Polymers and

Small Angle Scattering

Henrich Frielinghaus

Jülich Centre for Neutron Scattering

München - Garching

Polymers in daily life

Structure: Example Polymer

1Å

CC

Monomer

10Å

Chain

100Å

Coil

Domains

1000Å 1µm

Superstructures

Fractals

Small Angle (neutron/X-ray) Scattering

SALS

Conformation ?

What are Polymers ?

HH

HH

Polymer

HH

HH

HH

HH

HH

HH

HH

HH

H

HH

HH

HH

HH

HH

HH

HH

HH

HH

HH

HH

HH

HH

HH

H

HH

HH

H

HH

HH

H

HH

HH

Dimer

Monomer

Polymers in Biology

DNA

folding complicated structure

The Cytoskeleton

(A) Actin Filaments (Microfilaments)

(B) Microtubules

(C) Intermediate Filaments

Polymerization

Continuous process polymerization / depolymerization

End capping stabilizes microtubules

Coarse Graining

Polymer Models (freely jointed chain)

Random Walk

Sequence of steps

Limited step-length

Can be on a grid or in free space

Polymers: we look on the full path

Diffusion (Brownian motion): When is it where?

Transition to semiflexibility ???

Freely Jointed Chain

Bond length fixed, angle free

N

i

iee rR1

0

eeR

N

ji

jieeeeee rrRRR1,

2

Nji

ji

N

i

iee rrrR11

22 2

Nji

ji

N

i

iee rrrR11

22 2

2

1

22 02

NrRN

i

iee

Av. chain size much smaller than full extensionNRee 2

Semiflexible chains

10-2

10-1

100

10-3

10-2

10-1

100

6

12

R2

g

rigid rod

rigid rod

flexib

le ch

ain

R2

ee

flexib

le ch

ain

1/NK

R2 g

and

R

2 ee in

unit

s (N

K

2 K

2)

222

NRR gee

2222

NRR gee

Semiflexible chains

Summary of chain models

222

NRR gee

Chain:

NRR gee

22

2222

NRR gee

Rod:

NRR gee

22

Rubber elasticity

vulkanization of high molecular weigth polymers,

reversible strain of several 100%,

elastic modulus increases linearly with temperature,

at very low temperatures similar to normal solids.

Rubber elasticity

)(//)()( 2

0

2

11

CB MRTVnTk

topological crosslinks

Non-Gaussian

(rigidity)

Density of Monomers

NR

2/1

3

NR

N

Coils in melt penetrate

Coils in solution are dilute

CC

CC

C

C CC C

C

C C

C

CC

CC

H

H

H

HH H

HH

HH

H

H

H

H

H

H

H

H

H

H

H

H

H

H

H

H

H

H

H

H

HH

HH

HH

Lattice theory:

Element

Translational Entropy

N: lattice sites

k: species A

N-k: species B

BBAA

B kNk

N

k

S lnln

)!(!

!ln)ln(

BA 1

Translational Entropy of Polymer

N: lattice sites

k: species A

N-k: species B

NA: degree of polymerization

NB: degree of polymerization

B

BB

A

AA

B NNk

S lnln

Size Matters,

Not Flexibility

Gibbs Free Energy of Mixing

BA

B

BB

A

AA

B NNTk

G

lnln

G

ΦA

02

2

A

G

‚local‘ stability

Gibbs Free Energy of Mixing

BA

B

BB

A

AA

B NNTk

G

lnln

G

ΦA

global stability

Gibbs Free Energy of Mixing

BA

B

BB

A

AA

B NNTk

G

lnln

G

ΦA

not

stablestablestable

me

ta-s

tab

le

me

ta-s

tab

le

0.30 0.35 0.40 0.45 0.50 0.55 0.60

55

60

65

70

Tem

per

atu

re [癈

]

PB

Phase Diagram

G

ΦA

not

stablestablestable

me

ta-s

tab

le

me

ta-s

tab

le

sh

T

Susceptibility

BA

B

BB

A

AA

B NNTk

G

lnln

2

11)/(2

2

sus

BBAAA

B

B NN

TkG

Tk

TkQS

B

sus1 )0(

Scattering experiments

Scattering

211

)0(1

BBAA NNQS

sh

T

Measure χ

Critical fluctuations

What Q-dependence?

