Computer Simulation of Cyclic Polymers in Disordered Media
Kuriata Aleksander, Sikorski Andrzej *
Department of Chemistry, University of Warsaw
Pasteura 1, 02-093 Warsaw, Poland
∗E-mail: sikorski@chem.uw.eu.pl
Phone: +48 22 822 0211, Fax: +48 22 822 5996
Received:
Received: 21 November 2014; revised: 26 February 2015; accepted: 27 February 2015; published online: 18 March 2015
DOI: 10.12921/cmst.2015.21.01.003
Abstract:
In order to determine the structure and dynamical properties of cyclic polymers (rings) in a random environment we developed and studied an idealized model. All atomic details were suppressed, chains were represented as a sequence of identical beads and were embedded to a simple cubic lattice. A set of obstacles was also randomly introduced into the system and it can be viewed as a model of porous media. A Monte Carlo sampling algorithm using local changes of chain conformation was used to sample the conformational space. It was shown that the mean dimensions of the chain changed with the concentration of obstacles but these changes were non-monotonic. The long-time (diffusion) dynamic properties of the system were also studied. The differences in the mobility of chains depending on the obstacle density were shown and discussed.
Key words:
cyclic polymers, disordered media, lattice models, Monte Carlo method
References:
[1] A. Baumgärtner and M. Muthukumar, A trapped polymer
in random porous media, Adv. Chem. Phys. 94, 625-708
(1996).
[2] M. Ediger, Spatially Heterogeneous Dynamics In Super-
cooled Liquids, Annu. Rev. Phys.Chem. 51, 99-128 (2000).
[3] E. Eisenriegler, Polymers near Surfaces, World Scientific,
Singapore1993.
[4] A.Sikorski,PolymerChains inConfinementandPorousMe-
dia, Solid State Phenom. 138, 451-475 (2008).
[5] G.W.Slater and S.Y. Wu, Reptation, EntropicTrapping, Per-
colation, and Rouse Dynamics of Polymers in “Random”
Environments, Phys. Rev.Lett. 75, 164-167 (1995).
[6] V. Yamakov, D. Stauffer, A. Milchev, G.M. Foo and R.B.
Pandey, Crossover Dynamics for Polymer Simulation in
Porous Media, Phys.Rev. Lett. 79, 2356-2358 (1997).
[7] V. Yamakov andA. Milchev, Diffusion of a polymer chain
in porous media,Phys. Rev. E 55, 1704-1712 (1997).
[8] G.I. Nixon and W.G. Slater, Relaxation length of a polymer
chain in a quenched disordered medium,Phys. Rev. E 60,
3170-3173 (1999).
[9] P.M. Saville and E.M. Sevick, Collision of a Field-Driven
Polymer with a Finite-Sized Obstacle: A Brownian Dynam-
ics Simulation,Macromolecules32, 892-899(1999).
[10] S.H. Chern and R.D. Coalson, Entropic trapping of a flexi-
ble polymer inafixed networkof randomobstacles,J.Chem.
Phys. 111, 1778-1781 (1999).
[11] A.Duaand B.J. Cherayil, Theanomalous diffusionof poly-
mers inrandom media, J. Chem.Phys.112, 421-427 (2000).
[12] G.I. Nixon and W.G. Slater, Saturation and entropic trap-
ping of monodisperse polymers in porous media, J.Chem.
Phys. 117, 4042-4046 (2002).
[13] A. Bhattacharya, Conformation and drift of a telechelic
chain inporous media, J. Phys.:Condens. Matter16,5203-
5211 (2004).
[14] A.J. Moreno and W. Kob,Relaxationdynamics of a linear
moleculein a random staticmedium: Ascaling analysis, J.
Chem. Phys. 121,380-386 (2004).
[15] G.C. Randall and P.S. Doyle, Collision of a DNA Poly-
mer with a Small Obstacle, Macromolecules 39, 7734-7745
(2006).
