In conclusion, we have demonstrated that when narrowly confined, polymer chains intrinsically attain a certain degree of chain stiffness. Fundamentally, this change in chain flexibility stems from an entropic penalty imposed on polymer chains by geometric constraints. That is, polymer chains in 2D or narrowly confined systems naturally lose some portion of their original configurational states in 3D space; this is directly reflected by the change in the torsional probability distribution observed in this study. The entropic loss of the polymer chain consequently leads to an increase in chain stiffness.
Therefore, the original flexible polymer chains in 3D bulk systems tend to become semiflexible chains in narrowly confined or interfacial systems. Since the entropic penalty generally increases as the system becomes more restricted, chain flexibility decreases as the degree of confinement increases. A strictly 2D system represents an extreme case of confinement. Moreover, we extend this finding to ring polymer in two-dimensional monolayer systems, thus analyze in detail the effect of dimensional restriction and intrinsic closed-loop geometry on the structure and dynamics, in comparison to linear analogues. We find that both 2D linear and ring polymers in monolayer systems exhibit very similar static and dynamic characteristics, despite molecular architectural differences.
Also, we carried out a detailed analysis on the rheological response of ring polymers in a melt under both shear and elongational flows using atomistic NEMD simulations. Through direct comparison to the corresponding linear polymeric systems, we found that the ring polymers generally exhibit stronger structural resistance against an external flow field, leading to a lesser degree of chain deformation, which eventually give rise to quantitatively different rheological behavior for the ring melts as compared to their linear analogues.
These findings provide basic guidelines for future theoretical and experimental analyses of various polymeric systems as a function of various architectures of polymer.
70
References
(1) Nosé, S. A molecular dynamics method for simulations in the canonical ensemble. Mol. Phys.
1984, 52, 255–268.
(2) Nosé, S. A unified formulation of the constant temperature molecular dynamics methods. J. Chem.
Phys. 1984, 81, 511.
(3) Hoover, W.G. Canonical dynamics: equilibrium phase-space distributions, Phys. Rev. A 1985, 31, 1695–1697.
(4) Tuckerman, M.; Berne, B.J.; Martyna, G.J. Reversible multiple time scale molecular dynamics. J.
Chem. Phys. 1992, 97, 1990–2001.
(5) Siepmann, J.I.; Karaborni, S.; Smit, B. Simulating the critical behavior of complex fluids. Nature 1993, 365, 330–332.
(6) van der Ploeg, P.; Berendsen, H.J.C. Molecular dynamics simulation of a bilayer membrane. J.
Chem. Phys. 1982, 76, 3271-3276.
(7) Jorgensen, W.L.; Madura, J.D.; Swenson, C.J. Optimized intermolecular potential functions for liquid hydrocarbons. J. Am. Chem. Soc. 1984, 106, 6638-6646.
(8) Weeks, J.D.; Chandler, D.; Anderson, H.C. Role of repulsive forces in determining the equilibrium structrue of simple liquids. J. Chem. Phys. 1971, 54, 5237-5247.
(9) Grest, G.S.; Kremer, K. Molecular dynamics simulation for polymers in the presence of a heat bath. Phys. Rev. A 1986, 33, 3628-3631.
(10) Kremer, K.; Grest, G.S. Dynamics of entangled linear polymer melts: A molecular-dynamics simulation. J. Chem. Phys. 1990, 92, 5057-5086.
(11) Plimpton, S. Fast parallel algorithms for short-range molecular dynamics. J. Comp. Phys. 1995, 117, 1-19.
(12) Faller, R.; Kolb, A.; Müller-Plathe, F. Local chain ordering in amorphous polymer melts: influence of chain stiffness. Phys. Chem. Chem. Phys. 1999, 1, 2071-2076.
(13) Faller, R.; Müller-Plathe, F.; Heuer, A. Local reorientation dynamics of semiflexible polymers in the melt. Macromolecules 2000, 33, 6602- 6610.
(14) Baig, C.; Edwards, B. J.; Keffer, D. J.; Cochran, H. D. A proper approach for nonequilibrium
71
molecular dynamics simulations of planar elongational flow. J. Chem. Phys. 2005, 122, 114103.
