10_SAXS-SANS_2Outline
•contrast variation •mesoporous structure, pores and fractals
references
The scattering intensity is the FT of pair-correlation function p(R)
dR qR
qR RR
Real space vs reciprocal space
For concentred system (identical scatterers) and centrosymmetric
[ ] ∞→→−=−
2
0
2'2
π
ρr
If g(R) is the probability of finding the centre of any particle at a distance R from the centre of a given particle then for N particles in a volume V , (N/V )g(R)dV is the number of particles in volume element dV at a distance R from a given particle.
TF
If no correlation S(q)=1 (high dilution).
I(q) is proportional to the form factor of the scattering entity
For example a nanoparticle
•Pedersen, J.S. (1997), Adv. Colloid Interface Sci., 70, 171 •SASfit
…. And so on
Form factor for a distribution of spheres
For concentred system (identical scatterers) and centrosymmetric
[ ] ∞→→−+=
Structure factor
•Importance of contrast variation (SAXS / SANS) using same structural parameters, to eliminate the SF contribution as a first step, to discriminate between couples of structural parameters
•Importance of the dilution or swelling law as φ-1, φ-1/2, φ-1/3
•Check if the shape of the individual particles is similar whatever the dilution or concentration If not, reorientation, new molecular assembling, second or first order transition ..
Things to keep in mind when performing SAS experiments
Analyzing the scattering curves over a large range of q to obtain the best set of structural parameters (size, aggregation number, interaction distance)
It allows to understand the thermodynamic of the system
Limiting form of I(q)
For non-interacting particles and for qR<<1, then Guinier Law
In case of homogeneous particles with a sharp interface, it exist an asymptotic limit, qR>>1 that leads to the Porod law:
And finally, the invariant (for an incompressible and two-phase system)
) 3
)( exp(...)(
222 )1(2)( ρπ Φ−Φ== ∫ dqqIqQ
To perform scattering experiments
•To use always refererence sample and empty cell in similar conditions
•To known the sample thickness and other scattering parameters
• To get data in absolute units
• to the largest (suitable) q-range as possible (when necessary)
Common SAXS experiment
Detector efficiency
Sample thickness (homo)
)exp( eT µ−=
optimum of exp(-µe) for e=1/µ or a transmission T=1/e=0.37
, µ: linear attenuation coefficiant
Acquisition time photons in direction θ
Flux and efficiency has to be determined using calibrant (lupolen, water or other solvent..)
sacq abs etT
Invariant
• To extract S/V, ρ has to be known: Porod law using the absolute intensity and a
contraste estimated using a contraste variation method (S/V=25m2/cm3)
From SEM (S/V=28m2/cm3)
coh Acoh
salt = 0.15 M
sampleenvgrain VV=φ
gipSolid φφφ )1( −=
envinnerityinnerporos VV=φ
0
44 ** )()()( qqqIqIqI mesmes
Relationship between Imes and Iabs-mat
44 ** )()()( qqqIqIqI absabs
2
0
envinnerityinnerporos VV=φ
solidB Te µ/)ln(−=
envinnerityinnerporos VV=φ
envsolidityinnerporos VV=−φ1
2
0
44 ** )()()( qqqIqIqI mesmes
• Inner pores filled with a solvent
• Inner pores and inter grain filled with air
2
0
And for the envelop of grains in air
Pores with solvent
Pores fillled by air
Be careful: we have considered an homogeneous distribution of pores within the grain!
Other Techniques. • The surface area and porosity of spray-
dried powders were determined via N2
sorption • Scanning electron micrographs (SEM)
SAXS from CeO2 obtained by a slow evaporation of a colloidal suspension and then calcination at different temperature
d=2.68, r=2.17 1011 cm-2
Σe,G=0.32 m2/g
Σe,G=8.6 105 m-1
Altered Glass (SiO2/Na2O,B2O3/ZrO2)
Thank you and see the attached references 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, Academic Press, 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 : Scattering methods 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.