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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

tx x e: relaxation time

b. WLF (Tg )

Time Dependent Behavior – Example: Silly Putty

Stress relaxation

tt0tt0

exp /t

E t E t

t0 tt0 t/

1 ttD t D e /tt e

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)

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

Elasticity + Viscosity = Viscoelasticity ?

d ddt E dt

0 exp m

0 exp tt

For stress relaxation, d /dt = 0,

At time t = 0, = 0,

Relaxation time: = m/Em:

Model I: Maxwell Model

d ddt E dt

1 = Em 1

2 = m(d 2 dt)

ttE t E e

Maxwell Model fails to describe Creep

d ddt E dt

For creep, = 0,0 0 01

dddt E dt

the “creep” behavior of viscous liquids.

Model II - Voigt-Kelvin model

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

Relaxation time:Retardation time = m/Em:

/0 1 t

Model II - Voigt-Kelvin model

/1 tD t D e

/00 0/ / 1 t

t eECreep compliance

For creep recovery, = , 0 m mdEdt

Model III – Burger’s Model

E2, 22, 2

002( ) 1 t

For creep, = 0:

= m/Em

Dynamical Mechanics Analysis

0 sin t

= (d /dt)= 0/ sin( t- /2)

0 0 sinW t d t

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

' cosE 0

" sinE

E”"'E EE

' " "'*E EE iEE E

'an "t

* iE e

Complex Viscosity: As “Liquid”

0 0sin cos s ncos it tt0 sin t

0 cosd tdt

synchronizationasynchronism

Viscosity ofsynchronization part

Viscosity ofasynchronism part

' sin0

' cosE 0

" sinE

'" E "' E

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

d t dt i e dtE

2 2 2 2*1 1

m mt E EE it

= m/Em

k=2k=1 k=-1

Model II - Voigt-Kelvin model 0i te

Complex compliance 2 2 2 2

1 1** 1 1m m

D iE E E

d ddt E dt

m mdEdt

internal friction

single

polymer

tG t G e

G t G e

General Maxwell Model

iE t E e

tE t f e d

For stress relaxation relaxation time spectrum ( )

Mw: III>II>>I

/ lntE t H e d

single

polymer

Viscosity & Relaxation Modulus

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*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

i ii i i i

D iE E

General Maxwell Model

General Voigt Model

For creep recovery /2 2 lntD t L e d

Viscosity & Modulus & Relaxation Time

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.

Characteristic of polymer viscous flow

Bingham

Pseudoplastic

Dilatant

Newtonian

Shear thinning ( )

Time-dependent shear stress and primary normal stressdifference after start-up of steady shearing

Relaxation of shear stress and primary normal stressdifference after cessation of steady-state shearing

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.)

Polymer Processing

Characterization of viscous flow

a. ( )-shear viscosity s b. ( )-tensile viscosity t

c. ( )-bulk viscosity

for incompresible fluids b

velocity gradient2

Melt viscosities of polymers

apparent viscosity

/advdh

differential viscosityd

shear viscosity

dvdh c

complex viscosity ( )

0 sin t

sin / 2t

1-D2-D

extensional viscosity

low molecular weight

Measurement of shear viscosityCone-plate viscometer ( )- an example

tanh r rhdv r

As liquids: The flow curve of polymer melts

entanglementdisentanglementorientation

turbulence

0 dependent of M

W cMM M

Critical molecular weight at theentanglement ( ) limit: Mc

Theory1

0 ~ Rouse ModelM

3.3~3.40 ~

W cM M M

30 ~ Tube (Reptation) ModelM

As Solids: Time vs. Frequency vs. Temp at Low Deformations

0 1~Ne

1.2.3.

2. -Vogel-Fuchler0exp /A B T T

3.Brigid>Bflexible

Viscosity & Modulus in Polymer Melts

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.

