Statistical theory of polymeric lyotropic liquid crystals

  • Alexandr Yu. Grosberg
  • Alexei R. Khokhlov
Conference paper
Part of the Advances in Polymer Science book series (POLYMER, volume 41)


This article deals with some topics of the statistical physics of liquid-crystalline phase in the solutions of stiff chain macromolecules. These topics include: the problem of the phase diagram for the liquid-crystalline transition in the solutions of completely stiff macromolecules (rigid rods); conditions of formation of the liquid-crystalline phase in the solutions of semiflexible macromolecules; possibility of the intramolecular liquid-crystalline ordering in semiflexible macromolecules; structure of intramolecular liquid crystals and dependence of the properties of the liquid-crystalline phase on the microstructure of the polymer chain.


Triple Point Virial Coefficient Anisotropic Phase Triple Point Temperature Small Globule 
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List of Main Symbols


total number of particles in the system, i. e. in Sects. 2, 3 — total number of rods (segments) in the solution, in Sects. 4, 5 — total number of rods (segments) in the macromolecule


contour length of a macromolecule


width of a polymer chain

statistical segment length (in Sect. 5.5 — length of a rod-like side group)


assymmetry parameter, i. e. in Sect. 2 p=L/d, in Sect. 3 p=1/d


angle between the interacting rods (segments)


orientational distribution function for the rods (segments)

element of a spatial angle


concentration of rods (segments)


polymer volume fraction in the solution

ϑ(i) and ϑ(a)

polymer volume fractions in the isotropic and anisotropic phases coexisting at equilibrium


absolute temperature in energetic units


free energy


theta temperature


temperature corresponding to the triple point of the phase diagram


energy of interaction of two perpendicular rods (segments) in contact


lattice co-ordination number in lattice models


Flory interaction parameter


second virial coefficient


third virial coefficient

ξ (in Sect. 2), λ

see Eq. (2.21)


excluded volume of a monomer


characteristic distance between the ends of a flexible filament between the monomers


excluded volume parameter of the coil


polymer volume fraction inside the globule


parameter of liquid-crystalline order

R, r

radii of a toroidal globule (Fig. 12)

ξ (in Sect. 5)

set of parameters defining the state of a monomer


density of monomers in the state ξ

ψ(ξ) and ψ+(ξ)

distribution functions for the end monomers of the macromolecule

g(ξ, ξ′)

conditional probability that the monomer is in the state ξ′ provided the previous one is in the state ξ


integral operator with the kernel g(ξ, ξ′) (see Eq. (5.2a))


conformation entropy of a globule with the given density distribution n(ξ).


