13th Oxford School on Neutron Scattering, 12-13 September 2013

Neutrons in soft matter

João T. Cabral & Julia S HigginsDepartment of Chemical Engineering

Imperial College London

Lecture 2 – Reflectometry & Dynamics

David G BucknallGeorgia Tech, USA

Outline

Single chain polymer conformation(solution and blends)

Polymer blends: interactions, conformation & dynamics(equilibrium and phase separation)

Introductionsoft matter & relevance of neutron scattering

Single objects: spheres, coils, rods...

Lecture 1 – Structure & kinetics – SANS

Lecture 2 – Interfaces and dynamics

Reflectivity and diffusion

Dynamics in soft matter, QENS, BS, Spin-echo

Forster et al (2011)

• Interdiffusion, e.g., welding

Miscible systems

Immiscible systems

• Copolymers, e.g., di-blocks

• Reduce interfacial tension

• Entangle with homopolymers

smaller dispersed phase

increase strength

Reflectometry: study of interfaces

Reflectometry

CRISP (ISIS)

Significance of the interfacial width

Theoretical width

Infinite molecular weight limit

E Helfand & AM Sapse

J Chem Phys 62 (1975) 1327

Finite molecular weight limit

2

1

12

11

2

t NN66

a2w

M Stamm & DW Schubert

Ann Rev Mater Sci

25 (1995) 325

Measure interfacial width to find

wt =2a

(6)0.5

where a (statistical segment length)

Basics of Reflectivity

q < qcrit

q qcrit

q > qcrit

only reflection

critical angle

reflection and refraction

q

I(q)

qcrit

Reflection

Reflection

&

Refraction

The Reflectivity Profile

Information

Content

r

d

Simplest Case

Complex Case

r

d

What about lateral

information?

Off-specular !

Evaluating Reflectivity Data

qz (Å

-1)

0.00 0.05 0.10 0.15 0.20 0.25 0.30

Refl

ecti

vit

y,

R(q

z)

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

qz (Å

-1)

0.00 0.05 0.10 0.15 0.20 0.25 0.30

R(q

z)q

z

4

(Å-4

)

10-11

10-10

10-9

10-8

10-7

qc

q2

q1

(a)

(b)

r

d

fft

Ideal

r

d

Real world

r

21

21

22

zdj

bNn

nj+1

nj

refracted

beam

incident

beamreflected

beam

qj

qj+1

qj

11 qq jjjj cosncosn

11

11

1

qq

qq

jjj

jjj

j,jsinnsinn

sinnsinnr

*

j,jj,j rrR 11

q

sinkq

42

1

1

1

j,zj,z

j,zj,z

j,jqq

qqr

0

1

2

m-1

m

m+1

n0

n2

nm-1

nm

n1

nm+1

Air

layer

number

refractive

index

substrate

d1

d2

dm-1

dm

mm,mm,m

mm,mm,m

m,miexprr

iexprrr

21

2

11

11

1

q sindn mmm 2

mmm

mmm

mcossini

sinicosc

m

m

mMM

MMcM

0 2221

1211

2

12221011211

12221011211

mm

mm

MMMM

MMMMR

Snell’s Law

Fresnel’s law

Single layers and bilayers

Normalised Distance

-3 -2 -1 0 1 2 3

Vo

lum

e F

rac

tio

n,

0.0

0.2

0.4

0.6

0.8

1.0

Interfacial Width - Definition

Polymer 2Polymer 1

w

1

21

2tanh

z

w

z

Momentum Transfer, Q (Å-1

)

0.01 0.1

Refl

ect

ivit

y

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

w = 0 nm

w = 5 nm

w = 10 nm

Effect of Interdiffusion on

Reflectivity Profiles

Momentum Transfer, Q (Å-1

)

0.01 0.1

Refl

ect

ivit

y

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

Effect of Limiting

Q range on Observation Window

Momentum Transfer, Q (Å-1

)

0.01 0.1

Re

fle

ctiv

ity

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

q = 0.25

q = 0.65

q = 1.5

Effect of Angle on the Q Range

q

sin

4Q

q

0.05 < (nm) < 0.65

Horizontal Distance (m)

0 100 200 300 400

Ve

rtic

al D

ista

nce

(

m)

-0.04

-0.02

0.00

0.02

0.04

Roughness causes off-specular scattering

and increased resolution term.

