and −Ψ

_{2}values compared to linear melts. For higher shear rates, both Ψ

_{1}and −Ψ

_{2}exhibit large power law regions, similar to those for $\eta \left(\dot{\gamma}\right)$, decreasing by several orders of magnitude with shear rate. We also observe that the rate of decline of Ψ

_{1}and −Ψ

_{2}with $\dot{\gamma}$ is greater in the linear melts than in the rings. To quantify this difference, we fit the simulation data for Ψ

_{1}and −Ψ

_{2}with $\dot{\gamma}$ for ${Wi}_{\mathrm{C}}<100$ with a power law of the form ${\mathsf{\Psi}}_{1}~{\dot{\gamma}}^{-{b}_{1}}$ and ${\mathsf{\Psi}}_{2}~{\dot{\gamma}}^{-{b}_{2}}$. For PEO-1k the two exponents are similar: b

_{1}= 1.38 ± 0.05 and b

_{2}= 1.21 ± 0.05 for the ring, and b

_{1}= 1.38 ± 0.04 and b

_{2}= 1.41 ± 0.05 for the linear melt. With increasing chain size, the differences in the power law exponents between ring and linear melts increase. Thus, for PEO-2k, b

_{1}= 1.48 ± 0.04 and b

_{2}= 1.44 ± 0.09 for the ring melt, which should be compared to b

_{1}= 1.61 ± 0.03 and b

_{2}= 1.63 ± 0.03 for the linear melt. For the marginally entangled PEO-5k, b

_{1}= 1.55 ± 0.06 and b

_{2}= 1.53 ± 0.03 for the ring melt, while b

_{1}= 1.76 ± 0.05 and b

_{2}= 1.75 ± 0.04 for the linear melt. Our simulation findings are in reasonable agreement with the results of [14] for marginally entangled (Z = 6) linear and ring PE melts. In particular, the Ψ

_{1}and Ψ

_{2}exponents reported in [14] are b

_{1}= 1.45 ± 0.03 and b

_{2}= 1.49 ± 0.11 for ring PE, which increased to b

_{1}= 1.61 ± 0.03 and b

_{2}= 1.67 ± 0.12 for linear PE. The finite extensibility of rings, due to their loopy geometry and more compact arrangement of monomers around their center of mass, should be considered as the main factor responsible for the weaker dependence of Ψ

_{1}and Ψ

_{2}on shear rate compared to linear melts.

#### 3.2. Conformational Properties

**G**is defined as

_{xx}, G

_{yy}, G

_{zz}and G

_{xy}) between linear and ring melts. We focus our attention first on the xx component. At low shear rates (${Wi}_{\mathrm{C}}<1$), the magnitude of G

_{xx}remains practically unaffected by the flow for both types of melts. At intermediate shear rates, chains in the melt deform and at the same time align in the direction of the flow, which causes a considerable increase in the value of G

_{xx}. At even higher shear rates, the rate of increase of G

_{xx}with shear rate declines and G

_{xx}approaches constant values which are, however, different between ring and linear melts. The fact that G

_{xx}assumes eventually constant values is due to two factors: (a) the finite extensibility of the simulated chains due to bond stretching and bond-bending interactions, and (b) chain rotation and tumbling due to the nature of the applied flow (shear) [32]. An interesting point to notice in the curves of Figure 6 is that for all three PEO melts studied, the ratio of the asymptotic G

_{xx}values between linear and ring melts is very similar and approximately equal to 2, which seems to suggest that, as far as their fully extended conformations are concerned, rings can be considered as linear chains of half the length.

_{yy}and G

_{zz}display exactly the opposite behavior. At low shear rates, their values remain unaffected by the imposed flow field; however, as the strength of the flow increases, both decrease considerably due to chain alignment in the direction of flow. It is also true that G

_{yy}decreases more rapidly than G

_{zz}, which should have been expected given that z is the neutral axis. As already mentioned, due to the nature of the applied flow, the average melt velocity varies along the y-axis, and this can cause chain rotation and tumbling (see below), which is another reason why G

_{yy}decreases faster than G

_{zz}at higher shear rates. As with many other (rheological and conformational) properties discussed so far, the values of G

_{yy}and G

_{zz}for rings are smaller than for linear melts, which is another manifestation of the more compact structure of cyclic molecules due to the more symmetric arrangement of atoms around their center-of-mass.

