Tube Model Under Tension

Tube Model Under Tension VIEWPOINT Results from a new method of analyzing neutron-scattering data from polymer samples under deformation may challenge the prevailing ``tube model'' of polymer motion. by Daniel J. Read olymers are the extremely long, stringy molecules inside many of the materials around us. Sitting in my office I might glance down at my pen, my tele- P phone, my computer monitor, my chair, and so on. They all contain plastics, and so all contain polymers. Plas- tics are versatile and useful because they can be processed easily—they can be stretched, pulled, and squeezed into any shape we desire. But what happens to the stringy poly- mers in such processing? Understanding the shapes that the polymers are forced into when their host materials are pro- cessed is key to controlling the processing. In a new study, Yangyang Wang at Oak Ridge National Laboratory in Ten- nessee and co-workers [1] have used neutron scattering as a microscope to see the shape of polymers within samples under deformation. The results of their novel data analysis provide a “fingerprint” of the molecular deformation, and they claim that their results challenge the dominant “tube model” of polymer flow. The tube model is arguably the most successful theory Figure 1: To measure the size and shape of polymers within of modern soft-matter physics, having provided a picture samples under deformation, Wang and colleagues [1] red a beam for understanding polymer motion for about 50 years. Pi- of neutrons through stretched polymer samples that contained oneered by Pierre-Gilles de Gennes [2] and Masao Doi and labeled polymers. Scattering from the polymers gave a Sam Edwards [3], the theory now enables practical predic- two-dimensional diffraction pattern, which is a measure of the tions for manufacturing and processing of industrial plastic deformed polymer sizes and shapes. The blue dashed lines denote the entanglement ``tube'' conning the polymers. (Z. Wang materials [4]. The model is easy to visualise. In poly- et al., Phys. Rev. X (2017)) meric materials, the long molecules become tangled with each other—like intertwined spaghetti—and these “entan- glements” restrict the molecular motion. A molecule can move easily along the direction of its length, but motion per- terials under small deformations. For these conditions, it is sufficient to describe only the thermal Brownian motion pendicular to it is strongly suppressed because this involves of the polymer strings subject to the constraints from entan- traversing other molecules, which would break interatomic glements. This is the back-and-forth random motion of the bonds. The tube model pictures this motion by assuming polymers along their tube contour, which de Gennes called the polymer is trapped within a “tube” of entanglements, so that the molecule is only able to move along the con- reptation [2] in analogy to the motion of a snake. Early predictions of the response of the material to small deforma- tour of that tube. This inspired guess allows the polymer motion to be described mathematically in equilibrium and tions using this description were not perfect. It took many refinements to this basic model, such as accounting for fluc- strong-deformation conditions, predictions to be made, and materials to be successfully modeled. tuations of the length of the strings along the tube, before a fully quantitative theory was derived [5]. The tube model works well at describing polymeric ma- But the real difficulty in describing polymer flow comes when the material is strongly deformed. What then happens School of Mathematics, University of Leeds, Leeds LS2 9JT, to the entanglements and to the tube? Do these survive the United Kingdom deformation, and if so, how? The truth is, nobody knows. physics.aps.org 2017 American Physical Society 10 July 2017 Physics 10, 77 But this has not prevented further inspired guesses from be- ing made. Doi and Edwards [3] made one of the earliest such guesses. They claimed that the polymer strings, and the tubes surrounding them, would be stretched “affinely” with the material, that is, in direct proportion to the material deformation. As a result, the length of the polymers along the contour of their tubes must increase on average. Doi and Edwards predicted that the first response of the molecules after the deformation must be to “retract” back until they Figure 2: Wang and colleagues decomposed each of their are once again at their equilibrium length in the tube. Be- diffraction patterns into a sum over spherical harmonic contributions: an isotropic (circular) component plus higher-order cause the tube model assumes that all polymer motion is harmonics with increasing numbers of peaks and troughs. Shown constrained by entanglements, this retraction can only oc- here is a simulated pattern. (Z. Wang et al., Phys. Rev. X (2017)) cur along the tube contour. This gives a very characteristic and specific prescription for how the shapes of the polymers change after deformation: