Article pubs.acs.org/Macromolecules
Effect of Chain Orientation and Stretch on the Stress Overshoot of Entangled Polymeric Materials under Start-Up Shear Sohdam Jeong,# Jun Mo Kim,# and Chunggi Baig* School of Energy and Chemical Engineering, Ulsan National Institute of Science and Technology (UNIST), UNIST-gil 50, Eonyang-eup, Ulju-gun, Ulsan 689-798, South Korea S Supporting Information *
ABSTRACT: One of the most important nonlinear rheological phenomena in flowing polymeric materials is the stress overshoot under start-up shear at sufficiently high shear rates γ̇, i.e., γ̇ > τd−1, where τd is the terminal relaxation time of system. According to the well-known tube theory for entangled polymeric materials, a linear relationship between the anisotropy of the stress and that of the birefringence holds for strain rates in the τd−1 < γ̇ < τR−1 range, where τR is the Rouse time. This implies that in such a flow range the stress overshoot behavior of entangled polymers is accurately described in terms of the chain orientation, as confirmed by previous experiments. However, there has been some recent debate on this issue, following the study of Lu et al. [ACS Macro Lett. 2014, 3, 569−573], whose coarse-grained molecular dynamics simulations produced results in apparent conflict with existing theoretical and experimental predictions, which had suggested that even for γ̇ < τR−1, the stress overshoot is associated with the global stretch of the chains rather than with their segmental orientation. While several subsequent studies by other groups did not support the results of Lu et al., the issue remains open, mainly due to the use of mesoscopic coarse-grained models in all these simulations which may give rise to quantitatively inaccurate results especially at high flow rates. In order to resolve this issue and clarify the possible causes of the controversy, in this study we conducted direct atomistic nonequilibrium molecular dynamics (NEMD) simulations of an entangled polymer melt under start-up shear flow. Our results unambiguously show that the segmental orientation is the primary molecular mechanism leading to the stress overshoot. We also determine and discuss various structural properties relevant for the quantitative analysis of the stress response of flowing polymeric systems. rheological responses of polymer systems under flow.1−3 The analysis of birefringence measurements is thus considered to work remarkably well in exposing the basic characteristics (i.e., overshoot strain and magnitude) of stress overshoot in a broad range of flow strengths, except at very high flow fields where the accuracy of the SOR seriously deteriorates. According to the tube theory4,5 for entangled polymeric materials, the SOR is valid in the range of strain rates τd−1 < γ̇ < τR−1, where τd and τR are the terminal relaxation time and the Rouse time, respectively. This implies that in such flow range the stress overshoot and segmental orientation of entangled polymers are closely related to each other, as also confirmed by experimental studies.1,6 However, recent work by Lu et al.,7−9 based on coarse-grained (CG) Kremer−Grest molecular dynamics (MD) simulations,10 obtained results in apparent conflict with the above theoretical and experimental predictions: Lu et al.’s results indicated that even in the range of γ̇ < τR−1, the stress overshoot is directly associated with the global chain stretching (as represented by the variation of the primitive path (PP) contour length Lpp of a chain within the tube theory) rather than with the segmental (i.e., entanglement
S
tress overshoot is one of the most important nonlinear rheological phenomena exhibited by polymeric liquids undergoing start-up shear above a certain critical flow strength. In this phenomenon, the shear stress (or viscosity η) of the system initially increases with time, then reaches a maximum at a certain strain followed by a rather smooth decrease, and eventually attains a steady-state value. The overshoot peak is ≈ 2−3, irrespective of the polymer found at strain values γmax η type and the applied shear rate (a similar behavior has been observed for the first normal stress coefficient Ψ1, but for a higher value of critical shear rate and peak strain, γmax ψ1 ≈ 1−3 2γmax ). Flow birefringence, a very useful and noninvasive η rheo-optical technique, has been widely applied in the field of polymer rheology for characterizing transient rheological responses such as the one discussed above and for understanding the physical states of anisotropic polymeric materials under an external flow field. These studies provided invaluable information about structure−stress correlations in flowing polymeric materials.1 While it is generally known that the initial linear relationship between the stress and birefringence tensors (known as stress-optical rule, SOR) of polymeric materials under elongation or shear starts to break down above a certain intermediate flow strength, their intimate relationship remains valid even at strong flow fields and thus represents a useful tool for analyzing both transient and steady-state © XXXX American Chemical Society
Received: February 10, 2017 Revised: April 5, 2017
A
DOI: 10.1021/acs.macromol.7b00288 Macromolecules XXXX, XXX, XXX−XXX
Article
Macromolecules
applied a wide range of flow strengths, corresponding to Weissenberg numbers Wid and WiR (defined as the product of the characteristic relaxation time τ of the system and the imposed strain rate) satisfying 3.71 ≤ Wid ≡ τdγ̇ ≤ 928 or 0.91 ≤ WiR ≡ τRγ̇ ≤ 229, with τd = 218 ± 10 ns21,24 and τR = 53.7 ± 10 ns25,26 for the C400H802 PE melt system.27,28 Relevant information on the rheo-optical properties of the studied system was obtained by quantifying the relative contributions of segmental orientation and chain stretch to the stress at very strong flow fields (γ̇ ≫ τR−1), where the linearity of the SOR breaks down and the accuracy of coarse-grained models is largely deteriorated. Figure 1a presents the typical overshoot behavior of shear stress σxy of entangled polymer melts under start-up shear in sufficiently nonlinear flow regimes.
