Small-Angle X-Ray Scattering for the Study of Nanostructures and Nanostructured Materials

5

CHAPTER

Small-Angle X-ray Scattering for the Study of Nanostructures and Nanostructured Materials Giuseppe Portale1, Alessandro Longo2 1

Netherlands Organization for Scientific Research (NWO), DUBBLE beamline at the European Synchrotron Radiation Facility (ESRF), Grenoble, France Institute of Nanostructured Materials of the Italian National Research Council, ISMN-CNR, Palermo, Italy

2

Contents 1. Introduction 2. Theory of SAXS 2.1. General Equations: From the Scattering of a Single Electron to the Macroscopic Scattering Cross-Section 2.2. SAXS Invariants 2.2.1. Guinier Approximation 2.2.2. Porod Approximation 2.2.3. Examples

175 177 177 185 185 187 188

3. Application of SAXS Technique 3.1. Determination of the Shape, Size and Size-Distribution Function Via SAXS 3.2. Interparticles Interactions: The Structure Factor 3.3. Quantum Dots and Metallic Clusters 3.4. Nanoscale Heterogeneities in Amorphous Semiconductor Alloys Studied by SAXS and ASAXS 3.5. Surfactant-Coated Cobalt and Cobalt/Nickel Nanoclusters 3.6. Magneto-SAXS 3.7. Behind the 100-nm Resolution: Microradian Diffraction and Scattering Technique Acknowledgments References

191 191 198 201 209 212 215 219 225 225

1. INTRODUCTION In a general small-angle X-ray scattering (SAXS) experiment, an angular range of the order of 0.5e50 mrad is probed by a detector usually positioned some meters away from the sample and using wavelengths close to 1 A˚. In such a way, structures ranging in size from 1 nm up to about 100 nm can be investigated. Such a range is extremely important for the study of a variety of problems extending from bioscience to material science. Biological macromolecules have dimensions ranging from few up to tens of Characterization of Semiconductor Heterostructures and Nanostructures Second Edition (C. Lamberti and G. Agostini Eds.) ISBN 978-0-444-59551-5, http://dx.doi.org/10.1016/B978-0-444-59551-5.00005-4

Ó 2013 Elsevier B.V. All rights reserved.

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nanometers, metallic clusters have generally dimensions of few nanometers, and synthetic polymeric chains organize themselves in nanometer-sized domains in solutions, bulk and thin films. Moreover, the material properties strongly depend on the structure and the size of subunit components. Therefore, SAXS is an ideal technique to study nanostructured materials. Quantum dots, metallic nanowires, and inhomogeneities in semiconductor films are all structures in semiconductor science that can be studied by SAXS. Although SAXS can be successfully performed at a laboratory scale and powerful laboratory equipment using rotating anodes with both slit-like and point-like focused X-ray beams have been developed [1e4], high photon flux synchrotron SAXS offers the unique possibility of second and subsecond time resolution and in situ measurements using complex sample environments. The majority of SAXS experiments are performed using a monochromatic X-ray beam with fixed energy although some interesting examples of the use of energy-dispersive SAXS have been reported in literature [5,6]. Energy-dependent SAXS is usually referred to as anomalous SAXS or ASAXS in literature and is a powerful technique to investigate ion distribution and clustering of metallic atoms in solution and bulk [7e9]. The high penetration power of hard X-rays, the absence of sample preparation and the possibility of performing scattering experiments in solution or suspension are some of the advantages offered by SAXS with respect to microscopy techniques (see chapter 10 of this book) [10]. Moreover, SAXS provides information over a relative volume of the sample while microscopy results are limited by the intrinsically low statistic related to the small probed fraction of the sample. On the other hand, SAXS is unable to reach the spatial resolution typical of electron microscopy, making SAXS and microscopy complementary techniques. This chapter is organized as follows: in Section 2, the general SAXS theory is described starting from the most fundamental of the X-ray equations, in order to clarify the link between X-ray diffraction and SAXS equations. The Guinier and Porod laws, the so-called SAXS invariant, are also introduced in Section 2 as they allow one to extract information from a SAXS profile independently from any assumed structural model. The first part of Section 3 is dedicated to the understanding of the form and structure factor for the most common particle shapes. In Section 3, a series of recent experiments are selected and presented to exemplify the application of SAXS in the field. Among the selected examples are recent studies on metallic clusters and quantum dots in solution and their superlattice formation; investigation of semiconductor inhomogeneities; scattering from ordered molecular aggregates and their alignment in an external magnetic field. Finally, it will be briefly reported about the recent development of microradian X-ray diffraction and scattering that allows studying particles with dimensions larger than 100 nm (e.g. colloidal particles).

Small-Angle X-ray Scattering for the Study of Nanostructures and Nanostructured Materials

2. THEORY OF SAXS 2.1. General Equations: From the Scattering of a Single Electron to the Macroscopic Scattering Cross-Section X-rays are electromagnetic waves characterized by the wavelength l, or equivalent wave number 2p/l. Generally, a monochromatic X-ray beam characterized by a single wavelength (or a very narrow wavelength distribution) with dimension typically close to ˚ (10
10 m) is employed. 1A The polarization of the electric field in three dimensions and neglecting the magnetic ~ is written according to classical physics as a unit vector~ field H ε, the wave vector along the ~ ið k~ r
utÞ ~ ~ r; tÞ ¼ ^εe . Electromagnetic waves are propagation direction is written as k, E rad ð~ ~ ~ ~ ~ ~ transverse and thus, ~ εk ¼ 0 and kE ¼ kH ¼ 0. From a quantum mechanical perspective, a monochromatic beam is viewed as being quantized into photons, each having an energy E ¼ -u, and momentum Z~ k; consequently, the intensity of the beam is given by the number of photons passing through a given area per unit time. An X-ray photon interacts with an atom in two ways: it can be (i) scattered and (ii) absorbed. Only the scattering process will be treated in this chapter. The scattering of an X-ray photon by a single electron will be treated at first since the elementary scattering unit of an X-ray in an atom is the electron. Protons can be neglected as a result of their high mass with respect to the electron (z1000 times heavier). Classically, the electric field of the incident X-rays exerts a force on the electronic charge that subsequently accelerates and radiates the scattered wave (Fig. 5.1). If the wavelength of the scattered wave is the same as that of the incident one, the scattering is called elastic. This is not true in the general quantum mechanical description, in which

Figure 5.1 Description of the scattering event. The electric field of an incident plane wave induces oscillations on the electron, which then acts as a source and radiates like a small dipole. If the wavelength of the emitted wave is unchanged, the scattering is elastic, otherwise Compton scattering occurs. For color version of this figure, the reader is referred to the online version of this book.

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the energy can be transferred to the electron with the result that the scattered photon has energy lower than that of the incident one. This event is known as inelastic or Compton scattering. The inelastic scattering is less important than the elastic one, and it will be neglected in our discussion. ~ ¼ Z~ Momentum (Q kf
Z~ k) may be transferred even in an elastic scattering event 
2p ki j ¼ and this leads to the definition of the scattering vector ~ q as j~ kf j ¼ j~ l k, where ~ k and ~ kf are the initial and final wave vectors of the photon, ~ q ¼~ kf
~ ˚
1. respectively. The scattering vector ~ q is generally expressed in A According to the geometrical description of the ~ q vector depicted in Fig. 5.2, the ~ q modulus is given by: qj ¼ j~

4psinw l

[5.1]

The capability of an electron to scatter is expressed in terms of a scattering length or amplitude. A vibrating electron acts as a source and radiates like a small dipole antenna. The problem is to evaluate the radiated electric field at an observation point X, which is at a distance R from the source, and an angle j with respect to the direction of the incident beam. According to the classical derivation for unpolarized radiation (Fig. 5.3),
2
2  r0 2 2 1 þ cos j E [5.2] jERad ðR; T Þj ¼ R2 In 2

Figure 5.2 Definition of the wave vector transfer ~ q. ~ k and ~ k f are the incident and the scattered wave 2sinðqÞ vectors, respectively. Since the scattering is assumed elastic, j~ k f j ¼ j~ kj . The wave vector transfer l q ¼ ðk~f
~ kÞ and its modulus is given by Eqn [5.1]. is defined as the difference between ~ k and ~ kf , ~ For color version of this figure, the reader is referred to the online version of this book.

Small-Angle X-ray Scattering for the Study of Nanostructures and Nanostructured Materials

Figure 5.3 In the figure, the radiated field is considered at an observation point X at a distance R. Two distinct cases can be considered: (a) The point X lies in the same plane as the polarization of the incident wave. The observed acceleration has to be multiplied by a factor cosðjÞ and the scattered
     e2 eikR field is Erad R; T ¼
Ein cos j . (b) The observation point X lies in the plane normal 2 4pε0 mc R to the incident polarization and the acceleration of the electron is equally seen at all the scattering
   e2 eikR Ein angles, so that the scattered field is Erad R; T ¼
. For color version of this figure, 4pε0 mc2 R the reader is referred to the online version of this book.

 e2 , is known as the classical The prefactor denoted by the symbol r0, r0 ¼ 4pε0 mc 2 ˚. electron radius or the Thomson scattering length. Its value is 2.8179  10
5 A Generally, X-ray detectors count single photons, and the measured intensity Isc is then the number of photons per second recorded at the detector. This can be expressed as the energy per second, i.e. the power, flowing through the area of the detector divided by the energy of each photon. It is usual to normalize the scattered intensity Isc by both the incident flux and the solid angle of the detector, DU. This leads to the definition of the differential scattering cross-section.


ds ðnumber of scattered photons=sÞ ¼ dU ðIncident fluxÞðDUÞ Scattered energy through unit solid angle : ¼ Incident energy per unit area

[5.3]

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~

For an ingoing wave plane eik~r , the wave function far away from the scattering region   i~k~r  eikr r fe þ f q; 4 (asymptotic wave function), where q and must be expressed by j. ~ k r 4 are measured with respect to the ingoing direction. f (q,4) is the scattering length or scattering amplitude. It can be proved that the outgoing electron current corresponds to the original ingoing current flowing through a perpendicular area of size dsðq; 4Þ ¼ dUj f ðq; 4Þj2 , so that the differential cross-section is defined as follows: dsðq; 4Þ ¼ j f ðq; 4Þj2 dU

[5.4]

Let us now consider the scattering by an atom with Z electrons. The electron distribution of an atom is specified by the number density rð~ rÞ. Introducing V ð~ rÞ as the potential confined in the region, where rð~ rÞ is defined, it is possible to describe 2m rÞ is the reduced field potential Uð~ rÞ ¼ V ð~ rÞ 2 confined in the region where rð~ Z defined, so that the corresponding wave equation can be written as ½V2 þ k2
Uð~ rÞjð~ rÞ ¼ 0. The standard approach to solving the above equation is to transform it into an integral equation employing the functions of Green. If Uð~ rÞ is small, the integral equation can then be solved by iteration (time depending perturbation theory). Therefore, assuming that the detector is positioned at a distance r much larger than the range of the potential, it can be written as Z

 m eikr ~ j.k ~ r ¼ eik~r
2pZ2 r

 0 ~ 0  0 r dv e
ikf~ r V~ r j. ~ k

[5.5]

V

j~ r
~ r 0j

¼ r was made in the denominator. However, this The approximation approximation cannot be made in the exponential. ~ kf ~ r 0 , which is ^r$~ r 0 ¼ kr
~ kf $~ r 0 Þ, takes the different paths of the scattering  krðkj~ r
~ r 0 jykr
k~ vector in the potential region into account, so it represents the phase difference between waves scattered by different portions located in the potential region. This term cannot thus be neglected in the exponential function (Fig. 5.4). The first-order approximation to the scattering description is known as Born approximation and it is given by replacing j in the integral on the right of Eqn [5.5] with ~

the zeroth-order term eik~r . Thus, the wave function can be written as  m eikr ~ r ¼ eik~r
j. ~ k 2pZ2 r

Z V

e
i



~ k~ r
~ kf~ r0

   V~ r 0 dv

[5.6]

Small-Angle X-ray Scattering for the Study of Nanostructures and Nanostructured Materials

Figure 5.4 ~ r is the distance from the localized potential region to the detector. ~ r 0 (~ r 0 << ~ r ) represents the distance between two different portions of the potential which exhibit different paths and consequently different phases. For color version of this figure, the reader is referred to the online version of this book.

