Physics:Chain entanglement

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Short description: Topological interaction between long polymer chains that constrains their motion

Template:TopicTOC-Polymer In polymer physics, a chain entanglement is a topological interaction that occurs when polymer chains become long enough that they cannot pass through one another. The chains do not bond chemically, and they are not literally knotted, but their mutual uncrossability means that one chain effectively traps and constrains the motion of its neighbours. The result is a dramatic change in the mechanical and flow behaviour of the material: viscosity climbs steeply, elastic behaviour emerges, and the polymer starts to behave, at least transiently, like a rubber.[1][2]

Entanglements are central to nearly every aspect of polymer processing and performance[3][4]. They are why long-chain plastics can be drawn into fibres without snapping, why polymer melts are so resistant to flow, and why phenomena such as die swell and melt fracture occur during extrusion[5].

Nature of entanglements

The physical picture is simpler than the theory. Think of a bowl of cooked spaghetti, or a tangled ball of string. Individual strands cannot pass through each other; they can only slide around each other by threading past end-to-end. Polymer chains in a melt or concentrated solution are in exactly this situation, only at a molecular scale and in constant thermal motion.[6]

The key point is that entanglements are not fixed[7]. They are temporary constraints: a given chain will eventually wriggle free of its neighbours by diffusing along its own contour, a process called reptation[8]. But on timescales shorter than the reptation time, the entanglements behave like a network of physical crosslinks, giving the melt a rubbery, elastic character[9].

Short chains, below a threshold length, are simply too stubby to constrain each other this way. Only when chains exceed a certain minimum length do entanglements become significant[10]. This threshold is characterised by two related quantities[11]: the entanglement molecular weight Me and the critical molecular weight Mc[12][13].

Entanglement molecular weight

Me is defined as the average molecular weight of a chain segment between two successive entanglement points[14]. It is a material-specific constant, determined most reliably from the plateau modulus GN0 measured in dynamic mechanical experiments[15]:

Me=gρRTGN0

where ρ is the melt density, R is the gas constant, T is the temperature, and g is a numerical prefactor close to 1.[2][16]

Mc, the critical molecular weight observable as a kink in a log-log plot of melt viscosity versus molecular weight, is empirically close to 2Me. Below Mc, viscosity scales roughly linearly with molecular weight (η ∝ M). Above it, the scaling steepens sharply to η ∝ M3.4, reflecting how heavily entanglements penalise chain motion.[1][16]

Values differ considerably between polymer types, depending on chain stiffness and backbone structure. Stiffer, bulkier chains entangle at lower molecular weight because they cannot coil past each other as easily[17]. Flexible chains like polyethylene require longer runs before constraints become significant[18].

Reptation and chain relaxation

Because entanglements prevent lateral motion, a chain in an entangled melt is effectively confined to a tube-shaped region defined by its neighbours[19][20]. The dominant relaxation mechanism is reptation: the chain diffuses along this tube in a snake-like, back-and-forth motion until it escapes from its original tube entirely and finds a new environment.[6]

Pierre-Gilles de Gennes introduced the reptation concept in 1971. Sam Edwards and Masao Doi later formalised it into the tube model that remains the standard framework today.[1][21] The tube model[22] predicts that the longest relaxation time τ scales as M3; experiment gives M3.4, a discrepancy attributed to effects such as contour-length fluctuations and constraint release, where the escape of neighbouring chains allows the tube itself to rearrange.[1]

Effects on physical properties

Melt viscosity and elasticity

The steep M3.4 dependence of viscosity above Mc is one of the most practically important consequences of entanglement[23]. Doubling the molecular weight of an already-entangled polymer more than doubles its resistance to flow[24]. This is why processing high molecular weight plastics requires high temperatures or powerful screws, and why very high molecular weight grades, such as ultra-high molecular weight polyethylene (UHMWPE), are essentially impossible to process by conventional melt extrusion.[16]

