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Small-Angle Scattering in Materials SciencePrinciples and Applications
N. Sanjeeva Murthy
2017 Denver X-ray Conference
Polymer Workshop
New Jersey Center for Biomaterials
Rutgers University
1
OUTLINE
• Principles and Instrumentation
• Scattering from disordered structures (1D patterns)– Particles and voids
– Dilute and concentrated systems
• Scattering from ordered structures (2D patterns)– Lamellar structures
– Examples of 2D patterns and significance
– Deformation in semicrystalline polymers
– Analysis of 2D patterns
• Hydration in polymers (SANS and SAXS)– Partitioning of water
– Microphase separation
2
Systems, length scales and techniques
Source:Thiyagarajan/ANL 3
What is Small-Angle X-ray Scattering?
Amemiya and Shinohara. 2011 Cheiron School. University of Tokyo 4
Instrumentation
Not to scale
X-ray
source
Sample chamberDetector
From Rigaku U.S. A.
Statton camera (pinholes)
Beeman collimation (slits)
Kratky collimation
Franks’ focusing optics
Bonse-Hart collimation
• Collimation
• Long flight path
• Evacuated beam path (He)
• Alignment
5
6
Small- and Wide-Angle XRD
Nylon 6
Murthy and Grubb (2002) 7
The four limiting cases
Particulate
systems
Non-particulate
systems
Ordered structures
Dilute particles: Protein and polymer in solution, defects in crystals, inclusions in solids and void
Dense systems: Colloidal aggregates, catalysts, clusters of nanoparticles
Two-phase systems: Block copolymers, porous media, any microphase separated systems
Ordered structures: Block copolymers, semicrystalline polymers, multilayered structures,
liquid crystalline materials, nanostructures8
X-ray scattering
nmto
nm
d 1001
1.02
2 = 0.06 to 5
q~0.003 Å-1 to 0.5 Å-1
sin2sin4 sq
qd
2
sin2
Intensity often expressed as a function of a scattering vector q
(sometimes identified as Q, h or s)
Neutron (SANS) 0.1 < λ < 2 nm
X-ray (SAXS) 0.06 < λ < 0.2 nm
Light (LS) 400 < λ < 700 nm
The wavelength should be of the order of the size of the structure being studied.
The radiation chosen should maximize the contrast between the structure and its surroundings.
For diffraction only neutron and x-rays are useful
9
SAXS of particulate system
Scattering vector q
q-4 sphere
q-2 plate
q-1 rod
Larger structure Smaller structure
Inter-particle
interference
Guinier region
Radius of gyration
I(q) ~ exp(-q2Rg2/3)
Porod Region
I(q) ~ q-4
Fractal
dimensions
I(q) ~ q-(6-ds)
Inter-atomic
structure
(WAXD)
Scatt
ering inte
nsity I
(q)
Size Shape Interface Structure
10
Basic equations in SAXS
Define the Correlation Function (r):22 )(~))(()( VrVVr rjri
Pair Distance Distribution Function p(r) = r2 (r)
rji dVrrr )()()(~2 Autocorrelation function at r = ri -rj (a constant)
rjri
rriq
ji dVdVerrqI ji )()()()( Absolute scattered intensity
qr
qre iqr sin
, 1D scattered intensity isdrrdVr
24andSince drrqr
qrrqI 22 sin)(~4)(
dqqqr
qrqIrVdrr
qr
qrrVqI 2
2
2 sin)(
2
1)(4
sin)()(
11
I(q) and p(r) of particles
Flat
particle
Cylindrical particle
Spherical particle
Ellipsoids
Two ellipsoids - dimer
Courtesy of Dr. I.L.Torriani. From: Amemiya and Shinohara. 2011 Cheiron School. University of Tokyo 12
Dilute systems and Radius of Gyration (Rg)
0 2 4 6 8
2 (Degrees, Cu K)
Re
lative
in
ten
sity
0 4
8
1
2
233,600 Da, Dia 112 Å
29,200 Da, Dia 51 Å
1,900 Da, Dia 20 Å
1
100
10
Murthy, 1983
Rg is the mean square
distance to the center of
mass weighted by the
contrast of electron density.