Surface profile of i-PP

at RT

Momentum Transfer, Q (Å-1)

0.02 0.03 0.04 0.05 0.06

Ref

lect

ivit

y

0.001

0.01

0.1

1

Measured reflectivity

of d(i-PP) at 180 C

Calculated reflectivity

of i-PP assumimg small

surface roughness

Effect of Crystallinity

on reflectivity

Brewster angle micrograph

of surface of i-PP(bar 20m

Thermally Excited Capillary Waves

Mean field theory assumes thatthe interface is flat.

At equilibrium capillary wavesare thermally excited.

According to the equipartition theorem each mode increases the surface energy by 0.5 kT. The actual surface is roughened by a superposition

of all possible capillary wave modes.

wt

w0

w

z

x

y

z

Projection onto z-y plan

w = (wt2 + w0

2)0.5

Definition of w0 dominates derivation of wt

NR Measured Interfacial Width

s

As made

t = 0

Annealed

t > 0

s

Polymer Interdiffusion

s

s

Non-Fickian Diffusion - Case II Diffusion

8

s tn

n = 1

2

Non-Fickian Diffusion

t = 0 t > 0

s

The Tube Model

Polymer chains inthe melt

Each chain can be considered to beconstrained withina tube

Polymer Motion

t

t = te

Entanglement time

t = tR

t = td

Rouse relaxation

time

Reptation time

t = 0

Polymer Diffusion

t < te

te < t < tR

t > td

tR < t < td

aT

t (mins)

0.001 0.01 0.1 1 10 100 1000 10000

s (

Å)

10

100

te

tR t

d

A Karim et al, Phys Rev B 42 (1990) 6846

NR Results4

1

ts

41

ts

81

ts

21

ts

Momentum Transfer, Q (A-1)

0.01 0.1

Ref

lect

ivit

y

10-11

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

05

32

63

93

124

155

186212

t (mins)

Real Time ReflectivityMeasurements

Si / PS (50k) / dPS (40k) @ 115 C

‘225’

Dtw 4

Calculating a Diffusion Coefficient

For dPS-PS system:

D = (1.7 0.2) 10-17 cm2s-1

D

Nbr 2

2

3t

tr = 3223 363 s (dPS)= 4333 489 s (hPS)

Reptation time:

M Doi and SF Edwards

The Theory of Polymer Dynamics (1986)22

2

3 bN

dTkD TB

D = 2.81 10-17 cm2s-1

When (115C) = 0.199 dyne.s.cm-1

and dT = 5.7 nm

Rouse time:

D

dTR 2

2

9t

tR = 215 23 s

Polymer-Oligomer InterdiffusionReflectivity Cell

Motor

Heatable fixedtop plate

Heatable moveablebase plate

Polymer coatedsilicon

‘Fluid’ container

Neutron Reflectivity Melt Cell

heated brass

base plate

thermocouple

Neutron

beam

aluminium plate

dPS film

bulk PE layer

silicon block

retaining screw

heaters

Momentum Transfer, Q (Å-1

)

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

Refl

ect

ivit

y

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

Si

dPMMA

OMMAd

w

dPMMA(100k) / OMMA(510) @ 45 C

0.5

20.5

43.0

64.5

84.5106.0

127.5

t (mins)

Momentum Transfer, Q (Å-1)

0.00 0.02 0.04 0.06 0.08 0.10

Refle

ctivit

y

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

3

27

47

65

101

123

135

t (mins)

dPS (101k) / OSt (1100) Interdiffusion @ 65C

t (mins)

0 20 40 60 80 100 120 140

Inte

rface

Positio

n, x

0 (

nm

)

88

90

92

94

96

t (mins)

0 20 40 60 80 100 120 140

Inte

rfacia

l w

idth

, w

i (nm

)

0

2

4

6

10

12

14

16

dPS (101k) / OSt (1100) Interdiffusion @ 65C

w1

w

2

Off-specular reflection

(B) Soft Matter application: depth profiling of nanofilled polymer films

h

SiO2 / Si

h = 5-500 nm

1

nm

Fullerene C600-5% w/w

Polystyrene (PS)

CH2 CH

n

Rg=0.27 Mw 1/2

Rg = 2–20 nm

Phys Rev Lett 105, 038301 (2010); Macromolecules 43, 9578-9582

(2010); J. Mol. Liq 153 79-87 (2010); Macromolecules (2010 ASAP)

Polymer-fullerene ‘mixtures’

Electrical propertiesFullerene derivative (PCBM)

Heegar et al. Adv.

