Entangled linear, branched and network polymer systems — Molecular theories

  • William W. Graessley
Conference paper
Part of the Advances in Polymer Science book series (POLYMER, volume 47)

Abstract

This article deals with recent work on the theory of entanglement effects in polymer rheology, in particular, the reptation idea of deGennes and the tube model of Doi — Edwards. Predictions which depend on the independent alignment approximation are omitted. Attention is focussed primarily on linear viscoelastic properties, macromolecular diffusion, relaxation following step strains and network properties. Theoretical predictions and experimental observations are discussed for monodisperse linear and star-branched polymer liquids. Effects due to chain length distribution, relaxation behavior of unattached chains in networks, and the equilibrium elasticity of networks are also considered. The results suggest a need to consider relaxation mechanisms in addition to simple reptation as well as certain modifications in the tube model itself. The effect of fluctuations in chain density along the tube is probably quite important in branched chain liquids. Considerations about lifetimes of the tube defining constraints seem necessary to account for polydispersity effects and for differences between relaxation rate of chains in liquids and in networks. A modified tube model is proposed which gives a somewhat better description of elasticity in entangled networks while still producing the Doi-Edwards expression for stress in entangled liquids. Experimental results so far are qualitatively consistent with the picture which is presented here. Much work needs to be done however to test for quantitative consistency.

List of Symbols

a

primitive path step length

c

polymer concentration (mass/volume)

D

macroscopic diffusion coefficient

D*

diffusion coefficient along the primitive path

Dr

diffusion coefficient from the Rouse model

d

mesh size, length of a primitive segment (Pt. II)

E

displacement gradient tensor

F

free energy

F(t)

fraction of initial primitive path steps which are still occupied

f

branch point functionally, number of strands emanating from the same junction point

k

Boltzmann constant

L

average primitive path length

Lo

Polymer chain length

Lm

length of a primitive path with m primitive segments

M

molecular weight of the polymer

Mo

molecular weight of the monomer unit

m

number of path-occupying primitive segments

mo

total number of primitive segments for a molecule

no

number of monomer units in a chain

N

mean number of primitive path steps for a molecule

Nb

mean number of primitive path steps along a branch

Ns

number of primitive path steps occupied by the matrix chains in a mixture

Pm

path length probability function

p

probability that a primitive segment selected at random is part of the surplus (loop) population

Qm

probability that a chain selected at random contains at least m path-occupying primitive segments, \(\sum\limits_{n = m}^{mo} {P_m }\)

Rg

position of center-of-gravity of a molecule

RG

universal gas constant

〈R2〉, 〈r2

mean-square end-to-end distance for unperturbed chains and parts of chains

ri

end-to-end vector for a primitive path step

S

entropy

S1, S2, S3

numerical coefficients of order unity obtains from summations

T

temperature

u

unit tangent vector for a primitive path step

W

stored energy function

z

number of “suitably situated” constraints defining a primitive path step

zo

total number of constraint strands defining a primitive path step

γ

strain in simple shear

ε

Curtiss-Bird link tension coefficient

ζo

Monomeric friction coefficient

λ

stretch ratio in uniaxial extension

λi

eigenvalues in the Rouse model

ν

chains per unit volume

ϱ

undiluted polymer density

σ

shear stress or tensile stress (in context)

σ

α, β component of the stress tensor

τ

relaxation time

τd

Doi-Edwards disengagement time

τe

Doi-Edwards equilibration time

τr

longest relaxation time in the Rouse model

τw

mean waiting time for contraint release

τtr

time characterizing the linear viscoelastic transition region

φ

jump frequency

φ

volume fraction of polymer

Λ(z)