(15) Lees, A.W.; Edwards, S.F. The computer study of transport processes under extreme conditions.
J. Phys. C 1972, 5, 1921-1929.
(16) Kraynik, A.M.; Reinelt, D.A. Extensional motions of spatially periodic lattices. Int. J. Multiphase Flow 1992, 18, 1045-1059.
(17) Rädler, J. O.; Koltover, I.; Salditt, T.; Safinya, C. R. Structure of DNA–cationic liposome complexes: DNA intercalation in multilamellar membranes in distinct interhelical packing regimes. Science 1997, 275, 810-814.
(18) Maier, B.; Rädler, J. O. Conformation and self-diffusion of single DNA molecules confined to two dimensions. Phys. Rev. Lett. 1999, 82, 1911-1914.
(19) Weiss, H.; Cheng, H.-W.; Mars, J.; Li, H.; Merola, C.; Renner, F. U.; Honkimäki, V.; Valtiner, M.;
Mezger, M. Structure and dynamics of confined liquids: challenges and perspectives for the X‑ray surface forces apparatus. Langmuir 2019, 35, 16679-16692.
(20) Johner, A.; Joanny, J.-F.; Rubinstein, M. Chain statistics in adsorbed polymer solutions. Europhys.
Lett. 1993, 22, 591-596.
(21) Sukhishvili, S. A.; Chen, Y.; Müller, J. D.; Gratton, E.; Schweizer, K. S.; Granick, S. Surface diffusion of poly(ethylene glycol). Macromolecules 2002, 35, 1776-1784.
(22) Wang, Y.; Sun, J.; Li, L. What is the role of the interfacial interaction in the slow relaxation of nanometer-thick polymer melts on a solid surface? Langmuir 2012, 28, 6151-6156.
(23) Adrjanowicz, K.; Winkler, R.; Dzienia, A.; Paluch, M.; Napolitano, S. Connecting 1D and 2D confined polymer dynamics to its bulk behavior via density scaling. ACS. Macro Lett. 2019 8, 304-309.
(24) Van Alsten, J.; Granick, S. Molecular tribometry of ultrathin liquid films. Phys. Rev. Lett. 1998, 61, 2570-2573.
(25) Jones, R. L.; Kumar, S. K.; Ho, D. L.; Briber, R. M.; Russell, T. P. Chain conformation in ultrathin polymer films. Nature 1999, 400, 146.
(26) Jones, R. L.; Kumar, S. K.; Ho, D. L.; Briber, R. M.; Russell, T. P. Chain conformation in ultrathin polymer films using small-angle neutron scattering. Macromolecules 2001, 34, 559-567.
72
(27) Chandran, S.; Reiter, G. Segmental rearrangements relax stresses in nonequilibrated polymer films. ACS. Macro Lett. 2019, 8, 646-650.
(28) Li, S.; Ding, M.; Shi, T. Spatial distribution of entanglements and dynamics in polymer films confined by smooth walls. Polymer 2019, 172, 365-371.
(29) Salditt, T.; Koltover, I.; Rädler, J. O.; Safinya, C. R. Two-dimensional smectic ordering of linear DNA chains in self-assembled DNA-cationic liposome mixtures. Phys. Rev. Lett. 1997, 79, 2582-2585.
(30) Chen, Y.-L.; Graham, M. D.; de Pablo, J. J.; Randall, G. C.; Gupta, M.; Doyle, P. S. Conformation and dynamics of single DNA molecules in parallel plate slit microchannels. Phys. Rev. E 2004, 70, 060901.
(31) Choi, M. C.; Santangelo, C. D.; Pelletier, O.; Kim, J. H.; Kwon, S. Y.; Wen, Z.; Li, Y.; Pincus, P.
A.; Safinya, C. R.; Kim. M. W. Direct observation of biaxial confinement of a semiflexible filament in a channel. Macromolecules 2005, 38, 9882-9884.
(32) Reisner, W.; Morton, K. J.; Riehn, R.; Wang, Y. M.; Yu, Z.; Rosen, M.; Sturm, J. C.; Chou, S. Y.;
Frey, E.; Austin, R. H. Statics and dynamics of single DNA molecules confined in nanochannels.
Phys. Rev. Lett. 2005, 94, 196101.