In terms of scattering amplitude f (q,4), the Born approximation reads: Z   m eikr eiqr0 V r 0 dv fðqÞBorn ¼ 2 2ph r

[5.7]

where q is the wave vector whose modulus is given by Eqn [5.1]. According to Eqn [5.7], the scattering amplitude is the Fourier transform of the potential and the scattered radiation field is the superposition of contributions from different volume elements of the charge distribution correlated to the potential. For conventional working energies for SAXS (typically between 8 and 12 keV), the contribution of the exchange potential is small, so that the exchange scattering can be neglected (static exchange method). Thus, for any atom, only the static potential is important, i.e. the electrostatic interaction between the incident electron and the target atom, so that the potential V ð~ rÞ can be approximated to: Z  Z rð~ r2Þ V ~ r ¼
þ [5.8] d~ r2 r r 2j jr
~ v

where Z is the atomic number. The first term is related to the interaction with the nucleus, whereas the second one accounts for interactions with the electronic charge density of the atom. Using the Born approximation, the scattering amplitude fBorn becomes   Z 2 i~ q~ r rÞ d~ r [5.9] fðqÞBorn ¼ 2 Z
e rð~ q

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where rð~ rÞ is the electronic density of the atom that can be represented by the wellknown hydrogen-like function. R rÞ d~ r in Eqn [5.9] is defined as the atomic scattering factor. The term fatom ¼ ei~q~r rð~ Correspondingly, the atomic differential cross-section is: 2 ds 4 qÞ ¼ j f ðqÞj2Born ¼ 4 Z
fatom ð~ dU q

[5.10]

In the case of atomic hydrogen (Z ¼ 1) and r(r) ¼ exp(
2r)/p, fH(q) ¼ [1 þ (q/2)
2] and  2 2 q þ8 ds ¼ 4 dU ðq2 þ 4Þ4 The function j f ð~ qÞj2Born is generally defined as the scattering intensity Ið~ qÞ and it has the dimensions of a length squared. For a generic sample constituted by an ensemble of atoms, it is possible to define the scattering length as the Fourier transform of the electronic density distribution of the object, which contains the atoms: Z   fobject ~ q ¼ ei~q~r robject ð~ rÞ d~ r [5.11] However, experimentally the scattering amplitude given by Eqn [5.11] cannot be directly measured but rather the intensity, described by the following equation: Z Z       I~ q ¼ f ~ q f ~ q ¼ e
i~qð~r 1
~r 2 Þ rð~ r 2 Þrð~ r 1 Þ d~ r 1 d~ r2 [5.12] It should be noted that for a collection of discrete atoms, the scattering amplitude is P f ð~ qÞ ¼ i fi ei~q~r i , where fi is the atomic scattering factor relative to the atom i. The corresponding intensity is thus given by the equation XX   I~ q ¼ fi fj ei~qð~r i
~r j Þ i

[5.13]

j

For an isotropic system in which the value of ~ r ij ¼ ~ r i
~ r j is fixed but its direction varies randomly, the intensity is obtained by performing a spherical average, so that sin qrij hei~q~r ij i ¼ and the following equation is obtained: qrij XX   sinðqrij Þ fi fj I~ q ¼ qrij i j

[5.14]

Small-Angle X-ray Scattering for the Study of Nanostructures and Nanostructured Materials

Equation [5.14] is the well-known Debye equation for an isotropic system of N identical atoms. If the system is not discrete (this is the case of a particle made up of atoms and measured at small angles), the integral equation reported by Eqn [5.12] is obtained. It is mathematically convenient to perform the double integration in two different steps: at first, summarizing all pairs with equal relative distance (Patterson function), and integrating subsequently over the whole relative distances, including the phase factor (note that ~ r ¼ ð~ r 1
~ r 2 Þ). The first step is the mathematical operation of the convolution square or autocorrelation ~ r2 ð~ rÞ. The intensity distribution is thus given by Z   rÞ d~ r [5.15] I~ q ¼ e
i~q~r ~r2 ð~ The intensity distribution in the reciprocal space or~ q space is uniquely determined by the structure of the object as expressed by ~r2 ð~ rÞ. Conversely, the latter can be obtained from Ið~ qÞ by the inverse Fourier transformation as 1 3 Z   2 ~ e
i~q~r I ~ r ð~ rÞ ¼ q d~ r [5.16] 2p The main difficulty in structure analysis is the reconstruction of the density distribution from the measured intensity Ið~ qÞ according to Eqn [5.16]. It is worth noting that ~ r2 ð~ rÞ and Ið~ qÞ are related by the phase ~ q~ r only. The result would be the same when r is enlarged and q is diminished by the same factor, and thus, larger particles will give diffraction patterns at smaller scattering vectors, i.e. small angles. Let us now introduce two restrictions met in the majority of the cases studied by SAXS, which greatly simplify the problem: (1) the system is statistically isotropic; either if the isotropy arises from the structure itself or if it is a consequence of a variation upon time; (2) there exists no long-range order. According to restriction (1), the autocorrelation ~ r2 ð~ rÞ depends only on the magnitude r of the distance. In fact, the resolution is poor, so that two points inside the scattering object cannot be distinguished. If it is averaged in all directions, the intensity becomes Z sin ðqrÞ 2 ~r ðrÞ dr IðqÞ ¼ 4pr 2 [5.17] qr According to restriction (2), at large r, the electron density distribution should become independent, and it might be replaced by the mean value r. It is evident that the autocorrelation, as defined above, must tend to a constant value given by V r2, where V is the volume of the scattering object. For r ¼ 0, the autocorrelation is V r2 . It follows that the structure is defined in the finite region only where ~r2 deviates from the final value, being V r2 a constant value. The constant term V r2 can be discarded in this discussion or it might be considered equal to zero.

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It is convenient to divide the autocorrelation function by the volume where the electronic density is defined and thus the so-called correlation function is defined by ~2 ðrÞ
V ðrÞ2 ¼ gðrÞ ¼ r

hV ðrÞi 2 r V

[5.18]

where 1 hV ðrÞi ¼ 4p

Z

Z  du V ð~ r þ~ uÞV ð~ uÞ d~ u

[5.19]

Equation [5.19] is sometimes reported as g0(r). g(r) represents the probability of finding a point within a particle at distance r from a given point and it is equal to zero at r  D, where D is the largest diameter where the electronic density is defined. A simple geometrical construction was suggested by D. Wilson (1949) for g0 ðrÞ, which allows one to define this function as the common volume (with respect to all orientations) of a particle and its ghost shifted by the vector~ r. For a given particle shape, Eqn [5.19] should be integrated to obtain the corresponding correlation function. This is relatively simple for an object like a sphere, parallelepiped, cylinder and disk (Section 3.1). However, in the majority of cases, the analytical expressions are seldom available, so g0 ðrÞ is numerically evaluated. According to Eqns [5.17] and [5.18], the intensity can be written as Z sinðqrÞ IðqÞ ¼ 4p [5.20] gðrÞr 2 dr qr It is now clear that the intensity is related to the volume and the shape of the scattering object. This result suggests that theoretically, by means of an SAXS experiment, the shape of the scattering object can be obtained by inverse Fourier transform of the measured intensity. Note that this is true only when both the restrictions (1) and (2) are valid. So far, only the scattering of single particles has been considered. However, real systems include a great number of particles. For a dilute solution, it is assumed that the total diffraction pattern is simply the sum of the intensity scattered by individual particles. As concentration is increased, interparticle interference effects become more likely. This interference comes from two sources: pure geometric influence (impenetrability of the particles) and electrostatic Coulomb interaction. However, in this chapter, only the pure geometrical influence will be taken into account. The simplest case is a system of volume V, containing N identical particles. By increasing the particles concentration, each particle will “feel” more and more the existence of

Small-Angle X-ray Scattering for the Study of Nanostructures and Nanostructured Materials

its neighbors. The scattered intensity from a collection of discrete particles can be written as + * N  N X N 2 1 X dSðqÞ 1 ds 1 X
iq r
r  ð Þ k j fk ðqÞ þ fk ðqÞfj ðqÞe [5.21] ¼ ¼ dU V dU V k¼1 V k¼1 j¼1 jsk

where r k and rj are the centers of mass of particles k and j, respectively. For the special case of monodisperse spherical particles, f(q) ¼ fk(q) ¼ fj (q). According to this consideration, Eqn [5.21] can be factorized as follows: ( * +) N X N X dSðqÞ N e
iqðrk
rj Þ [5.22] ¼ jf ðqÞj2 1 þ dU V k¼1 j¼1

jsk

According to Eqn [5.22], the scattering intensity is composed of two terms: the first called the form factor, P(q) (intraparticle interference factor); and the second called structure factor, S(q), arising from the interparticle interference. The equation of the scattering intensity can thus be rewritten as follows:    dSðqÞ N ds ¼ ¼ NV 2 Dr2 Pðq S q dU V dU

[5.23]

Equation (5.23) represents the most general form of the SAXS equations. For a more detailed discussion about P(q) and S(q), the reader is referred to Sections 3.1 and 3.2.

2.2. SAXS Invariants 2.2.1. Guinier Approximation At very low q-values, the diffraction data are insensitive to details at atomic or molecular level. Under the assumptions (1) system statistically isotropic and (2) no long-range order as previously discussed, it is possible to derive for identical scatterers an approximation of the scattering curve without a priori information. This approximation is known as the Guinier equation. It can be derived as follows: at very low q, according to Maclaurin series, sinðqrÞ q2 r 2 q2 r 2 þ
. ¼ 1
6 120 qr Combining Eqns [5.20] and [5.24], it can be written as: ! q2 Rg2 IðqÞ ¼ Ið0Þ 1
3

[5.24]

[5.25]

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where I(0) is:

Z Ið0Þ ¼ 4p

gðrÞr 2 dr

[5.26]

and the gyration radius Rg is: Z 1 gðrÞr 4 dr 2 Rg2 ¼ R gðrÞr 2 dr

[5.27]

The term between brackets in Eqn [5.25] can be considered as the first two terms of q2 R2 
3g . Thus, to an accuracy of terms proportional the Maclaurin series of the function e to q4, it is possible to describe the scattering curve at low q values by:
IðqÞ ¼ Ið0Þexp

q2 Rg2

 [5.28]

3

Equation (5.28) is the well-known Guinier equation. In order to correlate the parameters Rg and I(0) to the structure of the scatterer, it can sinðqrÞ be substituted by the expansion of the directly in the autocorrelation function qr obtaining: Z Z d~ r 2 rð~ r2Þ d~ r 1 rð~ r1Þ hIðqÞi ¼ object




q2 6

object

d~ r2 object

The first term Ið0Þ ¼ j

R

Z

Z

    r 2 ð~ r 21 Þ2 þ/ d~ r 1r ~ r1 r ~

object

d~ rrð~ rÞj2 is the square of the total scattering mass. To

object

analyze the second term, the following relation can be used: 2 ¼ j~ r 2
~ r 1 j2 ¼ j~ r 1 j2 þj~ r 2 j2
2j~ r 1 j$j~ r21 r 2 j$cosðaÞ

By choosing the origin to be at the center of mass of the scattering density, the previous equation can be rewritten as follows: Z Z r 2 Þ$j~ r 1 j$j~ d~ rd~ r 1 rð~ r 1 Þrð~ r 2 j$cos ðaÞ ¼ 0 object

object

Small-Angle X-ray Scattering for the Study of Nanostructures and Nanostructured Materials