strand) orientation. This finding immediately stimulated several subsequent studies. For example, using different coarse-grained simulations (based on the primitive chain network model), Masubuchi and Watanabe11 showed that the occurrence of stress overshoot at τd−1 < γ̇ < τR−1 is strictly due to segmental orientation rather than chain stretch, contradicting the conclusions of Lu et al.7 Cao and Likhtmann12 also carried out a rheo-optical analysis of entangled polymer melts similar to those investigated by Lu et al.7−9 using the same Kremer− Grest MD simulations. Interestingly, they observed a strong correlation between the overshoot behaviors of stress and segmental orientation (rather than Lpp variation), corroborating the validity of the SOR while calling into question Lu et al.’s7−9 results for τd−1 < γ̇ < τR−1.13 In an effort to resolve these conflicting results and remove any ambiguity associated with the coarse-graining nature of the simulations (e.g., according to a previous study,14 a coarsegrained model is likely to result in an artificially large stretching tendency of polymer chains above a certain intermediate flow strength, via an increase in the effective CG bead friction), in this work we carried out direct atomistic nonequilibrium molecular dynamics (NEMD) simulations of an entangled polymer melt under simple shear flow. The present study not only analyzes the mutual relationships between stress and localto-global structure (i.e., birefringence, entanglement segmental orientation, chain end-to-end vector, and PP length) but also focuses on the structural properties relevant for the quantitative analysis of the stress response of flowing polymeric systems. The model employed in this work consists of a sufficient number (i.e., 198) of monodisperse C400H802 linear polyethylene (PE) chains enclosed in a rectangular simulation box with dimensions (318.8 Å, 86.96 Å, 86.96 Å) in (x, y, z) space (where x, y, and z represent the flow, velocity gradient, and neutral directions, respectively). The box was enlarged in the flow (x-) direction to reduce system size effects due to significant chain stretch and alignment, especially under strong flow fields. Canonical NEMD simulations were carried out at constant temperature T = 450 K and density ρ = 0.7640 g/cm3, using the well-known p-SLLOD algorithm15,16 implemented by a Nosé−Hoover thermostat17,18 and the standard Lees− Edwards sliding-brick boundary conditions19 (see the Supporting Information). The set of evolution equations was numerically integrated using the reversible reference system propagator algorithm (r-RESPA)20 with two different MD time steps: 0.48 fs for three bonded (bond-stretching, bond-bending, and bond-torsional) interactions and 2.39 fs for nonbonded inter- and intramolecular Lennard-Jones interactions, thermostat, and flow field (refer to the Supporting Information in ref 21 for the detailed RESPA formula). The well-known Siepmann−Karaborni−Smit22 united-atom potential model was utilized in the simulations, with the exception that the rigid bond adopted in the original model was replaced by a flexible one described through a harmonic potential. Based on the known experimental value,23 i.e., Me = 920 g/mol for HDPE (high density polyethylene) at T = 443 K and ρ = 0.768 g/cm3, the number Ne of carbon atoms per entanglement strand for the PE melt employed in this study is estimated as Ne ≈ 68 and the number of entanglements per chain as roughly equal to 6, indicating a rather weakly entangled melt system here; yet, we believe that the present system is still able to capture the essential features of stress-optical behaviors for entangled polymeric systems. For a thorough exploration of the structural and dynamical characteristics of stress overshoot, we
Figure 1. Transient behavior of (a) shear stress σxy, (b) birefringence nxy, and (c) entanglement segment orientation Sxy (eq 1), as a function of strain for the simulated C400H802 polyethylene melt under start-up shear at four different flow strengths: Wid = 18.6 and WiR = 4.57 (dark yellow solid lines), Wid = 92.8 and WiR = 22.9 (dark green dashed lines), Wid = 371 and WiR = 91.4 (black dotted lines), and Wid = 928 and WiR = 229 (orange dashed-dotted lines). The lines and symbols in panel c represent the orientation tensor results calculated from the entanglement network obtained with the Z1 code34,35 and with the number of carbon atoms per entanglement strand (Ne) set to 68, respectively (see the text for details). Panels d−f display the corresponding results for the (xx−yy) component of each property. For the quantitative comparison between different flow strengths, all properties are normalized (as indicated by the tilde symbols) by their steady-state value (see Table S1 in the Supporting Information).
