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Chapt. 5 Amorphous State of Polymers
5.1 Molecular motions of polymers
5.3 Glass transition of polymers
5.2 Viscous flow of polymers
,
, ,
, ,
1
5.1
1.
/
x
t
/0
tx x e: relaxation time
(1)
(2)
a.
b. WLF (Tg )
(3)
/0
E RTe
2
Time Dependent Behavior – Example: Silly Putty
3
4
Stress relaxation
ddt t
tt0tt0
1
0
00
exp /t
E t E t
E
5
Creep
E
t0 tt0 t/
0
1 ttD t D e /tt e
6
Relaxation Time Originates fromViscoelastic Properties of Polymers
Elasticity and viscosityHooke’s law describes the behavior of a linear elastic solid andNewton’s law that of a linear viscous liquid:
Spring as a model:Modulus:
Hooke’s law: = E
Dashpot as a model:Viscosity ( ):
: stress ( ); : strain ( )
Newton’s law: = (d dt)
7
Phenomenological models for linear viscoelasticity
= E = (d dt)
=+ Viscoelasticity ?
Elasticity Viscosity
Model I - Maxwell model Combining the spring and dashpot in series
Model II -Voigt-Kelvin model Combining the spring and dashpot in parallel
Model III – Burger’s Model Combining the Maxwell and Voigt elements in series
….8
Elasticity + Viscosity = Viscoelasticity ?
1
m m
d ddt E dt
dtEd
m
m
0 exp m
m
Et t
0 exp tt
For stress relaxation, d /dt = 0,
At time t = 0, = 0,
Relaxation time: = m/Em:
Model I: Maxwell Model
2
m
ddt
1 1
m
d ddt E dt
1 2
1 2
9
1 = Em 1
2 = m(d 2 dt)
/0
0
ttE t E e
Maxwell Model fails to describe Creep
10
1
m m
d ddt E dt
For creep, = 0,0 0 01
m m m
dddt E dt
the “creep” behavior of viscous liquids.
Model II - Voigt-Kelvin model
1
1
Em
2
2
m
1 1mE
22 m
ddt
Total stress: = 1 + 2;strain: = 1 = 2
1 2 m mdEdt
For stress relaxation, d /dt = 0, 0mEIt fails to describe the stress relaxation behavior.
For creep, = 0, 0 m mdEdt
At time t = 0, = , /0 1 m m t
m
Et eE
Relaxation time:Retardation time = m/Em:
/0 1 t
m
t eE
11
Model II - Voigt-Kelvin model
/1 tD t D e
/00 0/ / 1 t
m
t eECreep compliance
For creep recovery, = , 0 m mdEdt
/tt e
12
Model III – Burger’s Model
E1, 1
E2, 22, 2
3, 3
01
23
3
/
1
002( ) 1 t
Ett e
E
2
2
E
For creep, = 0:
where
1
1
3
2
2
3
13
-
14
= m/Em
Dynamical Mechanics Analysis
t
0 sin t
0 sin t
= (d /dt)= 0/ sin( t- /2)
0 0 sinW t d t
15
hysteresis and mechanics loss
Complex Modulus: As “Solid”
0 sin t0 0sin cos s ncos it tt
0 sin t
synchronization asynchronism
Modulus ofsynchronization part
Modulus ofasynchronism part
0
0
' cosE 0
0
" sinE
E’
E”"'E EE
' " "'*E EE iEE E
'an "t
EE
or
0
0
* iE e
16
Complex Viscosity: As “Liquid”
0 0sin cos s ncos it tt0 sin t
0 cosd tdt
synchronizationasynchronism
Viscosity ofsynchronization part
Viscosity ofasynchronism part
0
0
' sin0
0
" cos
0
0
' cosE 0
0
" sinE
'" E "' E
17
For dynamic mechanics
Model I - Maxwell model 0i te
0 0 0 0i t i t i t
m m m m
d ti e e i e
dt E E
0 0 i t
m m
d t dt i e dtE
2 2
2 2 2 2*1 1
m mt E EE it
= m/Em
lnE
()
ln
E’
E”
tan
=1
k=2k=1 k=-1
k=0
18
Model II - Voigt-Kelvin model 0i te