At the same time, entanglements give the melt an elastic character[25]. When deformed faster than the reptation time, the entanglement network stores energy like a rubber[26]. This stored energy is the direct cause of die swell and of the Weissenberg effect in which a polymer solution climbs a rotating rod.[2]

Solid-state mechanical properties

In the solid state, entanglements act as physical crosslinks in a way directly analogous to the chemical crosslinks in a vulcanised rubber[27]. They resist deformation, transmit stress between chains, and are responsible for the toughness and ductility of high-molecular-weight plastics. Below Mc, the same polymer is brittle[28]. The difference between candle wax (low molecular weight polyolefin) and a polyethylene milk jug is largely a difference in entanglement density: the wax has none to speak of, and it shatters; the jug has plenty, and it deforms without breaking.[2]

Polymer processing

Entanglements set practical limits in several areas:

  • Extrusion ː the long relaxation times of entangled melts mean that chains leaving a die still carry significant stored elastic stress, which drives die swell[29][30]. The higher the molecular weight, and the shorter the die, the more pronounced the effect[31].
  • Melt fracture ː above a critical wall shear stress, the entanglement network cannot relax fast enough during flow, and the extrudate surface or bulk breaks up.[32]
  • Fibre drawing ː entanglements provide the melt strength that allows a molten filament to be drawn down to a fine diameter without breaking. Short-chain materials simply neck and rupture[33].
  • Solid-state drawing ː the same network that constrains flow in the melt allows solid polymer to be drawn to high draw ratios, aligning chains and producing high-strength fibres such as high-performance polyethylene ropes and UHMWPE fibres[34][35].

Measuring entanglements

The standard measurement is oscillatory shear rheology[36][37]. A small-amplitude oscillatory deformation is applied over a wide range of frequencies, and the storage modulus G′ and loss modulus G″ are measured[38]. In an entangled polymer, G′ passes through a rubbery plateau at intermediate frequencies; the plateau modulus GN0 is read from this region and inserted into the formula above to give Me.[16]

An older but still used approach is to plot zero-shear viscosity against molecular weight for a series of fractions and locate the kink at Mc[39][40].

