Small-Angle Scattering Techniques

Chapter 2.6

Small-Angle Scattering Techniques N. Cohauta and D. Tchoubarb a b

CRMD, CNRS-Universite´ d’Orle´ans, F-45071 Orle´ans Cedex 2, France Expert CRT-Plasma Laser, Orle´ans, F-45160 Olivet, France

Chapter Outline 2.6.1. Principles 178 2.6.1.1. Scattering Length and Particle Scattering Length Contrast 179 2.6.1.2. Quantitative Definitions of the Interference Function P (q) 180 2.6.1.3. The Particular Case for Clay Minerals: Definitions 184 2.6.2. Hydration and Sol–Gel Transition (SGT) 185 2.6.2.1. Initial Hydration Stages and Osmotic Transition 186

2.6.2.2. Anomalous Small-Angle X-ray Scattering 190 2.6.2.3. Gel Structure and SGT 191 2.6.2.4. Modelling SGT in Laponite Dispersions 200 2.6.3. Organoclay Dispersion in Organic Solvent and Polymer Nanocomposites 203 2.6.3.1. Dispersions in Organic Solvents 203 2.6.3.2. Clay–Polymer Nanocomposites (CPN) 206 2.6.4. Conclusions 207 References 208

With the possible exception of those who live in very dry deserts, people living in any other region can see small-angle scattering (SAS) by water droplets when they observe a moon halo predicting rain for the next day. Indeed, any medium showing fluctuations in matter density can give rise to observable SAS when it is placed in the path of a radiation beam, provided the medium Developments in Clay Science, Vol. 5B. http://dx.doi.org/10.1016/B978-0-08-098259-5.00008-1 © 2013 Elsevier Ltd. All rights reserved.

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TABLE 2.6.1 Scattering Techniques in Relation to Particle Dimension (nm) 10

102

103

104

Small-angle scattering (SAXS and SANS) (X- rays and neutrons)

Ultra-small-angle scattering

Dimension d (nm)

Raleigh–Debye–Gans

Elastic light scattering Lorenz–Mile

Fraunhofer diffraction

Quasi-elastic light scattering From Glatter (1991).

is adequately transparent and the fluctuations in matter density are at appropriate scales with respect to the radiation wavelengths. The last 50 years have seen the development of SAS by X-rays (SAXS), neutrons (SANS) and light (SALS), and the use of these techniques to investigate a large number and variety of media (Guinier and Fournet, 1955; Bacon, 1975; Glatter and Kratky, 1982). Light scattering by colloidal particles was studied for over a century, starting with the theoretical works by Lord Rayleigh (1842–1919). Bohren and Huffman (1998) published a relatively modern review of light scattering by small particles. SAS patterns are observable if the dimension of the dispersed particles is comparable to the radiation wavelength (Table 2.6.1). The different techniques, shown in Table 2.6.1, are generally complementary, taking into account their respective wavelengths. For light scattering experiments, the choice of samples is more restricted by the necessity for transparency, and problems of turbidity and multiple scattering. In all cases involving elastic scattering (without changing the wavelength of the scattered beam), the experimental principles are approximately the same. A sample is placed on the path of a well-collimated beam of X-rays, neutrons or light. The special devices used are described in the references cited earlier. ‘Zooming’ on the central part of the diffraction patterns enables a profile analysis to be made.

2.6.1 PRINCIPLES The physical principles of scattering are the same for both wide-angle diffraction (WAD) and SAS. The latter technique consists of analyzing only the 000 reflection profile. Being insensitive to atomic structure, this feature is not taken into account in the classical WAD studies. The principal difference

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179

between WAD and SAS lies in the extent to which structural details can be investigated. Therefore, the meaning of the scattering unit has to be explained. The intensity of a scattered beam is recorded as a function of the scattering angle 2y. The modulus |q| of the scattering vector q is related to the scattering angle by q ¼ 4psiny/l, where l is the radiation wavelength. Some papers use the parameter s ¼ q/2p ¼ 1/d (where d is the Bragg distance) in place of q. For particles in dispersion, the scattered intensity is expressed by I ðqÞ ¼ I0 Fk2 PðqÞ

(2.6.1)

where I(q) is the number of photons (or neutrons) counted per unit of solid angle and time; I0 is the intensity of the incident beam, that is, the total number of photons (or neutrons) irradiating the sample per unit of time; F is the volume fraction of particles that are immersed in the incident beam; and k is the scattering length contrast of the particle (see later). The interference function P(q) is the most important parameter of Eq. (2.6.1) because its profile depends on the size and shape of particles, on their interactions, and, if the particles are heterogeneous, on their internal structure.

2.6.1.1 Scattering Length and Particle Scattering Length Contrast It is generally assumed that the unit domain size (scattering unit) is large in comparison with the interatomic distances. On this assumption, the angular extension of the SAS pattern is limited to a qmax value much lower than the first diffraction peak position. In this case, the scattering unit is considered to be homogeneous and characterized by its mean scattering length bu. Depending on the interaction mode of the radiation with the material, bu is specific to each technique.

2.6.1.1.1 Neutron Scattering The neutrons normally used (Cotton, 1991) have an energy of about 103 eV and their wavelengths lie between 0.1 and 2 nm. The neutrons interact with the nuclei of the atoms of the particles and the dispersion medium. For each nucleus, the scattering length b cannot yet be theoretically computed; it can only be experimentally evaluated. The b-values have been tabulated (Bacon, 1975). The values can be both positive and negative and may differ from one isotope to another. The difference in value between hydrogen (0.374  1012 cm) and deuterium (þ0.667  1012 cm) is an important point. Taking into account the atomic composition of the scattering unit, bu may be computed from the relation bu ¼ Sinibi, where ni indicates the number of atoms of type i and bi their scattering lengths. The summation (Si) is performed on all types of atoms constituting the scattering unit. If the particles are homogeneous, the particle scattering length density rp is defined as the ratio bu/vu, where vu is the scattering unit volume. For particles

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that are dispersed in a solvent, the solvent scattering length density rs must also be computed. The scattering length contrast k of the particle is defined by the difference (rprs), while the scattering power of the dispersed particle is given by
2 Fk2 ¼ F rp  rs

(2.6.2)

2.6.1.1.2 X-ray Scattering X-ray photons have an energy of about 104 eV and their wavelengths l are distributed between 0.05 and 0.5 nm (Guinier and Fournet, 1955). Their electromagnetic interactions with matter involve the Z electrons of the atom electronic shell. Therefore, the scattering length for one atom is b ¼ Zbe, where be (¼ þ 0.282  1012 cm) is the scattering power of an electron and can be computed theoretically. In that case, the scattering power of the dispersion depends only on the electron density of the particle and that of the solvent, and on the scattering power of free electron Ie(¼b2e ):
2 Fk2 ¼ FIe rp  rs (2.6.3) where rp and rs are, respectively, the particle and the solvent electron density.

