Characterization of complex fluids or materials using small angles scattering techniques

O. Diat

UMR 5257 (CEA/CNRS/UM2/ENSCM)

OutlineBrief and classical introduction to scattering methods

•Form and Structure factors •Porous materials, specific surface •Examples

references

Detector

Incident beam(planar wave)

rdtreeR

EtRdE rkkitRkiS

issrrr rrrrr

),(1

4

1),( ).().(

0 ρπ

ω −−−−=

Hyp Born approx : far field detection, weak scattering (s index for scattering, i for incident)

0r

θ

R

)(rrρ rd

r

ikr

skr

jj btrtr ),(),(rr

∑= ρρ ρρρ −=∆

=−=2

sin4

q and θ

λπ

is kkqrrr

Scattering vectorScatters densityScattering length

Classic:10-1<q(nm-1)<4

Special: 6.10-3<q(nm-1)<20

Fundamental equation of the instantaneous scattering amplitude :the FT of small heterogeneities ( of the dielectrique cte or electronic or nuclear) depending on the radiation, light, x-ray or neutron

∫−−− ∆==

V

rqitRkiSS rdetre

REtqEtRE s

rrrr rrrr.).(

0 ),(1

4

1),(),( ρ

πω

ρρρ −=∆

0r

θ

R

)(rrρ rd

r

ikr

skr

Integral over the irradiated volume

cmcm

eb

eTh

132

0

10.83,24

−==πε

3cm/cmin Thmolecular

rayX bV

Z=−ρ

FEDORS table, polymer engineering, 14 (2), 1974, 147-154

scattering length density for X-ray radiation

Do not work with too much hydrogenated compound in performing SANS!Incoherent scattering depends on energy!

(fm)

scattering length density for neutron radiation

3cm/cmin cohA

molecular

cohneutron b

M

dN

V

b ==ρ

ρH2O = -0.56 1010 cm-2

ρD2O = 6.38 1010 cm-2

ρpolystyrene = 1.41 1010 cm-2

ρD-PS = 6.47 1010 cm-2

Contrast enhanced in neutron scattering if possibility of deuteration!

scattering length density for neutron radiation

3cm/cmin cohA

molecular

cohneutron b

M

dN

V

b ==ρ

For an assembly of discrete particles:

[ ] rdrrRrrdr j

N

jjj

rrrr)()()(

1

ρδρ ∆+−=∆ ∑=

j

j

j Rqi

qG

jrqi

j

Vj

N

jS erderqE

rrrr

444 3444 21

rrr .

)(

).(

1

])([)( ρ∆∝ ∫∑=

∫−−− ∆=

V

rqitRkiS rdetre

REqE s

rrr rrrr.).(

0 ),(1

4

1)( ρ

πω

drjr j

Rj+1Rj

∑j

« Scattering intensity »or differential scattering cross-section

∑∑−−==

Ω j k

RRqikj

kjeqGqGEEd

d

V).(** )()(.

),(1 rrrrrλθσ

j

j

j Rqi

qG

jrqi

j

Vj

N

jS erdertqE

rrrr

444 3444 21

rrr .

)(

).(

1

])([),( ρ∆∝ ∫∑=

Ωd

d

V

),(1 λθσ

« Scattering intensity »or differential scattering cross-section

∑∑−−==

Ω j k

RRqikj

kjeqGqGEEd

d

V).(** )()(.

),(1 rrrrrλθσ

∫

∫

−∆

∆∝

=−∆∆=∆

RdeR

RqI

RrdRrrR

Rqi

S

V j

rr

rr

rrrrvr

rr.2

2

2

)(

)( of ansformFourier tr)( and

)()()()(

ρ

ρ

γρρρ

444 3444 21

rrr rr

jrqi

j

Vj

j rderqG j ).()()( ρ∆= ∫

When j=k

« Scattering intensity »or differential scattering cross-section

∑∑−−==

Ω j k

RRqikj

kjeqGqGEEd

d

V).(** )()(.

