Letter Cite This: ACS Macro Lett. 2018, 7, 190−195
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Molecular Dynamics Investigation of the Relaxation Mechanism of Entangled Polymers after a Large Step Deformation Wen-Sheng Xu,† Jan-Michael Y. Carrillo,†,‡ Christopher N. Lam,† Bobby G. Sumpter,†,‡ and Yangyang Wang*,† †
Center for Nanophase Materials Sciences and ‡Computational Sciences and Engineering Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, United States S Supporting Information *
ABSTRACT: The chain retraction hypothesis of the tube model for nonlinear polymer rheology has been challenged by the recent smallangle neutron scattering (SANS) experiment (Wang, Z.; Lam, C. N.; Chen, W.-R.; Wang, W.; Liu, J.; Liu, Y.; Porcar, L.; Stanley, C. B.; Zhao, Z.; Hong, K.; Wang, Y., Fingerprinting Molecular Relaxation in Deformed Polymers. Phys. Rev. X 2017, 7, 031003). In this work, we further examine the microscopic relaxation mechanism of entangled polymer melts after a large step uniaxial extension by using large-scale molecular dynamics simulation. We show that the unique structural features associated with the chain retraction mechanism of the tube model are absent in our simulations, in agreement with the previous experimental results. In contrast to SANS experiments, molecular dynamics simulations allow us to accurately and unambiguously determine the evolution of the radius of gyration tensor of a long polymer chain after a large step deformation. Contrary to the prediction of the tube model, our simulations reveal that the radius of gyration in the perpendicular direction to stretching increases monotonically toward its equilibrium value throughout the stress relaxation. These results provide a critical step in improving our understanding of nonlinear rheology of entangled polymers.
E
extension, their theory predicts a characteristic, nonmonotonic change of the radius of gyration (Rg) in the perpendicular direction to stretching during the stress relaxation. However, subsequent small-angle neutron scattering (SANS) studies of the relaxation behavior of deformed entangled polymers evolved into a seemingly confusing and controversial debate spanning over more than 30 years. The early work of Maconnachie et al.18 and Boué et al.19,20 found no evidence for the chain retraction mechanism, prompting de Gennes to put forward a “nonclassical” proposal in which the chains do not retract.21 Many years later, Blanchard et al.22 reported the long-awaited observation of chain retraction after a large step uniaxial extension, contradicting the conclusion of the previous investigations. The source of these conflicting reports has never been properly understood. As pointed out in our previous analysis,23 since testing the retraction hypothesis inherently requires high-molecular-weight, well-entangled polymers, it is very difficult to resolve the radius of gyration tensor via the model-independent Guinier analysis.24 This technical difficulty has further complicated the debate about the chain retraction hypothesis, raising the question of whether it is even possible to settle this issue with small-angle neutron scattering. Very
ntangled polymers exhibit fascinating nonlinear mechanical response under large and rapid deformation.1−5 Our current understanding of the nonlinear rheological behavior of entangled polymers is built upon the tube model pioneered by de Gennes,6 and Doi and Edwards.3,7−10 While de Gennes’ original model was intended to address the dynamics of entangled polymers in the equilibrium state, Doi and Edwards developed the tube idea into a full-blown molecular theory for nonlinear rheology by envisioning decoupled contour length and orientation relaxations for a deformed entangled polymer chain.8 This decoupled relaxation mechanism, often called the chain retraction hypothesis, states that the contour length of a deformed chain relaxes on the order of the Rouse time, whereas the orientation relaxation is dictated by the much slower reptation mechanism. This chain retraction hypothesis leads to the formulation of the so-called Q tensor,3,8 giving rise to the prediction of a unique nonaffine elastic deformation mechanism for entangled polymers.2,11 Despite the tremendous effort in the past several decades to expand and refine the original Doi− Edwards theory,12−17 the chain retraction mechanism remains the cornerstone of the nonlinear tube theory for entangled polymer rheology. In their very first paper on the nonlinear tube theory,8 Doi and Edwards pointed out that the chain retraction hypothesis can be directly checked by measuring the radius of gyration tensor after a step deformation. In particular, for uniaxial step © XXXX American Chemical Society
Received: November 15, 2017 Accepted: January 23, 2018
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DOI: 10.1021/acsmacrolett.7b00900 ACS Macro Lett. 2018, 7, 190−195
