Biomedical Engineering Reference
In-Depth Information
Simple and beautiful though this approach is, it became clear quite soon after its
development in the 1940s that a number of corrections would need to be applied. This is
particularly an issue when applying the theory of ideal (
) networks to real
systems. Corrections include those mentioned above, which compensate for various
more speci
'
virtual
'
, and also those arising from topological entanglements,
including ones which have become permanently
c forms of
'
wastage
'
during the cross-linking
process. Progress in improving the theory was relatively slow until the 1970s, because
the chemistry of network formation was not easily controlled, and so model samples were
not available. However, when samples of networks formed from end-linking PDMS were
prepared and tested, newer theories began to appear, including those due to Flory and
Erman, which included contributions from network junction constraints. Detailed dis-
cussions of these approaches lie outside the scope of this volume, but are well argued in
the monograph by Erman and Mark (
1997
).
Almost at the same time, Edwards and his co-workers (Doi and Edwards,
1986
; Edwards
and Vilgis,
1988
) published a number of papers that included the effects of entanglements,
in the so-called tube or slip-link model. Detailed observation of networks during deforma-
tion also became possible using the newly developing technique of neutron scattering. Such
modi
'
trapped
'
cations to the theory become important when considering large degrees of swelling,
discussed in the next section, since the assumptions behind the classic theory include that
of Gaussian chain behaviour, which becomes increasingly unlikely at high degrees of
swelling. Some workers have assumed limited extensibility Langevin chain behaviour.
These Langevin chains (strictly chains whose end-to-end distribution follows the inverse
Langevin function approximation) have the property that they behave like freely jointed
chains at low deformation but, unlike the Gaussian function, have a realistic and
nite
large-deformation limit, so a chain cannot be longer than its maximum contour length.
However, attributing deviations solely to
cult because,
for real networks, deformation-induced order (crystallization, even liquid crystal forma-
tion) can also contribute to deviations from ideal behaviour.
Finally we need to reassert one of the conclusions of
Chapter 3
, that the equilibrium
modulus of any gel depends essentially on the product of two factors
finite-chain effects has proved dif
-
the number of
'
per unit volume, which in turn is related to the percolation
degree of conversion parameter p, and the contribution of these to the overall modulus. For
idealized
cross-links
'
or
'
junction zones
'
or ideal rubber-like networks the latter term is small, of order unity. For
more rigid network structures such as agarose, elasticity is very unlikely to be entropic. In
other words, the contribution per cross-link comes from bending and twisting contribu-
tions, which raise the enthalpy of the system. Consequently we adopt the term
'
entropic
'
'
and assign its contribution to a generalized (i.e. non-ideal rubber-like or entropic) front
factor, which for agarose, for example, is estimated to be
'
enthalpic
10 (Clark and Ross-Murphy,
1987
). That said, separating these two contributions is not trivial, and for these physical gels
requires a number of additional assumptions, which are discussed throughout this topic.
≳
4.2.1
Reel chain models
For thermoreversible gels, which are a major topic of this volume, Nishinari et al.(
1985
)
formulated an approach to the modulus
-
temperature relationship (a direct proportionality
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