12th Oxford School on Neutron Scattering, 15-16 September 2011

Neutrons in soft matter

João T. CabralDepartment of Chemical Engineering

Imperial College London

Lecture 1 - Structure

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)

Soft Matter“molecular systems giving a strong response to

very weak command signal”

deGennes (1991)

Condensed matter: states are easily deformed by

small external fields, including thermal stresses and

thermal fluctuations.

Relevant energy scale comparable with room

temperature thermal energy.

Complex fluids: including colloids, polymers, surfactants, foams,

gels, liquid crystals, granular and biological materials. shear

Viscosity

Viscosity: proportionality coefficient

Simplest setupto measure viscosity

d

z S

f,

dz

d

S

f

d

Sf

Step-wise stress

starting at t =0

t

normal liquid

ln t

ln

polymer

ln

ln tln *

Viscoelasticity?t << * : elastic response

t >> * : viscous responseE

t

shear

angle

Movie: complex fluids are generally non-Newtonian... and structured

Neutron scattering is key in soft condensed matter

Neutron scattering is key in soft condensed matter

Atomic number

Scatt

eri

ng

len

gth

(fm

)

Effect of Mw on flow

Common soft matter

Effect of Mw on flow

Speciality

polymers

CH2 CH

Cl

n

PVC poly(vinylchloride)

CH2 CH

C O CH3

O

n

PMMA poly(methylmethacrylate)

CH2 CH CH CH2 n

PB poly(butadiene)

CH2 CH

n

PS poly(styrene)

CH2 CH2 n

PE poly(ethylene)

BPA-PC bisphenol-A polycarbonate

O C O C

CH3

CH3

n

O

Common polymers

Molecular weight, N

Polymer key properties

(size)

Tacticity

COOCH3

CH2 CH CH2

COOCH3

CH2 CH CH2

COOCH3

CH

COOCH3

CHCOOCH3

CH2 CH CH2

COOCH3

CH2 CH CH2

COOCH3

CH

COOCH3

CH

Polydispersity (distribution of sizes)

Glass transition (solid-liquid)

Crystallinity

Interaction parameter Combine properties to make new materials!

Soft matter: DNA

5 mm

Real space

Reciprocal space

Soft Matter: membranes, photovoltaics (BHJ)

Scattering theory reminderScattering cross section

coherent incoherent

Dynamic structure factor

Form factor

Structure factor

S(q)

Intermediate scattering function

FT (t,w)

Pair correlation function

FT (r,q)

Elastic structure factor wd

Reminder: Fourier Transforms

H(f) = - h(t) e2pift dt

h(t) = - H(t) e-2pift df

Fourier

transform:

F (H(f),t) = h(t)

F -1(h(t),f) = H(f)

H(f) = - h(t) e2pift dt

H(f) = 0 if (ft 1)

H(f) = A eif if (ft = 1)

t

h(t)

H(f)

f

h(t) = Aei(t+f)

Fourier Transform

f2 f3 f5

Time, space

Frequency,

reciprocal space

Reminder: Fourier Transforms

H(f) = - h(t) e2pift dt

h(t) = - H(t) e-2pift df

Fourier

transform:

F (H(f),t) = h(t)

F -1(h(t),f) = H(f)

Reminder: Fourier Transforms

Real space

Reciprocal space

SMALL-ANGLE NEUTRON SCATTERING

form factor

structure factor

incoherent background

contrast

number

density

volume

Absolute scattering intensity [cm-1]

Scattering length density

Relationship between q l q and d

( 2/sin4

ql

pq

qd

p2

small q ~ large d [large q ~ small d]

small l ~ large q ~ small d

[large l ~ small q ~ large d]

Radiation Wavelength

~ 500 nmlight

~ 1 ÅX-rays

neutrons ~ 5 Å

Bottom line:

radiation of small wavelength l can „see‟ smaller sample features d

(provided that contrast is sufficient).

Ångstrom:

1 Å = 10-10 m

1 nm = 10 Å

Example: crystalline structure of polymer

Semi-crystalline poly(ethylene) PE

lamella spacing is d ~ 20

nmq = 2p/d ~ 0.3 nm-1

Radiation Wavelength

~ 500 nmlight

~ 1 ÅX-rays

neutrons ~ 5 Å

( 2/sin4

ql

pq

Scattering angle

q ~ 0.27 degrees

sin(q/2) ~ 0.013, q ~ 0.7 degrees

sin(q/2) > 1 impossible!

the smallest dimension probed by wavelength l corresponds to largest angle q=180 degrees

(backscattering). For light dmin~ 0.25 mm, for X-rays or neutrons dmin~ 0.5 to 2.5 Å.

In typical experiments, scattering angles range from 0.1 < q < 70 degrees

A brief history of polymers

1920 Staudinger proposed plastics are long molecules with covalent bonds.

1940 Kuhn established they are usually flexible

1953 Flory described the shape and size of individual polymer molecules in solutions and melts.

1967 Edwards – polymer molecules in rubbers and glasses are trapped by neighbours in a “tube”. Chain confirmation ↔ trajectories quantum particles.

