{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,31]],"date-time":"2025-10-31T17:03:53Z","timestamp":1761930233133,"version":"build-2065373602"},"reference-count":54,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2022,1,25]],"date-time":"2022-01-25T00:00:00Z","timestamp":1643068800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The challenge of calculating nonequilibrium entropy in polymeric liquids undergoing flow was addressed from the perspective of extending equilibrium thermodynamics to include internal variables that quantify the internal microstructure of chain-like macromolecules and then applying these principles to nonequilibrium conditions under the presumption of an evolution of quasie equilibrium states in which the requisite internal variables relax on different time scales. The nonequilibrium entropy can be determined at various levels of coarse-graining of the polymer chains by statistical expressions involving nonequilibrium distribution functions that depend on the type of flow and the flow strength. Using nonequilibrium molecular dynamics simulations of a linear, monodisperse, entangled C1000H2002 polyethylene melt, nonequilibrium entropy was calculated directly from the nonequilibrium distribution functions, as well as from their second moments, and also using the radial distribution function at various levels of coarse-graining of the constituent macromolecular chains. Surprisingly, all these different methods of calculating the nonequilibrium entropy provide consistent values under both planar Couette and planar elongational flows. Combining the nonequilibrium entropy with the internal energy allows determination of the Helmholtz free energy, which is used as a generating function of flow dynamics in nonequilibrium thermodynamic theory.<\/jats:p>","DOI":"10.3390\/e24020175","type":"journal-article","created":{"date-parts":[[2022,1,25]],"date-time":"2022-01-25T20:38:58Z","timestamp":1643143138000},"page":"175","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Nonequilibrium Thermodynamics of Polymeric Liquids via Atomistic Simulation"],"prefix":"10.3390","volume":"24","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2378-5627","authenticated-orcid":false,"given":"Brian","family":"Edwards","sequence":"first","affiliation":[{"name":"Materials Research and Innovation Laboratory, Department of Chemical and Biomolecular Engineering, University of Tennessee, Knoxville, TN 37996, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2346-6797","authenticated-orcid":false,"given":"Mohammad","family":"Nafar Sefiddashti","sequence":"additional","affiliation":[{"name":"Materials Research and Innovation Laboratory, Department of Chemical and Biomolecular Engineering, University of Tennessee, Knoxville, TN 37996, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0091-2312","authenticated-orcid":false,"given":"Bamin","family":"Khomami","sequence":"additional","affiliation":[{"name":"Materials Research and Innovation Laboratory, Department of Chemical and Biomolecular Engineering, University of Tennessee, Knoxville, TN 37996, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,1,25]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1007\/BF01262690","article-title":"The thermodynamics of elastic materials with heat conduction and viscosity","volume":"13","author":"Coleman","year":"1963","journal-title":"Arch. Rat. Mech. Anal."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/BF00283864","article-title":"Thermodynamics of materials with memory","volume":"17","author":"Coleman","year":"1964","journal-title":"Arch. Rat. Mech. Anal."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Flory, P.J. (1969). Statistical Mechanics of Chain Molecules, Interscience.","DOI":"10.1002\/bip.1969.360080514"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1177\/004051754101100401","article-title":"Textile materials and the time factor. I. Mechanical behaviour of textile fibers and plastics","volume":"11","author":"Leaderman","year":"1941","journal-title":"Text. Res."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1007\/BF01517499","article-title":"Nonequilibrium thermodynamics and rheology of viscoelastic polymer media","volume":"15","author":"Leonov","year":"1976","journal-title":"Rheol. Acta"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"730","DOI":"10.1002\/pen.760141012","article-title":"Thermodynamics of dissipative materials with entropic elasticity","volume":"14","author":"Astarita","year":"1974","journal-title":"Polym. Eng. Sci."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"490","DOI":"10.1002\/pen.760160705","article-title":"An approach to thermodynamics of polymer flow based on internal state variables","volume":"16","author":"Astarita","year":"1976","journal-title":"Polym. Eng. Sci."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Beris, A.N., and Edwards, B.J. (1994). Thermodynamics of Flowing Systems: With Internal Microstructure, Oxford University Press.","DOI":"10.1093\/oso\/9780195076943.001.0001"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"\u00d6ttinger, H.C. (2005). Beyond Equilibrium Thermodynamics, Wiley Interscience.","DOI":"10.1002\/0471727903"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1007\/s003970050162","article-title":"Macroscopic thermodynamics of flowing polymeric liquids","volume":"38","author":"Dressler","year":"1999","journal-title":"Rheol. Acta"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"119","DOI":"10.1122\/1.4903498","article-title":"Individual chain dynamics of a polyethylene melt undergoing steady shear flow","volume":"59","author":"Edwards","year":"2015","journal-title":"J. Rheol."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1227","DOI":"10.1122\/1.4963800","article-title":"Steady shearing flow of a moderately entangled polyethylene liquid","volume":"60","author":"Edwards","year":"2016","journal-title":"J. Rheol."