[16] R.Chang andA.Yethiraj, Dynamics of Chain Moleculesin
Disordered Materials, Phys.Rev.Lett. 96, 107802(2006).
[17] O.A. Hickey and W.G. Slater, The diffusion coefficient of
apolymerin an arrayof obstaclesis a non-monotonic func-
tion of the degree ofdisorder in the medium, Phys. Lett.A
364,448-452 (2007).
[18] A. Balducci and P.S. Doyle, Conformational Precondition-
ing by Electrophoresis of DNA through a Finite Obstacle
Array, Macromolecules41,5485-5492(2008).
[19] B.J. Sung, R. Chang and A. Yethiraj, Swelling of polymers
in porous media,J. Chem. Phys.130, 124908 (2009).
[20] P. Romiszowski,A. Sikorski, Properties of Linear Polymer
Chainsin Porous Media, J.Non-Crystal. Solids 352, 4303-
4308 (2006).
[21] J.A. Semlyen, Cyclic Polymers(2nd edition),Kluwer,Dor-
drecht2000.
[22] T.C.B. McLeish, Polymers Without Beginning orEnd, Sci-
ence 297, 2005-2006 (2002).
[23] T.C.B. McLeish, Polymer dynamics: Floored by the rings,
Nature Mater.7, 933-935(2008).
[24] M. Kapnistos, M. Lang, D. Vlassopoulos, W. Pyckhout-
Hintzen, D. Richter, D. Cho, T.Chang and M.Rubinstein,
Unexpected power-law stress relaxation of entangled ring
polymers, Nature Mater. 7, 997-1002(2008).
[25] C.W. Bielawski, D. Benitezand R.H. Grubbs, An“Endless”
Route to Cyclic Polymers, Science297, 2041-2044(2002).
[26] G. Beaucage, A.S. Kulkarni, Dimensional Description
of Cyclic Macromolecules, Macromolecules 43, 532-537
(2010).
[27] V. Arrighi, S. Gagliardi, A.C. Dagger, J.A. Semlyen, J.S.
Higgins andM.J.Shenton, Conformation ofCyclics and Lin-
earChain Polymers in Bulkby SANS, Macromolecules 37,
8057-8065 (2004).
[28] R.M.Robertson and D.E. Smith, Strong effects of molecu-
lar topologyon diffusionof entangled DNAmolecules, Proc.
Natl. Acad. Sci.U.S.A. 104,4824-4827(2007).
[29] F. Baldelli Bombelli, F. Gambinossi, M.Lagi, D. Berti, G.
Caminati, T. Brown, F.Sciortino, B. NordenandP. Baglioni,
DNA Closed Nanostructures:AStructuraland MonteCarlo
Simulation Study, J. Chem. Phys. B 112, 15283-15294
(2008).
[30] S.P. Obukhov, M. Rubinstein and T. Duke, Dynamics of
a Ring Polymer in a Gel, Phys. Rev. Lett. 73, 1263-1266
(1994).
[31] J.Klein,Dynamicsof entangled linear, branched, andcyclic
polymers, Macromolecules 19, 105-118 (1986).
[32] G.Zifferer andW.Preusser,MonteCarlo Simulation Studies
of the Size andShapeof Ring Polymers, Macromol. Theory
Simul. 10, 397-407 (2001).
[33] J. Reiter, Monte Carlo simulations of linear and cyclic
chains on cubic and quadraticlattices, Macromolecules 23,
3811-3816 (1990).
[34] M. Bishop and J.P.J. Michels,The shapeof ring polymers, J.
Chem. Phys. 82, 1059-1061 (1985).
[35] M. Bishop, J.P.J.Michels, Scaling inthree-dimensional lin-
earand ring polymers, J. Chem.Phys. 84, 444-446 (1986).
[36] M. Bishop and C.J. Saltiel, Polymer shapesin two, four,and
fivedimensions,J. Chem.Phys. 88,3976-3982(1985).
