Structure Analysis using Small-Angle X-ray Scattering

Structure Analysis using Small-Angle X-ray Scattering

Bruker AXS Korea / Carbon Nanomaterials Design Lab.

@ X-선 회절 측정클럽, 한국표준과학연구원, 2013. 08. 27.

Bruker AXS Korea([email protected]) / 서울대학교 재료공학부 CNDL ([email protected])

김 세 훈

http://www.art-com.co.kr/online/ppt_gallery_1.htm

2013.08.27. X-선 회절 측정클럽 WORKSHOP @한국표준과학연구원

SAXS

Small-angle X-ray Scattering

Scattering vs

Diffraction

Small-angle vs

Wide-angle

Introduction


Introduction : Various structural analysis methods


Introduction : Various structure levels in materials (aramid fiber)


Introduction : X-ray Diffraction/Scattering Equipments

Source

: Properties of X-ray

Sample

: Interaction between X-ray & Materials

Detection

: X-ray Patterns

• X-ray Diffractometer (XRD/D8 Advance) • Small-angle X-ray Scattering (SAXS/Nanostar)

../../../


• Angle Range of SAXS - Length Scale of Small Angle Scattering : 10 – 2000 Å - Information on relatively large r is contained in I(q) at relatively small q - Bragg’s Law - Sample contains a scattering length density inhomogeneity of dimension

larger than 10Å (1nm), scattering becomes observable in small angle region.

d2sin

d = few Å = 1Å 2 = 20

d = 100 Å = 1Å 2 = 0.6

• Nano-structural Paramaters obtained from SAXS - Mean Size, Size Distribution - Shape (sphere, cylinder, etc.) - Orientation, Degree of Orientation - Mean distance between particles

SAXS determines the site, site distribution, orientations and structure arrangement of macromolecules or precipitants

in bulk materials. (Ref. http://www.bruker-axs.de)

Introduction


• Small Angle vs Wide Angle (scattering vs diffraction)

Introduction


Introduction

• Transmission SAXS vs. GI (Grazing Incidence) SAXS

- Transmission : X-rays are incident normal to the surface of the sample (liquid dispersion, gels, powders, sheets, etc.) - GI : Incident angle close to the critical angle (0.1 to 1 degree) (semiconductor quantum dot/island, porous films on substrate, condensed powder, nanoparticle embedded in polymer)

Transmission SAXS GI-SAXS


Introduction

• Transmission SAXS vs. GI (Grazing Incidence) SAXS

Transmission SAXS GI SAXS


Introduction

• GI (Grazing Incidence) SAXS vs. XRR


Introduction

• Statics and Kinetics

• Statics - Relatively Low Intensity (Lab SAXS) - DNA Cage - SDS micelles and protein-SDS complexes

• Kinetics - High Intensity (Synchrotron) - Fibrillation of glucagon (polypeptide) - SDS-a-synuclein complexes and fibrillation - Nanoporous silica synthesis directed by polymer

Phase transformation of Girl Group


Introduction

• Classification 1 : SAXS Experimental Setup / Smaple-to-Detector Distance


Introduction

• Classification 1 : SAXS vs. WAXS Example : Silver Behenate (AgBh) for system calibration

SAXS

WAXS


X-ray Sources using in SAXS


X-ray Sources : Synchrotron Radiation

• The European Synchrotron Radiation Facility (ESRF) • http://www.esrf.eu/


Detection : Bruker’s Commercial Detectors

LYNXEYE XE High-Resolution Energy-Dispersive 1-D Detector

Vantec 500 2-D Gas Detector

Vantec 2000 2-D Gas Detector

with large 14 x 14 cm2 active area

Platinum 135 135mm CCD Detector

APEX II CCD Detector

Vantec 1 Super Speed 1-D Gas Detector


SAXS in Korea

1. SAXS 4C1, 4C2 @Pohang Accelerator Laboratory : Synchrotron radiation / Open

2. Nanostar @HOMRC, Seoul National University : X-ray tube type / Open

3. SAXS w. GADDS @NICEM, Seoul National University : X-ray tube type / Open

4. Nanostar @Dong Woo Fine Chem : Incoatec Microfocus Source + VANTEC Detector + GI-SAXS Sample Stage / Not open

5. Anton Paar @KIST / @RIC, Wonkwang University


SAXS Software (FREE)


SAXS : Theory


Introduction

The Soluble Blend System

The Dilute Particulate System

The Nonparticulate

Two-Phase System The Periodic System

Four Models adopted in analysis

of SAXS data


Introduction

1. The Dilute Particulate System - Particles of one materials are dispersed in a uniform matrix of a second material. When the concentration is sufficiently dilute, the position of individual particles, far from from each other, are uncorrelated. - When the conc. is not sufficiently dilute, the interference effects becomes an important concern of the analysis