(33) Liu, J.; Gardel, M. L.; Kroy, K.; Frey, E.; Hoffman, B. D.; Crocker, J. C.; Bausch, A. R.; Weitz, D. A. Microrheology probes length scale dependent rheology. Phys. Rev. Lett. 2006, 96, 118104.
(34) Witz, G.; Rechendorff, K.; Adamcik, J.; Dietler, G. Conformation of circular DNA in two dimensions. Phys. Rev. Lett. 2008, 101, 148103.
(35) Wang, Y.; Tree, D. R.; Dorfman, K. D. Simulation of DNA extension in nanochannels.
Macromolecules 2011, 44, 6594-6604.
(36) Dai, L.; Jones, J. J.; van der Maarel, J. R. C.; Doyle, P. S. A systematic study of DNA conformation in slitlike confinement. Soft Matter 2012, 8, 2972-2982.
(37) Halverson, J. D.; Smrek, J.; Kremer, K.; Grosberg, A. Y. From a melt of rings to chromosome territories: the role of topological constraints in genome folding. Rep. Prog. Phys. 2014, 77, 022601.
(38) Rubinstein, M.; Colby, R. H. Polymer Physics, Oxford Univ. Press: London, 2003.
(39) de Gennes, P. G. Scaling Concepts in Polymer Physics, Cornell Univ. Press: Ithaca and London,
73 1979.
(40) Doi, M.; Edwards, S. F. The Theory of Polymer Dynamics, Oxford Univ. Press: New York, 1986.
(41) Duplantier, B. Statistical mechanics of polymer networks of any topology. J. Stat. Phys. 1989, 54, 581-680.
(42) Semenov, A. N.; Johner, A. Theoretical notes on dense polymers in two dimensions. Eur. Phys. J.
E 2003, 12, 469-480.
(43) Carmesin, I.; Kremer, K. Static and dynamic properties of two-dimensional polymer melts. J.
Phys. (France) 1990, 51, 915-932.
(44) Meyer, H.; Wittmer, J. P.; Kreer, T.; Johner, A.; Baschnagel, J. Static properties of polymer melts in two dimensions. J. Chem. Phys. 2010, 132, 184904.
(45) Meyer, H.; Semenov, A. N. Anomalous dynamics in 2D polymer melts. Phys. Rev. Lett. 2012, 109, 248304.
(46) Baschnagel, J.; Meyer, H.; Wittmer, J.; Kulić, I.; Mohrbach, H.; Ziebert, F.; Nam, G.-M.; Lee, N.- K.; Johner, A. Semiflexible chains at surfaces: worm-like chains and beyond. Polymers 2016, 8, 286.
(47) Polanowki, P.; Jeszka, J. K.; Sikorski, A. Monte Carlo studies of two-dimensional polymer–
solvent systems. J. Mol. Model 2017, 23, 63.
(48) Pressely, J. F.; Riggleman, R. A.; Winey, K. I. Increased polymer diffusivity in thin-flim confinement. Macromolecules 2019, 52, 6116-6125.
(49) Fang, Y.; Yang, J. Two-dimensional condensation of DNA molecules on cationic lipid membranes.
J. Phys. Chem. B 1997, 101, 441-449.
(50) Kumaki, J.; Kajitani, T.; Nagai, K.; Okoshi, K.; Yashima, E. Visualization of polymer chain conformations in amorphous polyisocyanide Langmuir-Blodgett films by atomic force microscopy. J. Am. Chem. Soc. 2010, 132, 5604.
(51) Sugihara, K.; Kumaki, J. Visualization of two-dimensional single chain conformations solubilized in a miscible polymer blend monolayer by atomic force microscopy. J. Phys. Chem. B 2012, 116, 6561-6568.
(52) Kumaki, J. Observation of polymer chain structure in two-dimensional films by atomic force
74 microscopy. Polym. J. 2016, 48, 3-14.
(53) Wang, X.; Foltz, V. J. Chain conformation in two-dimensional dense state. J. Chem. Phys. 2004, 121, 8158-8162.
(54) Ostrovsky, B.; Smith, M. A.; Bar-Yam, Y. Simulation of polymer interpenetration in 2D melts. J.
Mod. Phys. C 1997, 8, 931-939.
