Chapter 6 Condensed States of Polymers

6.1 Polymer Melts and Glasses6.2 Crystalline States of Polymers

6.3 Oriented States of Polymers

6.2.1 Chain Conformations in Crystals

6.2.2 Basic Morphology of Polymer Crystals

6.2.3 Polymer Crystallization

Models Thermodynamics Kinetics

6.4 Liquid Crystalline Polymers1

6.1 Polymer Melts and Glasses

(liquid)

liquid

glasses

(glasses)

slowly cooledfast cooled

Tg fastslow Tg

G

T

The transition from melt to glass is called glass transition ( )

Tg: glass transition temperature ( )

2

Tm: Melting point

Tc<Tm: crystallization temperature

The transition from melt to solid(crystal) is called solidification(crystallization)

T=Tm-Tc: undercooling

Glass Transition as a Relaxation Process

Thermal history dependence of Tg

liquid

V

crystal

glass 2

glass 1

1: fast cooling2: slow cooling

supercooledliquid

TTmTg1Tg2

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) 3

6.2 Polymer CrystallizationRequisites for polymer crystallization

Chemical regularityHomopolymerCopolymer

block copolymerrandom copolymer

Stereoregularity ( )Isotactic ( )Syndiotactic ( )Atactic ( )

4

6.2.1 Chain Conformations in Crystals

21 31 41

61525142= 21

62= 31 71 72 73

92918381

32 31

H31

Helix Conformations

Zig-zag Conformations

H21

Hydrogen bonding

syndiotactic

isotactic3 2 3CHCH CH CHCHrepeating unit

5

Why Helix ?

H H

0.25 nm

PE

0.12 nm

PP CH3: 0.2 nm 6

E

r

0.2nm

Chain packing in crystal lattices

LH RH

LHLH

LH

LH

RH RH

RH

7

Chain packing in crystal lattices

PE

orthorhombic8

triclinic

Hydrogen bonding

9

Trigonal

i-Poly-1-butene

Monoclinic10

6.2.2 Basic Morphology of Polymer Crystals

Crystal Habits:a: lamellar ( )b: platyc: tabulard: isometrice: prismaticf: acicularg: needle-likeh: fibrous ( )

“Single crystals” have one lattice

Macromolecular crystals are often lamellar or fibrous.

“Polycrystalline samples” are aggregates of many single crystals

11

(1) Lamellae ( )Crystallized from Dilute Solution

Polyethylene ( )

Tc ~ 70 CTc ~ 80 C

12

Orientations of polymer chains in lamellae

PE chains must be folded back andforth within the lamella!

Note: the lamellar thickness is ~10 nm,

PE chains are perpendicular to the lamellarsurface.

13

the chain length is > 1000 nm.

Polyoxymethylene ( )PEO( )-b-PS( )diblock copolymer

14

The Structure of Lamellae

15

Melting and Stretching of lamellae

16Melting Stretching

(2) Lamellae Crystallized from Melt

Spherulites ( ) of polymers

Under Polarized optical microscopy

17

Banded spherulites ( )

