Wang, Zhe; Lam, Christopher N.; Chen, Wei-Ren; Wang, Weiyu; Liu, Jianning; Liu, Yun; Porcar, Lionel; Stanley, Christopher B.; Zhao, Zhichen; Hong, Kunlun; Wang, Yangyang

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Selected for a Viewpoint in Physics PHYSICAL REVIEW X 7, 031003 (2017) 1,* 2 1 2 3 4,5 6 Zhe Wang, Christopher N. Lam, Wei-Ren Chen, Weiyu Wang, Jianning Liu, Yun Liu, Lionel Porcar, 1 3 2 2,† Christopher B. Stanley, Zhichen Zhao, Kunlun Hong, and Yangyang Wang Biology and Soft Matter Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA Center for Nanophase Materials Sciences, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA Department of Polymer Science, University of Akron, Akron, Ohio 44325, USA Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA Department of Chemical and Biomolecular Engineering, University of Delaware, Newark, Delaware 19716, USA Institut Laue-Langevin, B.P. 156, F-38042 Grenoble CEDEX 9, France (Received 23 March 2017; published 10 July 2017) The flow and deformation of macromolecules is ubiquitous in nature and industry, and an understanding of this phenomenon at both macroscopic and microscopic length scales is of fundamental and practical importance. Here, we present the formulation of a general mathematical framework, which could be used to extract, from scattering experiments, the molecular relaxation of deformed polymers. By combining and modestly extending several key conceptual ingredients in the literature, we show how the anisotropic single-chain structure factor can be decomposed by spherical harmonics and experimentally reconstructed from its cross sections on the scattering planes. The resulting wave-number-dependent expansion coefficients constitute a characteristic fingerprint of the macromolecular deformation, permitting detailed examinations of polymer dynamics at the microscopic level. We apply this approach to survey a long- standing problem in polymer physics regarding the molecular relaxation in entangled polymers after a large step deformation. The classical tube theory of Doi and Edwards predicts a fast chain retraction process immediately after the deformation, followed by a slow orientation relaxation through the reptation mechanism. This chain retraction hypothesis, which is the keystone of the tube theory for macromolecular flow and deformation, is critically examined by analyzing the fine features of the two-dimensional anisotropic spectra from small-angle neutron scattering by entangled polystyrenes. We show that the unique scattering patterns associated with the chain retraction mechanism are not experimentally observed. This result calls for a fundamental revision of the current theoretical picture for nonlinear rheological behavior of entangled polymeric liquids. DOI: 10.1103/PhysRevX.7.031003 Subject Areas: Fluid Dynamics, Materials Science, Soft Matter I. INTRODUCTION fixed obstacles. A few years later, in a series of seminal publications [6–8], Doi and Edwards illustrated how the The entanglement phenomenon is one of the most molecular motion under flow and deformation could be important and fascinating characteristics of long flexible explained with the aid of the tube concept. The advent of chains in the liquid state [1–3]. Our current understanding the tube model has revolutionized the field of polymer of the dynamics of entangled polymers is built on the tube dynamics, and the predictions of the model about both the theoretical approach pioneered by de Gennes [4] and Doi linear and nonlinear viscoelastic properties of entangled and Edwards [5–9]. In his 1971 paper [4], de Gennes polymers have been significantly improved over the years, demonstrated how the diffusion problem of a flexible chain by incorporating additional molecular mechanisms such as could be understood in terms of reptation in the presence of contour length fluctuation [10–12], constraint release [13–17], and chain stretching [18–20]. However, despite the remarkable success of the tube approach, particularly in zwang.thu@gmail.com the linear response regime, one of the central hypotheses of wangy@ornl.gov the model has thus far eluded experimental confirmation. Published by the American Physical Society under the terms of In an effort to account for the nonlinear rheological the Creative Commons Attribution 4.0 International license. behavior, Doi and Edwards [6] proposed a unique micro- Further distribution of this work must maintain attribution to scopic deformation mechanism for entangled polymers, the author(s) and the published article’s title, journal citation, and DOI. which asserts that the external deformation acts on the tube, 2160-3308=17=7(3)=031003(17) 031003-1 Published by the American Physical Society ZHE WANG et al. PHYS. REV. X 7, 031003 (2017) instead of the polymer chain [21]. The chain retraction scattering techniques, particularly small-angle neutron within the affinely deformed tube would lead to nonaffine scattering, provide a powerful experimental method for evolution of chain conformation beyond the Rouse time, this problem, because of their ability to retrieve micro- with entanglement strands being oriented but hardly scopic information about chain statistics over a wide range stretched. This hypothesis, being a keystone of the tube of length scales. The theoretical [6,40–51] and experimen- model, stands in stark contrast to the elastic deformation tal attention [25–30,32,34,35,52–59] in the past, however, mechanisms of many other alternative theoretical approaches has been focused primarily on the analysis of the radius of such as the transient network model [22–24], where the affine gyration tensor of a polymer under flow and deformation, deformation mechanism is adopted. While scattering tech- and a systematic approach for quantitative analysis of the niques, particularly small-angle neutron scattering (SANS), anisotropic scattering patterns has not emerged from the have long been envisioned as the ideal tool for critical previous investigations. The radius of gyration, being an examination of this key hypothesis of the tube model, the averaged statistical quantity, offers only a coarse-grained SANS investigations in the past several decades have not led picture of the molecular deformation on large length scales. to a clear conclusion, with many questioning the validity of In the case of entangled polymers, because of the large the nonaffine mechanism [25–30], some claiming support overall chain dimensions involved, it often becomes [31–33], and others being silent on this issue [34,35]. impractical to determine the radius of gyration R in a Moreover, recent experimental studies [36–39] have called model-independent manner via the Guinier analysis [60]. into question the basic premises of the tube model, including This difficulty has plagued research aimed at resolving the the picture of barrier-free Rouse retraction. Given the critical controversy regarding the chain retraction mechanism pro- role that chain retraction plays in the tube model, a