(55) Yethiraj, A. Computer simulation study of two-dimensional polymer solutions. Macromolecules 2003, 36, 5854-5862.
(56) Sung, B. J.; Yethiraj, A. Dynamics of two-dimensional and quasi-two-dimensional polymers. J.
Chem. Phys. 2013, 138, 234904.
(57) Hendricks, J.; Kawakatsu, T.; Kawasaki, K.; Zimmermann, W. Confined semiflexible polymer chains. Phys. Rev. E 1995, 51, 2658-2661.
(58) Köster, S.; Steinhauser, D.; Pfohl, T. Brownian motion of actin filaments in confining microchannels. J. Phys.: Condens. Matter 2005, 17, S4091-S4104.
(59) Cifra, P.; Benková, Z.; Bleha, T. Effect of confinement on properties of stiff biological macromolecules. Faraday Discuss 2008, 139, 377-392.
(60) Cifra, P.; Benková, Z.; Bleha, T. Persistence lengths and strucure factors of wormlike polymers under confinements. J. Phys. Chem. B 2008, 112, 1367-1375.
(61) Cifra, P.; Benková, Z.; Bleha, T. Persistence length of DNA moleucules cofined in nanochannels.
Phys. Chem. Chem. Phys. 2010, 12, 8934-8942.
(62) Liu, Y.; Chakraborty, B. Shapes of semiflexible polymers in confined spaces. Phys. Biol. 2008, 5, 026004.
(63) Steinhauser, M. O.; Schneider, J.; Blumen, A. Simulating dynamic crossover behavior of semiflexible linear polymers in solution and in the melt. J. Chem. Phys. 2009, 130, 164902.
(64) Hsu, H.-P.; Binder, K. Semiflexible macromolecules with discrete bond angles confined in nanoslits: a Monte Carlo test of scaling concepts. Macromolecules 2013, 46, 8017-8025.
(65) Huang, A.; Bhattacharya, A.; Binder, K. Conformations, transverse fluctuations, and crossover dynamics of a semi-flexible chain in two dimensions. J. Chem. Phys. 2014, 140, 2014902.
(66) Huang, A.; Hsu, H.-P.; Bhattacharya, A.; Binder, K. Semiflexible macromolecules in quasi-one-
75
dimensional confinement: discrete versus continuous bond angles. J. Chem. Phys. 2015, 143, 243102.
(67) Milchev, A.; Binder, K. How does stiffness of polymer chains affect their adsorption transition?
J. Chem. Phys. 2020, 152, 064901.
(68) At the moment, we can only compare for the basic structural characteristics of polymer chains in dense state between the present numerical work and some available interfacial experiments such as optical experiments of highly condensed 2D DNA molecules on membranes, monolayer Langmuir polymer films, and ultra-thin semiflexible polymer film.49,52,53 The present atomistic MD simulations show the structural characteristics of overall extended interpenetrated chain conformations that compare favorably with those observed in the aforementioned experiments. It is further noted that since the original semiflexible chain in 3D bulk system in experiment become even stiffer in 2D system, more ordered, extended interpenetrated chain configurations were detected in the experiments in comparison to that of the original flexible polymer chain studied here.
(69) Flory, P. J. Statistical Mechanics of Chain Molecules, Wiley: New York, 1969.
(70) Ramachandran, R.; Beaucage, G.; Kulkarni, A. S. Persistence length of short-chain branched polyethylene. Macromolecules 2008, 41, 9802-9806.
(71) Harmandaris, V. A.; Mavrantzas, V. G.; Theodorou, D. N.; Kröger, M.; Ramírez, J.; Öttinger, H.
C.; Vlassopoulos, D. Crossover from the Rouse to the entangled polymer melt regime: signals from long, detailed atomistic molecular dynamics simulations, supported by rheological experiments. Macromolecules 2003, 36, 1376-1387.
(72) Stephanou, P. S.; Baig, C.; Tsolou, G.; Mavrantzas, V. G.; Kröger, M. Quantifying chain reptation in entangled polymer melts: topological and dynamical mapping of atomistic simulation results onto the tube model. J. Chem. Phys. 2010, 132, 124904.