Mol. Wt. Rg (Å)
233,600 43.4
58,400 29.6
29,200 19.9
7,300 12.9
3.700 10.0
1,900 8.0
0 10 20 30
q2 *103 (Å2)
Ln
(I(
q)
Dendrimers
Guinier Plot
I(q)=I(0) exp (-Rg2q2 /3)
Ln(I(q)) vs. q2
Ln (Iq) = Ln (Io) – (q2 Rg2/3)
Babinet Principle 13
Rg for simple objects
and modified Guinier equations
Guinier Approximation:
I(q) = I(0) exp(-q2Rg2/3)
Ln[I(q)].vs.q2 plot
where qmax.Rg < 1.0
Rg = (3.slope),
(Rsphere=(5/3)Rg)
= Average square
of end to end
distance
Rod-like Particle
I(q) = (1/q) Ic(0) exp(-q2Rc2/2)
A modified Guinier plot
Ln[q.I(q)].vs.q2
where qmax.Rc < 0.8
Rc=(2.slope), (R= d/2=2.Rc)
Sheet-like Object
I(q) = (1/q2) It(0) exp(-q2Rt2/12)
A modified Guinier plot: Ln[q2.I(q)].vs.q2
where qmax.Rt < 0.8
Rt= slope) , (Thickness=12.Rt)14
Particle shape effects
for spherical particles:
I(q) ~ 1/q4
for disk shaped particles:
I(q) ~ 1/q2
for needle shaped particles:
I(q) ~ 1/q
C. Windsor, J. Appl. Cryst. (1987) 15
Modeling of SAS Data with Some Form Factors
Polydisperse Spheres Cylinder vs. nanotube
Spherical shell
Nanotubes
DiskCylinder
Source:Thiyagarajan/ANL 16
Concentrated systems
A peak in scattering intensity
functions observed as a
consequence of interference
effects produced by spatial
correlation of closely located
SiO2 - based clusters
embedded in the polymeric
matrix.
Synchrotron SAXS Studies of Nanostructured
Materials and Colloidal Solutions. A ReviewA.F. Craievich
Mat. Res. v.5 n.1 São Carlos ene./mar. 2002
www.scielo.br/img/fbpe/mr/v5n1/a02fig07.gif
17
Random two-phase systems – Porod’s law
Adapted from: Thiyagarajan/ANL
Two complementary structures produce
the same scattering (Babinet principle).
2
4)(
2)(Lim
S
qqI
q
q
I = k.q-4
For sharp
interfaces
18
BA
B
log (q)
q-4
log I(q)
Intercept Surface area
)(4))(2())(( 2 qLnSLnqILn
Invariant Q
Where ne is the number of excess electrons
and is the relative electron density = (r) -
22
2
2222
)(2
1)0(
)()(4)()0(
VdqqqIV
nVdrrrVI e
22
0
2 2)( VdqqIqQ
<2 > does not depend on the structure
19
dqqqr
qrqIrV
drrqr
qrrVqI
2
2
2
sin)(
2
1)(
4sin
)()(
The integrated intensity over scattering space, the invariant,
is equal to the total irradiated volume times the mean-square
electron density fluctuation- independent of domain shapeKratky plot
Invariant and the Porod’s law
V
S
Q
qIq
VQ
)1(
1)(lim
))(1(2
4
22
S/V is the surface to volume ratio.
Valid for single particles, for densely
packed systems, and two phase systems.
To avoid using absolute intensity, make use the invariant Q
2
4)(
2)(Lim
S
qqI
q
20
Porod scattering
Final slope and internal surface
Smaller objects have
larger surface area
Larger objects have
lower surface area
21
USAXS data from a fine powder borosilicate glass during corrosion
Source:Thiyagarajan/ANL and http://www-drecam.cea.fr/scm/lions/techniques/saxs/
1 Non altered glass; 2 Altered glass (2 weeks); 3 Altered glass (8 weeks)
Scattering exponents and Fractal dimensions
Where Dm and Ds are called the mass fractal and surface fractal dimensions,
and “6” is replaced by “2d” for a d-dimensional object.
4)( qqIPorod’s law:
Valid for two-phase system with sharp boundaries.
Diffuse boundaries reduce the exponent.