[37] J. Suzuki,A. Takano and Y. Matsushita, Topological effect
in ring polymers investigatedwith Monte Carlo simulation,
J. Chem. Phys.129, 034903 (2008).
[38] S. Brown and G. Szamel, Structure and dynamics of ring
polymers, J.Chem. Phys. 108,4705-4708(1998).
[39] S. Brown and G.Szamel,Computer simulation studyof the
structure and dynamics of ring polymers, J. Chem. Phys.
109,6184-6192(1998).
[40] N. Kanaeda and T. Deguchi, Diffusion of a ring polymer
in good solutionvia the Brownian dynamics with no bond
crossing,J. Phys. A: Math. Theor.41, 145004(2008).
[41] S. Brown, T. Lenczycki and G. Szamel, Influence of topo-
logical constraintsonthestaticsand dynamicsof ring poly-
mers, Phys. Rev. E63,052801 (2001).
[42] A. Vettorel, A.Y. Grossberg and K. Kremer, Statistics of
polymer rings in the melt: a numerical simulation study,
Phys. Biol.6, 025013(2009).
[43] G. Tsolou, N. Stratikis, C. Baig, P.S. Stephanou and V.G.
Mavrantzas, MeltStructure and Dynamics of Unentangled
Polyethylene Rings:RouseTheory, Atomistic MolecularDy-
namics Simulation, and Comparison with the Linear Ana-
logues,Macromolecules43, 10692-10713 (2010).
[44] J. Suzuki, A. Takano, T. Deguchi and Y. Matsushita, Di-
mension of ring polymers in bulk studied by Monte-Carlo
simulation and self-consistent theory, J. Chem. Phys. 131,
144902 (2009).
[45] T. Pakula and K. Jeszka, Simulation of Single Complex
Macromolecules. 1. Structure and Dynamics of Catenanes,
Macromolecules32, 6821-6830 (1999).
[46] J. Reiter, Monte Carlo study of diffusion of an ideal ring
polymer inanetwork ofobstacles on a cubic and asquare
lattice, J.Chem. Phys. 95, 1290-1294 (1991).
[47] D.Gersappe and M.Olvera de laCruz, A Monte Carlo Study
of Ring Polymers in Disordered Systems, Mol. Simulat. 13,
267-283 (1994).
[48] B.V.S. Iyer,A.K.Lele, V.A. Juvekar and R.A. Mashelkar,
Self-Similar Dynamics of a Flexible Ring Polymer in
a Fixed Obstacle Environment: A Coarse-Grained Molec-
ularModel, Ind. Eng.Chem. Res. 48, 9514-9522 (2009).
[49] J. Skolnick and A. Kolinski, Dynamics of Dense Polymer
Systems: Computer Simulationsand AnalyticTheories,Adv.
Chem. Phys. 77, 223-278(1990).
[50] A. Koli ́nski, M. Vieth i A. Sikorski, Collapse of Semiflexi-
ble Polymersin Two Dimensions.Monte CarloSimulations,
Acta Phys. Polon.A 79, 601-612 (1991).
[51] K.Binder,M. Müller,J.Baschnagel,Polymer Models onthe
Lattice, in: M. Kotelyanskii, D.N. Theodorou (eds.) Sim-
ulation Methods for Polymers, Marcel Dekker, NewYork-
Basel,p. 125-146, 2004.
[52] A.Takano,Y. Ohta,K. Masuoka,K. Matsubara, T. Nakano,
A. Hieno, M. Itakura, K. Takahashi, S. Kinugasa, D.
Kawaguchi,Y.Takahashi,Y.Matsushita, Radii of gyration
of ring-shapedpolystyrenes with high purity indilute solu-
tions, Macromolecules45,369-373 (2012).
[53] A.Y. Grosberg,Critical Exponentsfor RandomKnots, Phys.
Rev.Lett. 85, 3858-3861 (2000).
[54] A. Dobay, J. Dubochet, K. Millett,P.-E. Sottas, A. Stasiak,
Scaling behavior of random knots, Proc. Natl. Acad. Sci.
U.S.A. 100,5611-5615 (2003).