Consistent with previous numerical11,12,29 and experimental1,6,30−32 studies, the overshoot peak is observed to occur at strain values around 2−3, with the peak strain value (γp) slightly increasing with increasing strain rate (i.e., γp being equal to 2.03, 2.52, 2.99, and 3.21 for Wid = 18.6, Wid = 92.8, Wid = 371, and Wid = 928, respectively) and the height of the peak also increasing with the applied flow strength [γp scales with 1/ 6 power of the applied shear rate rather than 1/3 power reported by experimental studies for entangled polymeric systems;31,32 this difference is attributed to a weakly entangled system studied here]. A similar behavior is shown by the first B
DOI: 10.1021/acs.macromol.7b00288 Macromolecules XXXX, XXX, XXX−XXX
Article
Macromolecules normal stress difference σxx − σyy (Figure 1d), but with a relatively weak overshoot and the peak strain value being approximately twice as large as that of the shear stress; these trends are also in line with existing experiments1,6 (a similar overshoot feature was further observed in the second normal stress difference σyy − σzz, but with exhibiting rather large statistical fluctuations; see Figure S1). Most importantly, the comparison of the results in Figures 1a and 1b clearly reveals the well-known stress−optical relationship between the shear stress and the corresponding birefringence (also refer to Figure 3a for their linear relationship). Since the flow birefringence of polymer melts directly results from the average orientation of anisotropic monomer−monomer bond vectors (i.e., C−C bonds in the case of the polyethylene system studied here) along the chain,33 this result demonstrates the one-to-one correspondence between the stress and the overall chain orientation under structural deformation. Further examination of these results highlights that the breakdown of the linear relationship between the stress and the birefringence tensors at steady states under high flow strengths (i.e., Wid > 100) reported in previous simulation studies21 is consistently reflected in the transient stress−optical responses observed in this work. Even the decreasing trend of the stress−optical coefficient at high flow fields estimated in the steady-state regime21 is in quantitative agreement with the behavior in the corresponding transient conditions investigated here, denoting a quasi-steady nonequilibrium state of the polymeric system under transient flowing conditions. We mention in passing that while chain orientation and stretching can occur simultaneously under shear flow, the orientation mechanism is generally considered to be preferred to the overall structural deformation at a given flow strength because the orientation of chains takes place more easily than their stretching. This is partly related to the fact that the peak strain value is larger for the first normal stress difference than for the shear stress, as the former is significantly more affected by the chain stretch than the latter, which is affected primarily by the chain orientation. In addition to the birefringence, we analyzed the transient behavior of the order (or orientation) tensor S of entanglement strands along the chains in association with the tube theory for entangled polymers, defined as Sαβ =
1 Nseg
Nseg
∑ i=1
present atomistic NEMD results thus resolve the issue discussed in the introductory remarks, by showing that the chain orientation, rather than the overall chain stretching, is the dominant molecular mechanism behind the macroscopic stress overshoot. As a matter of fact, the reason for the discrepancy between the atomistic and the coarse-grained simulation results has been indicated in a previous study:14 in comparison with the actual atomistic chain, a corresponding coarse-grained chain is prone to give rise to quantitatively (and even qualitatively) incorrect structural and dynamic results. This is due to the unphysically large chain extension above a certain intermediate flow strength, which in turn results from an increase in the effective CG bead friction; the discrepancy would generally become larger for higher degrees of coarse-graining due to the larger variation in the bead friction coefficient in response to an external flow field. We also notice that the overshoot peak is almost completely absent in the Sxx − Syy plot: this is mainly ascribed to the definition of the orientation tensor eq 1, based on the unit end-to-end vector of an entanglement strand, as the magnitude of Sxx − Syy is considerably affected by the segment stretch. We now look into the rheological behavior of several important topological measures associated with the entanglement network of the simulated melt systems: the primitive path contour length Lpp, the number of entanglements per chain Zes, the end-to-end length of an entanglement strand des, and the number of carbon atoms per entanglement strand Nes. Figure 2 shows that as the imposed flow strength increases, the steady-state values of Lpp, des, and Nes increase while that of