clarifi- posed by Doi and Edwards. Moreover, the traditional R cation of the molecular relaxation mechanism of entangled analysis provides only an incomplete picture of the molecu- polymers after a large step deformation is an urgent need. lar deformation by examining a limited number of directions Here, we present a general approach for extracting in space. This method is evidently inadequate in the case of microscopic information about molecular relaxation in complex scattering patterns [61–71], such as “butterfly” and deformed polymers using small-angle scattering (SAS). “lozenge” shapes, where a full two-dimensional data analy- By combining and modestly extending the ideas of spheri- sis is clearly a more desired approach. cal harmonic decomposition in the literature, we demon- Motivated by the aforementioned scientific as well as strate how the fingerprint features of molecular relaxation technical challenges, we set out to explore a different can be obtained by a generalized Fourier analysis of the 2D approach to the rheo-SAS problem of polymers, by SAS spectrum. The application of this novel method to borrowing, combining, and extending the idea of spherical small-angle neutron scattering experiments on deformed entangled polymers permits, for the first time, quantitative harmonic expansion that has been introduced by several groups of authors in different contexts spanning over a and model-independent analysis of the full anisotropic 2D spectrum, and provides decisive and convincing evidence period of roughly half a century [72–97]. against the chain retraction mechanism conceived by the Building on the Taylor expansion treatment of earlier tube model. We show that the two prominent spectral researchers [98–100], Evans and co-workers [76,78– features associated with the chain retraction—peak shift of 80,82,87] were among the first who systematically inves- the leading anisotropic spherical harmonic expansion tigated the structural distortion of simple fluids under shear coefficient and anisotropy inversion in the intermediate by expressing the anisotropic pair distribution function in wave number (Q) range around Rouse time—are not terms of spherical harmonics. These computational studies experimentally observed in a well-entangled polystyrene inspired the discussion of the principles of group-theoretical melt after a large uniaxial step deformation. This result calls statistical mechanics for non-Newtonian flow [89–91], and for a fundamental revision of the current theoretical picture these concepts were also echoed by the experimental efforts for nonlinear rheological behavior of entangled polymeric of a number of research groups [77,81,88,92,93] around the liquids. The application of the spherical harmonic expan- same time. However, these investigations focused exclu- sion approach, as powerfully illustrated by the current sively on colloidal suspensions under small shear perturba- study of entangled polymers, opens a new venue for tion, whereas large extensional deformation is the preferred improving our understanding of macromolecular flow condition for probing polymeric systems. Additionally, and deformation via rheo-SAS techniques. while it is straightforward to perform spherical harmonic decomposition in computer simulations where three- II. HISTORICAL SURVEY OF THE FIELD dimensional real-space information of particle coordinates is readily available, small-angle scattering experiments can The central problem in the study of macromolecular deformation is to gain knowledge about the evolution of only access the two-dimensional reciprocal space cross conformational statistics of polymers under external per- sections on certain planes, for which the projected spherical turbation. It has long been recognized that small-angle harmonics may not necessarily form an orthogonal basis set. 031003-2 FINGERPRINTING MOLECULAR RELAXATION IN … PHYS. REV. X 7, 031003 (2017) The delicacy of this issue has not been fully appreciated until segment distribution function that describes statistically very recently [101]. the separation between beads i and j. We can define an In the polymer community, Roe and Krigbaum have intrachain pair distribution function gðrÞ as [104] already conceived the idea of spherical harmonic expansion N N XX of the orientation distribution function of statistical seg- gðrÞ¼ ψ ðrÞ; ð3Þ ij ments in deformed polymer networks and discussed the i¼1 j¼1 potential application of this technique in analyzing the variation of x-ray intensity of the amorphous halo observed which is related to the single-chain structure factor through for stretched polymers [72]. However, it was not until the the Fourier transform: work of Mitchell and co-worker almost 20 years later [84–86] that a more formal treatment of the measured −iQ·r SðQÞ¼ gðrÞe dr: ð4Þ scattering intensity in terms of Legendre expansion for the uniaxial extensional geometry was developed. Despite the We note that statistical distribution functions are the widespread use of this method, the polymer community has centerpiece of the kinetic theory of polymer fluid dynam- so far mainly looked at the problem of scattering of ics. The long tradition of kinetic theory for polymeric deformed polymers through the lens of rheology, where liquids was initiated by the celebrated paper of Kramers the major interest is to extract an order parameter to [105], developed by Kirkwood and co-workers compare with stress. Consequently, the previous works [98,106,107], Rouse [108], Zimm [109], Lodge and co- in this area fell short at recognizing the value of spherical workers [23,110,111], Yamamoto [24], and others [112– harmonic expansion as a general approach for character- 115], and epitomized in the classical book by Bird et al. izing Q-dependent deformation anisotropy and chain con- [102]. We see, from Eqs. (1)–(4), that the spatially formation at different length scales. anisotropic scattering intensity accessed by SAS techniques in the reciprocal space reflects nothing but the perturbation III. SPHERICAL HARMONIC EXPANSION of configuration distribution functions of the polymer chain APPROACH by external deformation. However, this seemingly obvious and yet fundamental viewpoint has not been fully appre- A. Philosophical shift ciated, as witnessed by the immense disparity between In this section, we present our general formulation of the theoretical development and experimental efforts by SAS. small-angle scattering problem of deformed polymers. We As we show below, a powerful weapon for analyzing SAS start the discussion by describing the angle from which we data can be forged by drawing upon the concept of approach this topic. As we demonstrate, our viewpoint spherical harmonic expansion. This new approach supplies represents a philosophical departure from the previous a convenient platform for connecting small-angle scattering method employed in the polymer community, where the experiments and statistical and molecular theories of primary concern was to extract a single