[55] A. Kolinski, J.Skolnick, R.Yaris, Monte Carlo studies on
the longtimedynamic properties of densecubiclattice mul-
tichain systems.I. The homopolymeric melt, J. Chem. Phys.
86, 7164-7173 (1987).
In order to determine the structure and dynamical properties of cyclic polymers (rings) in a random environment we developed and studied an idealized model. All atomic details were suppressed, chains were represented as a sequence of identical beads and were embedded to a simple cubic lattice. A set of obstacles was also randomly introduced into the system and it can be viewed as a model of porous media. A Monte Carlo sampling algorithm using local changes of chain conformation was used to sample the conformational space. It was shown that the mean dimensions of the chain changed with the concentration of obstacles but these changes were non-monotonic. The long-time (diffusion) dynamic properties of the system were also studied. The differences in the mobility of chains depending on the obstacle density were shown and discussed.
Key words:
cyclic polymers, disordered media, lattice models, Monte Carlo method
References:
[1] A. Baumgärtner and M. Muthukumar, A trapped polymer
in random porous media, Adv. Chem. Phys. 94, 625-708
(1996).
[2] M. Ediger, Spatially Heterogeneous Dynamics In Super-
cooled Liquids, Annu. Rev. Phys.Chem. 51, 99-128 (2000).
[3] E. Eisenriegler, Polymers near Surfaces, World Scientific,
Singapore1993.
[4] A.Sikorski,PolymerChains inConfinementandPorousMe-
dia, Solid State Phenom. 138, 451-475 (2008).
[5] G.W.Slater and S.Y. Wu, Reptation, EntropicTrapping, Per-
colation, and Rouse Dynamics of Polymers in “Random”
Environments, Phys. Rev.Lett. 75, 164-167 (1995).
[6] V. Yamakov, D. Stauffer, A. Milchev, G.M. Foo and R.B.
Pandey, Crossover Dynamics for Polymer Simulation in
Porous Media, Phys.Rev. Lett. 79, 2356-2358 (1997).
[7] V. Yamakov andA. Milchev, Diffusion of a polymer chain
in porous media,Phys. Rev. E 55, 1704-1712 (1997).
[8] G.I. Nixon and W.G. Slater, Relaxation length of a polymer
chain in a quenched disordered medium,Phys. Rev. E 60,
3170-3173 (1999).
[9] P.M. Saville and E.M. Sevick, Collision of a Field-Driven
Polymer with a Finite-Sized Obstacle: A Brownian Dynam-
ics Simulation,Macromolecules32, 892-899(1999).
[10] S.H. Chern and R.D. Coalson, Entropic trapping of a flexi-
ble polymer inafixed networkof randomobstacles,J.Chem.
Phys. 111, 1778-1781 (1999).
[11] A.Duaand B.J. Cherayil, Theanomalous diffusionof poly-
mers inrandom media, J. Chem.Phys.112, 421-427 (2000).
[12] G.I. Nixon and W.G. Slater, Saturation and entropic trap-
ping of monodisperse polymers in porous media, J.Chem.
Phys. 117, 4042-4046 (2002).
[13] A. Bhattacharya, Conformation and drift of a telechelic
chain inporous media, J. Phys.:Condens. Matter16,5203-
5211 (2004).
[14] A.J. Moreno and W. Kob,Relaxationdynamics of a linear
moleculein a random staticmedium: Ascaling analysis, J.
Chem. Phys. 121,380-386 (2004).
[15] G.C. Randall and P.S. Doyle, Collision of a DNA Poly-
mer with a Small Obstacle, Macromolecules 39, 7734-7745
(2006).
[16] R.Chang andA.Yethiraj, Dynamics of Chain Moleculesin
Disordered Materials, Phys.Rev.Lett. 96, 107802(2006).
[17] O.A. Hickey and W.G. Slater, The diffusion coefficient of
apolymerin an arrayof obstaclesis a non-monotonic func-
tion of the degree ofdisorder in the medium, Phys. Lett.A
364,448-452 (2007).
