The entanglement concept in polymer rheology

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

Nomenclature

A

Matrix in the Rouse theory

a

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

a

Exponent in Fox-Allen η0M correlation

a

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

b

Concentration exponent in Je0 and τm correlations

b

3 r02/〈r2

C

Total number of crosslinks (Part 7)

C(t)

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

c

Polymer concentration in gm/ml

c*

Cornet critical concentration

c[η]

Simha interaction parameter

cM

Bueche interaction parameter

D

Self-diffusion coefficient

d

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

E

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

E0

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

F

Viscosity structure factor

F

Frictional force

ΔF

Change in free energy with deformation

f

Nominal tensile stress, force/initial cross sectional area

fi

Fraction of chain pairs belonging to topological class i

f(β)

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

G

Equilibrium shear modulus

G0

Instantaneous shear modulus in stress relaxation

GN0

Shear modulus in the viscoelastic plateau region

G(t)

Shear stress relaxation modulus

G′(ω)

Shear storage modulus

G″(ω)

Shear loss modulus

[G′]ω

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

[G″]ω

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

g

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

g

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)

g(θ)

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

H(τ)

Relaxation time distribution

h

Intramolecular hydrodynamic interaction parameter

h*

h/N1/2

h(θ)

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

J(t)

Shear creep compliance

Je0

Steady state recoverable shear compliance

JeR

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

J*(ξ)

Steady state compliance for a monodisperse polymer of molecular weight ξ

J\(\dot \gamma\)

N1/2σ2

K

Unspecified proportionality constant

K

Spring constant in the bead-spring models

K(s)

Bueche entanglement slip function

k

Boltzmann constant

k′

Huggins constant

M

Forsman coupling matrix

M

Molecular weight

M(ξ)

Memory function in Lodge theory

Me

Molecular weight between entanglements in undiluted polymer

Mc

Characteristic molecular weight from η0 vs M behavior of undiluted polymers

M′c

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

Mx

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

N

Number of sub-molecules in spring-bead models

N

Number of primary molecules (Part 7)

Na

Avogadro's number

N1

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

N2

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

n

Number of main chain atoms

ne

Number of main chain atoms between entanglements

n1, n2

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

P

Degree of polymerization

P0

Unspecified isotropic pressure term in stress tensor p for incompressible materials

pij

Component of stress tensor in rectangular coordinates

R

Universal gas constant, kNa

Rg

Gel point radiation dose (Part. 2)

R0

Stokes radius

r

Position vector

r0

Contour length of polymer chain

r2

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

S

Mean radius of gyration, 〈S21/2

ΔS

Entropy change with deformation

s

Bueche slip factor

T

Absolute temperature

Te

Langley entanglement trapping factor

u

Unit vector

V

Pervaded volume of polymer coil

v

Velocity vector

v

Speed, velocity magnitude

v0

Volume per main chain atom in undiluted polymer

wg

Gel fraction

X

Structure parameter S2/v0 in Fox-Allen η0M correlation

α

Coil expansion ratio (Parts 2 and 5)

α

Extension ratio in tensile deformation (Part 7)

β

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

β′

Reduced frequency, (η0ηs) M ω/cRT

β0

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

γ

Extent of simple shear deformation from rest state

γ

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

γ0

Instantaneously imposed shear deformation

\(\dot \gamma\)

Shear rate

\(\dot \gamma _0\)

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

δ()

Dirac delta function

δij

Kronecker delta function

ɛ

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

ζ

Frictional coefficient

ζ0

Frictional coefficient per main chain atom

ζe

Frictional coefficient associated with an entanglement junction

η

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

η′

Dynamic viscosity, G′(ω)/ω

|η*|

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

η0

Viscosity at zero shear rate

ηs

Viscosity of solvent

ηm

Viscosity of monomeric fluid in Eyring's theory

ηc

Value of viscosity in undiluted polymer at M=Mc

[η]

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

[η]0

Intrinsic viscosity at zero shear rate (Part 8)

ϑ

Theta condition for a polymer-solvent system

ϑ

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

λi

Eigenvalues of transformation matrices

λ1, λ2, λ3

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

ν

Chain concentration, molecules per unit volume

ν

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

νe

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

ν0

Concentration of primary molecules prior to crosslinking (Part 7)

νc

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

ϱ

Polymer density, mass/volume

σ

Shear stress in simple shear flow

τ

Relaxation time

τ0

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

τm

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

τn

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

τw

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

ϕ

Volume fraction of polymer

Φ

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

χ

Polymer-solvent interaction coefficient (Part 2)

χ

Extinction angle in flow birefringence

χ0

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

ψ

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

ψ1

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

ψ2

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

Ω

Number of distinguishable configurations

ω

Frequency ω, rad/sec

ω

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

ω0

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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