the size of the molecules should aim at the tube model (see, for example, Ref. [8]); on many decrease in all directions during the retraction process, and occasions the model has withstood the criticism (see, for in- any experiment that can detect the molecular size should be stance, Ref. [9]). There is also the paper I co-authored [7], able to confirm this retraction. This is where the neutron- which did claim to observe retraction. Setting potential ex- scattering experiments of Wang and colleagues [1] come in. perimental artefacts aside, one distinction of our work was Neutron scattering is an excellent method to study the that our polymers had a significantly larger number of en- molecular size. It usually involves a mixture of molecules, tanglements, and there are several corrections to the theory in this case a mixture of polymers, in which most of the that may disrupt the observation of retraction for a smaller molecules have their hydrogen atoms replaced by deu- number of entanglements. Still, the challenge is there, and terium, while a small fraction of the molecules are labeled by it is clear: can tube theorists such as myself successfully ex- retaining their hydrogen atoms. The labeled and unlabeled plain why Wang and colleagues did not observe retraction, polymers scatter the neutron beam by different amounts, and can we predict the fine details revealed by their novel and the resulting two-dimensional diffraction pattern is a data analysis? Or, should we modify, or even jettison, the measure of the polymer size and shape. In their study, Wang tube theory? The next steps will most likely be to re-examine and co-workers stretched a series of such mixtures, waited the tube-model calculations and the approximations made for various times following the stretch before freezing the along the way to derive the final theoretical results. molecules in place by cooling, and finally fired the neutron beam through the materials (Fig. 1) to obtain a collection This research is published in Physical Review X. of diffraction patterns. They expected to see the character- istic decrease in polymer size from the retraction process, REFERENCES but they found no evidence of retraction, in contradiction to tube-model expectations. [1] Z. Wang, C. N. Lam, W.-R. Chen, W. Wang, J. Liu, Y. Liu, L. Porcar, C. B. Stanley, Z. Zhao, K. Hong, and Y. Wang, ``Fin- They are not the first group to attempt to see retraction gerprinting Molecular Relaxation in Deformed Polymers,'' Phys. using neutron-scattering data: Boué and co-workers tried Rev. X 7, 031003 (2017). but did not see the retraction [6]; and I was involved in [2] P. G. de Gennes, ``Reptation of a Polymer Chain in the Presence a collaboration [7] that did seem to observe the retraction. of Fixed Obstacles,'' J. Chem. Phys. 55, 572 (1971). What sets Wang and colleagues’ work apart is their novel [3] M. Doi and S. F. Edwards, The Theory of Polymer Dynamics data analysis. The researchers decomposed each of the (Clarendon Press, Oxford, 1986). two-dimensional diffraction patterns into a sum over spher- [4] D. J. Read, ``From Reactor to Rheology in Industrial Polymers,'' ical harmonic contributions (Fig. 2). This allowed them to J. Polym. Sci., Part B: Polym. Phys. 53, 123 (2014). make use of all the data to extract the important anisotropic [5] A. E. Likhtman and T. C. B. McLeish, ``Quantitative Theory components of the scattering for different scattering an- for Linear Dynamics of Linear Entangled Polymers,'' Macro- gles, giving information about the degree of deformation molecules 35, 6332 (2002). [6] F. Boué, M. Nierlich, G. Jannink, and R. Ball, ``Polymer Coil Re- of the polymers on different size scales. These scattering laxation in Uniaxially Strained Polystyrene Observed by Small components can be carefully compared with theoretical ex- Angle Neutron Scattering,'' J. Phys. 43, 137 (1982). pectations. Previous analyses focused primarily on the data [7] A. Blanchard, R. S. Graham, M. Heinrich, W. Pyckhout-Hintzen, parallel or perpendicular to the deformation direction, using D. Richter, A. E. Likhtman, T. C. B. McLeish, D. J. Read, E. a fraction of the available data. Straube, and J. Kohlbrecher, ``Small Angle Neutron Scattering What might this mean, and what will happen next? Wang Observation of Chain Retraction after a Large Step Deforma- and colleagues’ paper is one of several recent papers to take tion,'' Phys. Rev. Lett. 95, 166001 (2005). physics.aps.org 2017 American Physical Society 10 July 2017 Physics 10, 77 [8] P. E. Boukany, S.-Q. Wang, and X. Wang, ``Step Shear of En- meric Fluids Under Step Shear,'' Phys. Rev. Lett. 110, 204503 tangled Linear Polymer Melts: New Experimental Evidence for (2013). Elastic Yielding,'' Macromolecules 42, 6261 (2009). [9] O. S. Agimelen and P. D. Olmsted, ``Apparent Fracture in Poly- 10.1103/Physics.10.77 physics.aps.org 2017 American Physical Society 10 July 2017 Physics 10, 77 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Physics American Physical Society (APS)