(ri + 1 − ri)α (ri + 1 − ri)β (ri + 1 − ri)2
(1)
where ri denotes the end-to-end vector of the ith entanglement strand and Nseg the total number of entanglement strands. For this purpose, we generated the entanglement network corresponding to the modeled system via the well-known Z1 code34,35 (an efficient geometric procedure for minimizing the contour length of individual chains while respecting interchain topological constraints). In addition, in order to compare our results with the similar analysis performed by Lu et al.,9 we also calculated the order tensor based on the entanglement strands connecting the two centers of mass of neighboring subchains, formed by dividing a chain into six subchains based on the known experimental value for the number Ne of carbon atoms per entanglement strand (Ne = 68 for polyethylene23). The results are presented in Figures 1c and 1f. The two representations exhibit practically identical results at all shear rates applied in this study. Importantly, the transient behavior of the orientation tensor of entanglement strands appears very similar to that of the stress and the birefringence tensor. The
Figure 2. Transient behavior of different topological measures: (a) PP contour length Lpp, (b) number of entanglements per chain Zes, (c) end-to-end length of an entanglement strand des, and (d) number of carbon atoms per entanglement strand Nes, as a function of strain for the simulated C400H802 polyethylene melt under start-up shear at four different flow strengths, Wid = 18.6 and WiR = 4.57 (dark yellow solid lines), Wid = 92.8 and WiR = 22.9 (dark green dashed lines), Wid = 371 and WiR = 91.4 (black dotted lines), and Wid = 928 and WiR = 229 (orange dashed-dotted lines). All properties are normalized (as indicated by the tilde symbols) by their equilibrium values before exposure to the flow field.
Zes decreases, indicating a global structural deformation of the chains. The important thing to notice, however, is that none of the topological properties examined here exhibit a clear overshoot behavior even at very strong flow fields (i.e., Wid = 928 and WiR = 229), at variance with the behavior exhibited by the birefringence and the orientation tensor in Figure 1. This C
DOI: 10.1021/acs.macromol.7b00288 Macromolecules XXXX, XXX, XXX−XXX
Article
Macromolecules
The transient behavior of σxy, nxy, Sxy, See xy = ⟨uee,xuee,y⟩ (where uee is the unit chain end-to-end vector), and Lpp as a function of strain is compared in Figures 3b−3e. At relatively low strain rates (Wid = 18.6 and WiR = 4.57, Figure 3b), the σxy, nxy, and Sxy curves appear to overlap, highlighting the strictly linear relationships between these properties. Furthermore, the plot of the orientation tensor See xy, which reflects the structural feature of the chains corresponding to the largest length scale (i.e., the chain end-to-end vector), quantitatively matches the stress and birefringence curves. In contrast, Lpp does not manifest (even at a qualitative level) the transient overshoot behavior exhibited by the other properties. At stronger flow fields [i.e., Wid = 92.8 (WiR = 22.9) and Wid = 371 (WiR = 91.4); see Figures 3c and 3d], the nxy values for a given σxy appear to be systematically underestimated as compared to the corresponding values at lower flow strengths, corresponding to the breakdown of the SOR. However, it is important to notice that the overall evolution (i.e., uphill, maximum, and downhill) of the birefringence as a function of strain coincides exactly with that of the stress. On the other hand, See xy displays a trend of overestimation relative to that of the stress. More specifically, larger discrepancies in the overall predictions of the transient behavior as a function of strain are observed for See xy than for nxy, indicating that the orientation tensor based on the largest structural length scales is less capable of representing the stress response which involves all the relevant length scales of a system. Interestingly, the Sxy curve calculated from the entanglement strands turns out to match the σxy one even at rather high flow strengths. This result would support the use of the entanglement strand as a fundamental length unit in modeling entangled polymeric systems. In sharp contrast with this finding, however, the figure also shows that the plot of Lpp does not properly reflect the stress response. Overall, our results confirm the adequacy of rheo-optical tools for investigating the rheology of entangled polymers in the lowto-intermediate flow regimes and even at rather strong flow fields. At very high flow fields (i.e., Wid = 928 and WiR = 229; Figure 3e), some quantitative differences emerge