Abstract
Early investigations by Weber, R. and F. Kohlrausch, Maxwell, and Boltzmann of relaxation in viscoelastic solids are reviewed. A two-state model stress-tensor describing strain coupling to internal conformations of a polymer chain is used to derive a linear response version of the Boltzmann superposition principle for shear stress relaxation. The relaxation function of Kohlrauschφ(t)= exp[−(t/τ α)] is identical to the Williams-Watts empirical dielectric relaxation function and in the model corresponds to the autocorrelation function of a segment's differential shape anisotropy tensor. By analogy with the dielectric problem, exp[−(t/τ α)] is interpreted as the survival probability of a frozen segment in a swarm of hopping defects with a stable waiting-time distributionAt −α for defect motion. The exponent a is the fractal dimension of a hierarchical scaling set of defect hopping times. Integral transforms ofφ(t) needed for data analysis are evaluated; the cosine and inverse-Laplace transforms are stable probability densities. The reciprocal kernel for short-time compliance is discussed.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
H. Leaderman,Elastic and Creep Properties of Filamentous Materials and Other High Polymers (The Textile Foundation, Washington, 1943).
L. C. E. Struik, Abstracts of the 10th Europhysics Conference on Macromolecular Physics, p. 135 (1980).
L. C. E. Struik,Physical Aging in Amorphous Polymers and Other Materials (Elsevier-North-Holland, Amsterdam, 1978).
R. McCormmack,Historical Studies in the Physical Sciences X, p. 23 (University of Pennsylvania Press, Philadelphia, 1979).
R. Kohlrausch,Pogg. Ann. Phys. 91:198 (1854).
F. Kohlrausch,Pogg. Ann. Phys. 119:352 (1863).
J. C. Maxwell,Philos. Trans. R. Soc. (London) 157:52 (1867).
L. Boltzmann,Pogg. Ann. Phys., Erganzungsband VII.624 (1876).
F. Kohlrausch,Pogg. Ann. Phys. 158:337 (1876).
J. Hopkinson,Philos. Trans. R. Soc. (London) 167:599 (1877).
J. D. Ferry,Viscoelastic Properties of Polymers, 3rd ed. (Wiley, New York, 1980).
G. P. DeVault and J. A. McLennan,Phys. Rev. 137A:724 (1965).
R. Zwanzig,J. Chem. Phys. 43:714 (1965).
J. T. Bendler,Macromolecules 15:1325 (1982).
B. C. Eu, S. Rackovsky, and A. Eisenberg,J. Chem. Phys. 67:1344 (1977).
D. R. Squire, A. C. Holt, and W. G. Hoover,Physica 42:388 (1969).
A. R. Edmonds,Angular Momentum in Quantum Mechanics (Princeton University Press, Princeton, 1960), Chap. 5.
D. A. McQuarrie,Statistical Mechanics (Harper & Row, New York, 1976), Chap. 21.
G. Williams and D. C. Watts,Trans. Faraday Soc. 66:80 (1970).
A. A. Jones, J. F. O'Gara, P. T. Inglefield, J. T. Bendler, A. F. Yee, and K. L. Ngai,Macromolecules 16:658 (1983).
G. D. Patterson,Adv. Polym. Sci. 48:125 (1983).
J. E. Shore and R. Zwanzig,J. Chem. Phys. 63:5445 (1975).
P. Bordewijk,Chem. Phys. Lett. 32:592 (1975).
K. L. Ngai,Comments Solid State Phys. 9:127 (1979);9:141 (1980).
K. L. Ngai, A. K. Jonscher, and C. T. White,Nature 277:185 (1979).
J. L. Skinner and P. G. Wolynes,J. Chem. Phys. 73:4015 (1980).
J. L. Skinner,J. Chem. Phys. 79:1955 (1983).
E. W. Montroll and J. T. Bendler,J. Stat. Phys. 34:129 (1984).
M. F. Shlesinger and E. W. Montroll,Proc. Natl. Acad. Sci. 81:1280 (1984).
R. Zwanzig,J. Stat. Phys. 30:255 (1983).
H. Scher and E. W. Montroll,Phys. Rev. B 12:2455 (1975).
M. F. Shlesinger,J. Chem. Phys. 70:4813 (1979).
E. W. Montroll and M. F. Shlesinger,Proc. Natl. Acad. Sci. 79:3380 (1982).
S. H. Glarum,J. Chem. Phys. 33:639 (1960).
E. W. Montroll and B. J. West,Fluctuation Phenomena (North-Holland, Amsterdam, 1979).
E. W. Montroll and M. F. Shlesinger, Proc. NSF Workshop (1983).
N. G. McCrum, B. E. Read, and G. Williams,Anelastic and Dielectric Effects in Polymeric Solids (Wiley, New York, 1967).
A. Wintner,Duke Math. J. 8:678 (1941).
C. T. Moynihan, L. P. Boesch, and N. L. Laberge,Phys. Chem. Glasses 14:122 (1973).
D. R. Holt and E. L. Crow,J. Res. Natl. Bur. Stand. Sec. B 77B:143 (1973).
C. P. Lindsey and G. D. Patterson,J. Chem. Phys. 73:3348 (1980).
C. T. Moynihan and A. V. Lesikar,Ann. N.Y. Acad. Sci. 371:151 (1981), and references therein.
G. H. Weiss and M. Dishon, private communication.
K. L. Ngai and R. W. Rendell,Polym. Prepr. 23:46 (1982).
E. N. da C. Andrade,Proc. R. Soc. (London) A84:1 (1911);A90:329 (1914).
Author information
Authors and Affiliations
Additional information
Reference 1 contains a thorough discussion of historical background.
Rights and permissions
About this article
Cite this article
Bendler, J.T. Levy (stable) probability densities and mechanical relaxation in solid polymers. J Stat Phys 36, 625–637 (1984). https://doi.org/10.1007/BF01012927
Issue Date:
DOI: https://doi.org/10.1007/BF01012927