order parameter. polymers. Following the convention in the field of polymer dynamics [9,102], let us suppose that the polymer chain is modeled B. 3D decomposition and 2D reconstruction by a series of N beads, each located at r . In the context of In the context of our current investigation, the measured small-angle neutron scattering by isotopically labeled scattering signal is dominated by coherent scattering, i.e., deformed melts, the measured coherent scattering intensity I ≫ I . Thus, coh inc I , which is dependent on a scattering wave vector Q,is coh proportional to the single-chain structure factor (form .h i factor) SðQÞ [60,103]: SðQÞ¼ IðQÞ limI ðQÞ ; ð5Þ iso Q→0 IðQÞ¼ I ðQÞþ I coh inc where I ðQÞ is the scattering intensity from the isotropic iso 2 2 ¼ðb − b Þ fð1 − fÞnN SðQÞþ I ; ð1Þ D H inc sample. Because of this simple proportionality between SðQÞ and IðQÞ, we focus only on SðQÞ in the discus- 1 sions below. −iQ·ðr −r Þ i j SðQÞ¼ e ; ð2Þ Formally, the dependence of the single-chain structure i;j factor (form factor) SðQÞ on the magnitude (Q) and orientation (Ω) of the scattering wave vector can be where (b − b ) is the contrast factor due to the difference D H expressed in terms of spherical harmonics: in the coherent scattering length between hydrogen and deuterium, f is the fraction of the labeled species, n is the m m SðQÞ¼ S ðQÞY ðΩÞ; ð6Þ number density, and I is the incoherent background. coh l l l;m h i stands for ensemble average. Let ψ ðrÞ be the ij 031003-3 ZHE WANG et al. PHYS. REV. X 7, 031003 (2017) where S ðQÞ is the expansion coefficient corresponding to angle in the xy plane from the x axis with ϕ ∈ ½0; 2πÞ.For each real spherical harmonic function Y ðΩÞ. In this work, the uniaxial extension problem investigated herein, the our choice of the spherical coordinates follows the con- stretching is along the z axis and the incident neutron beam vention in physics [116], for which θ is the polar angle from is perpendicular to the xz plane [Fig. 1(a)]. Our real the positive z axis with θ ∈ ½0; π, and ϕ is the azimuthal spherical harmonic functions are defined as qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃ jmj ðl−jmjÞ! 2 ð2l þ 1Þ P ðcos θÞ sinðjmjϕÞðm< 0Þ ðlþjmjÞ! l pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ m m Y ðΩÞ¼ Y ðθ; ϕÞ¼ ð7Þ 2l þ 1P ðcos θÞðm ¼ 0Þ l l > qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ > pﬃﬃﬃ ðl−mÞ! 2 ð2l þ 1Þ P ðcos θÞ cosðmϕÞðm> 0Þ; ðlþmÞ! l 0 0 which differ from the classical definitions by a factor of functions. In particular, the term S ðQÞY ðθÞ represents the 0 0 pﬃﬃﬃﬃﬃﬃ 1= 4π. Because of the axial symmetry of the uniaxial isotropic part of the distorted structure, whereas the term 0 0 extension problem, it is easy to see that all the m ≠ 0 terms S ðQÞY ðθÞ is the leading anisotropic component that 2 2 and the odd l terms are forbidden [72,84,87,96]; namely, corresponds to the symmetry of uniaxial deformation. Equation (8) gives the spherical harmonic expansion of SðQÞ¼ SðQ; θÞ the anisotropic single-chain structure factor in three- 0 0 dimensional space. In order to obtain the expansion ¼ S ðQÞY ðθÞ l l coefficients S ðQÞ from SAS experiments, we must con- l∶even l pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ sider the cross section of SðQÞ on the two-dimensional 0 0 ¼ S ðQÞ 2l þ 1P ðcos θÞ: ð8Þ l l detector plane. In the case of shear, the low symmetry of l∶even this geometry makes the reconstruction of SðQÞ from 2D In other words, SðQÞ is independent of ϕ and could be scattering patterns rather complicated [101].However,the written as a linear combination of even order Legendre unique symmetry of uniaxial extension greatly simplifies FIG. 1. (a) Illustration of SANS measurements on uniaxially elongated samples: the stretching is along the z axis, whereas the incident SANS beam is perpendicular to the xz plane. (b) Evolution of the SANS spectrum with polymer relaxation. 031003-4 FINGERPRINTING MOLECULAR RELAXATION IN … PHYS. REV. X 7, 031003 (2017) the matter. It is evident from Eq. (8) that the cross section of 0 0 S ðQÞ¼ SðQ;θ;ϕ ¼ 0ÞY ðθÞsinθdθ l l SðQÞ on the xz plane is SðQ ;Q ¼ 0;Q Þ¼ SðQ; θ; ϕ ¼ 0Þ¼ SðQ; θÞ ¼ I ðQ;θÞY ðθÞsinθdθ; ð10Þ x y z xz 2lim I ðQÞ X 0 Q→0 iso 0 0 ¼ S ðQÞY ðθÞ: ð9Þ l l where I ðQ; θÞ is the detected scattering intensity on the l∶even xz xz plane. 1 0 0 Equation (9) and the fact that P ðxÞP ðxÞdx ¼ n m −1 In passing, we note that the spherical harmonic expansion ½2=ð2n þ 1Þδ indicate that Y ðθÞ form an orthogonal nm l approach is inclusive of the traditional data analysis method basis set not only in 3D space but also on the xz plane. that focuses on the scattering intensities along the parallel and Therefore, the expansion coefficient S ðQÞ can be straight- perpendicular directions: the projected structures in these forwardly computed from the small-angle scattering pattern directions could be expressed as linear combinations of the on the xz plane as expansion coefficients. For example, we have pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 0 0 S ðQÞ¼ SðQ ¼ 0;Q ¼ 0;Q Þ¼ SðQ; θ ¼ 0Þ¼ S ðQÞ 2l þ 1P ð1Þ ∥ x y z l l l∶even pﬃﬃﬃ pﬃﬃﬃ pﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃ 0 0 0 0 0 ¼ S ðQÞþ 5S ðQÞþ 9S ðQÞþ 13S ðQÞþ 17S ðQÞþ ; ð11Þ 0 2 4 6 8 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 0 0 S ðQÞ¼ SðQ ;Q ¼ 0;Q ¼ 0Þ¼ S Q; θ ¼ ¼ S ðQÞ 2l þ 1P ð0Þ ⊥ x y z l l l∶even pﬃﬃﬃ pﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃ 5 9 5 13 35 17 0 0 0 0 0 ¼ S ðQÞ − S ðQÞþ S ðQÞ − S ðQÞþ S ðQÞ þ ; ð12Þ 0 2 4 6 8 2 8 16 128 FIG. 2. Illustration of the spherical harmonic expansion approach with the simulated spectrum from the affine model, for a linear polymer that is uniaxially stretched to λ ¼ 3. (a) Angular dependence of the anisotropic single-chain structure factor at various Q’s. (b) Angular dependence of the projections of the spherical harmonic functions on the xz plane. As we discuss in the text, these are essentially Legendre functions. (c) The Q-dependent expansion coefficients S ðQÞ are given by Legendre expansion of SðQ; θÞ. (d) The spherical harmonic expansion decomposes the 2D SANS spectrum into contributions from different symmetries: the isotropic 0 0 0 0 0 0 0 0 component S ðQÞY ðθÞ, and the anisotropic components S ðQÞY ðθÞ, S ðQÞY ðθÞ, S ðQÞY ðθÞ, etc. 0 0 2 2 4 4 6 6 031003-5 ZHE WANG et al. PHYS. REV. X 7, 031003 (2017) where S ðQÞ and S ðQÞ are the cross sections of SðQÞ along question that we raised at the beginning of this article: ∥ ⊥ the parallel and perpendicular directions to stretching, how can we critically test the chain retraction hypothesis of respectively. the tube theory for entangled polymers? The investigations in the past have been focused on the analysis of the radius gyration tensor in step-strain relaxation experiments, fol- C. Fingerprinting molecular deformation lowing the original strategy outlined in the celebrated 1978 To further illustrate the idea of spherical harmonic paper of Doi and Edwards [6] [Fig. 3(a)]. Theoretically, expansion analysis, let