[18] A. Balducci and P.S. Doyle, Conformational Precondition-
ing by Electrophoresis of DNA through a Finite Obstacle
Array, Macromolecules41,5485-5492(2008).
[19] B.J. Sung, R. Chang and A. Yethiraj, Swelling of polymers
in porous media,J. Chem. Phys.130, 124908 (2009).
[20] P. Romiszowski,A. Sikorski, Properties of Linear Polymer
Chainsin Porous Media, J.Non-Crystal. Solids 352, 4303-
4308 (2006).
[21] J.A. Semlyen, Cyclic Polymers(2nd edition),Kluwer,Dor-
drecht2000.
[22] T.C.B. McLeish, Polymers Without Beginning orEnd, Sci-
ence 297, 2005-2006 (2002).
[23] T.C.B. McLeish, Polymer dynamics: Floored by the rings,
Nature Mater.7, 933-935(2008).
[24] M. Kapnistos, M. Lang, D. Vlassopoulos, W. Pyckhout-
Hintzen, D. Richter, D. Cho, T.Chang and M.Rubinstein,
Unexpected power-law stress relaxation of entangled ring
polymers, Nature Mater. 7, 997-1002(2008).
[25] C.W. Bielawski, D. Benitezand R.H. Grubbs, An“Endless”
Route to Cyclic Polymers, Science297, 2041-2044(2002).
[26] G. Beaucage, A.S. Kulkarni, Dimensional Description
of Cyclic Macromolecules, Macromolecules 43, 532-537
(2010).
[27] V. Arrighi, S. Gagliardi, A.C. Dagger, J.A. Semlyen, J.S.
Higgins andM.J.Shenton, Conformation ofCyclics and Lin-
earChain Polymers in Bulkby SANS, Macromolecules 37,
8057-8065 (2004).
[28] R.M.Robertson and D.E. Smith, Strong effects of molecu-
lar topologyon diffusionof entangled DNAmolecules, Proc.
Natl. Acad. Sci.U.S.A. 104,4824-4827(2007).
[29] F. Baldelli Bombelli, F. Gambinossi, M.Lagi, D. Berti, G.
Caminati, T. Brown, F.Sciortino, B. NordenandP. Baglioni,
DNA Closed Nanostructures:AStructuraland MonteCarlo
Simulation Study, J. Chem. Phys. B 112, 15283-15294
(2008).
[30] S.P. Obukhov, M. Rubinstein and T. Duke, Dynamics of
a Ring Polymer in a Gel, Phys. Rev. Lett. 73, 1263-1266
(1994).
[31] J.Klein,Dynamicsof entangled linear, branched, andcyclic
polymers, Macromolecules 19, 105-118 (1986).
[32] G.Zifferer andW.Preusser,MonteCarlo Simulation Studies
of the Size andShapeof Ring Polymers, Macromol. Theory
Simul. 10, 397-407 (2001).
[33] J. Reiter, Monte Carlo simulations of linear and cyclic
chains on cubic and quadraticlattices, Macromolecules 23,
3811-3816 (1990).
[34] M. Bishop and J.P.J. Michels,The shapeof ring polymers, J.
Chem. Phys. 82, 1059-1061 (1985).
[35] M. Bishop, J.P.J.Michels, Scaling inthree-dimensional lin-
earand ring polymers, J. Chem.Phys. 84, 444-446 (1986).
[36] M. Bishop and C.J. Saltiel, Polymer shapesin two, four,and
fivedimensions,J. Chem.Phys. 88,3976-3982(1985).
[37] J. Suzuki,A. Takano and Y. Matsushita, Topological effect
in ring polymers investigatedwith Monte Carlo simulation,
J. Chem. Phys.129, 034903 (2008).
[38] S. Brown and G. Szamel, Structure and dynamics of ring
polymers, J.Chem. Phys. 108,4705-4708(1998).
