Small Angle X-ray

Scattering

The last resort of the desperate!

Nick Terrill – Principal Beamline Scientist, Scattering, Diamond

SAXS at Diamond

Beamline Progress and Science highlights

Small-angle, non-crystalline

diffraction provides essential

information on the structure

and dynamics of large

molecular assemblies in low

ordered environments. These

are characteristic of living

organisms and many complex

materials such as polymers

and colloids.

• Used for Archaeology

Biology

Biomaterials

Ceramics

Colloids

Cultural heritage

Environmental science

Forensic science

Liquid crystals

Mineralised tissue

Polymers

Surfactants

SAXS at Diamond

Beamline Progress and Science highlights

Probing the Length Scales

/USAXS

Scattering

X-ray scattering is probing distances that are large

compared to inter-atomic distances. Characteristics

are:

• Random orientation of particles (i.e. no long-range

order) leads to scattering rather than diffraction

(determination of size and shape)

• Electron density variations at the particle-matrix

interface cause x-rays to scatter.

• The scattered intensity, I(q), is measured in terms of

the scattering vector, q.

Scattering by two point centres

h or q

S

S0

P

o

r

q q

From G. Porod, Ch2 in “Glatter and Kratky”

X-ray scattering

dVrerqAVr

qir

.)()( Amplitude: (Volume Integral)

Where (r) is the relates to the electron density

and q is the scattering angle

Particle in solution => thermal motion => Particles have

a random orientation/x-ray beam. The sample is isotropic.

Only the spherical average of the scattered intensity is

experimentally accessible.

Intensity: I(q) = <A(q).A*(q)>

Porod's law: specific surface and interface

• When two media are separated by a sharp interface, the

scattered intensity follows an asymptotic law in the high

q region:

I(q)=Aq-4+B. • This law is called the Porod's limit and has more

sophisticated expressions in the case of complicated

interfaces.

• The asymptotic value, when the electronic contrast of

the sample is known, and when the intensity is

expressed in absolute scale, allows the calculation of the

specific surface S (in cm2/cm3) of the particles.

Surface Fractal Laws

• For a smooth surface S(r) = r2, and for a rough

surface S(r) = rds, where ds is the surface fractal

dimension that varies from 2 to 3

– I(q) proportional to qds-6

• Surface fractals display power-law decays

weaker than Porod's Law and are termed

positive deviations from Porod's Law.

• From SAXS pattern:

• Particle size

• Particle shape

• Polydispersity

• Kinetics

• 2q = scattering angle, = radiation wavelength

• N = no. of scatterers in volume V of sample.

• Vparticle = volume of the individual scattering

entity.

• = density of the particle or the matrix

• P(q) = Form factor and S(q) = Structure Factor

Source.

Monochromator,

Optics

2q

=

Solution

I(c,q)

Motif (Protein)

P(0,q)

*

* Lattice

. S(c,q)

)()()()(

)( 22 qSqPVV

N

d

qdqI matrixsampleparticle

What do we mean by “size”?

Radius of gyration:

Rg2 is the average

squared distance of

the scatterers from

the centre of the

object 2

2

1 1

1 3

Rg2 =(12+ 12+ 12+ 22+ 22+ 32 )/6=20/6

Rg=√3.333 = 1.82

Form Factor for simple shapes Sc

atte

rin

g in

ten

sity

q

Sphere

Cylinder

Disk

Ellipsoid

RRg5

3

R

12

LRg

L

d

22

DRg

t

D

32

34

23

1a

b

b

aRg

For more complicated shapes see reviews by J. S. Pedersen – http://www.chem.au.dk/~jansp/resdescription.html

Solution SAXS: Rg, I0 and P(r)

0.001

0.01

0.1

1

0 0.5 1 1.5 2

Inte

nsi

ty I(

q)

q [nm-1] qmax

Experimental

Scattering Profile

λ

πsinθ4q

q min

distance r [A]

0

0.004

0.008

0.012

0.016

0 2 4 6 8 10

p(r

)

Dmax

Pair Distribution Function

inverse Fourier Transform of the

scattering density

Rg = slope = shape function

independent radius

I (at q=0) or intercept

proportional to number of

particles / volume of solution

PDF = shape and size info

0.6

0.7

0.8

0.9

1

0 0.02 0.04 0.06 0.08 0.1

q 2 [nm2]

Inte

nsi

ty I

(q)

I(0)