See also

References

  1. ↑ 1.0 1.1 1.2 1.3 Doi, M.; Edwards, S. F. (1986). The Theory of Polymer Dynamics. Oxford: Oxford University Press. ISBN 978-0-19-852033-7. 
  2. ↑ 2.0 2.1 2.2 2.3 Rubinstein, M.; Colby, R. H. (2003). Polymer Physics. Oxford: Oxford University Press. ISBN 978-0-19-852059-7. 
  3. ↑ Kang, Taewon; Dai, Jinyue; Huang, Yan; Kim, Haeji; Keten, Sinan; Kim, Junsoo (2025-11-26). "Entanglements and Fracture in Polymers". Chemical Reviews 125 (22): 11032–11057. doi:10.1021/acs.chemrev.5c00459. ISSN 1520-6890. PMID 41043043. 
  4. ↑ "Polymer entanglement | Milner Research Group" (in en-US). https://sites.psu.edu/stm9research/research/polymer-entanglement/. 
  5. ↑ Dynisco (2017-02-15). "Polymer Melts and Elastic Effects" (in en). https://www.azom.com/article.aspx?ArticleID=13578. 
  6. ↑ 6.0 6.1 de Gennes, P.-G. (1979). Scaling Concepts in Polymer Physics. Ithaca: Cornell University Press. ISBN 978-0-8014-1203-5. 
  7. ↑ Klein, J. (1978-01-12). "Evidence for reptation in an entangled polymer melt". Nature 271 (5641): 143–145. doi:10.1038/271143a0. ISSN 0028-0836. Bibcode: 1978Natur.271..143K. https://www.scopus.com/pages/publications/0000539616. 
  8. ↑ "Polymer Reptation". https://euler.phys.cmu.edu/widom/research/rep/repton.html. 
  9. ↑ Kang, Taewon; Dai, Jinyue; Huang, Yan; Kim, Haeji; Keten, Sinan; Kim, Junsoo (2025). "Entanglements and Fracture in Polymers". Chemical Reviews 125 (22): 11032–11057. doi:10.1021/acs.chemrev.5c00459. PMID 41043043. https://pubs.acs.org/action/cookieAbsent. Retrieved 2026-05-12. 
  10. ↑ Pawlak, Andrzej; Krajenta, Justyna (2024-07-20). "Entanglements of Macromolecules and Their Influence on Rheological and Mechanical Properties of Polymers". Molecules (Basel, Switzerland) 29 (14): 3410. doi:10.3390/molecules29143410. ISSN 1420-3049. PMID 39064989. 
  11. ↑ Huang, Qian; Mednova, Olga; Rasmussen, Henrik K.; Alvarez, Nicolas J.; Skov, Anne L.; Almdal, Kristoffer; Hassager, Ole (2013-06-25). "Concentrated Polymer Solutions are Different from Melts: Role of Entanglement Molecular Weight". Macromolecules 46 (12): 5026–5035. doi:10.1021/ma4008434. ISSN 0024-9297. Bibcode: 2013MaMol..46.5026H. 
  12. ↑ "Mc from L-M theory | Practical Rheology Science | Prof Steven Abbott". https://www.stevenabbott.co.uk/practical-rheology/L-M_Mc.php. 
  13. ↑ Cho, Kwang Soo; Ahn, Kyung Hyun; Lee, Seung Jong (2004-07-15). "Simple method for determining the critical molecular weight from the loss modulus" (in en). Journal of Polymer Science Part B: Polymer Physics 42 (14): 2724–2729. doi:10.1002/polb.20150. ISSN 0887-6266. Bibcode: 2004JPoSB..42.2724C. https://onlinelibrary.wiley.com/doi/10.1002/polb.20150. 
  14. ↑ Fetters, Lewis J. (1965). "Determination of the Intermolecular Entanglement Coupling Spacings in Polyisoprene by Viscosity Measurements". Journal of Research of the National Bureau of Standards. Section A, Physics and Chemistry 69A (1): 33–37. doi:10.6028/jres.069A.006. ISSN 0022-4332. PMID 31927846. 
  15. ↑ Bukowski, Cynthia; Zhang, Tianren; Riggleman, Robert A.; Crosby, Alfred J. (2021-09-17). "Load-bearing entanglements in polymer glasses". Science Advances 7 (38). doi:10.1126/sciadv.abg9763. ISSN 2375-2548. PMID 34533993. Bibcode: 2021SciA....7.9763B. 
  16. ↑ 16.0 16.1 16.2 16.3 Pawlak, A. (2024). "Entanglements of macromolecules and their influence on rheological and mechanical properties of polymers". Molecules 29 (14): 3410. doi:10.3390/molecules29143410. PMID 39064989. 
