Note pubs.acs.org/Macromolecules
Coupled Effect of Orientation, Stretching and Retraction on the Dimension of Entangled Polymer Chains during Startup Shear Yuyuan Lu,† Lijia An,*,† Shi-Qing Wang,*,‡ and Zhen-Gang Wang*,†,§ †
State Key Laboratory of Polymer Physics and Chemistry, Changchun Institute of Applied Chemistry, Chinese Academy of Sciences, Changchun 130022, People’s Republic of China ‡ Department of Polymer Science, University of Akron, Akron, Ohio 44325-3909, United States § Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, California 91125, United States
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the tube model. Finally, our work did not examine the behavior of the contour length, which is a more direct measure of stretching. Thus, the original purpose of this Note is 3-fold, (a) to perform a new set of BD simulation to higher values of strain under the condition of Wi > 1 and WiR < 1; (b) to directly examine the chain contour length and the coupled effects of stretching and orientation on ⟨Rg2⟩; (c) to compare the BD results on ⟨Rg2⟩ with a more refined version of the tube model without invoking the IAA. During the preparation of this Note, we learned of the publication of the Comment by Graham et al.,20 which raised critical comments on our earlier results in ref 19. We take this opportunity to give our response to these comments. We use the Kremer−Grest model21 consisting of purely repulsive Lennard-Jones monomers connected by the finitely extensible nonlinear elastic (FENE) potential between adjacent monomers with the same parameters as those in ref 19. The additional potential to prevent chain crossing22 used in ref 19 turned out to be unnecessary and was not included since under the strain rates used in the simulation, the probability of chain crossing induced by shear is negligibly small. The dimensions of the simulation box are increased from Lx = Ly = 4Rg0 and Lz = 3Rg0 in our previous study to Lx = Ly = 6Rg0 and Lz = 4Rg0,23 where x, y, and z refer respectively to the shear, gradient and vorticity directions, and Rg0 is the equilibrium radius of gyration. The system contains 423 chains of length N = 500 at a monomer density of ρ = 0.85d−3 where d is the LennardJones diameter. The simulation uses standard Brownian dynamics in the LAMMPS platform. To be free of possible wall effects, we use periodic boundary condition in the x and z directions and Lees−Edwards boundary condition in the y direction.24 We have verified that the actual velocity profile is linear in the gradient direction at all stages of the simulation. Length, energy, and time are nondimensionalized respectively by the molecular diameter d, energy ε, and the Lennard-Jones time scale τLJ = (md2/ε)1/2 where m is the mass of the particle. The simulation time step is chosen to be Δt = 0.001τLJ and the temperature is set at kBT = 1. The data reported are results of averaging over 16 independent samples. The number of monomers between entanglements Ne ≈ 36 in the quiescent state, is determined from the crossover time (from t1/2 to t1/4
he theoretical description of how entangled polymers undergo large deformation has been at the heart of polymer dynamics.1−3 Over the years, the polymer physics community has adopted the tube model as the standard paradigm for describing polymer deformation and flow in both the linear and nonlinear regimes.2−5 The tube model envisions an ensemble of chains that undergo reptation and fluctuation inside smooth confining tubes on time scales of the Rouse relaxation time. It provides a reasonable and appealing description of the equilibrium and near equilibrium dynamics of entangled polymer melts and concentrated solutions. Some of the predictions based on this paradigm are also in apparent agreement with macroscopic rheological measurements for large deformations.1,2 However, some discrepancies with the tube model have been noted in the literature. For example, the time for the onset of time-strain superposibility in experiments has been shown to be much longer than the Rouse time anticipated based on the tube model.3,6−8 More recently, S.