part ii. mechanical properties of polymers
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Part II. Mechanical Properties of Polymers
Chapter 5. The nature of rubber elasticity
• In 1805, Gough found that the length of rubber sample held under constant tension decreased as its temperature was increased and demonstrated that heat was evolved as a result of adiabatic extension.
• Until 1930, by that time the macromolecular character of the rubber molecules was completely accepted, the theory of rubber elasticity became possible to develope.
• The rubber specimendeparted from the Hooke’s law over such as an enormous range of extensions.
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I.
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Thermal elastic inversion
0<⎟⎠⎞
⎜⎝⎛∂∂
LTf
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VTU LUf ,)/( ∂∂=
VTS LSTf ,)/( ∂∂−=
The relationship between andPLT
f
,⎟⎠⎞
⎜⎝⎛∂∂
VLTf
,⎟⎠⎞
⎜⎝⎛∂∂
Since f is function of T and P, therefore
dPPfdT
Tfdf
TP⎟⎠⎞
⎜⎝⎛∂∂
+⎟⎠⎞
⎜⎝⎛∂∂
=
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II. The statistic thermodynamic theory of high etasticity
1. The statistics of freely-jointed chain
Space.
ψcosllx =
24)(sin2)(
lldldllP xx πψψπ
=
ψψdsin=
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Let −+ −= nnm ,!
2!
2
!21),(),(
2
⎟⎠⎞
⎜⎝⎛ −
⎟⎠⎞
⎜⎝⎛ +
⎟⎠⎞
⎜⎝⎛==−+ mnmn
nmnnn ωω
From Stirling’s approximation, nn enn /2! 21+≅ π
2/21
2
/1/112),(
mn
nmnm
nm
nmn ⎥⎦
⎤⎢⎣⎡
+−
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎠⎞
⎜⎝⎛−⎟
⎠⎞
⎜⎝⎛=
⎟⎠⎞
⎜⎝⎛ +
−
πω
If m/n<<1, ( )( )
( )mmm
nmnmnm
nmnm /1
/1/1
/1/1
2/
2
22/
−≅⎥⎦
⎤⎢⎣
⎡
−−
=⎟⎠⎞
⎜⎝⎛+−
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( ) ( )[ ] nmnmmnm enmnm /// 22
/1/1 −−− ≅−=−
Moreover,
( )[ ] ( )[ ] nmn
mn
nmn
enmnm 22
1
221
222
2
2
/1/1 ≅⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
−=−
⋅+
⎟⎠⎞
⎜⎝⎛−
+−
So, nm
nm
nm
en
een
mn 22/1
22/1 222
22),(−−
⎟⎠⎞
⎜⎝⎛=⎟
⎠⎞
⎜⎝⎛=
ππω
Since each time we increase one unit in n+, n- will be decreased by one unit.
If we use x to replace m,33
)( lmlnnx =−= −+
Thus,2
2
232/1
232),()( nl
x
elnx
mnx−
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛=
∆=
πωω
32lx =∆and
xnnl=− −+ )(
3Note , 13
<<⋅
=− −+
nlx
nnn
, 3nlx ⋅
<<
Also,dxedxx nlx
o22 2/3)( −= ωω
lno1
23 2/1
⎟⎠⎞
⎜⎝⎛==ππ
βωwhere
( ) ⎟⎠
⎞⎜⎝
⎛=ln
12/123β
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( ) lnr =⎟⎠⎞
⎜⎝⎛=
5.02/1
2
231
β
1/β
• The effect of bond angle restrictions
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If we assume
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• Chain with hintered rotation
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• Long range interactionTwo segments in a polymer chain can not occupy the same space.
lnro6.02/12 ~)(
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2. Persistent (wormlike) chains
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3. The equation of state for a single polymer chain
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• Langevin approximation
A segment of length a in a freely orienting chain will usually have no preferred direction or orientation. When the chain is subjected to a tension F, the segment will have differentf
f
f
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f
f
ff
f
f
On the other hand, the work spent to pull the polymer chain will be
( ) ( )∫ ∫=r r
drNaraKTfdr0 0
* //
If no force was applied to pull the polymer chain, the possibility for the chain end to appear in r and r+dr lcan be expressed by
drrKTfdrconstdrrr 2
04/exp)( πω ⎥⎦⎤
⎢⎣⎡ ⎟
⎠⎞⎜
⎝⎛−= ∫
f
f
f
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Assuming uNar =)/(* , thenu
uNar 1coth −=
24lnsinh
ln)(ln ru
uuNarNconstr πω +⎥
⎦
⎤⎢⎣
⎡+⎟
⎠⎞
⎜⎝⎛−=
2642
4ln...35099
209
23 r
Nar
Nar
NarNconst π+
⎥⎥⎦
⎤
⎢⎢⎣
⎡+⎟
⎠⎞
⎜⎝⎛+⎟
⎠⎞
⎜⎝⎛+⎟
⎠⎞
⎜⎝⎛−=
rNar
Nar
Nar
lrr
T
2...175297
5931)(ln 53
+⎥⎥⎦
⎤
⎢⎢⎣
⎡+⎟
⎠⎞
⎜⎝⎛+⎟
⎠⎞
⎜⎝⎛+⎟
⎠⎞
⎜⎝⎛=⎟
⎠⎞
⎜⎝⎛
∂∂ ω
VTVT rTrZKT
rFf
,,
),(ln⎥⎦⎤
⎢⎣⎡
∂∂
−=⎟⎠⎞
⎜⎝⎛∂∂
=
⎥⎦
⎤⎢⎣
⎡−⎟
⎠⎞
⎜⎝⎛
∂∂
−=rr
rKTT
2)(lnω
⎥⎥⎦
⎤
⎢⎢⎣
⎡+⎟
⎠⎞
⎜⎝⎛+⎟
⎠⎞
⎜⎝⎛+⎟
⎠⎞
⎜⎝⎛= ...
175297
593
53
Nar
Nar
Nar
aKT
When r/Na >0.3, the polymer chain does not follow the Hook’s law.
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• The stress/strain equation of a network
Affine deformation: Upon deformation, the crosslink junctions in the network transform affinely, i.e., in the same ratio as the macro-scopic deformation ratio of the rubber sample. If the
23or
rKTf =(Since )
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III. Practical aspects of elasticity theory1. Testing the stress/strain equation
Eq. 69 indicates that the stress is functions of temperature, strain, molecularweight, degree of crosslinking, et al.
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2.
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3.
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4. The effect of molecular weight
⎟⎟⎠
⎞⎜⎜⎝
⎛−
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛⎟⎠⎞
⎜⎝⎛ −= 22
2121γ
γo
ic
co r
rMM
MLmRTf
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4. Behavior of elastomers at large deformations
6. The effect of crystallization
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7. Fillers
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