Transcript Заголовок слайда отсутствует

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.
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.
Typical stress-strain curves

For steel

2109 Pa
A
B
For rubber
3 107 Pa
C
B
C
A
0.01
l l
5
l l
A - upper limit for stress-strain linearity
B - upper limit for reversibility of deformations
C - fracture point
• Characteristic values for deformation l / l are
much larger for rubber.
• Characteristic values for strain  are much
larger for steel.
• Characteristic values for Young moduli are
enormously larger for steel ( E  210 Pa )
than for rubber (E  106 Pa ).
• For steel linearity and reversibility are lost
practically simultaneously, while for rubbers
there is a very wide region of nonlinear reversible deformations.
• For steel there is a wide region of plastic deformations (between points B and C) which is
practically absent for rubbers.
11
Elasticity of a Single Ideal Chain

f
l
For crystalline solids the elastic response appears, because external stress changes the
equilibrium inter-atomic distances and
increases the internal energy of the crystal
(energetic elasticity).

R

f
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.
According to Boltzmann, the entopy


S ( R)  k ln W N ( R)

Where k is the Boltzmann constant and W N (R)
is the number of chain conformations compa
tible with the end-to-end distance R .


W N ( R)  const  PN ( R) 


S ( R)  k ln PN ( R)  const
  3 3 2
 3R 2 
But, PN ( R)  

 exp 
 2Ll 
 2 Ll 

3kR 2
Thus, S ( R)  
 const
2 Ll
The free energy F :
3kTR 2
F  E  TS  TS 
 const
2 Ll
 F 3kT 
 
R
fdR  dF  f   
R
Ll
 F 3kT 
f   
R
R
Ll
• The chain is elongated in the direction

 
of f and f  R (kind of a Hooke law).
3kT
:
• “Elastic modulus”
Ll
1) is proportional to 1 L , 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.

Limitations: P (R) should be Gaussian which
N
is the case for not too strongly elongated chains.
Elasticity of a Polymer Network (Rubber)
y
x
z
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  x ,  y ,  z , i.e the sample
dimensions along the axes are a x   x a x0 ,
a  a ,a  a .
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 R0 x , R0 y , R0 z ,
in the deformed state its coordinates are
Rx   x R0 x , Ry   y R0 y , Rz  z R0 z .
y
y
y0
z
z
z0
Thus, the change of the free energy of the chain
between two crosslink points upon extension is
3kT  2  2 3kT
R  R0   Rx2  R02x  
f 
2 Ll
2 Ll
3kT 2 2
2
2
2
2
R0 x  x  1 
 R y  R0 y   Rz  R0 z  
2 Ll
 R02y 2y  1  R02z 2z  1
For the whole sample F  V f
where  is the number of chains per unit
volume and V is the volume of the sample.


3kT
V 2x  1 R02x  2y  1 R02y 
2 Ll
 2z  1 R02z .
F 
R
Ll



3
3
2
But,
R
2
0x
 R
2
0y
 R
2
0z
0
1
F  kTV 2x  2y  2z  3
2
1
F  kTV 2x  2y  2z  3
2
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.
Let us apply the general formula (see above)
for the case of uniaxial extension ( x    1)
or compression ( x    1) along the axis x.
Since V   x a0 x  y a0 y  z a0 z   x  y  zV0 we
have from the uncompressibility condition
 x  y z  1
(at characteristic values of stress applied to
rubbers  105  106 Pa
the intermolecular
distances practically do not change: 1%
change at 107 Pa ).
1
2
 y   z   y  1   y   z 

kT
2
V 2   3
2
 

1 F
1
F 1 F



a0 y a0 z a x a0 y a0 z a0 x  V 
F 
  kT   

1
2
 
  kT   

1
2
 
• Modulus of elasticity is E  3kT .
For loosely crosslinked networks it is small
(from the incompressibility condition N  1
where  is the volume of a monomer unit,
thus   1 N ). This is just the origin of the
high elasticity of rubbers.
• The final formula predicts not only modulus,
but also nonlinear elasticity.

1

• 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.
1

  kT    2 
 

• Main assumption in the above derivation:
i) the chains are Gaussian;
ii) the chain entanglements are neglected.
• If   const  0 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:

experiment
theory
1

0.4    1.2 - very good agreement
1.2    5 - theory slightly overestimates stress at a given
strain.
Reason: chain entanglements
  5 - theory significantly underestimates stress, at a
given strain.
Reason: finite extensibility of the chains.