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Nafar Sefiddashti, M.H., Edwards, B.J., and Khomami, B. (2019). Individual molecular dynamics of an entangled polyethylene melt undergoing steady shear flow: Steady-state and transient dynamics. Polymers, 11.","DOI":"10.3390\/polym11030476"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"567","DOI":"10.1122\/1.2838250","article-title":"Atomistic simulation of energetic and entropic elasticity in short-chain polyethylenes","volume":"52","author":"Ionescu","year":"2008","journal-title":"J. Rheol."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"6310","DOI":"10.1021\/ma9714878","article-title":"Atomistic simulation of polymer melt elasticity: Calculation of the free energy of an oriented polymer melt","volume":"31","author":"Mavrantzas","year":"1998","journal-title":"Macromolecules"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"132","DOI":"10.1063\/1.1723668","article-title":"Statistical thermodynamics of rubber","volume":"10","author":"Wall","year":"1941","journal-title":"J. Chem. Phys."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1272","DOI":"10.1063\/1.1699180","article-title":"A theory of linear viscoelastic properties of dilute solutions of coiling polymers","volume":"21","author":"Rouse","year":"1953","journal-title":"J. Chem. Phys."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"4571","DOI":"10.1063\/1.447210","article-title":"The energy storage in the Rouse model in an arbitrary flow field","volume":"80","author":"Booij","year":"1984","journal-title":"J. Chem. Phys."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Larson, R.G. (1988). Constitutive Equations for Polymer Melts and Solutions, Butterworth-Heinemann.","DOI":"10.1016\/B978-0-409-90119-1.50012-9"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"625","DOI":"10.1122\/1.2240308","article-title":"A comparison of simple rheological models and simulation data of n-hexadecane under shear and elongational flow","volume":"50","author":"Baig","year":"2006","journal-title":"J. Rheol."},{"key":"ref_21","unstructured":"Doi, M., and Edwards, S.F. (1986). The Theory of Polymer Dynamics, Oxford University Press."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"6886","DOI":"10.1021\/ma100826u","article-title":"Flow effects on melt structure and entanglement network of linear polymers: Results from a nonequilibrium molecular dynamics simulation study of a polyethylene melt in steady shear","volume":"43","author":"Baig","year":"2010","journal-title":"Macromolecules"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1122\/1.550114","article-title":"Poisson bracket formulation of incompressible flow equations in continuum mechanics","volume":"341","author":"Beris","year":"1990","journal-title":"J. Rheol."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"503","DOI":"10.1122\/1.550094","article-title":"Poisson bracket formulation of viscoelastic flow equations of differential type: A unified approach","volume":"34","author":"Beris","year":"1990","journal-title":"J. Rheol."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"111101","DOI":"10.1063\/5.0056547","article-title":"A method for calculating the nonequilibrium entropy of a flowing polymer melt via atomistic simulation","volume":"155","author":"Edwards","year":"2021","journal-title":"J. Chem. Phys."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"36003","DOI":"10.1209\/0295-5075\/89\/36003","article-title":"Atomistic simulation of flow-induced crystallization at constant temperature","volume":"89","author":"Baig","year":"2010","journal-title":"Europhys. Lett."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"992","DOI":"10.1016\/j.jnnfm.2010.04.007","article-title":"Atomistic simulation of crystallization of a polyethylene melt in steady uniaxial extension","volume":"165","author":"Baig","year":"2010","journal-title":"J. Non-Newton. Fluid Mech."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"184906","DOI":"10.1063\/1.3415085","article-title":"Analysis of the configurational temperature of polymeric liquids under shear and elongational flows using nonequilibrium molecular dynamics and Monte Carlo simulations","volume":"132","author":"Baig","year":"2010","journal-title":"J. Chem. Phys."},{"key":"ref_29","unstructured":"Green, H.S. (1952). The Molecular Theory of Fluids, North-Holland."},{"key":"ref_30","first-page":"254","article-title":"The entropy of systems with random number of particles","volume":"1","author":"Stratonovich","year":"1955","journal-title":"Sov. Phys. JETP"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1365","DOI":"10.1063\/1.1744724","article-title":"Expression in terms of molecular distribution functions for the entropy density in an infinite system","volume":"29","author":"Nettleton","year":"1958","journal-title":"J. Chem. Phys."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"015701","DOI":"10.1103\/PhysRevLett.119.015701","article-title":"Enhancing entropy and enthalpy fluctuations to drive crystallization in atomistic simulations","volume":"119","author":"Piaggi","year":"2017","journal-title":"Phys. Rev. Lett."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"114112","DOI":"10.1063\/1.4998408","article-title":"Entropy based fingerprint for local crystalline order","volume":"147","author":"Piaggi","year":"2017","journal-title":"J. Chem. Phys"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"10487","DOI":"10.1021\/acs.macromol.0c02144","article-title":"A thermodynamically inspired method for quantifying phase transitions in polymeric liquids with application to flow-induced crystallization of a polyethylene melt","volume":"53","author":"Edwards","year":"2020","journal-title":"Macromolecules"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"330","DOI":"10.1038\/365330a0","article-title":"Simulating the critical properties of complex fluids","volume":"365","author":"Siepmann","year":"1993","journal-title":"Nature"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"3376","DOI":"10.1063\/1.469211","article-title":"Calculation of the shear viscosity of decane using a reversible multiple time-step algorithm","volume":"102","author":"Mundy","year":"1995","journal-title":"J. Chem. Phys."