Structural characterization

Part 2

Determining partial pair distribution functions

• X-ray absorption spectroscopy (XAS). • Atoms of different elements have absorption edges at

different energies. Structure from interference pattern of scattered electron waves from neighboring atoms.

• Neutron scattering using different isotopes • Different isotopes have different scattering lengths bi • Measurements on three samples of different isotopic

composition all three partial structure factors can be obtained

Extended X-ray Absorption Fine Structure

• EXAFS is an element specific technique • Probes the local structure atound each atom type • X-ray absorption spectrum is measured close to an X-ray absorption

edge of a particular element • Pre-edge region. • Absorption edge: Steep increase in X-ray absorption coefficient,

µ(E) • Post-edge region: Decreasing µ(E) with small oscillations • An X-ray photon is absorbed by an atom • A photoelectron is ejected and backscattered by neighboring atoms

X-ray absorption experiments • Performed at beam line of

a synchrotron • Transmission geometry

Source: Aksenov et al. 2006

Absorption process • Direct detection • Fluorescence detection • Auger electrons

Source: Rehr and Albers 2000 Source: Aksenov et al. 2006

Schematic picture of EXAFS

• Absorption of X-rays • Photoelectron ejected from

central atom • Electron waves scattered from

neighboring atoms interfere Source: Elliott

• Interference pattern extends 400-1000eV from the edge

Theoretical interpretation

Theory, continued • EXAFS equation can be

generalized to represent contributions from NR multiple scattering contributions of path length 2R.

• Electrons lose energy as they travel in the material – mean free path

• Limited range of tens of Å in EXAFS measurements

• A Fourier transform of χ(k) gives an effective reduced radial distribution function

• Peaks close to nearest and next-nearest neighbor distances

• After the first peak multiple scattering contributions are of increasing importance

• Should be taken into account in fitting to experimental data

Source: Ravel 2005

Analysis of EXAFS data • Obtain the EXAFS function

χ(k) and its Fourier transform χ(r) from fits to the experimental spectrum.

• Generate a simulated model structure consistent with the experimental data

• Reverse Monte-Carlo modelling and fitting to data

• Molecular dynamics simulations of amorphous structure for comparisons

• Model structures further analyzed

What do we learn?

• Partial pair distribution functions

• Interatomic spacings for nearest and next nearest neighbors, maybe further out

• Average coordination numbers • Coordination distributions • Bond angle distributions • Mean square deviations σ

(from Debye-Waller factor)

Reverse Monte-Carlo Modeling

• Choose interatomic potential • Minimize the energy • Minimize difference between experimental data and

simulation by varying the atomic configuration • Combine data from X-ray, neutron, EXAFS…. • Gives ”optimized” structural model that is consistent with

experiments • Not necessarily the ”true” structure • Shows important structural features of the material • Wide range of applications: Liquids, glasses, polymers,

crystals, magnetic materials

Example: Amorphous TiO2

Source: Carlos Triana

Neutron diffraction

2. Sample

2θ

3. Detector

Scan as a function of 2θ

fixed , ii kr

λ

ff kr

,λ

Isotope substitution • For a two component system the total structure factor S(Q) is made

up of 3 different partial structure factors Sij(Q) (~ scattering amplitude)

1 2 S11 S22 S12

( ) ( ) ( )1)(21)(1)()( 1221212222

2211

21

21 −+−+−= QSbbxxQSbxQSbxQS

Sij(Q) - partial structure factors ↓

gij(r) - partial pair distribution functions

Determining partial structure factors

• Different isotopes have different scattering lengths bi

• Measurements on three samples of different isotopic composition all three partial structure factors can be obtained

• Inversion to partial pair distribution functions or partial radial distribution functions

• Alternative: Combine ordinary and magnetic neutron scattering with X-ray scattering

Ex: Amorphous alloy Ni81B19

Source: Elliott: Physics of Amorphous materials

Gij(r)=4πrn0(g2,ij(r)-1)

Reverse Monte-Carlo Modeling

• Choose interatomic potential • Minimize the energy • Minimize difference between experimental data and

simulation by varying the atomic configuration • Combine data from X-ray, neutron, EXAFS…. • Gives ”optimized” structural model that is consistent with

experiments • Not necessarily the ”true” structure • Shows important structural features of the material • Wide range of applications: Liquids, glasses, polymers,

crystals, magnetic materials

-2 0 2 4 6 8 10 12 14 16 18 20 22 24 26

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

0.05

0.10

Q/Å-1

F N(Q

)

0 2 4 6 8 10 12 14 16

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

FX(Q

)

Q/Å-1

2 4 6 8 10

-2

0

2

FAg

(Q)

Q/Å-1 2 4 6 8

-4

-2

0

2

4

FI(Q

)

Q/Å-1

Neutron X-ray

Ag K EXAFS I LIII EXAFS

Ex: Glassy (AgI)x(AgPO3)1-x

(From R. McGreevy)

FSDP

(AgI)x(AgPO3)1-x

x=0

x=0.5

x=0

x=0.5

Ag+I P+O

AgI pushes apart phosphate chains -> FSDP

(From R. McGreevy)

Small angle scattering

• The study of structures on larger length scales

• Composites, particle aggregates • Porous materials

• X-rays, neutrons, light • SAXS, SANS, SALS

Scattering angle

• Crystalline materials – Bragg’s law: Scattering vector Q ~ d-1, where d is interplanar distance

• Q has dimension [m-1], hence large Q (large scattering angles) corresponds to small length scales

• At large Q we can resolve atomic distances • Small Q larger length scales • With small scattering angles (small Q) we can

study clustering on the nano-scale.