22
for mass fractals:
)6()( sD
qqI
mDqqI
)( Where 1 ≤ Dm < 3
and for surface fractals: Where 2 ≤ Ds < 3
In general,
Summary IDisordered structures: particulate and random two-phase
systems
• Particulate systems (Guinier and Kratky plots, modeling)
− Size and shape
− Colloids, globular proteins, voids
• 2-phase systems (Kratky and Porod plots)
− Surface area, volume and S/V ratio
− Fractal dimensions
• Other parameters (I(0) and simulation)
− Molecular weight
− Particle/pore-size distribution
− Long-range organization
23
Organized structures
and 2D patterns
24
Small Angle Scattering/Diffraction
incident
beam
Sample Detector
x
y
I(q) = A (p-s)2n V12 P(q) S(q)
Radiation and
material
dependent
Spatial arrangement
of atoms
V1 - scatterer volume
n - scatterer concentration (g/mL)
d - scatterer density
bi, Mi - scattering length and atomic weight of elements of scatterer
NA – Avogadro’s number
P(q) - form factor of the scatterer
S(q) - structure factor
A - instrument const (scale factor)V: irradiated sample volume
=NAd(∑bi/∑Mi)
2
21 2 1
1
1( )
iQrdQ e dr
d V V
q4sin/
q=2/L; L – repeat distance
Source:Thiyagarajan/ANL 25
Fibers: Self-assembled, functionally optimized, unidirectional, nanostructures
Model for a semicrystalline fiber
(b)(a)
Fringed micelle
Folded chain
lamellae
26
Model fitting (Profile analysis)
I(q) = Ib + Jd(q) + Jl(q)/q2
I = IB + ID + IL
cm
Ib is linear background
Jd is the intensity from independent scatterers, represented by [(sinq)/q)]2
JL is the intensity of the lamellar peak generated by [(sinq)/q)*(ae (q-b)/c1 + ae (q-2b)/c]2
Where a, b and c are related to the volume fraction, lamellar spacing and the size
Murthy et al., Macromolecules 31:142-152 (1998)
Wang et al., J< Appl. Cryst 33:690-694 (2000) 27
Correlation function analysis
dqqrqIqr )cos()()(0
2
For 1D scans from lamellar structures
L
lc
dtr
Amemiya and Shinohara. 2011 Cheiron School. University of Tokyo 28
Correlation function is a plot of the degree
of over lap (correlation) between the object
and its copy as the copy is translated.
Analysis of 2D Small-angle Patterns
Nylon 6Murthy and Grubb (2002) 29
Interpretation of the Meridional Reflections
50 100 150 200 250
50
100
150
200
250
(a)
Murthy and Grubb, J. Polym. Sci. Polym. Phys.(2006) 44: 1277-1286
Keller’s group, J. mat. Sci. (1968) 3:646-654
Matyi and Crist, Jr. , J. Polym. Sci. Polym. Phys. (1978) 16:1329-1354
Tilted lamellae
L
Lamellar
spacing
= 2/q
Stack height
50 100 150 200 250
50
100
150
200
250
(b)
2*Tilt angle
30
Lamellar stack Checkerboard pattern
Stack diameter
Tilt-angle and Fiber Properties (PET)
6
7
8
9
50 60 70 80 90 100 110 120
Tilt Angle, (Degrees)
Ten
acit
y (
g/d
en
ier)
Murthy and Grubb, J. Polym. Sci. Polym. Phys.(2006) 44: 1277-1286 31
0
4
8
12
16
0 200 400 600 800
Stress (MN/m2)
Str
ain
in L
(%
)
Four-Point Strain
Modulus = 5.6 GPa
100 200 300 400 500
100
200
300
400
500100 200 300 400 500
100
200
300
400
500
100 200 300 400 500
100
200
300
400
500100 200 300 400 500
100
200
300
400
500
100 200 300 400 500
100
200
300
400
500
100 200 300 400 500
100
200
300
400
500
Murthy and Grubb (2003) J. Polym. Sci. Polym. Phys. 41:1538-1553.
Murthy and Grubb, J. Polym. Sci. Polym. Phys. (2002) 40: 691-705.
