cross-linked polymers and rubber elasticity 5/13/20151
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Cross-linked Polymers and Rubber Elasticity
04/18/23 1
Definition
• An elastomer is defined as a cross-linked amorphous polymer above its glass transition temperature.
1. Capability for instantaneous and extremely high extensibility
2. Elastic reversibility, i.e., the capability to recover the initial length under low mechanical stresses.when the deforming force is removed.
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Crosslinking effect
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Defects in crosslinks
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For the purpose of the theoretical treatments presented here, the elastomer network is assumed to be structurally ideal, i.e., all network chains start and end at a cross-link of the network.
Force and Elongation
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Hookian
Rubber elasticity Stress
induced crystallinity
Rubber Elasticity and Force
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The origin of the forceAt constant V
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Under isothermal conditions
Entropy origin
Eneregy origin
Entropy change or internal energy change is important?
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Since F is a function of state:
The change in internal energy in effect of l change
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Experimental data
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Experimental data
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Thermodynamic Verificationat constant p
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According to the first and second laws of thermodynamics, the internal energy change (dE) in a uniaxially stressed system exchanging heat (dQ) and deformation and pressure volume work (dW) reversibly is given by:
The Gibbs free energy (G) is defined as:
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The partial derivatives of G with respect to L and T are:
The partial derivative of G with respect to L at constant p and constant T
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The derivative of H with respect to L at constant p and constant T
Experiments show that the volume is approximately constant during deformation, (V /L)p,T= 0 . Hence,
Statistical Approach to the Elasticity
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Elasticity of a Polymer Chain
relates the entropy to the number of conformations of the chain Ω
Entropy of the chain
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the probability per unit volume, p(x, y, z)
<r2>o represents the mean square end-to-end distance of the chain
The entropy decreases as the end-to-end distance increases
The work required for change in length
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It can be concluded that (1)is proportional to the temperature, so that as T increases the force needed to keep the chain with a certain value of r increases, and (2)the force is linearly elastic, i.e., proportional to r.
Elasticity of a Netwrok
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Assumptions
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l. The network is made up of N chains per unit volume. 2. The network has no defects, that is, all the chains are joined by both
ends to different cross-links. 3. The network is considered to be made up of freely jointed chains,
which obey Gaussian statistics. 4. In the deformed and undeformed states, each cross-link is located
at a fixed mean position. 5. The components of the end-to-end distance vector of each chain
change in the same ratio as the corresponding dimensions of the
bulk network. This means that the network undergoes an affine deformation.
Model of deformation
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And the chain
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The entropy change
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For N chain
And
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the work done in the deformation process or elastically stored free energy per unit volume of the network.
The total work;
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True and Nominal stress
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The Phantom Model
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When the elastomer is deformed, the fluctuation occurs in an asymmetrical manner. The fluctuations of a chain of the network are independent of the presence of neighbor in chains.
Other quantities:Young Modulus
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RTr
rnE
LLE
i
VT
20
2
,
3
RTr
rnE
LLE
i
VT
20
2
,
3
RTr
rnG
EG
i2
0
2
5.0),1(2/
RTr
rnG
EG
i2
0
2
5.0),1(2/
?
Statistical Approach to the Elasticity
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a) For a detached single chain
VTVT r
TrkT
r
Ff
TrkTconsF
TSUF
,,
),(ln
),(ln.
VTVT r
TrkT
r
Ff
TrkTconsF
TSUF
,,
),(ln
),(ln.