Then, only the two following terms are present: Z Z Z Z     r 2 $j~ r 1 j2 ¼ d~ r 1 d~ r 2r ~ r1 r ~ d~ r 2 rð~ r2Þ object

object

object

  2 r 1j d~ r1r ~ r 1 j~

object

Z ¼ Ið0Þ

  2 r1j d~ r 1r ~ r 1 j~

object

The term on the right is analogous to the well-known expression for the gyration R radius of a particle. Rg ¼ object d~ r 1 rð~ r 1 Þj~ r 1 j2 represents the root mean square distance of the scattering density from its center of mass: the role of mass, in this case, is represented by the electrons. It is important to stress that according to the restrictions (1) and (2) adopted in the derivation, the Guinier approximation is only valid for dilute solutions of monodisperse objects and cannot be used in concentrated or in polydisperse systems. 2.2.2. Porod Approximation Another useful approximation involves the asymptotic behavior of the intensity at large q values, which is related to the Porod invariant. Large q values are correlated to small r values of the correlation function gðrÞ. According to Eqn [5.19] and according to the geometrical interpretation described above, for small r values, the common volume differs from itself only by the shift of the surface S. The contribution of a surface element dS to this shell is dSr cos(q), where q is the angle between r and the normal to the surface. Let us first take the average of this contribution for all the directions of r, which involves the average of the jcos(q) (jcos(q)j2 ¼ ½). Furthermore, only the part of r directed inward makes a contribution, i.e. where the electronic density is different from zero, which results in a second factor 1/2. Therefore, integrating over the surface and averaging for all r directions, gðrÞ ¼

S hV ðrÞi 2 r ¼ 1
rþ/ 4V V

[5.29]

This approximation can now be substituted in the intensity expression given by Eqn [5.20] and after integration by parts and considering only the first term (bearing in mind that only an approximation to the final slope is intended here for q/N) is obtained: lim IðqÞ ¼ Dr2

q/N

2p S q4

[5.30]

Equation [5.30] was derived without any particular assumption and it is valid not only for single particles but also for densely packed systems and for nonparticulate structures,

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as the final slope of the intensity is not influenced neither by the large-scale properties of the particles nor by their mutual arrangement. Furthermore, the contributions of different particles will simply be added, so that the asymptotic value of I(q) must be proportional to the total interface S. The use of Eqn [5.30] requires absolute intensity measurements in cm
1. However, it is also possible to use the relative intensity when normalization by the invariant Q is performed, as normalizes for the total squared number of electrons per scattering volume: kp IðqÞ 1 S 1 ¼ 4 ¼ 4 q Q p V q where

Z Q ¼

  q2 I q dq ¼ V Dr2

[5.31]

[5.32]

In the limit for q / N of IðqÞq4 =Q is a constant, bearing information about the specific surface S/V of the scatterer, and it is known as the Porod constant kp. 2.2.3. Examples The aim of the two following examples is to give the reader the basic guidelines on how to perform Guinier and Porod analysis. 2.2.3.1. Determination of the Guinier Radius of the Lysozyme in Water Solution

Equation [5.28] can be used when a monodisperse (in size and shape) system of scatterers is investigated. Aqueous solutions of proteins, such as lysozyme, are typical examples of systems that can be investigated using Guinier analysis. Lysozyme is a globular protein with mean radius 24.5 A˚. Figure 5.5 shows SAXS data from a lysozyme solution. The best fit using the Guinier equation is shown as a solid red line. The agreement between this approximation and the experimental data is good over a limited q region corresponding to the Guinier region for lysozyme. An empirical rule, Rg q < 1, defines the maximum q value for which the Guinier equation applies. As already mentioned above, the accuracy of Rg calculation depends on several factors such as interparticle interference and possible aggregation of the scatterers. In order to eliminate these undesirable effects, it is advisable to extrapolate Rg to infinite dilution. If these effects are absent, the Rg should be independent of concentration. Otherwise, if Rg decreases linearly with concentration, its value at zero concentration can be deduced by extrapolation. Figure 5.6 (left panels) shows the scattering intensities of a lysozyme water solution (pH 7) at different concentrations. The Guinier analysis is performed by plotting the natural logarithm versus the square of the modulus of the momentum transfer q. According to the empirical rule, the highest value of square

Small-Angle X-ray Scattering for the Study of Nanostructures and Nanostructured Materials

Figure 5.5 SAXS curves and Guinier approximation of a 50 mg/ml lysozyme solution (left panel) and corresponding Guinier plot (right panel). For color version of this figure, the reader is referred to the online version of this book.

Figure 5.6 SAXS data of different lysozymeewater solutions (pH 7) (left panel). Guinier plot of the proteinewater solutions (up right panel) and the corresponding “infinite dilution extrapolation” (low right panel). For color version of this figure, the reader is referred to the online version of this book.

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˚ 2. Figure 5.6 (right momentum transfer considered for Rg determination is 0.05 A upper panel) reports the Guinier plot and the linear regression for the different protein solutions. The infinite dilution analysis (showed in the lower right panel of Fig. 5.6) indicates a Guinier radius of 18.0 A˚. Considering that lysozyme is a globular protein, it can be approximated as a sphere, so its Rg can be calculated using the following relation. Using the following relation: 3 Rg2 ¼ R2 [5.33] 5 According to Eqn [5.33], a radius of 23.23 A˚ is obtained which is very close to the value expected from theoretical considerations. 2.2.3.2. Determination of the Specific Surface of Commercial Alumina Powder

As reported previously, the “final slope” of the small-angle scattering curve allows one to calculate the total surface of the scattering objects using the Porod law. Assuming a constant background, the specific surface can be obtained without an absolute intensity measurement by using the invariant as defined in Eqn [5.31]. Figure 5.7 shows a Porod plot obtained by plotting IðqÞq4 vsq4 . The intercept of the straight line with the ordinate axis correspondss to the Porod constant. The specific surface is obtained after dividing by the invariant Q and multiplying the result by p: S=V ðm2 =gÞ ¼

kp p 10; 000 Qdb

[5.34]

Figure 5.7 Porod plot for the commercial alumina. For color version of this figure, the reader is referred to the online version of this book.

Small-Angle X-ray Scattering for the Study of Nanostructures and Nanostructured Materials

where kp is the Porod constant, db is the bulk density and 10,000 is the conversion factor, so that the result is in square meters per grams. According to Eqn [5.34] and using the calculated value of the Porod constant (kp ¼ 19,753), the bulk density of dp ¼ 2.0 g/cm3 and the invariant Q (Q ¼ 814,202), a surface of 381.0 m2/g for commercial alumina is obtained. This value represents the total surface for all the pores present in the solid matrix (closed and open pores).

3. APPLICATION OF SAXS TECHNIQUE 3.1. Determination of the Shape, Size and Size-Distribution Function Via SAXS Quantitative analysis of SAXS profiles provides information not only on the average nanoparticles size, but more interestingly gives the opportunity to determine the objects’ shape as well as calculate the size-distribution function. According to Eqn [5.23], the SAXS intensity can be described as the product between the form and the structure factors. In a dilute solution, i.e. in a solution where the objects are far enough apart to avoid any interaction between them, S(q) ¼ 1 and can be neglected. According to Eqn [5.20], the scattering cross-section is related to the objects’ shape via the autocorrelation function g(r) or alternatively, via the one-dimensional pair distance-distribution function (PDDF) p(r) that describes the objects’ shape: ds ¼ 4p dU

ZN gðrÞr 0

qr dr ¼ 4p qr

2 sin

ZN pðrÞ

sin qr dr qr

[5.35]

0

where pðrÞ ¼ gðrÞr 2. A large number of form factors are reported in books and articles ranging from simple to very complex equations. This chapter will be only focused on the three classical shapes (sphere, cylinder and disk) since they are the most easy to understand and can be readily used to explain the information contained in an SAXS intensity profile. References for more complex form factors are also given below in this section. In the case of monodisperse objects with spherical symmetry, Eqn [5.35] can be analytically solved and is equal to the well-known Rayleigh function: " #2 ds 3ðsin qR
qRcos qRÞ [5.36] ¼ N ðDrÞ2 V 2 Psph ðq; RÞ ¼ N ðDrÞ2 V 2 dU ðqRÞ3 where N is the number of spheres in solution, ðDrÞ2 is the electron density difference between the spherical objects and the solvent (or matrix) and R is the radius of the spheres.

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Figure 5.8 Semilog plot of the form factor for spheres with different radius R.

The term between squared brackets is the sphere form factor Psph(q,R) and q is the module of the scattering vector as defined in Eqn [5.1]. In Fig. 5.8, the intensities resulting from Eqn [5.36] for different values of R are plotted. From inspection of Fig. 5.8, the following observations are made: (1) For monodisperse spherical objects, the form factor is characterized by a series of zeros at well-defined positions that can be used to calculate R: qR ¼ 4.493, 7.725, etc. Note that this procedure is only correct for homogeneous spheres with no or very little polydispersity. (2) The entire curve and the position of the zeros shift toward smaller q values with ˚
1 and the first minima of the scattering increasing R. The distance between q ¼ 0 A function is inversely proportional to R and it was rationalized by Guinier and described as the zero-order peak [11]. Size-monodisperse spherical colloidal particles with diameters larger than 200 nm are often available. On the contrary, monodisperse nanoparticles with diameters smaller than 100e50 nm are more difficult to obtain and size polydispersity has to be considered. For an ensemble of spherical objects, size polydispersity can be imagined as the discrete sum of numbers of spheres with different R. Figure 5.9 shows the SAXS intensity for a mixture of 10 spheres: three of radius 2.8 nm, five of 3 nm and two of 3.2 nm. The relative SAXS profile for a sphere with R ¼ 3 nm is plotted for comparison.

Small-Angle X-ray Scattering for the Study of Nanostructures and Nanostructured Materials

Figure 5.9 (Continuous line) Simulated intensity for a mixture of 10 spheres: three of radius 2.8 nm, five of 3 nm and two of 3.2 nm. (Broken line) Intensity for a single sphere with R ¼ 3 nm.

It is striking just how much the deep minima are smeared out in the scattered intensity from the mixture. The higher the polydispersity, the less marked the minima. Moreover, it is important to notice that in Eqn [5.36], the intensity is weighted for the V2 term, which means that objects with larger dimension will contribute more to the total measured intensity. This is very important when one is interested in extracting the size-distribution function from the experimental data. Let us now consider an ensemble of particles with a continuous number size distribution described by the function DN(R). The measured intensity is given by the sum over all the contributions from the different sizes, the probability of which is described by DN(R): ds ¼ ðDrÞ2 dU

ZN DN ðRÞV ðRÞ2 Pðq; RÞ dR

[5.37]

0

The probability functions usually used to successfully describe size distributions in most of the practical cases are the Gaussian, Schultz and Weibull distribution functions. Gaussian distribution:   1 ðR
RÞ [5.38] DN ðRÞ ¼ pffiffiffiffiffiffi exp 2s2 s 2p

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where R is the mean sphere radius value (position of the peak) and s is the standard deviation (controls the width of the peak). Schultz distribution:    
z þ 1 zþ1 z zþ1 1 R DN ðRÞ ¼ R exp
[5.39] Gðz þ 1Þ R R where z is the parameter related to the width of the distribution. Weibull distribution:
b
 R R b
1 R exp DN ðRÞ ¼ b R R

[5.40]

where b is the parameter related to the width of the distribution. The polydispersity index can be calculated for the three different functions according to Table 5.1. The Gaussian function is the most used due to its simplicity although several studies showed that the Schultz or Weibull distributions work better for real systems. The Schultz distribution is able to describe the polydispersity in microemulsion droplets [12], while the Weibull distribution was found to work better for alloys and systems with hardsphere interaction potential [13e15]. The advantage of using the Weibull or Schultz distributions is that they can account for asymmetry in the shape of the size-distribution function, which is often the case in real systems. In Fig. 5.10, the polydispersity analysis on Au nanoparticles (AuNPs) in solution is presented. Experimental data show that the system is polydisperse (shallow minima) and simulation using a monodisperse spherical function does not describe properly the system (Fig. 5.10(a)). Using Eqn [5.37] for a polydisperse system of spheres, the SAXS intensity can be successfully simulated. Figure 5.11(b) shows the calculated Weibull distribution for the AuNPs system. The mean radius for the nanoparticles is 27 nm and the polydispersity index calculated according to Table 5.1 is 1.1. Note that the value for the mean radius of AuNPs as estimated using the monodisperse equation is inexact and is around 28.4 nm, with an error of about 5%. In some occasions, DN(R) bimodal distributions can be found in catalytic systems [16], in cluster growth [17] and in bimodal pore systems [18e20]. In such bimodal

Table 5.1 Polydispersity index for different distribution functions Gaussian Schultz

Polydispersity index

s R

1 ð1 þ zÞ1=2

Weibull


 
2 1 2
G 1þ R G 1þ b b

Small-Angle X-ray Scattering for the Study of Nanostructures and Nanostructured Materials

Figure 5.10 (a) SAXS profile of Au nanoparticles at the end of an in situ reduction reaction of Au3þ to Au0. (o) Experimental curve, (d) simulated intensity using a polydisperse system of spheres, (- - -) is the intensity of monodisperse spheres with R ¼ 28.4 nm. Data have been acquired at BM26B at the ESRF. (b) Resulting DN(R) calculated from the model with polydispersity according to Eqn [5.37] and using a Weibull distribution.

