The entanglement concept in polymer rheology

  • William W. Graessley
Part of the Advances in Polymer Science book series (POLYMER, volume 16)



Matrix in the Rouse theory


Exponent in Mark-Houwink equation: [η]=KMa


Exponent in Fox-Allen η0M correlation


Parameter of order unity in the continuous form of the Rouse spectrum


Concentration exponent in Je0 and τm correlations


3 r02/〈r2


Total number of crosslinks (Part 7)


Auto-correlation function for end-to-end vector

C1, C2

Moduli in the Mooney-Rivlin elasticity equation

Cij, Cij−1

Components of the Cauchy-Green strain tensor and its inverse, with the current configuration as the reference configuration


Polymer concentration in gm/ml


Cornet critical concentration


Simha interaction parameter


Bueche interaction parameter


Self-diffusion coefficient


Power law exponent in\(\eta \propto |\dot \gamma |^{ - d}\)


Average number of entanglement points per molecule; E=M/Mc−1


Value of E at\(\dot \gamma = 0\)(Part 8)


Viscosity structure factor


Frictional force


Change in free energy with deformation


Nominal tensile stress, force/initial cross sectional area


Fraction of chain pairs belonging to topological class i


Reduced steady shear viscosity function (ηηs)/(η0ηs)


Equilibrium shear modulus


Instantaneous shear modulus in stress relaxation


Shear modulus in the viscoelastic plateau region


Shear stress relaxation modulus


Shear storage modulus


Shear loss modulus


Intrinsic storage modulus,\(\mathop {\lim }\limits_{c \to 0} \left( {\frac{{G'(\omega ,c)}}{c}} \right)\)


Intrinsic loss modulus,\(\mathop {\lim }\limits_{c \to 0} \left( {\frac{{G''(\omega ,c) - \omega \eta _s }}{c}} \right)\)


Pair correlation function in molecular theory of liquids (Part 6)


Front factor in modulus equation from rubber elasticity theory (Part 7)

g1, g2

Fraction of configurations of free chains which are consistent with specified end-to-end coordinates (Part 7)


Fractional reduction in entanglement density due to steady shear flow; g(θ)=E/E0 (Part 8)


Relaxation time distribution


Intramolecular hydrodynamic interaction parameter




Fractional reduction in energy dissipation rate per molecule due to dis-entanglement in steady shear flow (Part 8)


Shear creep compliance


Steady state recoverable shear compliance


Reduced steady state compliance; JeR=Je0cRTη02/M(η0ηs)2


Steady state compliance for a monodisperse polymer of molecular weight ξ

J\(\dot \gamma\)



Unspecified proportionality constant


Spring constant in the bead-spring models


Bueche entanglement slip function


Boltzmann constant


Huggins constant


Forsman coupling matrix


Molecular weight


Memory function in Lodge theory


Molecular weight between entanglements in undiluted polymer


Characteristic molecular weight from η0 vs M behavior of undiluted polymers


Characteristic molecular weight from Je0 vs M behavior of undiluted polymers

\(\bar M\)n

Number-average molecular weight

\(\bar M\)w

Weight-average molecular weight

\(\bar M\)z

z-average molecular weight

\(\bar M\)z+1

z+1-average molecular weight


General designation for characteristic molecular weights in the rheological behavior of undiluted polymers


Number of sub-molecules in spring-bead models


Number of primary molecules (Part 7)


Avogadro's number


First normal stress function, p11p22 at steady state in steady simple shear flow


Second normal stress function, p22p33 at steady state in steady simple shear flow


Number of main chain atoms


Number of main chain atoms between entanglements

n1, n2

Moles of solvent and polymer respectively in solution thermodynamics (Part 2)


Degree of polymerization


Unspecified isotropic pressure term in stress tensor p for incompressible materials


Component of stress tensor in rectangular coordinates


Universal gas constant, kNa


Gel point radiation dose (Part. 2)