Funct. Mater (2005) 15,

1617Tuteja et al. Nat. Mat. (2003) 2, 762

Transport properties

Mackay et al. Science (2006) 311, 1740

Polystyrene (PS)

Fullerene C60

500 μm

a b

PS PS + 5%C60

2K

, 30

nm

c d

500 m

2K

, 15

0 n

m

PS+5% C60 h = 100nm

140oC

180oC

Polymer layer

Diffuse BB layer ~ 5nm

• Neutron reflection from PS containing

5% fullerene on Si

• Fullerenes not present at air surface

(confirmed water contact angle)

• Segregation of 2-5 nm thick diffuse

layer of fullerenes on Si

-6

-5

-4

-3

-2

-1

0

Lo

g R

q ( nm-1

)0 0.5 1.0 1.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0 20 40 60 80 100 120

Depth Z (nm)

full

ere

ne

500 μm

a b

PS PS + 5%C60

2K

, 30

nm

c d

500 m

2K

, 15

0 n

m

PS+5% C60 h = 100nm

‘Spinodal Clustering’

140oC

180oC

180oC

1 min 2 min 2.9 min 3.8 min

5 min 30 min22 min 61 min30 μm

2.5 min

10 min

0 min

a b c d e f

g h i j k

10 μm61 min

l

Coarsening Kinetics

PS+5%C60

170 oC

h=160 nm

PS-C60 film

‘photoresist’

pattern

Annealing above Tg

500 m(a)

High Mw

dewe

t

stabl

e

Low Mw

light

dark

?

Coupling of self-assembly & patterning!

Summary

• study and design interfaces

Reflectivity

• investigate diffusion mechanisms

• engineer ‘functional’ surfaces / devices

13th Oxford School on Neutron Scattering, 12-13 September 2013

Neutrons in soft matter

João T. CabralDepartment of Chemical Engineering

Imperial College London

Lecture 2 (II) – Dynamics

Scattering theory reminderScattering cross section

coherent incoherent

Dynamic structure factor

Intermediate scattering function

FT (t,w)

Pair correlation function

FT (r,q)

RHH

R

vibrations

rotations

motion decomposition

)]0()([)]0()([)]0()([1),( RtRiQTtTiQ

i

VtViQ

self eeeN

tQI

vibrations

CM

translationrotations

single-particle dynamics

RHH

R

vibrations

rotations

single-particle tools

motion decomposition)]0()([)]0()([)]0()([1

),( RtRiQTtTiQ

i

VtViQ

self eeeN

tQI

22

31 uQ

e

frozen for polymers T<<Tg.

vibrations

CM

translationrotations

2210rot2/3

2/31)Q(A)()Q(A),Q(S

wt

t

ww

)3Qr(j21)Q(A 031

0

)Q(A1)Q(A 01 with

Methyl protons 3-fold jumps

relevant proton reorientations: methyl and phenyl rotations about group’s axis.

CM translation

Proton delocalisation DW factor:

2210rot/2

/21)Q(A)()Q(A),Q(S

wt

t

ww

)Qr2(j1)Q(A 021

0

RHH

R

with

Phenyl proton 2-fold jumps

R

R 1.032 Å

R 2.28 Å

22

31 uQ

e

frozen for polymers T<<Tg.CM translation:

Proton delocalisation: DW factor:

2210rot2/3

2/31)Q(A)()Q(A),Q(S

wt

t

ww

)3Qr(j21)Q(A 031

0

)Q(A1)Q(A 01 with

Methyl protons 3-fold jumps

RHH

RR 1.032 Å

3-fold CH3 potential

Side group rotations:

motion decomposition in the glass

vibrations

CM

translationrotations

single-particle dynamics

example:

distribution tcorrelation

RT

EA

e

0

2E

20i

2

)EE(

E

i e2

1)E(g

s

s

glassy polymers: no single relaxation time

variety local molecular

environments

Eo: average barrier height

s: distribution width

V()

V3

intra-

inter-

(Gaussian) distribution of potential barriers:

(log-Gaussian) distribution of reorientation times:

if

2

0i2

2

)(ln

i e2

1)(lng s

s

wwwN

1iii10rot )(Lg)Q(A)()Q(A),Q(SDynamic structure factor:

Case study: Polycarbonates

C OC

OCH3

CH3

O

BPA-PC

Bisphenol-A polycarbonate

thermoplastic polymer with remarkable

• optical clarity

• mechanical properties

• commercial applications

– high Tglass transition

– large impact strength

– ductility.

depend strongly on architecture

impact strength

BPA-PC: ~2400 J/m

TMPC: ~70 J/m

(Fried et al. 1990)C

Rp

Rp Rp

Rp

OC

ORc

Rc

O

PC modified

Toughness

C

CH3

CH3 CH3

CH3

OC

OCH3

CH3

O

TMPC

C OC

OCH3

CH3

O

BPA-PC

TMPC

BPA-PCStress(N/m2)

Strain (m/m)

Polycarbonates

CH2 CH

PS

r(PC) = 1.198g/cm3

r(TMPC) =1.084g/cm3

Glassy BPAPCtough co-operative phenyl motion,

involve 1 monomer

(account for dielectric/mechanical activity).