2. The Nonparticulate Two-Phase System - Two different materials are irregularly intermixed and neither of them is considered the host matrix or the dispersed phase. - The analysis leads to determination of parameters characterizing the state of dispersion of the materials - The correlation length, the specific interphase boundary area and the thickness of the phase boundaries

3. The Soluble Blend System - A single phase material in which two components are dissolved molecularly as a homogeneous solution - Miscible polymer blend, block copolymer in a disordered state and polymer solution

4. The Periodic System - Semicrystalline polymers consisting of stacks of lamellar crystals - Block copolymers having ordered, segregated microdomains - Micellar aggregates of organic and inorganic substances


Two-Phase System

Dilute Particulate System

Guinier Law

Porod’s Law


SAXS : Dilute Particle System

• SAXS – Smaller sample vs Larger sample



• Even when the shape is unknown, or irregular and not describable in simple terms, the scattering function still follows a certain universal form, in the limit of small q, that is given by

where I(q) is the intensity of independent scattering by a particle

which called the Guinier law, -> allows determination of the radius of gyration of a particle

• The Guinier law is valid provided that - q is much smaller than 1/Rg - the system is dilute, so that the particles in the system scatter independently of each other - the system is isotropic as a result of the particles assuming random orientations - the matrix in which the particles are dispersed is of constant density and is devoid of any internal structure that can by itself give scattering in the interested range of q

)3

1exp()( 2222

0 gRqvqI



• The Guinier law suggests that

when the logarithm of I(q) is plotted against q2, the initial slope gives .

• For the purpose of determining the radius of gyration it suffices to have the intensity determined in relative units.

• If, on the other hand, the intensity is measured in absolute units by means of

an instrument suitably calibrated, it is possible to determine the value of

as well.

• Since the value of , the average scattering length density in the particle, is

usually known from its chemical composition, this provides the means of

evaluating the particle volume .

• Knowledge of both the radius of gyration and the particle volume

provides a clue about the shape of the particle.

22

03

1ln2)(ln qRvqI g

2

3

1gR

22

0 v

0

v

gR v



• Pair-Distance Distribution Function



• Examples of Rg of well-defined geometric shape, - for solid sphere of radius R - for solid ellipsoid of half axes a, b, c - for solid rod with length L, radius R - for thin rod of length L - for disk of radius R

RRg5

3

2

1

222 )(5

1cbaRg

212

222 RL

Rg

12

LRg

2

RRg



Case Study : Au Nanoparticles in Liquid Suspension



Case Study : Au Nanoparticles in Liquid Suspension



Case Study : Curve Simulation by Dummy Residue Model

pH = 7.2 PH = 5.4



Case Study : Curve Simulation by Dummy Residue Model


SAXS : TWO Phase System

• Definition of ideal two-phase system

(1) The system contains only two different regions (or phases),

each of constant scattering length density or .

(2) The boundary between these two regions is sharp with no measurable thickness.

(3) These two phases are irregularly intermixed,

so that the system as a whole is isotropic and there is no long range order.

• If the scattering length densities of the two phases are known from knowledge of their

chemical compositions, the experimental value of Q can be used provide the relative

amounts of the two phases.

1 2

2211

211

122

21

2

2

2

21

2

1

2 )()( VVVQ



2

4

2

4

222

654

222

6

22

32

44&02cos2)(

largefor)2cos1(

8)(

2cos12sin2)2cos1(8)(

)(

)cos(sin9

3

4)()(

RNRSqRq

S

qq

qRRN

q

qR

q

qRR

q

qRRN

qR

qRqRqRRNqI

Intensity of scattering from a system containing N solid spheres of radius R

Intensity of scattering from a solid spheres of radius R

6

22

32

0

2

3

3

0

0

0

0

2

)(

)cos(sin9

3

4)()(

)(

)cos(sin3

3

4

)sin(4sin

4)()(

qR

qRqRqRRqAqI

qR

qRqRqRR

drqrrq

drqr

qrrrqA

R

Rrfor

Rrforr

0)(

0

R

0

Porod Law



Two phase system with finite boundary

pl

rV

V

SV

SrVr

1

4

11

4)(

2

2

2

2

2

2

2

21

2

1

2

2

4

S

Vlp

p

p

lrforl

rVr

exp)( 2

223

3

2

0

22

1

8

sin4exp)(

ql

lV

drqr

qrr

l

rVqI

p

p

p

qasql

VqI

p

4

2

18)(

qasq

SqI

4

2)(2)(



4

2 2)()(

q

SqI

Intensity of scattering from a system containing N solid spheres of radius R

Ideal two phase system

1l

2l

1l

2l

2l4

21

4

2

12)(

)(2)(

qV

S

Q

qI

qasq

SqI

11

2211

111

4,4

lll

V

Sl

V

Sl

p

Porod’s length of inhomogeneity

Sl

V

dr

rd

pr

2

2

04

1)(

pl

pl

rVr exp)( 2

VdrrrQ 2)0()()(



Case Study : Glassy Carbon (Porous Material)