us consider a simulated single-chain immediately after a fast step deformation, the radius of structure factor for a polymer uniaxially elongated to a the gyration tensor hR i is equal to the affinely deformed g αβ stretching ratio λ of 3.0 (Fig. 2), calculated using the affine one [6,46]: model [35,117]. At a given magnitude of the scattering wave vector Q, SðθÞ is a periodic function of θ with a 2 2 hR i ¼hR i hðE · uÞ · ðE · uÞ i; ð13Þ g αβ g 0 period of π [Fig. 2(a)]. Because of the orthogonality of α β 0 0 Y ðθÞ, SðθÞ can be decomposed in terms of Y ðθÞ, and the l l 0 where hR i is the equilibrium mean-square radius of g 0 expansion coefficient S can be obtained by angular 0 gyration, E is the deformation gradient tensor, and u is averaging SðθÞ with the weighing factor Y ðθÞ [Eq. (10) a unit vector of isotropic distribution. The averaging h i ad Figs. 2(b) and 2(c)]. Carrying out this procedure for all for ðE · uÞ · ðE · uÞ is taken over the equilibrium distri- α β the different Q’s, we translate the anisotropic 2D scattering bution of u. The chain retraction along the tube around the pattern [Fig. 2(d)] into a 1D plot of Q-dependent expansion Rouse time would reduce all components of hR i by a coefficients S ðQÞ [Fig. 2(c)]. g αβ factor of hjE · uji: While Figs. 2(c) and Fig. 2(d) contain the same amount of information mathematically, the plot of expansion hðE · uÞ · ðE · uÞ i coefficients is much more convenient to analyze in great α β 2 2 hR i ¼hR i : ð14Þ g αβ g 0 detail. Moreover, by isolating the spectral contributions hjE · uji from different symmetries [Fig. 2(d)], the spherical har- monic decomposition approach provides a new means to After the retraction, the chain continues to relax towards the study the molecular relaxation and deformation mecha- equilibrium state through reptation. In the case of uniaxial nisms of polymers, as we see in Secs. IV and V. extension geometry, the above-mentioned mechanism is Mathematically, our treatment of the small-angle scatter- expected to lead to a nonmonotonic change of radius of ing spectrum can be regarded as a generalized Fourier gyration in the perpendicular direction during the stress expansion approach. The anisotropic single-chain structure relaxation. factor is decomposed by spherical harmonic functions and Figure 3(b) gives an example for the evolution of the resynthesized from the 2D patterns in small-angle scattering radius of gyration in the parallel and perpendicular direc- experiments. This approach helps to distill the “hidden” tions to stretching, calculated according to the modified information about molecular deformation from the distorted tube model proposed by Graham, Likhtman, Milner, and 2D spectrum. At this point, it is useful to draw an analogy to McLeish [20], i.e., the GLaMM model. The GLaMM the ideas of “rheological fingerprinting” of complex fluids model is widely considered the state-of-the-art version of using large-amplitude oscillatory shear [118–132]. In par- the tube theory, as it incorporates the effects of reptation, ticular, it has been proposed that the Fourier or Chebyshev chain stretch, and convective constraint release on the expansion coefficients for the stress response could be used microscopic level through a stochastic partial differential to define unique fingerprints of nonlinear rheology of soft equation for the contour dynamics. From Fig. 3(b), we see viscoelastic materials and reveal properties that are typically that the qualitative feature of chain retraction—the non- obscured by conventional test protocols. It has also been monotonic change of R in the perpendicular direction—is recognized that the model-independent nature of the har- well captured by the GLaMM model. In addition, the monic analysis not only enables quantitative characterization ⊥ ⊥ magnitude of retraction, i.e., R ðt ¼ 0Þ=R , is also con- g g of materials but also allows one to challenge constitutive sistent with the expectation from the original Doi-Edwards pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ relations. From this perspective, our spherical harmonic theory, which predicts the ratio to be hjE · uji. expansion approach to SAS and the widely used (general- In principle, one should be able to critically test the chain ized) Fourier analysis in the complex fluids community share retraction hypothesis by performing SANS experiments on a similar philosophical root. uniaxially stretched entangled polymer melts and compar- ing the measured R with theoretical predictions. In reality, IV. MOLECULAR FINGERPRINTS experimentalists have encountered tremendous difficulty in OF CHAIN RETRACTION following this approach. First, due to inevitable elastic Having laid the foundation for the spherical harmonic breakup after a large step deformation [133], stress relax- expansion technique, let us now return to the central ation experiments of this kind are typically restricted to 031003-6 FINGERPRINTING MOLECULAR RELAXATION IN … PHYS. REV. X 7, 031003 (2017) FIG. 3. (a) Illustration of the molecular relaxation mechanism envisioned by the Doi-Edwards theory. The chain conformation immediately after the step-strain deformation can be described by the affine transformation. The chain retraction around Rouse time quickly equilibrates the contour length and leads to a reduction of all components of the radius of gyration tensor. The molecular relaxation continues via reptation after chain retraction. (b) Evolution of the radius of gyration in the parallel and perpendicular directions to stretching, as predicted by the GLaMM model for an entangle polystyrene (Z ¼ 34) after a step deformation of λ ¼ 1.8, performed with a constant crosshead velocity v ¼ 40l =τ . (c) Upper panel: Expansion coefficient S ðQÞ before and after the chain 0 R retraction, calculated using the affine model and Doi-Edwards (DE) model, respectively. Lower panel: Predictions from the GLaMM model. Our choice of the GLaMM parameters follows the standard practice in the literature [20]. relatively small strains. This constraint means the magni- expansion technique. The upper panel of Fig. 3(c) presents tude of the chain retraction would be rather small and thus the major component of the deformation anisotropy S ðQÞ require highly accurate SANS experiments. On the other before and after the chain retraction for a step strain of hand, however, it is practically impossible to reliably λ ¼ 1.8, calculated using the affine model and the Doi- determine the radius of gyration tensor through model- Edwards model [6], respectively. We see that the chain independent Guinier analysis, because of the limited Q retraction would lead to a horizontal shift of S ðQÞ towards range and flux of existing SANS instruments and the large higher Q. This prediction is consistent with the physical molecular size of entangled polymers. As a result, exper- picture offered by the tube model: the chain retraction reduces imentalists in the past have had to resort to using the the overall dimension of the chain, causing the