Post on 26-Sep-2020
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• …….