Rheologica Acta

, Volume 7, Issue 4, pp 379–392

# Constitutive equations from molecular network theories for polymer solutions

• A. S. Lodge
Article

DOI: 10.1007/BF01984856

Lodge, A.S. Rheol Acta (1968) 7: 379. doi:10.1007/BF01984856

## Summary

In this mainly expository paper, constitutive equations based on the network models ofYamamoto,Lodge, andKaye are re-derived in a common notation involving the use of base vectors embedded in the deforming macroscopic continuum. The derivations are thereby simplified in some respects and the differences of detail between the models are clarified. InLodges theory, the sub-network superposition assumption is replaced by alternative assumptions concerning the creation and loss of network segments, and the theory is extended to non-Gaussian networks.Kayes theory is extended to allow for the presence of entanglement junctions of different complexities.

### Notation

bn

= 3/(2n l2)

n

= number of freely-jointed links, each of lengthl, in a strand

r

= time-average (or ensemble-average) end-to-end vector for a typical network segment;r = magnitude ofr

A

=Helmholtz free energy per unit volume of solution

k

=Boltzmanns constant

T

= absolute temperature

F(x, n, t) d3x=F(x1,x2,x3,n, t) dx1dx2dx3

= concentration at timet ofn-segments whoser-vectors lie in the range (xiui, (xi +dxi)ui)

u1,u2,u3

= orthonormal base vectors, fixed in space;r =xiui

e1,e2,e3

= linearly independent, time-dependent base vectors, embedded in the macroscopic continuum

ξi

= convected components ofr: r =ξiei

γjj

=ei· ej (scalar product)

cn

= concentration ofn-segments

γ

= det [γij]

γij

= element of matrix reciprocal to matrix

N*0

= concentration of segments

πij

= stress components referred to basise1,e2,e3

IIij

=πij+p γij,p arbitrary

x

= a parameter (of values 1, 2,...) labeling segments according to the complexity of their junctions with the network

τxn−1

= constant probability per unit time that an (n, x)-segment will leave the network

Ψ(ξ, n, x, t' ¦t) d3ξ dt'

= concentration at timett' of (ξ, n, x)-segments which were created in the interval (t', t' + dt')

Lxn

= rate of creation, per unit volume, of (n, x)-segments

N* (t− t') dt'

= concentration at timet≥t' of segments which were created in the interval (t', t'+dt')

CdV

= density of configurations available to a strand having one end fixed at a given point and the other end within a volume elementdV about a second given point

p11p22,p22p33

= primary and secondary differences of normal cartesian stress components for a liquid in steady shear flow in which the velocity components arev1=Gx2,v2=v3=0

G

= shear rate

Φ (r, n, T)

= contribution toA from a typical (r, n)-segment at timet

β (r, n)

= probability per unit time that a given (r, n)-segment will leave the network

G(r, n, t) d3x dt

= concentration of (r, n)-segments created during (t, t + dt)

Ψ (ξ, n, t′¦t) d3ξ dt′

= concentration at timet≥t′ of (ξ, n)-segments which were created during (t′, t′ + dt′)

r′

=r(t′) r″ =r(t″)

Pij

= components of extra stress tensor referred to basisu1,u2,u3

c (t)

= a cartesian space tensor defined byr(t) =e(*) · r(0)

h

=r(0) in [7.22],r(t′) in [7.24]

e+

= transpose ofe

e−1

= reciprocal of e

g

=g[Qi(t), Q2(t)]= stress-dependent probability per unit time at timet of the loss of any given network junction

Q1(t), Q2(t)

= functions of invariants of stress at timet, defined by [8.11], [8.12]

g″xn ≡ g [Qi(t″), Q2(t″), x,n]

= probability per unit time at timet″ of the loss of any given (n, x)-segment

N*xn

= concentration of (n, x)-segmentsN0 = concentration of junctions

s

=N*0/N0, a number whose value is about 1 or 2