2.6.1.1.3 Light Scattering Visible light photons have an energy of about 10 eV. Their electromagnetic interaction with matter is macroscopic and induces an electric field giving rise to electric dipoles in the medium. The scattering length is expressed as a function of the polarizability a of the medium, which, in turn, is related to the refractive indices of the particles and the solvent. The theoretical expression of the scattering power of the dispersion depends on the size of the particles in comparison with the wavelength of the light. This expression, therefore, depends on the approximations used in the theoretical treatment of the patterns. Table 2.6.1 illustrates the range of validity of the different methods of SALS. The corresponding theories have been discussed by Hofer (1991).

2.6.1.2 Quantitative Definitions of the Interference Function P (q) 2.6.1.2.1 Dilute Dispersion of Homogeneous Particles Without any Interaction In the case of a statistically isotropic dispersion, the particles can adopt all possible orientations. The interference function P(q) (Eq. 2.6.1) is expressed by the following Fourier transform:

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PðqÞ ¼ 4p

ð1

pð r Þ

0

sin qr dr with pðr Þ ¼ r 2 Vp gðr Þ qr

(2.6.4)

For monodisperse particles, p(r) is the distribution function of distances inside the particle and depends on Vp, the particle volume and g(r), the characteristic function of the particle (Porod, 1982). y(r) is defined as the probability of finding two points at the distance r within a particle. The intensity i(q) scattered by only one particle is called the particle shape function: ð1 sin qr 2 dr (2.6.5) iðqÞ ¼ F ðqÞ ¼ 4p r 2 gðr Þ qr 0 i(q) is the square of the orientationally averaged particle form factor F(q). One property is of interest: for q ¼ 0, i(0) ¼ Vp and therefore P(0) ¼ (Vp)2. If the parameter k and the particle volume fraction F are known, the particle mass can be determined from I(0) in the general expression of Eq. (2.6.1). Scattering by dilute dispersions of monodisperse or weakly polydisperse particles may be analyzed by comparing the experimental curve profiles with the theoretical i(q). Expressions for i(q) have been determined for all geometrical shapes, including such complex configurations as polymer blobs (Guinier and Fournet, 1955; Glatter and Kratky, 1982).

2.6.1.2.2 Properties of the Shape Function i(q) The i(q) profile for homogeneous particles is characterized by three ‘q-ranges’ (Fig. 2.6.1). Range 1 depends only on particle size. As shown by Guinier and Fournet (1955), i(q) can be approximated by

i(q)

(1) Size range Guinier’s law

(2) Shape range Shape function

Rg−1

(3) Surface range Porod’s law

p/H

q (Å−1)

FIGURE 2.6.1 The three ranges of the SAS scheme related to the geometrical properties of dilute particle dispersions.

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iðqÞ ffi 1 

q2 R2g 3

þ   ffi exp

q2 R2g

!

3

(2.6.6)

where Rg is the gyration radius of the particle. Range 1 is limited by the condition qRg ¼ 1. The corresponding Guinier plot ln i(q) versus q2 shows a linear decrease whose slope is equal to Rg 2 =3. Extrapolation to q ¼ 0 yields i(0). Range 2 depends on particle shape. For a cylinder of diameter 2R, height 2H and volume V, ð p=2

f 2 ðq,fÞsin fdf iðqÞ ¼ VF ðqÞ ¼ 4V 0 ! ! ð p=2 sin 2 ðqH cos f J12 ðqR sin fÞ ¼ 4V sin fdf ðqH cos fÞ2 ðqR sin fÞ2 0 2

(2.6.7)

where ’ is the angle between the scattering vector q and the cylinder axis. The structure factor f(q,’) of the particle is the product of two terms, a radial and an axial component, the radial component containing the first-order Bessel function J1:For very thin disks, Eq. (2.6.7) is approximated by iD ðqÞ ffi

2 q2 H 2 exp 3 q2 R 2

(2.6.8a)

If the thickness of the disk is very small in comparison with its radius, i D ð qÞ ffi

K q2 R 2

(2.6.8b)

This applies to montmorillonite (Mt) layers. If the clay mineral is totally dispersed, the plot of log i(q) versus log q shows a linear decrease, with a slope of 2 within the q ranges 2 and 3 (Fig. 2.6.1). Auvray and Auroy (1991) discussed the scattering by thin microemulsion layers. Their first conclusion was that layer curvature did not change the scattering behaviour and always decreased as q2. The q range 3 is essentially sensitive to the external surface of the particle, obeying Porod’s (1982) law (Guinier and Fournet, 1955; Glatter and Kratky, 1982). For compact particles, the external part of scattering i(q) varies as sq4, where s is the specific surface area.

2.6.1.2.3 Shape Function of a Core–Layer Disk In some studies on the structure of organoclay dispersion, a core–layer disk model is adopted. The scattered intensity by a disk composed of a central core with the mean length density rc and covered on each side by a layer with a mean length density rl, was given by Hanley et al. (1997)

Chapter

2.6

Small-Angle Scattering Techniques

h iðqÞ ¼ ðrl  rs ÞðVt Ft ðqÞ  Vc Fc ðqÞ þ ðrc  rs ÞVc Fc ðqÞ2

183

(2.6.9)

Fc(q) and Ft(q) are the disk form factors for the core and for the total particle, respectively, as defined in Eq. (2.6.7). rc is the mean scattering density in the solvent.

2.6.1.2.4 Particle Size Distribution If the particles are not equal (polydisperse), i(q) is replaced by its average hi(q)i: X am i m ð q Þ (2.6.10) hiðqÞi ¼ where am is the particle weight fraction of size m and im(q) is its respective scattering. The mean particle volume hVpi replaces Vp in Eq. (2.6.4). This case was illustrated by the SANS study of Hall et al. (1985) on dilute dispersions (0.8%, m/v) of allophane particles. The spherical shape of these particles was particularly well suited for determining their size distribution by SAS techniques. The analysis was performed by comparing experimental SANS spectra with the simulated ones.

2.6.1.2.5 Heterogeneous Particles The p(r) function depends not only on the particle size and shape but also on its internal structure. In this case, the product k2i(q), indicating the scattering power of one particle, is replaced by the structure factor F(q)2. If the particles are fractal clusters, their mass fractal dimension a is generally a non-integer, varying within 1 < a < 3 (Mandelbrot, 1967, 1977). The scattered intensity varies as the power law i(q)  qa within the q ranges 2 and 3 in Fig. 2.6.1 (Schaefer et al., 1985). The limiting value of a ¼ 3 for non-compacted particles, observed for some clay mineral gels, has not yet been clearly and quantitatively explained. In the case of homogeneous particles but with fractal surfaces, a generalized Porod’s law was derived by Bale and Schmidt (1984), as i(q)  qds6, where ds is the fractal dimension of the surface. For a smooth surface, ds ¼ 2 and the classic Porods’ law is recovered. 2.6.1.2.6 Interacting Identical Particles The function P(q) also takes into account interparticle interactions (Guinier and Fournet, 1955). In the simplest case of spherical particles, P(q) ¼ F(q)2G(q) and the scattered intensity is expressed by I ðqÞ ¼ I0 FFðqÞ2 GðqÞ

(2.6.11)

where F(q) is the structure factor of one particle and G(q) the interparticle interference function, which depends only on particle–particle interactions.