Radius of Gyration

Guinier Plot

q 4 p q

sin

SAXS studies on silica templated with

polyelectrolyte-surfactant complexes

3

4

56

10-10

2

3

4

56

10-9

2

Inte

ns

ity

(c

m-1

)

3 4 5 6 7 8 9

0.12 3 4 5

q (angstrom-1

)

4

6

810

-9

2

4

6

810

-8

2

4

Inte

ns

ity

(c

m-1

)

0.012 3 4 5 6 7

0.12 3 4 5

q (angstrom-1

)

13.9 Å 10 Å

17.2 Å 12.5 Å

Oblate form CTAB micelles

10min 26min

41 Å

47 Å

10-8

10-7

10-6

Intensity(a.u.)

0.50.40.30.20.10.0

Q(angstrom)-1

40003000

20001000

Time(s)

0.037CTAB/40LPEI/0.025TMOS

10-13

10-12

10-11

10-10

10-9

10-8

10-7

10-6

Intensity(a.u.)

0.50.40.30.20.10.0

Q(angstrom)-1

40003000

20001000

Times(S)

Core radius 10.14 Å

Shell thickness 11.8 Å

Core length 836 Å

Bin Yang, Robben Jaber, and Karen J. Edler, Langmuir, 2012, 28 (22), 8337, DOI: 10.1021/la3014317

Melanie C. O’Sullivan, Johannes K. Sprafke, Dmitry Kondratuk, Corentin Rinfray, Timothy D. W. Claridge, Alex Saywell,

Matthew O. Blunt, James N. O’Shea, Peter H. Beton, Marc Malfois, Harry L. Anderson,

Nature Letters Volume: 469, Pages: 72–75, 2011

Vernier templating and

synthesis of a 12-porphyrin

nanoring.

Size polydispersity

2

2

2

3

0

2

2

)(exp

2

1),(

)(

)]cos()[sin(3),(

),(),()(

R

a

R

R

R

RRRD

qR

qRqRqRVRqP

dRRDRqPqI

p

R

D(R,R)

Size polydispersity

Smeared form factor P(q) for a sphere vs q showing the damping of

Porod oscillation with increasing polydispersity (eff). The oscillations

disappear for eff > ~0.21. The mean particle diameter a0 = 100nm for

all calculations. Note the overall q-4 power law for q >0.01nm-1. The

calculations terminate in the Guinier regime at low q.

Gold colloid

Gold colloids

q [Å-1

]

0.01 0.1 1

I(q

) [c

m-1

]

0.0001

0.001

0.01

0.1

1

10

Data

FitSize distribution

R [Å]

0 10 20 30 40 50

D(R

)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Rav

= 25.5 Å

= 4.4 Å

The spherical gold colloidal

particles coated with thiols

can be dissolved in an organic

solvent like toluene

The Stability of Silver

Nanoparticles in a Model of

Pulmonary Surfactant

Bey Fen Leo, Shu Chen, Yoshihiko Kyo, Karla-Luise Herpoldt, Nicholas J. Terrill, Iain

E. Dunlop, David S. McPhail, Milo S. Shaffer, Stephan Schwander, Andrew Gow,

Junfeng Zhang, Kian Fan Chung, Teresa D. Tetley, Alexandra E. Porter, and Mary P.

Ryan, Environ. Sci. Technol. 2013, 47, 11232−11240, DOI: 10.1021/es403377p front

of the Huns local

Iron oxyhydroxides in the environment Thermodynamics vs. Kinetics

• Most stable phases are goethite

(FeOOH) and hematite (Fe2O3)

• Poorly-ordered iron oxyhydroxide

(ferrihydrite) very common in soils and

sediments

Iron oxyhydroxide

bearing

contaminant plume

(Restronguet

Creek, Cornwall)

Acid mine

drainage

Fe2+ (aq)

Fe2+ (aq)

Fe2+ (aq)

Fe2+ (aq)

Fe2+ (aq)

Fe2+ (aq)

oxidation &

hydrolysis

Formation of ferric oxyhydroxide

nanoparticles

Poorly–ordered

nanoparticles Dissolved ferrous iron

(Janney et al., 2000)

Schwertmannite Fe16O16(OH)12(SO4)2

Ferrihydrite FexOyOHz.nH3O

Shaw et al

Newcastle University

8/11/05

Pair/Size distribution function (pure)

(PDF/SDF)

0.E+00

1.E-11

2.E-11

3.E-11

4.E-11

5.E-11

0 10 20 30 40 50Radius (Å)