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The simulations are performed with the GPU-accelerated LAMMPS package.38−40 To prepare equilibrated initial configurations for the NEMD simulation, the polymer chains are randomly placed in a cubic box at a low density of ρ = 0.001. After equilibration of 2 × 104τ0 (τ0 = σ m/ε ) at a constant temperature of T = 1ε/kB under periodic boundary conditions using a Langevin thermostat with a friction coefficient of Γ = 0.5, the system is compressed to a density of ρ = 0.85σ−3 within a period of 104τ0. To speed up the equilibration of the polymer melt at this density, a Monte Carlo bond-swap algorithm41 is subsequently implemented at three stages, each lasting 6 × 105τ0, and the coefficient kb for the FENE potential is correspondingly increased from 10 to 20 to 30. The total time for the Monte Carlo moves is thus 1.8 × 106τ0, which should be sufficient for proper equilibration of the entanglement network according to a recent study.42 After the Monte Carlo bond swaps, further equilibration is performed for 106τ0, followed by a production run of 2 × 106τ0, in which the equilibrium properties of the melt are analyzed. A velocity Verlet algorithm with a time step of Δt = 0.01τ0 is used to integrate the equations of motion in all simulations. A total of 10 independent simulations are performed to improve the statistics. We check a number of quantities to ensure that the configurations obtained from the aforementioned procedure have been well equilibrated (see section S3 of the Supporting Information for additional details). In particular, the meansquare internal distance ⟨R2(s)⟩ is analyzed as a function of the difference in indices of two beads i and j, s = |i − j|, which is a standard test for judging the quality of the equilibration.41 Figure 1 shows that the ⟨R2(s)/s⟩ of the chain of length N =
recently, we successfully circumvented this intrinsic problem with SANS by employing a spherical harmonic expansion technique.23 Our SANS measurements on well-entangled polystyrenes show that the unique features associated with the chain retraction hypothesis are not experimentally observed, lending support to earlier SANS studies.18−20 Despite this recent development in small-angle neutron scattering experiments, it is still necessary to examine the chain retraction hypothesis of the tube model by using a different technique to corroborate the findings. When experimentalists disagree,22,23,25 computational simulations can offer tremendous insights that potentially help resolve the issues. Looking back, it is a bit astonishing that computer simulations have not been employed to critically and directly examine the chain retraction hypothesis, in a manner originally envisioned by Doi and Edwards in their 1978 paper.8 Historically, the seminal molecular dynamics simulation by Kremer and Grest26 played a key role in enthroning the linear tube theory as the current paradigm of entangled polymer dynamics in the equilibrium state. Should not a similar study using nonequilibrium molecular dynamics (NEMD) simulation for entangled polymers likewise serve to help clarify the chain retraction hypothesis of the tube model? We note that a large body of the scientific literature is devoted to mesoscopic simulations with the slip-link type approaches that are inspired by the tube theory.27−31 However, those simulations were inherently incapable of addressing the basic premise of the tube approach. As a result, we resort to molecular dynamics simulations where no assumptions about the entanglement dynamics are introduced. The existing microscopic NEMD studies based on either the coarse-grained bead−spring model32,33 or the united-atom model34 have been limited to mildly entangled systems that are unsuitable for critically examining the chain retraction hypothesis (see section S2 of the Supporting Information for a further explanation). In this work, we break this boundary in the previous investigations by simulating wellentangled polymer chains (number of entanglements per chain Z≅33). Taking advantage of the Titan Cray XK7 supercomputer at the Oak Ridge Leadership Computing Facility, we perform large-scale nonequilibrium molecular dynamics simulations with the explicit goal of critically testing the chain retraction hypothesis of the tube theory. We adopt the coarse-grained bead−spring model pioneered by Kremer and Grest,26 which has been extensively used to study the dynamics of entangled polymers in both equilibrium and nonequilibrium states.33,35,36 The pair interactions between any two beads are described by the repulsive part of the Lennard-Jones potential, that is, the so-called Weeks-ChandlerAndersen (WCA) potential: UWCA = 4ε[(σ/r)12 − (σ/r)6] + ε for r < 21/6σ and UWCA = 0 for r ≥ 21/6σ, where r is the distance between two beads, σ is the effective diameter, and the depth of the potential well ε sets the energy scale of the system. Attractive interactions37 should not play a significant role in the current problem of the polymer melt rheology and therefore are not considered. The bond connectivity of the polymer chain is maintained by the finitely extensible nonlinear elastic 1 (FENE) potential: UFENE = − 2 k bR 02 ln[1 − (r /R 0)2 ], where kb = 30ε/σ2 and R0 = 1.5σ. To simulate well-entangled polymer melts, our system consists of M = 250 linear chains, each with N = 2000 beads, and the number density is fixed at ρ = MN/V = 0.85σ−3 with V being the volume of the simulation box.
