1

2

1.

2.

4.

5.

( )

3.

Some Suggested Topics (I)

3

Electrophoretic actuatorsUltra strong polymers for ballistic protectionDendritic polymers for drug deliveryConjugated Polymer SensorsBirefringence as a Measure of Chain OrientationPolymeric Coatings on Optical FibersPlastic Contact LensesHydrogelsSelf Assembled Photonics3d Interference Lithography – positive and negative resists2D Lithographic Masks via Self Assembled BCPs - Flash MemoryMorphology of Immiscible BlendsIonomer ClustersBlock Copolymers

noncrystallineliquid crystallinecrystalline

Some Suggested Topics (II)

4

Block Copolymer-Homopolymer BlendsDetermination of Polymer Surface EnergySingle Walled Carbon Nanotubes – an example of a class ofpolymersAdhesion of PolymersOptical Properties of Liquid Crystalline PolymersPolymer WaveguidesPolymers for Optical StoragePhotoelastic Determination of StressesCrystalline PolymersFibersBiodegradable FibersPolyelectrolytesGelsPolyeletrolyte GelsDefects in MesophasesRole of Defects in Controlling Properties

Some Suggested Topics (III)

5

Additives to Polymers and Sequestration of Nanoparticles in BlendsTargeted Location and Orientation of Nanoparticles in Block CopolymersPolymerization Induced Phase Separation (PIPS)Orientation Methods for controlling polymer structure

Flow FieldsElectric, Magnetic FieldsSubstrates

Epitaxy in PolymersThin Film Patterning - terracing and dewettingPolymer FoamsSegmented copolymers (polyurethanes, polyetheresters)Inorganic and metal-containing polymers

Some Suggested Topics (IV)

6

Techniques applied to Polymer MorphologyScattering

Light scatteringWide angle X-ray and/or neutron scattering

Small angle X-ray and/or neutron scatteringElectron diffraction

Microscopy TechniquesTEMSEMAFM

Thermal AnalysisDSCDMA

,,

, ;,

——Lord Kelvin( )7

“When you can measure what you are talking aboutand express it in numbers you know sth. about it,but when you cannot, your knowledge is of ameagre and unsatisfactory kind: it may be thebeginning of knowledge, but you have scarcely inyour thoughts advanced to the state of Science.”

8

9

1.

…“I am inclined to think that the development ofpolymerization is, perhaps, the biggest thing chemistryhas done, where it has had the biggest effect on everydaylife. The world would be a totally different place withoutartificial fibers ( ), plastics ( ), elastomers ( ),etc. Even in the field of electronics, what would you dowithout insulation? And there you come back to polymersagain.”

Lord Todd, president of the Royal Society ofLondon, quoted in Chem. Eng. News 1980,58(40), 29, in answer to the question, What doyou think has been chemistry’s biggestcontribution to science, to society?

10

11

1.

——

12

13

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

20000

1940 1960 1980 1998

2

1/7~1/8

20 20

14

15

20118161 7400

40170

(PE PP PVC PS ABS)90%

160

200

400

600

800

1000

1200

1996 1997 1998 1999 2000 2001 2002 2003 2010 0

2 0 0

4 0 0

6 0 0

8 0 0

1 0 0 0

1 2 0 0

1 4 0 0

1 6 0 0

1 8 0 0

1 9 9 6 1 9 9 7 1 9 9 8 1 9 9 9 2 0 0 0 2 0 0 1 2 0 0 2 2 0 0 3 2 0 1 0

PE PP 20121.5 . 60% 50%

2010 3100 1950 (62%),2000 50%;

60% .

PP PE

17

604%

500

GDP 4%

Why Polymers/Plastics?

18

Why Polymers/Plastics?

19

20

Carbon Fibers & Composite MaterialsPAN

Composite Materials in Commercial Aircrafts

21

20 30

C/C

Composite Materials in Fighters

22

60 80

40 60

Milestones of Polymer Materials

23

)

–

Staudinger 1920-1930

(1).Kuhn, Mark 1930-1935

24

2.

(2). DNA 1935-1965Flory Rouse-ZimmWatson-Crick d-Helix DNA, Natta

25

(3). 1965-de Gennes Edwards (reptation)

, (scaling concept)

- (soft condensed matter)(soft matter) (complex fluids)26

27

3 Unique Features of Polymers

N>>1

N~102-104

N~109-1010 DNA RNA Protein,…

(1) Large Spatial Extent

Polydispersity

28

The Mean Size of a Polymer Chain

29

~R h CN l

Freely Jointed Chainand Kuhn Segment le

2 2

0 0/ 6 / 6eR h L l

Unperturbed Chain:

=0.5 Ideal Chain Random Walk Chain=0.6 Real Chain Self-avoiding Chain

12 2 2

1 1

2N N

i ji j i

h Nl Nlr r

Root Mean Square of end-to-end Distance

L: Contour Length of a Chain

2 1/2h h N l

Radius of gyration

DNA, RNA & Virus

30

DNA leaking from bacteriophages

31

DNAle = 50 nmL = 16,500 nm (330 le )