A Spherical Shell and the End of the Chain in it
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The probability for finding the chain end in the spherical shell between r and r+r
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2
0
4
),(
),()( r
drTr
drTrdrrW
2
0
4
),(
),()( r
drTr
drTrdrrW
Recall=>
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)2(3
4)(
20
2
2
3
21
22
r
rerW r
)2(3
4)(
20
2
2
3
21
22
r
rerW r
20
3
r
kTrf 2
0
3
r
kTrf Retractive force for a
single chain
Gaussian distribution
Recall again =>
b) For a Macroscopic Network
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3
1izyx rr 22222
3
1izyx rr
The Stress-Strain Relationship
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2
12
212
)(
)(20
3 r
rel
el
i
rdrr
nRTF
FW
2
12
212
)(
)(20
3 r
rel
el
i
rdrr
nRTF
FW
20
20
20
2222
0
2
2 zyxzyxi
el r
rnRTF 2
02
020
2222
0
2
2 zyxzyxi
el r
rnRTF
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22
2
,
1
o
i
VT r
rnRT
F
22
2
,
1
o
i
VT r
rnRT
F
21
20
20
20
1,
1
1
zyx
zyx
zxyx
21
20
20
20
1,
1
1
zyx
zyx
zxyx
3
2
22
2
2
o
iel r
rnRTFW
3
2
22
2
2
o
iel r
rnRTFW
We have:
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22
2
,
1
o
i
VT r
rnRT
F
22
2
,
1
o
i
VT r
rnRT
F
And the stress-strain eq. for an elastomer
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Eq.Hookean -non a 1
2
G Eq.Hookean -non a
12
G
Equibiaxial tension
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such as in a spherical rubber balloon, assuming ri2/r 2
0 = 1, and the volume changes of the elastomer on biaxial extension are nil.
The Carnot Cycle for an Elastomer
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2
0
0 L
L
L
LnRT
2
0
0 L
L
L
LnRT
Work and Efficiency
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dLe dLe
PdVg PdVg
II
IIIg q
II
IIIg q
II
III
IIe Q
Q
dL
II
III
IIe Q
Q
dL
A Typical Rubber Network
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Vulcanization with sulfur
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Radiation Cross-linking
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Using Multifunctional Monomers
Comparison between Theory and Experiment
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Thermodynamic Verification
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At small strains, typically less than = L/ L0 < 1.1 (L and L0are the lengths of the stressed and unstressed specimen, respectively), the stress at constant strain decreases with increasing temperature, whereas at λ values greater than 1.1, the stress increases with increasing temperature. This change from a negative to a positive temperature coefficient is referred to as thermoelastic inversion. Joule observed this effect much earlier (1859). The reason for the negative coefficient atsmall strains is the positive thermal expansion and that the curves are obtained at constant length. An increase in temperature causes thermal expansion (increase in L0 and also a corresponding length extension in the perpendicular directions) and consequently a decrease in the true λ at constant L. The effect would not appear if L0 was measured at each temperature and if the curves were taken at constant λ (relating to L0 at the actual temperature). The positive temperature coefficient is typical of entropy-driven elasticity as will be explained in this section.
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Stress at constant length as a function of temperature for natural rubber.
Thermodynamic Verification
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The reversible temperature increase that occurs when a rubber band is deformed can be sensed with your lips, for instance. It is simply due to the fact that the internal energy remains relatively unchanged on deformation, i.e. dQ=-dW (when dE=0). Ifwork is performed on the system, then heat is produced leading to an increase in temperature. The temperature increase under adiabatic conditions can be substantial. Natural rubber stretched to λ=5 reaches a temperature, which is 2-5 K higher than thatprior to deformation. When the external force is removed and the specimen returns to its original, unstrained state, an equivalent temperature decrease occurs.
At constant V and T
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TSUF TSUF
VLVT
VTVTVT
T
f
L
S
L
ST
L
U
L
Ff
,,
,,,
VLVT
VTVTVT
T
f
L
S
L
ST
L
U
L
Ff
,,
,,,
Wall’s differential mechanical mathematical relationship
VLVT L
fT
L
Uf
,,
VLVT L
fT
L
Uf
,,
Thermodynamic eq. of state for rubber elasticity
A Similar Equation
Analysis of Thermodynamic Eq.
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Stress-Temperature Experiments
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se fff se fff
End of Chapter 9
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