Figure 1. Mean-square internal distance ⟨R2(s)⟩ divided by the chemical distance s for various chain lengths. s = |i − j| is the difference in indices of two beads i and j within a chain. The results for N = 40, 120, and 500 are included here as references. Error bars represent the standard deviations over ten independent simulations. The gray solid line is the benchmark function in the literature.41
2000 exhibits the expected behavior and plateaus at large separations, in excellent agreement with the literature data41 as well as our own simulations of shorter chains. Figure 1 also demonstrates that 10 trajectories of the current system (M = 250 and N = 2000) are adequate for achieving good statistics. We further confirm the quality of the equilibration procedure by examining the end-to-end distance, radius of gyration, and static single-chain structure factor, and comparing these to benchmarks in the literature.26,36 We determined the Rouse relaxation time τR of the polymer chain with N = 2000 from the scaling relation: τR = τe(N/Ne)2, where the entanglement strand relaxation time τe = 3290τ0 and 191
DOI: 10.1021/acsmacrolett.7b00900 ACS Macro Lett. 2018, 7, 190−195
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Figure 2. (a) Snapshot of the simulated entangled polymers in the equilibrium state. The picture in the bottom right corner shows the conformation of one of the chains in the isotropic melt. (b) The corresponding spherical harmonic expansion coefficients S0l (Q) of the single-chain structure factor S(Q) for the equilibrated melt. The inset is the cross section of S(Q) on the xz-plane (the z-axis is parallel to the uniaxial extension direction, whereas the x-axis is perpendicular to it). (c) Snapshot of the simulated polymers immediately after the uniaxial extension (λ = 1.8). (d) The corresponding spherical harmonic expansion coefficients.
and computational cost, five out of the ten independently equilibrated melts are used in the NEMD simulations. Figure 3 shows the decay of the normalized tensile stress Σ/ Ge during the relaxation, where Ge is the rubbery plateau
the number of beads per entanglements Ne = 60, following the work of Cao and Likhtman.32 For Ne = 60, the number of entanglements per chain (Z) is approximately 33 for the simulated polymer melt, which is very close to the estimated entanglement level of 34 for the experimental system.23 While there is no consensus in the literature with regard to the exact value of Ne for the “standard” coarse-grained model, we point out that even if we take a larger value from the Primitive Path Analysis,43 that is, Ne = 85, the main conclusions of this work are not affected. Additionally, the quantitative agreement between experiment and simulation, which we shall demonstrate below, suggests that our choice of Ne and other simulation parameters are reasonable. For the NEMD simulations, the equilibrated polymer melt is uniaxially elongated in the z-direction with a constant engineering strain rate (Figure 2), and the equilibrium pressure of the melt (4.88ε/σ3) is imposed in the x- and y-directions via a Nosé−Hoover barostat. The same protocol has been used in a number of NEMD simulations44−46 (see section S4 of the Supporting Information for a brief discussion of the validity of this protocol). To mimic the SANS experiment,23 the melt is deformed to a final stretching ratio of λ = Lz,f/Lz,i = 1.8, where Lz,f and Lz,i are the final and initial lengths of the simulation box in the z-direction, respectively. The stretching procedure is performed within an interval of t1 = 7 × 104τ0, which gives an initial Rouse Weissenberg number of WiR,i = (λ − 1)τR/t1 = 41.8, a value that is comparable with the experimental condition. After the deformation, molecular dynamics simulations with a Langevin thermostat are carried out at T = 1ε/kB to study the relaxation behavior up to approximately 2.7τR, which is much shorter than the estimated terminal relaxation time47 but sufficiently long for testing the chain retraction hypothesis. To achieve a reasonable balance between statistics
Figure 3. Stress relaxation behavior in NEMD simulation of the polymer melt with N = 2000. Σ = σzz − (σxx + σyy)/2 is the tensile stress defined by the difference in the diagonal components of the stress tensor σαβ. Ge is the rubbery plateau modulus. Error bars represent the standard deviations over five independent simulations. The adaptive interval procedure of Gao and Weiner is used in the time averaging of stress.48 Solid line: experimental data.23