Stokes radius


Position vector


Contour length of polymer chain


Mean-square end-to-end distance of polymer chain


Mean radius of gyration, 〈S21/2


Entropy change with deformation


Bueche slip factor


Absolute temperature


Langley entanglement trapping factor


Unit vector


Pervaded volume of polymer coil


Velocity vector


Speed, velocity magnitude


Volume per main chain atom in undiluted polymer


Gel fraction


Structure parameter S2/v0 in Fox-Allen η0M correlation


Coil expansion ratio (Parts 2 and 5)


Extension ratio in tensile deformation (Part 7)


Reduced shear rate, (η0ηs) M\(\dot \gamma\)/cRT


Reduced frequency, (η0ηs) M ω/cRT


Characteristic reduced shear rate locating the onset of shear rate dependence in the viscosity


Extent of simple shear deformation from rest state


Crosslink index, fraction of mers participating in crosslinks multiplied by DPn of primary chains (Part 7)


Instantaneously imposed shear deformation

\(\dot \gamma\)

Shear rate

\(\dot \gamma _0\)

Characteristic shear rate locating the onset of shear rate dependence in the viscosity


Dirac delta function


Kronecker delta function


Parameter characterizing the internal viscosity of chain molecules (Part 8)


Frictional coefficient


Frictional coefficient per main chain atom


Frictional coefficient associated with an entanglement junction


Steady state shear viscosity,\(\sigma (\dot \gamma )/\dot \gamma\)


Dynamic viscosity, G′(ω)/ω


Absolute value of the complex viscosity [G′(ω)2+G″(ω)2]1/2/ω


Viscosity at zero shear rate


Viscosity of solvent


Viscosity of monomeric fluid in Eyring's theory


Value of viscosity in undiluted polymer at M=Mc


Intrinsic viscosity,\(\mathop {\lim }\limits_{c \to 0} \left[ {\frac{{\eta - \eta _s }}{{\eta _s c}}} \right]\)


Intrinsic viscosity at zero shear rate (Part 8)


Theta condition for a polymer-solvent system


Argument (\(\dot \gamma\)τ0/2) (η/ηo) in g(ϑ) and h(ϑ) functions of Graessley's theory (Part 8)


Eigenvalues of transformation matrices

λ1, λ2, λ3

Intermolecular distances in Eyring's viscosity theory (Part 6)


Chain concentration, molecules per unit volume


Concentration of elastically effective strands in crosslinked network (Part 7)


Twice the concentration of entanglement junctions in a system prior to crosslinking (Part 7)


Concentration of primary molecules prior to crosslinking (Part 7)


Twice the concentration of crosslinks in a system (Part 7)


Polymer density, mass/volume


Shear stress in simple shear flow


Relaxation time


Characteristic relaxation time associated with the onset of shear rate dependence in the viscosity


Characteristic “maximum” relaxation time determined from the terminal region of the viscoelastic spectrum


Number-average relaxation time of the terminal viscoelastic region, η0/GN0


Weight-average relaxation time of the terminal viscoelastic region, η0Je0


Volume fraction of polymer


Flory constant, [η]M/〈r23/2 for linear flexible chains at high molecular weights


Polymer-solvent interaction coefficient (Part 2)


Extinction angle in flow birefringence


Effective polymer-solvent interaction coefficient in determining chain dimensions in concentrated systems


Probability density distribution function for bead positions in the spring-bead molecular models


First normal stress coefficient, N1/\(\dot \gamma\)2


Second normal stress coefficient, N2/\(\dot \gamma\)2


Number of distinguishable configurations


Frequency ω, rad/sec


Number of distinguishable configurations available to a free chain (Part 7)


Characteristic frequency at which η′(ω) begins to depart from η0


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Copyright information

© Springer-Verlag 1974

Authors and Affiliations

  • William W. Graessley
    • 1
  1. 1.Chemical Engineering and Materials Science DepartmentsNorthwestern UniversityEvanstonU.S.A.

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