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Figure 2 : a) Example of scattering intensity profile measured
between qmin and
qmax.
b) Binary sample and "q-window" corresponding to a measurement
at a given q0
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What is measured in a Small Angle X-ray Scattering (SAXS) ?
X-rays are used to investigate the structural propertieof
solids, liquids or gels. Photons interact wielectrons, and provide
information about thfluctuations of electronic densities in the
matter. typical experimental set-up is shown on Figure 1 :
monochromatic beam of incident wave vector k
selected and falls on the sample. The scattereintensity is
collected as a function of the so-callescattering angle 2. Elastic
interactions acharacterised by zero energy transfers, such that
th
al wave vectork is equal in modulus to k . The relevant
parameter to analyse the interaction is the momentum transfer or
scattering vector q=k -kefined by :
he scattered intensity I(q) is the Fourier Transform of g(r),
the correlation function of the electronic density r(r), which
corresponds to the probability td a scatterer at position r in the
sample if another scatterer is located at position 0 : elastic
x-ray scattering experiments reveal the spatial correlationthe
sample. Small angle scattering experiments are designed to measure
I(q) at very small scattering vectors q(4p/l)q, with 2q ranging
from few micrdians to a ten of radians, in order to investigate
systems with characteristic sizes ranging from cristallographic
distances (few ) to colloidal sizes (up w microns).
Electronic contrast. The number of photons scattered by one
sample is proportional to its total volume V and to its electronic
contrast . In the simplse of a binary system for instance, like
scattering objects of density r embedded in a solvent of density r
, the electronic contrast is Dr=r -r . Th
gher the contrast between particles and solvent, the greater the
signal.
Absolute intensity. The experimental intensity is usually fitted
in order to determine the density r(r), the size, the shape and the
internal structure of onementary scatterer, as well as the
structure and the interactions between scatterers. The
determination of physical quantities, such as molecular
weigharticle volume, specific surface or osmotic compressibility,
is feasible only if I(q) is measured on absolute scale. As shown on
figure 1, a part of thcident red beam is absorbed in the material.
The number of photons scattered in the solid angle DW in the
direction 2q have to be normalised witspect to the number of
photons transmitted through the sample (green beam). This imply to
control several parameters : the sample thickness e (cm
nd transmission T, the incident flux of photons f (photons/s)
and the solid angle of the experiment DW. The absolute intensity
can then be measured bsolute units (cm ).
What does "q-range" mean ?
typical small angle scattering intensity profileis shown on
figure2a. The intensity isotted versus q, in the range qmin-qmax
defined by the experimental set-up andually fixed by geometric
limitations
tuitively, a measurement made at a given q allows to investigate
the densityctuations in the sample on a distance scale D =2p/q . It
is equivalent to observe thestem through a 2p/q diameter "window"
in real space, as shown in figure 2b. Thed circle is the
observation window. A scattering signal is observed if the contrast
Drside the circle is different from zero. To study objects much
smaller or much largeran D =2p/q , another "window" has to be
chosen. The smallest (largest) observationndow is given by D =2p/ :
it determines the size of the smaller (bigger)
articles that can be observed with the instrument.
High q domain :
The window is very small : there is a contrast only at the
interface between the two media. This domaicalled the Porod's
region, gives information about the surfaces.
Intermediary zone :
The window is of the order of the elementary bricks in the
systems. The form factorP(q) can be measure(size, shape and
internal structure of one particle).
Low q domain :
When the observation window is very large, the structural order
can be obtained : it is the so-called structurfactorS(q), which
allows to calculate the interactions in the system.
orod's law : specific surface and interface
hen two media are separated by a sharp interface, the scattered
intensity follows an asymptotic law in the high q region :
I(q)=Aq-4+B. This law is callee Porod's limit (and is not verified
any more for more complicated interfaces). The asymptotic value,
when the electronic contrast of the sample
own, and when the intensity is expressed in absolute scale,
allows to calculate the specific surface S of the particles.he
q-range of validity of a Porod's law can cover several decades,
giving information about the particle sizes. When various types of
particles, wit
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Figure 3 : USAXS on borosilicated glasses.
Figure 4 : USAXS determination of the form factor of Silica
Spheres.
fferent characteristic scales, exist in the sample, it is
possible to measure different Porod's law, as shown ine following
example. Borosilicated glass are alterated and studied as a
function of time, in order tovestigate the alteration process. The
initial glass before alteration can be represented as a
two-mediastem, with large grains distributed in the solvent. The
scattering signal shown in red is a Porod's lawsulting from the
sharp interface between the solvent and the grain, ranging from
10-4 to 0.1 -1. When themple is alterated a second Porod's law
appears at larger q : small pores are created in each grain. It is
thearp interface pore-grain that gives rise to that second law. As
the pore's size increases with time, the
orod's law after 8 weeks is shifted to smaller q respect to the
2 weeks alteration glass signal. In this particularse, measuring
the intensity on absolute scale is critical to calculate the pore's
surface (and radius, if aherical shape is assumed), as well as the
grain's surface.
hen two media are separated by a sharp interface, the scattered
intensity follows an asymptotic law in thegh q region : I(q)=Aq +B.