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If F(q) is known, G(q) can be obtained from the relation GðqÞ ¼ I ðqÞ=ðI0 FFðqÞÞ2

(2.6.12)

W(r), the inverse Fourier transform of G(q), is the radial distribution function giving the probability of finding a particle at a distance r from any other particle, taken as the origin. From W(r), it is also possible to compute g(r), the first neighbour particle distance distribution. Such an operation is quite delicate because it can give rise to an ‘artefact’ due to the limited extension of the scattering curves. It can be applied only if the external parts of the experimental scattering can be extrapolated such as by Porod’s law for the largest q values, or by the Guinier’s plot for the lowest ones. This allows the computation limits to be widened to such an extent that the cut-off effects do not influence the experimental results. Case of the stacking of core–layer disks. The interference function G(q) arising from a particle formed by the parallel stacking of N disks was calculated by Kratky and Porod (1949) as G ð qÞ ¼ 1 þ

N h 2X ðN  kÞcos ðkd cos ’Þ exp kðq cos ’Þ2 s2d =2 N k¼1

(2.6.13)

where N is the number of disks inside a stack, d represents the distance between adjacent disks in the stack and sd the Gaussian standard deviation, and ’ is the angle between q and the major axes of the disks. The scattered intensity by a heterogeneous particle composed by the stacking of N core–layer disks embedded in a solvent is the following: I ðq Þ  N

ð p=2 0

½ðrl  rs ÞðVt Ft ðq,’Þ  Vc Fc ðq,’ÞÞ þ ðrc  rs ÞVc Fc ðq, ’Þ2 GðqÞsin ’d’ (2.6.14)

with parameters as defined for Eqs. (2.6.7 and 2.6.9).

2.6.1.3 The Particular Case for Clay Minerals: Definitions For natural smectites, the situation is not really simple, even when we deal with dilute dispersions, for the following reasons: i. Because of the layer structure of particles and their great anisotropy, the condition of statistical isotropy for the dispersions is not so evident and needs to be specified in each case. For locally oriented systems, Eqs. (2.6.10 and 2.6.11) have to be resolved taking into account the particle orientation. ii. Because of the small layer thickness, a continuum of scattering could exist between the 000 and the 001 reflections. The interlayer distances can be distributed within a wide range of dimensions from about 1 nm

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185

to several micrometres. Indeed, the limit between WAD and low-angle scattering is not well defined in some cases. iii. The particles in the same sample have different thicknesses. In the case of interacting anisotropic particles, the correlation of orientation with position cannot be directly determined from G(q) except in special experiments with well-oriented samples. The best methods consist in simulating the G(q) functions (Eq. 2.6.13).

2.6.1.3.1 Definitions The ‘basal spacing’ d is the distance from the centre of the oxygen plane of one layer to the centre of the oxygen plane of an adjacent layer, as directly deduced from the position of the 001 reflection in the XRD spectrum. W(r) is the probability of finding a layer at a distance r from a layer taken as the origin. The ‘interlayer spacing’ h is the distance between two neighbouring parallel layers, Thus, h ¼ d  e, where e is the layer thickness. The ‘interference function’ G(q) is replaced by S(q), the ‘structure factor of dispersions’, in many publications. Following Me´ring (1949), we prefer using the notation G(q), the ‘interference function’, to avoid confusion with the ‘particle structure factor’ F(q).

2.6.2

HYDRATION AND SOL–GEL TRANSITION (SGT)

For many years, SAS experiments were used to describe the dispersion state of clay minerals or, more recently, organoclays in different media: aqueous and organic solvents or melt polymers. The first studies based on the SAS structural description of clay minerals are related to the hydration mechanisms by which a powder of smectite is transformed into a concentrated gel or a paste. The SAS investigations consist in observing the shift in reflections that occurs during clay mineral hydration, that is, monitoring the profile change of the (001) reflections from the dry state (corresponding to d  1 nm) to the gel state. During this process, the interlayer distances can increase to several tens or even several hundreds of nanometres. This evolution was followed as a function of various parameters such as clay concentration, nature of interlayer cation, ionic strength, and external pressure or temperature. As already mentioned, most investigations are concerned with natural di- and tri-octahedral smectites and vermiculites. The development of powerful devices combining larger scale range and higher flux such as ultra SAS of neutrons and X-rays and synchrotron radiation facilities has enabled investigation of the SGT mechanisms, the relationship between gel structure and rheological behaviour and thermodynamic models of the swelling clay mineral dispersions. In order to minimize the experimental difficulties associated with earlier studies, most of the secondgeneration researchers used Laponite as the clay mineral. The small diameter

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of Laponite particles, in comparison with natural smectites, allows interparticle interactions to be simplified. Laponite particles also have a narrow particle size distribution, and hence a ‘mean’ size may be used. The diameter of Laponite particles is much smaller than the size that is accessible to SAS techniques. In the case of Mt, even ultra SAS of X-rays or neutrons cannot include the whole range of Mt particle diameters. Unlike Mt, the structure and behaviour of Laponite dispersions can be thoroughly studied. The light scattering scale could be suitable even though it is generally available for highly dilute dispersions.

2.6.2.1 Initial Hydration Stages and Osmotic Transition 2.6.2.1.1 SAS Analysis by Fourier Transform of Intensity Rausell-Colom and Norrish (1962) and Norrish and Rausell-Colom (1963) used SAXS to investigate the swelling in salt water solutions of orientated flakes of Naþ-Mt (Wyoming) and single crystals of Liþ-vermiculite (Kenya). The increase in interlayer spacing was studied as a function of the ionic strength. The scattering intensity was recorded along the direction perpendicular to the flake (or single crystal) plane using a low-angle diffractometer and within an angular range that allowed interlayer distances to be measured just after the transition to ‘osmotic’ swelling, that is, from 3 to 50 nm. The scattered intensity can be expressed by Eq. (2.6.10), in which the q vector is replaced by its component qz perpendicular to the flakes. I ðqz Þ / F2 ðqz ÞGðqz Þ The 1D Fourier transform of G(qz) is given by ð dm 1 W ðr Þ  1 ¼ Gðqz Þ cos ðrqz Þdqz p 0

(2.6.15)

(2.6.16)

In this case, W(r) is the 1D distance distribution. The most probable basal distance d corresponds to the maximum of W(r). Because the structure of these clay mineral–water systems is heterogeneous and disordered, W(r) gives poor information. Nevertheless, the following conclusions can be made. Interlayer (‘crystalline’) swelling occurs by a process that is highly dependent on the hydration energy of the interlayer cations. With some monovalent interlayer cations such as Naþ and Liþ, the interlayer distance ‘jumps’ at a high water content. Gel formation can be explained in terms of repulsive osmotic forces resulting from the entry of water into the interlayer space and the development of ionic double layers. After passing the osmotic transition threshold, the increase in interlayer distance is proportional to the water content, as is the case for swelling in pure water. In salt solutions, the interlayer distance increases linearly with C1/2, where C is the electrolyte concentration.