N(R

)

60 seconds

300 seconds

600 seconds

840 seconds

180 seconds

0.0E+00

5.0E-09

1.0E-08

1.5E-08

2.0E-08

0 20 40 60 80Radius (Å)

P(r

)

60 Seconds

120 seconds

240 seconds

180 seconds

300 seconds

480 seconds

600 seconds

840 seconds

pH = 4

Pair distribution function (Rg more accurate: based on full scattering

pattern not only low q)

• Equant particle shape at

beginning of reaction

• Elongated particles forming by

end of reaction

Size distribution function (degree of polydispersity)

• Monodispersed system at

beginning of reaction

• Slight increase in

polydispersity with time

Shaw et al

SDS micelle (Soap!)

20 Å

Hydrocarbon

core

Headgroup/counterions

SDS 1wt%

q [Å-1

]

0.1 0.2 0.3

I(q

) [c

m-1

]

0.001

0.01

0.1

Data

Fit

(r) for SDS micelle model

r (Å)0 10 20 30 40 50

(r)

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

core

Headgroup/counterions

ASAXS example from BESSY (Berlin)

A. Hoell et al, Scripta mater 44 (2001) 2335

Fibrillin Protein Fragment 13

Wess et al – Cardiff

What if you have a Non-Dilute system?

• Scattering (Interference)

determined by spatial

dimensions

• Form Factor - P(q) -

particle size and shape

(intraparticle)

• Structure Factor - S(q)

interparticle correlations

function of local order and

interaction potential;

complex if correlation

between position and

orientation

Concentration effects (Sq)

Semicrystalline Block Copolymers

Spherulite

O(10 mm)

Lamella

O(10-100 nm)

Unit cell

O(1-10 Å)

• Few commercial examples

• Crystallisable end blocks

• PE-PEP-PE (hPB-hPI-hPB)

• Low crystallinity PE look-alike

• Metallocenes for multi-blocks

• Very complicated phenomenology

• Break-out & confined crystallisation

depending on morphology and

Tg of noncrystallising material.

Iq2

q / Å-1q1 q2

Scattering at

Small Angle

d

L

semicrystalline lamellar stacks

electron density profile

||

Bragg’s law gives an estimate of interference

function d=2p/q* but how do we get the

degree crystallinity and hence L?

The scattering invariant

Q = (1-)2

Q = ∫ I(q)q2dq with limits 0≤ q≤ ∞

But the SAXS pattern has data in a range of q

SR School 2010

Kinetics from

time-resolved

(static) scattering

0

0.2

0.4

0.6

0.8

1

1.2

0 500 1000 1500 2000 2500 3000 3500

t /s

Inte

gra

ted I

nte

nsity (

Arb

)

SAXS WAXS

-10

-8

-6

-4

-2

0

2

4

6

5 6 7 8 9

Ln (t /s)

Ln

(-L

n(X

-1))

SAXS Invariant

Q = (1-)2

and in a lamellar stack

do not change during crystallisation

but the crystalline volume increases so

Q = Xs

the volume fraction of spherulites WAXS degree of crystallinity

Xc = Ac/(Ac+ Aa)

Avrami

Kinetics

1-X=kexp(tn)

SR School 2010

Correlation

Function Analysis 0.001

0.01

0.1

1

10

100

0

0.002

0.004

0 0.1 0.2 0.3 0.4 0.5

I(q)

Lorentz Corrected I(q)q 2

q / Å -1

Damped Porod

I(q) = I b + K exp (q 2 2 ) / q 4

0 g

Guinier

I(q) = I exp(q 2 R

2 /3)

g 1 ( r ) 1

Q q 2

0

I ( q ) cos ( qr ) dq

L p

L

D tr

-0.4

0

1.0

0 300 100 200

r / Å

g 1 (r)

The correlation function

For Lamellae

there is more!

How does it work?