Small angle scattering

( ) ( ) ( ) ( )nm

i

V nm ddeV

QI nm rrrr rrQ −∫ ∫= ρρ1

( ) ( )∑ −=nm

inm

nmeffV

QI,

1 rrQ

At higher scattering angles/Q-values we get scattering from each atom.

At small angles/Q-values we have low resolution for individual atoms but see clusters of atoms in a volume V with scattering length density ρ(r).

Two-component material

• Consider as particles in a matrix • Define the scattering contrast by ρ(r)-ρ0 • Particle form factor

• Intensity per unit volume

• Structure factor S(Q) (assume isotropic particles)

( ) ( ) rr rQ deQf i

Vp

p

•∫ −= )( 0ρρ

)()()(2

QSQfVN

QI pp=

Spherical particles • Define

• Spheres of radius R – asymptotic expressions

• Radius of gyration often used for other shapes as well as for aggregates

• Radius of gyration Rg2=3R2/5 for spheres

• S is the surface area

2)()( QfQP p=

1)()(29)(2)(

1)5/1()()(

420

24

20

2220

2

>>−=−

=

<<−−=

− QRQRVQ

SQP

QRRQVQP

ρρρρπ

ρρ

Pair distribution function

• Number density np=Np/V • Relations between S(Q) and particle pair

distribution function analogous to those for atomic systems

• Isotropic materials

( )dQQrQrQSQnrg

drQrQrrgrnQ)S

p

p

)/))((sin1)((4)8(1)(

)/)((sin1)(41(213

2

22

−+=

−+=

∫∫

− ππ

π

Experimental techniques

• Limits: λ(nm) Q(Å-1) r (nm) • Light 400-600 5 10-5-3 10-3 200-10000 (SALS) • X-rays 0.1-0.4 10-2-15 0.05-50 (SAXS) • Neutrons 0.1-3 10-3-15 0.05-500 (SANS) • Complementary techniques

Small angle neutron scattering

Source: Per Zetterberg

Limiting expressions • Low Q: Guinier approx. • High Q: Porod approx.

• S is the total surface area • Influenced by particle

shape, size distributions: average of Rg

• Aggregation: correlation length ξ.

1/)(2)(

1)3/exp()()(42

0

2220

2

>>−=

<<−−=

g

gg

QRQSQP

QRRQVQP

ρρπ

ρρ

QR

Porod approx. compared to P(Q) for a sphere

Source: J. Teixeira in On Growth and Form

Fractal surfaces

• Smooth surface: S~r2

• Fractal surface: S~rDs

• Porod: P(Q)~(Qr)2/Q6 • Fractal surface:

• Slope between 3 and 4 • Proportionality constant is

a function of Ds.

sDQVQP −−∝ 620

2 /)()( ρρ

Lignite coal Ds=2.5

Bale and Schmidt, PRL 53 (1984) 586

Volume fractals • Pair distribution function g2(r)-1~rDf-3 • Structure factor

• S(Q)~1 at large Q and I(Q)=npP(Q) • Smaller Q: Fractal region

• Small Q: Guinier type law with correlation length ξ instead of Rg

( )∫ −+= drQrQrrgrnQ)S p )/)((sin1)(41( 22π

fff

f

DDD

D

QdyyyQQS

drQrQrrQ)S−−−

−

∫∫

~sin~)(

)/)((sin~(2

1

Gold colloidal aggregates • Model for g2(r)

• Slope between 1 and 3: Volume fractal Df~2

• SAXS exp. vs model

Source: P. Dimon et al, PRL 57 (1985) 598

Examples of porous materials

• Rocks, sandstones • Clays • Soils • Coals • Cement • Cellulose, cotton • Biomolecules, protein

aggregates • Food

• Some porous materials are built up of connected fractal aggregates

• Fractal surfaces are often present also in cases where the solid is non-fractal

• Examples of these two cases

Volume fractals: Silica aerogel • Extremely porous

continuous SiO2 solid network strucutre

• Combination of light and X-ray scattering data

• Df=2.1

• Smooth surfaces

Source: Schaefer et al, 1984

”Greige” Cotton • SAXS data • Guinier type cutoff

at low Q • Df=2.13 • Different kinds of

cotton have values in the range 2.1 to 2.7

• Aggregation of cellulose microcrystals

Q(nm-1)

Source: Lin et al, ACS Symp. Ser. 340 (1987) 233

Surface fractals: Sandstones • Sedimentary rocks • Structure and

properties interesting for oil industry

• ”Toy sandstones”: sand, crushed glass

• Example shows fractal surfaces in sandstones and shales.

Source: Po-zen Wong, Phys. Today 41 (1988)

Small angle neutron scattering (SANS)

Cement: A complex case • Calcium-silicate-hydrate

(CSH) aggregates • Volume fractal • D ~ 1.8 to 2.7 depending

on C/S and preparation

• Ordinary Portland cement during hydration

• Seems surface fractal

Source: Adenot et al. C.R. Acad. Sci. II, 317 (1993) 185. Source: Häussler et al. Phys. Scr. 50 (1994 )210.

Local porosity analysis

• Sintered glass beads • Diameter 250 µm

Source: R. Hilfer, Transport and relaxation phenomena in porous media

• Works for both fractal and non-fractal structures!

Example: Berea sandstone

• Local density function for different cell size L

• Local percolation probabilities for different L

Other techniques

• Nitrogen and water adsorption isotherms • Mercury porosimetry Pore size distributions

• X-ray microtomography for porous

structures