horizontal affinelike model to determine R by averaging over an shift, but the orientation anisotropy is not relaxed, as the peak opening angle along the principal axes [26,28,30,32,34,55]. amplitude remains the same. This analysis shows that there are two distinct spectral features associated with the chain Putting the ambiguity in model fitting aside, this approach does not even seem to be logically self-consistent: it is not retraction in a step-strain relaxation experiment: the peak shift of S ðQÞ and the increase of anisotropy in the intermediate Q possible to critically test a nonaffine model (tube model) by fitting the experimental data with the affinelike model, range. We term the latter feature “anisotropy inversion”— which assumes the same transformation rule for chain instead of relaxation of deformation anisotropy, the chain conformation at all length scales. retraction is expected to give rise to an increase of anisotropy To circumvent the dilemma with the traditional R analy- in the intermediate Q range. sis, here, we propose a different approach to examine the Having made this qualitative analysis with the original chain retraction hypothesis by using the spherical harmonic Doi-Edwards theory, we now turn to the more sophisticated 031003-7 ZHE WANG et al. PHYS. REV. X 7, 031003 (2017) the chain retraction hypothesis of the tube model, by using GLaMM model for quantitative predictions [Fig. 3(c),lower small-angle neutron scattering. Our experimental system is panel]. First, the GLaMM model still faithfully captures the based on the mixture of protonated and deuterated poly- two unique features of chain retraction, i.e., peak shift and styrene (PS) homopolymers that are synthesized by anionic anisotropy inversion. Moreover, the original Doi-Edwards polymerization in benzene with sec-butyllithium as the model and the GLaMM model produce consistent calcu- initiator (h-PS: M ¼ 450 kg=mol, M =M ¼ 1.06; d-PS: lations about the magnitude of the peak shift. Beyond Rouse w w n M ¼ 510 kg=mol, M =M ¼ 1.04). The h-PS and d-PS time, the GLaMM model predicts that S ðQÞ continues to w w n are dissolved at an h=d ratio of 5=95 in toluene, fully relax towards the equilibrium state without much change in mixed, and precipitated in excess methanol. The resulting the peak position, in agreement with the idea that relaxation blend is dried in a vacuum oven first at room temperature after chain retraction is orientational. and then at 130 °C to completely remove the residual The above calculations and analyses powerfully dem- solvents. onstrate that the spherical harmonic expansion technique The linear viscoelastic properties of the blend are char- allows us to directly translate the physical idea of chain acterized on an HR2 rheometer (TA Instruments) by small retraction into unique and intuitive spectral patterns. More amplitude oscillatory shear measurements in the frequency importantly, it provides a platform for us to bring together range 0.1–100 rad=s and at temperatures between 200 °C theory and experiment, and to critically test, for the first and 120 °C. Figure 4(a) shows the master curve for the time, the retraction hypothesis in a model-independent, 0 00 dynamic moduli (G and G )at 130 °C, constructed by using nonlinear-fitting-free manner. the time-temperature superposition principle [134].The average number of entanglements per chain Z is estimated V. NEW RESULTS AND DISCUSSIONS to be 34 for this system (Z ¼ G M =ρRT, with G being the e w e A. Experimental methods plateau modulus and ρ the polymer density). We evaluate Equipped with new insight from spherical harmonic the Rouse relaxation time τ using three different methods: expansion analysis, we carry out a critical examination of the classical tube model formula (τ ¼ τ=3Z,with τ FIG. 4. (a) Linear viscoelastic properties of the mixture of d-PS and h-PS at 130 °C. (b) Stress relaxation behavior after a step deformation of λ ¼ 1.8, performed with a constant crosshead velocity v ¼ 40l =τ . (c) Expansion coefficients S ðQÞ at t ¼ 0, τ , and 0 R R 10τ . The solid lines are computed according to the affine deformation model for λ ¼ 1.8. The SANS data are collected on the NGB30 SANS beam line at NIST. 031003-8 FINGERPRINTING MOLECULAR RELAXATION IN … PHYS. REV. X 7, 031003 (2017) being the reptation time) [135], the Osaki formula B. Spherical harmonic expansion analysis 2 2.4 [τ ¼ð6M η=π ρRTÞð1.5M =M Þ ,with η being the R w e w Figure 4(c) presents spherical harmonic expansion coef- zero-shear viscosity and M the entanglement molecular ficients S ðQÞ (l ¼ 0, 2, 4, 6) immediately after the step weight] [136,137], and the Likhtman-McLeish theory [12], deformation (t ¼ 0), and at τ and 10τ . As a reference, we R R which yield 251, 592, and 715 s, respectively at 130 °C. In also plot the coefficients of the affine deformation model this work, we choose to use Osaki’s formula, as it overcomes for λ ¼ 1.8. First, Fig. 4(c) nicely illustrates the benefit of the well-known problem with the classical tube model performing spherical harmonic decomposition. The iso- formula and is yet much more straightforward than the 0 tropic component S ðQÞ, which does not change signifi- Likhtman-McLeish theory. cantly from t ¼ 0 to t ¼ 10τ , makes a major contribution The specimens for the SANS measurements are prepared to the 2D spectrum. On the other hand, the relaxation of the on an RSA-G2 Solids Analyzer from TA Instruments 0 0 anisotropic coefficients S ðQÞ and S ðQÞ is clearly visible 2 4 (Fig. 4(b)]. The temperature is controlled by the forced during the same period of time. Therefore, it makes sense to convection oven of the RSA-G2, using nitrogen as the gas separate these different components via the spherical source. Rectangular samples are uniaxially stretched at harmonic decomposition technique, rather than directly 130 °C to λ ¼ 1.8, with a constant crosshead velocity perform analysis on the composite 2D spectra [Fig. 1(b)], v ¼ 40l =τ , where l is the initial length of the sample. 