Complex compliance 2 2 2 2
1 1** 1 1m m
D iE E E
1
m m
d ddt E dt
m mdEdt
internal friction
tan
Tg Tf
tan
log
tan
19
20
single
polymer
/0
tG t G e
/ iti
i
G t G e
General Maxwell Model
/ iti
iE t E e
/
0
tE t f e d
For stress relaxation relaxation time spectrum ( )
H t f
Mw: III>II>>I
/ lntE t H e d
single
polymer
21
Viscosity & Relaxation Modulus
22
/ iti
iE t E e
i i iE/ iti iE t Ee
/ /
0 0 0i it t
i i i i i iE t dt Ee dt E e dt E
0 0 0i i ii i i
E t dt E t dt E t dt
0,T E T t dt
General Voigt Model
For creep retardation time spectrum ( )
2 2
2 2 2 2*1 1
i i i i
i ii i
E EE i
/1 1 1 lntD t L e d
For dynamic mechanics
2 2 2 2
1*1 1
i
i ii i i i
D iE E
or
General Maxwell Model
General Voigt Model
For creep recovery /2 2 lntD t L e d
23
Viscosity & Modulus & Relaxation Time
24
'"
,,
ET
T"'
,,
ET
T
0,T E T t dt
1. oscillatory shear
2. static shear
/,, lntH TE T t e d
Modulus vs Relaxation Time
Viscosity vs Modulus
i= i/Ei i-th movement mode
/, iti
iE T t E T e
relaxation time spectrum
Viscoelasticity of Polymer
Solid Elasticity(short) Liquid Viscosity(long)= E = (d dt)
5.2 Viscous Flow of Polymers
The rheological properties ( ) ofpolymers is extremely important forpolymer processing
StressStrain
Velocity
Rheology: The study of the deformation
and flow of matter.
25
Characteristic of polymer viscous flow
1.
ddt
nK
Bingham
Pseudoplastic
Dilatant
Newtonian
2.
26
Shear thinning ( )
27
Time-dependent shear stress and primary normal stressdifference after start-up of steady shearing
28
3.
Relaxation of shear stress and primary normal stressdifference after cessation of steady-state shearing
29
3.
Consider a steady simple shear flow
Shear force Without
Shear force
Die-swell ( ) Extrudate swellobserved for a melt ofPS for various shearrates and temperatures.(Burke, J. J.; etc.Characterization ofMaterials in Research,Syracuse Univ. Press,1975.)
30
Polymer Processing
31
32
Characterization of viscous flow
( )
a. ( )-shear viscosity s b. ( )-tensile viscosity t
c. ( )-bulk viscosity
for incompresible fluids b
velocity gradient2
dvdx
33
Melt viscosities of polymers
apparent viscosity
/advdh
differential viscosityd
dc
shear viscosity
tt
/ss s
dvdh c
a
complex viscosity ( )
'* "i
0 sin t
0
0
sin
sin / 2t
t
1-D2-D
34
extensional viscosity
low molecular weight
35
Measurement of shear viscosityCone-plate viscometer ( )- an example
M
H
R
tanh r rhdv r
dh r
3
32
MR
323a
M Rbb
r
36
As liquids: The flow curve of polymer melts
s
N1
N2p
d
entanglementdisentanglementorientation
turbulence
log s
log
N1
N2
0
0
t
p
00/
/
37
0 dependent of M
10 ~
W cMM M
Critical molecular weight at theentanglement ( ) limit: Mc
Exp.
Theory1
0 ~ Rouse ModelM
38
3.3~3.40 ~
W cM M M
30 ~ Tube (Reptation) ModelM
As Solids: Time vs. Frequency vs. Temp at Low Deformations
39
0 1~Ne
GM
0NG
0NG
1.2.3.
2. -Vogel-Fuchler0exp /A B T T
Tf
1.
3.Brigid>Bflexible
4. -
404.