Rg = 380 nm

Rcapsid = 28 nm

g 6eL lR

T2 phage osmotically shocked by saltKleinschmidt, A. 1962

DNA Condensation (DNA )

32

a. Tacticity –

b. Polymer Topology – Homo, Hetero, Copolymers

(2) Connectivity

33

c. Flexible vs. Rigid Polymers

34

Static Flexibility

Persistence Lengthlp=l exp( /kT)

// kTt p e

Dynamical Flexibility

1

pt

e

E

Persistence Timep= exp( e/kT)

/e kTr e11~ time ~ 10 ps

d. Multiple Configurations(Confirmations)

35

lnS k4 N

ln / 'S k

Total number of chain conformations

Entropy of Chain Conformations

l

'The number of chain conformations in a state

The Probability DistributionFunction of Conformations (R)

36

32 2expR R

h1, (R1)

…

h2, (R2) h3, (R3)

22

32 e eN l

ln / 'S k

(3) Multiple InteractionsVan der Waals, Hydrogen bond, Electrostatic

Multi-scale ….

37Glass, Gel

”

Condensed Matters

38Crystal Liquid Crystal

Blend

Triblock CopolymerHIPS & ABS Amorphous & Melt

39

(4) Uncrossibility, Entanglement

3.30 M

0NG

(5) Responsive Molecules - Viscoelasticity

40

a. Large-scale Relaxation Time Spectrumln

E(

)

ln

E’

k=2

k=1 k=-1E”

b. Temp, Rate and Time Dependent Behavior

b. Temp, Rate and Time Dependent BehaviorAn Example: Silly Putty

41

2 Unique Features of Polymers

42

(1) Large Spatial Extent

(2) Connectivitya. Tacticity –

b. Polymer Topology

c. Flexible vs. Rigid

d. Multiple Confirmations (Entropy)

(3) Multiple Interactions (Enthalpy)

(4) Entanglement

Thermodynamics:

Dynamics:

(5) Responsive Moleculesa. Large-scale Relaxation Time Spectrum

b &c . Temp, Rate and Time Dependent Behavior

43

Polymers are Complex yet Simple

Large number of degrees of freedomand multiple confirmations (Entropy)

Multiplicity of interactions (Enthalpy)

Allows averaging – Mean-field Theory

Universal properties – Scaling behavior

Osmotic, Viscosity, Viscoelasticity …

Shortcomings of Polymer Materials

Mechanical PropertiesModulus, Strength, Hardness, Creep, ……Surface PropertiesElectrical & Optical PropertiesAging & Stability……

44

Open Discussion

45

(1).

46

(2).

47

(3).

48

37%

21%

16%

1%

3%

13%

9%

(3).

49

(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%

iPP

50

51

2 714 3

314 4782969 4.8 106

10 4.8 105 131050 (~3.8 )

52

53

(Monomer) (Segment) (Blob) (Chain)

( )

( )

( )

WLF Eq. Rouse-Zimm Model

Tube Model (entanglement & disentanglement time)

))

terminal relaxation timeMaster ( ) RelaxationSecondary ( ) Relaxation

( (4)) 2008( )

( , )Introduction to Polymer Physics (Oxford,1995, M. Doi)Polymer Chemistry (2nd, CRC Press, 2007, Hiemenz,P. C., Lodge, T. P.)Polymer Physics (Cambridge, 2003, Rubinstein)The Physics of Polymers (3nd) (Springer, 2007, G. R.Strobl)Introduction to Physical Polymer Science (Wiley,2006, L. Sperling)Principles of Polymer Chemistry (Flory)

( Mark, Tobolsky et. al.)

54

Primary( ):

Secondary( ):

Tertiary( ):

Quaternary( ):

55

2.1 (local)

2.1.1 (Constitutions or Architectures)

-CH2-CH2-

2.2.1.1

2.2.1.2

56

57

2.2.1.3

Branched polymer Cross-linked polymer

58

Star ( ) polymers

homo-armed( )

hetro-armed( )

Hyperbranched ( ) Polymersand Dendrimers ( ):

The composition of dendrimerscan be varied throughout themolecule in a systematic way

AFM of branched Polymers

59

2.1.1.4

Homopolymers ( ):

Linear polymer

Branched polymer

Copolymers ( ):

Linear random copolymer

Linear alternating copolymer

Linear block copolymer

Graft copolymer

: two different monomersand

60

61

Cross-linked polymer

Homopolymers ( ): Copolymers ( ):

2.1.2 (Configurations)

Arrangements fixed by the chemical bonding in themolecule, such as cis ( ) and trans ( ), isotactic( ) and syndiotactic ( ) isomers. Theconfiguration of a polymer cannot be altered unlesschemical bonds are broken and reformed.

