Abstract
The influence of topological constraints on the local dynamics in cross-linked polymer melts and their contribution to the elastic properties of rubber elastic systems are a long standing problem in statistical mechanics. Polymer networks with diamond lattice connectivity (Everaers and Kremer 1995, Everaers and Kremer 1996a) are idealized model systems which isolate the effect of topology conservation from other sources of quenched disorder. We study their behavior in molecular dynamics simulations under elongational strain. In our analysis we compare the measured, purely entropic shear moduli G to the predictions of statistical mechanical models of rubber elasticity, making extensive use of the microscopic structural and topological information available in computer simulations. We find (Everaers and Kremer 1995) that the classical models of rubber elasticity underestimate the true change in entropy in a deformed network significantly, because they neglect the tension along the contour of the strands which cannot relax due to entanglements (Everaers and Kremer (in preparation)). This contribution and the fluctuations in strained systems seem to be well described by the constrained mode model (Everaers 1998) which allows to treat the crossover from classical rubber elasticity to the tube model for polymer networks with increasing strand length within one transparant formalism. While this is important for the description of the effects we try to do a first quantitative step towards their explanation by topological considerations. We show (Everaers and Kremer 1996a) that for the comparatively short strand lengths of our diamond networks the topology contribution to the shear modulus is proportional to the density of entangled mesh pairs with non-zero Gauss linking number. Moreover, the prefactor can be estimated consistently within a rather simple model developed by Vologodskii et al. and by Graessley and Pearson, which is based on the definition of an entropic interaction between the centers of mass of two loops in a conserved topological state.
Polymer networks are the basic structural element of systems as different as tire rubber and gels. They are not only technically important but also commonly found in biological systems such as the cytoskeleton. Networks of flexible macromolecules display an elastic and thermoelastic behaviour quite different from ordinary solids. (Treloar 1975) Crystals, metals, ceramics, or glasses can be stretched only minimally. Small deformations of the sample extend down to atomic scales and lead to an increase of the internal energy. Rubber-like materials reversibly sustain elongations of up to 1000% with small strain elastic moduli that are four or five orders of magnitude smaller than for other solids. Most importantly, the tension induced by a deformation is almost exclusively due to a decrease in entropy. As a consequence, the underlying mechanism has to be different from the case of conventional solids.
Preview
Unable to display preview. Download preview PDF.
References
Barsky, S. J., Plischke, M., Zhou, Z. and Joós, B. (1996): Phys. Rev. E, 54, 5370
Barsky, S. J. and Plischke, M. (1996): Phys. Rev. E, 53, 871
Bird, R.B., Armstrong, R. C. and Hassager, O. (1977): Dynamics of Polymeric Liquids, volume 1. Wiley, New York
Curro, J. G. and Pincus, P. A. (1983): Macromolecules, 16, 559
Curro, J. G., Pearson, D. S. and Helfand, E. (1985): Macromolecules, 18, 1157
Curro, J. G., Schweizer, K. S., Grest, G. S. and Kremer, K. (1991): J. of Chem. Phys., 91, 1357
Deam, R. T. and Edwards, S. F. (1976): Phil. Trans. R. Soc. A, 280, 317
Doi, M. and Edwards, S. F. (1986): The Theory of Polymer Dynamics. Claredon Press, Oxford
Duering, E. R., Kremer, K. and Grest, G. S. (1991): Phys. Rev. Lett., 67, 3531
Duering, E. R., Kremer, K. and Grest, G. S. (1992): Progr. Colloid Polym. Sci., 90, 13
Duering, E. R., Kremer, K. and Grest, G. S. (1993): Macromolecules, 26, 3241
Duering, E. R., Kremer, K. and Grest, G. S. (1994): J. of Chem. Phys., 101, 8169
Edwards, S. F. (1967a): Proc. Phys. Soc., 92, 9
Edwards, S. F. (1967b): Proc. Phys. Soc., 91, 513
