Polymer network consists of long polymer chains which are crosslinked with each other and form a continuous molecular framework. All polymer networks (which are not in the glassy or partially crystalline states) exhibit the property of high elasticity,

1
High Elasticity of Polymer Networks
Polymer network consists of long polymer chains
which are crosslinked with each other and form a
continuous molecular framework. All polymer
networks (which are not in the glassy or
partially crystalline states) exhibit the
property of high elasticity, i.e the ability to
undergo large reversible deformations at
relatively small applied stress.
2
High elasticity is the most specific property of
polymer materials it is connected with the most
fundamental features of ideal chains considered
above. In everyday life, highly elastic polymer
materials are called rubbers. Molecular picture
of high-elastic deformations
Elasticity of the rubber is composed from
the elastic responses of the chains crosslinked
in the network sample.
3
Typical stress-strain curves
For rubber
For steel
0.01
A - upper limit for stress-strain linearity B -
upper limit for reversibility of deformations C -
fracture point

• Characteristic values for deformation
  are
• much larger for rubber.
• Characteristic values for strain are much
• larger for steel.
• Characteristic values for Young moduli are
• enormously larger for steel (
  )
• than for rubber ( ).
• For steel linearity and reversibility are lost
• practically simultaneously, while for rubbers
• there is a very wide region of nonlinear rever-
• sible deformations.
• For steel there is a wide region of plastic
  defor-
• mations (between points B and C) which is
• practically absent for rubbers.

4
Elasticity of a Single Ideal Chain
For crystalline solids the elastic response
ap- pears, because external stress changes
the equilibrium inter-atomic distances and
increases the internal energy of the
crystal (energetic elasticity).
Since the energy of ideal polymer chain is equal
to zero, the elastic response appears by purely
entropic reasons (entropic elasticity). Due to
the stretching the chain adopts the less probable
conformation its entropy decreases.
5
According to Boltzmann, the entopy
Where k is the Boltzmann constant and is the
number of chain conformations compa- tible with
the end-to-end distance .
But,
Thus,
The free energy F
6

• The chain is elongated in the direction
• of and (kind of a Hooke law).
• Elastic modulus

1) is proportional to , i.e very small for
large values of L. Long polymer chains are very
susceptible to external actions. 2) is
proportional to kT which is the indication to
entropic nature of elasticity.
7
Elasticity of a Polymer Network (Rubber)
Let us consider densely packed system of
crosslinked chains (freely jointed chains of
contour length L and Kuhn segment length l
). Flory theorem the statistical properties of a
polymer chain in the dense system are equivalent
to those for ideal chains. Let the deformation of
a the sample along the axes x, y, z be
, i.e the sample dimensions along the
axes are
Affinity assumption the crosslink points are
deformed affinely together with the network
sample. I.e if in the initial state the
end-to-end vector has the coordinates in the
deformed state its coordinates are
8
Thus, the change of the free energy of the
chain between two crosslink points upon extension
is
For the whole sample where is the number of
chains per unit volume and V is the volume of
the sample.
But,
9
It is interesting that the answer does not depend
on the parameters L and l that describe an
individual subchain. This indicates that the
theory is universal. It works whatever is the
particular structure of the subchains (regardless
of whether they are freely jointed or wormlike),
for whatever contour length and Kuhn lengths, and
so on. If we glance again at our calculations, we
can see that basically all we needed to draw the
main conclusion was just to regard the subchains
as ideal.
10
Since
we have from the uncompressibility
condition
(at characteristic values of stress applied to
rubbers the intermolecular
distances practically do not change 1 change at
107 Pa ).
11

• Modulus of elasticity is
• For loosely crosslinked networks it is small
• (from the incompressibility condition
  where is the volume of a monomer unit, thus
  ). This is just the origin of the
  high elasticity of rubbers.
• The final formula predicts not only modulus, but
  also nonlinear elasticity.

• Analogous formula can be obtained for other
  kinds of deformation (shear, twist etc).
• The final formula is universal, i.e independent
  of specific chain model.
• Reason entropic elasticity is caused by large -
  scale properties of polymer coils.

12

• Main assumption in the above derivation
• i) the chains are Gaussian
• ii) the chain entanglements are neglected.
• If and T increases, the
  value of
• should decrease, i.e the rubber shrinks upon
  heating (contrary to gases) and vice versa.
• Also at adiabatic extension the rubber is heated
  ( contrary to gases). This is the consequence of
  entropic character of elasticity.
• Correlations with experiment

- very good agreement - theory slightly
overestimates stress at a given strain.
Reason chain entanglements - theory
significantly underestimates stress, at a
given strain. Reason finite extensibility of the
chains.