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"5615","DOI":"10.1063\/1.473582","article-title":"Modified non-equilibrium molecular dynamics for fluid flows with energy conservation","volume":"106","author":"Tuckerman","year":"1997","journal-title":"J. Chem. Phys."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1016\/S0377-0257(00)00132-4","article-title":"A reversible problem in non-equilibrium thermodynamics: Hamiltonian evolution equations for non-equilibrium molecular dynamics simulations","volume":"96","author":"Edwards","year":"2001","journal-title":"J. Non-Newton. Fluid Mech."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"114103","DOI":"10.1063\/1.1819869","article-title":"A proper approach for nonequilibrium molecular dynamics simulations of planar elongational flow","volume":"122","author":"Baig","year":"2005","journal-title":"J. Chem. Phys."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"114106","DOI":"10.1063\/1.2035079","article-title":"An examination of the validity of non-equilibrium molecular dynamics simulation algorithms for arbitrary steady-state flows","volume":"123","author":"Edwards","year":"2005","journal-title":"J. Chem. Phys."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"194104","DOI":"10.1063\/1.2192776","article-title":"A validation of the p-SLLOD equations of motion for homogeneous steady-state flows","volume":"124","author":"Edwards","year":"2006","journal-title":"J. Chem. Phys."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1006\/jcph.1995.1039","article-title":"Fast parallel algorithms for short-range molecular dynamics","volume":"117","author":"Plimpton","year":"1995","journal-title":"J. Comput. Phys."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"1045","DOI":"10.1016\/0301-9322(92)90074-Q","article-title":"Extensional motions of spatially periodic lattices","volume":"18","author":"Kraynik","year":"1992","journal-title":"Inter. J. Multiph. Flow"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"8124","DOI":"10.1021\/acs.macromol.9b01099","article-title":"Elucidating the molecular rheology of entangled polymeric fluids via comparison of atomistic simulations and model predictions","volume":"52","author":"Edwards","year":"2019","journal-title":"Macromolecules"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"6847","DOI":"10.1021\/ma990620o","article-title":"Packing length influence in linear polymer melts on the entanglement, critical, and reptation molecular weights","volume":"32","author":"Fetters","year":"1999","journal-title":"Macromolecules"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"5054","DOI":"10.3390\/ijms10115054","article-title":"Combined molecular algorithms for the generation, equilibration and topological analysis of entangled polymers: Methodology and performance","volume":"10","author":"Karayiannis","year":"2009","journal-title":"Int. J. Mol. Sci."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"209","DOI":"10.1016\/j.cpc.2005.01.020","article-title":"Shortest multiple disconnected path for the analysis of entanglements in two-and three-dimensional polymeric systems","volume":"168","year":"2005","journal-title":"Comput. Phys. Comm."},{"key":"ref_48","doi-asserted-by":"crossref","unstructured":"Hughes, B.D. (1996). Random Walks and Random Environments: Vol. 1: Random Walks, Oxford University Press.","DOI":"10.1093\/oso\/9780198537892.001.0001"},{"key":"ref_49","unstructured":"Flory, P.J. (1953). Principles of Polymer Chemistry, Cornell University Press."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1080\/00031305.1976.10479172","article-title":"Aesthetic frequency classifications","volume":"30","author":"Doane","year":"1976","journal-title":"Am. Stat."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"141103","DOI":"10.1063\/1.5026792","article-title":"Communication: A coil-stretch transition in planar elongational flow of an entangled polymeric melt","volume":"148","author":"Edwards","year":"2018","journal-title":"J. Chem. Phys."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"247802","DOI":"10.1103\/PhysRevLett.121.247802","article-title":"Configurational microphase separation in elongational flow of an entangled polymer liquid","volume":"121","author":"Edwards","year":"2018","journal-title":"Phys. Rev. Lett."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"013035","DOI":"10.1103\/PhysRevResearch.2.013035","article-title":"Flow-induced crystallization of a polyethylene liquid above the melting temperature and its nonequilibrium phase diagram","volume":"2","author":"Edwards","year":"2020","journal-title":"Phys. Rev. Res."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1007\/BF01535441","article-title":"A general solution to the necklace model problem in the rheology of macromolecules","volume":"5","author":"Booij","year":"1971","journal-title":"J. Eng. Math."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/2\/175\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T22:07:15Z","timestamp":1760134035000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/2\/175"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,1,25]]},"references-count":54,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2022,2]]}},"alternative-id":["e24020175"],"URL":"https:\/\/doi.org\/10.3390\/e24020175","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2022,1,25]]}}}