Case Study : Pore Characterization (Two-Phase) by microbeam SAXS

• Characterization of pore distribution in activated carbon fibers(ACFs) by microbeam SAXS D . Lozano-Castello et al., Carbon, 40, 2002, 2727

• The experiments done with CO2 and steam ACFs have demonstrated the suitability of this technique to characterize a single ACF. The experiments show that scattering intensity increases with the burn-off degree, which agrees with SAXS experiments carried out using bigger amounts of fibers.

• The use of an X-ray microbeam of 2 μm diameter allows the characterization of different regions of the same fiber with microscopic position resolution.

• CO2 ACFs : the scattering is high in different regions across the fiber diameter, confirming that CO2 activation takes place within the fibers, generating a quite homogeneous development of porosity.

• Steam ACFs : the scattering is much higher in the external zones of the fibers than in the bulk, which means that steam focuses the activation in the outer parts of the fibers.



Case Study : Pore Characterization (Two-Phase) by microbeam SAXS

• Two phases : the carbonaceous matrix and the pores. -> an increase in the scattering corresponds to an increase in the porosity

• Before plotting the curves, it must be taken into account that the fibers have cylindrical shape and, volume correction is needed.

• The results indicate a higher concentration of pores in the outer zone.

• One useful parameter for the analysis of porous materials is

Porod Invariant (PI), defined as


Dimension / Mass Density

Surface Fractal

Mass Fractal

Surface Roughness (2D)


SAXS : Fractal Dimension Analysis

Fractal objects : Definitions

• At large q the intensity I(q) of scattering from a sphere decays as q-4,from a thin disk as q-2, and from a thin rod as q-1. related to the dimensionality of the scattering object

• The inverse power-law exponents can be explained in terms of the concept of a fractal.

• A fractal possesses a dilation symmetry, that is, it retains a self-similarity under scale transformations. In other words, if we magnify part of the structure, the enlarged portion looks just like the original.

The Koch curve

The Sierpinski triangle



Fractal objects

• A fundamental characteristic of a fractal is its fractal dimension.

Mass fractal

• Suppose we draw a sphere of radius r around a point in the object.

• If the fractal object is a line, the mass M(r) within the sphere will be proportional to r. If it is a sheet, then proportional to r2 and if a solid three-dimensional object, proportional to r3.

• In a fractal, where d is the fractal dimension : 1 < d < 3

• Since the volume of the sphere is proportional to r3, the density of actual material embedded in it is

drrM )(

3)(~ drr

)(~ r

rrM )( 2)( rrM 3)( rrM

open structure

close structure



Surface fractal

• Some object possess a surface that is rough and exhibit fractal properties.

• Imagine we cover the island completely with square tiles of edge length l, and we mark those tiles that at least partially overlap the coastline.

• N(l ) : the number of marked tiles

• If the coastline is smooth and nearly straight, N(l ) will be proportional to l -1.

• If the coastline is irregular and fractal, N(l ) depends more strongly on l , and is proportional to l -ds where ds is a number larger than 1.

• The length L(l ) of the coastline : l N(l ) or

• ds : the fractal dimension of two-dimensional surface fractal.

• In a three-dimensional surface fractal, where S(r) is the surface area measured with a tool of area r2

sdllL

1

)(

sdrrS

2)(



Case Study : Surface Fractal Dimension

• Fractal dimension analysis of polyacenic semiconductive (PAS) materials

K . Tanaka et al., Carbon, 39, 2001, 1599– 1603

• The fractal dimensions D of the pristine and the Li-doped PAS materials have been analyzed by SAXS and compared with that of graphite.

• D of the PAS powder increases according to decrease in [H]/[C] molar ratio -> the dehydrogenation process with the raise of pyrolysis temperature.

• Introduction of binder for the fabrication of the battery electrode generally makes D smaller. The D value depends on the binder species.

• Li doping into the PAS sheet causes smaller D, which signifies the existence of Li atoms makes the nanoporous structure more plane-like at least from the fractal aspect.



