Fourier Transform Perspective on Order-Disorder in Polymeric Systems

Fourier Transform Perspective on

Order-Disorder in Polymeric

Systems

CCAST, 2007.10.26

F U T S

F U T S

U T S U T S U T S

( ) ( )

( ) ( )

f R F R

U R T S R

From Order to Disorder

2

0

1( ) lim

x

x e

( ) ( )lik Rlk k

r R e r

1( )

2ikxx e dk

,

,

exp ( )

exp ( )

l

l s

l k kR

l s R Rk BZ

i k k R N

ik R R N

Convolution and Correlation

*

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

g x h x g x h x x dx g x x h x dx

g x h x G f H F

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )


g x h x G f H F

Convolution

Correlation

Self-Correlation ( ) ( )g x g x

Models of Polymer Chain

4 510 10n

Degree of Polymerization


0

0

1exp

e

exp

x

p

p

B

B

B

p

p

l

L N k T

k T

l

E

lk T


1

1 1

2

2

2

1

2

1

2

0,

N N

i ii

i i

i i j ij

N

i

i

N N

i i j

i i j

R b R R

b b b b

R b

b b b

Nb

R N 1/ 2 3/ 5 1/ 2 R N

Freely joint chain (random walk)


1/ 2 2

Gaussian c

ha

i

n

2 f

R N N

d

R

3/ 5 5/ 3

Self-avo

iding chain

5/ 3 1.7 f

R N N R

d

2

1,

1( )

( )

2, ( ) const.

3, ( ) 1/ 0

d d

R

R

d C R R

d C R

d C R R

MC R

R R


11

322

11 1

33

2

1

2

1

( , )

1exp

3 3( , ) ex

2

p2 2

NN

i iii

N N N

i i i ii i

N N

i ii i

P b p b

P N R db R b P b

R

RP N R

b N

b d k

b

ik R b

2 3 2 2( )R d RR P R Nb

2( ) (1/ 4 ) fixe d i i ip b b bb b

322

22

2

( ) ( )

3 3( ) ex fip xed

4 2

i i

ii i

p b p b

bp b b

b b

2 2 2 2

1, long wave approximatio

sin1 exp

6

n

6

Nkb k b Nk b

k

b

b

k


2 2 20( )F NR R r r

Root-mean-square

End-to-end distance RF

Radius of gyration RF

2 2

0

22

, 0

1( )

1

1 ( )

2( 1)

N

g i Gi

N

i ji j

R r rN

r rN

2 2

Gaussian cha

1

6

in

g FR R

Wiener-Edwards integral2

22 2

2 22 2

2 2 2 211

2

2

( ) exp 2 2

{( )} exp exp2 2 2 2

{( )} exp2

d

i i i i j

d dN N

i i iii

d

i

d dp b b b R R

b b

d d d dP b b b

b b b b

dP b

b

==

æ ö æ ö÷ ÷ç ç@ - = -÷ ÷ç ç÷ ÷ç çè ø è ø

æ öæ ö æ ö æ ö ÷÷ ÷ ÷ çç ç ç= - = - ÷÷ ÷ ÷ çç ç ç ÷÷ ÷ ÷ç ç ç ÷çè ø è ø è ø è ø

æ ö÷ç= ÷ç ÷çè ø

åÕ

( )

( )

2

021

2

0 21

2

2 0

2

exp ({ })2

({ })2

( ) /1 /

{( )} exp2

( )exp2

N

i j ii

N

i i ji

i j

N

i

dR R H R

b

dH R R R

b

R R

d RZ DR s

b

R s

d Rb ds

b

s

Ps

=

=

é ù é ùê ú- - = -ê úë ûê úë û

= -

- ® ¶ ¶

é ùæ ö¶ê ú÷ç ÷@ - çê ú÷ç ÷ç¶è øê úë û

æ ö¶ ÷ç= - ççç¶è ø

å

å

ò2

0all paths ( )

N

R s

ds

é ùê ú÷ê ú÷÷ê úë û

ò ò A B

(0,r0)

(s,r)

s

Wiener-Edwards integral2

2

22

22

22

( , ) 0 0, 02

. .

(

, ) 02

( , ; ,

( , ; , ) ( ) ( )2

. . 2

hih t R

t m

hi

bP N R R N

N d

c p

bG R R s s R R s s

s d

c h tt m

p G R R

æ ö¶ ÷ç ÷- Ñ = " ¹ >ç ÷ç ÷¶è ø

æ ö¶ ÷ç ¢ ¢ ¢ ¢÷- Ñ = - -ç ÷ç ÷¶è ø

æ ö¶ ÷ç ÷- - Ñ =ç ÷ç ÷¶è ø

æ ö¶ ÷ç ¢ ¢÷+ Ñç ÷ç ÷¶è ø) ( ) ( )t ih R R t t

¢ ¢=- - -

22( , ) ( , ) ( ) ( , )

6p p p p

bq r t q r t r q r t

t

Self-consistent Field Theory (SCFT)

Self-avoiding chain2

2 0 0 0

3

2

2 0 0 0

1({ ( )}) ( ) ( )

2 2

( ) ( ) ( )