62

2.1.2.1

63

CH2 CH2

C C CH2CH2

C C

CH2

CH2 CH2

C C

C C

CH2

0.816ÅCH2=CH-CH=CH2

CH2

CH2C

C CH2

C

C

CH2

CH2

C

C

CH2

CH2

C

C

CH2

CH2

C

C

0.48Å

2.1.2.2

: C*?? 64

C*

COOH

OHCH3

H

*C

COOH

HOCH3

H

isotactic

syndiotactic

atactic

66

67

(diads) (triads)

, mm

, rr

, mr

m

r

68PMMA 60MHz

CH2-C*CH3

C=OOCH3

2 1 0 ppm

mm rr

mrrm

69

CH2-C*H

Cl

mm

rm

rr

56 55 54 ppm

PVC 13C

70

(tetrads)

mmm

mmr

rmrrrr

rrm

mrm

71

(pentads)mmmm

rmrr

rmrm

mmrr

mmrmrmmr

mmmr

rrrr

rrrm

mrrm

( )

( )

( )

Hotta, H., et al., PNAS, 103(42), 15327 (2006)

72

73

PP-RHIPP

PE

….

74

75

76

77

( )

()

( )

BOPP

78

79

BOPP

BOPP BOPP1999

973 3010

PE

2:

80

(TM)(ET)

3:

81

-PAN

82

83

PAN

ITA( )

ITA

ITA

AN!

2.1.2 (Configurations)

,

!!!

84

2.2 (glouble)

85

Molecular Weight (MW) and Molecular Weight Distribution(MWD) Degree of polymerization (DP)

~R h N l= 0.6 0.5

2.2

86

(1)

Distribution of molecular weights in a typical polymer

87

5g 4 8g 5 10g 3?

: 12 ; : 90g ii

ii

nNn

4/12

5/12

3/12

ii

ii

wWw

4 5/90

5 8/90

3 10/907.5n i i

iM N M 8w i i

iM W M

88

89

Definitions of Average MWs and MW Distribution

ii

n n ii

n Nn

1ii

Ni

iw w ii

w Ww

1ii

W

n: total mole number; ni: molenumber of ith molecule with molemass of Mi; Ni: mole fraction ofith molecule;

(1) Number-average molecularweight ( ):

(2) Weight-average molecularweight ( ):

nMnw ii

iw

ii

MwM

w

ii iw n M ii

in w

M

=dd

=d

>

w: total weight of the sample; wi:weight of the ith molecule; Wi: weightfraction of ith molecule.

=1

d =d

d

=

= / d = 1= 1

=

= /

ii iW N M ii

iN W

M

= = = = =

90

(3) Z-average molecular weight(Z ):

(4) Viscosity-average molecularweight ( ):

3

2

2i

i

ii ii

zi

i

ii

i

i

i ii

i

i

w M

w

z MM

z

n M

n MM

1 1// 1i i

i i

i iii

i i

i

nw M

w

M

nM

M

wz nM MM M

=

=

=d

d

=

: Mark-Houwink =kMK-M-

MW Distribution & Polydispersity( )

91

Schulz

Tung

Logarithm normaldistribution

2 2( )n n nM M

2d

dw

M MN M

N MM

M M

22 1wn

nn M

MM

w

n

MdM

Width of molecular weight distribution ( ):

Polydispersity ( ):

MW Distribution Function/,

/

,

i i i i

i i ii i

i i i i

i i ii i

n n w MN Mn n w M

w w n MW Mw w n M

=d

d=

= ( ) =

Bimodal Distribution

92

Measurement of Mn

93

End group analysis ( )molecular weight determination through group analysis requiresthat the polymer contain a known number of determinablegroups per molecule.measure the number-average molecular weight, Mn 2.5 104

Colligative property ( ) measurement:measure the number-average molecular weight Mn

vapor-pressure lowingboiling-point elevation (ebulliometry)freezing-point depression (cryoscopy)osmotic pressure (osmometry)

Tb, f, and are the boiling-point elevation,freezing-point depression, and osmotic pressure.

is the density of the solvent. Hv and Hf arethe enthalpies of vaporization and fusion.

nv

bc MH

RTcT 1lim

2

0

nf

f

c MHRT

cT 1lim

2

0

nc M

RTc0

lim

94

Vapor-phase osmometry (VPO, )

Small temperature difference resulting fromdifferent rates of solvent evaporation from andcondensation onto droplets of pure solvent andpolymer solution maintained in an atmosphereof solvent vapor.Measure Mn that is too lower for membraneosmometry method.Calibrated with low-molecular weightstandards: a relative method.Quasi-steady-state phenomena. care must betaken to standarize such variables as time ofmeasurement and drop size betweencalibration and sample measurement.

11

22

1

2

21

2

//MwMwA

nnA

nnnAT

Measurement chamber of VPO(Pasternak, 1962). Droplets of solventand polymer solution are placed, withthe aid of hypodemic syringer, on the“beads” of two theristors used astemperature-sensing elements andmaintained in equilibrium with anatmosphere of solvent vapor.