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8 References

  1. 1.
    Papkov, S. P., Kulichikhin, V. G.: Liquid-Crystalline State of Polymers, Khimia, Moscow 1977Google Scholar
  2. 2.
    Papkov, S. P.: Vysokomol, Soedin. A 19, 3 (1977)Google Scholar
  3. 3.
    Shibaev, V. P., Plate, N. A.: Ibid. A 19, 923 (1977)Google Scholar
  4. 4.
    Miller, W. G., Rai, J. H., Wee, E. L.: in Liquid Crystals and Ordered Fluids (Eds. J. F. Johnson, R. S. Porter) Plenum Press, New York 1974, Vol. 2, p. 243Google Scholar
  5. 5.
    Ciferri, A.: Polym. Eng. Sci. 15, 191 (1975)Google Scholar
  6. 6.
    Aharoni, S. M.: J. Polym. Sci., Polym. Phys. Ed. 17, 683 (1979), Macromolecules 12, 94 (1979), 12, 537 (1979); Aharoni, S. M., Walsh, E. K.: J. Polym. Sci., Polym. Lett. Ed. 17, 321 (1979), Macromolecules 12, 271 (1979)Google Scholar
  7. 7.
    Onsager, L.: Ann. N. Y. Sci. 51, 627 (1949)Google Scholar
  8. 8.
    Flory, P. J.: Proc. Roy. Soc. (London) 234, 60 (1959)Google Scholar
  9. 9.
    Flory, P. J.: ibid. 234, 73 (1956)Google Scholar
  10. 10.
    Flory, P. J., Abe, A.: Macromolecules 11, 1119, 1122 (1978)Google Scholar
  11. 10a.
    Flory, P. J., Frost, R. S.: ibid. 11, 1126, 1134 (1978)Google Scholar
  12. 10b.
    Flory, P. J.: ibid. 11, 1138 (1978)Google Scholar
  13. 11.
    Flory, P. J.: Macromolecules 11, 1141 (1978)Google Scholar
  14. 12.
    Flory, P. J., Ronca, G.: Mol. Cryst. Liq. Cryst. 54, 289 (1979)Google Scholar
  15. 13.
    Marrucci, G., Ciferri, A.: J. Polym. Sci., Polym. Lett. Ed. 15, 643 (1977)Google Scholar
  16. 14.
    Krigbaum, W. R., Salaris, F.: J. Polym. Sci., Polym. Phys. Ed. 16, 883 (1978)Google Scholar
  17. 15.
    Khokhlov, A. R.: Phys. Lett. A 68, 135 (1978)Google Scholar
  18. 16.
    Grosberg, A. Yu.: Biofizika 24, 32 (1979)Google Scholar
  19. 17.
    Khokhlov, A. R.: Vysokomol. Soedin. A 21, 1981 (1979)Google Scholar
  20. 18.
    Khokhlov, A. R.: ibid. B 21, 201 (1979)Google Scholar
  21. 19.
    Khokhlov, A. R.: Int. J. Quant. Chem. 16, 857 (1979)Google Scholar
  22. 20.
    Grosberg, A. Yu.: Vysokomol. Soedin A 22, 90 (1980)Google Scholar
  23. 21.
    Grosberg, A. Yu.: ibid. A 22, 96 (1980)Google Scholar
  24. 22.
    Grosberg, A. Yu.: ibid. A 22, 100 (1980)Google Scholar
  25. 23.
    Lifshitz, I. M.: Zh. Eksp. Teor. Fiz. 55, 2408 (1968)Google Scholar
  26. 24.
    Lifshitz, I. M., Grosberg, A. Yu., Khokhlov, A. R.: Rev. Mod. Phys. 50, 683 (1978)Google Scholar
  27. 25.
    Lifshitz, I. M., Grosberg, A. Yu., Khokhlov, A. R.: Usp. Fiz. Nauk 127, 353 (1979)Google Scholar
  28. 26.
    Birshtein, T. M., Skvortzov, A. M., Sariban, A. A.: Vysokomol. Soedin. B 17, 607 (1975)Google Scholar
  29. 27.
    Birshtein, T. M., Skvortzov, A. M., Sariban, A. A.: ibid. A 17, 1962 (1975)Google Scholar
  30. 28.
    de Gennes, P. G.: The Physics of Liquid Crystals, Clarendon, Oxford 1974Google Scholar
  31. 29.
    Lasher, G.: J. Chem. Phys. 53, 4141 (1970)Google Scholar
  32. 30.
    Kayser Jr., R. F., Raveche, H. J.: Phys. Rev. A 17, 2067 (1978)Google Scholar
  33. 31.
    Landau, L. D., Lifshitz, E. M.: Statistical Physics, Part I, Nauka, Moscow 1976Google Scholar
  34. 32.
    Khokhlov, A. R.: J. Phys. 38, 845 (1977)Google Scholar
  35. 33.
    Khokhlov, A. R.: Polymer 19, 1387 (1978)Google Scholar
  36. 34.
    Khokhlov, A. R.: Vysokomol. Soedin. A 20, 2754 (1978)Google Scholar
  37. 35.
    Saito, N., Takahashi, K., Yonoki, Y.: J. Phys. Soc. (Japan) 22, 219 (1967)Google Scholar
  38. 36.
    Freed, K.: J. Chem. Phys. 54, 1453 (1971)Google Scholar
  39. 37.
    Freed, K.: Adv. Chem. Phys. 22, 1 (1972)Google Scholar
  40. 38.
    Lifshitz, I. M., Grosberg, A. Yu.: Zh. Eksp. Teor. Fiz. 65, 2399 (1973)Google Scholar
  41. 39.
    Sobolev, V. I.: Lectures on Additional Topics of Mathematical Analysis, Nauka, Moscow 1968Google Scholar
  42. 40.
    Lerman, L. S.: in Physico-Chemical Properties of the Nucleic Acids (Ed. J. Duchesme), Academic Press, New York 1973, p. 61Google Scholar
  43. 41.
    Akimenko, N. M. et al.: FEBS Lett. 38, 61 (1973)Google Scholar
  44. 42.
    Evdokimov, Yu. M. et al.: Nucleic Acids Res. 3, 2353 (1976)Google Scholar
  45. 43.
    Naghizadeh, J., Massih, A. R.: Phys. Rev. Lett. 40, 1299 (1978)Google Scholar
  46. 44.
    Frisch, H. L., Fesciyan, S.: J. Polym. Sci. Polym. Lett. Ed. 17, 309 (1979)Google Scholar
  47. 45.
    Post, C., Zimm, B.: Biopolymers 18, 487 (1979)Google Scholar
  48. 46.
    Grosberg, A. Yu., Erukhimovich, I. Ya., Shakhnovich, E. I.: Biofizika 25 No. 3 (1980)Google Scholar

Copyright information

© Springer-Verlag 1981

Authors and Affiliations

  • Alexandr Yu. Grosberg
    • 1
  • Alexei R. Khokhlov
    • 2
  1. 1.Institute of Chemical PhysicsUSSR Academy of SciencesMoscowUSSR
  2. 2.International Centre for Theoretical PhysicsTriesteItaly

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