proportionality constant relating τw and τd for linear chains

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References

  1. 1.
    P. G. de Gennes, J. Chem. Phys. 55, 572 (1971); see also Chap. VIII of P. G. de Gennes, “Scaling Concepts in Polymer Physics“, Cornell University Press, Ithaca, 1979.Google Scholar
  2. 2.
    M. Doi and S. F. Edwards, J. Chem. Soc. Faraday Trans. 2 74, 1789 (1978)Google Scholar
  3. 3.
    M. Doi and S. F. Edwards, J. Chem. Soc. Faraday Trans. 2 74, 1802 (1978)Google Scholar
  4. 4.
    M. Doi and S. F. Edwards, J. Chem. Soc. Faraday Trans. 2 74, 1818 (1978)Google Scholar
  5. 5.
    M. Doi and S. F. Edwards, J. Chem. Soc. Faraday Trans. 2 75, 38 (1979)Google Scholar
  6. 6.
    M. Doi, J. Polymer Sci.: Polymer Phys. Ed. 18, 2055 (1980)Google Scholar
  7. 7.
    M. Doi, J. Polymer Sci.: Polymer Phys. Ed. 18, 1891 (1980)Google Scholar
  8. 8.
    M. Doi, J. Polymer Sci.: Polymer Phys. Ed. 18, 1005 (1980)Google Scholar
  9. 9.
    M. Doi and N. Y. Kuzuu, J. Polymer Sci.: Letters 18, 775 (1980)Google Scholar
  10. 10.
    M. Doi, J. Polymer Sci.: Letters 19, 265 (1981)Google Scholar
  11. 11.
    W. W. Graessley, J. Polymer Sci.: Polymer Phys. Ed. 18, 27 (1980)Google Scholar
  12. 12.
    J. D. Ferry, “Viscoelastic Properties of Polymers”, 3rd Ed., John Wiley and Sons, New York, 1980Google Scholar
  13. 13.
    W. W. Graessley, Adv. Polymer Sci. 16, 1 (1974)Google Scholar
  14. 14.
    H.-C. Kan, J. D. Ferry and L. J. Fetters, Macromolecules 13, 1571 (1980)Google Scholar
  15. 15.
    C. R. Taylor, H.-C. Kan, G. W. Nelb and J. D. Ferry, J. Rheology 25, 507 (1981)Google Scholar
  16. 16.
    S. Granick, S. Pederson, G. W. Nelb, J. D. Ferry and C. W. Macosko, ACS Polymer Preprints 22(2), 186 (1981)Google Scholar
  17. 17.
    P. G. de Gennes, J. Phys. 36, 1199 (1975)Google Scholar
  18. 18.
    W. W. Graessley, T. Masuda, J. E. L. Roovers and N. Hadjichristidis, Macromolecules 9, 127 (1976)Google Scholar
  19. 19.
    W. W. Graessley, Accts. Chem. Research 10, 332 (1977)Google Scholar
  20. 20.
    W. E. Rochefort, G. G. Smith, H. Rachapudy, V. R. Raju and W. W. Graessley, J. Polymer Sci.: Polymer Phys. Ed. 17, 1197 (1979)Google Scholar
  21. 21.
    P. G. de Gennes, Macromolecules 9, 587 (1976)Google Scholar
  22. 22.
    Ref. 13, Eq. 6.33Google Scholar
  23. 23.
    M. Fukuda, K. Osaki and M. Kurata, J. Polymer Sci.: Polymer Phys. Ed. 13, 1563 (1975), and earlier articlesGoogle Scholar
  24. 24.
    K. Osaki and M. Kurata, Macromolecules 13, 671 (1980)Google Scholar
  25. 25.
    W. W. Graessley, Macromolecules 8, 186 (1975)Google Scholar
  26. 26.
    M. Gottlieb, C. W. Macosko, G. S. Benjamin, K. O. Meyers and E. W. Merrill, Macromolecules 14, 1039 (1981) and referencesGoogle Scholar
  27. 27.
    J. E. Mark, Rubber Chem. Techn. 48, 495 (1975)Google Scholar
  28. 28.
    J. D. Ferry and H.-C. Kan, Rubber Chem. Tech. 51, 731 (1978)Google Scholar
  29. 29.
    G. Marrucci and G. de Cindio, Rheologica Acta 19, 68 (1980); G. Marrucci and J. J. Hermans, Macromolecules 13, 380 (1980); G. Marrucci, Macromolecules 14, 434 (1981)Google Scholar
  30. 30.
    C. F. Curtiss and R. B. Bird, J. Chem. Phys. 74, 2016 (1981); ibid., 74, 2026 (1981)Google Scholar
  31. 31.
    K. E. Evans, Doctoral Dissertation, The Cavendish Laboratory, University of Cambridge (1980); K. E. Evans and S. F. Edwards, J. Chem. Soc. Faraday Trans. 2, 77, 1891 (1981); ibid. 77, 1913 (1981); ibid. 77 1929 (1981)Google Scholar
  32. 31a.
    R. B. Bird, O. Hassager, R. C. Armstrong and C. F. Curtiss, “Dynamics of Polymeric Liquids” Vol. 2. “Kinetic Theory”, John Wiley and Sons, New York, 1977Google Scholar
  33. 31b.
    P. J. Flory, “Principles of Polymer Chemistry”, Cornell University Press, Ithaca, 1953Google Scholar
  34. 32.
    R. E. Cohen and N. W. Tschoegl, Intern. J. Polymeric Mater. 3, 3 (1974)Google Scholar
  35. 33.
    K. E. Evans, J. Chem. Soc. Faraday Trans. 2, to be publishedGoogle Scholar
  36. 34.
    W. W. Graessley and J. Roovers, Macromolecules 12, 959 (1979)Google Scholar
  37. 35.
    G. Marin, E. Menezes, V. R. Raju and W. W. Graessley, Rheologica Acta 19, 462 (1980)Google Scholar
  38. 36.
    V. R. Raju, E. V. Menezes, G. Marin, W. W. Graessley and L. J. Fetters, Macromolecules 14, 1668 (1981)Google Scholar
  39. 37.
    R. A. Orwoll and W. Stockmayer, Adv. Chem. Phys. 15, 305 (1969)Google Scholar
  40. 38.
    J. Klein, Macromolecules 11, 852 (1978); M. Daoud and P. G. de Gennes, J. Polymer Sci.: Polymer Phys. Ed. 17, 1971 (1979)Google Scholar
  41. 39.
    J. Klein, Phil. Mag. A43, 771 (1981)Google Scholar
  42. 40.
    L. Léger, H. Hervet and F. Rondelez, Macromolecules 14, 1732 (1981)Google Scholar
  43. 41.
    J. Klein, ACS Polymer Preprints (March, 1981) 22, 105 (1981)Google Scholar
  44. 42.
    J. D. Ferry, private communicationGoogle Scholar
  45. 43.
    W. W. Graessley and S. F. Edwards, Polymer 22, 1329 (1981)Google Scholar

Copyright information

© Springer-Verlag 1982

Authors and Affiliations

  • William W. Graessley
    • 1
  1. 1.Cavendish LaboratoryCambridgeEngland
  2. 2.Chemical Engineering DepartmentNorthwestern UniversityEvanstonU.S.A.

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