2 iN-1

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

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

33 exp exp2 2

b b k Trr

3 / 20

3 1exp2

3 exp2

ub k T

b b Gaussian Segment

32n B n nu k Tb

n B n nn

U k Tb

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

U RdR k T k T R R R fdt b R b

2 i N-1

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

f: Random force of Brownian Motion

Normal Coordinates of Rouse Model

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

dR k R R fdt

dR k R R R fdt

k R R R

dR k R R fdt

1 12i i i ii k R R R

23 Bk Tk

b P156 of Chapt 1-2

Applications of Gaussian Chain Model: Stretching of an Ideal Chain

3 3, exp2 2

NN l N l

3 3, , exp2 2g g g g

N N N NN l N l

3 3 3( , ) ln( ) ln2 2 2g B B B g

S N k k k NN l N lhh

23( , ) Bg

k TN l

3( , ) ( )2g B g

G N U TS k T G NN lhh

, , /g g gN N Nh h

lnS k h

ln / 'S k

X TQ1TZT

1dT k T T T Tfdt

Q Z Q I

3 Bk Tddt bX X

0 expp pp

tX t X2

24sin2ppN

1 23 B

N bk T

Terminal relaxation time2

23 4sin2

bpk TN

30 exp Bk Tt tb

N bk T p 1 2

3n 3n 3n,

The Zimm Model

nnmH m

Rouse model

1 ˆ ˆ Zimm model8

nm nmnm

Oseen Tensor

Rouse Model Zimm Model

nmH m bead n bead

n m n mm

v r H r r F r

Oseen Tensor

The Momentum Equation of Fluids – Navier-Stokes Equation

Stokes Approximation:2

0Pv F r

0i P F

ik k k

k v kk v

In Fourier Space:

1 ˆ ˆvk k kI kk F H k Fk

Doi, Edwards, The Theory of Polymer Dynamics, p89

2 0k kk v F

' ' 'dv r r H r r F r

1 1 1ˆ ˆ ˆˆ82

k rH r k I kk I rrk

ˆ ˆT I kk

P Ptv F r v F r

22i ik k

Estimating The Longest Relaxation Times

f v Bk TDFriction coefficient Einstein Relation

1/22 1/20 6t Dtr r

Stokes Law 6 s hRR=l

206R D

0 6 6 B

R lD k T

Brownian Motion

Stokes-Einstein Relation6

k TDR 6

Rouse Model:

k TD R N

gR N l

Zimm Model:

Z s gRBZ

l Nk T

Relaxation times of Rouse-Zimm Model

1,Rouse

N b pk Tp

Rouse model

1,zimm

N b pk Tp

Zimm model ingood or solvent

1,Rouse 2

N bk T

Terminal relaxation time3 3

1,zimmB

N bk T

3 / 23 9 / 5

3/2~ M9/5~ M

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

( ) exp( )

exp 2 /

tG t nk T

c k T dp tpN

c k TtN

G nk T

1,2, , .p p

Relaxation time for the pth mode:For N >> 1

Stress relaxation modulus andcomplex modulus (Maxwell-element model):

Prediction of Viscoelasticity by Rouse-Zimm ModelRouse: Zimm:

G nk T 2 21

G nk T

1<<12 2 2

'( ) ~p

G p 11

"( ) ~p

1>>12 2

1 1/1/

1/ 1/1

~2 sin / 2

1/ 1/1

~2 cos / 2

1 1: '( ) "( )G G 1 terminal relaxation time3.

( =2 or 11/5) ( =3/2, solvents) or ( =9/5 , good solvents)

Dynamic Modulus of Polymer in Dilute Solution

Rouse ( =2) Zimm ( =3/2, solvents) Experiments

2'( ) ~G"( ) ~G

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

k=1 k=-1E”

Linearsuperpose ofMultipleRelaxationTimeSpectrum

Single RelaxationTime

Maxwell Model Rouse-Zimm Model

MultipleRelaxation Time

Relaxation Modulus and Dynamic Modulus of Rouse Model

( ) exp( )

exp 2 /

tG t nk T

c k T dp tpN

c k TtN

(t< R)

( ) exp /RB R

cG t k T tN t (t>> R)

2a xe dx a

aNote:

exp 2 /

c dt tpNk T

c pNk Tc Nb M (M<Mc)

RouseModel

ZimmModel

G”~ 1

G’~ 2

G”~ 1

G’~ 2

' co sG 0

" sinG

Dynamics Modulus of Polymer Melts (M>Mc)

0 1~Ne

Plateaumodulus

???G”

Relaxation Modulus of Polymer Melts

Relaxation Modulus vs Dynamic Modulus in Melts

Tube & Reptation Model for Entanglement

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.