0 R 0 which do not exhibit any characteristic features. Second, The samples are allowed to relax for different amounts of the affine model seems to give a satisfactory description of time (from 0 to 20τ )at 130 °C and then immediately the molecular deformation during the step uniaxial stretch- quenched by pumping cold air into the oven. At 130 °C, the ing [Fig. 4(c), left-hand panel], although upon closer Rouse time of the sample is about 10 min, whereas the examination, we do find that the affine model slightly terminal relaxation time is on the order of 7 h. Furthermore, overestimates the anisotropy at high Q’s (not visible on the since the test temperature is only about 30 °C above T , the scale of the current plot). This result should be expected, relaxation time increases sharply with decreasing temper- because the step deformation is performed with a high ature. In our experiment, it takes less than 10 s for the strain rate—the initial Rouse Weissenberg number (τ v=l ) R 0 temperature to drop from 130 °C to 125 °C, at which point in this case is 40. the chain relaxation is already exceedingly slow. Therefore, Our analysis in Sec. IV reveals that the chain retraction we are able to effectively freeze the conformation of the mechanism should give rise to two distinct spectral features polymer chain with negligible stress relaxation during the for the leading anisotropic component S ðQÞ: peak shift quenching procedure. and anisotropy inversion. Now let us turn to this critical test Small-angle neutron scattering measurements of the of the retraction hypothesis and examine the evolution of quenched glassy polystyrene films are performed on the S ðQÞ during the stress relaxation. Figure 5 shows the NGB30 SANS diffractometers at the Center for Neutron 0 expansion coefficients S ðQÞ at t ¼ 0, 0.5τ , τ , 3τ , 10τ , R R R R Research of NIST. Twowavelengths of incident neutrons, 6.0 and 20τ . The black dashed line marks the peak position and 8.4 Å, are used to cover a range of scattering wave vector immediately after the step deformation (t ¼ 0), whereas the −1 Q from 0.001 to 0.1 Å . The measured intensity is corrected gray dashed line indicates the theoretically expected peak for detector background and sensitivity, and placed on an position after full chain retraction, at t ≈ τ . As we point absolute scale using a direct beam measurement. out in Sec. IV, the “simple” Doi-Edwards model and the FIG. 5. Evolution of the expansion coefficient S ðQÞ. The vertical gray dashed line indicates the theoretically expected peak position after full chain retraction. The SANS data are collected on the NGB30 SANS beam line at NIST. 031003-9 ZHE WANG et al. PHYS. REV. X 7, 031003 (2017) more sophisticated GLaMM model give the same predic- tion for the peak position after retraction. It is evident from Fig. 5 that the unique scattering patterns [Fig. 3(c)] associated with chain retraction are not experimentally observed. The peak shift is negligible up to 20 times the Rouse relaxation time, suggesting there is no strong decoupling of stretch and orientation relaxa- tion. Furthermore, there is no anisotropy inversion either: the anisotropy decays monotonically with time at all Q’s. Here, we emphasize the model-independent nature of the spherical harmonic expansion analysis—it is simply a different way of presenting the “raw” 2D data. Unlike the previous investigations, there is no ambiguity associ- ated with model fitting and no room for human bias. Therefore, our critical test clearly demonstrates that the chain retraction hypothesis of the tube model is not supported by small-angle neutron scattering experiments. C. Analysis of radii of gyration Having reviewed the evidence from spherical harmonic expansion analysis, let us now return to the traditional FIG. 6. (a) Evolution of the radius of gyration in the parallel and analysis of the radius of gyration tensor. As we explain in ∥ perpendicular directions to stretching. R and R are the g;0 g;0 Secs. II and IV, it is not feasible to extract R from the equilibrium radius of gyration in the parallel and perpendicular SANS measurement using model-independent Guinier directions, respectively. Obviously, R ¼ R . (b) Evolution of g;0 g;0 analysis due to the large size of the polymer chain. the absolute scattering intensity in the perpendicular direction. Following the common procedure in the literature, we The SANS data are collected on the NGB30 SANS beam line apply a modified Debye function to determine the R in the at NIST. parallel and perpendicular directions to stretching: with increasing relaxation time for all the data points we ∥;⊥ ∥;⊥ ∥;⊥ −ðQ R Þ 2 4 ∥;⊥ g I ¼ 2I ½e þðQ R Þ − 1=ðQ R Þ þI ; g g 0 ∥;⊥ ∥;⊥ inc have collected; for the sake of clarity in presentation, only the data for t ¼ 0, τ , and 10τ are shown here. Therefore, R R ð15Þ as long as the fitting procedure is consistently applied over a relatively wide Q range, we should obtain only a where I is the forward scattering intensity and I is the 0 inc monotonic trajectory for the R in the perpendicular incoherent background. However, we stress that Eq. (15) direction. should only be taken as an approximate form for the In this context, we further point out a troubling feature in scattering intensity in the intermediate- and low-Q range. the paper of Blanchard et al. Their Fig. 2(a) shows that in The purpose of our analysis is to put our current results in the parallel direction R at 0.4τ is larger than that at perspective with the existing reports in the literature. g R 0.005τ (t ≈ 0). Upon closer examination, it appears that Figure 6(a) shows the evolution of the radius of gyration R the difference is slightly greater than the uncertainty during the stress relaxation for both parallel and represented by the error bars. If so, it would imply that perpendicular directions. While the tube theory predicts the sample was further stretched during the stress relaxa- that chain retraction would lead to a nonmonotonic change tion, which apparently violates the second law of thermo- of radius of gyration in the perpendicular direction to dynamics. This puzzling trend suggests the work of stretching [Fig. 3(b)], experimentally, we observe that the Blanchard et al. might have some experimental issues, R in both perpendicular and parallel directions relax as we briefly discuss below, in Sec. VE. monotonically towards the equilibrium value. This result is consistent with the findings of Maconnachie et al. [25] D. Discussion of possible explanations and Boué et al. [26,27], but at odds with the report of Blanchard et al. [32]. To further demonstrate that the fitting Since the analyses of the spherical harmonic expansion by Eq. (15) does faithfully capture the qualitative behavior coefficient S ðQÞ and the radii of gyration both reject the of the radius of gyration, we present the absolute scattering characteristic signature of chain retraction, we are now intensity in the perpendicular direction during the stress confronted by the inevitable question: what is the explan- relaxation in Fig. 6(b). We do observe a systematic and ation for the observed SANS results, if the chain retraction monotonic “shift” of scattering profile towards lower Q hypothesis does not hold? First, while the tube model is 031003-10 FINGERPRINTING MOLECULAR RELAXATION IN … PHYS. REV. X 7, 031003 (2017) founded on the assumption of affine deformation of the tube [6,138], the idea of nonaffine tube deformation has been floating around for quite some time, particularly in the case of cross-linked systems [139,140]. However, incor- porating this idea into a dynamic theory of polymeric liquids is still an uncharted territory. Furthermore, the major discrepancy between theory and experiment occurs during the stress relaxation rather than the stress growth, as the chain conformation immediately after the step deformation can be approximated by the affine model [Fig. 4(c), t ¼ 0]. Therefore, without an alternative mechanism for molecular relaxation, the idea of nonaffine deformation alone