),(1 rrrrrλθσ

444 3444 21

rrr rr

jrqi

j

Vj

j rderqG j ).()()( ρ∆= ∫

When j=k

dRqR

qRRR

)sin()(~4

nsorientatio allaver averagean with

functionon distributidistancepair p(R)

22

043421

=

∞

∆∝ ∫ ρπ

j

k

Rik=50

R50 100

p(R)

The scattering intensity is the FT of pair-correlation function p(R)

j

k

Rik=50

FT

« Scattering intensity »or differential scattering cross-section

For diluted system (uncorrelated scatterers and identical)

)(...),(1

)( 2 qPVd

d

VqI partS ρλθσ ∆Φ=

Ω=

For concentred system (identical scatterers) and centrosymmetric

[ ] ∞→→−+=

∆Φ=+=

∫∞

qasdRqR

qRRRg

V

NqS

qSqPVqIV

NqG

V

NqI

S

partSS

s

1sin

1)(41)(factor structure

)()(...)()()(

2

0

2'2

π

ρr

10-2 cm-1

1cm-1

Latex sphere, O. Spalla

Polydispersity effect

)(cm ...).(

),(1 1exp −

∆Ω=

Ω=

sacqabs etT

I

d

d

VI

λψλθσ

22

0

2 )).(1(2. ρϕϕπ ∆−== ∫∞

ssabs dqqIQ

( )2

4

)(2

lim

ρπ ∆

=Σ ∞→qabsqI

Porod law

Invariant (for 2-phase system)

Specific surface

salt = 0.15 M

(O. Spalla, S. Lyonnard Langmuir 02)

Relationship between Imes and Iabs-mat

44** )()()( qqqIqIqI absabs

Corrabs −=

/es

SAXS from CeO2 obtained by a slow evaporation of a colloidal suspension and then calcination at different temperature

50 nm

X-ray scattering

0.01 0.1 1

0.01

0.1

1

10

100

Inte

nsity

(cm

-1)

q (Å-1)

q-4

Mesoporous materials, MCM or SBA type

J. Cambedouzou et al, JAC 2012

Modèle pour le calcul du diagramme de poudre de mésoporeux hexagonaux

w arp

Rg

Rn,m

Rr

rh

n,mpore

uv

q

z

x φ

qh

ψx

u

∑−=Np

pg qAqAqA )()()(

+−= ∑ ∑

i jiijppippgggg

b qRJqRJRqRJqRJRqRJRqRJRqL

qI

,0

21

2011

21

23

23

)()()()()(2)(4)( ρπ

Autre exemple : MCM-41

0.1 0.2 0.3 0.4 0.5 0.6

0.01

0.1

1

10

100

Inte

nsity

(cm

-1)

q (Å-1)

Rp=16.5 ÅRp=15 Å

expRp=18 Å

Caractérisation structurale :

-a = 44.2 Å

-Rp = 16.5 Å

-Désordre 20% paracristal

Estimation surface spécifique

0.1 0.2 0.3 0.4 0.5 0.6

1E28

1E29

1E30

Iq4 (c

m-5)

q (Å-1)

Σ = 8.74e5 cm-1

pour la poudre

Soit:

S=400 m²/g

À comparer avec S BET = 1100 m2/g…

Geopolymerization followed by SAXS

(P. Steins PhD program, CEA/DTCD – Macromolecules 2012 )

•To use always refererence sample and empty cellin similar conditions

•To known the sample thickness and otherscattering parameters

• To get data in absolute units

• to the largest (suitable) q-range as possible (whennecessary)

•If possible to do SAXS and SANS (differentcontrast with same scale!)

To perform scattering experiments

Thank you and see the attachedreferences for more details

• 1. Guinier, A. and Fournet, G. (1955) Small-Angle Scattering of X-rays, Wiley, New York.

• 2. Glatter, O. and Kratky, O., Eds., (1982), Small-Angle X-ray Scattering, AcademicPress, London.

• 3. Feigin, L.A. and Svergun, D.I. (1987) Structure Analysis by Small-Angle X-ray and Neutron Scattering, Plenum Press, New York.

• 4. Brumberger, H., Ed., (1995) Modern Aspects of Small-Angle Scattering, Kluwer Academic, Dordrecht.

• 5. Lindner, P. and Zemb, T., Eds., (2002) Neutrons, X-rays and Light : Scatteringmethods applied to soft condensed matter, Elsevier, Amsterdam.

• 6. Schmidt, P.W. (1995) in Modern Aspects of Small-Angle Scattering, Brumberger, H., Ed., p. 1, Kluwer Academic, Dordrecht.

• 7. « Soft matter characterization: scattering, imaging and manipulation », Pecora, Borsali edited by Springer,

• 8. X-ray data booklet LBL, California• 9. Neutron Data Booklet, ILL/ITU• 10. SASfit software package, PS Institure• 11. O. Spalla et al, JAC 36 (2003) 338 + présentation Bombannes, école d’été• 12. J. Teixeira, JAC 21 (1988), 781• …….