Tube Model Under Tension

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Abstract

VIEWPOINT Results from a new method of analyzing neutron-scattering data from polymer samples under deformation may challenge the prevailing ``tube model'' of polymer motion. by Daniel J. Read olymers are the extremely long, stringy molecules inside many of the materials around us. Sitting in my office I might glance down at my pen, my tele- P phone, my computer monitor, my chair, and so on. They all contain plastics, and so all contain polymers. Plas- tics are versatile and useful because they can be processed easily—they can be stretched, pulled, and squeezed into any shape we desire. But what happens to the stringy poly- mers in such processing? Understanding the shapes that the polymers are forced into when their host materials are pro- cessed is key to controlling the processing. In a new study, Yangyang Wang at Oak Ridge National Laboratory in Ten- nessee and co-workers [1] have used neutron scattering as a microscope to see the shape of polymers within samples under deformation. The results of their novel data analysis provide a “fingerprint” of the molecular deformation, and they claim that their results challenge the dominant “tube model” of polymer flow. The tube model is arguably the most successful theory Figure 1: To measure the size and shape of polymers within of modern soft-matter physics, having provided a picture samples under deformation, Wang and colleagues [1] red a beam for understanding polymer motion for about 50 years. Pi- of neutrons through stretched polymer samples that contained oneered by Pierre-Gilles de Gennes [2] and Masao Doi and labeled polymers. Scattering from the polymers gave a Sam Edwards [3], the theory now enables practical predic- two-dimensional diffraction pattern, which is a measure of the tions for manufacturing and processing of industrial plastic deformed polymer sizes and shapes. The blue dashed lines denote the entanglement ``tube'' conning the polymers. (Z. Wang materials [4]. The model is easy to visualise. In poly- et al., Phys. Rev. X (2017)) meric materials, the long molecules become tangled with each other—like intertwined spaghetti—and these “entan- glements” restrict the molecular motion. A molecule can move easily along the direction of its length, but motion per- terials under small deformations. For these conditions, it is sufficient to describe only the thermal Brownian motion pendicular to it is strongly suppressed because this involves of the polymer strings subject to the constraints from entan- traversing other molecules, which would break interatomic glements. This is the back-and-forth random motion of the bonds. The tube model pictures this motion by assuming polymers along their tube contour, which de Gennes called the polymer is trapped within a “tube” of entanglements, so that the molecule is only able to move along the con- reptation [2] in analogy to the motion of a snake. Early predictions of the response of the material to small deforma- tour of that tube. This inspired guess allows the polymer motion to be described mathematically in equilibrium and tions using this description were not perfect. It took many refinements to this basic model, such as accounting for fluc- strong-deformation conditions, predictions to be made, and materials to be successfully modeled. tuations of the length of the strings along the tube, before a fully quantitative theory was derived [5]. The tube model works well at describing polymeric ma- But the real difficulty in describing polymer flow comes when the material is strongly deformed. What then happens School of Mathematics, University of Leeds, Leeds LS2 9JT, to the entanglements and to the tube? Do these survive the United Kingdom deformation, and if so, how? The truth is, nobody knows. physics.aps.org 2017 American Physical Society 10 July 2017 Physics 10, 77 But this has not prevented further inspired guesses from be- ing made. Doi and Edwards [3] made one of the earliest such guesses. They claimed that the polymer strings, and the tubes surrounding them, would be stretched “affinely” with the material, that is, in direct proportion to the material deformation. As a result, the length of the polymers along the contour of their tubes must increase on average. Doi and Edwards predicted that the first response of the molecules after the deformation must be to “retract” back until they Figure 2: Wang and colleagues decomposed each of their are once again at their equilibrium length in the tube. Be- diffraction patterns into a sum over spherical harmonic contributions: an isotropic (circular) component plus higher-order cause the tube model assumes that all polymer motion is harmonics with increasing numbers of peaks and troughs. Shown constrained by entanglements, this retraction can only oc- here is a simulated pattern. (Z. Wang et al., Phys. Rev. X (2017)) cur along the tube contour. This gives a very characteristic and specific prescription for how the shapes