(73) Halverson, J. D.; Lee, W. B.; Grest, G. S.; Grosberg, A. Y.; Kremer, K. Molecular dynamics simulation study of nonconcatenated ring polymers in a melt. II. dynamics. J. Chem. Phys. 2011, 134, 204905.
(74) Sefiddashti, M. H.; Edwards, B. J.; Khomami, B. Individual chain dynamics of a polyethylene melt undergoing steady shear flow. J. Rheol. 2015, 59, 119.
(75) Karayiannis, N. C.; Mavrantzas, V. G.; Hierarchical modeling of the dynamics of polymers with a nonlinear molecular architecture: calcuation of branch point friction and chain reptation time of
76
H-shaped polyethylene melts from long molecular dyanmics simulation. Macromolecules 2005, 38, 8583-8596.
(76) Tsolou, G.; Stratikis, N.; Baig, C.; Stephanou, P.S.; Mavrantzas, V.G. Melt structure and dynamics of unentangled polyethylene rings: Rouse theory, atomistic molecular dynamics simulation, and comparison with the linear analogues, Macromolecules 2010, 43, 10692–10713.
(77) Cates, M. E.; Deutsch, J. M. Conjectures on the statistics of ring polymers. J. Phys. (Paris) 1986, 47, 2121–2128.
(78) Geyler, S.; Pakula, T. Conformation of ring polymers in computer simulated melts. Makromol.
Chem., Rapid Commun. 1988, 9, 617–623.
(79) Müller, M.; Wittmer, J. P.; Cates, M. E. Topological effects in ring polymers: A computer simulation study. Phys. Rev. E 1996, 53, 5063–5074.
(80) Brown, S.; Szamel, G. Structure and dynamics of ring polymers. J. Chem. Phys. 1998, 108, 4705–
4708.
(81) Brown, S.; Lenczycki, T.; Szamel, G. Influence of topological constraints on the statics and dynamics of ring polymers, Phys. Rev. E 2001, 63, 052801.
(82) Arrighi, V.; Gagliardi, S.; Dagger, A. C.; Semlyen, J. A.; Higgins, J. S.; Shenton, M. J.
Conformation of cyclics and linear chain polymers in bulk by SANS. Macromolecules 2004, 37 8057–8065.
(83) Hur, K.; Winkler, R. G.; Yoon, D. Y. Comparison of Ring and Linear Polyethylene from Molecular Dynamics Simulations. Macromolecules 2006, 39, 3975−3977.
(84) Hur, K.; Jeong, C.; Winkler, R. G.; Lacevic, N.; Gee, R. H.; Yoon, D. Y. Chain Dynamics of Ring and Linear Polyethylene Melts from Molecular Dynamics Simulations. Macromolecules 2011, 44, 2311−2315.
(85) Suzuki, J.; Takano, A.; Matsushita, Y. Topological effect in ring polymers investigated with Monte Carlo simulation. J. Chem. Phys. 2008, 129, 034903.
(86) Suzuki, J.; Takano, A.; Deguchi, T.; Matsushita, Y. Dimension of ring polymers in bulk studied by Monte-Carlo simulation and self-consistent theory. J. Chem. Phys. 2009, 131, 144902.
(87) Vettorel, T.; Grosberg, A. Y.; Kremer, K. Statistics of polymer rings in the melt: A numerical simulation study. Phys. Biol. 2009, 6, 025013.
77
(88) Halverson, J. D.; Lee, W. B.; Grest, G. S.; Grosberg, A. Y.; Kremer, K. Molecular dynamics simulation study of nonconcatenated ring polymers in a melt. I. Statics. J. Chem. Phys. 2011, 134, 204904.
(89) Rosa, A.; Everaers, R. Ring polymers in the melt state: The physics of crumpling. Phys. Rev. Lett.
2014, 112, 118302.
(90) Pasquino, R.; Vasilakopoulos, T. C.; Jeong, Y. C.; Lee, H.; Rogers, S.; Sakellariou, G.; Allgaier, J.; Takano, A.; Bras, A. R.; Chang, T.; Gooßen, S.; Pyckhout-Hintzen, W.; Wischnewski, A.;
Hadjichristidis, N.; Richter, D.; Rubinstein, M.; Vlassopoulos, D. Viscosity of ring polymer melts.