between the Sxy and σxy curves; however, the Sxy data still provide the best representation of the stress response of the system, among all the structural measures examined. In summary, whereas the orientation properties appear to reflect reasonably well (at least semiqualitatively) the transient stress response of entangled polymeric materials under shear flow, the PP contour length associated with the global chain stretch does not represent an appropriate structural measure for the stress of flowing polymeric materials. Finally, we carried out a more detailed analysis of the stress overshoot behavior in terms of chain conformation. Figure 4a shows the probability distribution function (PDF) of the yy component of the gyration tensor G̃ αβ = 3∑Ni=1(Ri − Rc)α(Ri − Rc)β/⟨Rg2⟩eq (where ⟨Rg2⟩eq is the mean-square chain radius of gyration at equilibrium, and Ri and Rc are the position vectors of the ith atom of the chain and of the chain center of mass, respectively) for the simulated C400 PE melt at equilibrium before subjecting the system to an external field. As shown in the figure, the PDF follows a Poisson distribution under equilibrium conditions. In order to examine in higher detail the transient behavior of individual chain structures in response to the applied flow, we further divided the PDF into three representative regimes (marked as 1, 2, and 3 in the figure) with respect to the magnitudes of the equilibrium G0yy. Figure 4b shows the temporal variation of the chain structure in
finding contradicts the results of the coarse-grained MD simulations performed by Lu et al.7−9 As mentioned above, the peculiar overshoot behavior of Lpp observed by Lu et al.7−9 presumably arises from the unphysically large chain deformation predicted by their coarse-grained representation. In this respect, it seems that the transient response of the topological characteristics that are particularly associated with chain stretch rather than chain orientation does not show a strong quantitative correlation with the stress response of the system. Figure 3a shows the trend of the birefringence nxy vs the shear stress σxy for the melt subjected to start-up shear flow at
Figure 3. (a) Plot of birefringence nxy versus shear stress σxy at four different flow strengths: Wid = 18.6 and WiR = 4.57 (dark yellow circles), Wid = 92.8 and WiR = 22.9 (dark green diamonds), Wid = 371 and WiR = 91.4 (black triangles), and Wid = 928 and WiR = 229 (orange squares). (b−e) Plots of σxy (red solid lines), nxy (dark yellow long-dashed lines), Sxy (eq 1, black short-dashed lines), and See xy = ⟨uee,xuee,y⟩ (dark brown dotted lines, where uee is the unit chain end-toend vector), as a function of strain for the simulated C400 PE melt under start-up shear at four different flow strengths; each property is normalized by its steady-state value (see Table S1). For comparison purposes, the corresponding result of Lpp (dark green dashed-dotted lines) is also shown in each plot.
four different shear rates in the intermediate-to-high flow regime. The stress and birefringence tensors maintain their linear relationship fairly well up to intermediate flow strengths (i.e., Wid = 91.4 and WiR = 22.9); this is consistent with the results obtained for the same C400H802 PE system under steady shear flow in a previous study,21 where the SOR was found to be valid up to Wid ≈ 100. The stress−optical coefficient C evaluated from the data in Figure 3a (using Δn = CΔσ for the linear part of the nxy vs σxy plot) is (1.13 ± 0.52) × 10−9 Pa−1 and in good agreement with the (1.48 ± 0.35) × 10−9 Pa−1 value previously obtained under steady shear21 (see Figure S2 for the corresponding result of nxx−nyy and σxx−σyy). Following a further increase in the flow strength [i.e., Wid = 371 (WiR = 229) and Wid = 928 (WiR = 229)], not only the plot exhibits a nonlinear trend (which becomes more evident at higher fields) but also the values of nxy at a given σxy appear to be different from the corresponding ones at lower flow strengths. D
DOI: 10.1021/acs.macromol.7b00288 Macromolecules XXXX, XXX, XXX−XXX
Article
Macromolecules
likely to overpredict the chain stretch especially at strong flow fields due to variation in the coarse-grained bead friction with the applied field strength. Therefore, the peculiar overshoot behavior of the PP contour length of a chain Lpp observed by Lu et al.7−9 is considered to result from the unphysically large chain stretch predicted by their coarse-grained representations. On the basis of the present results, we conclude that the chain orientation is the primary cause for the macroscopic stress overshoot of entangled polymeric systems under start-up shear. Further detailed investigations targeting the underlying molecular mechanisms leading to the stress overshoot could represent a possible interesting development of this work.