18

Lamellae in a spherulite

1. Lamellae2. Tie Molecules3. Amorphous

19

intercrystalline links in spherulites

20

21

Growth of spherulites

22heterogeneously nucleatedhomogenously nucleated

The mechanism of Maltese cross extinction pattern

QE

OP: Polarizer

OA: Analyzer

QE: Vector of polarized light

I0

QR

QT

projectingon OA

OPQM

QN

2QNQM I

23

Theoretical analysis of Maltese cross extinction pattern

0QE i tE e

0 0

2 22 2 2 2 2//sin 2 siQ n ~ sin 2 / 4

2QM N E E n nI

0QR sin i tE e

0QT cos i tE e

projectingon OA

0QM sin cos i tE e

0QN cos sin i tE e

0 0

20 0

sin 2sin cos2

sin 2 2sin 2sin cos sin 2 sin sin cos2

QNQM cos si

2 2 2 2 2

1

2

n1ii t i t

i t i t

E e E e

E e

e

e

i

i E i

////

//

positive spherulite~

negative spheruliten nn nn n

24

Relations of POM and SALS (small angle lightscattering) on spherulites

M

/ / n E E nn E n nnM2

~I M OA

POM

SALS

2

2

22

0

22

/ /

/ /

2

/ /

/ /0

cos cos2

sin 2 / 4

n E n n OA

n

n

n

n E n n OA

E n n OA

En

nE

n

~ diS M OA e q rq r

*I q S q S q

M

25

SALS patterns of spherulites: Stein Formula

Vv

Hv qmax~1/R

Positive spherulites negative spherulites

2 22 2

Hv 0 3

3 cos sin cos 4sin cos 3Si2r tI AV q q q q

q

22 32

Vv 0 Hv30

3 1 cos2sin cos Si Si sin3 sint s r sqI AV q q q q q q I

q AV

4 sin2

Rq0

sinSi dU xq x

x

26

27

Foldedchain

Extendedchain

(3) Formation of shish-kebab crystallite ( ) undershear flow

28

6.2.3 Polymer Crystallization

How do the polymer chains pack in the lamellar crystals?

Random coil with the chain contourlength ( ) longer than 1 m

Lamella with the thickness of 5-50 nm and lateral size of microns.

Crystallization

29

Macroconformations ( ):Extended chains

( )

Folded chains( )

Random coils( )

Fringed micelle( )

6.2.3.1 Models

30

50K

300K

Yamamoto, T., Adv in PolymSci, 191, 37(2005)

31

32

Chain-Folded Lamellae of Polymers

Chain-Folded Lamellae of Polymers

Random coil

Crystallization

TC or T

Chain folding ( ) concept

Chain-folded Lamellae

e

Folded chain ( )

Fold surface( )

Lateral surface( )

TC: crystallization temperature ( )T: supercooling ( ), T = Tm

0 - TC

e: fold surface free energy ( ): lateral surface free energy ( ), e 10

l

Lamellar thicknessor fold length (

)

33

6.2.3.2 Process of Polymer CrystallizationTwo steps: (1) Nucleation & (2) Linear Growth

(1) Nucleation ( ) process

Schematic representation of thechange in free energy as afunction of size illustrating thenucleation process

Homogeneous ( ) and heterogeneous ( ) nucleation34

Phase Separation MechanismsNucleation and growth ( ) mechanism of Phase Separation

* **

In metastableregion, separationcan proceed onlyby overcomingthe barrier with alarge fluctuationin composition.

Nucleation Growth

23 43

4)( rgrrG )''()( 0 gggNucleation barrier: with

r: radius of the nuclear; : excess free energy per unit surface area.

droplet-dispersedphase

35

(1) Nucleation ( ) process of Crystallization

G = H – T S, G = Gcrystal – Gmelt

a b cTypes of crystal nuclei. (a) primary ( ), (b) secondary ( ),(c) tertiary ( ) nucleus.

36

Gcrystal = Gbulk + A

G = Gbulk - Gmelt + A = Gf + A

A is the suface area and Gf is thebulk free energy change.

Classic Nucleation Theory - Estimate thecritical nucleus size

* 4 4e e m

f f

Tlg h T

2 24 2c f c eG T a l g T al a

0f m f m fg T h T s

For primary nucleus

For secondary nucleus

* 2 2e e m

f f

Tlg h T

2 4 4 0f c eG al g T l aa

2 4 0f cG a g T al

4*f c

a g T

4* e

f cl g T

2 2f eG abl g bl ab

22* m

f f

Tag h T

4* m

f

Tah T

0Ga

0Gl

f c f c fg T h T s

37

ff

m

hs

T

fm

ThT

* 4 e mII

f

b TGh T

*expIIB

Gv kk T

Chain sliding diffusion model of primary nucleation

Hikosaka, M., Adv in Polym Sci, 151, 137 (2005) 38

(2) Linear Growth of Polymer Crystallization

Temperature dependence of lineargrowth rate ( )

Temperature dependence of the radial growthrate u of spherulites in isotactic polystyrene(left), polyamid 6 (center) and poly(tetramethl-p-silpheylene siloxane) (right). Tf : equilibriummelting temperature.