Scattering of Chains (Gaussian Chain)

)1)(exp()(

)(

)exp(1

)(

2

2

22

0 0

22

61

xxxf

RQfN

jiQdjdiN

QS

xD

gD

N N

K

Debye-Function

N

ji

ji RRQN

QS0,

))(exp(1

)(

i

N

ji

ji RRQN

QS0,

2

21 ))((exp

1)(

N

ji

ji RRQN

QS0,

22

61 )(exp

1)(

Scattering of Chains (Gaussian Chain)

QRQN

QRQNQS

g

g

largefor)/(2

smallfor)1()(

22

22

31

0.0 0.5 1.0 1.5 2.0 2.5 3.00.0

0.5

1.0

f D (

QR

g)

QRg

Random Phase Approximation

211

)0(1

BBAA NNQS

2)(

1

)(

1)(1

gBDBBgADAA QRfNQRfNQS

Limit OK!

Other Limit: χ0, NA = NB, ΦA = ΦB = ½

)(4

1)( gD QRfNQS for instance: H and D-chains

0.00 0.02 0.04 0.06 0.08 0.10 0.120

500

1000

1500

2000

2500

3000

S(Q*)

Q

Q*

PEP-PDMST=170癈P=1bar

S(Q

) [c

m3 m

ol-1

]Q [臸

0.00 0.02 0.04 0.06 0.080

1000

2000

3000

-1

S(0)

S(Q

) [c

m3 m

ol-1

]

Q [ -1]

Scattering functions of melts:

SusceptibilityCorellation lengthDomain spacing

2

21 )/(1

)0(d

VGd

TkS

B

Q [Å-1]

PB(1,4) / PS

T=104°C

P=500bar

T

0.30 0.35 0.40 0.45 0.50 0.55 0.60

55

60

65

70

Tem

per

atu

re [癈

]

PB

Phase diagrams of Polymers:

ΓV

2.9 3.0 3.10.03

0.06

0.09

2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.40.0

0.1

0.2

0.3

0.4

0.5

TODT

Meanfield

1bar

515bar

1365bar

S-1

(Q*)

[10

-3 m

ol/c

m3]

T-1 [10-3/K]

0

1

2

3

4

5

6

7

2.6 2.7 2.8 2.9

CMF 1 C+

1.24

d-PB(1,4)/PS

500bar

Meanfield Crossover 3d-Ising

1/T [10-3/ K]

S-1

(0)

[10

-4 m

ol/

cm

3]

Susceptibilities of melts:

Mean field:

enthalpyentropy

)()(2

1)(*

01

TS h

stranslSQQ

Q

Fluctuation corrections:Temperature change: TMF -> TS

Ginzburg number: Gi ~ (TMF-TS)/TS

Fluctuations modify the free energy(or simply the mean field picture)

Gi ~ Probability of AB-contact in a certain point:GiHom~ √N-1•√N-1 = N-1 GiDibl~ √N-1

This is not true: Also the segmental entropy plays a role -> pressure experiments

Generally: compressibility

-> interaction parameter χ is pressure dependent

Applications of poylmer blends:

Stabilize domains mechanically,

Control domain sizes,

Cotrol domain structure

Blends of A/B Homopolymers and A-B Diblock

V. Pippich,

D. Schwahn

10 10020

40

60

80

100

120

140

160Disorder

Ordering Transition

Disorder

Lamel

laDE

Lifshitz Line

BE

Two-

Phase

Tem

per

atu

re [

°C]

Concentration Diblock [%]

Ordering of Copolymer

Blends of A/B Homopolymers and A-B Diblock

V. Pippich,

D. Schwahn

Summary:

Fluctuations are important !

Polymeric microemulsion interesting for applications.

Find substitute for diblock copolymer.