Figure 5.11 Simulated form factors for different particles shape: (d) sphere with R ¼ 10 nm; (e e) cylinder with R ¼ 10 nm and L ¼ 300 nm; (- -) disk with t ¼ 20 nm and R ¼ 300 nm. Representation of geometrical shapes is reported on the left for clarity.

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systems, coupling of BET, TEM and SAXS is often necessary to infer the exact size distribution function. So far, only the scattering from spherical particles has been considered. As mentioned above, SAXS is a powerful tool for the study of particles with different shapes, especially in systems where the polydispersity remains limited. The scattered intensity for monodisperse homogeneous anisotropic particles (cylinders and disks of finite thickness t) with circular cross-section of radius Rc and length L can be written as [Fournet 1949]: ds ¼ N ðDrÞ2 V 2 Pcyl ðq; RÞ dU  Zp=2 2J1 ðqR sin aÞ sinðqL cos a=2Þ 2 2 2 ¼ N ðDrÞ V sin ada qR sin a qL cos a=2

[5.41]

0

where a is the angle between the cylinder axis and the q direction and J1 is the first-order Bessel function. For infinitively thin disks of radius R, the intensity can be written as [21]:   ds J1 ð2qRÞ 2 2 2 2 2 ¼ N ðDrÞ V Pthin-disk ðq; RÞ ¼ N ðDrÞ V 2 2 1
[5.42] dU q R qR Figure 5.11 shows the form factors for three different particles: sphere with R ¼ 10 nm; cylinder with R ¼ 10 nm and L ¼ 300 nm and disks with t ¼ 20 nm and R ¼ 300 nm. Inspection of Fig. 5.11 allows the reader to understand how the measured SAXS intensity can be used to discriminate the particle shape. The slope in a logelog plot of I(q) in the Guinier region (q < 1/Rg) is directly related to the particle shape. For cylindrical or rod-like objects, a slope of q
1 is found. For flat objects, the slope is equal to q
2. Different slopes of I(q) are also found for more complex shapes like worm-like micelles or systems characterized by fractal arrangement of the subunits. According to Eqn [5.35], the scattered intensity is related to the p(r) or pair distance distribution function PDDF by a Fourier transform. The PDDF can be extracted from the measured intensity via inverse Fourier transform: ZN 1 ds ðqÞqr sinðqrÞ dq pðrÞ ¼ [5.43] 2p2 dU 0

For practical reasons, the scattering curve is measured only in a limited q range from qmin to qmax. This experimental limitation complicates the calculation of the p(r). A dedicated approach using indirect Fourier transformation IFT was developed by Glatter in order to extract p(r) from the measured intensity [22].

Small-Angle X-ray Scattering for the Study of Nanostructures and Nanostructured Materials

For a homogeneous sphere of radius R, an analytical expression for the PDDF has been reported:
 3 r2 3r r3 pðrÞ ¼ 2
[5.44] þ 4p R2 2 R 8R3 This function is zero for r > 2R. Once extracted, the p(r) can be used to determine the particles shape for homogeneous systems (Fig. 5.12). Spherical, rod-like and lamellar particles can be readily discriminated by comparing the resulting PPDFs, without a priori information about the systems. In this section dealing with form factors of homogeneous particles, it was shown how it is possible by inspection of the SAXS curves and the extracted PDDFs to distinguish between different particle shapes and to extract the average particles size and size distribution. As it will be shown in the coming sections, the three classical morphologies (spheres, cylinders and disks) are the most commonly found for semiconductors nanostructures and nanoparticles. However, other interesting and more complex form factors have been introduced in the past years: form factors for polymeric chains in different solvents, worm-like micelles, coreeshell spheres, ellipses and cylinders and many others. It is behind the scope of this chapter to deal with these form factors, mostly found in soft-condensed matter and the reader is referred to the following works for a more detailed description on different form factors: Pedersen [23] and Forster [24,25].

Figure 5.12 Pair distance-distribution function p(r) calculated for (a) a monodisperse sphere with R ¼ 10 nm and (b) a monodisperse cylinder with R ¼ 10 nm and L ¼ 300 nm. The p(r) has been calculated using GIFT program [115].

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3.2. Interparticles Interactions: The Structure Factor The case for particles interacting with a hard-sphere potential will be first described in this section. Moreover, due to the recently discovered importance of ordered quantum dots superlattices (examples given in Section 3.3), the equations for lattices of ordered nanoparticles systems will be described as well. Analytical expressions for the structure factor S(q) can be calculated only in few cases. Analytical expressions have been obtained using the liquid state theory for particles with spherical symmetry interacting with a spherically symmetric potential. For a homogeneous and isotropic system of spherical particles, the static structure factor S(q) is: Z ZN N 4pN sinðqrÞ iqr SðqÞ ¼ 1 þ r 2 ðgðrÞ
1Þ e ðgðrÞ
1Þ dr ¼ 1 þ dr [5.45] V V qr 0

It is important to note that the static structure factor is related to the pair or radial distribution function g(r) via a Fourier transform. This equation gives the link between the static structure factor S(q) measured in the scattering experiment and the radial distribution function g(r) representing the statistical mechanical description of the microstructure and thermodynamic of the system. The pair distribution function is indeed related to the thermodynamic properties, such as pressure or isothermal compressibility, of the fluid. The pair distribution function, g(r), can be calculated using the liquid state theory by solving the OrnsteineZernike (OZ) equation. To obtain information on systems of interacting colloids, it is necessary to model the scattered intensity by calculating the form and structure factors (Eqn [5.23]). This is easily done for monodisperse, spherical particles. For a homogeneous, isotropic fluid of spheres, the OZ equation is: Z hðrÞ ¼ gðrÞ
1 ¼ cðrÞ þ r cð~ r
~ r 0 Þhð~ r 0 Þ d~ r0 [5.46] The physical meaning of the OZ equation is the “total” correlation function, h(r), between two particles is due in part to the “direct” correlation function c(r) between them, but also to the “indirect” correlation propagated via increasingly large numbers of intermediate particles. With this physical picture in mind, it is plausible to suppose that the range of c(r) is comparable with that of the pair potential v(r), while the range of h(r) is generally much longer due to indirect correlations. The structure factor S(q) depends directly on c(r) via its Fourier transform: SðqÞ ¼

1 1
r~c ðqÞ

[5.47]

Unfortunately, c(r) and h(r) are both unknown functions, and the OZ equation can only be solved if there is an additional relation between them. This additional equation is

Small-Angle X-ray Scattering for the Study of Nanostructures and Nanostructured Materials

an approximation, called the closure relation, which relates h(r) with c(r). A deeper understanding of the meaning of c(r) can be obtained by diagrammatic and density functional derivative methods (Hansen and McDonald, 2006). The most popular closure relation is the PercuseYevick closure [26]:  
vðrÞ [5.48] cðrÞ ¼ gðrÞ 1
exp kB T Equation [5.48] provides a good description of fluids with very short-ranged interactions. For hard spheres, the potential v(r) can be approximated by vðrÞ ¼ N

r
vðrÞ ¼ 0

r>s

With this definition, c(r) is zero for r > s, where s is the diameter of the particles. The structure factor for interacting spheres with a hard-sphere radius RHS can be calculated analytically using the PercuseYevick approximation for the closure relation and is equal to: SðqÞ ¼

1 1 þ 24hGð2RHS qÞ=ð2RHS qÞ

[5.49]

where h is the hard-sphere volume fraction and     aðsin A
Acos AÞ b 2Asin A þ 2
A2 cos A
2 GðAÞ ¼ þ A2 A3       g
A4 cos A þ 4 3A2
6 cos A þ A3
6A sin A þ 6 þ A5 2

a ¼

ð1 þ 2hÞ

4

ð1
hÞ

;

b ¼

2
6h 1 þ h2 4

ð1
hÞ

;

g ¼

ha 2

The PercuseYevick approximation works well for systems characterized by a shortrange interaction potential. Figure 5.13 shows the calculated SAXS intensity using Eqns [5.23], [5.36], and [5.49] for a system of hard spheres with different effective sphere concentration ceff. Note that the presence of the structure factor reduces the intensity for low q values even for very low ceff. At high ceff, a clear interaction peak is present, whose position is related to the average distance between neighboring particles (qmax ¼ 2p/d). A detailed presentation of different structure factors is found in Ref. [23].

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Figure 5.13 SAXS profiles of a system of spheres with R ¼ 10 nm interacting with a hard-sphere potential and different effective volume fractions ceff. The hard-sphere radius is RHS ¼ 15 nm. Simulations have been performed using the PercuseYevick approximation. The thin line is the Porod law with slope q
4.

A general approach for ordered nanoparticles structures was given by Fo¨rster et al. [25]. This approach is based on the theory developed by Ruland [27] and uses the decoupling approximation [28]. According to the decoupling approximation, the intensity can be considered as the product of the particles form factor and the interparticles structure factor. However, this approximation is not valid for high polydispersity [29]. The intensity for an ordered system of particles can be written as follows: IðqÞ ¼ ðDrÞ2 rN PðqÞð1 þ bðqÞ½hZðqÞi
1Þ

[5.50]

where (Dr)2 is the electron density difference, or contrast factor, between the particles and the surrounding media, rN is the number density of the particles, P(q) is the form factor and Z(q) is the lattice factor. The angular brackets h.i denote the averaging with respect to the particle size distribution for P(q) and to the spatial distribution for Z(q). The different form factors have been discussed in the previous section. b(q) describes the influence of the particle polydispersity in radius to the interference term and has to be calculated according to the ratio: bðqÞ ¼

hFðqÞi2 hF 2 ðqÞi

where F(q) is the scattering amplitude of the particle or the Fourier transform of the particle form and hF 2 ðqÞi ¼ PðqÞ. b(q) for different geometries are reported in Ref. [25].