95

Membrane osmometry ( )

0cc Van’t Hoff relation

Polymer solution pure solvent

P P2 , , 0, ,s sP T P T

1

1 0cV c2

2 30

1c

RT A c A cM

RTM

96

Based on the Flory-Huggins Theory of Polymer Solutions

21 1

2

2

1 1 12

1

cRTc V c M

T cA

V

RM

22 2

1

12V

Asecond Virialcoefficent

11

mixFn

Chemical potentialof solvents

Flory-Hugginsparameter

0 0ic ci

i

i i

cRTM

i

i i

ii

cMRTcc

ii

i ii

nRTc

n M

1

n

RTcM

97

Membrane osmometry ( )

Diagram of the Zimm-myerson osmometer(Zimm 1946). A typical diameter for themeasuring and reference (solvent cell)capillary is 0.5 - 1 mm. The closure of thefilling tube is a 2-mm metal rod. A mercuryseal is used at the top to ensure tightness.

Two membrane are heldagainst a glass solution cellby means of perforatedmetal plates

The assembled instrument issuspended in a large tube partlyfilled with solvent

The success of the osmotic experiment depends on the availability of amembrane through which solvent but not solute molecules can passfreely. Existing membranes only approximately ideal semipermeability.Measurable molecular weight: 2 104 < Mn < 106.

0 0ic ci

i

i i

cRTM

i

i i

ii

cMRTcc

ii

i ii

nRTc

n M

1

n

RTcM

Measurement of Mw – light scattering

98

General set-up of a light scattering ( ) experiment

Incident beamIntensity I0

Scattered beamIntensity I(q)

Detector

Sample

if kkq 2if kk

4 sin2

q

Scattering vector ( ):

where

: wavelength of the radiation

Tyndall ( ) phenomenon

Theory of Light Scattering

The intensity Is of the wave scattered by a single molecule:

42

4 2

16iI cN I

M r

Rayleigh ( )derived:

iI

sI

: polarizability ( )

The averaged intensity I of the wave scattered by many molecules:

42

4 2

16isI I

r

100

The change in the square of the refractive index ( ) n is linear

2 20 4 cNn n

M 2n M dn

N dc4

24 2

16iI cN I

M r

22 2

4 24

in dn

r dcMIN

c I

Rayleigh ratio( )

2

i

r KcMI

R I 22 2

4

4 n dnKN dc

i ii

i i i wi i i

i

c MR R K c M Kc KcM

c

Hence, dilute scattering measurementsgive the weight-average molar mass of apolydisperse polymer sample.

1

w

KcR M

101

For small particles d< /20The contrast required for scattering from polymer solutionsprimarily comes from concentration fluctuations

22

2

12

mix mixmix mix

F FF F

1 1

22

2

1 ...2

mix mix

mix

F F F n

F

122

2mixFI kT

c

2I

F kT

1 1mixF

n

Based on the Flory-Huggins Theory of Polymer Solutions:I c

2I c

102

21 1 2Kc A c

R RT c M 2 21 2

12AV

22

1 21 2

KcMR KcM A cMA cM

2 ?AwM M

2,1 2i i i i ii i

R K c M A c M

At low concentrations, polymer interactions make acontribution to the scattering proportional to thesecond virial coefficient, just as in the case ofosmotic pressure.

2

2mixF

c

1: 11

note xx

large particles d> /20 ?

103

11 1 2

, 1I

g Nq

q

21 2Kc A c

R Mp

2

i

rI

RI

r

I0

I

q

22/ 1

3 gp g N Rqq qStructure Factor &Form Factor

Light scattering intensitybecomes dependent ofthe shape of the molecule.

Total Intensity of LightScatteringRayleigh ratio

104

11 1 2I

Npq

q

22

21 1 2

3gRKc A c

R Mq

00

1 11 x

xx 2

221 2

3gRKc cY A c

R Mq

' 1 1 2K c cR Np q

22 1

,c MNV

2 21 2

12AV

21 2Kc A c

R Mp q

2 22

22

1 81 sin 29 2

h A cM

4 sin2

q

221

3 gp Rqq

22

6ghR

Measurement of M

105

dd

F vA y

/ /dv dy d dt

For Newtonian fluids

1-dimension

: Stress( ): Viscosity( )

:Velocity Gradientdd

vy/d dt :Flow Rate

=

106

Nomenclature of solution viscosity

r: ( ); sp: ( ); red: ( );inh: ( ); [ : ( )

: viscosity of polymer solution at temperature T

0: viscosity of pure solvent at T.

