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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