Small-Angle X-ray Scattering for the Study of Nanostructures and Nanostructured Materials

The undistorted lattice factor can be written using the Miller indices (hkl) for a given plane of the crystal lattice with unit cell of dimension d (¼ 1,2,3) as: ð2pÞd
1 c X 2 mhkl fhkl Lhkl ðqÞ Z0 ðqÞ ¼ nvd Ud qd
1 fhklg

[5.51]

where n is the number of particles per unit cell, fhkl is the factor that takes into account symmetry-related extinction rules [25], Ud is the d-dimensional solid angle, Lhkl is the normalized peak shape function and mhkl is the corresponding multiplicity. vd is the volume for three-dimensional unit cells (bcc, fcc, etc.), the surface for two-dimensional unit cells (2-D hexagonal lattice), and the long period for one-dimensional unit cells (lamellar lattice). The translational disorder can be taken into account by assuming a Gaussian lattice point distribution around zero and with a variance sa: ZðqÞ ¼ Z0 ðqÞGðqÞ þ ð1
GðqÞÞ G(q) is the so-called DebyeeWaller factor for thermal disorder:   GðqÞ ¼ exp
s2 a2 q2

[5.52]

[5.53]

where a is the nearest neighbor distance between particles. The term (1
G(q)) gives the diffuse scattering contribution. The width d of the peak shape function Lhkl is related to the finite crystal domain size L by the DebyeeScherrer equation, L ¼ 2p/d. For diffraction peaks described using a Lorentian function of full-width at half-maximum (fwhm) d: Lhkl ðqÞ ¼

d=2p q2 þ ðd=2Þ2

[5.54]

Examples of simulated 3-D and 2-D crystals are given in Fig. 5.14, pointing out the effect of the form factor, and the domain size on the scattered intensity of ordered nanoparticles structures. The form-factor oscillations are observable in the total intensity curve. When the coherent domain size is decreased, the lattice reflections become broader and weaker and the form factor oscillations are more noticeable. For anisotropic particles, the slope in the Guinier region is close to that expected for the form factor.

3.3. Quantum Dots and Metallic Clusters A quantum dot (Q-dot) is a semiconductor nanoparticle with typical dimensions ranging from 1 nm to some tens of nanometers. Q-dots have recently attracted a lot of interest due to the change of electrical and optical properties as a function of their size [30e32]. The properties of such small particles can be very different from those of the bulk material mainly for two reasons: a) the surface-to-volume atomic ratio is much

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Figure 5.14 (Left) SAXS intensity for a bcc lattice of spherical particles with radius of 15 nm. (Right) SAXS intensity for a hexagonal lattice of cylindrical particles with radius of 15 nm and length of 100 nm. Lattice spacing is 50 nm in both cases. L is the coherent domain size. For color version of this figure, the reader is referred to the online version of this book.

higher than in the bulk, b) strong variation in the density of the electronic energy levels is found with reducing the crystal size. Research for Q-dots science is focused in producing semiconductor nanoparticles with the same bulk-bonding geometry and with passivated surface. Passivation is achieved by bonding the cluster surface to another material of much larger band gap, eliminating all of the energy levels inside the energetically forbidden gap. Passivation is thus very important since it avoids degradation of electrical and optical properties. Inorganic or organic materials can be used as passivating agents. Today, there is a lot of enthusiasm centered on the possibility of manipulating single Q-dots to produce three-dimensional quantum-dot superlattices. “Quantum dot solids” can be formed by self-organization of single quantum dots into 2-D and 3-D macroscopic assemblies [33e36]. Moreover, the incorporation of Q-dots into a matrix with self-assembly properties seems to be a promising way of forming nanocomposites with improved properties [37]. In order to construct these semiconductor nanocrystals, high tunability of the nanocrystal building blocks is needed, which requires control over size, polydispersity and shape. As discussed in the previous section, SAXS is the perfect tool for the investigation of all of these morphological properties. Very small ZnS nanoparticles can be obtained using solidesolid reaction in confined space as reported by Calandra et al. [38]. The synthesis is based on a solidesolid reaction between small nanoparticles of Na2S/AOT and ZnSO4/AOT

Small-Angle X-ray Scattering for the Study of Nanostructures and Nanostructured Materials

Figure 5.15 SAXS profiles for (a) (D) ZnSO4/AOT nanoparticles and (b) (B) Na2S/AOT nanoparticles suspended in n-heptane at Rs ¼ 0.13. (,) is the profile for salt-free AOT/n-heptane micelles. Solid lines are best fits using Eqn [5.37]. (Reprinted from Ref. [39] with permission. Copyright (2001) by Springer Science.)

(AOT ¼ bis(2-ethyl-hexyl)sulphosuccinate) [39]. After solvent evaporation from an Na2S or ZnSO4 water/oil microemulsions, Na2S/AOT and ZnSO4/AOT nanoparticles are formed due to water evaporation from the salt/water micellar core. This can be seen as a confined crystallization process. Figure 5.15 shows SAXS profiles for (a) the ZnSO4/AOT and (b) the NaS2/AOT nanoparticles in n-heptane suspensions at Rs ¼ 0.13, where Rs is the [salt]/[AOT] molar ratio [39]. The profiles can be described using the intensity for a polydisperse spherical system described by Eqn [5.37] and the spherical form factor in Eqn [5.36]. From best fit of Eqn [5.37] to the experimental profiles in Fig. 5.15, an average dimension of about 1 nm for ZnSO4 and 1e1.2 nm for Na2S salt nanoparticles is calculated independently to the Rs used. These dimensions are very close to the saltfree AOT reverse micelles in n-heptane (R ¼ 1 nm) as one would expect from the growth of a salt nanocrystals inside the AOT reverse micelles. However, the polydispersity of the salt/AOT nanoparticles is larger than the one observed from salt-free AOT (Fig. 5.16). One of the most important advantages of SAXS over TEM is highlighted in this work. TEM of the same samples used to obtain Fig. 5.15 showed spherical nanoparticles of about 12 nm average radius for Na2S/AOT and about 2 nm for ZnSO4/AOT. These values are much larger than those given by the SAXS data from suspensions. Since TEM requires solvent evaporation prior to the measurements, salt nanoparticles aggregate leading to larger particles. In this case, in situ SAXS on suspensions is the only tool available for particle characterization. By further mixing equal amounts of the ZnSO4/AOT and the Na2S/AOT nanoparticles produced with the same Rs, the author reported the synthesis of ZnS ultrasmall

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Figure 5.16 Size-distribution functions calculated for (
) salt-free AOT micelles; (e e) ZnSO4/AOT; (- -) Na2S/AOT. (Adapted from Ref. [39].)

nanoparticles [38]. At the end of the reaction (usually few seconds), ZnS nanoparticles capped with AOT molecules are formed. The structure of the synthesized nanoparticles was studied as a function of the ZnS-to-AOT molar ratio (Rs) and different AOT concentrations. SAXS profiles in logelog scale are reported for ZnS nanoparticles obtained with two different Rs (Fig. 5.17(a)) and with two different [AOT] (Fig. 5.17(b)). In agreement with the explanation given above, all the profiles in Fig. 5.17 can be described as the scattering from an ensemble of dilute spherical objects (Eqn [5.37]). Depending on salt and surfactant concentration, the average core radius of the AOT micelles containing the ZnS quantum dots varies between 1 and 1.3 nm. This means that ZnS Q-dots synthesized via a solidesolid nanoparticles reaction are smaller or equal to 1 nm in radius. The synthetic route discussed in the example above does not allow one to obtain Q-dots with very low polydispersity. Low polydispersity is needed to obtain Q-dot superlattices. An effective method to produce high-quality CdS, CdSe and CdTe Q-dots with a narrow-size distribution was proposed by Murray et al. [40,41]. The synthesis is based on the pyrolysis of organometallic reagents by injection into a hot coordinating solvent. The key points of this process are (i) temporally discrete homogeneous nucleation and (ii) controlled growth and annealing of the nanocrystallites by fine temperature tuning in response to the growth process. The growth temperature determines the size

Small-Angle X-ray Scattering for the Study of Nanostructures and Nanostructured Materials

Figure 5.17 SAXS profiles for ZnS Q-dots synthesized via solidesolid reaction in confined space (a) at different ZnS/AOT molar ratio (Rs) and (b) at different AOT concentration. (Reprint from Ref. [38] with permission. Copyright (2002) by the American Chemical Society.)

and polydispersity of crystallites. The growth process has to be maintained in a stable condition in order to achieve low-size polydispersity. If the temperature is increased, the average crystallite size grows and the size distribution sharps. On the contrary, the temperature is decreased as soon as the size distribution begins to widen. The nanocrystallites obtained following this route are of relatively low polydispersity. Moreover, nanoparticles produced in such a way are sterically stabilized by a capping agent, i.e. alkyl molecules chemisorbed on their surface. The organic-molecules shell determines the stability of nanocrystals in suspension. By adding a nonsolvent, stability is reduced and size-selective flocculation occurs [40]. Larger particles tend to precipitate as a consequence of the greater attractive forces while smaller particles remain in the supernatant. As a result, the size distribution of particles in both the supernatant and the precipitate becomes sharper. Precipitation can be repeated a number of times in order to produce narrower size distributions. Following this route, almost monodisperse CdSe nanoparticles with diameter ranging from 11.5 nm down to 1.2 nm can be obtained (Fig. 5.18(a)). Almost monodisperse CdSe clusters are reported to self-organize after solvent evaporation (Fig. 5.18(b)) [36]. The crystalline structure of CdSe nanocrystals produced in this way is predominantly the wurtzite form as in the bulk material [36]. The size and polydispersity of CdSe nanocrystals in suspension and the local structure of closely packed CdSe Q-dots solids with different sizes and very low polydispersity can be characterized by SAXS in a similar way that reported above for ZnS Q-dots [37] and making use of the SAXS theory for the form factor reported in Section 3.1. Standard deviations <4.5% are reported for CdSe Q-dots with average diameter of 3.8 nm or 6.2 nm assuming that the diameter has a Gaussian distribution.

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Figure 5.18 (a) Optical absorption spectra for CdSe nanocrystals dispersed in hexane with tuned diameter (Reprint from Ref. [40] with permission. Copyright (1993) of the American Chemical Society); (b) TEM picture for an fcc superlattice of CdSe Q-dots with 4.8 nm diameter. The figure is relative to the (101) superlattice orientation. Inset shows electron diffraction pattern. (Reprint from Ref. [36] with permission. Copyright (1995) by the American Association for the Advancement of Science.)

Size-selective precipitation has been efficiently used to obtain self-organization into superlattices in several systems. Korgel et al. reported superlattices formation using low-polydispersity hydrophobic dodecanethiol-capped silver nanoparticles [38]. Figure 5.19(a) shows SAXS profiles of dodecanethiol-capped Ag nanoparticles obtained using consecutive size-selective precipitation and redispersed in hexane. Each size-selective precipitation step is reported to be able to isolate Ag particles differing by one lattice plane width in radius. The standard deviation for the size distributions of the clusters in Fig. 5.19(a) is about 8%. Such low-polydisperse dodecanethiol-capped Ag nanoparticles are able to form face-center-cubic (fcc) superlattices upon solvent evaporation (Fig. 5.19(b)). Size polydispersity plays a crucial role in the Q-dots superlattice formation. The same dodecanethiol-capped Ag nanocrystals of Fig. 5.19 but with larger polydispersity (s  12%) do not form superlattices upon solvent drying, but rather liquid-like solids [42]. As first reported by Alder et al. [43], a disordereorder phase transition is predicted at 0.49 hard-sphere volume fraction. For the lattices reported in Ref. [44], the Ag core volume fraction is around 0.4, lower than the predicted value. This can be understood considering the capped Ag nanoparticles as

Small-Angle X-ray Scattering for the Study of Nanostructures and Nanostructured Materials

Figure 5.19 (a) Dodecanethiol-capped Ag nanoparticles obtained by consecutive size-selective precipitation using chloroform/ethanol as solvent/nonsolved pair. Numbers are indicative of the mean nanoparticles radius; (b) 3-D superlattices for the Ag nanoparticles of (a). (Adapted from Ref. [44] with permission.)

“soft spheres,” where the dodecanethiol corona around the metallic clusters contributes to the effective volume fraction Feff playing a role in the formation of the ordered fcc phase. A theory for disordereorder transition of soft spheres was derived by McConnel et al. for diblock copolymers using the input from SAXS and SANS data and it has proved to be successful in the prediction of fcc polymeric micelles structure [45,46]. The disordereorder transition is governed by the corona layer that offers medium-range repulsion necessary for the ordering. What counts in determining Feff is not only the core radius Rc but the effective radius of the nanoparticles/corona (capping ligand for the Ag example) layer: 3 p  [5.55] Feff ¼ r 2Reff 6 where r and Reff are the density and the effective radius of the nanoparticles/corona “soft sphere.” This concept was recently applied by Fisher et al. to a system of nanoparticles with a functionalized layer of polystyrene (PS) as superficial ligand [47]. Interaction between functionalized nanoparticles depends on the attached polymer layer and establishes even at very low-volume fractions when increasing the molecular weight (MW) of the PS layer. Superlattice formation for CdSe nanoparticles with core radius of 2.1 nm and PS layer with increasing MW has been reported and was investigated by SAXS [47].