This law is called the Porod's limit (and is not verified any more
for moremplicated interfaces). The asymptotic value, when the
electronic contrast of the sample is known, and whene intensity is
expressed in absolute scale, allows to calculate the specific
surface S of the particles.
he q-range of validity of a Porod's law can cover several
decades, giving information about the particle sizes. When various
types of particles, witfferent characteristic scales, exist in the
sample, it is possible to measure different Porod's law, as shown
in the following example. Borosilicated glase alterated and studied
as a function of time, in order to investigate the alteration
process. The initial glass before alteration can be represented as
o-media system, with large grains distributed in the solvent. The
scattering signal shown in red is a Porod's law resulting from the
sharp interfac
etween the solvent and the grain, ranging from 10 to 0.1 . When
the sample is alterated a second Porod's law appears at larger q :
small pores aeated in each grain. It is the sharp interface
pore-grain that gives rise to that second law. As the pore's size
increases with time, the Porod's law after eeks is shifted to
smaller q respect to the 2 weeks alteration glass signal. In this
particular case, measuring the intensity on absolute scale is
critical tlculate the pore's surface (and radius, if a spherical
shape is assumed), as well as the grain's surface.
Form factor P(q) : size and shape of particles
Generally, the size and the shape of the particles is of
fundamental interest understand a material. X-ray small angle
scattering gives valuable informatio
providing that the contrast is sufficient. If the particles are
of the order of one microor larger, the results can be checked by
light scattering. If the system is complex (thredifferent media for
instance), it can be completed by neutron scattering, which offethe
possibility of contrast variation methods.
Experimentally, form factors can only be measured in the dilute
regime where particlecan be considered as independent scatterers
without any interactions. In this case, thintensity is directly
proportional to :
the contrast Dr.
the volume fraction f and the volume of one particle V .
the form factor of a single particle P(q).
Elementary shapes. Form factors are easily calculated for
spheres, cylinders, disks, rods, micelles, lamellas or Gaussian
polymers, which are the basapes encountered in soft condensed
matter. However, the only analytical expression is the sphere form
factor that is represented on Figure 5.
Example. The experimental intensity scattered by a diluted
solution of silica spheres of radius R=3000 is shown above on
figure 4. The experiment haeen performed on our laboratory
Bonse/Hart camera. The data are compared to the theoretical form
factor P(Q). Experimental data are smoothe
mpared to the model because of the instrument resolution (and
some polydispersity). The location of the first minimum gives the
radius of the particlesR=4.5. A very good fit is obtained in this
case with R=3000 . The extrapolation of the absolute intensity at
q=0 allows to calculate the volume fraction articles in the
solvent.
Structure factor S(q) : interactions between particles
omplex systems are described through interaction potentials. Are
the interactionstractive or repulsive, electrostatic or not ? is
there any long range order in the systemWhat is the effect of
temperature, salt or pressure on the equilibrium ? SAXS
xperiments combined with adapted models bring some answers,
provided theructure factor S(q) can be extracted from the scattered
intensity I(q).
or instance, in the case of centrosymmetric identical particles
in solution, the signale to one single particle can be dissociated
from the signal arising from the
teractions between particles. The intensity is then
proportionnal to the productq)*S(q).
he limit S(q=0) gives the osmotic compressibility of the sample,
but this quantity isnly accessible if absolute intensities can be
measured. When there is no interaction, in the previous example of
Silica spheres in diluted regime, the structure factor isual to 1.
Repulsive electrostatic nteractions appear between these charged
silicaheres when one concentrates the sample. The experimental
signal is reported in Figure 5: at large q, where the "observation
window" mainly shows th
article itself, the form factor of a sphere is predominant. The
intensity profile in this region is identical to the one mesaured
in diluted regime. Howeven interaction peak appears at small q, the
position of which is related to the main first neighbour distance D
between the silica spheres. At low q thgnal results from the
competition of P(q) and S(q).
y dividing I(q) by P(q), the experimental structure factor is
obtained. A model taking into account electrostatic repulsions and
Van der Waals attractivotential is shown in blue. Fitting the data
with the model allow to quantify the interaction potential.
echnical potential : a view of our possibilities
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