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187

Andrews et al. (1967) carried out the same analysis with an SAS device that can extend to smaller scattering angles, that is, much larger particle sizes than in the above experiment (about 100 nm). In agreement with RausellColom and Norrish (1962) and Norrish and Rausell-Colom (1963), they underlined the inaccuracy of using a direct Fourier transform for such samples.

2.6.2.1.2 Geometrical Simulation of SAS The description of the gel structure can be improved by simulation of the scattering curves. Previous experiments performed with standard laboratory SAS equipment were not easy to model, because a precise collimation of X-ray sources was impossible to realize while maintaining the intensity of scattering. It was then necessary to correct the experimental data by using a heavy de-smearing method or to introduce a complex optical transfer function in the spectra simulation. Nowadays, this problem is solved by using new generations of laboratory SAXS devices that are able to deliver a point-like collimation with an intensity flux convenient for many applications and, if necessary, by using radiation supplied by a synchrotron, allowing point-like collimation with a high intensity to be achieved. Pons et al. (1981, 1982a) performed SAX experiments with a synchrotron beam of the Laboratoire d’Utilisation du Rayonnement Synchrotron (LURE, Orsay, France). They studied the swelling of Wyoming Naþ-Mt and Liþsaponite in pure water, starting from a gel containing 17 mass% clay mineral. The scattering evolution was observed in the course of alternate freezing at 70  C followed by slow warming up to room temperature (RT). The timeresolved diffraction patterns were recorded during temperature increase. The theoretical spectra were simulated using the methods previously elaborated for the simulation of 001 reflections from disordered or interstratified layer stacking systems (see Section 2.6.1). The simulation model of the dispersion consisted of particles (layer stackings) that were disoriented with respect to each other. The intensity is given by IðqÞ ¼

F200 ðqÞ X aðMÞGðq, M, pi , di Þ Oq2 M

(2.6.17)

where Ω is the unit cell area, F00(q) is the structure factor of the unit layer along the 001 direction, q is the vector modulus, M is the number of layers in a particle, a(M) is the weight distribution of particles containing M layers and G(q, M, pi, di) is the interference function (or modulation function), which depends on M and on the interlayer distance di with their respective probabilities pi. The spectra confirmed the increase in basal spacing from 2 nm to more than 3 nm at the osmotic transition threshold as well as the reversibility of this

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I(s) 0 ⬚C

10 Room temperature

−10 ⬚C

5

−3 ⬚C −6 ⬚C

s(Å−1) 0.01

0.03

0.05

0.07 d(Å)

100

40

20

15

FIGURE 2.6.2 Time-resolved experiments showing the evolution of experimental SAXS intensity I(s) versus basal distance d or s ¼ 1/d, of a gel after freezing at 70  C and then slowly heated to room temperature. From Calas et al. (1984).

transition. But the primary aim of the study was to compare the scattering evolution between 0  C and RT (Fig. 2.6.2). The peak position was the same at both temperatures, but the narrow peak at 0  C transformed into a diffuse modulation at RT, indicating that swelling was not impeded at 0  C. At this temperature, the system was constrained by the presence of residual ice in equilibrium with water. At RT, a large disorder in layer stacking took place but with the same most probable basal distance. This would indicate that the energy spent in establishing a swelling equilibrium contains an entropy term inducing a dispersion of unit layers and the formation of smaller particles (stacked layers) that are disoriented with respect to each other. Comparison of experimental and theoretical scattering curves showed that the Naþ-Mt gel at 17 mass% formed at RT consisted of particles containing between one and eight parallel layers with interlayer distances of 4–20 nm, giving an average value of 10 nm. The particles themselves are completely disoriented with respect to each other. The proposed structural model was checked by HRTEM (Pons et al., 1982b). Because of the high clay mineral

Chapter

0.04

2.6

c = 0.12 M p = 55 g/cm2

0.03

r (d )

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Small-Angle Scattering Techniques

c = 0.037 M p = 33.76 g/cm2

0.02 c = 0.02 M p = 20.9 g/cm2

c = 0.035 M p = 37.2 g/cm2

0.01

0.00 50

100

150

200

250

300

350

400

d (Å)

FIGURE 2.6.3 Vermiculite wafer swelling. Comparison of basal spacing distributions computed from diffuse double-layer interaction (broken line) and inferred from XRD data (continuous line). c, ionic concentration of solvent; p, pressure applied on the wafer to balance the swelling pressure. From Rausell-Colom et al. (1989).

concentration, other explanations may be proposed such as the formation of nematic structures (see below). The same simulation and fitting method was used by Rausell-Colom et al. (1989) to analyze the swelling of single crystals of Santa Olalla vermiculite in aqueous solutions of L-ornithine hydrochloride. They compared the basal spacing distributions obtained from SAS with those derived from the DLVO theory, taking into account a Boltzmann distance distribution and a layer of water molecule on the clay mineral surface (Cebula et al., 1980). Figure 2.6.3 illustrates the fit between the two types of plots for different ionic concentrations of the swelling solution. Following Callaghan and Ottewill (1974), Ben Rhaiem et al. (1987) studied the hydration–dehydration behaviour of Naþ- and Ca2þ-Mt in 103 M NaCl and CaCl2 by synchrotron SAS. The samples were examined during drying and rewetting in an ultrafiltration cell by varying the suction. Their results again confirmed an osmotic transition and its reversibility for NaþMt but not for Ca2þ-Mt. They further showed that the gel, at least at high clay mineral contents, is formed by stackings of expanded layers as in nematic domains with variable interlayer distances that are disoriented from each other. These gel domains alternate with pores whose hierarchical size distribution can be determined by comparing SAS spectra and permeability results of pressure experiments. Keren and Klein (1995) used SALS to determine the clay mineral concentration in aqueous dispersions of bi-ionic Naþ/Ca2þ-Mt. The samples were carefully prepared by selecting the particle size and stirring the two

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homo-ionic clay mineral dispersions for long periods. From experiments with various Naþ/Ca2þ ratios at 0.1% clay mineral content, they concluded that the SALS technique is not appropriate for such systems because of the broad particle-size heterogeneity. A theory for mixtures is required to take into account the scattering of particles with different sizes (cf. the light scattering theories in Table 2.6.1). Indeed, it is always perilous to analyse heterogeneous media without any possibility to perform contrast-matching in SAS experiments. As described later, there are techniques that can be used for particular cases.