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Ordering of nanoparticles occurs at volume fraction as low as 0.01 when nanoparticles are covered by a layer of long PS chains (15.6 kg/mol). As reported above for the Ag-capped nanoparticles, CdSe Q-dots capped with lowMW ligand (alkyl phosphine) form fcc lattices. On the contrary, Q-dots covered by a layer of PS chains tend to form body-centered cubic (bcc) lattices. This can be understood by a change in the interaction potential between particles: strong hard-sphere repulsive potential induce fcc formation; particles with longer corona layers experience a much softer repulsion and bcc lattices are preferentially formed [45,48,49]. Moreover, by changing the MW of the PS in the layer, the interparticles distance in the CdSe bcc lattice can be increased from 5 nm up to 20 nm, making it possible to tune the physical properties of the lattice. The dynamics of Q-dot superlattice formation can be monitored by synchrotron SAXS in real time and the role of the solvent on superlattice formation can be elucidated. It seems that solvent molecules are necessary initially to form the superlattice and they are incorporated in the first expanded superlattice. Subsequently, the incorporated solvent molecules are slowly released by evaporation [50]. It has been discussed above how the capping agent or the organic layer is important for steric stabilization of Q-dots nanocrystals suspensions as well as for superlattice formation. Using surface exchange methods, i.e. capping ligand exchange, the crystallite surface can be modified as desired. Matoussi et al. presented a detailed SAXS study about interparticle interactions in suspensions of CdSe Q-dots as a function of the capping ligand and solvent [51,52]. The authors isolated the main factors influencing the stability of Q-dots dispersions: (I) nature and length of the capping ligand; (II) degree of surface coverage; (III) core size; and (IV) solvent. By using a long enough capping ligand coupled with a solvent with a favorable ligand/solvent interaction, attraction between nanocrystalline cores is screened and stable suspensions are obtained. On the other side, when shorter ligands are used with a less favorable interaction with the solvent, the suspension is not stable and strong attraction causes precipitation. Stable nanoparticles suspensions but with slight attractive potential between particles can be obtained when short ligands are used in a favorable solvent. This is the case of pyridinecapped CdSe nanocrystals with about 2 nm radius dispersed in pyridine [52]. The capping agent, in this case, is the same as the solvent molecules, and is much shorter than the tryoctil phosphine. The presence of an attractive potential in the suspension is evidenced in SAXS by a positive deviations at lower q values (intensity upturn for ˚
1) as shown in Fig. 5.20. Nevertheless, as the solvent (pyridine) is a good q < 0.8 A solvent for the capping agent, the suspension is still stable over a reasonable timescale. Once pyridine-capped CdSe Q-dots are dispersed in toluene, the screening effect of the capping layer is not enough to shield the core-to-core attractive potential and the nanoparticles experience stronger attraction (larger intensity upturn). The suspension is unstable and precipitation occurs.

Small-Angle X-ray Scattering for the Study of Nanostructures and Nanostructured Materials

Figure 5.20 SAXS profiles for CdSe Q-dots with different capping agent and in different solvents. TOP ¼ tryoctil phosphine; PYR ¼ pyridine. Lines represent fits using polydisperse spherical model. (Reprint from Ref. [52] with permission. Copyright (1996) by the American Institute of Physics.)

3.4. Nanoscale Heterogeneities in Amorphous Semiconductor Alloys Studied by SAXS and ASAXS In the previous section, it has been discussed how Q-dots (semiconductor nanostructures) can be essentially considered as fragments of a particular crystalline material. On the other hand, noncrystalline semiconductors films based on amorphous alloys are of immense technological interest. A metal:insulator transition is often described using the Anderson-type transition, where the electrons at the Fermi level become delocalized and extended-state conduction can occur [53]. This description assumes that the impurity atoms are homogenously incorporated into the alloy structure, which undergoes a continuous structural transition as the metal concentration is increased. On the other hand, depending on the materials used and the deposition conditions, amorphous alloys may phase separate into small amorphous particles and may incorporate small voids. If phase separation of metal atoms in the amorphous matrix occurs, the electrical conductivity is better described in terms of percolation theory [54]. Usually, the characteristic size of such a phase separation is very small (<2e3 nm) and because of low contrast, the studies of amorphous alloys using microscopy techniques are difficult. Fine-size composition fluctuations can be successfully studied by SAS techniques. For systems with large size of metal clusters, TEM can be also employed [55].

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Figure 5.21 SAXS profiles for a-SixNi1
x:H obtained by reactive cosputtering on mica at different metal concentrations. (Reprint from Ref. [56] with permission. Copyright (1999) by the Materials Research Society.)

Amorphous hydrogenated silicon/metal alloys were investigated using SAXS by Ridgen and Newport [56]. a-SixSn1
x:H and a-SixNi1
x:H alloys can be deposited by reactive cosputtering. Figure 5.21 shows the SAXS pattern for these alloys. SAXS data suggest that the technique is extremely sensitive to the formation of metal clusters. In the a-SixNi1
x:H alloys, islands of nickel silicide segregate within the native tetrahedral random network of the host silicon and an SAXS peak is visible even at the lowest metal concentration. The peak intensity is concentration-dependent suggesting the growth of clusters with increasing metal concentration. On the other end, a-SixSn1
x:H alloys do not show an SAXS peak, suggesting that tin atoms are incorporated in the amorphous semiconductor with tetrahedral coordination in a homogeneous manner. Unfortunately, it has proved difficult to model the experimental SAXS data resulting from atomic-phase separation in a matrix. Thus, the use of SAXS is limited to obtaining qualitatively information about nanoscale heterogeneities in metal/semiconductor alloys. However, when SAXS is coupled with anomalous scattering (ASAXS), a more quantitative description of the systems is possible. ASAXS has been used to investigate sputtered amorphous MoeGe, FeeGe and FeeSi films [57e59]. The principles of anomalous scattering and diffraction are treated in details in Chapter 4 (Schu¨lli et al. in this book) [60]. When X-rays have energy close to the absorption edge of an atom, the atomic scattering factor is energy-dependent and is equal to: f ðEÞ ¼ Z þ f 0 ðEÞ þ if 00 ðEÞ

[5.56]

Small-Angle X-ray Scattering for the Study of Nanostructures and Nanostructured Materials

where Z is the number of electrons far from the resonance, f 0 ðEÞ and f 00 ðEÞ are, respectively, the energy-dependent anomalous scattering factors related to the scattering and the absorption phenomena. The scattering of the resonant atoms (electrons) can be decreased by several electron units when the X-ray energy is close to the absorption edge of a particular atom, where f 0 ðEÞ has a minimum. By measuring the ASAXS profiles at two different energies, E1 far from and E2 close to the absorption edge, it is possible to obtain information on the composition of the amorphous alloy and on the atoms that are inhomogeneously distributed in the alloy. The origin of the SAXS signal can thus be revealed. The method of calculating the alloy composition will not be discussed here as it is described in detail by Schu¨lli et al. in detail for SiGe alloys in Chapter 4 [60]. The origin of the SAXS signal in amorphous alloys can be studied by using the change in the scattering cross-section D½ds=dU: ds ds  ðq; E1 Þ
ðq; E2 Þ ds dU dU D ¼ dU jf ðE1 Þ
f ðE2 Þj 

[5.57]

In Fig. 5.22 (left part), the change of the SAXS pattern is reported as a function of the incident angle of the sample for different alloys. The presence of the SAXS peak is

Figure 5.22 (left) ASAXS profiles as a function of different incident angles on the samples for three different alloys: (a), (b) and (c). (d)e(f) are the change in the scattering cross-section at different absorption edges: Fe K-edge ¼ 7112 eV, Mo K-edge ¼ 20,000 eV, Ge K-edge ¼ 11,103 eV. (Reprint from Ref. [58] with permission. Copyright (1994) by the American Physical Society.) (right) Rotation of the sample with respect to the incident beam allows one to move the scattering vector from in-plane to out-of-plane direction. The scattering vector q is depicted in red. For interpretation of the references to color in this figure legend, the reader is referred to the online version of this book.

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associated with the fluctuation in composition within the alloy produced during the sputtering process. The correlation length estimated by Bragg’s law is 2e3 nm, depending on the alloy. By tuning the angle of incidence between the beam and the film surface, the scattering vector is moved from the in-plane direction toward the direction of deposition (Fig. 5.22 (right part)). The change in the peak position as a function of the incident angle denotes a strong anisotropy in the deposited structure. The insets in Fig. 5.22 (right part) report the change in the ASAXS scattering crosssection for the different alloys. Large changes in D½ds=dU are found for the metal atoms (Fe and Mo), while little or no change in intensity is observed at the Ge edge. It is straightforward to conclude that the inhomogeneous distribution of the metal atoms in the alloy is responsible for the SAXS signal.

3.5. Surfactant-Coated Cobalt and Cobalt/Nickel Nanoclusters Recently, there has been an increasing interesting in coupling morphological and structural information obtained by SAXS with short-range local information obtained via X-ray absorption spectroscopy techniques (XANES and EXAFS) [61e64]. Dedicated synchrotron beamlines with configurations able to simultaneously perform SAXS/ WAXS/XAS have been described recently [65]. The formation of small metallic nanoparticles can be successfully studied with both SAXS and XAS. The size and structure of cobalt nanoparticles (CoNP) formed in confined media can be studied by coupling mesoscale SAXS data with local atomic scale information from EXAFS [66]. The confined reaction (Scheme 5.1) is mediated by metal/surfactantreversed micelles. One of the most used surfactant is Na(AOT) (AOT ¼ bis(2-ethylhexyl) sulfosuccinate). In the liquid phase, reversed water-in-oil micelles are formed. The micelle size is controlled by the water fraction w ([H2O]/[Na(AOT)]). Small reversed Co(AOT)2

(a) O O O

O S O O

+

Na

O

(b) Co(NO3)2 + 2NaAOT

H2

H 2O

N-heptane

Co(II)

NaBH4

AOT-capped Co(0) nanoparticles

(c)

Co(AOT)2 + 2NaBH4

Co + 2NaAOT + B2H6 + H2

Scheme 5.1 (a) bis(2-ethylhexyl)sulfosuccinate (AOT) surfactant molecule. (b) Reaction path used to obtain Co nanoparticles capped with AOT surfactant. (c) Reduction reaction formula.

Small-Angle X-ray Scattering for the Study of Nanostructures and Nanostructured Materials

micelles can be obtained in oil at low water contents (w < 4) [67]. After mixing Co(AOT)2 micelles in n-heptane with NaBH4, a reduction reaction takes place and the mixture turns black. The black color indicates that Co(0) metallic powder is formed during the reaction. After some time, a precipitate is formed. In a reduction process, such as the one described above, some important questions need to be addressed: (i) Has the reduction reaction worked? Do the products of the reaction correspond to those expected? (ii) Has the surfactant layer worked as a template for the nanoparticles? (iii) What is the nature of the precipitate and is it structured? XANES analysis at the Co K-edge (7.7 keV) can be used to determine the valence state of cobalt atoms while EXAFS can be used to study the atomic arrangement and provide a rough estimation of the average nanocrystal dimension when its diameter is smaller than 1 nm. The average cluster size can be estimated according to the Borowski relation [68] as   3 

3 ri 1 ri theo Ni ¼ Ni 1
þ [5.58] 4 Rcluster 16 Rcluster where Ni is the contracted coordination number of the shelli and Ntheo is the bulk value i of the corresponding shelli. The contraction of the coordination shells is due to nanometric dimensions of the clusters. Rcluster is the cluster dimension. The theory of XANES and EXAFS applied to the study of the nanostructures is treated in detail in Chapter 9. Figure 5.23 reports (left) the XANES spectra and (right) the k2-weighted EXAFS profiles for samples collected right after the reaction was completed (A), aged 1 h in the reacting solution (B) and aged 1 day in air after the reduction (C). All the samples have been dried before analysis to remove the solvent. By comparing the XANES features of

Figure 5.23 (left) XANES profiles and (right) k2-weighted EXAFS data for samples A, B and C. Solid lines in the right figure are best fits for EXAFS data obtained using CoeCo amplitude and phase of the firstshell metallic cobalt calculated with FEFF. For comparison, the measured data for a standard Co(0) foil are reported as well. (Reprint from Ref. [66] with permission. Copyright (2009) by the American Institute of Physics.)