Viscosity & Modulus in Polymer Melts
41
log s
log
N1
N2
0
0
p
log log log logssViscosity
Modulus
1 3.30 0 orM M
Molecular Theory of Viscous Flow andViscoelasticity of Polymer – How to get
Rouse model
Rouse-chain:The chain is subdivided in N ‘Rouse-sequences’, eachsequence being sufficiently long so that Gaussian propertiesare ensured.Each Rouse sequence is substituted by a bead and a spring.The springs are the representatives of the elastic tensileforce, while the beads play the role of centers whereonfriction forces apply.
1
2 iN-1
N
Rouse-chain composed of N+1 beadsconnected by springs
When a bead is displaced from itsequilibrium position there are two types offorces acting on it: (1) those that resultfrom the viscous interaction with thesurrounding molecules, and (2) those thatrepresent the tendency of the molecularchains to return to a state of maximumentropy by Brownian diffusion movement.42
Potential of a Spring
0F R k R R2
012
U R k R R
Hook’s law
Boltzmann Factor~ exp / BU R k T
Partition Function~ exp /i B
iZ U R k T
Free Energy lnBF k T Z 43
A Brief Review of Gaussian Model
3/ 2 20
2 2
33 exp exp2 2
nn
B
uA
b b k Trr
kk k
kk
3 / 2
0211
3 / 20
2
3 1exp2
3 exp2
nn n
nnn B
nn
B
ub k T
U
b k T
R r
r
rn
b b Gaussian Segment
20 12
32n B n nu k Tb
r R R
k
k
20 12
1
32
n
n B n nn
U k Tb
r R R
Rn
Rn-1
44
Rouse model
Consideration of the restoring force when a bead is displaced from its equilibriumposition leads to the expression
dx/dt: the time differential of the displacement of ith bead: the friction coefficient of a bead
l: the length of each link in a chainN: the number of the links in a macromolecule(for N submolecules there are 3N of these equations)
1 12 2
3 3 (2 )ii B Bi i i i
i
U RdR k T k T R R R fdt b R b
1
2 i N-1
N
For the Brownian motion of a harmonic oscillator21
2U R kR
dR U f F f kR fdt R
For the Brownian motion of the bead-spring model2 2
1 112i i i i iU R k R R R R
Langevin equation
45
f: Random force of Brownian Motion
Normal Coordinates of Rouse Model
46
d k fdtQ ZQ I
1 11 2 1 0
0 1 2 1...
0 1 2 11 1
Z Rouse-ZimmMatrix
11 2 1
21 2 3 2
1 1
1
0
2 0
0 2 0
0
i i i
nn
i
n
i
n
dR k R R fdt
dR k R R R fdt
k R R R
dR k R R fdt
tR f
1 12i i i ii k R R R
tfR
23 Bk Tk
b P156 of Chapt 1-2
1
n
R
RQ
47
Applications of Gaussian Chain Model: Stretching of an Ideal Chain
3/22
2 2
3 3, exp2 2
g g
gg g
NN l N l
hh3/2
2
2 2
3 3, , exp2 2g g g g
g g
N N N NN l N l
hh h
2
2 2
3 3 3( , ) ln( ) ln2 2 2g B B B g
g g
S N k k k NN l N lhh
23( , ) Bg
g
G Nh
k TN l
hf h
2
2
3( , ) ( )2g B g
g
G N U TS k T G NN lhh
, , /g g gN N Nh h
lnS k h
=???
kf x
ln / 'S k
/' '/
( )
48
X TQ1TZT
1dT k T T T Tfdt
Q Z Q I
2
3 Bk Tddt bX X
: T
Z :
1
2
...