2.6.2.2 Anomalous Small-Angle X-ray Scattering Anomalous small-angle X-ray scattering (ASAXS) refers to an extension of standard SAXS experiments in which the energy of the probing X-rays are tuned near the K, L or M absorption edges of an element in the sample. By performing SAXS experiments near one of the characteristic absorption edges of any given atom, it is possible to vary the scattering contrast of that atom. This systematic variation in contrast yields the partial scattering functions of the specific atomic species. In general, the atomic scattering can be expressed as f ðq ; EÞ ¼ f0 ðqÞ þ f 0 ðq ; EÞ þ if 00 ðq ; EÞ 0

(2.6.18)

00

where E is the energy of the probing X-rays and f and f the real and imaginary parts of anomalous scattering. The variation of f 0 is responsible for the change in contrast seen in the ASAXS signals. Near the absorption edge, the scattering intensity is given by   (2.6.19) I ðq ; lÞ ¼ In ðqÞ þ f 0 ðlÞIc ðq ; lÞ þ f 02 ðlÞ þ f 00 ðlÞ Ir ðqÞ where In(q) is the normal non-resonant scattering, Ic is a cross term reflecting scattering between the specific element of interest and the remainder of the material and Ir corresponds to the distance correlations of the resonant scattering centres. This matching method is easily accessible to elements having Z values between 20 and 36 for K edges, and between 50 and 82 for L edges. Since f 0 and f 00 are sharply varying functions near the edge, these experiments require the highest possible energy resolution (Dl/l  104). To solve Eq. (2.6.16), it is necessary to perform several measurements with different wavelengths far from and near the edge. Such a technique can reveal the distribution of specific species within a multi-component matrix. Carrado et al. (1998) exploited ASAXS to monitor the solvation behaviour of transition-metal and lanthanide ions within the interlayer space of Mt. The experiments were performed at the Stanford Synchrotron Radiation Laboratory (SSRL, USA). The variations of scattering intensities, as a

Chapter

2.6

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Small-Angle Scattering Techniques

function of the absorption energy, were monitored for Cu2þ-, Er3þ- and Yb3þMt as a function of hydration.

2.6.2.3

Gel Structure and SGT

The second series of investigations using ultra-small-angle X-ray scattering (USAXS), neutrons and light scattering were devoted to studying the structure of clay mineral gels and the SGT.

2.6.2.3.1 Gel Structure in Natural Smectite Dispersions Cebula et al. (1980) performed SANS experiments at the Institute LaueLangevin (ILL) in Grenoble, France, on aqueous dispersions, comparing the effect of exchangeable Liþ, Kþ, and Csþ ions on the structure of dilute Mt sols (0.8 mass%). Figure 2.6.4 illustrates the Guinier plots ln (q2I) versus q2. The particle thickness may be derived from the slope of the linear part of each curve, using Eq. (2.6.8a). The extrapolation of these straight lines to q ¼ 0 gives (Iq2)0. This last value was used to control the exchange H2O ! D2O. By changing the relative proportions between H2O and D2O, it was possible to reach the condition (Iq2)0 ¼ 0 when the contrast disappears. Matching allows even a very low exchange between D2O and H2O layers at the clay mineral surface to be detected and compared with the free H2O molecules of the bulk solvent. Matching enhances the scattering contribution of these water layers and allows their thickness (of about two water layers) to be evaluated. The measurement of particle thickness, using the plots of Fig. 2.6.4, showed that Liþ-Mt was completely delaminated with a layer thickness of ca. 1.10 nm

In (q2I)

−3.0

−4.0

−5.0 0.0

0.0065

0.013

q2 (Å-2) FIGURE 2.6.4 Guinier plots obtained from SANS experiments for aqueous dispersions of Liþmontmorillonite (□), Kþ-montmorillonite (●) and Csþ-montmorillonite (○). Clay mineral concentration was 0.8% (w/w). From Cebula et al. (1980).

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including the water layers. In Kþ-Mt, the average particle thickness was ca. 2.6 nm, corresponding to two parallel clay mineral layers. In Csþ-Mt, aggregation was more pronounced with a broad distribution of particles around a thickness of about 4.2 nm corresponding to three associated layers with their hydration layers. Matching of the particle–solvent scattering length contrast was obtained with a solvent containing 65% D2O. This experimentally defined amount of D2O cannot be obtained theoretically by considering only the scattering length of the clay mineral and the solvent. This experimental value also depends on the very low exchange rate between hydrogen and deuterium in water within the interlayer space, even if this water is very mobile (self-diffusion coefficient  1010 m2/s). Pinnavaia et al. (1984) used SANS (ILL, Grenoble) to study particle aggregation and particle–pore distribution of Spur homoionic Mt (in the Liþ, Naþ, Kþ or Csþ forms). For 1% dispersions, Liþ-Mt obeyed Guinier’s law, while Csþ-Mt followed Porod’s law (I(q) / q4), indicating particle aggregation in the presence of Csþ ions. Naþ-Mt dispersions showed some aggregation of clay minerals layers, while Kþ-Mt gave a pattern similar to Csþ-Mt. The dynamics of water and other intercalated molecules were also investigated by quasi-elastic neutron scattering.

2.6.2.3.2 Comparing Mt with Laponite Morvan et al. (1994) performed USAXS and SAXS experiments (CEA, Saclay, France) on Laponite in pure water and in salt solutions. Figure 2.6.5 shows the log–log plots of scattering curves at 1.8, 3.5 and 10 mass%. 108

I(q)

106

104

102 100 10−4

10−3

10−2

10−1

100

q (Å-1) FIGURE 2.6.5 Experimental USAXS intensity I(q) versus the modulus of scattering vector q of aqueous Laponite dispersions presented as a log–log plot. The corresponding clay concentrations were (■ ■ ■) 1.8% (w/w), (—) 3.5% (w/w) and (□ □ □) 10% (w/w). From Morvan et al. (1994).

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1010 -2.15

109

-2

108

I(q)

107 106

-4

105 104 103 102 101 10−4

10−3

10−2

10−1

1

q(Å−1) FIGURE 2.6.6 Experimental USAXS intensity I(q) versus the modulus of scattering vector q of aqueous montmorillonite dispersions in 0.1 M NaCl, presented as a log–log plot. Clay mineral concentration is 4% (w/w). The slope values (2.15, 2, 4) represent scale laws characteristic of the different structures. From Faisandier-Cauchois et al. (1995).

The 1.8 and 3.5 mass% samples give curves that are typical for ideally dispersed disk-like particles. Their profile was simulated using the i(q) intensity expression of Eq. (2.6.7) for randomly oriented cylinders of axial length 2H and cross-sectional radius R, and Eq. (2.6.3) taking into account the scattering length contrast. A plateau is observed for the Guinier plot domain. Thus, the Laponite layers scatter independently without any interactions. For the 10 mass% concentration, the increase in intensity at small q values has an exponent a ¼  3.1 indicative of a heterogeneous medium. Figure 2.6.6 shows the USAXS curves for a 4 mass% Mt dispersion in 0.1 M NaCl. (Faisandier-Cauchois et al., 1995; Faisandier et al., 1998). Both plots are typical of well-dispersed Mt obeying scaling laws over the whole q range. Such behaviour is also typical of scattering by non-interfering, very large membranes with a diameter of D ¼ p/qm  1 mm (where qm is the lowest experimental q value) and a thickness of ca. 1 nm. This diameter is clearly larger than the usual diameter of one Mt unit layer. For the sample prepared at RT, the exponent 2.15 indicates that some layers are parallel. For the sample under pressure (in sealed ampoules at 200  C), the slope is 2, as expected for ideally dispersed layers. These results show that, at this clay mineral concentration and ionic strength, the unit layers form large bands by edge-to-edge contacts, as mentioned by Low (1991). Ramsay et al. (1990) compared neutron diffraction and SANS (PLUTO Harwell, UK) data for Laponite RD and Wyoming Mt in D2O. The structural changes investigated concerned samples in which the clay mineral particles were initially aligned in the dry state. The authors followed the development of orientational disorder and eventually a randomly oriented isotropic structure, over a wide range of D2O concentration (D2O/clay ratio x ¼ 0.2–31 (m/m)).