■
ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.7b00288. Table S1: steady-state values of rheological properties: stress (σ), birefringence (n), entanglement segment orientation (S), primitive path contour length (Lpp), and unit chain end-to-end vector orientation (See xy); Figure S1: transient behavior of (yy−zz) component of (a) stress and (b) birefringence as a function of strain for the simulated C400H802 polyethylene melt undergoing startup shear; Figure S2: plot of birefringence nyy−nxx versus the second normal stress σxx−σyy at four different flow strengths (PDF)
Figure 4. (a) Probability distribution function P(G0xy) of the yy component G0yy of the gyration tensor for the simulated C400 PE melt at equilibrium before exposure to the flow field (three different regimes, based on the magnitude of G0yy, are specified by two vertical dashed lines). (b) Plot of G0xy for the chains corresponding to each G0yy regime as a function of strain for the same PE melt under start-up shear at intermediate flow strength. The results were calculated by averaging over five independent system configurations, each of which was fully equilibrated for more than the longest relaxation time of system (τd = 218 ns). The average number of polymer chains belonging to each regime was equal to 26, 136, and 36 for regime 1, 2, and 3, respectively.
■
terms of Gxy (which is closely associated with σxy) measured for each regime upon subjecting the system to start-up shear. The Gxy values as a function of strain appear significantly different for the different regimes. In particular, polymer chains with larger G0yy give rise to larger Gxy values and can thus be considered as making larger contributions to σxy. Furthermore, the overshoot peak of Gxy becomes more prominent for chains with originally (before exposure to the flow field) larger values of G0yy. It should be noted that the chains with relatively smaller dimension in the y direction exhibit a simple monotonic growth of Gxy, with no apparent overshoot; this indicates that polymer chains with a small y-dimension would make an insignificant contribution to the stress overshoot exhibited by the system. The relatively larger contribution to Gxy by the chains with larger initial values of G0yy can be understood based on a rather simple argument: molecules with a large y-dimension would experience a large amount of torque imposed by the external shear flow and thus a high degree of chain rotation on the vorticity (xy) plane (a similar argument has been applied for dilute polymer solutions under shear flow36). To resolve the recent debate (raised by the already mentioned CG molecular dynamics simulations) on the microscopic interpretation of the stress overshoot phenomenon displayed by polymeric materials under start-up shear above a certain critical flow strength, we carried out direct atomistic NEMD simulations of entangled (C400H802) polyethylene melts under shear flow. The atomistic NEMD results obtained in this study unambiguously show that the transient behavior of the orientation tensors (namely, the birefringence and the order tensor) matches well (at least qualitatively) that of the stress for a wide range of flow strengths; in contrast, the PP contour length Lpp generally does not capture such transient behavior accurately. As previously reported,14 a coarse-grained model is
AUTHOR INFORMATION
Corresponding Author
*E-mail
[email protected]; Tel +82-52-217-2538; Fax +82-52217-2649 (C.B.). ORCID
Chunggi Baig: 0000-0001-8278-8804 Author Contributions #
These authors contributed equally.
Notes
The authors declare no competing financial interest.
■
ACKNOWLEDGMENTS This work was supported by the National Research Foundation of Korea (2016R1D1A1A09916994 and 2016R1C1B1014515). The simulations of this work were carried out on the computational resources of the UNIST Supercomputing Center.