modulus and is estimated as Ge = ρkBT/Ne = 0.014εσ−3. There is remarkable agreement between the NEMD data and our rheological measurement on an entangled polystyrene melt.23 This highly nontrivial result confirms that the present coarsegrained model is capable of capturing some essential physics of entangled polymers even in the nonequilibrium state, as others have also suggested.32,33 Our previous SANS investigation23 of the chain retraction hypothesis of the tube model consisted of two critical analyses. The first one involves examination of the fine features of the 2D 192
DOI: 10.1021/acsmacrolett.7b00900 ACS Macro Lett. 2018, 7, 190−195
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dimensions of the polymer without significant relaxation of the orientation anisotropy. This physical picture translates to a horizontal shift of the S02(Q) peak around the Rouse time, accompanied by an increase of anisotropy in the high-Q region (Figure 4a). These unique spectral features of chain retraction are not observed in our previous SANS experiments on well-entangled polystyrenes (Figure 4b).23 This finding immediately raises questions of the validity of the central hypothesis of the nonlinear tube theory. Considering the gravity of the issue, as well as the presence of contradictory SANS results in the literature,22,25 we naturally wonder what “computer experiments” have to say. It turns out that implementing the spherical harmonic expansion analysis of the single-chain structure factor S(Q) is quite straightforward in MD simulations. Figure 2b,d show two examples of such a decomposition for the isotropic and deformed states, respectively. Focusing on the expansion coefficient S02(Q) during the stress relaxation, we find that our NEMD simulation reproduces all the key features of the SANS experiment (Figure 4c and its insets): there is no peak shift up to 2.7τR and the deformation anisotropy relaxes monotonically at all Qs. Moreover, there is excellent quantitative agreement between experiment and simulation for the peak position and magnitude of S02(Q). The second critical analysis in our prior SANS study involves examination of the radius of gyration in the parallel and perpendicular directions to stretching. Since there is no ideal method for determining Rg in this case, as discussed above and elsewhere,23 we previously followed the traditional protocol and extracted the Rg values by fitting the intensities in the parallel and perpendicular directions with the Debye function.24 In contrast to scattering techniques, computer simulations have the power of accurately and unambiguously determining the anisotropic radius of gyration tensor in real space. Therefore, our NEMD simulations provide us the opportunity to directly carry out the critical test that Doi and Edwards proposed in their 1978 paper.8 While the tube theory predicts a nonmonotonic change of the radius of gyration in the perpendicular direction (Figure 5a), both the SANS experiment23 and our NEMD simulations suggest a different picture (Figure 5b,c). In the NEMD simulation, R⊥g evolves monotonically toward the equilibrium value, showing no sign of decrease up to 2.7τR. The error bars for the simulation, estimated from five independent runs, are on the order of 2.6%. In contrast, the tube theory anticipates approximately a 10% reduction in R⊥g within the Rouse time, a feature that is well beyond the error bars of the NEMD simulation. Our additional analysis of the mean-square internal separation in both the parallel and perpendicular directions indicates the absence of nonmonotonic features at all length scales. Considering the unmistakable, monotonic trend revealed by the current simulation, we reason that it is unlikely for NEMD simulations on a larger, more entangled system to produce qualitatively different results.25 Lastly, it is worth noting that a previous NEMD simulation study of unentangled polymer melts also reported a monotonic change of the radius of gyration tensor during the stress relaxation after a large step deformation.50 In summary, we have performed large-scale nonequilibrium molecular dynamics simulations on well-entangled polymers to critically test the chain retraction hypothesis of the nonlinear tube theory. The results of our NEMD simulations are qualitatively consistent with the recent SANS study,23 but at