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The Porod’s part of the SAS curves followed a Qa law with the same exponent for both clays. As the heavy water concentration increased, the exponent varied from 4 in the compacted system to 2 for the highly dilute dispersions. Because of the small size of Laponite particles (in comparison with Mt), the SAS in the lowest q range allows particle diameters (25–30 nm) to be determined, using Eqs. (2.6.8a and 2.6.8b).

2.6.2.3.3 SANS and SAXS Studies Under Shear Rheo-SAXS and Rheo-SANS experiments were recently developed in order to study the relationship between the structure/orientation of clay mineral particles and complex rheological behaviour of systems under shear flow conditions. These studies under shear were promoted by the emerging of higher flux sources such as third-generation synchrotrons, together with the development of 2D detectors. Thereby, rheological small-angle instruments are now available on many neutron reactors and synchrotron sources. In principle, the Couette-type cell of a rheometer is placed under a neutron or an X-ray beam. The incoming beam is oriented according to the radius of the cell (radial geometry) or laterally translated to direct the beam according to a tangent direction of the cell (tangential geometry) (Fig. 2.6.7). This last configuration is possible only with a very small size of the X-ray beam compared to the width of the cell. The interest of these two configurations is to reconstruct the 3D orientation distribution of anisotropic particles. Using SANS (ILL, Grenoble), Ramsay and Lindner (1993) compared the in situ scattering of Laponite and Mt dispersions under static conditions and under shear in a Couette-type cell placed in the radial position under the X-ray beam. The range of clay contents was chosen within 0.5–6.5 mass% where timedependent gelation and thixotropic behaviour occur. Spatial and orientation correlations between the particles developed in dispersions of low ionic strength (<103 mol/L). Such self-organized structures are limited to domains of restricted size, as indicated by the anisotropic scattering behaviour, and are influenced by the clay mineral concentration, particle size and particle shape (Fig. 2.6.8). Under a high shear rate, reduction of interferences gives evidence for a disruption at short-range order. For Mt, the shear induces an anisotropic SANS pattern at low shear rates (25 s1), demonstrating a preferential alignment of particles in the direction of flow whereas spatial correlations are maintained. At high shear rates (ca. 104 s1) the 3D structure broke down and only the preferential alignment was observed. The timeresolved SANS studies of Mt dispersions showed a high degree of thixotropy. Some preferential alignment may persist at distances higher than 102 nm under equilibrium conditions. Similarly, Hanley et al. (1994) carried out a SANS study of a 1 mass% dispersion of Naþ-Mt using the instrument at CNR (Cold Neutron Research)

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A w

Stator X-ray

Rotor X-ray X-ray Tangential

Radial

B

FIGURE 2.6.7 (A) Schematic representation of a Couette cell and the two configurations for SAXS measurements. (B) Commercial rheometer installed on the beamline SWING at the SOLEIL Synchrotron.

facility of the National Institute of Science and Technology (NIST), USA. The spectra were recorded on a 2D detector. At equilibrium, the plot of log I versus log q showed the classical scaling law with a slope of 2.2, indicating a high degree of dispersion with scarcely detectable anisotropy. The behaviour of the dispersion under shear was as expected for disk-shaped particles. The disks aligned under shear with the normal parallel to the velocity gradient. Schmidt et al. (2002) studied the effect of shear on the viscoelastic behaviour of clay mineral–polymer hybrids using Rheo-SANS. The sample was a 3 mass% dispersion of Laponite LRD in a 1.5% aqueous solution of polyethylene oxide (PEO). With increasing shear rate, an anisotropic scattering pattern was developed as a result of orientation of clay mineral particles in the shear field. The shear-induced orientation of the macromolecules and the particles were measured in D2O. SANS measurements of contrastmatched samples revealed the orientation of the polymer molecules alone. As the shear rate increased, the clay mineral particles orient first, and then

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D 2 C

In I(q)

2 B

0

Q−2

A

−2 −6

−4

−2

In q (Å−1)

FIGURE 2.6.8 SANS curves for aqueous Laponite dispersions: (A) static and (B) sheared. For montmorillonite dispersions: (C) static, (D) sheared. Clay mineral concentration, 0.05 g/cm3; shear rate, g ¼ 1.2  104s1. Data are radially averaged. From Ramsay and Lindner (1993).

the PEO chains become more and more stretched. The recovery from anisotropy is much faster than expected from simple Brownian motion of only the clay mineral particles in a medium of the same viscosity. This behaviour would indicate dynamic coupling of the polymer chains to the clay mineral. Rheo-SAXS studies of diluted beidellite dispersions under different conditions with   320 nm, C ¼ 1.02 vol%, m ¼ 105 M and   210 nm, C ¼ 0.4%, m ¼ 105 M and of Wyoming Mt dispersion with   240 nm, C ¼ 0.64 vol%, m ¼ 104 M were carried out by Philippe et al. (2011) on the beamline SWING at the Soleil synchrotron facilities (Saclay, France). In tangential geometry, SAXS intensities are more concentrated along the horizontal direction of the 2D pattern when the shear rate increases, which reflects the alignment of the particles in the shear plane. Besides, an anisotropy in the radial geometry is also observed, indicating a biaxial orientation of particles (Fig. 2.6.9). For the smallest particles (210 and 240 nm), the tangential anisotropy is lower and less sensitive to shear rate than for the 320-nm particles, but, in contrast, the radial anisotropy increases more with shear rate. Tangential and radial plots were compared to intensities calculated for a set

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197

FIGURE 2.6.9 Relation between the anisotropy of the radial and tangential pattern and the orientation of particles under shear. From Philippe et al. (2011).

of disk-like particles with a preferred orientation assuming a biaxial ellipsoid to describe the probability density function of the normals. They showed that the shear-thinning behaviour of the dispersion (viscosity vs. shear rate) may be described by taking into account the evolution under shear of the effective volume of particles, which is affected by the preferred alignment (Fig. 2.6.10).