A sequence of SAXS patterns
from PET as it is being stretched
from zero strain to ~ breaking
elongation
Mechanical strength
determined by the lamellar structure
32
0
4
8
12
16
Fib
er
Str
ain
(%
)
Fiber Strain
Modulus = 4.4 GPa
Changes in the Lamellar Structure During Deformation:Coexistence of two families of lamellae (Bi-stable lamellar orientation)
0
100
200
300
400
500
Azi
muth
al I
nte
nsi
ty (
Counts
)
0
100
200
300
Azi
muth
al I
nte
nsi
ty (
Counts
)
0
100
200
300
0 100 200 300 400 500
Distance Along x (Pixels)
Azi
mu
tha
l In
ten
sity
(C
ou
nts
)
(a)
(b)
(c)
Murthy and Grubb, J. Polym. Sci. Polym. Phys. (2002) 40: 691-705.
Stretch
Relax
33
34
2D Patterns From Amorphous Polystyrene
Macromolecules (1982) 15:880-882
Hadziioannou, Wang, Stein and PorterSANS scattering studies on amorphous polystyrene
oriented by solid-state extrusion
Anisotropy of Rg agrees qualitatively with that predicted
from affine model
35
Azimuthal Asymmetry is Elliptical
• Polystyrene– Summerfield and Mildner;J. Appl. Cryst. (1983) 16:384-389
• Crazed plastics – [Steger and Nielsen, 1978]
• Voids in pyrolitic graphite
– [Hamzeh and Bragg, 1976; Bose and Bragg, 1978]
• Precipitates in Ni-based alloys – [Schwahn, Kesternich and Schuster, 1981]
• BCC lattices of spherical domains in in triblock copolymers BCC– Brandt and Ruland, Acta Polymer. (1996) 47: 498-506
– Styrene-isoprene-styrene, 30 wt% styrene, undergo affine deformation on uniaxial stretching and give
rise to elliptical interference patterns
• Stretched macromolecules– Hammouda et al. [J. Appl. Cryst. (1990) 23:1-5] – Macromolecules follow external stretching affinely, but small chain portions relax more rapidly than
large ones (eccentricity of the ellipse decreases and approaches 1 with increasing Q)
36
Reflections fall on an Elliptical trajectory
Murthy, Zero and Grubb, Polymer (1997) 1021-1028
From: X-ray diffraction From Polymers (N.S. Murthy)
In: Polymer Morphology (Qipeng Guo)
o
EM
o
o
o
z
xLLL
abz
bxa
bz
b
z
a
x
tan;tan
tan111
1
2222
2
222
22
2
2
22
200
160
120
80
80 120 160 200 240
37
Elliptical Cylindrical coordinates
Solution: Describe the intensities in elliptical cylindrical coordinate
system so that variation along one of the coordinates (u) does not affect
the intensity along the other (v); i.e., using two orthogonal functions
Problem: SAS data from oriented specimens (e.g., polymers) do not fall on
either a Cartesian or polar grid
)sin(
)cos(
)()()0,0(),(
22
vuy
vuAx
byscoordinateCartesiantorelatedarevanduwhere
vgufIvuI
Instead of
I =I(0,0) f(x) g(y) or I = I(0,0) f(r) g()
usex
y
u
v
38
Lamellar peaks fitted in elliptical coordinates
using a single function with 4 parameters: u and v (for position), and u and v (for widths)
50 100 150 200 250
50
100
150
200
250
(c) Final Load
50 100 150 200 250
50
100
150
200
250
(d) Relax
50 100 150 200 250
50
100
150
200
250
(c) Final Load
50 100 150 200 250
50
100
150
200
250
(d) Relax
Observed Fitted
Murthy and Grubb, J. Polym. Sci. Polym. Phys. (2002) 40: 691-705. 39
Androsch et al. J. Polym. Sci. Polym. Phys. (2002) 40:1919-1930
Wang, Murthy and Grubb, (2007) Polymer 48:3393-3399
Elliptical Fit to “Butterfly” Patterns
40
Contour map
Elliptical fit to the peak maxima
Equatorial streak: Surfaces or Voids?