Theoretical and Experimental basis

107

[ ] aKMMark-Houwink equation:

Flory-Fox equation3/22

g hR VM M

2 2gR M 3 1M

Experiments:

Theory

Size of the polymer chain :

empirical formula

Solution viscosity as a measure of polymer molecular weight

108

Viscosity measurement

The small molecular liquids and dilutepolymer solutions are Newtonian flow:the viscosity does not change with theshear stress and shear rateMeasurements of solution viscosity areusually made by comparing the effuxtime t ( ) required for a specifiedvolume of polymer solution to flowthrough a capillary ( ) tube withthe corresponding effux time t0 of thepure solvent.

tBAt

ltVm

lVtghR

88

4

000 //

tBAttBAt

r

0tt

r0

01t

ttrsp

If B/t is much small than At and0

109

Treatment of viscosity data

aKM][

ckcsp 2'

...32

1lnln32spsp

spspr

232 '31

21'ln ckck

cr

ckc

r 2"ln

Huggins equation:

If sp < 1, r can be expressed byTaylor series expansion

Mark-Houwink equation:

Viscosity-average molecular weight( ):

= == = =

Measurement of MWD

110

Gel permeation chromatography ( )Size exclusion ( ) chromatography

111

High MW

Elution volume

Am

ount

ofpo

lym

erel

uted Mark of

injection

Low MWGPC curve:

ebVaMlogVe: elution volume

Samplemixture

Separation

begins

Partialseparation

separation

complete

separatedsamples

leave column

112

MW and MWD

ii

ii

hWh

hi

1/ /

i i ii i

i i i ii i

ni

i

n M w

n w MM

W M

113

Universal calibration ( )log A A

A eM A B V3/22

~ ~g hR VM M

,hA hB A BA BV V M M

Universal calibration parameter

1 1log log log1 11 1log log1 1

11

1 1 log1 1

A AB A

B B B

A AA AB e

B B B

AAe

B

Be

AA A

B B B

B

a K

a KM Ma a Ka KM A B Va a K

a BAa a K

A

Va

B V

,A Ba aA A B BA B

K M K M

log B BB eM A B V

Flory-Fox equation

Mark-Houwink equation

2.2.2

trans ( ) and gauche ( ) conformations:Potential energies associatedwith the rotation of centralC-C-bond for ethane (dashedline) and butane (solid line).The sketches show the twomolecules in views along theC-C-bond.

Rotational Isomeric States ( RIS)

2 2N

114

Random coil ( ) of polymer

i-1i+1

i

In melt or solution

: ( )

Zigzag conformation of PE (21 helix) Random coil

ComputerSimulation of aSingle Chain inSolutions

115

(zigzag) ( )

Polyproplenes in Crystals

116

2.2.3 (flexibility)

2.2.3.12.2.3.1 Static Flexibility

1 ( )

lp=l exp( /kT)

117

/kTp e

pt

// kTt p e

lp l

2.2.3 (flexibility)2.2.3.2 Dynamical Flexibility

1

2t ( )

p= exp( E/kT)

E p

2.2.3.3

118

/E kTr e 11~ time ~ 10 ps

t p+

pt

2.2.4

2.2.4.1

<h2>

<Rg2>

119

)

L=Nl ??

mean square radius of gyration( )

120

Rs

2 22

20 4

0+4s s s

gs s

R R dRRR dR

2

2 0

0

N

i ii

g N

ii

m RR

m

2 2

0/

N

g ii

R R N

2 22 20

2

0

4 354

s

s

R

g sR

R R dRR R

R dR

2

2 0

0

N

i ii

g N

ii

m RR

m

2.2.4.1 <h2> <Rg2>

Schematic representation ofhomopolymer chain of N+1mass points and N bondvectors.

N

iirR

1

2

1 1 1 1

N N N N

i j i ji i i j

h r r r r

End-to-end vector ( ):

Mean-squared end-to-enddistance ( ):

Mean-squared radius of gyration(homopolymer chain) ( ):

2 2

0/

N

g ii

R R Nmi: mass of each pointRi: vector from the

common center ofmass to i-point

<h>=<R>=0

121

122

<h2>

l: the bond lengthei: the unit vector

(1)

(2)

N n!!!

1 2 3 1... N Nh r r r r r 2h

i ilr e

2h

cosi j ije e1

2 2 2

1 12 cos

N N

iji j i

h Nl l

N=2n-1 2n

= = + 2

= + 2 = + 2

(1) (freely-jointed) <h2>

12 2 2

fj1 1

2 cosN N

2ij

i j i

h Nl l Nl

h2fj N

ij ~ 0-180 cos 0ij

123

Nn!!!

N=2n-1 2n

< >=< > / ? ? ?

??

??

