chapter 6 condensed states of polymersjpkc.fudan.edu.cn/_upload/article/files/76/8c/96a7...6.2...
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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