n
0 expp pp
tX t X2
3pB p
bk T
24sin2ppN
2 2
1 23 B
N bk T
Terminal relaxation time2
23 4sin2
p
B
bpk TN
2
30 exp Bk Tt tb
X X
( )
X:
2 2
2 2
13 B
N bk T p 1 2
1p
2
42pN
49
50
3n 3n 3n,
C2H4
51
The Zimm Model
,mm
nnmH m
ttUk
RfR
Rouse model
1 ˆ ˆ Zimm model8
nm
nm nmnm
Ir
I
Hr r
Oseen Tensor
Rouse Model Zimm Model
nmH m bead n bead
n m n mm
v r H r r F r
Oseen Tensor
5252
The Momentum Equation of Fluids – Navier-Stokes Equation
Stokes Approximation:2
0Pv F r
v
2
0i P F
ik k k
k
k v kk v
In Fourier Space:
2
1 ˆ ˆvk k kI kk F H k Fk
Doi, Edwards, The Theory of Polymer Dynamics, p89
2 0k kk v F
k
vk
kv
k
//kv
' ' 'dv r r H r r F r
3 2
1 1 1ˆ ˆ ˆˆ82
id er
k rH r k I kk I rrk
ˆ ˆT I kk
2
0
P Ptv F r v F r
v
22i ik k
Estimating The Longest Relaxation Times
53
f v Bk TDFriction coefficient Einstein Relation
1/22 1/20 6t Dtr r
Stokes Law 6 s hRR=l
206R D
2 2
0 6 6 B
R lD k T
Brownian Motion
Stokes-Einstein Relation6
B
s h
k TDR 6
Bh
s
k TRD
Rouse Model:
BR
R
k TD R N
2g
RR
RD
gR N l
Zimm Model:
Z s gRBZ
Z
k TD
2g
ZZ
RD
2g
B
NRk T
21 2
B
l Nk T
2Z g
B
Rk T
3s g
B
Rk T
33s
B
l Nk T
sl
31 2s
B
l Nk T
3s
B
lk T
Relaxation times of Rouse-Zimm Model
54
2 22
2
1,Rouse
3
2
pB
N b pk Tp
Rouse model
3 33
1,zimm
pB
N b pk Tp
Zimm model ingood or solvent
2 2
1,Rouse 2
2
3~
B
N bk T
M
Terminal relaxation time3 3
1,zimmB
N bk T
3 / 23 9 / 5
3/2~ M9/5~ M
or
relaxation time of different mode
or11/5~ Mor
Rouse-Zimm model
The first three normal modes of a chain
R-Z model is good for M < MC3
0.425 AN N aM
230(RZ)
230 exp
0.425 2.56 10
2.2 ~ 2.87 10AN
1
2
01/2
( ) exp( )
exp 2 /
2 2
m
Bp p
B R
RB
tG t nk T
c k T dp tpN
c k TtN
2 2
2 21
'( )1
mp
Bp p
G nk T
2 21
"( )1
mp
Bp p
G nk T
1
1,2, , .p p
p m
Relaxation time for the pth mode:For N >> 1
Stress relaxation modulus andcomplex modulus (Maxwell-element model):
55
Prediction of Viscoelasticity by Rouse-Zimm ModelRouse: Zimm:
2 2
2 21
'( )1
mp
Bp p
G nk T 2 21
"( )1
mp
Bp p
G nk T
1<<12 2 2
11
'( ) ~p
G p 11
"( ) ~p
G p
1>>12 2
12 20
1
1 1/1/
1 20
1/ 1/1
'( )1
11
~2 sin / 2
pG dp
p
xdxx
12 20
1
1/1/
1 20
1/ 1/1
"( )1
11
~2 cos / 2
pG dp
p
xdxx
1 1: '( ) "( )G G 1 terminal relaxation time3.
56
( =2 or 11/5) ( =3/2, solvents) or ( =9/5 , good solvents)
Dynamic Modulus of Polymer in Dilute Solution
57
Rouse ( =2) Zimm ( =3/2, solvents) Experiments
2'( ) ~G"( ) ~G
1<<1
1>>1 1/"( )G
1/'( )G1
5/9 or 2/3
1<<12'( )G1"( )G
From Maxwell Model to General Maxwell Model orRouse-Zimm Model
58
lnE
()
ln
E’
k=2
k=1 k=-1E”
Linearsuperpose ofMultipleRelaxationTimeSpectrum
Single RelaxationTime
Maxwell Model Rouse-Zimm Model
MultipleRelaxation Time
Relaxation Modulus and Dynamic Modulus of Rouse Model
59
1
2
01/2
( ) exp( )
exp 2 /
2 2
m
Bp p
B R
RB
tG t nk T
c k T dp tpN
c k TtN
(t< R)
1/2
( ) exp /RB R
cG t k T tN t (t>> R)
2 2
0 0
2a xe dx a
aNote:
0 0
2
01
2
1
2
exp 2 /
2
36
RpB
R
pB
dtG t
c dt tpNk T
c pNk Tc Nb M (M<Mc)
60
,
M<Mc
RouseModel
ZimmModel
G”~ 1
G’~ 2
G”~ 1
G’~ 2
0
0
' co sG 0
0
" sinG
Dynamics Modulus of Polymer Melts (M>Mc)
61
M
0 1~Ne
GM
Plateaumodulus
2
1/ 2
???G”
G’
Relaxation Modulus of Polymer Melts
62
M<Mc
M>Mc
Relaxation Modulus vs Dynamic Modulus in Melts
63
Tube & Reptation Model for Entanglement
64
Contour length of the primitive tube: Lpr
Length between entanglement: apr
Microscopic Dynamical Model of Polymers
Reptation modelReptation model:Decomposition of the tuberesulting from a reptationmotion of the primitivechain. The parts which areleft empty disappear.