(2) (freely-rotating) <h2>

22 cos

cos

cos

i i

mi i m

i ji j

e e

e e

e e

2

cos 0 cos / 2 cos0 1 0 sin cos

sin 0 cos sin sinie

2 22 cos sin sini ie e

O’x’ O’y’ O’z’

Ox cos 0 cos( /2+ )

Oy 0 1 0

Oz sin 0 cos

O’y’ , O’z’

1 cosi ie e

-

i

i+1 i+2

xy

zx’

y’

z’

1 0 0ie

O

O’

124

(freely-rotating) <h2>2 1 1

12 2

1 1 2 21

2 2

1 1

12

cos cos ... cos 1 cos2 cos cos ... cos 1 cos

2 2 cos ...... ... ...2 1 cos... cos

1 coscos

cos 1 cos2 1 cos1 cos 1 cos

N NN N

Nij

i j i

N N

i ji j i

N

ll l

l e e

l N

2 22

2 2 12

2

1 cos 2 cos1 cos 1 cos

2 cos 1 cos2 cos1 cos 1 cos

N

Nl Nlh

ll

2 2

fr

1 cos1 cos

h Nl

j

2

fr

2

fh h

125

<h2> <Rg2>

22

6g

hR

126

2 2fr

1 cos1 cos

h Nl

2 2

fjh Nl

2 2ideal

h CNl

???

<h2>=?

Radius of gyration

127

2

1 1 1 1

12

N N N N

ij ij iji j i i j

r r r

jij iR Rr

2 2

1 1 1 1 1 1

1 1 1 22 2 2

N N N N N N

ij ij j i j i i j i ji j i j i j

r r R R R R R R R R

2 2

1/

N

i gi

N RR 2 2 2

1 1

N N

j gi j

N RR0 0 0 0

0n n n n

i j i ji j i j

R R R R

2 2 2

1 1/

N N

g iji j i

R Nr

rij i j

Ri im

rij =hij: i j

2 2ij Ch j i l

Ri

Rj

rij

m

2 2

1 1/

N N

iji j i

h N 2 2

1 1

/N N

i j i

l NC j i

2 2 2

1 1

N N

ij gi j i

N Rr

=1 + cos1 cos

= 1 or

2 2 2

0 0 0/ /

N N N

g i i i ii i i

R m R m R N

128

1 1 11 2 1

N N N

i j i ij i N i N i

1

12

N

i

N i N i

2 2

1 2 2 2

N

i

N i N iiN

3 1 11 2 1 12 12 2

N N N N N N N

3

6N

2 2 2

1 1/

N N

gi j i

R NC j i l 2 2/ 6 / 6Nl hC

??

1.

2.129

2 6/5h N

2h N

(2)(1)

When does the freely jointed chain works& Kuhn segment le

(1)

130

<h2>0=C Nl2<h2>~N6/5

C

- > -- << -

131

“Coarse-grained” ( ) picture:

R

: -

Ne le (Kuhn segment)

chainhead

chain end

PE Ne le

max

2 20e

e

e

eN l

N l

L

h

Lmax (Contour Length)

m2

0 axeLh l

<h2>0=C Nl2N’=C Nl’2=C l2

<h2>0=N’l2 ?<h2>0=Nl’2 ?

orLmax=Nl Lmax’

2max

2

0/e LN h

2

0 max/e h Ll

Lmax’=N’l or Nl’(2)

PE Ne le

PE1/22

0 0.11nmhAM

22 -90 0.11 10h M

2 2

0

max

e e

e e

N l h

N l L

22

0 max 1/2

2

2

7.2/ 8.82 13.52 / 3

7.2108.28

e

e

Nll h L l nmNl

Nl NNl

n:

N: =2n

132

2-90.11 10 28n2-90.11 10 / 2 28N

90.154 10 ml

27.2Nl C =7.2

Kuhn Segment

133

(2) C2 2 2 2

0 fj 0/ /h h h NlC

(3)1/ 21/ 2

2 2 2 2

0 fr 0

1 cos/ /1 cos

h h h Nl

(1) A1/ 2

2

0/ MA h

(4) le2

max0e /l h L134

(5) a (persistence length)

11 1

1 1 coscos1 cos 1 cos

ni

ii i

la l l l ll

l1 a

135

/ 2ea lFor flexible chain:

2.2.4.2

136

lnS k

4 N

ln / 'S k /Probability

' '/

Total number of chain conformations

Entropy of Chain Conformations

l

'

The number of chain conformations in a state

h (h)

h1, (h1)

…

h2, (h2) h3, (h3)

142

=

138

2 2

2 2gg

h h h dh

R R h dh

138ln / 'S k R R

h

h'

(h) of 1-Dimension Random Walk

139

0

h

N -stepsN-

N+

h=(N+-N-)l

N=(N++N-)

/2

N lN h /2

N lN h

Total Numbers of ways of making steps and stepsTotal numbers of all possible ways

h N N

12Nh

!! !N

N N /2

/2

1 !2 ! !

N N h l N h lN

140

Using Stirling Approx.( ) ln ! lnN N N N

1 !2 !

2/ /!

2

N N h N h lh

lN 1 exp ln ! /ln /ln!

2!