Added Value from the sample

Phase Transition Kinetics Rheology Physical Properties

Processing Reaction Kinetics Alignment

WAXS

SAXS

Monochromator Mirrors

Block Copolymer Self Assembly

free energy = separation - stretching

- interfacial energy

T

At high T thermal motion

overcomes unfavourable

interactions between blue

and red segments

Lamellar phase

Peak order – q0, 2q0, 3q0, 4q0

S. Förster

Hexagonal phase

Hex Rods q0 √3q0 √4q0

S. Förster

106

107

108

109

1010

1011

0 0.05 0.1 0.15 0.2

1107100

q/Å-1

Body Centred Cubic liquid-like

spheres

0-0.13 0.13

q / Å-1

qv / Å-1

0

0.13

-0.13

1 2

single crystal bcc spheres

BCC Cubic q0 √2q0 √3q0 √4q0

1.0

10.0

100.0

1000.0

10000.0

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14

0.0

q/ Å-1

211

220

420

332

422611 543

0

1

2

3

0

Ia3d

Gyroid

Khandpur et al Macromolecules 1995

√6q0 √ 8q0 √ 14q0 √ 16q0 √ 20q0 √ 22q0

A real time SAXS study of oriented block

copolymers during fast cyclical deformation, with

potential application for prosthetic heart valves

Joanna Stasiak, Adriano Zaffora, Maria Laura Costantino and Geoff D. Moggridge, Soft Matter, 2011

• Study of a range of thermoplastic elastomers with all rubbery components in

the block copolymers

• Time resolution of RAPID 10ms used (although trials of <1ms also proved

successful)

• Cycling used to mimic conditions for a prosthetic heat valve (10,000 cycles)

Time resolved X-ray diffraction studies

of drug induced membrane degradation

F

G

H

I

J

K

A

ED

B

Water circulation

Temperature sensor well

Sample closure

X-ray path C X-ray beam

sample

15˚

300μm

21˚

L

c. X-ray path & diffraction angles

b. Sample closure type

O-ring type Packing ring type

a. Partly exploded cross section of high pressure cell

Mylar window

carrier

a. Spacer Sample Holder

& carrier

b. Capillary Sample Holder

& carrier

plug

glass bead

plastic capillary

sample

glass bead

plug

carrier

adhesive

adhesive

sample

plastic spacer

Mylar window0

69

70

71

72

1 2 3 4

Time / seconds

Layer

spacin

g /

Å

Automated high pressure cell for pressure jump x-ray diffraction, Nick Brooks, Beatrice Gauthe, Nicholas

Terrill, Sarah Rogers, Richard Templer, Oscar Ces, John Seddon (2010) Review of Scientific Instruments

81:064103 DOI: 10.1063/1.3449332

Robert Greasty, Jana Heuer, Susanne Klein, Claire Pizzey, and Robert Richardson

Mol. Cryst. Liq. Cryst., 545, 133

Scattering pattern obtained from Permanent Rubine in

dodecane (30 wt%) using short camera length with (a)

zero field (b) Electric field applied (4V=mm) giving nematic

phase.

Electric Field Induced Orientational

Order in Suspensions of Anisotropic

Nanoparticles

Microfocus End Station

FWHMv = 10.4mm

dx/dyf=y0+a*exp(-.5*((x-x0)/b)^2)

X Data

25.70 25.75 25.80 25.85 25.90 25.95

Y D

ata

-2000

-1000

0

1000

2000

3000

4000

sorted unique x vs dy/dx

x column vs y column

dx/dyf=y0+a*exp(-.5*((x-x0)/b)^2)

X Data

-3.75 -3.70 -3.65 -3.60 -3.55 -3.50 -3.45

Y D

ata

-500

0

500

1000

1500

2000

2500

3000

sorted unique x vs dy/dx

x column vs y column

FWHMh = 12.7mm 2.86Å distance

between amino

acids in Type I

collagen helix

650Å first order

of dry Type I

collagen

Nanoscale Fracture Mechanisms in Fibrolamellar

Bone in Bending and Compression

d spacing (Angstroms)

0.07 0.08 0.09 0.10 0.11 0.12 0.13

Inte

nsit

y (

a. u

.)

50

100

150

200

250

300

Q (Inverse Nanometers)0.090 0.095 0.100 0.105 0.110

Inte

ns

ity

100

150

200

250

300

350

T = 0 s

T = 240 s

a

b c d

Angelo Karunaratne, Christopher R Esapa, Jennifer Hiller, Alan Boyde, Rosie Head, JH Duncan Bassett, Nicholas J Terrill, Graham R

Williams, Matthew A Brown, Peter I Croucher, Steve DM Brown, Roger D Cox, Asa H Barber, Rajesh V Thakker, and Himadri S Gupta

, Journal of Bone and Mineral Research, Vol. 27, 2012, 876.