_{xy}initially increases with applied shear rate, goes through a maximum at an intermediate value of shear rate, and then starts decreasing. It is known [32] that the overall change of G

_{xy}with respect to applied shear rate can be understood by considering two competing effects: (a) more open molecular conformations due to flow stretching in the x-direction and the spatial correlations between the x and y components of the chain end-to-end vector, leading to an increase in G

_{xy}, and (b) chain orientation along the flow direction, leading to a decrease in G

_{xy}. It is the competition between these two effects that gives rise to the maximum. Overall, and despite the obvious similarities in the overall qualitative behavior of G

_{xy}with shear rate, certain differences are observed. For example, for linear melts, G

_{xy}increases rapidly with shear rate even for small shear rates whereas for rings the increase is slower and is initiated at higher shear rates. This is another manifestation of stronger resistance to the applied flow, and can be explained again by the more compact structure of rings compared to linear melts which can attain more open and wider conformations. Similar results have been reported by Yoon et al. [14] for PE.

_{1}> G

_{2}> G

_{3}) of the radius-of-gyration tensor for all three different PEO melts simulated. The results found for the PEO-5k are reported in Figure 10. Also reported in Figure 10 is the ratio of the largest eigenvalue G

_{1}with the smallest G

_{3}. Admittedly, the changes of the three eigenvalues with shear rate follow (to a large degree) the corresponding changes of the xx, zz and yy components of the radius-of-gyration tensor

**G**.

**A**denotes a second order tensor, e.g., the radius-of-gyration tensor, the stress tensor or the birefringence tensor. In the present work, we choose to work with the radius-of-gyration tensor

**G**. Figure 11, then, shows how the alignment angle changes with applied shear rate between rings and linears. At low shear rates, θ tends to the value of 45° for both types of melts. The limiting value of 45° has already been observed in other simulation studies [14,32] and is also explained theoretically. Specifically, at low shear rates, the chains start to align, leading to a nonzero value of <A

_{xy}> but with negligible chain deformation (i.e., negligible normal stresses). At intermediate shear rates, the chains align in the flow direction and, also, stretch considerably; as a result, normal stresses develop in the melt, which in turn cause a steep rise in the value of θ. At high shear rates, chains reach their maximum allowable stretch, thus θ reaches a plateau value. Comparing the results between ring and linear melts at sufficiently high shear rates, we notice that rings are characterized by a lower degree of alignment. Once more, this demonstrates the stronger resistance of rings to the applied flow in comparison to linear chains [14] as a result of their shorter spatial extent due to their closed structure.

#### 3.3. Terminal Relaxation

_{C}= 1000) undergoing tumbling and tank-treading motion, respectively. The interested reader can also visualize the two types of motion for the chosen pairs of ring molecules in the two videos that we prepared and uploaded as Supplementary Material to this manuscript from the NEMD simulation with the R-5k melt at Wi

_{C}= 1000.

**u**(t)·

**u**(0)>

_{L}of the unit vector

**u**directed along the chain end-to end vector (

**R**

_{ee}), and its time decay in the course of equilibrium MD and NEMD runs. For ring melts, terminal relaxation is quantified by looking at the OACF <

**u**(t)·

**u**(0)>

_{R}of the unit vector

**u**directed along the diameter vector (

**R**

_{d}) of the ring, averaged over all possible such vectors for a given ring molecule. The rate with which <

**u**(t)·

**u**(0)> approaches the zero value is a measure of how fast the chain forgets its initial configuration, i.e., of the rate of the overall orientational relaxation of the chain.

**u**(t)·

**u**(0) functions for all chains in the melt have dropped to zero. Surprisingly, our MD simulations reveal that, for both molecular architectures studied (ring and linear), the orientational dynamics are highly heterogeneous, since the individual

**u**(t)·

**u**(0) curves deviate significantly from the average <

**u**(t)·

**u**(0)> curve. In fact, this dynamic heterogeneity is more pronounced in the case of linear melts where, in addition, chains require much longer times to relax. For the PEO-1k system examined in Figure 13, this behavior is counter-intuitive, since it is unentangled and, in addition, excess free volume around chain ends should accelerate chain relaxation in the case of the linear melt [12]. However, we can explain this peculiar behavior if we recall that, due to their looped structure, ring chains assume conformations that are spatially extended to shorter distances than chains in the linear melt. That is, terminal relaxation for linear chains involves the decorrelation of the unit vector along the end-to-end chain vector

**R**

_{ee}which corresponds to twice the molecular length covered by the diameter

**R**

_{d}vector used to define the OACF function in the case of rings. Under strong flow conditions (${Wi}_{\mathrm{C}}=10$ and ${Wi}_{\mathrm{L}}=10$), both types of melts relax faster but the main features of the decorrelation still remain. With increasing molecular weight (see Figure 14), the effect of excess free volume in the linear melts becomes less and less important, and this is reflected in the enhanced dynamic heterogeneity of linear chains. Similar behavior was observed for the entangled R-5k melt (see Figure 15).

