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The entanglement concept in polymer rheology

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The Entanglement Concept in Polymer Rheology

Part of the book series: Advances in Polymer Science ((POLYMER,volume 16))

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Abbreviations

A:

Matrix in the Rouse theory

a:

Exponent in Mark-Houwink equation: [η]=KM a

a:

Exponent in Fox-Allen η 0−M correlation

a:

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

b:

Concentration exponent in J 0e and τ m correlations

b:

3 r 20 /〈r 2〉

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, C −1ij :

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/M c−1

E 0 :

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 (η−η s)/(η 0−η s)

G:

Equilibrium shear modulus

G 0 :

Instantaneous shear modulus in stress relaxation

G 0N :

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/E 0 (Part 8)

H(τ):

Relaxation time distribution

h:

Intramolecular hydrodynamic interaction parameter

h*:

h/N 1/2

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

J 0e :

Steady state recoverable shear compliance

J eR :

Reduced steady state compliance; J eR=J 0e cRTη 20 /M(η 0−η s)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

M e :

Molecular weight between entanglements in undiluted polymer

M c :

Characteristic molecular weight from η 0 vs M behavior of undiluted polymers

M′ c :

Characteristic molecular weight from J 0e 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

M x :

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)

N a :

Avogadro's number

N 1 :

First normal stress function, p 11−p 22 at steady state in steady simple shear flow

N 2 :

Second normal stress function, p 22−p 33 at steady state in steady simple shear flow

n:

Number of main chain atoms

n e :

Number of main chain atoms between entanglements

n1, n2:

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

P:

Degree of polymerization

P 0 :

Unspecified isotropic pressure term in stress tensor p for incompressible materials

p ij :

Component of stress tensor in rectangular coordinates

R:

Universal gas constant, kN a

R g :

Gel point radiation dose (Part. 2)

R 0 :

Stokes radius

r:

Position vector

r 0 :

Contour length of polymer chain

〈r 2〉:

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

S:

Mean radius of gyration, 〈S 2〉1/2

ΔS:

Entropy change with deformation

s:

Bueche slip factor

T:

Absolute temperature

T e :

Langley entanglement trapping factor

u:

Unit vector

V:

Pervaded volume of polymer coil

v:

Velocity vector

v:

Speed, velocity magnitude

v 0 :

Volume per main chain atom in undiluted polymer

w g :

Gel fraction

X:

Structure parameter S 2/v 0 in Fox-Allen η 0−M correlation

α:

Coil expansion ratio (Parts 2 and 5)

α:

Extension ratio in tensile deformation (Part 7)

β:

Reduced shear rate, (η 0−η s) M \(\dot \gamma\)/cRT

β′:

Reduced frequency, (η 0−η s) M ω/cRT

β 0 :

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 DP n of primary chains (Part 7)

γ 0 :

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

ζ 0 :

Frictional coefficient per main chain atom

ζ e :

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

η 0 :

Viscosity at zero shear rate

η s :

Viscosity of solvent

η m :

Viscosity of monomeric fluid in Eyring's theory

η c :

Value of viscosity in undiluted polymer at M=M c

[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 (\(\dot \gamma\) τ 0/2) (η/η o) in g(ϑ) and h(ϑ) functions of Graessley's theory (Part 8)

λ i :

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)

ν e :

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

ν 0 :

Concentration of primary molecules prior to crosslinking (Part 7)

ν c :

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

τ 0 :

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

τ m :

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

τ n :

Number-average relaxation time of the terminal viscoelastic region, η 0/G 0N

τ w :

Weight-average relaxation time of the terminal viscoelastic region, η 0J 0e

ϕ:

Volume fraction of polymer

Φ ∞ :

Flory constant, [η]M/〈r 2〉3/2 for linear flexible chains at high molecular weights

χ:

Polymer-solvent interaction coefficient (Part 2)

χ:

Extinction angle in flow birefringence

χ 0 :

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

ψ 1 :

First normal stress coefficient, N 1/\(\dot \gamma\) 2

ψ 2 :

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

Ω:

Number of distinguishable configurations

ω:

Frequency ω, rad/sec

ω:

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

ω 0 :

Characteristic frequency at which η′(ω) begins to depart from η 0

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  1. Chemical Engineering and Materials Science Departments, Northwestern University, 60201, Evanston, Illinois, U.S.A.

    William W. Graessley

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Graessley, W.W. (1974). The entanglement concept in polymer rheology. In: The Entanglement Concept in Polymer Rheology. Advances in Polymer Science, vol 16. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0031037

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  • DOI: https://doi.org/10.1007/BFb0031037

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