Microfocus spot obtained on I22

with CRL (90 lenses) at 14 keV

Nanoscale Fracture Mechanisms in

Fibrolamellar Bone in Bending and

Compression

• Collagen Fibrillar strain

• Micro damage and

cracking

• Mineral crystallite

orientation

A. Karunaratne, J. Hiller, C. Esapa, A. Boyde, N.J. Terrill, A.H. Barber, R.V. Thakker, H.S. Gupta, Journal of Bone

and Mineral Research, 2012

Tablet Compaction and its effect

on pharmaceutical activity

0.94

0.93

0.920.91

0.90

0.89

0.88

0.87

0.86

0.85

0.84

0.83

0.82

RD scale a

b

c

(a) (b)

(c) (d)

5.76 mm 7.60 mm

6.02 mm 5.70 mm

5.72 mm

Tablet Test shapes including representative indentation types

Compaction/ Morphology

affects (a)

Range used for

azimuthal scan

(b)

Beam-stop

shadow

1500

2000

2500

3000

3500

4000

4500

5000

0 90 180 270 360

Azimuthal angle, lab (°) Inte

nsity (

arb

itra

ry u

nits)

(c)

Experimental data Model

Intensity in transverse direction

(d)

(e)

0

1000

2000

3000

4000

5000

6000

0 50 100 150 200 250 300 350

Azimuthal angle, lab (°)

Inte

nsity (

arb

itra

ry u

nits)

(f)

azimuthal scan, showing spikes

‘typical’ variation (fitted by model)

2

1cos3 2

H

180

0

180

0

2

2

.sin.

.sin.cos.

cos

dI

dI

(a)

(b)

(d)

(c)

(a)

(b)

0.94

0.93

0.92

0.91

0.90

0.89

0.88

0.87

0.86

0.85

0.84

0.83

0.82

Relative

density

Orientation around

compaction site

Data Collection via GDA

Normalisation

1D/2D I0/It

Detector

Corrections

1D/2D

Dark Current,

Detector Response,

etc.

Background

Subtractions

1D/2D

Sample cell, Porod,

amorphous (variable)

Absolute

intensity

1D/2D

By Cross calibration

or directly

Radial Averaging

1D

Sector and full

circle with masking

Data Reduction

Data Calibration &Reduction in Dawn

Data Analysis

• Peaks and single parameter

values are being integrated

into DAWN.

• More detailed analysis is

available via a range of

packages that have been

developed to focus on

particular areas of science or

experiment.

I22 – Small Angle X-ray

Scattering for Diamond

Beamline:

Energy Range 3.7-20keV

d/Å range 1-5000Å (probably >1mm)

Beam size:

70mm (V) – 330mm(H)

6mm x 7mm with microfocusing

End Station:

Flexible Sample platform

Inline sample viewing (Microfocus)

Flux @ 1Å at Sample (Measured):

6.07 x 1012

Detectors:

Fast photon counting to <1ms in 1- and 2D

Software:

EPICS, GDA and LabVIEW™ (for some

sample environments)

(48.4m)

B21 – Solution SAXS for

Diamond

Beamline:

Energy Range 6-23keV

d/Å range 1-2000Å

Beam size:

250mm (V) – 350mm(H)

End Station:

BioSAXS Robot for Solution Samples

Flux @ 1Å at Sample (Measured):

8 x 1011

Detectors:

Fast photon counting to 30ms with P-2M

Software:

EPICS, GDA and ISPyBB plus analysis

pipelines

Small Angle Scattering (SAXS) • O. Glatter and O. Kratky – “Small Angle X-Ray Scattering”

(http://physchem.kfunigraz.ac.at/sm/Software.htm)

• A. Guinier, G. Fournet, Chapman & Hall 1955- “Small Angle Scattering of X-

Rays” (Good University Libraries!)

• L.A. Feigun and D.I. Svergun - “Structure Analysis by Small Angle X-Ray &

Neutron Scattering” Nruka, Moscow, 1986, English Translation Ed. G.W.

Taylor, Plenum, New York, 1987

• H. Brumberger – “Modern Aspects of Small Angle Scattering”, NATO ASI

Series, Kluver Academic Press 1993

• P. Linder, Th Zemb - “Neutron, Xray & Light Scattering, Introduction to an

Investigative tool for Colloidal and Polymeric Systems”

• Ryong-Joon Roe – “Methods of X-ray and Neutron Scattering in Polymer

Science” Oxford University Press 2000

• Norbert Stribeck – “X-ray Scattering of Soft Matter” Springer 2007

• Wilfred Gille – “Particle and Particle Systems Characterisation” CRC Press

2014

Thanks for your attention