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)


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


primitive path step length


polymer concentration (mass/volume)


macroscopic diffusion coefficient


diffusion coefficient along the primitive path


diffusion coefficient from the Rouse model


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


displacement gradient tensor


free energy


fraction of initial primitive path steps which are still occupied


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


Boltzmann constant


average primitive path length


Polymer chain length


length of a primitive path with m primitive segments


molecular weight of the polymer


molecular weight of the monomer unit


number of path-occupying primitive segments


total number of primitive segments for a molecule


number of monomer units in a chain


mean number of primitive path steps for a molecule


mean number of primitive path steps along a branch


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


path length probability function


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


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


position of center-of-gravity of a molecule


universal gas constant

〈R2〉, 〈r2

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


end-to-end vector for a primitive path step



S1, S2, S3

numerical coefficients of order unity obtains from summations




unit tangent vector for a primitive path step


stored energy function


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


total number of constraint strands defining a primitive path step


strain in simple shear


Curtiss-Bird link tension coefficient


Monomeric friction coefficient


stretch ratio in uniaxial extension


eigenvalues in the Rouse model


chains per unit volume


undiluted polymer density


shear stress or tensile stress (in context)


α, β component of the stress tensor


relaxation time


Doi-Edwards disengagement time


Doi-Edwards equilibration time


longest relaxation time in the Rouse model


mean waiting time for contraint release


time characterizing the linear viscoelastic transition region


jump frequency


volume fraction of polymer


proportionality constant relating τw and τd for linear chains


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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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