does not seem to be able to explain the experimental observation. Second, is it possible that the chain retraction does take place, but some other nonlinear effects, unanticipated by the original Doi-Edwards theory, are responsible for the absence of the scattering signature of retraction in SANS? For example, the interplay between test chain motion and topological constraints has been widely recognized [16,20,141–145]. Could constraint release (CR) during FIG. 7. Stress relaxation behavior at two different strains λ ¼ 1.14 and λ ¼ 1.80. Here, following the approach of retraction lead to the observed scattering patterns? There Ref. [39], the measured engineering stress σ is normalized is no easy answer to this question. Despite the recent eng by G ½λ − 1=λ , where λ is the imposed strain. (a) Theoretical herculean effort by Sussman and Schweizer [141–145] to predictions from the GLaMM model. (b) Experimental results. model the topological constraints in a self-consistent manner, their theory has not yet produced any predictions about SAS behavior for us to compare with our experi- which clearly contradicts the overwhelming shear-thinning ments. At this moment, the only available option for us to data in the literature. quantitatively explore the “constraint release” effect is the What about the work by Viovy [148], who, upon hearing GLaMM model, in which the CR can be controlled by the results of Boué, also proposed his own explanation for tuning the model parameter c . Our calculations show, ν the absence of chain retraction in SANS experiments? however, that varying c from 0.1 to 1.0 does not change ν Viovy’s proposal consists of two crucial components: one the model prediction of the SANS spectrum for the current is the loss of topological constraints on one chain due to the step-strain experiment in a substantial way. Additionally, retraction of neighboring chains, and the other is the contrary to the prediction of the tube model [37,39,146], screening of retraction due to contour length fluctuation. the stress relaxation of our sample (Fig. 7) is in agreement First, as we discuss above, the idea of loss of entanglements with the “quasilinear” behavior previously identified by at λ ¼ 1.8 does not seem to be compatible with the Cheng et al. [39]—the stress relaxation curves at small quasilinear experimental stress data, which already lacks strain (λ ¼ 1.14) and large strain (λ ¼ 1.80) can be col- the feature of accelerated relaxation due to chain retraction lapsed by applying an affine scaling to stress. The chain (Fig. 7). Second, contour length fluctuation is already retraction mechanism, on the other hand, would produce a incorporated in the calculations with the GLaMM model, two-step relaxation for λ ¼ 1.8 [Fig. 7(a)]. It thus seems but its effect on the qualitative behavior of the SANS unlikely that introducing an additional strong nonlinear CR spectrum is minimal. Overall, the ideas of Viovy have not effect into the tube model would account for the quasilinear been fully developed to yield a complete microscopic model for entangled polymers. This makes it difficult for stress relaxation behavior observed experimentally. Our us to thoroughly evaluate his proposal. In particular, since GLaMM calculations indeed confirm that increasing c R is not an ideal quantity for comparison of theory and would result in more pronounced stress drop during relaxation, which is inconsistent with experiment. experiments, analysis of SðQÞ through the spherical har- It is interesting to point out a “nonclassical proposal” that monic expansion technique is the preferred approach. de Gennes [147] made many years ago upon learning the Unfortunately, this has not been done in Ref. [148]. SANS investigations by Boué et al. [26,27]. The fact that So what could be the explanation for the experimental the signature of retraction was sought but not found in result (Fig. 5)? We are currently not in a position to propose Boué’s studies prompted de Gennes to suggest that the our own theory. However, we add a few more comments chain may indeed not retract at all. However, the proposal before leaving this section. First, it should be emphasized that although a stretched chain obviously needs to “retract” by de Gennes leads to the prediction that “there is no strong shear dependence of the viscosity for a monodisperse melt,” in order to return to its equilibrium state, the concept of 031003-11 ZHE WANG et al. PHYS. REV. X 7, 031003 (2017) “chain retraction” is a construct of the tube model. It refers discussions of the existing SANS experiments on deformed specifically to the restoration of the arc length of the polymers. The consistency of our stretching experiments on primitive chain defined by the “tube.” Therefore, the lack of RSA-G2 is demonstrated by the recorded stress response evidence for chain retraction does not imply that the chains [Fig. 4(b)]—the stress rise and relaxation data of different do not relax. On the contrary, both the stress measurements runs essentially collapse onto the same envelope. [Fig. 4(b)] and SANS patterns (Fig. 5) suggest that the Fourth, the width and thickness of our quenched samples system does continuously relax towards the equilibrium are carefully measured. We verify that the dimensions of state. Therefore, the issue of chain retraction is about the the samples are consistent with the applied macroscopic pathway through which the chain relaxes. In other words, it strain. It is important to note that accurate sample thickness is a matter of the particular molecular relaxation mecha- is critical for determination of the absolute scattering nism that an entangled polymer undergoes after a step intensity. The work of Blanchard et al. did not describe deformation. Second, the signature patterns of chain how the sample thickness was obtained—this is a nontrivial retraction in SANS experiments are the consequence of issue for their soft, compressible polyisoprene sample. In the assumption of “decoupled” stretch and orientation contrast, the thickness measurement for the high-T poly- relaxation in the tube model. The peak shift and anisotropy styrene is rather straightforward. inversion are directly tied to the physical picture that the Fifth, we confirm that our quenched samples have contour length equilibrates on the time scale of τ , while it uniform stress distribution by performing birefringence takes τ to completely relax the orientation through repta- measurements. It appears that none of the previous studies tion. Our analysis with a network-type phenomenological [25,26,32] conducted such a test to verify the qualities of model [149] suggests that it is possible to simultaneously their samples. This was particularly a challenge for describe both the SANS spectrum and stress by assuming Blanchard et al., as their low-T sample could be examined coupled stretch and orientational dynamics. The details of only in situ, i.e., on the SANS beam line. our quantitative analysis will be published in a future paper. Sixth, we