  17. ↑ "Three Things that Make Polymers Different" (in en). https://pslc.ws/macrog/three.htm. 
  18. ↑ Galuska, Luke A.; McNutt, William W.; Qian, Zhiyuan; Zhang, Song; Weller, Daniel W.; Dhakal, Sujata; King, Eric R.; Morgan, Sarah E. et al. (2020). "Impact of Backbone Rigidity on the Thermomechanical Properties of Semiconducting Polymers with Conjugation Break Spacers". Macromolecules 53 (14): 6032–6042. doi:10.1021/acs.macromol.0c00889. Bibcode: 2020MaMol..53.6032G. https://pubs.acs.org/action/cookieAbsent. Retrieved 2026-05-12. 
  19. ↑ Lee, Nam-Kyung; Diddens, Diddo; Meyer, Hendrik; Johner, Albert (2017-02-08). "Local Chain Segregation and Entanglements in a Confined Polymer Melt" (in en). Physical Review Letters 118 (6). doi:10.1103/PhysRevLett.118.067802. ISSN 0031-9007. PMID 28234517. Bibcode: 2017PhRvL.118f7802L. https://link.aps.org/doi/10.1103/PhysRevLett.118.067802. 
  20. ↑ Qin, Jian; Milner, Scott T. (2014). "Tubes, Topology, and Polymer Entanglement". Macromolecules 47 (17): 6077–6085. doi:10.1021/ma500755p. Bibcode: 2014MaMol..47.6077Q. https://pubs.acs.org/action/cookieAbsent. Retrieved 2026-05-12. 
  21. ↑ Goldenfeld, Nigel (2016-01-05). "Samuel Frederick Edwards: Founder of modern polymer and soft matter theory". Proceedings of the National Academy of Sciences of the United States of America 113 (1): 10–11. doi:10.1073/pnas.1523001113. ISSN 1091-6490. PMID 26699498. Bibcode: 2016PNAS..113...10G. 
  22. ↑ Likhtman, Alexei E.; Talib, Mohamad S.; Vorselaars, Bart; Ramirez, Jorge (2013). "Determination of Tube Theory Parameters Using a Simple Grid Model as an Example". Macromolecules 46 (3): 1187–1200. doi:10.1021/ma302103p. Bibcode: 2013MaMol..46.1187L. https://pubs.acs.org/action/cookieAbsent. Retrieved 2026-05-12. 
  23. ↑ Edwards, S F; Grant, J W V (1973-08-01). "The effect of entanglements on the viscosity of a polymer melt". Journal of Physics A: Mathematical, Nuclear and General 6 (8): 1186–1195. doi:10.1088/0305-4470/6/8/013. ISSN 0301-0015. Bibcode: 1973JPhA....6.1186E. https://iopscience.iop.org/article/10.1088/0305-4470/6/8/013. 
  24. ↑ Haward, R. N.; Young, R. J., eds (1997). "The Physics of Glassy Polymers". Springer. doi:10.1007/978-94-011-5850-3. ISBN 978-94-010-6472-9. https://link.springer.com/book/10.1007/978-94-011-5850-3?error=cookies_not_supported&code=3e646c07-2493-43d0-a748-b1129b6dd1f1. 
  25. ↑ (in en) Sage Journals: Discover world-class research. doi:10.1177/87560879211058768. https://journals.sagepub.com/action/cookieAbsent. Retrieved 2026-05-12. 
  26. ↑ Kang, Taewon; Dai, Jinyue; Huang, Yan; Kim, Haeji; Keten, Sinan; Kim, Junsoo (2025). "Entanglements and Fracture in Polymers". Chemical Reviews 125 (22): 11032–11057. doi:10.1021/acs.chemrev.5c00459. PMID 41043043. https://pubs.acs.org/action/cookieAbsent. Retrieved 2026-05-12. 
  27. ↑ Wang, Yueqiong; Liu, Hongchao; Yu, Heping; Zhao, Pengfei; Wang, Qifang; Liao, Lusheng; Luo, Mingchao; Zheng, Tingting et al. (2022-02-15). "New insight into naturally occurring network and entanglements induced strain behavior of vulcanized natural rubber". Polymer 241. doi:10.1016/j.polymer.2022.124545. ISSN 0032-3861. https://www.sciencedirect.com/science/article/pii/S0032386122000325. 