1971 de Gennes – molecules in tube move like snakes and eventually escape

1. Number of monomer units in the chain, N is large N >> 1.

synthetic polymers DNA42 1010 -N109 1010 -N

Main factors governing polymer physical properties :

3. Polymer chains are generally flexible.

2. Monomer units are connected in the chain. no freedom of

independent motion (unlike systems of disconnected particles, e.g. low molecular gases and liquids). Polymer systems are poor in entropy.

Image single molecule (AFM)

Magonov

Form factor: P

sizeshape

orientation

„sample‟

Form factor P

distance

histogram

Interference between scattered radiation

from different parts of the same object

(analytical solutions for common shapes)

Self-correlations

Multiple lengthscales

2p/q1

2p/q22p/q3

q3q2q1

I(q)

scattering spectrum corresponds to

different “magnifications”, thus several

approximations may be relevant

Scattering from a sphere

Incident beam

forward

scattering

(out of phase)

(in phase)

0.000001

0.00001

0.0001

0.001

0.01

0.1

1

0.001 0.01 0.1

I

q (Å-1)

R=100 Å

1/R

0.000001

0.00001

0.0001

0.001

0.01

0.1

1

10

0.001 0.01 0.1

Scattering from a (tiny) sphere

I

q (Å-1)

R=10 ÅIncident beam

forward

scattering

(nearly in phase)

(in phase)

1/R

Scattering from a random coil Debye form factor

0

0.2

0.4

0.6

0.8

1

0 0.05 0.1 0.15 0.2 0.25

0.001

0.01

0.1

1

0.001 0.01 0.1 1

I I

q (Å-1) q (Å-1)

-2

a=10 Å

N=100

N=1000

N=100

N=1000

N=100 → Rg≈4nm

N=1000 → Rg≈13nm

0

0.001

0.002

0 0.1 0.2 0.3

Scattering from a random coil Debye form factor

0

0.2

0.4

0.6

0.8

1

0 0.05 0.1 0.15 0.2 0.25

I Iq2

q (Å-1) q (Å-1)

a=10 Å

N=100

N=1000

N=100

N=100 → Rg≈4nm

N=1000 → Rg≈13nm

Kratky

depends

directly on a

5Rg-1

Polydisperse random coils Polydisperse debye form factor

0

0.2

0.4

0.6

0.8

1

0 0.05 0.1 0.15 0.2 0.25

a=10 Å

N=100 → Rg≈4nm

Nw=100, Nw/Nn=5

PDI=5

PDI=1

q (Å-1)

Mn/Mw gD

(normalised to PDI)

A polydispersity model

p

M

(Schultz-Zimm)

Useful form factors

S King

http://www.ncnr.nist.gov/resources/

distance

Ordered

structure

„crystal‟

Disordered

structure

Structure factor: S

distance

histogram

Radial distribution function,

provides information about their

relative position

Interference between radiation

scattered by distinct objects

Square well,

Percus-Yevick...

Interactions: Polymers in solution and melt

1953 Flory described the shape and size of individual polymer molecules in solutions and melts.

Ideal chains occur in q-solutions (ie, neutral solvent, A2=0) or in melt. Chains expand or contract depending on interactions: A2 (Second Virial coeff, for solutions) or (for polymer mixtures)A2>0 A2=0 A2<0

A2>0

A2<0

Polymer miscibility (1)

Flory-Hugginslattice

Binary mixture

Thermodynamics mmm STHG -

Combinatorial entropy

Enthalpy

Combinatorial entropy

BBAA lnnlnnR

Sff

-

Enthalpy

Boltzmann law

ABBABm TKH ff

ABBAB

B

BA

A

A

B

m

NNTK

Gfff

ff

f

lnln

0

0

mixing occurs

mmm STHG - only at high T

f,T

Phase boundaries?

Binodal

Spinodal

mff

21 B

m

B

m GG

( ( min21 BmBm GG ff

02

2

fmG

‘minima’

inflection points

Combinatorial entropy Enthalpy

ff

ff

ff

vNvNvTk

G

BBAAB

m )1()1ln(

)1(ln

--

-

0 1B1 B2

S1 S2

Composition

Binodal

To

T

To

Free Energy

of mixing

0 LCST

Spinodal

Thermodynamics mmm STHG -

Polymer miscibility (2)

Isotopic polymer mixture

Approximations: Guinier & Zimm

where

for a polymer coil

Interacting polymer mixtures

Zimm

where

Random Phase ApproximationRPA (de Gennes, 1979):

ovqSqSqS

12

21

~2

)(

1

)(

1

)(

1 -

spinodal

1/S(0) = G’’

TsTs determined by extrapolation of G’’ to 0

Ornstein-Zenike:

At low angle (qRg 1),

221

)0()(

q

SqS

spinodal

Linear in the Zimm representation as:

2)0(/1)(/1 AqSqS

22

FS q)0(S

)(2)q(S

1 -

q (A-1

)

0.01 0.11

10

100

246

243

240

235

220

200

185

o

AB

wBDnBBBwADnAAAAB vqgNvqgNvqS

ff

~2

)(

1

)(

1

)(

1-

Equilibrium: SANSTMPC/PSd 50/50

I (cm-1)

T

Random Phase Approximation

q2 (A

-2)

0.0002 0.0004 0.0006 0.0008 0.0010

0.05

0.10

0.15

0.20185

200

220

235

240

243

246

0

1/I (cm)

22

FS q)0(S

)(2)q(S

1 -

Orstein-Zernike

(

-

BB

B

wB

zB

AA

A

wA

zA

ABS v

a

N

N

v

a

N

Nv

ff

22

02

~~36

1/T (K-1

)

0.0015 0.0020 0.0025 0.0030

-X/v

o (

mo

l/cm

3)

0.0000

0.0001

0.0002

0.0003

X/vo=-0.578/T+0.00113

Xs/vo=2.277 10-5

mol/cm3

Ts=249oC

Interaction FH1-phase scattering

O TMPC (v/v)

0.0 0.5 1.0

Kra

tky a

sym

pto

te (

cm-1

A-2

)

0.000

0.005

0.010

0.015

150oC

250oC

Equilibrium: Kratky asymptoteTMPC/PSd 50/50

0.005

0.010

0.015

TMPC/PSd 50/50

0.005

0.010

0.015

q (A-1

)

0.05 0.10 0.15 0.20

I q

2 (

cm-1

A-2

)

0.005

0.010

0.015

TMPC/PSd 30/70

TMPC/PSd 70/30

0

0

00

22

21

ˆ

12)(

a

v

qqS off

22

2

2

11

2

1

0

2

21ˆ

v

a

v

a

v

a

ffff

Kratky asymptote and segment length

Non-equilibrium: Fluctuations & Phase separation

Concentration fluctuations

fo

+f

-f

Stable:equilibrium

Metastable:nucleation & growth

G 0Unstable:spinodal decomposition

0 1B1 B2

S1 S2

Composition

Binodal

To

T

To

Free Energyof mixing

0 LCST

Spinodal

300 s 500 s

700 s 1000 s

5 mm

Phase separation: spinodal decomposition

T

f’ f’’

t

10 mm

5 mm

200 nm

a

b

c

Phase separation

TMPC/PSd 70:30 MM

258oC

Q (A-1

)

0.002 0.004 0.006

Inte

nsi

ty (

cm-1

)

0

2000

4000

6000

8000 330 s

30 s

m: characteristic length of phase separation ~10s-100s nm

Composition

Free Energy

of mixing

B'

B''

C 0

Nucleation &

Growth

T0Comparatively SLOW, since activated process

Nucleation & Growth

Nucleation & Growth

r (au)

0 1 2 3

En

erg

y (

au

)

-15

0

15

0

50

100

150

- Bulk energy

Interfacial energy

Gm (r)

rcritical

Gbarrier

pp

- 23 r4gr3

4)r(G

Energy balance

bulk interface

Critical droplet

rc 2/g2

3

3

16

gGbarrier

p

Free Energyof mixing

CompositionC 0

B'

B''T0

Spinodal decomposition

FAST, spontaneous

concentration fluctuations

fff~

o

Spinodal decomposition

l

42

2

2

)/1()/1()( ll

l B

GARate -

F-

diffusion Sharp interfaces

lm Rate

growth

0

time

Cahn-Hilliard Cook theory

Spatially resolved “rheology”

T

f’ f’’

t

Opportunities & recent developments

nanocompositesH2C

HC

n

Bulk

Thermodynamics & phase separation

Concentration (wt%)

Ts

Temperature Spinodal

Binodal

Thin Films

Morphology

structure dynamics

Tute

jaet

al.

Nat

. Mat

. (2

00

3)

2, 7

62

Flow fields (and microfluidics?)

CTAC/ Pentanol/Water

Periodic constrictions: contraction/expansion flows

Periodic constrictions: contraction/expansion flows

Periodic constrictions: contraction/expansion flows

10 mm

i

ii

iii

a b

c d e

a b c d e

Optical micrographs: mixture of SDS / Octanol / Brine

q= 1 ml/h; extension rate of 104 s-1.

Periodic constrictions: contraction/expansion flows

Conclusion: Microstructure alignment & orientation

Spatio-temporal mapping:

Coil-to-globule transition

(Pollack, Austin, etc)

expanded

ms

collapse

b-lactoglobulin (BLG)

Spatio-temporal mapping:

Entangled polymers under flow

Soft colloids under flow

Forster et al. PNAS 2011

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

Design 3D composites

PRL 2010

GISANS & reflecometry

Summary

Intro

CH2 CH

C O CH3

O

n

HistorySoft matter

Mixtures & design

Outlook

0 1Composition

Binodal

To

TSpinodal

1

2

3

4

5

Form and structure