ACS Macro Lett. 2013, 2, 874–878.
(91) Robertson, R. M.; Smith, D. E. Self-diffusion of entangled linear and circular DNA molecules:
Dependence on length and concentration. Macromolecules 2007, 40, 3373–3377.
(92) Halverson, J. D.; Lee, W. B.; Grest, G. S.; Grosberg, A. Y.; Kremer, K. Molecular dynamics simulation study of nonconcatenated ring polymers in a melt. II. Dynamics. J. Chem. Phys. 2011, 134, 204905.
(93) Gooßen, S.; Bras, A. R.; Krutyeva, M.; Sharp, M.; Falus, P.; Feoktystov, A.; Gasser, U.; Pyckhout- Hintzen, W.; Wischnewski, A.; Richter, D. Molecular scale dynamics of large ring polymers.
Phys. Rev. Lett. 2014, 113, 168302.
(94) Tsalikis, D. G.; Koukoulas, T.; Mavrantzas, V. G.; Pasquino, R.; Vlassopoulos, D.; Pyckhout- Hintzen, W.; Wischnewski, A.; Monkenbusch, M.; Richter, D. Microscopic structure, conformation, and dynamics of ring and linear poly(ethylene oxide) melts from detailed atomistic molecular dynamics simulations: dependence on chain length and direct comparison with experimental data. Macromolecules 2017, 50, 2565-2584.
(95) Wong, C. P. J.; Choi, P. On the diffusivity of ring polymers, Soft Matter 2020, 16, 2350.
(96) Rubinstein, M. Dynamics of Ring Polymers in the Presence of Fixed Obstacles. Phys. Rev. Lett.
1986, 57, 3023-3026.
(97) Obukhov, S.P.; Rubinstein, M.; Duke, T. Dynamics of a Ring Polymer in a Gel, Phys. Rev. Lett.
1994, 73, 1263-1266.
(98) Grosberg, A.; Rabin, Y.; Havlin, S.; Neer, A. Crumpled Globule Model of the Three-Dimensional Structure of DNA. Europhys. Lett. 1993, 23, 373-378.
78
(99) Halverson, J. D.; Smrek, J.; Kremer, K.; Grosberg, A. Y. From a melt of rings to chromosome territories: the role of topological constraints in genome folding. Rep. Prog. Phys. 2014, 77, 022601.
(100) Lang, M. Ring conformations in bidisperse blends of ring polymers. Macromolecules 2013, 46, 1158−1166.
(101) Smrek, J.; Grosberg, A.Y. Minimal surfaces on unconcatenated polymer rings in melt. ACS Macro Lett. 2016, 5, 750−754.
(102) Jeong, S. H.; Cho, S.; Roh, E. J.; Ha, T. Y.; Kim, J. M.; Baig, C. Intrinsic surface characteristics and dynamic mechanisms of ring polymers in solution and melt under shear flow. Macromolecules 2020, 53, 10051−10060.
(103) Lee, E.; Kim, S.; Jung, Y. Slowing down of ring polymer diffusion caused by inter-ring threading. Macromol. Rapid Commun. 2015, 36, 1115-1121.
(104) Roovers, J. Melt properties of ring polystyrenes. Macromolecules 1985, 18, 1359–1361.
(105) Kapnistos, M.; Lang, M.; Vlassopoulos, D.; Pyckhout-Hintzen, W.; Richter, D.; Cho, D.;
Chang, T.; Rubinstein, M. Unexpected power-law stress relaxation of entangled ring polymers.
Nature Mater. 2008, 7, 997-1002.
(106) Vargas–Lara, F.; Hassan, A. M.; Mansfield, M. L. Knot Energy, Complexity, and Mobility of Knotted Polymers. Sci. Rep. 2017, 7, 13374.
(107) Vargas–Lara, F.; Pazmiño Betancourt, B. A.; Douglas, J. F. Influence of knot complexity on glass-formation in low molecular mass ring polymer melts. J. Chem. Phys. 2019, 150, 101103.
(108) Yoon, J.; Kim, J.; Baig, C. Nonequilibrium molecular dynamics study of ring polymer melts under shear and elongation flows: A comparison with their linear analogs. J. Rheol. 2016, 60, 673- 685.