Case Study : Surface Fractal Dimension


SAXS : Periodic System

Scattering from Ideal two-phase lamellar structure

• Let us first consider an ideal two-phase lamellar structure in which lamellae of phase A, of thickness da and uniform scattering length density ρa, alternate with lamellae of phase B, of db and ρb.

• The scattering length density profile :

• a one-dimensional lattice of period d (=da+db)

• The density distribution within a single period can be represented by

)/(*)()( dxzxx u

n

nxxz )()( :)/( dxz

abu dxx /)( ba

2

10

2

11

xfor

xforx



Scattering from Ideal two-phase lamellar structure

• By taking the absolute square of the Fourier transform of , the intensity of scattering I1(x) is obtained as

• With , we obtain

• Bragg peaks occur at a series of q values satisfying or , and that the height of (or, more exactly, the integrated area under) the nth order peak is proportional to , where is the volume fraction of phase A.

• From the measurement of the relativity heights of successive peaks, the relative volumes of the two phases can be determined.

• When the volumes of the two phases are equal, all even order peaks are reduced to zero heights.

)(x

)2/()()(2

dqzqFqI

)(xu 2

2

2

22sin

4)(

qda

qqF

ndq 2/ dnq /2

22 /sin nn a

)/( ddaa



Case Study : Liquid Crystal (Plulonic P84 / Water / p-xylene)


Case Study


SAXS Case Study : Ordered Porosity (Two Phase System)

• SAXS and EM Investigation of Silica and Carbon Replicas with Ordered Porosity Francoise Ehrburger-Dolle, et al, Langmuir, 19, 2000, 4303

• SAXS investigations of ordered porous carbon materials Francoise Ehrburger-Dolle, et al, Proceeding of Carbon Conference, 2003

• Ordered nanoporous carbons can be prepared by a replica technique starting from an organized silica template, SBA-15, which contains an hexagonal array of mesopores interconnected by micropores.

• Two routes for introducing carbon into the pores of the silica matrix, liquid impregnation by a solution of sucrose followed by carbonization and chemical vapor infiltration (CVI) of propylene.

• After dissolution of the silica template by hydrofluoric acid treatment, a carbon material is obtained.

High-resolution image (HRTEM) of pores with hexagonal symmetry for the silica SBA-15 (left) and the carbon replica R15AC-52 (right)



• The SAXS intensity curve obtained for the ordered mesoporous silica SBA-15 clearly shows three Bragg peaks. -> hexagonal packing

• In the low q domain (q < 2×10.2 nm.1), the data can be fitted (red line) by the Guinier relation I = I0 exp[-(qRG )

2/3]. RG = 180 ± 10 nm.

• For long cylindrical particles of uniform density and small cross section, this corresponds to a length L equal to (12)1/2RG=623 nm, which is comparable to the sizes observed on TEM images.

• In the intermediate q domain, I(q) scales as q-4, as expected from the Porod law. This result also indicates that the external surface of particles is smooth.




SAXS Case Study : Time Dependency of Scattering Curves (Kinetics)

Moitzi, Ch., Guillot, S. Fritz, G., Salentinig, S. Glatter, O. Advanced Materials (2007) 19, 1352-1358


SAXS Case Study : SAXS on Microfluidic Analysis

Thomas Pfohl, Department Chemie, Universitat Basel


SAXS Case Study : SAXS on Microfluidic Analysis


Tahseen Kamala, Soo-Young Park, et. Al. Polymer (2012) 53, 3360-3367

SAXS Case Study : An in-situ simultaneous SAXS and WAXS survey


SAXS Case Study : GI-SAXS from Liquid to Solid Surface

P. Siffalovic, Institute of Physics SAS, Slovakia


SAXS Case Study : GI-SAXS from Liquid to Solid Surface


SAXS Case Study : Particle Aggregates and Surface

• Characterizing dispersion and fragmentation of Fractal, Pyrogenic silica nanoagglomerates by SAXS, Langmuir 23 (2007) 4148-4154

• 압력을 가해 nanoagglomerate를 분산시키는 과정에서 primary particle size, surface area, polydipersity 등을 SAXS를 통해 구하고 TEM과 비교함으로써 분산 거동을 살펴봄

• 이때 기존의 Guinier law의 modification을 통해 보다 정확한 값을 구해냄

sf

f

D

Gp

G

D

GGG

q

qRerfC

RqG

q

qRerfRqB

RqGqI

63

1

2

1

2

1

3

2

2

1

2

2

2

2

2

2

)6/(

3exp

)6/(

3exp

3exp)(

1G

pCsfD

Primary particle

2G

sfD

fD)( 2BP

Agglomerate

xt dtexerf

0

22)(





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Structure Analysis using Small-Angle X-ray Scattering

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