Post on 10-Jun-2020
transcript
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.