1. Small-Angle Scattering Part I: Theory N
Small-Angle Scattering Part I: Theory N. Sanjeeva Murthy Rutgers University Piscataway, NJ 08854

3. Small angle scattering and Bragg’s Law
Scattering vector
Recall Bragg’s Law: λ=2d sinθ
small q – large d
large q – small d
qmin=(4p/lmax) sinqmin
qmax =(4p/lmin)sinqmax
q~ Å-1 to Å-1

4. Radiations used for SAS
Neutron (SANS) < λ < 2 nm
X-ray (SAXS) < λ < 0.2 nm
Light (LS) 400 < λ < 700 nm
For diffraction only neutron and x-rays are useful

5. Source:Thiyagarajan/ANL

6. Particulate
systems
Semicrystalline
Non-particulate
systems

7. Dilute system Rg Particle weight Particle volume Particle shape
Source: Wikipedia etc.

8. Theory Define autocorrelation function at r = ri -rj (a constant)
Define correlation function γ(r):
Define pair-correlation function P(r) for a homogeneous particle:

9. SAS probes fluctuations in chemical composition & density
SAS: Radius of gyration, Rg
Pair-distance distribution function, P(r)
P(r)
r (Å) D
max
Probability distribution of
distances reflects the size and
shape of the particle
I(q)
q (Å - 1 )
Fourier Transformation
P(r) = FT ([qr*I(q)]
I(q)=I(0)exp (Rg2q2 /3)
Histogram al such distancs r lm ij D
max
Rg = RMS distances
from center of mass -
q = 2p/D
q=4psinq/l
Limited to q < 1/Rg
Source:Thiyagarajan/ANL

10. Typical Particulate SAXS data
Murthy (1983)

11. Guinier Plots of particles of different Radius of gyration
I(h) = I(0) exp (-q2 Rg2/3)
Aharoni and Murthy, Polymer Commun. 24: (1983)

12. Source:Thiyagarajan/ANL

13. Guinier and Modified Guinier Analysis
Whole Particle
Guinier Approximation: I(q) = I(0) exp(-q2Rg2/3)
Ln[ I(q)].vs.q2 plot where qmax.Rg < 1.0
Rg = Ö(3.slope), (Rsphere=(5/3)Rg) M=(1000.I(0).d2.Na)/(C.Dr2)
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=Ö(2.slope), (R=Ö2.Rc) M/L = (1000.Ic(0). d2.Na)/(p.C.Dr2)
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= Ö(12.slope) , (Thickness=Ö12.Rt) It(0) µ Mt (Mass per unit area)
Source:Thiyagarajan/ANL

14. Modeling of SAS Data with Some Form Factors
Spherical shell
Polydisperse Spheres
Cylinder vs. nanotube
Disk
Cylinder
Cylindrical Nanotubes
Source:Thiyagarajan/ANL

15. The form factor
The form factor describes how d (q) / d is modulated by
interference effects between radiation scattered by different
parts of the same scattering body, i.e., it is depends on the
particle morphology or shape:
Scattered intensity:
I (q) = ||2 |dr e-iq · r|2 = ||2 |F(q)|2
where F(q) is the shape of the particle
For a sphere where V = 4(r3)/3
F(q) = 3[sin (qr) - qr cos (qr)]
(qr)3
Source:Thiyagarajan/ANL

16. Small Angle Scattering/Diffraction
incident
beam
Sample
Detector 2q x y f
I(q) = A (rp-rs)2n V12 P(q) S(q)
r=NAd(∑bi/∑Mi)
q=4psinq/l
q=2p/L; L – repeat distance
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)
Radiation and material dependent
Spatial arrangement of atoms
V: irradiated sample volume
Source:Thiyagarajan/ANL

18. Hammouda/NIST

19. Small angle scattering by particles or voids
General formula:
Inverse Fourier Transform:
Values at q=0 and r=0:
The Invariant, Q
Integrated intensity over scattering space:

20. The integral scattering, the invariant, is equal to the total irradiated volume times the mean-square electron density fluctuation - independent of domain shape

21. Interference effects
The interparticle structure factor is a function that describes how
d (q) / d is modulated by interference effects between
radiation scattered by different scattering bodies. rj rk
rj - rk
Np Np Np
d∑(q)/dΩ = (1/V)∑ |Fk(q)|2 + (1/V) ∑∑Fk(q)Fj*(q)eiq(rk-rj))
k= k=1 j=1
Source:Thiyagarajan/ANL

