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Synthesis and Degradation Rheology and Extrusion pp 67–117Cite as

Entangled linear, branched and network polymer systems — Molecular theories

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Part of the book series: Advances in Polymer Science ((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.

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Abbreviations

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

ζ o :

Monomeric friction coefficient

λ:

stretch ratio in uniaxial extension

λ i :

eigenvalues in the Rouse model

ν:

chains per unit volume

ϱ:

undiluted polymer density

σ:

shear stress or tensile stress (in context)

σaβ :

α, β component of the stress tensor

τ:

relaxation time

τ d :

Doi-Edwards disengagement time

τ e :

Doi-Edwards equilibration time

τ r :

longest relaxation time in the Rouse model

τ w :

mean waiting time for contraint release

τ tr :

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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Author notes

  1. William W. Graessley

    Present address: Chemical Engineering Department, Northwestern University, 60201, Evanston, Illinois, U.S.A.

Authors and Affiliations

  1. Cavendish Laboratory, Madingley Road, CB 3 OHE, Cambridge, England

    William W. Graessley

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© 1982 Springer-Verlag

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Graessley, W.W. (1982). Entangled linear, branched and network polymer systems — Molecular theories. In: Synthesis and Degradation Rheology and Extrusion. Advances in Polymer Science, vol 47. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0038532

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  • DOI: https://doi.org/10.1007/BFb0038532

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  • Print ISBN: 978-3-540-11774-2

  • Online ISBN: 978-3-540-39483-9

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