Surface nucleation on substratewith length L with a rate iLinear growth rate G: the growthrate of crystal perpendicular tothe substrate

l: fold length; a: width of stem; b:thickness of the stem; g: substratecompletion rate

39

Crystal Growth Rate of PEO in melt

40

A.J. Kovacs, C. Straupe, and A. Gonthier. J. Polym. Sci., Polym.Symp. Ed., 59:31, 1977

Regime I, II & III of Linear Growth

41

(3) Microscopic model - Lauritzen-Hoffman (LH) Theory

42

(1) Energy barrier for secondary nucleation

(1)(2)(3)

(3) Microscopic model - Lauritzen-Hoffman (LH) Theory

Regime III:,

Hoffman, J. D., Polymer, 38(13), 3151(1997) 43

(2) Linear growth rate G /i flux L

Experimental Evidence of regimes I, II, and III

44

2g I g III g IIK K K

*

exp exp gD

B B

KC QGn k T k T T

, ,I II III

04 e mg I

f

b TKh

02 e mg II

f

b TKh

*DQ

gK Regime

p.38

* 41exp exp e m

B B f

b TGik T k T h T

6.2.3.3 Overall Polymer Crystallization Rate

Overall crystallizationrate ( )

Tm0Tg TMAX

Overallrate

‘Bell’ curve ofoverallcrystallizationrate

Total volume or density: or

Crystallinity: or

1 (1 ) or 1

or

a ca a c c

a c

c cc c

c cc c

a ca c

c c a a

c a a c

W WW V V V

W Vw vW V

w w v v

v vwv v

usually 0.7 0.2

c a

c a

c

v

w

wc: weight fraction; vc: volume fractionW: weights

: density (g/cm3): specific volume (cm3/g) 1/

Definition of crystallinity ( )wc Wc/Wtotal; vc Vc/Vtotal

Time

Crystallinity

Isothermal crystallization

T1

T2

T3

t1/2

IIIIIIIV

45

Overall Crystallization Kinetics

h0

h

0

0 0

1 1t c ath h v vh hh h

h h

1/2

tt1/2

1/ 2

ln 2nk

t

lnt

0

ln ln tv vv vht

ht’

1/t t tv hDilatometric ( )

46

Avrami equation

0

nkttv v ev v

0

ln ln ln ln 1

ln ln

ctv v vv v

n t k

t

47

(1) Phenomenological models ( ) of overallcrystallization - based on classic nucleation theory

(1) Free growth model

R~vt

3crystal

total

43

ic V Vv N vt

V V

b. new nucleus generated - homogeneous

a. constant numbers of nucleus- heterogeneous

3*

0

43

ct

I v tv

t3*

34I tt v I* constant nucleation rate

3*0 4 03

It v t

t

48

3* * 3 4

0

4 d / 33

tI v t I v t

* ?I3*

3 + 4+ I v t

3*... 43 iI v t

Poisson’s ( ) raindrop problemIf raindrops fall randomly on a pond creatingexpanding circular waves, what is the chance thatthe number of waves created by differentraindrops which pass over a representative pointP up to time t is exactly n?

(2) Phenomenological models of overall crystallization -based on classic nucleation theory –Avrami equation(2) Impingement model

P

solution : / !E nnP t e E n Poisson distribution

What is the chance that no wavereaches a given point P in space ? 1 cv

0 1E cP e v iV tE t

V

heterogeneous

homogeneous

34 /31 N vtcv e

* 3 4 /31 c I v tv e

The Avramiequation for 3D

E(t): wavespreadingarea during t

While t or v is very smallcv E t1xe x * 3 4 / 3I v t

343

N vtor

49

50

(1) Nucleation and Growth – First-order phase transition

isotropic anisotropic

(2) Spinodal Decomposition – Second-order phase transition

?Two Phase Equilibrium

Phase Transition:

continuous phase transition

(3) Phenomenological models of overall crystallization -based on spinodal decomposition ?