22. Inter-particle structure factor
S(q) depends on the degree of local order in the sample.
SAS can be used to gain information about the relative
positions of the scattering bodies through the
radial distribution function:
G(r ) = 4 Npr2g(r )/V
Source:Thiyagarajan/ANL

23. Pair correlation function
If isotropically distributed, we can average over orientation: 
S(q) = 1 - 4Np  [g(r) - 1] (sin(qr)/qr) r2 dr
g(r)
1 - 2
Source:Thiyagarajan/ANL

24. C. Windsor, J. Appl. Cryst. (1987)
. . . In the Porod limit
Valid for single particles, and
for densely packed systems.
for needle shaped particles:
I(q) ~ 1/q
for disk shaped particles:
I(q) ~ 1/q2
for spherical particles:
I(q) ~ 1/q4
C. Windsor, J. Appl. Cryst. (1987)

26. Random two-phase systems Babinet’s principle
(q)
Source:Thiyagarajan/ANL

27. Porod scattering Final slope and internal surfaceB A
log (q)
q-4
log I(q)
Smaller objects have larger surface area
Larger objects have lower surface area
Source:Thiyagarajan/ANL

28. Scattering Exponents and Fractal Dimensions
Porod’s law:
Valid for two-phase system with smooth boundaries
Deviations from this law can be taken into account by writing
Ds is called the dimensionality of porous fractal with a minimum value of 1 for Euclidean systems, and a maximum of 3.
In general 6 is 2d for a d-dimensional object

29. SAS from semicrystalline polymers
Nylon 6
Murthy and Grubb (2002)

30. Fibrillar model to explain the 2D SAS patterns
SAXS and SANS patterns
2D SANS (above) and SAXS (below) patterns shown as isointensity contours
Equatorial scans
Above: SANS
Below: SAXS
Fibrillar model to explain the 2D SAS patterns
Murthy Archives

31. Morphology Information from Correlation Function
Theory
Experiment (PE at 37oC)
Φc=0.85
Si=0.065 nm-1

32. Murthy, Grubb and Zero Macromolecules 33:1012-1021 (2000)

33. Elliptical Cylindrical Coordinates
SAS data from oriented specimens (e.g. polymers) are
not well Described in Cartesian or polar coordinates.
Would like to obtain a separable function
that describe the scattering
I = f(x) g(y) or I = f(r) g()
SAS data can be well described in
elliptical cylindrical coordinate system

35. Elliptical Features in SAXS Patterns3 4 5
Tan 2 f L
(nm ) 20 40 60 80
100
120
140 1 x y o χ z
Macromolecules 33: (2000) and J. Polym. Sci. Polym. Phys. 44: (2006)

36. A scheme for explaining the observed elliptical patterns
(a)
(b)
(c)
(d)
(e) X 
Murthy and Grubb, J. Polym. Sci. Polym. Phys. 44: (2006)

37. Three classes of discrete SAS patterns in semicrystalline polymers
(b)
(c)
Wang, Murthy and Grubb, Polymer 48: (2007)

38. Elliptical fit to bar-shaped and 4-point curved-in SAS patterns
Z L
LM Zo
Wang, Murthy and Grubb, Polymer 48: (2007)

39. Elliptical Fit to “Butterfly” Patterns
(d)
Wang, Murthy and Grubb, Polymer 48: (2007)

40. Conclusions (1/2): Information and Phenomena
SAS (High sensitivity: nm)
Size
Shape
Molecular weight
Particle/Pore size distribution
Interaction
Fractal Dimension
Crystallinity
Long-range organization
Phenomena
Self Assembly
Phase transition
Thermodynamics/kinetics
Process-performance correlations
Source:Thiyagarajan/ANL

41. Conclusions (2/2) Small angle scattering – versatile to study . . .
Polymer materials
Conformation of polymer molecules in solution and in bulk
Structure of microphase-separated block copolymers
Factors affecting miscibility of polymer blends
Biomaterials
Organization of biomolecular complexes in solution
Conformational changes of proteins, enzymes, complexes, membranes, . . .
Pathways in protein and RNA folding
Chemistry
Colloidal suspensions, microemulsions, surfactant micelles
Molecular self-assembly in solution and on surfaces
Metals and ceramics
Crystallization, Deformation microstructures, precipitation
Source:Thiyagarajan/ANL

42. References 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)