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13th Oxford School on Neutron Scattering, 12-13 September 2013
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!
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)
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
2
2
FSq
)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)
2
2
FSq
)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