■
REFERENCES
(1) Janeschitz-Kriegl, H. Polymer Melt Rheology and Flow Birefringence; Springer-Verlag: New York, 1983. (2) Baig, C.; Edwards, B. J.; Keffer, D. J. A molecular dynamics study of the stress-optical behavior of a linear short-chain polyethylene melt under shear. Rheol. Acta 2007, 46, 1171−1186. (3) Baig, C. Torsional Linearity in Nonlinear Stress-Optical Regimes for Polymeric Materials. ACS Macro Lett. 2016, 5, 273−277. (4) De Gennes, P. G. Reptation of a Polymer Chain in the Presence of Fixed Obstacles. J. Chem. Phys. 1971, 55, 572−579. (5) Doi, M.; Edwards, S. The Theory of Polymer Dynamics; Oxford University Press: Oxford, UK, 1988. (6) Pearson, D. S.; Kiss, A. D.; Fetters, L. J.; Doi, M. Flow-Induced Birefringence of Concentrated Polyisoprene Solutions. J. Rheol. 1989, 33, 517−535. E
DOI: 10.1021/acs.macromol.7b00288 Macromolecules XXXX, XXX, XXX−XXX
Article
Macromolecules
PE melt at T = 450 K and P = 1 atm. The relationship τd/τR = 3Z predicted by the tube model is valid only for an entangled polymer system without any contribution from CLF and CR effects. If we account for the CLF effect on the present system based on the Doi’s approximate expression (ref 5) τd(F) ≃ τd(NF)(1 − X/√Z)2, where τd(F) and τd(NF) are the terminal relaxation time with and without CLF effects, respectively, and X is a numerical constant whose value is theoretically estimated to be larger than 1.47, we find τd(F)/τR ≃ 3Z(1 − X/√Z)2 ≤ 2.88. Taking further into account an additional CR contribution (as well as quantitatively more accurate CLF contribution), we consider the ratio of τd = 218 ± 10 ns and τR = 53.7 ± 10 ns for the present system to be with the tube model prediction. (28) Stephanou, P. S.; Baig, C.; Tsolou, G.; Mavrantzas, V. G.; Kröger, M. Quantifying chain reptation in entangled polymer melts: Topological and dynamical mapping of atomistic simulation results onto the tube model. J. Chem. Phys. 2010, 132, 124904. (29) Mohagheghi, M.; Khomami, B. Molecularly based criteria for shear banding in transient flow of entangled polymeric fluids. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 2016, 93, 062606. (30) Osaki, K.; Inoue, T.; Uematsu, T. Stress overshoot of polymer solutions at high rates of shear: Semidilute polystyrene solutions with and without chain entanglement. J. Polym. Sci., Part B: Polym. Phys. 2000, 38, 3271−3276. (31) Ravindranath, S.; Wang, S. Q. Universal scaling characteristics of stress overshoot in startup shear of entangled polymer solutions. J. Rheol. 2008, 52, 681−695. (32) Boukany, P. E.; Wang, S. Q.; Wang, X. Universal scaling behavior in startup shear of entangled linear polymer melts. J. Rheol. 2009, 53, 617−629. (33) The birefringence tensor was calculated by using the generalized formula of the Lorentz−Lorentz equation in optics: α = (3ε0/Ñ )(n·n + 2I)−1·(n·n − 2I), where ε0, Ñ , α, I, and n denote the permittivity of vacuum, the polymer number density, the polarizability tensor, the second-rank unit tensor, and the birefringence tensor, respectively. The derivation and calculation details can be found in ref 2. (34) Kröger, M. Shortest multiple disconnected path for the analysis of entanglements in two-and three-dimensional polymeric systems. Comput. Phys. Commun. 2005, 168, 209−232. (35) Jeong, S. H.; Kim, J. M.; Yoon, J.; Tzoumanekas, C.; Kröger, M.; Baig, C. Influence of molecular architecture on the entanglement network: topological analysis of linear, long- and short-chain branched polyethylene melts via Monte Carlo simulations. Soft Matter 2016, 12, 3770−3786. (36) Schroeder, C. M.; Teixeira, R. E.; Shaqfeh, E. S. G.; Chu, S. Characteristic periodic motion of polymers in shear flow. Phys. Rev. Lett. 2005, 95, 018301.