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the sample with those from the Co foil, it is evident that all the three analyzed samples have Co atoms in the metallic state. This is important information especially for sample C aged 1 day in air since this indicates that the surfactant molecules act as a shield to prevent Co oxidation by air. From EXAFS analysis (line fit in Fig. 5.23 right plot), the coordination number N, the main CoeCo distance R and the disorder factor s2 are calculated. The Co atoms in the nanoparticles have a mean first neighbor coordination number of 5/6 cobalt atoms. The absence of visible higher order shells and the high disorder factor (s2 > 1.0) suggests that the particles are small. Assuming a spherical shape for the CoNP and using Eqn [5.58], an average diameter of 0.8  0.2 nm is calculated [69]. As reported above, the cluster size is better studied by SAXS. Figure 5.24 reports the corresponding SAXS profiles for the samples A, B and C. By inspection of Fig. 5.24, a linear slope of q
1 typical for elongated objects is found (Section 3.1). One could naturally interpret such a slope as the presence of Co clusters of cylindrical shape. On the contrary, it is well known that spherical CoNP are obtained via reduction with NaBH4 at similar conditions of the data showed here [70]. Indeed, the shape of the metallic clusters is dominated by the structure of the initial metal ions/surfactant structure [67], which is spherical in this case. A model of spherical subunits preferentially organized in one direction can be successfully used to explain the data. The intensity for an ensemble of n identical spheres of radius R constituting the subunits of the elongated aggregate positioned as in a line grating is # " n
1 X n sinðqrij Þ 2X 2 2 [5.59] IðqÞ ¼ nðDrÞ V Pðq; RÞ 1
n i¼1 j¼iþ1 qrij

Figure 5.24 SAXS profiles for the samples A, B and C. (Reprint from Ref. [66] with permission. Copyright (2009) by the American Institute of Physics.)

Small-Angle X-ray Scattering for the Study of Nanostructures and Nanostructured Materials

R

Scheme 5.2 Illustration of a linear ensemble of spheres of radius R assembled on a line grating. The arrows represent the deviations of positions from the ideal points (structural disorder).

where P(q,R) is Psph(q,R), the spherical form factor reported in Section 3.1. rij is the distance between the spheres i and j defined according to their position on the line grating (Scheme 5.2). The structural disorder of the 1-D chain of spheres is considered as the displacement of the spheres positions along the line grating in the perpendicular direction. Should there be polydispersity in the radius of the spheres then,
2 ZN 4 Pðq; RÞ ¼ DN ðRÞR6 Psph ðq; RÞ dR p 3 0

where DN(R) is the distribution function (Gaussian, normalized Weibull, etc.). The SAXS data for the CoNP/AOT nanocomposite can be fitted with a chain of spheres model (Fig. 5.25). The average radius obtained is around 1 nm with a structural disorder of about 5%.

3.6. Magneto-SAXS In the previous examples, the ability of SAXS to determine shape, average dimensions and size distributions of nanostructures has been illustrated. Unfortunately, it is not always possible to easily determine the nanoparticles shape and size. This is the case when the average particle size is larger than the largest size observable (above the SAXS resolution), and/or when the shape is complex and the particles polydispersity is large. The first case can be solved by using specially designed experimental configurations allowing one to perform SAXS on very large particles (ultra-SAXS or microradian scattering, see Section 3.7) or by using a different technique as light scattering or microscopy (AFM, TEM, confocal microscopy, etc.). The latter case can be solved sometimes by coupling in situ synchrotron 2-D-SAXS with external forces acting on the sample (i.e. mechanical or magnetic forces). The basic principle used in this case is that the objects can be oriented or not by the applied external force depending on their anisotropy and additional structural information can thus be obtained. Sexithiophene (T6) molecules are composed by a p-conjugated core carrying two chiral oligo(ethyleneoxide) side chains at both the a- and u-position [71]. T6 can form different aggregates in protic polar solvents, with morphologies ranging from spherical structures [72], to needles [73], or rod-like helical assemblies and flat “creˆpes” [74], where the morphology and resulting properties depend on the solvent and the fabrication method used.

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Figure 5.25 (left) SAXS profile from sample A (broken line) and best fit using Eqn [5.59]. The size distribution D(r) obtained using a Weibull function is reported as well; (right) structure of the elongated CoNP/AOT domains. (Adapted from Ref. [66].)

Figure 5.26 shows (a) the 2-D-SAXS image and (b) the relative 1-D-SAXS curve for self-assembled sexithiophene aggregates in a mixture of o-dichlorobenzene and n-butanol [75]. From the inspection of the 1-D-SAXS profile at high q values, information about the internal structure of aggregates can be obtained. The relatively sharp peak found at q ¼ 1.3 nm
1 related to the stacking of the molecules, periodicity of about 4.8 nm, inside the aggregates. Since the length of the molecule is about 6.5 nm, it is reasonable to suggest that the molecules stack in a lamellar-like structure with a tilt angle of about 30e35 between the molecular axis and the diffraction plane. A simple lamellar-like stacking of the molecules is depicted in the inset of Fig. 5.26(b).

Small-Angle X-ray Scattering for the Study of Nanostructures and Nanostructured Materials

(a)

(b)

o

o

o

o

o

o o

s

s

s

s

s

s

o o

o

o

o

o

o

Intensity (a.u.)

1000 100 10

q-4

4.8 nm

1 0.1

B = OT

D=8m 0.1

D=1.5m

q (nm-1)

1

Figure 5.26 (a) 2-D-SAXS measured at 1.5 m S-to-D and (b) corresponding 1-D-SAXS for T6 aggregates in n-butanol/ODCB solution. For color version of this figure, the reader is referred to the online version of this book. (Adapted from Ref. [75].)

Unfortunately, any conclusion about the size and the shape of the aggregates can be drawn by inspecting the profile at lower q values. Only a q
4 slope is observed, indicating that T6 molecules form aggregates with dimension much larger than 150 nm. Due to its anisotropic chemical structure, sexithiophene molecules possess large anisotropy in their magnetic susceptibility c. It is known that aromatic molecules such as benzene and thiophene can be aligned by an external magnetic field with their aromatic plane parallel to the field minimizing the angle q between the aromatic plane and the magnetic field direction. The degree of alignment of the objects is given by ! ðck
ct ÞNB2 2 f ðqÞ ¼ exp
[5.60] cos q 2m0 NA kb T where ck and ct are the magnetic susceptibilities parallel and perpendicular to the aromatic molecular plane, N is the number of molecules in the aggregate and B is the magnetic field strength. NA, m0 and kb are the Avogadro’s number, the magnetic constant (vacuum permeability) and the Boltzmann constant, respectively. The orientation is easier for ordered aggregates (i.e. symmetry axis of the molecules are nearly aligned) in solutions containing a larger number N of molecules, so that high degrees of orientation can be achieved. The orientation under an external magnetic field can thus be used in combination with in situ 2-D-SAXS in order to observe the aggregates in the aligned state and obtain additional information about the possible shape of the T6 aggregates. Figure 5.27(a) and (b) shows the 2-D-SAXS images for aggregates oriented perpendicular to the X-ray beam direction using a 4-T magnetic field. Figure 5.27(b) shows the high orientation of the diffraction molecular planes at 90 with respect to the B-field axis (indicated with the white arrow in the figure). At the same

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Figure 5.27 2-D-SAXS images of T6 aggregates in 4-T magnetic field measured using (a) 8-m S-to-D and (b) 1.5-m S-to-D. The X-ray direction is perpendicular to the magnetic field direction. For color version of this figure, the reader is referred to the online version of this book. (Adapted from Ref. [75].)

time, inspection of the image close to the beamstop provides information about the aggregates orientation. The intensity distribution close to the beamstop is better shown in Fig. 5.27(a) (intensity observed at smaller angles, i.e. larger distances). The strong equatorial streaks are indicative of elongated objects that strongly align with their long axis parallel to the B-field direction. Two possible structures for T6 molecular aggregates can describe the results: (i) a lamellar-like structure or (ii) a multiwalled cylindrical-like structure. In order to understand which one of the two possible structures is the most probable one formed by T6 in ODCB/butanol, the measurements were repeated with the X-rays parallel to the B-field direction. In this case, any anisotropy in the SAXS intensity is detected in the probed plane (plane containing the aggregate short axis). This implies that the aggregates have radial symmetry along their short axis (oriented perpendicular to the field) suggesting that the multiwalled cylindrical structure is the most probable structure (Fig. 5.28(b)). Such a conclusion can be confirmed using in situ birefringence

Figure 5.28 (a) Confocal microscopy images for elongated T6 tubular aggregates. (b) Scheme of the multiwalled structure inferred by the SAXS analysis. For color version of this figure, the reader is referred to the online version of this book.

Small-Angle X-ray Scattering for the Study of Nanostructures and Nanostructured Materials

and in situ confocal microscopy in a magnetic field. Confocal microscope images of 3-months-aged solutions confirm the rod-like nature of T6 aggregates. The aggregates are found to be 5e10 mm long with an average diameter ranges from 0.5 mm to few microns (Fig. 5.28(a)). Similar to X-ray diffraction, the DebyeeScherrer method can be used here to estimate the total cylinder-shell thickness L from the angular width D(2q) at half-maximum of the 4.8-nm peak in the absence of preferential orientation. After correction for the instrumental spread function, the formula L ¼ l/[cos qMax D(2q)] can be applied, where l is the X-rays wavelength (l ¼ 1.24 A˚). For the studied T6 aggregates, the minimum calculated total-shell thickness is about L ~ 72.4 nm, which means that around 15 T6 layers stacks in a radial manner forming the shell in the cylindrical aggregate. T6 tubular aggregates grow in time by the continuous addition of new molecules. Depending on aging, one can go from T6 nanotubes to microtubes with diameters ranging from hundreds of nanometers up to 1 or 2 mm. Such dimensions are comparable to some lipid nanotubes [76] and they could be of interest for accommodating different guest molecules. The big challenge in this field remains the ability to control the molecular self-assembling in order to obtain the desired morphology/dimension. Manipulation of objects in the X-ray beam by using external fields is also widely used for anisotropic inorganic particles (like clays, metallic rods, etc.).