2 20 pr prR Nb L a
Define the contour length of theprimitive path Lpr:
apr is the associated sequence length,which describes the stiffness of theprimitive path and is determined by thetopology of the entanglement network.
65
2 /pr prL Nb a
1/ 22 2 20n n B e prt k Tb aR R4
2pr
eB
ak Tb
entanglement time
2pr
d
LD
Curvilinear diffusion coefficient D:
B
P
k TD (Einstein relation)
P bN ( b: friction coefficient of bead)In order to get disentangled, chainhave to diffuse over a distance lpr,and this requires a time:
3~d b NTherefore,
Lpr
Relation of entanglement andreptation model
apr
disentanglement time
6666
Time correlation function of End-to-end Vector
0 0
2
0t A C CD DB AC CD DB
CD a t
P P
3 4 22 2
2 2 2
1 3/ bd R
B
N b NbL Dk Ta a
20
exp / dt t p t
Doi, M., Edward, S. F., The Theory of Polymer Dynamics, Oxford, 1986, p.194
Comparison of relaxation times
67
40
2 ~eB
a Mk Tb
23
2
3 ~d RNb Ma
2 22
1,Rouse 2 ~3 B
N b Mk T
Effects of Entanglement on Relaxation ModulusDynamically ShearStep Shear
68
0
1/2
2
2
2
2
1~
e N
RB
e
B
e e
G t G
c k TN
cb k TacbN b M
Tg
0NG t G
1~
1/2
2 2R
BcG t k T
tN
Viscosity
69
0 0
2
01
2
1
2
exp 2 /
2
36
RpB
R
pB
dtG t
c dt tpNk T
c pNk Tc Nb M
Rouse Tube
0 02
0
3
=12
~
N d
dtG t
G
M
cM McM M
Note, ~ ~ Mwith 3.3 – 3.4for molecularweight higherthan Mc.
69
2 2
23RB
N bk T
23
2
3 ~d RNb Ma
Experiments & Simulations
Series of image of a fluorescent stained DNA chain embeddedin a concentrated solution of unstained chains. (Chu. S. etc.Science 1994, 264, 819.)
Initial conformation
Stretched
Reptationstarts
70
Normal stress difference and Elastic effects onviscous flow of polymers
11
21
22
33
F/A= s= 21
N1= 11- 22>0
N2= 22- 33<0
1122
33
Weissenberg effect
For polymer melts
F
For lowmolecularweight liquids
N1= 11- 22=0
N2= 22- 33=0
A
71
Rod-climbing
72
Extrudate swell
73
tubeless siphon ( )
74
toroidal eddy ( )
75
Instability in Processing
76
77
Instability in Processing
Gross Melt Fracture
Sharkskin
78
Stick-slip
79
GMF
80
Stick-Slip Transition
81
cSample is oriented
Stick-Slip Transition
82
1c
2 /sV b
Vs
b c
b
c enF 1/2B B
epr e e
k T k TFa N l
Lpr
apr
Flow instability and melt fracture
Shark skin
melt fracture
83
5.3
Tg Tf
T
Terminal flow< ~ >~
Tm Tm
Linear and Cross-link
Semi-crystalline orcrystalline Tm<Tf
Crystalline Tm>Tf
84
5.2
1.( )
( ) ( )
( - ) ( - )
2.