[39] S. Brown and G.Szamel,Computer simulation studyof the
structure and dynamics of ring polymers, J. Chem. Phys.
109,6184-6192(1998).
[40] N. Kanaeda and T. Deguchi, Diffusion of a ring polymer
in good solutionvia the Brownian dynamics with no bond
crossing,J. Phys. A: Math. Theor.41, 145004(2008).
[41] S. Brown, T. Lenczycki and G. Szamel, Influence of topo-
logical constraintsonthestaticsand dynamicsof ring poly-
mers, Phys. Rev. E63,052801 (2001).
[42] A. Vettorel, A.Y. Grossberg and K. Kremer, Statistics of
polymer rings in the melt: a numerical simulation study,
Phys. Biol.6, 025013(2009).
[43] G. Tsolou, N. Stratikis, C. Baig, P.S. Stephanou and V.G.
Mavrantzas, MeltStructure and Dynamics of Unentangled
Polyethylene Rings:RouseTheory, Atomistic MolecularDy-
namics Simulation, and Comparison with the Linear Ana-
logues,Macromolecules43, 10692-10713 (2010).
[44] J. Suzuki, A. Takano, T. Deguchi and Y. Matsushita, Di-
mension of ring polymers in bulk studied by Monte-Carlo
simulation and self-consistent theory, J. Chem. Phys. 131,
144902 (2009).
[45] T. Pakula and K. Jeszka, Simulation of Single Complex
Macromolecules. 1. Structure and Dynamics of Catenanes,
Macromolecules32, 6821-6830 (1999).
[46] J. Reiter, Monte Carlo study of diffusion of an ideal ring
polymer inanetwork ofobstacles on a cubic and asquare
lattice, J.Chem. Phys. 95, 1290-1294 (1991).
[47] D.Gersappe and M.Olvera de laCruz, A Monte Carlo Study
of Ring Polymers in Disordered Systems, Mol. Simulat. 13,
267-283 (1994).
[48] B.V.S. Iyer,A.K.Lele, V.A. Juvekar and R.A. Mashelkar,
Self-Similar Dynamics of a Flexible Ring Polymer in
a Fixed Obstacle Environment: A Coarse-Grained Molec-
ularModel, Ind. Eng.Chem. Res. 48, 9514-9522 (2009).
[49] J. Skolnick and A. Kolinski, Dynamics of Dense Polymer
Systems: Computer Simulationsand AnalyticTheories,Adv.
Chem. Phys. 77, 223-278(1990).
[50] A. Koli ́nski, M. Vieth i A. Sikorski, Collapse of Semiflexi-
ble Polymersin Two Dimensions.Monte CarloSimulations,
Acta Phys. Polon.A 79, 601-612 (1991).
[51] K.Binder,M. Müller,J.Baschnagel,Polymer Models onthe
Lattice, in: M. Kotelyanskii, D.N. Theodorou (eds.) Sim-
ulation Methods for Polymers, Marcel Dekker, NewYork-
Basel,p. 125-146, 2004.
[52] A.Takano,Y. Ohta,K. Masuoka,K. Matsubara, T. Nakano,
A. Hieno, M. Itakura, K. Takahashi, S. Kinugasa, D.
Kawaguchi,Y.Takahashi,Y.Matsushita, Radii of gyration
of ring-shapedpolystyrenes with high purity indilute solu-
tions, Macromolecules45,369-373 (2012).
[53] A.Y. Grosberg,Critical Exponentsfor RandomKnots, Phys.
Rev.Lett. 85, 3858-3861 (2000).
[54] A. Dobay, J. Dubochet, K. Millett,P.-E. Sottas, A. Stasiak,
Scaling behavior of random knots, Proc. Natl. Acad. Sci.
U.S.A. 100,5611-5615 (2003).
[55] A. Kolinski, J.Skolnick, R.Yaris, Monte Carlo studies on
the longtimedynamic properties of densecubiclattice mul-
tichain systems.I. The homopolymeric melt, J. Chem. Phys.
86, 7164-7173 (1987).