(109) Jeong, S.; Cho, S.; Kim, J. M.; Baig, C. Interfacial Molecular Structure and Dynamics of Confined Ring Polymer Melts under Shear Flow. Macromolecules 2018, 51, 4670-4677.
(110) Tsamopoulos, A. J.; Katsarou, A. F.; Tsalikis, D. G.; Mavrantzas, V. G. Shear Rheology of Unentangled and marginally entangled ring polymer melts from large-scale nonequilibrium molecular dynamics simulations. Polymers 2019, 11, 1194.
(111) Hsiao, K.-W.; Schroeder, C. M.; Sing, C. E. Ring polymer dynamics are governed by a coupling between architecture and hydrodynamic interactions. Macromolecules 2016, 49, 1961-
79 1971.
(112) Young, C. D.; Qian, J. R.; Marvin, M.; Sing, C. E. Ring polymer dynamics and tumbling- stretch transitions in planar mixed flows. Phys. Rev. E 2019, 99, 062502.
(113) Chen, W.; Chen, J.; An, L. Tumbling and tank-treading dynamics of individual ring polymers in shear flow. Soft Matter 2013, 9, 4312-4318.
(114) Huang, Q.; Ahn, J.; Parisi, D.; Chang, T.; Hassager, O.; Panyukov, S.; Rubinstein, M.
Vlassopoulos, D. Unexpected stretching of entangled ring macromolecules. Phys. Rev. Lett. 2019, 122, 208001.
(115) O’Connor, T. C.; Ge, T.; Rubinstein, M.; Grest, G. S. Topological linking drives anomalous thickening of ring polymers in weak extensional flows. Phys. Rev. Lett. 2020, 124, 027801.
(116) Witz, G.; Rechendorff, K.; Adamcik, J.; Dietler, G. Confromation of Ring polymers in 2D constrained environments. Phys. Rev. Lett. 2011, 106, 248301.
(117) Kim, J.; Kim, J. M.; Baig, C. Intrinsic chain stiffness in flexible linear polymers under extreme confinement. Polymer 2021, 213, 123308.
(118) Drube, F.; Alim, K.; Witz, G.; Dietler, G.; Frey, E. Excluded volume effect on semiflexible ring polymers. Nano Lett. 2010, 10, 1445-1449.
(119) Sakaue, T.; Witz, G.; Dietler, G.; Wada, H. Universal bond correlation function for two- dimensional polymer rings. EPL 2010, 91, 68002.
(120) Zerko, S.; Polanowski, P.; Sikorski, A. Percolation in two-dimensional systems containing cyclic chains. Soft Matter 2012, 8, 973.
(121) Polanowski, P.; Jeszka, J. K.; Sikorski, A. Dynamics properties of linear and cyclic chains in two dimensions. Computer simulation studies. Macromolecules 2014, 47, 4830-4839.
(122) Lee, E.; Jung, Y.; Segregated structures of ring polymer melts near the surface: a molecular dynamics simulation study. Soft Matter, 2015, 11, 6018.
(123) Meddah, C.; Milchev, A.; Sabeur, S. A.; Skvortsov, A. M. Molecular weight effects on interfacial properties of linear and ring polymer melts: A molecular dynamics study. J. Chem.
Phys. 2016, 145, 194902.
(124) Zhang, T.; Winey, K. I.; Riggleman, R. A. Conformation and dynamics of ring polymers under symmetric thin film confinement. J. Chem. Phys. 2020, 153, 184905.
80
(125) Pachong, S. M.; Chubak, I.; Kremer, K.; Smrek, J. Melts of nonconcatenated rings in spherical confinement. J. Chem. Phys. 2020, 153, 064903.
(126) Watanabe, H.; Inoue, T.; Matsumiya, Y.; Transient conformation change of beadspring ring chain during creep process. Macromolecules 2006, 39, 5419–5426.
(127) Maritan, A.; Micheletti, C.; Trovato, A.; Banavar, J. R. Optimal shapes of compact strings.
Nature 2000, 406, 287–290.
(128) Ostermeir, K.; Alim, K.; Frey, E. Buckling of stiff polymer rings in weak spherical confinement. Phys. Rev. E 2010, 81, 061802.