**u**(t)·

**u**(0)>

_{R}and <

**u**(t)·

**u**(0)>

_{L}for all chains in the melt, one obtains a measure of the characteristic time constants for relaxation for ring and linear melts, respectively. The corresponding histograms of these times are displayed in Figure 16 and Figure 17. For all PEO systems studied, characteristic relaxation times for chains in the linear melt span a much wider range than in the ring melt. With increasing molecular length, the distributions for both types of melts become broader and, of course, their peaks are shifted to larger times. The applied flow has a strong effect on the distributions causing their width and average value to decrease.

#### 3.4. Topological Analysis

_{k}per chain is proportional to the number of underlying entanglements Z per chain. For ring melts, the same concepts apply but the subsequent identification of threading events between rings requires a more elaborate analyis based on 3D vector calculus. All details regarding the implementation of the CReTA algorithm for melts of polymer rings and the geometric analysis for identifying ring–ring threadings can be found elsewhere [5,40].

_{k}> remains practically unaffected by the flow. However, as the shear rate increases, chains orient in the direction of flow (see Figure 5a), which causes the average number of kink points per chain <Z

_{k}> to drop. This indicates a significant change of the underlying topological network with the applied flow, exactly as has been reported in other computational studies [32,42,43,44] for linear PE melts.

## 4. Conclusions

_{e}of linear PEO. Our simulation results provide useful information concerning the effect of imposed shear rate on viscometric or material functions, conformational properties, terminal relaxation and topological interactions, and how these differentiate between ring and linear melts.

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Steady-state shear viscosity η as a function of imposed strain rate or ${Wi}_{\mathrm{C}}$ number for the simulated: (

**a**) PEO-1k, (

**b**) PEO-2k and (

**c**) PEO-5k ring and linear melts.

**Figure 2.**First normal stress coefficient Ψ

_{1}as a function of imposed strain rate or ${Wi}_{\mathrm{C}}$ number for the simulated: (

**a**) PEO-1k, (

**b**) PEO-2k and (

**c**) PEO-5k ring and linear melts.

**Figure 3.**Second normal stress coefficient, −Ψ

_{2}, as a function of imposed strain rate or ${Wi}_{\mathrm{C}}$ number for the simulated: (

**a**) PEO-1k, (

**b**) PEO-2k and (

**c**) PEO-5k ring and linear melts.

**Figure 4.**The ratio $-{\mathsf{\Psi}}_{2}/{\mathsf{\Psi}}_{1}$ as a function of applied shear rate or ${Wi}_{\mathrm{C}}$ number for the simulated: (

**a**) PEO-1k, (

**b**) PEO-2k and (

**c**) PEO-5k ring and linear melts.

**Figure 5.**Representative atomistic configurations from the NEMD simulations with the linear (

**a**) and ring (

**b**) PEO-5k melt at various shear rates or ${Wi}_{\mathrm{C}}$ numbers.

**Figure 6.**Variation of the xx component of the radius-of-gyration tensor with shear rate or ${Wi}_{\mathrm{C}}$ number for the simulated: (

**a**) PEO-1k, (

**b**) PEO-2k and (

**c**) PEO-5k ring and linear melts.

**Figure 10.**Variation of the three eigenvalues of the radius-of-gyration tensor with applied shear rate or ${Wi}_{\mathrm{C}}$ number for the simulated ring and linear PEO-5k melts: (

**a**) the largest eigenvalue G

_{1}, (

**b**) the second largest eigenvalue G

_{2}, (

**c**) the smallest eigenvalue G

_{3}, and (

**d**) the ratio between G

_{1}and G

_{3}.

**Figure 12.**A series of instantaneous snapshots of two PEO-5k ring chains from the NEMD simulation at Wi

_{C}= 1000, undergoing: (

**a**) tumbling (times between t = 1.40 ns and t = 1.72 ns), and (

**b**) tank-treading (times between t = 0.20 ns and t = 0.83 ns). Chain segments have been colored red and green for better visualization of the two types of motion.

**Figure 13.**Decay of the time autocorrelation function <

**u**(t)·

**u**(0)> for each individual chain in: (

**a**) the L-1k melt at equilibrium, (

**b**) the L-1k melt under strong flow (${Wi}_{\mathrm{L}}=10$), (

**c**) the R-1k melt at equilibrium, and (

**d**) the R-1k melt under strong flow (${Wi}_{\mathrm{C}}=10$). The thick black lines represent the average over all chains.