({ ( )}) ( ) ( )2 2

N N N

N N N

d RH R s ds ds dsV R s R s

b s

V R v R b R

d R vH R s ds ds ds R s R s

b s

æ ö¶ ÷ç é ù¢ ¢÷= + -ç ÷ ê úë ûç ÷ç¶è ø

= µ

æ ö¶ ÷ç é ù¢ ¢÷= + -ç ÷ ê úë ûç ÷ç¶è ø

ò ò ò

ò ò ò

2 2 1

2

size of polymer

connectivity term

excluded volume

2 1 2

3

2

d

R N

N

N

d

d

- +

-

~

~

- = -

=+

1 1

2 3/ 4

3 3/ 5 (0.589)

4 1/ 2

d

d

d

d

= =

= =

= =

= =

Pair Correlation Function

1

1 1 1

3

1 1

( ) ( )

( ) exp (

( ) ( )

1

)

1( )

1( )

N

n m nm

N N N

nn n m

N Niq

nn

m n

mr

m

g r r R R

S q iq R

g r r R R

g rN N

d re g rN

R

2

3 2

2

1( )

( )

rm

b

mg r

r rb

S k k

Scattering experiments

Laser light scattering (LLS),

Small/Wide angle x-ray scattering (SAXS/WAXS),

Neutron scattering

Scattering experiments

4sin

2

nq

Bragg'slaw

2qd m

Structure factor

( ) ( )

0

2

0

*

( ) ( )2

0

( )

,

2

( )

,

2

2

( ) , ( )

1( ) | ( ) |

1

1

1

j k

j j

j k

j k

i

s

iq R

i t i t

j i sj

s s

i t i ti

j k

i

j

R

j k

ik

E t E Ae E t E A e

I q dt E t

I A dt

e

e e

I A e

AN

I

NI N

(0)

(0)

( )

( ( )

1

) 1)

(s

s

S q

P q S q

NI q

NI

Structure factor( )

,

1 1 1 1 1( )

i j ij ijiq R R iq R iq R

i j i j j i j

ij i j

P q e e eN N N N N

R R R

0

2

22

2

,

,

4

22

sin( )1cos( ) cos cos sin

2

sin1 1

3!

sin( )1( )

1( ) 1 1

5!

1 36

ij

i j

ijij ij

ij

iji j

ij

g g

q

qRP q

N qR

P q q

Rq R qR d

qR

x x x

R qR R

xx

q

N

Structure factor

3/ 2 2

32 2

22

22

0 0

3 3exp exp( ) exp

2 | | 2 | |

exp | |6

1( ) exp | | ( )

6

n m

N N

g

riq R R d r iq r

n m b n m b

qn m b

qS q dn dm n m b Nf qR

N

24

2 2

2 ( ) exp( ) 1

1

2 ( )

g

g

f x x xx

qR

NS q

q R

2

Debye function2

2 0,

3/ 2 2

2 2

2 2

2,

2 2

2 0 0

2

sin( )1( ) (| |, )4

33(| |, ) exp

2 | | 2 | |

1 | |( ) exp

6

1 | | exp

6

2 exp( )

ij

ij

ijij ij

i j ij

ij

i j

N N

qRP q P i j R R dR

N qR

RP i j R

i j b i j b

q i j bP q

N

q u v bdu dv

N

QQ

2 2 2 2

2 2 2 222 2

2( ) exp(

1 ,

)

/

1

6

g g

g

g

P

Q Q q Nb

q

q

R q

R

q Rq R

PMMA

Structure factor2 2

2 2

11 1

3( )

2 1

g g

gg

N q R qR

S qN

qRq R

Gaussian chain

2 22

2

1

1 1 16 /sin

( ) 3 2

g

s g

qR

I q N NR

Scattering experiments2 2| ( ) |

R

d dF q

d d

2( ) - Form factorF q

2 3 2 /( ) ( ) i q r hF q d r r e

2 2 22

1 ( ) 1

6( 2ex.

/ )F q q r

h

2

2

Experiment | ( ) |

T

:

: S.E.heor ( )

y ()

(

)F q

d

r

Fd

r

q

Form factor and Structure factor

1

( )exp(

( )exp( )

)

( ) ( )

exp( )

e

( )exp( )

xp( )

j

G Gcell

s

j jj

j jGj

jG

j j

jj

F N dVn r iG r NS

n r n r r

S iG r r r

S iG

dVn iG

f dVn iG

f r

2 sin4 ( )j j

Grf drr n r

Gr

Convolution and Correlation

*

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )


g x h x G f H F

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )


g x h x G f H F

Convolution

Correlation

Self-Correlation ( ) ( )g x g x

Lei Zhu, Stephen Z. D. Cheng, etal., Macromolecules 2002, 35, 3553.

−50ºC Tc −10ºC 0ºC Tc 40ºC

Diblock Copolymers: Possible StructuresDiblock Copolymers: Possible Structures

Controlling parameters:

N

NfandN A

Ordered Phase: Lamellar

Equilibrium: 3/132

61

19.1,03.10 NTk

FNad

d

F

B

LL

a

d

NX

X

XN

kTNa

d

kT

F ABL

3/26/1

23/1

2

3

1

88

3

d

Na

a

kTAB

3

2

2chainper area linterfacia

6 tensionlinterfacia

1 2 3

1

2

Diblock Copolymers - Why Ordered Phases?

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Fourier Transform Perspective on Order-Disorder in Polymeric Systems

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