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Rubber elasticity of polymer networks: Theories

  • G. Heinrich
  • E. Straube
  • G. Helmis
Conference paper
Part of the Advances in Polymer Science book series (POLYMER, volume 85)

Abstract

The present state of development of the statistical mechanics of rubber elasticity is reviewed and analysed, starting from some problems and controversial results drawn from recent experimental progress in this area. Attention is focused on the tube model as a mean field approach to the statistical mechanics of polymer systems with topology conservation. In particular, a new model for simulating the topological constraints in polymer networks and melts is presented which allows the order and the deformation dependence of the tube dimensions to be calculated. Conclusions resulting from the description of large-strain and small-strain behaviour of dry, completely crosslinked networks are discussed and compared with experimental data where all modes of deformation usually employed can be described with similar accuracy. Further, the concept of relaxed microscopic deformation much smaller than the macroscopic deformation of the sample is introduced, which allows explanation of mechanical and thermodynamic properties as well as the scattering results for networks at higher swelling degrees. Similarly, constraint release effects are expected to be responsible for the different experimental results collected for end-linked networks and for networks prepared by cross-linking of long primary chains. The different degrees of completeness of crosslinking have to be considered as the main reason for these differences.

List of Symbols

A

microstructure factor

a1, c

activity of the solvent over a swollen cross-linked polymer

a1, u

activity of the solvent over an uncrosslinked polymer system

C1, C2

Mooney-Rivlin parameters

c1, c2

reduced Mooney-Rivlin parameters

d0

undeformed tube radius

dμ

deformed tube radius (μ=x, y, z)

F

elastic free energy

FT

elastic free energy of a network with a special topology

f

functionality of the crosslinks

G

shear modulus

GC

shear modulus contribution arising from chemical crosslinks

Ge

maximum possible contribution of entangled chains to the modulus

GN

shear modulus contribution arising from topological constraints

GN0

plateau modulus of an uncrosslinked polymer system

g

front factor

gr

reduced shear modulus

H

Hamiltonian

Iαβ

topological invariant

I1, I2, I3

invariants of the deformation tensor

Ĩ1, Ĩ2

generalized invariants of a generalized deformation tensor

k

scattering vector

k

Boltzmann constant

L

contour length of a macromolecule

Lc

contour length of a network strand

l

statistical segment length

M

number of chemical crosslinks

Mc

number-average molecular weight of a network chain

Mn

number-average molecular weight of a primary chain

Mw

weight-average molecular weight of a primary chain

Ms

molecular weight of a statistical segment

m

Gauss linking number

N

Number of primary chains

NA

Avogadro number

Nk

number of network chains

Ns

number of statistical segments per macromolecule

Nc

number of statistical segments per network chain

NF

Flory number [= number of network chains in the volume (lLc)3/2]

Nsl

number of slip-links

n

number of spatial neighbours of a network junction

np

number density of polymer chains

ns

number density of statistical segments

n1

number of moles of solvent in the swollen network

R(s), r(s)

configurations of a macromolecule

\(\hat R(s),\hat r(s)\)

mean configurations of a macromolecule

Rc

end-to-end distance of a network chain

Rgi

radius of gyration of a chain in the reference (undeformed) state

Rg∥

radius of gyration of a network chain parallel to the stretching

Rg⊥

radius of gyration of a network chain perpendicular to the stretching

S(k)

scattering function

s

chain arc length

T

temperature, topology

Te

trapping factor

V, V0

volume

\(\bar v\)1

molar volume of the diluent

v2

polymer volume fraction in a swollen gel

v2*

polymer volume fraction corresponding to the equilibrium swelling degree

v2**

cross-over volume fraction from Gaussian to excluded-volume behaviour of the network chains

v20

volume fraction of polymer in the solution prior to crosslinking

wel

elastic potential

ws

sol fraction

wT

probability of a special topology

Z

canonical partition integral

α (or β)

parameter of the constraint release effect in polymer networks

Λ

dimensionless parameter of the strength of topological constraints

ζ

parameter of the Flory-Kästner theory characterizing the departures from affine transformation of the shapes of domains

η

“memory” factor

κ

parameter of the Flory-Kästner theory characterising the strength of restrictions on junction fluctuations

λ

deformation ratio of the sample, deformation tensor

λi

linear isotropic extension ratio of swollen networks

λmic

microscopic deformation ratio

μc

chemical crosslink density

μ1, el

elastic contribution of the chemical potential of the diluent in swollen gels

vc

network chain density

ξ

cycle rank (= number of independent circuits in the network)

ϱp

polymer density (g/mol)

σ

nominal stress (equilibrium value of the elastic force measured in uniaxial deformation divided by the undeformed cross-sectional area of the sample)

σM

Mooney stress [=σ/(λ − λ−2)]

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Copyright information

© Springer-Verlag 1988

Authors and Affiliations

  • G. Heinrich
    • 1
  • E. Straube
    • 2
  • G. Helmis
    • 1
  1. 1.Technische Hochschule „Carl Schorlemmer“ Leuna-Merseburg Sektion PhysikMerseburgGermany
  2. 2.Sektion Werkstoff- und VerarbeitungstechnikMerseburgGermany

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