Abstract.
We study the properties of the classical Ising model with nearest-neighbor interaction for spins located at the monomers of long polymer chains in 2 and 3 dimensions. We compare results for two ensembles of polymers with very different single chain properties: 1) swollen, self-avoiding linear polymer chains in good solvent conditions and 2) compact, space-filling randomly branching polymers in melt. By employing a mean-field approach and Monte Carlo computer simulations, we show that swollen polymers cannot sustain an ordered phase. On the contrary, compact polymers may indeed produce an observable phase transition. Finally, we briefly consider the statistical properties of the ordered phase by comparing polymer chains within the same universality class but characterized by very different shapes.
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References
W. Lenz, Phys. Z. 21, 613 (1920)
E. Ising, Z. Phys. 31, 253 (1925)
S.G. Brush, Rev. Mod. Phys. 39, 883 (1967)
K. Huang, Statistical Mechanics, 2nd edition (John Wiley & Sons, 1987)
J.J. Hopfield, Proc. Natl. Acad. Sci. U.S.A. 79, 2554 (1982)
C. Schinckus, Physica A 508, 95 (2018)
B.K. Chakrabarti, S. Bhattacharya, J. Phys. C: Solid State Phys. 16, L1025 (1983)
S. Bhattacharya, B.K. Chakrabarti, Z. Phys. B: Condens. Matter 57, 151 (1984)
B.K. Chakrabarti, S. Bhattacharya, J. Phys. A: Math. Gen. 18, 1037 (1985)
B.K. Chakrabarti, A.C. Maggs, R.B. Stinchcombe, J. Phys. A: Math. Gen. 18, L373 (1985)
T. Garel, H. Orland, E. Orlandini, Eur. Phys. J. B 12, 261 (1999)
A. Rosa, R. Everaers, J. Chem. Phys. 145, 164906 (2016)
A.R. Khokhlov, S.K. Nechaev, Phys. Lett. A 112, 156 (1985)
M. Rubinstein, Phys. Rev. Lett. 57, 3023 (1986)
S.P. Obukhov, M. Rubinstein, T. Duke, Phys. Rev. Lett. 73, 1263 (1994)
A.Y. Grosberg, Soft Matter 10, 560 (2014)
A. Rosa, R. Everaers, Phys. Rev. Lett. 112, 118302 (2014)
A. Rosa, R. Everaers, PLOS Comput. Biol. 4, e1000153 (2008)
J.D. Halverson, J. Smrek, K. Kremer, A.Y. Grosberg, Rep. Prog. Phys. 77, 022601 (2014)
A. Colliva, R. Pellegrini, A. Testori, M. Caselle, Phys. Rev. E 91, 052703 (2015)
D. Jost, P. Carrivain, G. Cavalli, C. Vaillant, Nucl. Acids Res. 42, 9553 (2014)
D. Jost, Phys. Rev. E 89, 010701 (2014)
J.D. Olarte-Plata, N. Haddad, C. Vaillant, D. Jost, Phys. Biol. 13, 026001 (2016)
D. Jost, C. Vaillant, Nucl. Acids Res. 46, 2252 (2018)
M. Aertsens, C. Vanderzande, J. Phys. A: Math. Gen. 25, 735 (1992)
K.G. Wilson, Phys. Rev. B 4, 3174 (1971)
K.G. Wilson, Phys. Rev. B 4, 3184 (1971)
S. Ma, Modern Theory of Critical Phenomena (Benjamin London, 1976)
N. Goldenfeld, Lectures on Phase Transitions and the Renormalization Group (Addison-Wesley, 1992)
M. Rubinstein, R.H. Colby, Polymer Physics (Oxford University Press, New York, 2003)
R. Everaers, A.Y. Grosberg, M. Rubinstein, A. Rosa, Soft Matter 13, 1223 (2017)
A. Rosa, R. Everaers, Phys. Rev. E 95, 012117 (2017)
A. Rosa, R. Everaers, https://doi.org/arxiv.org/abs/1808.06861 (2018)
N. Madras, A.D. Sokal, J. Stat. Phys. 50, 109 (1988)
A. Rosa, R. Everaers, J. Phys. A: Math. Theor. 49, 345001 (2016)
W.A. Seitz, D.J. Klein, J. Chem. Phys. 75, 5190 (1981)
U. Wolff, Phys. Rev. Lett. 62, 361 (1989)
W. Krauth, Statistical Mechanics Algorithms and Computations (Oxford University Press, Oxford, 2006)
M. Bishop, J.P.J. Michels, J. Chem. Phys. 84, 444 (1986)
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Papale, A., Rosa, A. The Ising model in swollen vs. compact polymers: Mean-field approach and computer simulations. Eur. Phys. J. E 41, 144 (2018). https://doi.org/10.1140/epje/i2018-11752-2
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DOI: https://doi.org/10.1140/epje/i2018-11752-2