Effects of Shear Modulus of Polymer Gels

  • Yong Li
  • Toyoichi Tanaka


Gels exhibit mechanical properties characteristic of both solid and liquid-gas systems. From the compressibility (bulk modulus) point of view, a gel network system behaves like liquid-gas system. The value of the bulk modulus of a gel is small in the dilute (corresponding to the gas) phase and is large in the dense (corresponding to the liquid) phase. However, the shear modulus of a liquid-gas system is zero, but is finite for a gel (although much smaller than solids). Thus, the gel network behaves like a solid from the rigidity (shear modulus) point of view. These special properties make the gel system a unique material for theoretical study and practical applications. Compared with the liquid-gas, the free energy of a gel is
$$ dF = - SdT - PdV - YdX. $$
Where Y and X are the shear stress and the shear deformation. The bulk (K) and shear (µ) moduli are related with the second derivatives of F with respect to V and X, respectively [1]. The shear modulus given by Flory’s theory is µ = v eRT, with v e the effective number of chains per unit volume at 6 temperature [2]. In most of the experiments, the gel is free of macroscopic shear constraints and hence the last term in Eq.(l) can be neglected when the system is far from the critical point. For this reason the analogy between the phase transition of the gel system and liquid-gas system is often used in discussion of the gel phase transitions. Near the critical point, this term will suppress the density fluctuations of the system.


Shear Modulus Bulk Modulus Critical Exponent Universality Class Scattered Light Intensity 


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  1. [1]
    L. D. Landau and E. M. Lifshitz, Theory of Elasticity, 1986 Pergamon Press, Oxford.Google Scholar
  2. [2]
    P. J. Flory, Principles of Polymer Chemistry(Cornell Univ. Press., Ithaca and London, 1953).Google Scholar
  3. [3]
    A. Onuki, Phys. Rev. A. 38(4), 2192(1988).PubMedCrossRefGoogle Scholar
  4. [4]
    A. Onuki, Formation, Dynamics, and Statistics of Patterns, (edited by K. Kawasaki et al., World Science, 1989).Google Scholar
  5. [5a]
    A. Onuki, J. Phys. Soc. Jpn. 57(3), 699(1988) andCrossRefGoogle Scholar
  6. [5b]
    A. Onuki, J. Phys. Soc. Jpn. 57(6), 1868(1988)CrossRefGoogle Scholar
  7. [6a]
    Y. Li and T. Tanaka, J. Chem. Phys. 92(2), 1365(1990);CrossRefGoogle Scholar
  8. [6b]
    Y. Li and T. Tanaka, Dynamics and Patterns in Complex Fluids, (edited by A. Onuki, Springer, in print).Google Scholar
  9. [7]
    T. Tanaka, L. O. Hocker, and G. B. Benekek, J. Chem. Phys. 59(9), 5151(1973).CrossRefGoogle Scholar
  10. [8]
    T. Tanaka, S. Ishiwata, and Coe, Ishimoto, Phys. Rev. Lett. 38(14), 771(1977).CrossRefGoogle Scholar
  11. [9]
    T. Tanaka and D. Fillmore, J. Chem. Phys. 70(3), 1214(1979).CrossRefGoogle Scholar
  12. [10a]
    A. Peters and S. J. Candau, Macromolecules 19(7), 1952(1986);CrossRefGoogle Scholar
  13. [10b]
    A. Peters and S. J. Candau, Macromolecules 21(7), 2278, (1988). We believe the calculation in the second reference is not correct. In the first reference, due to a mistake made in the initial condition, the coefficients A n given by Eq. (6) are not correct. The correct answer should be \( {A_n} = 4\frac{{(\alpha_n^2 - 3)\sin {\alpha_n} + 3{\alpha_n}\cos {\alpha_n}}}{{2\alpha_n^2 + {\alpha_n}\sin (2{\alpha_n}) - 4{{\sin }^2}{\alpha_n}}}. \) The boundary condition of spherical gel, (Inline)2(/Inline).CrossRefGoogle Scholar
  14. [11a]
    E. Geissler and A. M. Hecht, Macromolecules 13, 1276(1980) andCrossRefGoogle Scholar
  15. [11b]
    E. Geissler and A. M. Hecht, Macromolecules 14, 185(1981).CrossRefGoogle Scholar
  16. [12]
    P. Heller, Rept. Prog. Phys. 30, 731(1967).Google Scholar
  17. [13]
    See, for instance, S-K. Ma, Modern Theory of Critical Phenomena, (Benjamin/CummingsGoogle Scholar
  18. Publishing Compand, Inc., 1976).Google Scholar
  19. [14]
    A. Hochberg, T. Tanaka, and D. Nicoli, Phys. Rev. Lett. 43(3), 217(1979).CrossRefGoogle Scholar
  20. [15]
    Y. Li and T. Tanaka, J. Chem. Phys. 90(9), 5161(1989). The critical density given inCrossRefGoogle Scholar
  21. this paper is not correct. The correct value should be 0.568mg/cc.Google Scholar
  22. [16]
    L. Golubovic and T. C. Lubensky, Phys. Rev. Lett. 63(10), 1082(1989).PubMedCrossRefGoogle Scholar
  23. [17]
    see H. E. Stanley, Introduction to Phase Transition and Critical Phenomena, (OxfordGoogle Scholar
  24. University, Oxford and New York, 1971).Google Scholar
  25. [18]
    C. Bervillier, Phys. Rev. B 34, 8141(1986), and the references cited therein.CrossRefGoogle Scholar
  26. [19]
    G. Sanchez, M. Meichle, and C. W. Garland, Phys. Rev. A. 28, 1647(1983).CrossRefGoogle Scholar
  27. [20]
    In the analysis presented in reference [11], we used the relaxation time of acrylamideGoogle Scholar
  28. reported by Tanaka and Fillmore in reference [6].Google Scholar
  29. [21]
    T. Tanaka, Dynamic Light Scattering(ed. Pecorra, Plenum Press, New York, 1985).Google Scholar
  30. [22]
    P. W. Anderson, Symmetries and Broken Symmetries, (ed. N. Bocarro, Idset, Paris,Google Scholar
  31. [23]
    A. Onuki, Phys. Rev. B. 39(16), 12308(1989).CrossRefGoogle Scholar
  32. [24]
    H. Zabel and H. Peisl, Phys. Rev. Lett. 42(8), 511(1979).CrossRefGoogle Scholar
  33. [25]
    S. Hirotsu, preprint.Google Scholar
  34. [26]
    S-T. Sun, Y. Li, E. Sato-Mutsuo, and T. Tanaka, to be published. Y. Li and T. Tanaka, to be published.Google Scholar
  35. [27]
    S. Hirotsu and A. Onuki, J. Phys. Soc. Jpn. 58, 1508(1989).CrossRefGoogle Scholar

Copyright information

© Plenum Press, New York 1991

Authors and Affiliations

  • Yong Li
    • 1
  • Toyoichi Tanaka
    • 1
  1. 1.Department of Physics and Center for Material Science and EngineeringMassachusetts Institute of TechnologyCambridgeUSA

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