3.7. Behind the 100-nm Resolution: Microradian Diffraction and Scattering Technique SAXS resolution is usually expressed as the maximum distance in real space detectable with the used experimental configuration and it is commonly reported to be around 100 nm for normal SAXS. Alternatively, it can be expressed as the minimum angular value for which the intensity can be measured, i.e. 10
3 radian at a reference wavelength ˚ for standard synchrotron SAXS configurations. In some cases, this value becomes of 1 A a limit and it is necessary to go behind this limit. This is the case for self-assembled colloidal crystals and colloidal suspensions of particles with large dimensions (R > 100 nm). Colloidal crystals are formed by the spontaneous organization of monodisperse particles in 3-D or 2-D structures with long-range order. Three-dimensional crystals of good quality can be produced by gravity sedimentation from colloidal dispersions of monodisperse (or very low polydisperse) spheres [77e81]. Colloidal crystals with micrometer and submicrometer periodicity are very important for their potential applications as photonics materials [82e85]. Several recently reviewed [86] methods can be used to produce highly ordered 2-D colloidal monolayers. Closely and nonclosely packed 2-D colloidal crystals with large domain sizes can be obtained by direct assembly at a watereair interface [87,88]. Suspensions of nonspherical colloidal particles are interesting because of their complex phase behavior and their ability to form different liquid crystalline phases [89e91]. Colloidal glasses can also be formed when

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mixtures of colloids with different sizes are used and equilibrium states are difficult to reach. In all these systems, the particle dimension is often much larger (up to hundreds or thousands of A˚) than X-ray wavelength employed so an angular resolution of 10
4 radian or higher is needed. Angular resolution higher than SAXS can be achieved by using ultrasmall-angle X-ray scattering (USAXS) [92e94]. USAXS have been performed using the BonseeHart configuration on systems with nanostructures or particles up to 1 mm [95e99]. Recently, ultrahigh angular resolution, called microradian X-ray diffraction (mradXRD) or scattering (mradXRS), has been achieved using an alternative method. This technique is relatively simple to implement on a beamline and allows 2-D images acquisition. A brief description of the potentialities of the technique and some examples are given here. The reader can find detailed information about mrad-XRD in the recent works of Petukhov et al. [100] and Thijssen et al. [101]. The experimental setup uses compound beryllium refractive lenses (CRLs) [102] in order to focus the direct beam at the detector plane. The setup is depicted in Fig. 5.29. Depending on the sample-to-detector distance and the wavelength used, high angular resolution of about 1e5 mrad can be obtained. In order to calculate the correlation length L over which positional order is preserved, the breadth of the Bragg reflection (Dq ¼ 2pk/L, where k is a constant of value close to 1) need to be estimated correctly. In colloidal crystals, L can be up to thousand times larger than the lattice spacing a, and Bragg peaks with a breadth of Dq ~ 10
6 nm
1 are possible. Interference of diffracted waves is needed over distances comparable to L [100,103,104]. For this reason, the CRLs are usually placed just after the sample, so that they focus the transmitted and the diffracted X-ray beams avoiding any loss in coherence. If the system is not as ordered as in colloidal crystals, L is small enough (i.e. Bragg reflections intrinsically large) that such a high coherence is not needed and the sample can be placed before or after the CRLs. This is the case for colloidal glasses for example. In order to acquire high-resolution microradian images, high-resolution 2-D detectors have to be utilized. Usually, CCD detectors with pixel size as small as 7  7 mm are used for high-resolution measurements, while CCDs with pixel size of 20  20 mm

Figure 5.29 Typical microradian configuration using compound refractive lenses (CRL) positioned after the sample [100]. S is the X-ray source, H1 are the slits used as secondary X-ray source, and Si-111 is the monochromator. For color version of this figure, the reader is referred to the online version of this book.

Small-Angle X-ray Scattering for the Study of Nanostructures and Nanostructured Materials

are used for low-resolution images. The possibility of acquiring 2-D images in microradian diffraction and scattering is one of the main advantages over USAXS using BonseeHart system, making possible the study of anisotropic structures. Figure 5.30 shows 2-D diffraction pictures for a colloidal single crystal made of silica sphere with 1.4 mm diameter obtained by assisted sedimentation in an electric field [105]. The crystal has a body-centered tetragonal (bct) structure and consists of hexagonal layers parallel to the glass substrate. The X-ray beam is normal to the hexagonal planes. The structure can be indexed using orthogonal vectors of length b1 ¼ 4p/(61/2a) and b3 ¼ 2p/a, where a is the nearest neighbor distance. The crystal has been oriented so that h ¼ k and only even values of l can be seen. Small-lattice imperfections cause the presence of forbidden reflections (dashed arrows) and non-body-centered tetragonal peaks (solid arrows). The 110 peak position is related to the spacing in the hexagonal layer and gives the largest distance observable. d110 is equal to 1.35 mm. This demonstrates the possibility of using microradian diffraction to go well beyond 100 nm resolution. Microradian resolution can be also used to perform experiments closer to the purpose of this chapter, i.e. SAXS. Recently, Kleshchanok et al. reported an extensive study on the nature of attractive glass formation in mixtures of colloids with anisotropic shape [106,107]. Gels and glasses formed by anisotropic colloids are of fundamental importance in many fields, ranging from food science to construction industry. Recently, Eckert and Bartsch showed experimentally that glass formation can be induced via depletion interaction [108e110].

Figure 5.30 mrad-XRD image for a colloidal crystal of 1.4 mm silica spheres with bct structure. For color version of this figure, the reader is referred to the online version of this book. (Reprint from Ref. [100] with permission. Copyright (2006) by the International Union of Crystallography.)

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Depletion attraction was first observed by Asakura and Oosawa in 1954 who considered two bodies in a solution of macromolecules [111]. The same is true for two large spheres (colloids) in a sea of small spherical particles (depletants). The depth of the depletion attraction depends on the depletant concentration f2, while its range on the depletant diameter D2 [111,112]: 8 N; r < D1 > > > > < veff ðrÞ ¼ vdepl ðrÞ; D1 < r < D1 þ D2 > > > > : 0; r > D1 þ D2 with
vdepl ðrÞ ¼
kB T f2

pD31

D2 1þ D1 6

3 2

3

6 7 3r r3 6 7
 þ
61
3 7 D2 4 5 2 1þ D1 2 1 þ D2 D31 D1 D1 [5.61]

where 1 and 2 denote colloidal and depletant particles, respectively. Thus, in a binary mixture of two different colloidal particles, depletion interactions can be used to alter the effective interactions between particles by varying the size and concentration of the depletant. Moreover, anisotropic colloids form arrested states at lower volume fractions than spherical colloids due to their large excluded volume and the particle shape can both influence the strength of attraction. Colloidal aqueous suspensions of inorganic gibbsite g-Al(OH)3 platelets can form upon sedimentation of isotropic (I), nematic (N) or columnar (C) liquid-crystalline phases with increasing platelet concentration. When gibbsite platelets (diameter D ¼ 232.5 nm and thickness L ¼ 8.4 nm) are mixed with small-silica spheres as depletant (diameter D ¼ 16.8 nm), phase separation occurs into a top and a bottom phase. The kinetic of the phase separation is determined by the quantity of the added depletant. The top phase does not show any birefringence between cross-polarizers and is isotropic. The bottom phase is also not birefringent and appears very rigid and incompressible, with characteristics typical of a colloidal glass. The phase diagram of the aqueous gibbsite/silica suspension can be explored with mrad-SAXS. Figure 5.31 shows the mradSAXS images and relative 1-D profiles for the bottom phase resulting from two mixtures made of 8% gibbsite platelets with different concentration of silica spheres (3.4% and 6.7%). For the sample with 3.4% silica spheres, four peaks are detected in the equatorial pffiffiffi pffiffiffi pffiffiffi direction. Their positions follow the relationship q : 3q : 4q : 7q and are

Small-Angle X-ray Scattering for the Study of Nanostructures and Nanostructured Materials

(a)

(b)

(c)

(d)

Figure 5.31 2-D-mrad-SAXS images and relative sectorial integrated profiles for (a) and (b) 8% gibbsite þ 3.4% silica suspension and for (c) and (d) 8% gibbsite þ 6.7% silica suspension. For color version of this figure, the reader is referred to the online version of this book. (Adapted from Ref. [106].)

attributed to the (100), (110), (200) and (210) Bragg reflections of a hexagonal lattice. The hexagonal lattice aD spacing can be calculated from the plot of qhkl vs pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 3p (inset of Fig. 5.31(b)) ðh2 þ hk þ k2 Þ. The slope of the linear trend is equal to 3aD and a value of about 248 nm. At large q values, a broad peak related to the face-to-face stacking of the gibbsite platelets is found. This peak corresponds to the (001) peak of the structure and the lattice spacing aL was calculated to be 27 nm using the Bragg relation 2p/aL. The observed reflections are in agreement with a columnar hexagonal packing of the platelets, observed for pure gibbsite suspensions [91,113]. Using the full width of the (100) reflection, a correlation length of about 1300 nm is calculated. This means that

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the columnar domains are very small (only five platelets diameter), compared to the equilibrium columnar phase, and are immersed in a disordered glassy structure. Due to the kinetical arrest caused by the depletion attraction, once formed, the small columnar domains cannot grow further by taking in new platelets from the surrounding suspension or by annealing with neighbor domains. By further increasing of the silica sphere concentration, depletion attraction becomes even stronger and only a glass phase is formed with a totally disordered structure (Fig. 5.31(d)). The estimated side-to-side aD and face-to-face aL distances for the glassy state are 253.7 and 23.3 nm, respectively. Note that in the glassy state, aD is larger than that observed from suspensions containing small columnar domains. This is related to the higher disorder of the system and the inability of the platelets to fill up the space effectively due to the kinetical arrest. A simple expression for the depletion potential for mixtures of platelets þ spheres was derived [106]: pffiffiffi
 Dplatelet 2 9 3 WdeplðcontactÞ ¼
kB T [5.62] f 4p sphere Dsphere where kB is the Boltzmann constant, kBT is the thermal energy of the system, fsphere is the sphere volume fraction, Dplatelet and Dsphere are the diameter of the platelets and spheres, respectively. This equation shows that the depletion potential depends on the temperature, on fsphere and, most importantly, on the squared of the ratio Dplatelet/Dsphere. The depletion potential for the given example where Dplatelet/Dsphere is equal to 13.8 is rather strong, up to 19 kBT. When smaller platelets are used and the ratio is equal to 5.7, the depletion attractive potential is so weak that the formation of glassy state is not observed [114]. The formation of the partial or full glassy-state occurs in a short timescale (hours to minutes). Due to the precipitation of the rigid glassy bottom phase, the top isotropic phase remains enriched in silica spheres. However, the upper isotropic phase is not stable and due to gravity slow sedimentation of platelets toward the bottom of the capillary occurs. A gradient of platelet concentration grows along the vertical direction in the capillary and the system undergoes further phase separate into an isotropic and a columnar phase [107]. Such long timescale evolution is schematically depicted in Fig. 5.32. The further sedimentation of platelets into the columnar region continues until it is counterbalanced by the osmotic pressure gradient due to the platelet concentration profile. Figure 5.32(d) shows that, after a month, a clear hexagonal structure is formed on top of the bottom glassy phase. A side-to-side aD distance of 266.6 nm and a face-to-face of 29.3 nm are calculated. These values are larger than those found in the small columnar domains generated few minutes after mixing platelets and spheres. This last observation means that the columnar phase formed upon slow sedimentation is less compressed. Moreover, the peaks appear more intense and narrower and the correlation length calculated by DebyeeScherrer equation is larger than 10 times their diameters.

Small-Angle X-ray Scattering for the Study of Nanostructures and Nanostructured Materials

Figure 5.32 Evolution of kinetically arrested precipitate of a suspension of 8% gibbsite platelets with 3.4% silica spheres. (a) The glassy precipitate at the bottom of the capillary is formed within minutes. The top phase is isotropic. (b) After a month, the isotropic phase further phase separate into isotropic þ columnar phase (green color). (c) mrad-XRD image for the columnar phase. (d) Intensity profile from the columnar phase. For interpretation of the references to color in this figure legend, the reader is referred to the online version of this book. (Adapted from Ref. [107].)

It is worth noting that with a standard SAXS configuration, the first order of the hexagonal face would be difficult to observe and only higher orders and the face-to-face reflection could be studied properly. The reported examples show how recently developed mrad-XRD extends the application of scattering and diffraction techniques toward significantly larger distances, well above 100 nm.

ACKNOWLEDGMENTS We acknowledge all the BM26B users for the interesting experiments and results discussed in this chapter. Martin Dulle is acknowledged for the p(r) calculations using the GIFT programs. We thank Jim Torbet and Daniel Hermida Merino for their suggestions and improvements. NWO and ESRF are acknowledged for granting the synchrotron beam time.

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