(7) Lu, Y.; An, L.; Wang, S.-Q.; Wang, Z.-G. Origin of stress overshoot during startup shear of entangled polymer melts. ACS Macro Lett. 2014, 3, 569−573. (8) Lu, Y.; An, L.; Wang, S.-Q.; Wang, Z.-G. Coupled Effect of Orientation, Stretching and Retraction on the Dimension of Entangled Polymer Chains during Startup Shear. Macromolecules 2014, 47, 5432−5435. (9) Lu, Y.; An, L.; Wang, S.-Q.; Wang, Z.-G. Molecular mechanisms for conformational and rheological responses of entangled polymer melts to startup shear. Macromolecules 2015, 48, 4164−4173. (10) Kremer, K.; Grest, G. S. Dynamics of entangled linear polymer melts: A molecular-dynamics simulation. J. Chem. Phys. 1990, 92, 5057−5086. (11) Masubuchi, Y.; Watanabe, H. Origin of stress overshoot under start-up shear in primitive chain network simulation. ACS Macro Lett. 2014, 3, 1183−1186. (12) Cao, J.; Likhtman, A. E. Simulating startup shear of entangled polymer melts. ACS Macro Lett. 2015, 4, 1376−1381. (13) The normal stress overshoot can also be predicted by the tube model with allowing for the stretch of the primitive path length (Lpp) beyond its equilibrium length at high flow strengths (i.e., γ̇ > τR−1) (ref 5). But in such case, the orientation-based stress−optical (linear) rule is no longer strictly valid, as the stress−optical coefficient is not a constant but a function of the imposed strain rate. (14) Baig, C.; Harmandaris, V. A. Quantitative analysis on the validity of a coarse-grained model for nonequilibrium polymeric liquids under flow. Macromolecules 2010, 43, 3156−3160. (15) Baig, C.; Edwards, B. J.; Keffer, D. J.; Cochran, H. D. A proper approach for nonequilibrium molecular dynamics simulations of planar elongational flow. J. Chem. Phys. 2005, 122, 114103. (16) Baig, C.; Edwards, B. J.; Keffer, D. J.; Cochran, H. D. Rheological and structural studies of liquid decane, hexadecane, and tetracosane under planar elongational flow using nonequilibrium molecular-dynamics simulations. J. Chem. Phys. 2005, 122, 184906. (17) Nosé, S. A molecular dynamics method for simulations in the canonical ensemble. Mol. Phys. 1984, 52, 255−268. (18) Hoover, W. G. Canonical dynamics: Equilibrium phase-space distributions. Phys. Rev. A: At., Mol., Opt. Phys. 1985, 31, 1695−1697. (19) Evans, D. J.; Morriss, G. P. Statistical Mechanics of Nonequilibrium Liquids; Academic Press: New York, 1990. (20) Tuckerman, M.; Berne, B. J.; Martyna, G. J. Reversible multiple time scale molecular dynamics. J. Chem. Phys. 1992, 97, 1990−2001. (21) Baig, C.; Mavrantzas, V. G.; Kröger, M. Flow Effects on Melt Structure and Entanglement Network of Linear Polymers: Results from a Nonequilibrium Molecular Dynamics Simulation Study of a Polyethylene Melt in Steady Shear. Macromolecules 2010, 43, 6886− 6902. (22) Siepmann, J. I.; Karaborni, S.; Smit, B. Simulating the critical properties of complex fluids. Nature 1993, 365, 330−332. (23) Fetters, L. J.; Lohse, D. J.; Milner, S. T.; Graessley, W. W. Packing length influence in linear polymer melts on the entanglement, critical, and reputation molecular weights. Macromolecules 1999, 32, 6847−6851. (24) Nafar Sefiddashti, M. H.; Edwards, B. J.; Khomami, B. Steady shearing flow of a moderately entangled polyethylene liquid. J. Rheol. 2016, 60, 1227−1244. (25) Kim, J. M.; Keffer, D. J.; Kröger, M.; Edwards, B. J. Rheological and entanglement characteristics of linear-chain polyethylene liquids in planar Couette and planar elongational flows. J. Non-Newtonian Fluid Mech. 2008, 152, 168−183. (26) Baig, C.; Mavrantzas, V. G. Multiscale simulation of polymer melt viscoelasticity: Expanded-ensemble Monte Carlo coupled with atomistic nonequilibrium molecular dynamics. Phys. Rev. B: Condens. Matter Mater. Phys. 2009, 79, 144302. (27) As is well-known (ref 5), such rather weakly entangled polymeric systems as the present system are considered to have strong influences by additional relaxation mechanisms such as contour length fluctuation (CLF) and constraint release (CR), whose effects was also reported in a previous simulation work (ref 28) for monodisperse C400 F
DOI: 10.1021/acs.macromol.7b00288 Macromolecules XXXX, XXX, XXX−XXX