Edwards S. F. (1968): J. of Phys. A, 1, 15
Edwards, S. F. and Vilgis, T. A. (1986): Polymer, 27, 483
Edwards, S. F. and Vilgis, T. A. (1988): Rep. Progr. Phys., 51, 243
Erman, B. and Flory, P. J. (1978): J. of Chem. Phys., 68, 5363
Erman, B. and Flory, P. J. (1982): Macromolecules, 15, 806
Everaers, R. and Kremer, K. in preparation
Everaers, R. and Kremer, K. (1994): Comp. Phys. Comm., 81, 19
Everaers, R. Kremer, K. and Grest, G. S. (1995): Macromol. Symposia, 93, 53
Everaers, R. and Kremer, K. (1995): Macromolecules, 28, 7291
Everaers, R. and Kremer, K. (1996a): Phys. Rev. E, 53, R37
Everaers, R. and Kremer, K. (1996b): J. Mol. Mod., 2, 293
Everaers, R. (1998): Eur. J. Phys. B, 4, 341
Flory, P. J. and Rehner, J. (1943): J. of Chem. Phys., 11, 512
Flory, P. J. (1976): Proc. Royal Soc. London Ser. A., 351, 351
Flory, P.J. (1977): J. of Chem. Phys., 66, 5720
Flory, P. J. and Erman, B. (1982): Macromolecules, 15, 800
Frank-Kamenetskii, M. D. Lukashin, A. V. and Vologodskii, A. V. (1975): Nature, 258, 398
Gao, J. and Weiner, J. H. (1995): J. of Chem. Phys., 103, 1614
de Gennes, P. G. (1971): J. of Chem. Phys., 55, 572
Gottlieb, M., Macosko, C. W., Benjamin, G. S., Meyers, K. O. and Merrill, E. W. (1981): Macromolecules, 14, 1039
Graessley, W. W. and Pearson, D. S. (1977): J. of Chem. Phys., 66, 3363
Graessley, W. W. (1982): Adv. Pol. Sci., 47, 67
Grest, G. S. and Kremer, K. (1990a): J. de Physique (France), 51, 2829
Grest, G. S. and Kremer, K. (1990b): Macromolecules, 23, 4994
Grest, G. S., Kremer, K. and Duering, E. R. (1992): Europhysics Lett., 19, 195
Grest, G. S., Kremer, K. and Duering, E. R. (1993): Physica A, 194, 330
Heinrich, G. Straube, E. and Helmis, G. (1988): Adv. Pol. Sci., 85, 34
Helfand, E. and Tonelli, A. E. (1974): Macromolecules, 7, 832
Herrmann, H. J., Hong, D. C. and Stanley, H. E. (1984): J. of Phys. A, 17, L261
Iwata, K. (1982): J. of Chem. Phys., 76, 6363
Iwata, K. (1985): J. of Chem. Phys., 83, 1969
James, H. (1947): J. of Chem. Phys., 15, 651
James, H. and Guth, E. (1947): J. of Chem. Phys., 15, 669
Kästner, S. (1981): Colloid Polym. Sci., 259, 499 and 508
Kremer, K. and Grest, G. S. (1990): J. of Chem. Phys., 92, 5057
Kremer, K. and Grest, G. S. (1991): J. of Chem. Phys., 94, 4103
Kremer, K. and Grest, G. S. (1995): In K. Binder, editor, Monte Carlo and Molecular Dynamics Simulations in Polymer Science. Oxford University Press, New York and Oxford
Larsson, I. and Kramer, O. (1993): Makromol. Chem., Macromol. Symp., 76, 117
Lazár, M., Rado, R. and Rychly, J. (1990): Adv. Polym. Sci., 95, 149
Leung, Y. K. and Eichinger, B. E. (1984): J. of Chem. Phys., 80, 3877 and 3885
Mark, J.E. (1982): Adv. Pol. Sci., 44, 1
Oeser, R., Ewen, B., Richter, D. and Farago, B. (1988): Phys. Rev. Lett., 260, 1041
Opperman, W. and Rennar, N. (1987): Prog. Colloid Polym. Sci., 75, 49
Patel, S. K. Malone, S., Cohen, C. and Gillmor, J. R (1992): Macromolecules, 25, 5241
Pearson, D. S. and Graessley, W. W. (1980): Macromolecules, 13, 1001
Queslel, J. P. and Mark, J. E. (1984): Adv. Pol. Sci., 65, 135
Ronca, G. and Allegra, G. (1975): J. of Chem. Phys., 63, 4990
Schweizer, K. S. and Curro, J. G. (1994): Adv. Polym. Sci., 116, 319
Shy, L. Y. and Eichinger, B. E. (1986): Macromolecules, 19, 2787
Straube, E., Urban, V., Pyckhout-Hintzen, W. Richter, D. and Glinka, C. J. (1995): Phys. Rev. Lett., 74, 4464
Tonelli, A. E. and Helfand, E. (1974): Macromolecules, 7, 59
Treloar, L.R.G. (1975): The Physics of Rubber Elasticity. Clarendon Press, Oxford
Wiegel, F. W. (1986): Introduction in Path Integral Methods in Physics and Polymers Science. World Scientific, Philadelphia
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1999 Springer-Verlag
About this paper
Cite this paper
Everaers, R., Kremer, K. (1999). Entanglement effects in model polymer networks. In: Pękalski, A., Sznajd-Weron, K. (eds) Anomalous Diffusion From Basics to Applications. Lecture Notes in Physics, vol 519. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0106845
Download citation
DOI: https://doi.org/10.1007/BFb0106845
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-65416-2
Online ISBN: 978-3-540-49259-7
eBook Packages: Springer Book Archive