perform the stretching and relaxation experi- ments at the same temperature for all the samples. This design avoids the potential complications in the previous E. Comments on the previous work studies [26,32] that utilized the time-temperature super- At this point, it seems imperative for us to comment on position principle [134]. the previous work of Blanchard et al. [32], which is the Last, but not least, as we repeatedly stress in this article, only paper in the literature that claims direct observation of our spherical harmonic expansion approach makes full use of chain retraction by SANS. Their report contradicts not only the entire 2D SANS spectrum, as opposed to the traditional the current work, but also at least two other independent method based on partial information along parallel and studies [25,26]. Rather than speculating what might have perpendicular directions. Moreover, the model-independent gone wrong in the work of Blanchard et al., we instead nature of the method allows us to circumvent the ambiguity emphasize the steps we take to improve the execution of the associated with R analysis. experiments and data analysis. First, our stretching experiments are conducted in the forced convection oven of the RSA-G2 Solids Analyzer VI. CONCLUDING REMARKS AND SUMMARY (TA Instruments), which is a well-tested commercial In summary, building on the idea of spherical harmonic sample environment. We further verify the uniformity decomposition, we develop a new framework to fingerprint and stability of the temperature by monitoring the built- macromolecular deformation from small-angle scattering in upper and lower resistance temperature detectors of the experiments. The spherical harmonic expansion analysis oven, as well as an additional resistance temperature permits a direct and unambiguous comparison of SANS detector that we attach to the lower sample clamp. experiments with the theoretical picture of the tube model. Second, prior to the final SANS experiments at NIST, we The chain retraction hypothesis of the tube model is not cross-examine the performance of three beam lines (the supported by the new SANS measurements of well- NGB30 SANS beam line at NIST, the EQ-SANS at SNS, and entangled polystyrenes after a large step uniaxial extension. the D22 at ILL) for 2D data analysis, where every “pixel” Since the tube theory is of paramount importance for our counts. We confirm that all three beam lines give consistent results for the same quenched samples and rule out any current understanding of the flow and deformation behavior uncertainty due to the performance of the instrument. of entangled polymers, the invalidation of the chain Third, we take care to provide a complete characteriza- retraction hypothesis has immense ramifications. It should tion of both the linear and nonlinear viscoelastic properties be emphasized, however, that the current investigation is of the sample. Detailed rheological information was not only concerned with the tube approach in the nonlinear available in most of the previous investigations on this rheological regime. In other words, our work does not topic [25,26,32]. This lack of sufficient information on question the linear part of the tube theory. Conversely, viscoelastic behavior, in our opinion, has hampered studies of contour length fluctuations in the equilibrium 031003-12 FINGERPRINTING MOLECULAR RELAXATION IN … PHYS. REV. X 7, 031003 (2017) [5] M. Doi and S. F. Edwards, Dynamics of Concentrated state [150] should not be used to infer the validity of chain Polymer Systems Part 1.—Brownian Motion in the Equi- retraction in the nonequilibrium state. librium State, J. Chem. Soc., Faraday Trans. 2 74, 1789 Finally, although the application of small-angle scatter- (1978). ing in deformed polymers has a long history, the full power [6] M. Doi and S. F. Edwards, Dynamics of Concentrated of the rheo-SAS technique is yet to be unearthed. The Polymer Systems Part 2.—Molecular Motion under Flow, spherical harmonic expansion approach we employ in this J. Chem. Soc., Faraday Trans. 2 74, 1802 (1978). work is surely not limited to entangled polymeric liquids, [7] M. Doi and S. F. Edwards, Dynamics of Concentrated but is also applicable to a wide variety of complex fluids Polymer Systems Part 3.—The Constitutive Equation, J. and soft solids. The spectrum decomposition method not Chem. Soc., Faraday Trans. 2 74, 1818 (1978). only provides a convenient way for comparing experimen- [8] M. Doi and S. F. Edwards, Dynamics of Concentrated tal results with the predictions from statistical and molecu- Polymer Systems Part 4.—Rheological Properties, J. lar models, but also allows many new questions to be asked, Chem. Soc., Faraday Trans. 2 75, 38 (1979). including the affineness, symmetry, and heterogeneity of [9] M. Doi and S. F. Edwards, The Theory of Polymer Dynamics (Oxford University Press, Oxford, 1986). macromolecular deformation. [10] M. Doi, Molecular Rheology of Concentrated Polymer Systems. I, J. Polym. Sci., Polym. Phys. Ed. 18, 1005 (1980). ACKNOWLEDGMENTS [11] S. T. Milner and T. C. B. McLeish, Reptation and Contour- This research was sponsored by the Laboratory Directed Length Fluctuations in Melts of Linear Polymers, Phys. Rev. Lett. 81, 725 (1998). Research and Development Program of Oak Ridge [12] A. E. Likhtman and T. C. B. McLeish, Quantitative Theory National Laboratory, managed by UT Battelle, LLC, for for Linear Dynamics of Linear Entangled Polymers, the U.S. Department of Energy. W.-R. C. acknowledges the Macromolecules 35, 6332 (2002). support by the U.S. Department of Energy, Office of [13] M. Daoud and P. G. de Gennes, Some Remarks on the Science, Office of Basic Energy Sciences, Materials Dynamics of Polymer Melts, J. Polym. Sci., Polym. Phys. Sciences and Engineering Division. J. L. and Z. Z. are Ed. 17, 1971 (1979). thankful for the financial support by the NSF Polymer [14] J. Klein, Dynamics of Entangled Linear, Branched, and Program (DMR-1105135). The polymer synthesis and Cyclic Polymers, Macromolecules 19, 105 (1986). characterization were carried out at the Center for [15] G. Marrucci, Dynamics of Entanglements: A Nonlinear Nanophase Materials Sciences, which is a DOE Office Model Consistent with the Cox-Merz Rule, J. Non- of Science User Facility. We acknowledge the support of Newtonian Fluid Mech. 62, 279 (1996). the National Institute of Standards and Technology, U.S. [16] D. W. Mead, R. G. Larson, and M. Doi, A Molecular Department of Commerce, for providing the neutron Theory for Fast Flows of Entangled Polymers, Macro- molecules 31, 7895 (1998). research facilities used in this work. The statements, [17] S. T. Milner, T. C. B. McLeish, and A. E. Likhtman, findings, conclusions, and recommendations are those of Reptation and Contour-Length Fluctuations in Melts of the authors and do not necessarily reflect the view of NIST Linear Polymers, J. Rheol. 45, 539 (2001). or the U.S. Department of Commerce. Access to [18] G. 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