of the polymers change after deformation: the size of the molecules should aim at the tube model (see, for example, Ref. [8]); on many decrease in all directions during the retraction process, and occasions the model has withstood the criticism (see, for in- any experiment that can detect the molecular size should be stance, Ref. [9]). There is also the paper I co-authored [7], able to confirm this retraction. This is where the neutron- which did claim to observe retraction. Setting potential ex- scattering experiments of Wang and colleagues [1] come in. perimental artefacts aside, one distinction of our work was Neutron scattering is an excellent method to study the that our polymers had a significantly larger number of en- molecular size. It usually involves a mixture of molecules, tanglements, and there are several corrections to the theory in this case a mixture of polymers, in which most of the that may disrupt the observation of retraction for a smaller molecules have their hydrogen atoms replaced by deu- number of entanglements. Still, the challenge is there, and terium, while a small fraction of the molecules are labeled by it is clear: can tube theorists such as myself successfully ex- retaining their hydrogen atoms. The labeled and unlabeled plain why Wang and colleagues did not observe retraction, polymers scatter the neutron beam by different amounts, and can we predict the fine details revealed by their novel and the resulting two-dimensional diffraction pattern is a data analysis? Or, should we modify, or even jettison, the measure of the polymer size and shape. In their study, Wang tube theory? The next steps will most likely be to re-examine and co-workers stretched a series of such mixtures, waited the tube-model calculations and the approximations made for various times following the stretch before freezing the along the way to derive the final theoretical results. molecules in place by cooling, and finally fired the neutron beam through the materials (Fig. 1) to obtain a collection This research is published in Physical Review X. of diffraction patterns. They expected to see the character- istic decrease in polymer size from the retraction process, REFERENCES but they found no evidence of retraction, in contradiction to tube-model expectations. [1] Z. Wang, C. N. Lam, W.-R. Chen, W. Wang, J. Liu, Y. Liu, L. Porcar, C. B. Stanley, Z. Zhao, K. Hong, and Y. Wang, ``Fin- They are not the first group to attempt to see retraction gerprinting Molecular Relaxation in Deformed Polymers,'' Phys. using neutron-scattering data: Boué and co-workers tried Rev. X 7, 031003 (2017). but did not see the retraction [6]; and I was involved in [2] P. G. de Gennes, ``Reptation of a Polymer Chain in the Presence a collaboration [7] that did seem to observe the retraction. of Fixed Obstacles,'' J. Chem. Phys. 55, 572 (1971). What sets Wang and colleagues’ work apart is their novel [3] M. Doi and S. F. Edwards, The Theory of Polymer Dynamics data analysis. The researchers decomposed each of the (Clarendon Press, Oxford, 1986). two-dimensional diffraction patterns into a sum over spher- [4] D. J. Read, ``From Reactor to Rheology in Industrial Polymers,'' ical harmonic contributions (Fig. 2). This allowed them to J. Polym. Sci., Part B: Polym. Phys. 53, 123 (2014). make use of all the data to extract the important anisotropic [5] A. E. Likhtman and T. C. B. McLeish, ``Quantitative Theory components of the scattering for different scattering an- for Linear Dynamics of Linear Entangled Polymers,'' Macro- gles, giving information about the degree of deformation molecules 35, 6332 (2002). [6] F. Boué, M. Nierlich, G. Jannink, and R. Ball, ``Polymer Coil Re- of the polymers on different size scales. These scattering laxation in Uniaxially Strained Polystyrene Observed by Small components can be carefully compared with theoretical ex- Angle Neutron Scattering,'' J. Phys. 43, 137 (1982). pectations. Previous analyses focused primarily on the data [7] A. Blanchard, R. S. Graham, M. Heinrich, W. Pyckhout-Hintzen, parallel or perpendicular to the deformation direction, using D. Richter, A. E. Likhtman, T. C. B. McLeish, D. J. Read, E. a fraction of the available data. Straube, and J. Kohlbrecher, ``Small Angle Neutron Scattering What might this mean, and what will happen next? Wang Observation of Chain Retraction after a Large Step Deforma- and colleagues’ paper is one of several recent papers to take tion,'' Phys. Rev. Lett. 95, 166001 (2005). physics.aps.org 2017 American Physical Society 10 July 2017 Physics 10, 77 [8] P. E. Boukany, S.-Q. Wang, and X. Wang, ``Step Shear of En- meric Fluids Under Step Shear,'' Phys. Rev. Lett. 110, 204503 tangled Linear Polymer Melts: New Experimental Evidence for (2013). Elastic Yielding,'' Macromolecules 42, 6261 (2009). [9] O. S. Agimelen and P. D. Olmsted, ``Apparent Fracture in Poly- 10.1103/Physics.10.77 physics.aps.org 2017 American Physical Society 10 July 2017 Physics 10, 77

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