2.6.2.3.4 Small-Angle Light Scattering and Dynamic Light Scattering Avery and Ramsay (1986) investigated an Naþ-Laponite dispersion ( 10 g/L) at low ionic strength ([Naþ]  103 mol/L) by static SALS, dynamic light

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100 f = 0.73% f = 0.65% f = 0.64% f = 0.38%

Viscosity (Pa.s)

10

1

0.1

0.01

0.001 0.01

0.1

1

10

100

Shear stress (Pa) FIGURE 2.6.10 Modelling of the viscosity of suspensions of Wyoming montmorillonite. Various symbols: rheological measurements. Open circles: effective models. From Philippe et al. (2011).

scattering (DLS) and SANS (PLUTO, AERE, Harwell, UK). This dispersion is discussed below. SALS. The data were evaluated using the scattering equation based on the Rayleigh–Debye theory. Because of the difficulty in measuring absolute intensities, the Rayleigh ratio Rq is generally used with Rq ¼ Iq/I0, where I0 and Iq are the intensity of the incident and scattered beam, respectively.   h i c ¼ hMw iPðqÞ1 þ 2Bc (2.6.20) K Rq where c is the particle number density, q( ¼ 4p siny/n˜l) the scattering vector and P(q) the particle scattering factor. Taking into account the size of the particle in comparison with the radiation wavelength (l ¼ 546 nm), P(q) is a constant  1, B is the second osmotic virial coefficient and hMwi is the weight-average molecular weight of the dispersed particles. K* is the optical constant, equivalent to the scattering length for neutrons or X-rays and given by  2 de n l4 N 1 (2.6.21) K  ¼ 2p2 ne20 dc where n˜0 is the refractive index of the solvent, dn˜/dc is the refractive index increment and N is the Avogadro number. For interacting particles, the Rayleigh ratio can be expressed by Rðq, cÞ ¼ K  cMPðqÞGðq,cÞ

(2.6.22)

where G(q) is the interference function, which depends on particle concentration.

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199

For q ! 0, G(0, c) depends on the isothermal osmotic compressibility (dp/dc)T by the relation Gð0, cÞ ¼

kT ðdp=dcn ÞT

(2.6.23)

where T is the absolute temperature, k is the Boltzmann’s constant and cn is the particle number density. Rq is plotted as a function of q for concentrations varying from 10 to 2.5 g/L. This method permits the molecular mass hMmi of Laponite particles to be determined, and their diameter (ca. 25 nm) and thickness (ca. 1 nm) to be evaluated. There is evidence to indicate that the particles are completely dispersed at these clay concentrations. DLS. This technique provides information on the Brownian motion in a sample by analysing the fluctuation of the scattered light intensity (Berne and Pecora, 1990). The DLS data yield the autocorrelation function of intensity g(t) in a normalized form, given by gð tÞ ¼

hI ð0ÞI ðtÞi hI i2

(2.6.24)

where I(0) is the intensity measured at some arbitrary time, I(t) is that after a delay of t and hIi is the time-averaged intensity. I(t) is measured at a fixed angle. gðtÞ  1 ¼ e2Dq

2

t

(2.6.25)

In the case of monodisperse or weakly polydisperse dispersions, g(t) is related to the translational diffusion D (¼kT/f) of the particles, where k is the Boltzmann constant and f (¼6pRH) is the collective friction factor.  is the solvent viscosity and RH is the hydrodynamic effective particle radius. Experiments were performed by Avery and Ramsay (1986) for three clay contents (10, 15 and 30 g/L) at different times after preparation (2.5, 8 and 29 h). In order to extract the translational diffusion D from the correlation function, ln g(t) is plotted versus q2t. If the particles move independently without any interaction and without changing the dispersion viscosity, the decay is almost linear over the whole q range. This was observed for the lowest clay mineral content and after 2.5 h. For higher contents and at longer t values, a certain deviation from linearity indicated a progressive decrease of the effective translational coefficient D. These authors suggested that the particles became spatially constrained in an equilibrium structure due to the mutual interaction of their double layers. This brief survey clearly indicates that, for natural smectites such as Mt, the conditions for investigating SGT by SAS techniques are relatively restricted. Because of the high anisotropy of the clay mineral layers and the large size distribution of particles, the determination of the smallest q values (to derive the overall size of particles and their interaction distance range)

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involves making a series of simplifications. The majority of studies on SGT mechanisms were therefore performed with Laponite dispersions.

2.6.2.4 Modelling SGT in Laponite Dispersions Mourchid et al. (1995) combined USAXS, SAXS (CEA, Saclay, France) and rheology experiments on Laponite dispersions with cryofracture and flow birefringence observations. The experimental SAS spectra were compared with spectra computed for hypothetical dispersion structures deduced from the equilibrium properties of dispersions modelled by Monte Carlo simulations. The general purpose of these numerical simulations was to evaluate the organization of clay particles induced by excluded volume effects and electrostatic repulsion between the diffuse double layers. The most interesting parts of this work are the phase diagram (ionic strength vs. clay concentration) describing the SGT and the state equations (Fig. 2.6.8). The curves of osmotic pressure versus Laponite concentration showed a break corresponding to the SGT followed by a pseudo-plateau separating the liquid and the gel phase. This was explained by an isotropic–nematic (I–N) transition (Onsager transition). However, the phase separation as predicted by this theory was not observed along the short pseudo-plateaus of Fig. 2.6.11. In explanation, the authors admitted the possibility of a 5000

4000

P (Pa)

3000

2000

1000

0

0

2

4

6

8

C (%) FIGURE 2.6.11 Experimental equations of state for Laponite dispersions. The plots show the variation of osmotic pressure P with clay concentration C at different ionic strengths I of NaCl: (●) I ¼ 104 M; (○): I ¼ 5  103 M; (▲): I ¼ 102 M. From Mourchid et al. (1995).

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201

thermodynamic transition coinciding with a mechanical phase transition resulting in a frustrated I–N transition. Another important result, deduced from the structure simulation, was that the most energetically favourable configuration between the two first neighbouring Laponite particles was the perpendicular orientation without any edge-to-face contact. This could be compatible with a certain number of like card-house structures (van Olphen, 1977) observed in coagulated dispersions by electron microscopy. The USAXS spectra indicate repulsive effects between particles and the absence of interparticle contacts. Dijkstra et al. (1997) also discussed a Monte Carlo model for the structure and gelation of smectite dispersions. They confirmed the edge-to-face particle configuration without any contacts. Kroon et al. (1996) made a DLS study of SGT in Laponite dispersions at various clay mineral concentrations (1–3.5%) on both sides of the gelling time Tg. Measurements of the static part of scattering combined with the DLS results showed the slowing down of the dynamics of a homogeneous sol until Tg. Thus, the SGT threshold was marked by a drastic change in the static part of the scattered intensity. In the gel phase, the function showed a power-law decay, with a concentration-dependent scaling exponent. The authors showed the strong similarity to the scenarios predicted by the mode-coupling theory of the structural glass transition. However, transition should lead to a fragile glass and should not be far off an I–N transition. Using SANS (ILL, Grenoble), SALS and USAXS (CEA, Saclay), Pignon et al. (1997) showed that Laponite XLG dispersions have fractal structures over very large length scales when the gels are observed over time. Parallel rheometric measurements reveal two characteristic scale lengths in the gel behaviour. The authors suggested that the gel is composed of subunits measuring a few tens of nanometres. The subunits combine to form dense aggregates measuring ca. 1 mm. At larger scale lengths, these micrometre-sized aggregates form isotropic fractal arrangements of dimension D, which increase with the particle volume fraction. The isotropic fractal arrangement may control the macroscopic mechanical behaviour. Figure 2.6.12 illustrates the structural evolution of a Laponite dispersion as a function of gelation time, derived from static light scattering patterns. Mourchid and Levitz (1998) used SALS and USAXS to study long-term gelation of Laponite dispersions. A plot of log I(q) versus log q showed a linear decrease with a slope 2 < a < 3 for the smallest q values. The authors concluded that this phenomenon involved a long-term mechanism leading to the formation of glassy systems with a fractal structure. However, such a structure is not observed if the starting chemical parameters are stable during the experiment. For example, a decrease in pH causes a release of divalent and trivalent cations, which, in turn, promotes aggregation. Therefore, chemical modification of the colloid particles can play a central role in the evolution of the system.