22NN hN hl N l

/ / /ln

/ / /ln

2

ln1 e

2x

22

2p

2

2

N

N h l N h

N h l N h

l NN N

l N h l

N h l

/ /ln2

ex / /ln2

p22N

N h l N h lN hC l N h l

h Nl

141

2

2"exp2N

hh CNl

1 / 1 /l 1 / 1 /ln2

x n2 22

e pNh Nl h NlN Nh Nl h NlNC N

h Nl

21ln 12

y y y O

/y h Nl 1 11 ln'ex 12 2

p2

ln2N

N y N yy NNC y

1h dh

1/2 2

2 2

1 exp2 2

hhNl Nl

Normalization

Gaussian Distribution

1 ln 12

1 l'exp ln2

n 12N

N y y y yNC NN

22 112

11p2 2

x2

"eNN y y y N y yC y 2"exp

2NNC y

Central Limit Theorem( )

142

N stochastic independent variables: x1, x2, x3 …. xN

Mean

Stand Deviation(variance)

0x22 2

x x x

In the limits of N

x ih xDistribution for h becomes Gaussian with variance

2 2xh xN

Irrespective of the distribution for r !!!1/2 2

2 21 exp

2 2x

xx x

hhN N

2 2 2x xx l

Law of Large Numbers

Apply Central Limit Theorem to 3-dFreely Jointed Chain Model

143

2 2 2 2 2 2 23i i i i x rr l x y z

x ih x

ih r

y ih y z ih z

For x-component:1/2

2

2

2

1 exp2 2

x

xx

x

hP hN N

Similarly for y- and z- components

1

2

2

2

/2

exp32 2

3 xhN Nl l

Overall Distribution for h

144

, ,x y z x y zh P h h h P h P h P h

Generalized to d-dimension:

3 d???

1/2 2

2 2, ,

3 3exp2 2

x y zR h h h Nl NR

l

3

2 2

/2 23 3exp2 2

hNl Nl

2 2 2 2x y zh h h h

145

2. hi

l

2

1=4

d 1

i i

i i

ll

h h

h h

1 00 0

xx

x

ex ???pd ik hhh

&

146

d exp ih k h h

02 0 0

21 d d sin exp cod s4

ikh ll

hh h

22 0

1

0 1

1 d4

d d expx ih h l khhl

x

22 0

1 d2

1 ikh ikhe ei

h lkh

h hl

22 0

1 d 2 i2

s n kh hh h ll kh

22

1 2 sin2

l kll kl

sin klkl -30 -20 -10 0 10 20 30

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

sin(

kl)/k

l

kl

2

1=4

ll

h h

d exp ih k h h

= cos + ( )

({hN})

147

1...N Nh h h

1

N

N ii

h h

1h2h

ih Nh……

2

0 or

14

N

l

21=

4i i ll

h h2

0 or1

4 l

=h (h, N)

148

:1

N

ii

h h

h, (h, N)

1 00 0

xx

x 31 d

2ie k xk=

1

N

ii

h h, Nh d . . . dd ih

Fourier

x

sin(

x) FT

f(x)

x

FT

iFT

iFT

f(x)=cos(ax)

F= ( )d if x F e x

149

F( )=f(x)=1

31 d

2ix e k xk

1 d2

iF f x xe x2

a aF

a

h (h, N)

150

31

1 21 d expd d ... d,

2

N

iN iiN ik kh h h hh hh

3 1 1

1 d e d ... p(2

d x )N i

Ni

iie ik h h k hh hk

31 d p

2 d ex

Nie ik h h hk k h

sind exp

kli

klh k h h

2

1=4i i l

lh h

1

N

N iih h

31 sind

2,

Ni kle

kN

lk hkh

h (h, N)

151

31 sind

2,

Ni kle

kN

lk hkh

2 2 2 2sin 1 exp6 6

NNkl k l Nk lkl

2

1/2 2

exp

exp4

ax bx dx

ba a

=1

2 d exp 6

, ,

=1

26

exp3

2

, ,

=3

2 exp3

2

12 d exp 6 x y zd dk dk dkk

, ,x y zdk

h

h

152

h , 0Nh

,

3/2 2

2 2

3 3, exp2 2

NNl Nl

hh

???

(h, N)

= ,

=h (h, N)

Path 1,

Path 2,

… Path i,

2D(random walk)

153

3D(random flight)

W(h, N)

RR’

W(h, N)

h

22

22/3

2 423exp

23, h

Nlh

NlNhW(R, N)

154

h

h

(h, N)

3/2 1/222

2 20

3 3 2 2exp 4 d2 2 3

h Nh h lNl N

hl

h

W(h, N)

h

,0

W h Nh

h*=(2N/3)1/2 l

155

2 210

2 1 !!d

2ax m

m m

me x x

a a

10

!dp axp

px e xa

< > < > /

3/ 2 22

2 20

2 2 23 3exp 4 d2 2

h h hNl N

h h Nll

>

< >=< > / ? ? ?