3.
85
Glass Transition as a Relaxation ProcessThermal history dependence of Tg
Temperature dependence of the specificvolume of PVA, measured during heating.Dilatometric ( ) results obtainedafter a quench to –20 C, followed by 0.02 or100 h of storage. (Kovacs, A. J. Fortschr.Hochpolym. Forsch. 1966, 3, 394)
86
liquid
Vcrystal
glass 2
glass 1
1: fast cooling2: slow cooling
supercooledliquid
TTmTg1Tg2
4.
(1) -(2) -DSC(3)a. -b. -DMA
(4) -NMR,
Tg0 sin t
0 sin tTTg
tan
87
Polymer Melts and Glasses
88
The deepest and most interesting unsolved problem in solid statetheory is probably the theory of the nature of glass and the glasstransition. (Anderson, P.W. Science 1995, 267, 1615.)
(liquid)
liquid
glasses
(glasses)
slowly cooledfast cooled
Tg fastslow Tg
G
T
The transition from melt to glass iscalled glass transition ( )
Tg: glass transition temperature ( ) 88
Why Glass Transition belongs to Segment Relaxation?
89
0NG
secondary eg
secondary g elenglength l hth engt
90
Liquid vs Glass
Solid vs Glass
5. - Free Volume Theory
Hole theory of liquid: the liquid consists of matter and holes. The larger volumeof liquid when compared to the crystal is represented by a number of holes of afixed volume. The holes represent a quantized free volume, which can beredistributed by movement or collapse in one place and creation in another.
The segmental motion of polymerchain requires more volume
Free volume: a concept useful in discussing transportproperties such as viscosity and diffusion in liquids.
91
Occupied volume: filled circles; free volume:hole
Free Volume Theory
The volume-temperature relationship for atypical amorphous polymer
The coefficient of thermal expansion(CTE, ) is constant for theoccupied volume for both temperaturebelow and above Tg
Assume that at the temperature below Tg,the free volume is constant; and the freevolume will increase with temperaturewhen temperature exceed Tg
0g f gg
dVV V V TdT
gr
gr TTdTdVVV
0f r f g gTg r g g
f gr g
dV dV dV dVV V V T V T T T TdT dT dT dT
dV dVV T TdT dT
ggrgf dT
dVVdT
dVV
11 )( ggfgT TTTTff
f TT g
g
Vf T T
V
fg g
g
Vf T T
V
Vg
0g
dVV TdT
Vr
Vf: free volume at T < Tg(Vf)T: free volume at T TgVr: total volume at T TgV0: occupied volume (determined by
van der Waals interaction + vibration)dV/dT): CTE of the glass- and rubber-state
92
(Vf)T
V0
Vf
Free Volume Theory (cont.)
T g f gf f T T
Relation of the molecular mobility to freevolume: Doolittle equation
fVBVA /exp 0
)()(
)()()(
00 TVTV
TVTVTV
f f
f
fT
( ) 1 1log( ) 2.303g T g
T BT f f
17.442.303 g
Bf
51.6g
f
f %5.2025.0gf
Kf /108.4 4
17.44( )(51.
)lo6
g( ) ( )
g
g g
T TTT T T
)(/)(lnln 0 TVTBVAT f
)(/)(lnln 0 gfgg TVTBVAT )()(
)()(
)()(ln 00
gf
g
fg TVTV
TVTVB
TT}
Normalized free volume:
WLF equation:
Nearlyequal to 1
2.303 / f
g
gg g
T TBTff T
( )log( )g
TT
93
Appendix: Doolittle equation & Einstein equation
fVBVA /exp 0
Doolittle equation
0fV V
01 / 1fA BV V A B
In solution and the volume fraction of suspensions0 0/ fV V V
For the solution of impenetrable spheres of radius R, Einstein derived theEffective viscosity of suspensions
0 1 2.5
Einstein equation
94
1xe x
In glass or melt 0fV V
fVBVA /exp 0
0 / fV V
Applications of WLF Eq.