(129) Liu, Y.; Chakraborty, B. Shapes of semiflexible polymers in confined spaces. Phys. Biol. 2008, 5, 026004.
(130) Roh, E. J.; Kim, J. M.; Baig, C. Molecular dynamics study on the0 structure and relaxation of short-chain branched ring polymer melts. Polymer 2019, 175, 107-117.
(131) de Gennes, P. G. Reptation of a polymer chain in the presence of fixed obstacles, J. Chem.
Phys. 1971, 55, 572–579.
(132) Chen, W.; Li, Y.; Zhao, H.; Liu, L.; Chen, J.; An, L. Conformations and dynamics of single flexible ring polymers in simple shear flow. Polymer 2015, 64, 93-99.
(133) Lang, P. S.; Obermayer, B.; Frey, E. Dynamics of a semiflexible polymer or polymer ring in shear flow. Phys. Rev. E 2014, 89, 022606.
(134) Baig, C.; Harmandaris, V. A. Quantitative analysis on the validity of a coarse-grained model for nonequilibrium polymeric liquids under flow. Macromolecules 2010, 43, 3156-3160.
(135) Baig, C.; Mavrantzas, V. G.; Kröger, M. 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. Macromolecules 2010, 43, 6886-6902.
(136) Kim, J. M.; Edwards, B. J.; Keffer, D. J.; Khomami, B. Single-chain dynamics of linear polyethylene liquids under shear flow. Phys. Lett. A 2009, 373, 769–772.
(137) LeDuc, P.; Haber, C.; Bao, G.; Wirtz, D. Dynamics of individual flexible polymers in a shear flow. Nature 1999, 399, 564-566.
(138) Smith, D. E.; Babcock, H. P.; Chu, S. Single-polymer dynamics in steady shear flow. Science 1999, 283, 1724-1727.
81
(139) Baig, C.; Edwards, B. J.; Keffer, D. J. A molecular dynamics study of the stress-optical behavior of a linear short-chain polyethylene melt under shear. Rheol. Acta 2007, 46, 1171-1186.
(140) Irving, J. H.; Kirkwood, J. G. The statistical mechanical theory of transport processes. IV. The equations of hydrodynamics. J. Chem. Phys. 1950, 18, 817-829.
(141) Kröger, M.; Loose, W.; Hess, S. Rheology and structural changes of polymer melts via nonequilibrium molecular dynamics. J. Rheol. 1993, 37, 1057-1079.
(142) Kröger, M.; Luap, C.; Muller, R. Polymer melts under uniaxial elongational flow: stress- optical behavior from experiments and nonequilibrium molecular dynamics computer simulations.
Macromolecules 1997, 30, 526-539.
(143) Matsumoto, T.; Bogue, D. C. Stress birefringence in amorphous polymers under nonisothermal conditions. J. Polym. Sci., Polym. Phys. Ed. 1977, 15, 1663-1674.
(144) van Meerveld, J. Validity of the linear stress optical rule in mono-, bi- and polydisperse systems of entangled linear chains. J. Non-Newtonian Fluid Mech. 2004, 123, 259-267.
(145) Luap, C.; Müller, C.; Schweizer, T.; Venerus, D. C. Simultaneous stress and birefringence measurements during uniaxial elongation of polystyrene melts with narrow molecular weight distribution. Rheol. Acta 2005, 45, 83-91.
(146) Janeschitz-Kriegl, H. Polymer melt rheology and flow birefringence, Springer-Verlag: New York, 1983.
(147) Roovers, J. Viscoelastic properties of polybutadiene rings. Macromolecules 1988, 21, 1517- 1521.
(148) McKenna, G. B.; Hostetter, B. J.; Hadjichristidis, N.; Fetters, L. J.; Plazek, D. J. A study of the linear viscoelastic properties of cyclic polystyrenes using creep and recovery measurements.
Macromolecules 1989, 22, 1834-1852.
(149) Baig, C.; Edwards, B. J.; Keffer, D. J.; Cochran, H. D.; Harmandaris, V. A. Rheological and structural studies of linear polyethylene melts under planar elongational flow using nonequilibrium molecular dynamics simulations. J. Chem. Phys. 2006, 124, 084902.