Applications of Gaussian Chain Model: Stretching of an Ideal Chain

156

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

/' '/

157

ri-1ri

hi

hi

ri-1

ri+1

ri hi+1

3/ 2 2

2 2

33, exp2 2

ii l l

hh

3/2 2

2 21

3 3exp2 2

gNi

iil l

hh

3/22

2 2

3 3, exp2 2

g g

gg g

NN l N l

hh

2 2 2 2 2

0i gh h l b l

2 2 2 2g g g g

h Nl N l N h

g

NN

31

1 21 d expd d ... d,

2

N

iN iiN ik kh h h hh hh

Two Definitions of Segment

(1) Kuhn Segment (Kuhn )le

lg le158

2maxeh l L

(2) Effective Gaussian Segment ( )lg

,

lg le

159

G H ST

1

3.

2.

I

160

2 1RW ~R N

3/2 2

2 21

33 exp2 2

gNi

ii l l

hh

23 /21

2 21

33 exp2 2

g gN Ni i

nb bR R

2

2 0

3exp2

gN nconst dnb n

R

2 2l b

Path Integral( )Edward’s Minimum Model

interaction energyih H Mean-field Free Energy

,=

6,, =

32

exp3

2

Appendix. Diffusion Equation

161

1Nh

h

has a probability of 1/zof occurring

, =1

, 1

=

, 1 = ,, ,

+12

,

1= 0

1=

13

2 ,= 6

,

, =3

2 exp3

2

Diffusion Equation

Each l has a directions of zl

h>>l, N>>1

II: blob

162162 162

2 2 6/5SAW ~ ~R N N

3/2 2

2 2

3 3, exp2 2

NNl Nl

hh2 1RW ~R N

???

(Blob)

163

blob

Blob

blob

~ BNG k Tg

2 1RW ~R N 2 2

SAW ~R N

2.2.5 -2.2.5.1

164

1 1

sin1 N Ni j

i j i j

gN

k r rk

k r r

2 2

1 1

1 16

N N

i ji jN

k r r

2

1 1

21 26

N N

i ji j i

N NN

k r r2 2 2

1 1

N N

i j gi j i

N Rr r

22 221

3 gN RNN

k

2 2sin 16

kl k lkl

l12

sing klkkl

22

13 gN Rk

sin j iij

j i

gk r r

kk r r

1/k r

2.2.6.2

165

2 2gi j i j lr r

i j|i-j|

2 2

0 0

1 d d 16

N N

i jg i jN

kk r r

0

2g

2

0

1 d d 16

N Ni j li j

Nk

2 2g

0 0

1 d d exp6

N N li i jj

Nk

2 2

00 0

1 d d expN N

gi j RN

i jk

g

Debye Function

166

gp

Nk

k

/ , /s j N t i N

2 2

00 0

1 d d expN N

gg i j R i jN

k k

2 2 2 2 2 2

0/ 6g g gQ R N R l Nk k k

1 1

0 0exp exp

t

tQ t s ds Q s t ds dt

1 1

0 0exp exp exp exp

t

tQt Qs ds Qt Qs ds dt

1

0

1 exp exp 1 exp exp expQt Qt Qt Q Qt dtQ

1

0

1 2 exp exp expQt Q Qt dtQ

exp 1 exp 11 2 expQ Q

QQ Q Q

22 exp 1 DQ Q f Q

Q

Form Factor:1 1

0 0exp Q s t ds dt

Debye

167

2 2D D gg Nf Q Nf Rk k

22 1Q

Df Q e QQ

2 2(1) 1gRk2 3

2

2 1 / 2! / 3! 1 1 / 3Df Q Q Q Q Q QQ

221

3D gg Nf Q N Rkk

22 2 2/ 1

3D g gp g N f R Rkk k kForm Factor:

2 2gQ Rk

168

2 2(2) 1gRk

2

2 21QDf Q e Q

Q Q

22 2 2

22 2 2 2

1 / 3 1

2 / 1

g g

D

g g

N R Rg Nf Q

N R R

k kk

k k k

Ornstein-Zernike

2 21 / 2g

NgR

kk

2 2 2 2 / 6g gQ R l Nk k

2 2

11 / 2g

pR

kkor

Structure Factor Form Factor

Polymers with a simple geometry and their form factors

169

170

Sphere

Rod

Form Factor ( )of a Real Chain

171

2 2

11 / 2g

pR

kk

???

2p kk1gkR

172

1/1

~ BhG k T

N l1/1

2, ~ exp ~ expR

G hh NkT h

2h h N l

2, ~R

hh Nh

2h h0.28

0.28 2.50.278 exp 1.206R x x x2/x h h

:

=3

2exp

32

=3

2 exp 1.5

173

1.

2.

(1)

(2)

(3)

(5)

(4)

-

174174

Gaussian Chain Model

Entropy of Chain Conformations

Mechanic Properties

Scattering Theory (R> /20)

h, (h, N)

h

h

2

2

3( , )2conf g B

g

S N k CN lhh

20 eR l L

2 2 2~R N l

Scaling Concept & Blob Model

2 20

R NlIn solution:

2 2

11 / 2g

pR

kk

23 Bk TNlhf

~ BNG k Tg

, =3

2exp

32