-Q. Wang’s group using particle-tracking velocimetry reported a number of nonlinear rheological phenomena9−12 that questioned the validity of the tube-model assumptions. The theoretical implications of the reported phenomena have been controversial.13−18 Using Brownian dynamics (BD) simulation, we have recently studied the evolution of chain conformation and entanglements during startup shear19 to specifically test the validity of the tube model description under conditions of Wi = γ̇τd > 1 and WiR = γ̇τR < 1, where WiR, Wi, τR, and τd are the Rouse−Weissenberg number, Weissenberg number, Rouse time and reptation time, respectively. Our results demonstrated that the mean-square radius of gyration ⟨Rg2⟩ increases with shear significantly beyond its equilibrium value ⟨Rg02⟩ and follows the limiting behavior prescribed by the affine deformation up to a strain of order one, from which we infer that there is significant chain stretching, which is not anticipated by the tube model in this rate regime. Up to the end of the simulation corresponding to a total shear strain γ of 3, ⟨Rg2⟩ increases monotonically. However, for sufficiently large strain, we expect that the stretched chains will retract, leading to a nonmonotonic dependence of the coil dimensions on the strain. Furthermore, we made comparison with a simplistic version of the Doi− Edwards model using the independent alignment approximation (IAA).2 Although the IAA is a reasonable approximation for some purposes (such as in the calculation of stress), the assumption of uniform and independent orientation of the tube segments yields a constant ⟨Rg2⟩ = ⟨Rg02⟩. The comparison is therefore not a convincing demonstration of the deficiencies in © 2014 American Chemical Society
Received: January 17, 2014 Revised: July 9, 2014 Published: July 18, 2014 5432
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of the Ne ≈ 36 monomers, so that L/L0 = rge/rge0; the results are very similar. The results for the normalized contour length L/ L0 are shown in Figure 1b. The pronounced peaks, spanning many Rouse times, offer clear evidence of chain stretching and retraction on time scales much longer than τR. In contrast, the GLaMM theory predicts again nearly imperceptible peaks for WiR = 1/6 and 1/4, and a rather weak peak for the larger WiR = 1/2 on the scales of the figure. Especially, for WiR = 1/6, the calculation using GLaMM only produces a 2% increase in the contour length, whereas the simulation yields 13% increase in the contour length at the peak. Thus, the simulation result of significant stretching followed by retraction on time scales of many Rouse times is not captured by the tube model in this rate regime. We note that the maximum of the chain dimension ⟨Rg2⟩ occurs at a later time than the contour length L. This is because while chain stretching has reached a maximum, the chain continues to be aligned by the shear. The maximum in ⟨Rg2⟩ is due to combined effects of orientation, stretching, and retraction. To gain further insight, we present the normalized meansquare radius of gyration ⟨Rg2⟩/⟨Rg02⟩ vs the applied strain in Figure 2, where the filled symbols (blue color online) are the
scaling) in the monomer mean-square displacement.19,21 The Rouse time τR = 2.3 × 105 and the reptation time τd = 3.3 × 106 are obtained by the method as described in ref 19 and are numerically in agreement with previous results.19,21 We first examine the evolution of ⟨Rg2⟩ with time. As seen in Figure 1a, ⟨Rg2⟩ exhibits nonmonotonic dependence on the
Figure 1. (a) Mean square radius of gyration and (b) contour length of the primitive chain, as a function of time t/τR. The red dash lines are predictions by the GLaMM theory.20,25,26 The short vertical bars indicate the locations of the maxima.
Figure 2. Mean-square radius of gyration as a function of the elapsed strain. The short vertical bars indicate the locations of the maxima.