Kaji, K., Adv in Polym Sci, 191, 185 (2005)51

Olmsted PD, Phys Rev Lett, 81:373–376 (1998)

52

6.2.4 Thermodynamics of Crystallization:

53

1. Melting point of Lamellar Crystals

54

(1). Effects of Chain Structure on Tm

mm

m

HTS

1) Symmetry andAsymmetry

2) Flexible and Rigid

3) Molecular Interaction 55

0m m m mG H T S

Thermodynamics of Crystallization:((2). Melting point of Lamellar Crystals

0 0 0f m f m fg T h T s

0mT

0f

fm

hs

T

0 21 em mT T

l hMelting data for lamellar polyethylenegrown from the melt and from solution

Tm = 414.2(1-0.627/l)

Lauritzen-Hoffman equation

0

0

2 e m

m m

Tlh T T

Similar to critical nucleus size

Gibbs-Thomson extrapolation

2 22 4f eG a l g a al

0G

mTl

a

0

0m m

f m f m f fm

T Tg T h T s hT

56

4 0al

Typical molecular process of thickeningin the thin lamella

57

Thermodynamics of Crystallization:(3). Melting point of Polymer mixture

Chemical Potential of crystallizable polymer in amorphous phase20

2 2 2 1 2ln 1 1RT x x

Consider a system of Polymer/Impurity mixture:

Chemical Potential of one monomer unit of polymer in amorphous phase

Polymer is compatible with impurity but can’t co-crystallize with impurity

0 0u

Cuu u

I

20 21 2

ln 11 1u u RTx x 58

Effects of “Impurities” on Tm

Condition of Phase Equilibrium

0 0u

I Cu uu

Superscript of C: Crystalline Phase

0u

C

uu

u u

m

GH T S

0

0u

m

m

m

m

uHS

T

HTS

0

221 2

ln 11 1

Iu u

RTx x

221 20

ln 11 11 mu

m

RTTHx xT

022

1 2ln 11 11 1

m m uT T HR

x x

Superscript of I: Amorphous Phase

01 mu

m

THT

59

Effects of “Impurities” on Tm

221 20

ln1 1 11 1m m u

RT T H x x

For x2 , x1=1

21 10

1 1

m m u

RT T H

For end-group effects: 1=2/N, =0

0

1 1 2

m m u

RT T H N

2 2 21 1 1 1 2

1 1 ln2 2u u

R RH H

21 1 1 2

1 ln 1 ln2Note:

221 20

2 1 2

ln1 1 1 1 1m m u

RT T H x x x

polymer/small molecule polymer/polymer

For polymer blends

61

Tm>Tc Tm<Tc

220

1 1 1m m u

RT T H

61

Crystallization vs Phase Separation ?

Phase Boundary ofLiquid-Liquid Phase

Phase Diagram of Crystal / Amorphous Polymer Mixture

A BT

2

c 1/2 1/21 2

1 1 12 x x

cc

A BT

Tm

Liquid + Solid Phase Equilibrium

Typical Phase Diagram of Mixture III:Liquid-Solid & Solid-Solid

62

& Liquid-Liquid

There may exit Solid Phase in Polymer System

For copolymer

63

0

1 1 lnm m u

R PT T H

alternating copolymer: P<<XA

Random copolmer: P=XA

Block Copolymer: P>>XA 1

P: Probability of crystallineunit with sequential connection

Crystallization of PEO-b-PS in Lamellae Crystallization of PEO-b-PS in Columnar

Quasicrystal?

64

D. Shechtman, Nobel Prize 2011

Roge Penrose

6.3 Oriented States of PolymersDefinition

21 3 cos 12

F

<cos > <cos2 >fully oriented

fully disorder

1

0perpendicular 0

1

1/30

F1

0-1/2

: 0~180°

65

one or two dimensional order

ApplicationsFiber, BOPP, BOPET, BOPE, …

Orientation function

iPP

66

67

(MPa) (g/cm3) (Mpa) (GPa) ( )iPP 230 0.94 244 4.1 0.9%

iPP 30-40 0.94 32-42 1.5-2 >50%400-800 7.8 51-102 ~200 ~100%

6.4 Liquid Crystalline Polymers

“Liquid crystals stand between the isotropic liquid phase and thestrongly organized solid state. Life stands between completedisorder, which is death, and complete rigidity, which is deathagain.”