SANS spectra using the spherical harmonic expansion technique.23,49 In the case of uniaxial extension, we have demonstrated that the anisotropic single-chain structure factor S(Q) can be expanded as a series of even-order Legendre functions Pl(cos θ): S(Q ) =
∑ l :even
Sl0(Q ) 2l + 1 Pl(cos θ )
(1)
where S0l (Q) is the Q-dependent expansion coefficient. The details of this technique and the definition of the relevant terms can be found in our previous work. 23 The physical interpretation of this expansion is that each coefficient S0l (Q) represents the Q-dependent deformation anisotropy associated with the spherical harmonic (Legendre) function. S00(Q) is the isotropic component of S(Q), whereas S02(Q) corresponds to the leading anisotropic component. In the spherical harmonic expansion analysis, the chain retraction hypothesis produces two distinct features for S02(Q) during the stress relaxation after a large step uniaxial deformation (Figure 4).23 Chain retraction within the Rouse time is expected to reduce the overall
Figure 4. (a) Theoretical predictions from the GLaMM model12 about S02(Q) during stress relaxation. The chain retraction hypothesis gives rise to a characteristic peak shift, as indicated by the black arrow. Additionally, the anisotropy at high Q increases during the retraction. The details of the tube model calculations are given in section S1 of the Supporting Information. (b) Results from previously reported SANS experiments,23 where the unique features associated with chain retraction are not observed. Solid lines: Guide to eye. (c) Results from NEMD simulations up to 2.7τR. Here, the scattering wavenumbers Q in theory and simulation are converted to real units by mapping the equilibrium radius of gyration in these calculations to that of the isotropic polystyrene sample determined by SANS measurement. The equilibrium Rgs are 24.5 σ and 170 Å in simulation and experiment, respectively. Therefore, σ = 6.94 Å in the mapping procedure. The insets in (a) and (c) provide a zoom-in view of the data in the Q range 0.008−0.02 Å−1. 193
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Letter
ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsmacrolett.7b00900. Details of the tube model calculations; Explanations for the necessity of studying well-entangled polymers; Additional information about the equilibration of the polymer melt; A plot of the stress growth during the uniaxial deformation (PDF).
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. ORCID
Wen-Sheng Xu: 0000-0002-5442-8569 Jan-Michael Y. Carrillo: 0000-0001-8774-697X Bobby G. Sumpter: 0000-0001-6341-0355 Yangyang Wang: 0000-0001-7042-9804 Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This research was sponsored by the Laboratory Directed Research and Development Program of Oak Ridge National Laboratory, managed by UT Battelle, LLC, for the U.S. Department of Energy. The data analysis was performed at the Center for Nanophase Materials Sciences, which is a DOE Office of Science User Facilities. This research used resources of the Oak Ridge Leadership Computing Facility at the Oak Ridge National Laboratory, which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC05-00OR22725. We thank Yexin Zheng (U. Akron) for helpful discussions.
Figure 5. (a) Theoretical predictions from the GLaMM model12 about the evolution of the radius of gyration in the parallel (R∥g ) and perpendicular (R⊥g ) directions during the stress relaxation. Chain retraction is expected to lead to a nonmonotonic change of R⊥g . R∥g,0 and R⊥g,0 are referred to as the equilibrium values of the radius of gyration in the parallel and perpendicular directions, respectively. 1 ∥ ⊥ = R g,0 = 3 R g,0 , with Rg,0 being the equilibrium Apparently, R g,0 (total) radius of gyration. Note that the elongation rate of the step deformation does not significantly affect the theoretical prediction when the Rouse Weissenberg number is much larger than unity. (b) Results from SANS experiments,23 where R⊥g increases monotonically during the relaxation. (c) Results from MD simulations. The horizontal (gray) dashed lines in (a) and (c) indicate the level of the expected R⊥g immediately after chain retraction according to the original Doi−Edwards tube model (DE). The horizonal (orange) dash-dotted line in (c) represents the R⊥g after chain retraction according to the GLaMM model, that is, the minimum value of R⊥g displayed in (a). Note that these horizontal lines do not represent the temporal evolution of R⊥g .
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REFERENCES
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