elapsed time for all the three shear rates used, contrary to the impression of monotonic increase from our previous study19 that involved shorter simulation times. The time corresponding to the maximum of ⟨Rg2⟩ is significantly longer than τR. This nonmonotonic variation of ⟨Rg2⟩ with time arises from the combined effects of orientation, stretching and retraction of the primitive chain. For comparison, we include results from the GLaMM theory calculation. For the two smaller rates WiR = 1/ 6 and 1/4, the GLaMM theory yields two faint peaks that are nearly imperceptible. For the larger shear rate WiR = 1/2, it yields a visible peak, which is, however, significantly off in its position and height compared to the simulation data. In particular for WiR = 1/6, which is the closer test of the WiR < 1 regime, the peak in the GLaMM result, occurring at t/τR ≈ 27, can only be recognized by going to the third decimal point. To directly address the issue of chain stretching and retraction, we measure an average contour length of the primitive chain by L = ΣsZ=−1 1⟨|R⃗ s+1 − R⃗ s|⟩, where R⃗ s is the center of mass position of a tube segment, obtained by coarse-graining the chain into Z = N/Ne tube segments, each with Ne consecutive monomers. At equilibrium L = L0. We have also computed the contour length by using the radius of gyration rge
results of the computer simulation and the open symbols (red color online) are the results of the GLaMM model calculation. As both chain orientation and stretching cause ⟨Rg2⟩ to increase, a pronounced maximum as shown in Figures 1 and 2 is possible only if chain retraction occurs beyond the peak in the chain contour. Our simulation shows that the peak of ⟨Rg2⟩ shifts to higher strain at a higher rate, reflecting the fact that the chain retraction occurs at a higher strain for a higher value of WiR. In contrast to the pronounced maxima in ⟨Rg2⟩ from the simulation for all three rates, a significant peak appears only for the largest rate WiR = 1/2 in GLaMM. For the two smaller rates, WiR = 1/4 and 1/6, which fall more closely into the WiR < 1 regime, there is only imperceptibly small nonmonotonicity in GLaMM, which can hardly be considered numerically significant. If we have to characterize the GLaMM in finer detail, i.e., accepting its ability to depict any nonmonotonicity, then we note that the locations of these invisible peaks occur at strains all higher than those from the simulation. Contrary to the physical picture of chain retraction as the origin of the decline in ⟨Rg2⟩, which occurs upon disentanglement from the 5433
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stretched entanglement network,27 the cause of the immeasurably small decrease of ⟨Rg2⟩ cannot be due to chain retraction because the times at the “peaks” are respectively 20τR and 27τR for WiR = 1/4 and 1/6, well after the perceived chain retraction at ∼ τR by GLaMM. Thus, although one could argue that both simulation and the GLaMM show nonmonotonicity in ⟨Rg2⟩, the underlying physics are entirely different. The physics behind the faint nonmonotonicity at such high strains in the GLaMM is unclear. We have also included in Figure 2 the curve representing affine deformation. We note that the initial curvature of the simulation data follows closely the affine deformation curve whereas the GLaMM calculation does not. This difference further indicates there is strong chain stretching well past the point of the perceived chain retraction by the tube theory. Because the chains become anisotropic under shear, it is of interest to follow the development of anisotropy as a function of shear strain. To this end, we separate the overall ⟨Rg2⟩ into its three Cartesian components. Figure 3 shows the evolution
et al.;20 some of these comments (b and c) are contained in the Supporting Information of their article. (a). “... the GLaMM model is able to account for the degree of deformation of ⟨Rg2⟩ seen in Lu et al.’s simulations via orientation ef fects alone.” The comparison between results from our extended simulations and the GLaMM calculations as shown in all the figures of this Note clearly contradicts this characterization. (b). “Lu et al. did not apply periodic boundary conditions in the f low gradient direction and instead they graf ted chains to impenetrable, moving plates at the upper and lower walls of their simulation box. Over half of all chains in their simulation are graf ted, meaning that each f ree chain will be in close proximity to a graf ted chain. This should lead to signif icant surface effects.” All simulation data reported in the current work are obtained using a larger simulation box size with the Lees−Edwards boundary condition. The difference between the results with wall-driven shear and the results using the Lees−Edwards periodic boundary condition is small. The surface has only a minor quantitative effect, which does not alter the qualitative behavior reported in our earlier work. (c). “Furthermore, the simulations do not report the most direct measure of chain stretching, the primitive path length. Finally, the bulk stress predictions for the simulations are not shown.... A further inconsistency with stress measures is that Lu et al.’s simulations appear to reach steady