95
1
2
lg lg sT
s s
C T Ta
C T T
Two principles for linear viscoelasticity(1) Time-Temperature Equivalence and Superposition
Time-temperature equivalence ( ) in its simplest form implies thatthe viscoelastic behavior at one temperature can be related to that atanother temperature by a change in the time-scale only.
1
1
/0
tTE t E e
1 2
2
2
/0
tTE t E e
lnaT
10 1ln ln / ln /TE E t t20 2ln ln / ln /TE E t t
+ln( 1/ 2)=lnaT
-ln( 1/ 2)=-lnaT
E(t)
ln(t)
T2T1
/0
E RTe
T1 T2 T3 …..
1 2 3 ….. 96
Synthesized master-curve ( )
Schematic creep plots atdifferent temperatures
Superpose
Master curve of creep from superposingplots of the left figure
97
Synthesized master-curve
1
2
lg lg sT
s s
C T Ta
C T T
In both the glass-transitionrange and terminal flowregion, the modes ofmotions vary greatly intheir spatial extensions,which begin with thelength of a Kuhn segmentand go up to the size of thewhole chain, and vary alsoin character, as theyinclude intramolecularmotions and diffusivemovements of the wholechain. Nevertheless, allmodes behave uniformly.
98
(2) The Boltzmann Superposition PrincipleThe Boltzmann superposition principle ( )In 1876, Boltzmann proposed:
1. The creep is a function of the entire past loading historyof the specimen;
2. Each loading step makes an independent contribution tothe final deformation, so that the total deformation canbe obtained by the addition of all the contribution.
1 1 2 2 3 3( ) ( ) ( ) ( )t J t J t J t
( ) ( ) ( )t
t J t d t( )( ) ( )
tt J t d
1 1 2 2 3 3( ) ( ) ( ) ( )t G t G t G t ( )( ) ( )t
t G t d
Creep
Stress relaxation
990
G s ds0
( ) ( )dt G s dsdIn steady shear: t s
d d s
Viscosity & Relaxation Modulus
100
/ iti
iE t E e
i i iE/ iti iE t Ee
/ /
0 0 0i it t
i i i i i iE t dt Ee dt E e dt E
0 0 0i i ii i i
E t dt E t dt E t dt
0,T E T t dt
mm
m
HTS
Tm vs. Tg
Chain rigidity: More rigidchain present higher Tg
2/1/ mg TT
3/2/ mg TT
For symmetrical backbone,
For asymmetrical backbone,
Molecular weight dependence Tg vs. Mn
Fox equation:n
gg MKTT )(
Tg vs. Tb
0g bT T
Heating rate dependence
T
slow
fast
ts<< 1 tf<<< 1T1
Tg1 Tg2
Tg1 ts ~ g1 tf < g1
T1
Tg2 tf ~ g2
/0
1/
E RT
fs
et v t t
v
t
0g bT T101
Effects of film thickness on Tg
Tsui, OKC, Macromolecules, 34, 5535 (2001)102
103
Glass transition of polymer mixtures
104
Glass transition of polymer mixtures
Some polymer blends exhibit partial miscibility. They have a mutual, limitedsolubility indicated by a shift in the two Tg’s accompanying a change in thephase composition of the blend. More uncommon is the type of miscibilityindicated by the presence of only single Tg.
2
2
1
11ggg T
wTw
TFox-Flory equation:
Tg of compatible blend PPOand PS as a function of PPOcontent. (Bair, H. E. Polym.Eng. Sci. 1970, 10, 247.)
Plasticization of PVC: Tg asfunction of di(ethylhexyl)-phthalate content. (Wolf, D.Kunststoffe 1951, 41, 89.)
Partial miscible blend
DSC curves of 50 mass-% blends ofPS and poly( -methyl styrene) at aheating rate of 10 K/min. (Lau, S. F.;etc. Macromolecules 1982, 15, 1278.)
105
Miscible blend
Effects of Tg on Morphology of Polymer Blends
Cheng SZD, Keller A, Ann Rev Mater Sci, 28, 533 (1998)Tanaka H, J Phys Condens Matter,
12, R207 (2000)106
Nucleation & Growth Spinodal Decomposition
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