  28. ↑ Garbelini, Maria Luísa; Le Gac, Pierre-Yves; Le Gall, Maelenn; Le Dain, Guillaume; Haller, Servane; Sotta, Paul; Fayolle, Bruno (2026-05-01). "Oxidative degradation of rosslinked EPDM elastomers: Impact on the entanglement network" (in en). Journal of Polymer Science. doi:10.1002/pola.70144. ISSN 2642-4150. https://onlinelibrary.wiley.com/doi/10.1002/pola.70144. 
  29. ↑ Hou, Ji-Xuan; Svaneborg, Carsten; Everaers, Ralf; Grest, Gary S. (2010-08-05). "Stress Relaxation in Entangled Polymer Melts" (in en). Physical Review Letters 105 (6). doi:10.1103/PhysRevLett.105.068301. ISSN 0031-9007. https://link.aps.org/doi/10.1103/PhysRevLett.105.068301. 
  30. ↑ Dynisco (2017-02-15). "Polymer Melts and Elastic Effects" (in en). https://www.azom.com/article.aspx?ArticleID=13578. 
  31. ↑ Hou, Ji-Xuan; Svaneborg, Carsten; Everaers, Ralf; Grest, Gary S. (2010-08-05). "Stress Relaxation in Entangled Polymer Melts" (in en). Physical Review Letters 105 (6). doi:10.1103/PhysRevLett.105.068301. ISSN 0031-9007. PMID 20868018. Bibcode: 2010PhRvL.105f8301H. https://link.aps.org/doi/10.1103/PhysRevLett.105.068301. 
  32. ↑ Denn, M. M. (2001). "Extrusion instabilities and wall slip". Annual Review of Fluid Mechanics 33: 265–287. doi:10.1146/annurev.fluid.33.1.265. Bibcode: 2001AnRFM..33..265D. 
  33. ↑ Hufenus, Rudolf; Yan, Yurong; Dauner, Martin; Kikutani, Takeshi (2020-09-26). "Melt-Spun Fibers for Textile Applications". Materials (Basel, Switzerland) 13 (19): 4298. doi:10.3390/ma13194298. ISSN 1996-1944. PMID 32993085. Bibcode: 2020Mate...13.4298H. 
  34. ↑ Walker, Julian; Melaj, Mariana; Giménez, Rocio; Pérez, Ezequiel; Bernal, Celina (2019-11-07). "Solid-State Drawing of Commercial Poly(Lactic Acid) (PLA) Based Filaments" (in English). Frontiers in Materials 6. doi:10.3389/fmats.2019.00280. ISSN 2296-8016. Bibcode: 2019FrMat...6..280W. 
  35. ↑ Mai, Fang; Tu, Wei; Bilotti, Emiliano; Peijs, Ton (2015-12-01). "The Influence of Solid-State Drawing on Mechanical Properties and Hydrolytic Degradation of Melt-Spun Poly(Lactic Acid) (PLA) Tapes" (in en). Fibers 3 (4): 523–538. doi:10.3390/fib3040523. ISSN 2079-6439. 
  36. ↑ Ahmad, Nurul Hawa; Ahmed, Jasim; Hashim, Dzulkifly M.; Manap, Yazid Abdul; Mustafa, Shuhaimi (2015). "Oscillatory and steady shear rheology of gellan/dextran blends". Journal of Food Science and Technology 52 (5): 2902–2909. doi:10.1007/s13197-014-1330-x. ISSN 0022-1155. PMID 25892789. 
  37. ↑ Townsend, Adam K.; Wilson, Helen J. (2018-11-01). "Small- and large-amplitude oscillatory rheometry with bead–spring dumbbells in Stokesian Dynamics to mimic viscoelasticity". Journal of Non-Newtonian Fluid Mechanics 261: 136–152. doi:10.1016/j.jnnfm.2018.08.010. ISSN 0377-0257. Bibcode: 2018JNNFM.261..136T. https://www.sciencedirect.com/science/article/pii/S0377025718300351. 
  38. ↑ GmbH, Anton Paar (2020-02-05). "Oscillatory Measurements: Back and Forth to the Result" (in en). https://www.azom.com/article.aspx?ArticleID=18928. 
  39. ↑ Mead, D. W.; Larson, R. G.; Doi, M. (1998). "A Molecular Theory for Fast Flows of Entangled Polymers". Macromolecules 31 (22): 7895–7914. doi:10.1021/ma980127x. Bibcode: 1998MaMol..31.7895M. https://pubs.acs.org/action/cookieAbsent. Retrieved 2026-05-12. 
  40. ↑ Locati, G.; Pegoraro, M.; Nichetti, D. (1999). "A model for the zero shear viscosity" (in en). Polymer Engineering & Science 39 (4): 741–748. doi:10.1002/pen.11462. ISSN 0032-3888. Bibcode: 1999PESci..39..741L. https://4spepublications.onlinelibrary.wiley.com/doi/10.1002/pen.11462. 



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