state by γ = 3, ...” Here in Figure 1b, we show that the primitive path length is obviously stretched and is nonmonotonic in time (strain). Recently, we have also studied the bulk stress in start-up shear27 to demonstrate that the origin of stress overshoot is caused by retraction of the significantly stretched chains in the entanglement network rather than orientation; the variation of the contour length with strain is shown to peak at roughly the same strain where the shear stress overshoots. The present longer-time simulation shows that ⟨Rg2⟩ peaks in the range of γ = 3−4 (depending on shear rates) as shown in Figure 2 due to a combination of orientation, stretching and retraction effects. Our earlier work only reported the behavior of ⟨Rg2⟩ up to γ = 3, where its decline has not yet shown up. Graham et al.20 also pointed out discrepancy between our results and those of Baig et al.:28 that in ref 28 there was essentially no change in the contour length of the primitive chain and there was only weak deformation in ⟨Rg2⟩. First, as the authors themselves acknowledged, the study by Baig et al. was on steady shear, whereas our work was on start up shear. The end points of our simulation had not reached the steady state, so there is no obvious basis for comparison. Second, the chains in these two studies have very different rigidities. Even though the degree of entanglements are similar (≈14), our 500monomer polymer chain has 287 Kuhn segments while the C400H802 linear polyethylene (PE) chains simulated in ref 28 has only 37 Kuhn segments. Therefore, our chains are considerably more flexible than those in ref 28. The response of ⟨Rg2⟩ and L to shear can be quite different between the two systems due to differences in chain rigidity. In conclusion, using BD simulation, we have studied the mean-square radius of gyration under conditions Wi > 1 and WiR < 1. Our results show that there is significant chain stretching as measured by the contour length of the primitive
Figure 3. Components of the gyration tensor of the entangled polymer chains for WiR = 1/6: (a) x-component; (b) y- and zcomponents.
of the three separate components with strain for WiR = 1/6. As expected, the chain dimension is expanded in the shear (x) direction and compressed in the gradient (y) and vorticity (z) directions. The corresponding results from the GLaMM prediction are also included for comparison. We see that the simulation data exhibit more expansion in the x direction and less compression in the y and z directions than results from the GLaMM prediction. In a recent Comment by Graham et al.20 directed at an article by Wang et al.,18 the authors also commented on our work published in ref 19. They raised issues questioning the validity of our results and conclusions in ref 19. Below we give our brief response to the specific comments (in italic) made by Graham 5434
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(23) To ensure that our results are not due to finite size effects, we have also carried simulations using two larger box sizes (9 × 6 × 4) and (6 × 9 × 4), where the dimensions are in unit of Rg0. These largerbox simulations confirm the validity of the results. (24) Lees, A. W.; Edwards, S. F. J. Phys. C 1972, 5, 1921. (25) We used the number of chain entanglements Z = 14, set the CCR parameter to the standard value20,26 of αd = 1.15, cv = 0.1 and 9 s = 2.0 for the contour length fluctuations, constraint release and retraction terms, respectively. Our calculation of the contour length uses the same definition as in ref 20; i.e., L = Σs Z= 1(Trf(s,s))1/2. (26) Graham, R. S.; Likhtman, A. E.; McLeish, T. C. B.; Milner, S. T. J. Rheol. 2003, 47, 1171. (27) Lu, Y. Y.; An, L. J.; Wang, S.-Q.; Wang, Z.-G. ACS Macro Lett. 2014, 3, 569. (28) Baig, C.; Mavrantzas, V. G.; Kröger, M. Macromolecules 2010, 43, 6886.
chain path, which makes a substantial contribution to ⟨Rg2⟩. Chain stretching persists over many Rouse times. As a result of combined chain stretching and orientation, the peak in ⟨Rg2⟩ occurs after the peak in the contour length. The peak strain for ⟨R2g ⟩ increases with increasing shear rate. Comparison with the state-of-the-art tube theory−the GLaMM theory−reveals marked differences in the behavior of both ⟨Rg2⟩ and the contour length L. We believe that these differences reflect a deficiency in the physical assumption of the tube model that a polymer chain undergoes simple Rouse dynamics inside a smooth confining tube when subjected to startup deformation. Finally, the more extensive simulations of the present study with larger box size and Lees-Edwards boundary condition reaffirm the validity of our previous results.
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AUTHOR INFORMATION
Corresponding Authors
*(L.A.) E-mail:
[email protected]. *(S.-Q.W.) E-mail:
[email protected]. *(Z.-G.W.) E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work is supported, in part, by the National Natural Science Foundation of China (Nos. 21120102037, 21334007, and 21304097) and is further subsidized by the Special Funds for National Basic Research Program of China (No. 2012CB821500).
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REFERENCES
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