2 /pr prL Nb a

1/ 22 2 20n n B e prt k Tb aR R4

entanglement time

Curvilinear diffusion coefficient D:

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,

Relation of entanglement andreptation model

disentanglement time

Time correlation function of End-to-end Vector

0t A C CD DB AC CD DB

CD a t

3 4 22 2

1 3/ bd R

N b NbL Dk Ta a

exp / dt t p t

Doi, M., Edward, S. F., The Theory of Polymer Dynamics, Oxford, 1986, p.194

Comparison of relaxation times

a Mk Tb

3 ~d RNb Ma

1,Rouse 2 ~3 B

N b Mk T

Effects of Entanglement on Relaxation ModulusDynamically ShearStep Shear

c k TN

cb k TacbN b M

0NG t G

BcG t k T

Viscosity

exp 2 /

c dt tpNk T

c pNk Tc Nb M

Rouse Tube

cM McM M

Note, ~ ~ Mwith 3.3 – 3.4for molecularweight higherthan Mc.

N bk T

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

Normal stress difference and Elastic effects onviscous flow of polymers

F/A= s= 21

N1= 11- 22>0

N2= 22- 33<0

Weissenberg effect

For polymer melts

For lowmolecularweight liquids

N1= 11- 22=0

N2= 22- 33=0

Rod-climbing

Extrudate swell

tubeless siphon ( )

toroidal eddy ( )

Instability in Processing

Gross Melt Fracture

Sharkskin

Stick-slip

Stick-Slip Transition

cSample is oriented

Stick-Slip Transition

2 /sV b

c enF 1/2B B

epr e e

k T k TFa N l

Flow instability and melt fracture

Shark skin

melt fracture

Terminal flow< ~ >~

Linear and Cross-link

Semi-crystalline orcrystalline Tm<Tf

Crystalline Tm>Tf

( ) ( )

( - ) ( - )

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)

liquid

Vcrystal

glass 2

glass 1

1: fast cooling2: slow cooling

supercooledliquid

TTmTg1Tg2

(1) -(2) -DSC(3)a. -b. -DMA

(4) -NMR,

Tg0 sin t

0 sin tTTg

Polymer Melts and Glasses

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

The transition from melt to glass iscalled glass transition ( )

Tg: glass transition temperature ( ) 88

Why Glass Transition belongs to Segment Relaxation?

secondary eg

secondary g elenglength l hth engt

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.

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 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

11 )( ggfgT TTTTff

f TT g

Vf T T

dVV TdT

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

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

( ) 1 1log( ) 2.303g T g

T BT f f

17.442.303 g

f %5.2025.0gf

Kf /108.4 4

17.44( )(51.

g( ) ( )

T TTT T T

)(/)(lnln 0 TVTBVAT f

)(/)(lnln 0 gfgg TVTBVAT )()(

)()(ln 00

fg TVTV

Normalized free volume:

WLF equation:

Nearlyequal to 1

2.303 / f

T TBTff T

( )log( )g

Appendix: Doolittle equation & Einstein equation

fVBVA /exp 0

Doolittle equation

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

In glass or melt 0fV V

fVBVA /exp 0

0 / fV V

Applications of WLF Eq.

lg lg sT

C T Ta

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.

tTE t E e

10 1ln ln / ln /TE E t t20 2ln ln / ln /TE E t t

+ln( 1/ 2)=lnaT

-ln( 1/ 2)=-lnaT

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

Synthesized master-curve

lg lg sT

C T Ta

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.

(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

Stress relaxation

G s ds0

( ) ( )dt G s dsdIn steady shear: t s

Viscosity & Relaxation Modulus

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

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

ts<< 1 tf<<< 1T1

Tg1 Tg2

Tg1 ts ~ g1 tf < g1

Tg2 tf ~ g2

et v t t

0g bT T101

Effects of film thickness on Tg

Tsui, OKC, Macromolecules, 34, 5535 (2001)102

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.

11ggg T

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.)

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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