**Figure 14.**Same as with Figure 12 but for the PEO-2k system.

**Figure 15.**Decay of the time autocorrelation function <

**u**(t)·

**u**(0)> for each individual molecule in the R-5k melt: (

**a**) at equilibrium, and (

**b**) under flow (${Wi}_{\mathrm{C}}=10$). The thick black lines represent the average over all chains.

**Figure 16.**Histograms of the characteristic orientatonal relaxation times at equilibrium and under flow for the simulated: (

**a**) R-1k, (

**b**) R-2k and (

**c**) R-5k ring PEO melts.

**Figure 17.**Same as with Figure 16 but for the two linear PEO melts.

**Figure 18.**Number of kinks per chain normalized with the corresponding number at equilibrium for the L-5k PEO melt as a function of applied shear rate (in dimensionless units).

**Figure 19.**Histogram (in log-linear coordinates) of the number of ring–ring threadings in the R-5k melt for various shear rates.

**Figure 20.**Example of a multiple threading event from the NEMD simulation with the R-5k melt under strong shear flow (${Wi}_{\mathrm{C}}=1000$). The green ring molecule is simultaneously threaded by the blue, the orange and the yellow ring molecules. For clarity, we have included parts of the periodic images of the blue and green rings in the direction of flow.

**Table 1.**Some technical details (degree of polymerization, molecular weight, number of chains in the simulation cell, mean equilibrium radius of gyration, maximum diameter length and box length in the direction of flow) concerning all pure poly(ethylene oxide) (PEO) melts studied in this work.

System | M_{w} (g/mol) | Number of Molecules | ${\mathit{R}}_{\mathbf{g}}^{\mathbf{eq}}(\AA )$ | ${\mathit{R}}_{\mathbf{d}}^{\mathbf{max}}{,\mathit{R}}_{\mathbf{ee}}^{\mathbf{max}}(\AA )$ | L_{x} (nm) |
---|---|---|---|---|---|

R-1k | 1320 | 3750 | 9 ± 1 | 54 | 187 |

R-2k | 1844 | 3125 | 11 ± 1 | 74 | 177 |

R-5k | 5324 | 1458 | 18 ± 1 | 220 | 207 |

L-1k | 1322 | 3750 | 13.5 ± 1 | 108 | 193 |

L-2k | 1846 | 3750 | 15.5 ± 1 | 148 | 202 |

L-5k | 5326 | 1920 | 26 ± 1 | 440 | 295 |

**Table 2.**Nonequilibrium molecular dynamics (NEMD)-based zero-shear rate viscosities extracted by fitting the simulation data to the Carreau model, along with experimental data [19,33]. For comparison, we also show the viscosities as predicted by the Rouse model expressions for ring and linear polymer melts, with the ring and linear chain orientational relaxation times obtained from independent equilibrium molecular dynamics (MD) runs.

System | Carreau Model | Rouse Theory | Experimental Data | |||
---|---|---|---|---|---|---|

η_{0,R} (mPa·s) | η_{0,L}(mPa·s) | η_{0,R} (mPa·s) | η_{0,L}(mPa·s) | η_{0,R}(mPa·s) | η_{0,L}(mPa·s) | |

PEO-1k | 16.8 ± 1 | 17.3 ± 1 | 19 ± 1 | 19 ± 0.5 | - | - |

PEO-2k | 32 ± 3 | 66 ± 5 | 24 ± 1 | 50 ± 5 | 30.2 ± 2 | 62.7 ± 5 |

PEO-5k | 96 ± 10 | 269 ± 12 | 65 ± 5 | 553 ± 20 | 122.3 ± 20 | - |

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**MDPI and ACS Style**

Tsamopoulos, A.J.; Katsarou, A.F.; Tsalikis, D.G.; Mavrantzas, V.G.
Shear Rheology of Unentangled and Marginally Entangled Ring Polymer Melts from Large-Scale Nonequilibrium Molecular Dynamics Simulations. *Polymers* **2019**, *11*, 1194.
https://doi.org/10.3390/polym11071194

**AMA Style**

Tsamopoulos AJ, Katsarou AF, Tsalikis DG, Mavrantzas VG.
Shear Rheology of Unentangled and Marginally Entangled Ring Polymer Melts from Large-Scale Nonequilibrium Molecular Dynamics Simulations. *Polymers*. 2019; 11(7):1194.
https://doi.org/10.3390/polym11071194

**Chicago/Turabian Style**

Tsamopoulos, Alexandros J., Anna F. Katsarou, Dimitrios G. Tsalikis, and Vlasis G. Mavrantzas.
2019. "Shear Rheology of Unentangled and Marginally Entangled Ring Polymer Melts from Large-Scale Nonequilibrium Molecular Dynamics Simulations" *Polymers* 11, no. 7: 1194.
https://doi.org/10.3390/polym11071194