Dervichian D. G. Mol. Cryst. Liq. Cryst. 1977, 40, 19.

States of matter:Solids, liquids, and gases“Liquid crystals” (LCs) represent a number of different states ofmatter in which the degree of molecular order lies intermediatebetween the perfect long-range positional and orientational orderfound in crystalline solid and the statistical long-range disorderfound in an isotropic liquid. Phenomenologically, LCs exhibit bothsolid-like anisotropic features and liquid-like fluidity. On the basisof these characteristics, the term “mesomorphic phases” or“mesophases”, may be a more appropriate name than liquidcrystals.

68

1961 1968, 1971 , 1980

Pierre-Gilles De Gennes

Pierre-Gilles De Gennes" "

199169

Pierre Gilles de Gennes (1932-2007.5.20)

What are Liquid CrystalsAnisotropic molecular shape of liquid crystals

Rod-like or ellipsoid-like

Space filling modelof molecule 7S5

Plate-like or disk-like

Lyotropic liquid crystals ( ):The liquid crystal phase is dependent on the concentration of onecomponent in another.

Thermotropic liquid crystals ( ):The liquid crystal phases of pure substance are caused bytemperature change.

70

Nematic Phase ( )

Director N

Long-range orientationalorder of molecules

Molecules

N

n n

n n

n n

n

Molecular arrangement in a nematic phase

Polydomain structure in a nematic phase.Local directors are represented by n, and theglobal director is represented by N.

71

Smectic Phases ( )

Director N

Layers

Molecules

Layer structure in smectic phases

Smectic A Smectic C

Tilt Angle

Director N

Layers

Molecules

72

Liquid Crystal Phase of Chiral Molecules

Structure of chiral smectic or smectic C* phase.The planes represent the smectic layers. Thedirector always makes the same angle with thesmectic planes, but the orientation of thedirector rotates about the line perpendicular tothe planes in going from one layer to the next.

Cholesteric phase ( )(chiral ( ) nematic phase)

Chiral smectic phase

Molecules in the cholesteric phase. Thedirector n rotates in a helical fashion. Becauseno physical quantities depend on the sign of n,the physical pitch of the cholesteric phase is P= /k0 rather than 2 /k0.

Cholesterol nonanoatemolecule:

73

Lyotropic liquid crystals:1) surfactants or amphiphilic block copolymers

74

Lyotropic liquid crystals:2) rigid-rod polymers

Cc2*~M-1

c1* c2

*

c1*~M-2 75

POM Used in the Study of LC PhasesFor different LC phases, the textures ( )under POM are different

Nematic: schlieren ( )texture

Smectic A: focal-conic fan( ) texture

Semctic C: schlieren texture

76

POM Used in the Study of LC Phases

Cholesteric (fingerprint) texture

77

Liquid Crystalline Polymers

hybrid: combination ofmain-chain and side-chain:

Common architectures for liquid crystalline polymers (LCPs):some examplesmain-chain rigid-rod: lyotropic

main-chain with flexible spacers:

side-chain with terminal attachment:

side-chain with lateral attachment:

mesogen-jacketed LCP

Applications:Ultra-high-strength fibers: Kevlar®, Xydar®, Vectra®, Ultrax®

membrane,Electro-optic (low molecular weight thermotropic LCs)……78

Thermodynamics of thermotropic liquid crystal transition

G-T diagram:Gibbs free energy as function oftemperature

G

T

LN

S

TS N TN I

21 3 cos 12

S

Order parameter( )

<S>

T/TNI

79

Maier-Saupe Theory for LC Transition

21cos 3cos 12i iS

cosij b ijU r SInteraction Potential

cosi ib SU S

/

/

cos sin

sin

cos exp cos sin

exp cos sin

i

i

b

b

U kTi i i

U kTi i

S e dS

e d

S S dk

Td

ST

kSS

Self-consistence method

Mean-fieldApproximation

b/kTNI=4.541

21cos 3cos 12ij ijS

cosi b ijj

U r Sij

i

80

Phase Transition Temp.

<S>

T/TNI

b/kTNI=4.541 TNI=4.541 b/k

81