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