Glassy TMPC most brittle PC substituted CH3 hinder backbone mobility;

poor chain packing (large free volume).

Packing

C OC

OCH3

CH3

O

BPA-PC

QENS:characterise dynamics of local reorientation.

C

CH3

CH3 CH3

CH3

OC

OCH3

CH3

O

TMPC

quantitative window scans

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 100 200 300 400 500

Temperature (K)

S(Q

,w~

0)

(T)/

S(Q

,w~

0)

(T~

0)

Q=1.82A-1

C

CH3

CH3 CH3

CH3

OC

OCH3

CH3

O

TMPC

Elastic scans

w res

00 arctan)Q(A12

)Q(A)0~,Q(S

RT

EA

e

0

0

')'()',()0~,(

w

wwwww dRQSQS

for a Lorenztian resolution

PARAMETERS

• <u2>(T) initial slope

• distribution: EA and s

• o

ASSUMED

• geometry EISF

• activation ansatz:

TMPC

-1.8

-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0 100 200 300 400 500

Temperature (K)

ln [

S(Q

,w~

0)

(T)/

S(Q

,w~

0)

(T~

0)]

Q=1.82A-1

Q=1.43A-1

Q=0.86A-1

124

2

KA106dT

ud .

Ea1~6

kJ/mol s1~1

Ea2=15

kJ/mol s1~5

C

CH3

CH3 CH3

CH3

OC

OCH3

CH3

O

TMPC

T

g

0

200

400

600

800

1000

-0.01 -0.005 0 0.005 0.01energy (meV)

S(Q

, w)

elasticquasielasticdatafit

Q=1.82Å-1

0.00

0.01

0.10

1.00

10.00

2 6 10 14

1000/T (K-1)

(

me

V)

IN10

1e

V

Mibemol

50eV

low temperature relaxation

E (meV)-0.005 0.000 0.005

Inte

nsity (

au

)

0

4K

11

TMPC first relaxation step:

3-fold CH3 potential

)()(9

))Qr(jo1(2)(

9

)Qr(jo45),Q(S ttrot wwww

w

w

(Colmenero et al, PRL 1998)

TMPCBPA-PC

• very low T low Eo

• rather sharp narrow

s

Mathiew equation: inelastic lines

candidate: rotational tunneling

AE4/3

At eE

wwith

Distribution of EA

highly asymmetric distribution of wt

-2.5

-2

-1.5

-1

-0.5

0

0 100 200 300 400 500

Temperature (K)

ln [

S(Q

,w~

0)

(T)/

S(Q

,w~

0)

(T~

0)]

BPA-PC

C OC

OCH3

CH3

O

BPA-PC

Tg

0

0.2

0.4

0.6

0.8

1

0 100 200 300 400 500

Temperature (K)

I(w

~0

,T)/

I(w

~0

,T=

0K

)

TMPC

BPA-PC

Q=1.82Å-1

0

0.05

0.1

0.15

0.2

0 200 400 600Temperature (K)

Apparent <u2>/Å

2

Ephenyl~37

kJ/mol s1~6

Ech3=15 kJ/mol

s1~3

(after Spiess et al. 1987)

compatible with TMPC

Distribution?

C

CH3

CH3 CH3

CH3

OC

OCH3

CH3

O

TMPC

CH2 CH

PS

TMPC: only PC miscible with PS, large FH

1st step: no resolvable perturbation

2nd step: broadened distribution

Glassy

polymers:

backbone chain conformation

Structural disorder

Blending

inter-

intra-

0

0.2

0.4

0.6

0.8

1

0 100 200 300 400 500

Temperature (K)

I(w

~0

,T)/

I(w

~0

,T=

0K

)

TMPC

TMPC/PSd

50:50

Q=1.82Å-1

molecular potential.

intramolecular environment

•average EA

•architectural

considerations

intermolecular limited effect on s

Conclusions: CASE STUDY

Characterisation local dynamics of PCs:

two architectures toughest (BPA-PC) & most brittle (TMPC)

exhibits two methyl relaxations of rather

different distribution of potentials

Phenyl + methyl

combined backscattering window scans, inelastic BS & TOFTechnique

TMPC

BPA-PC

Blending affects s(EA)

E