D.T. Grubb and N.S. Murthy, Macromolecules (2010) 43:1016-1027 (2010) 41
Central diffuse scattering
The equatorial streak
sx
sz
O
A
(sx,sz)B
Isotropic
Scattering
Equatorial Streak
D
C
(a) (c)(b)
Equatorial scattering can be
described by two components:
1. Isotropic scattering
2. Extended equatorial streak
Diamond, fan and propeller shaped patterns can be generated by the
same equations but different parameters suggesting that they arise from
the same features in the sample
42
Summary -II
• Discrete reflections arise from organized structures, typically lamellae in semicrystalline polymers
• They can be analyzed by using correlation functions or by modeling (profile fitting)
• In many instances 2D patterns from these structures fall on elliptical grid. They can be best analyzed in elliptical coordinates
• Analysis of 2D patterns gives valuable insight to the relation between structure and deformation
• Surface scattering is the major contributor to the equatorial streak
43
SANS and Study of Hydration
• Distribution of the amorphous phase in semicrystalline fibers
• Hydration that eventually leads to degradation in
copolymers with hydrophilic blocks
44
Distribution of the amorphous phase in fibers:
Partitioning of water in the amorphous regions
45
2D SANS pattern (isointensity contours)
Fibrillar model to explain the 2D SANS patterns
1H -0.374 (10-12 cm)2H(D) 0.66712 C 0.66514 N 0.99316 O 0.58019 F 0.55628 Si 0.415
Cl 0.958
Murthy et. al.
J. Appl. Polym. Sci. (1990) 40: 249-262
J. Polym. Sci. Poly. Phys. (1994) 32: 2695-2703
J. Polym. Sci. Polym. Phys.(1996) 821-835.
Textile Res. J. 67:511-520 (1997)
Meridional scan
Non heat set
Suessen, Dry heat set
Superba, wet heat set
Equatorial scan NHS
Suessen
Superba
Lamellae
with tilted
fold
surface
Extended
interfibrillar chains
45
Distribution of water in amorphous polymers:
SANS experiments
HN
O
O
O
O
O
O
O
O
O
x
m
1-x
Time ~ 0 h
Time = 1 h
125 Ang
dm
180 Ang
dm
SANS snapshots at 20 s intervals
SANS: Water molecules are not molecularly dispersed
throughout the matrix. 46
Structural changes upon hydration:
SAXS experiments
0
0.01
0.02
0.03
0.04
0.05
0 0.5 1 1.5
q (1/nm)
Inte
nsit
y
Dry D6 078T_19
Wet D6 190 T_19
SAXS
E0015 (2k)
Hydrophilic segments
(e.g., PEG)
Hydrophobic
segments
(e.g., DTE)
Dry
Hydration causes phase separation of PEG segments into PEG-rich
domains. Water resides in these domains. Thus, water molecules are not
molecularly dispersed throughout the matrix.
~15 nm
Wet
Hydrated,
PEG-rich domains
47
Conclusions
• Diffuse scattering can be analyzed to determine the size (Rg), shape (DDF and form factor fitting), volume (invariant), surface area (Porod limit), and fractal nature of the scattering entities
• Discrete reflections from anisotropic structures can be analyzed in elliptical coordinates to study the deformation in ordered structures, e.g., lamellar structures in semicrystalline polymers
• SANS can be used to study the distribution of water and the phase separation behavior
48
Small-angle Scattering of X-rays. Guinier and Fournet, Wiley (1955)
X-ray Diffraction. Guinier, Freeman and Co. (1963)
X-ray Diffraction Methods in Polymer Science. L.E. Alexander, Wiley (1969)
Small-angle X-ray Scattering. O. Glatter and O. Kratky, Academic Press
(1982)
Methods of X-ray and Neutron Scattering in Polymer Science. Ryoong-Joon
J. Roe, Oxford University Press (1999)
X-Ray Scattering of Soft Matter. N. Stribeck, Springer Publishers (2007)
X-ray Diffraction from Polymers. Chapter 2. Polymer Morphology (Ed. Qipeng
Guo, Wiley 2016)
References
49
Acknowledgements
• Karl Zero and David Grubb – Elliptical analysis
• Wenjie Wang – Work on “Butterfly" patterns,
CDS and polycarbonates
• Steven Weigand at the APS
• P. Thiyagarajan, S.V. Pingali, D. Wozniak and
W. Heller for SANS
• J. Kohn and Kohn Group members at Rutgers
for the tyrosine-derived polycarbonate work
• Various websites for some of the graphics
50