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q−1.8

I(q)

107

106

f v = 1.2 %

q−1

200 days < tp < 500 days

105

tp= 180 days tp= 130 days

q−3

tp= 10 days 4

10 10−5

tp= 5 days

10−4 q

(Å−1

)

FIGURE 2.6.12 Static light scattering for Laponite dispersions at different equilibrium gelation times. Volume fraction of clay mineral, 1.2%; ionic strength, 103M NaCl; and pH 9.5. From Pignon et al. (1997).

Pignon et al. (2000) investigated the structures of Laponite dispersions and the deposits that form during filtration at a transmembrane pressure of 0.5 bar using SANS (ILL, Grenoble) and SALS. They compared the gel structures to the respective fractal structures of the deposits. Levitz et al. (2000) showed that, at low ionic strength (I ¼ 3  105 M) and low-clay mineral contents (0.82%, 1.02% and 1.85%), the Laponite dispersions gave USAXS spectra with correlation peaks that are compatible with long-range electrostatic stabilization. Close inspection of these correlation peaks indicates that the individual particles are not homogeneously dispersed in the space. Such results contrast strongly with the behaviour at high ionic strength (I > 104 M). To evaluate the nematic order in dilute dispersions, Lemaire et al. (2002) performed SAXS experiments on oriented samples of Laponite gel at the European Synchrotron Radiation Facilities in Grenoble. They used a 3 mass% dispersion, obtained by slow evaporation of a 1% dispersion. The spectra were recorded on a 2D detector. The nematic order parameter S was derived from the scattered intensity of an oriented sample. At a clay content of 3%, S is similar to that found for classical liquid crystals. Cousin et al. (2002) followed a new way to determine the gel structure of Laponite dispersions along the transition pseudo-plateau of Fig. 2.6.11. Magnetic spherical colloidal particles of maghemite were added to the Laponite dispersions for SANS studies (ILL, Grenoble, France). The scattered intensity is given by IT ðqÞ ¼ IL ðqÞ þ Ip ðqÞ þ ILP ðqÞ

(2.6.26)

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203

where IT(q) is the total scattering by the dispersion, IL(q) is the scattering by the Laponite particles, IP(q) is the scattering of the dispersed magnetic maghemite particles and ILP(q) is an intensity contribution due to the coupling between Laponite and maghemite particles. By varying the ratio of D2O/water, the contrast between Laponite particles and the solvent was extinguished and IT(q) was reduced to IP(q). As the maghemite particles are spherical, the interference function G(q) was obtained from Eqs. (2.6.11 and 2.6.12). Thus, the distributions of the maghemite particles in the gel could be determined for different Laponite contents. Concentrations and ionic strengths of dispersions were selected to follow the pseudo-plateau of the SGT (Fig. 2.6.11). SANS and viscosity measurements showed that the magnetic probes were homogeneously dispersed in Laponite dispersions except in the region of the plateau where they were enriched in domains of low Laponite contents, indicating that the Laponite dispersions are microscopically biphasic. The authors concluded that this structure could be consistent with a frustrated I–N transition or a spinodal decomposition. DLS and shear rejuvenation experiments on a dilute Laponite dispersion (3 mass%) in pure water were performed by Bonn et al. (2002). Shear rejuvenation is given as the decay of the correlation function g(t)1 versus time (as measured by DLS) when different shear rates are applied to a dispersion after ageing (e.g. 1 h) for different shearing times. The reversion of the dispersion state after shearing is compared to the dispersion states previously observed after different ageing times. The results suggest that the dynamic behaviour of such a colloid is very close to that of a glassy system. The overall conclusion that can be drawn is that the SGT in Laponite dispersions may be explained in terms of a glassy transition with some specific characteristics probably due to particle anisotropy.

2.6.3 ORGANOCLAY DISPERSION IN ORGANIC SOLVENT AND POLYMER NANOCOMPOSITES 2.6.3.1

Dispersions in Organic Solvents

The dispersion state of organoclays in organic solvents was extensively studied by SAXS and SANS in relation to the nature of the clay (CEC, aspect ratio, concentration) and polarity of the solvent (benzene, toluene, p-xylene, chloroform). SAS profiles of organoclay dispersions may be fitted by the scattering model of N core–layer disk stackings dispersed in a solvent (Eq. 2.6.14). In this model, the silicate layer constitutes the core of a disk with diameter 2R, whereas the cationic surfactant adsorbed on each side forms a uniform layer with a thickness 2H. In all work mentioned below, the interference function G(q) (Eq. 2.6.13) used to calculate I(q) (Eq. 2.6.14), assumes that the distance d between two adjacent core–layer disks inside a stack is

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two times the organic layer thickness, that is, 4H. When applied to a dry organoclay, 2H corresponds to the surfactant thickness but when applied to particles dispersed in organic solvent it means that the solvent carbon chains are combined with the carbon chains of the surfactant inside the interlamellar space forming a homogeneous 2H layer. This model was used in different studies mentioned below. Ho et al. (2001) used WAXS and SANS to study the structure of organophilic clay minerals and their interaction with various organic solvents. Both types of experiments were performed at the NIST, NCNR (USA). Naþ-Mt was exchanged with ditalloyl dimethylammonium ions, consisting of a mixture of long-chain alkylammonium ions (ca. 65% C18, 30% C16 and 5% C14). The organic solvents chosen covered a range of solubility parameters. The organoclay samples (modelling SGT in Laponite dispersion Cloisite 15A) were used as received and after purification. In both cases, the layers were fully exfoliated in chloroform, while the particles retained their layer structure and expanded to a similar extent in benzene, toluene and p-xylene. However, the extracted (purified) material had a stronger tendency to gel. At the same clay mineral concentration, the swollen particles of this sample were also thinner and, therefore, more numerous than those of the unpurified counterpart. The SANS curves in the lowest q range (from 0.04 nm1) followed a type of scale law (qa), with a of ca. 2.2. The number of layers per particle varies from hNi ¼ 1 in chloroform to hNi ¼ 3 in benzene, with layer thickness of ca. 1.6–1.86 nm, depending on the solvent. Vaia et al. (2003) investigated the dispersion state in toluene and toluene– acetone blends of different commercial organically modified clay minerals: Cloisite 6A, 15A, 20A (prepared from Naþ-Mt and dimethyl dehydrogenated tallow ammonium chloride with different exchange ratio) and Cloisite 30B. In their SAXS fitting model, they took into account the fraction of isolated layer w. In toluene, the larger hNi value was 3–4, the distance periodicity was at ˚ , and w could reach 70%. The fraction of individual the maximum 49 A layers decreases as the organoclay concentration increases and as the surfactant exchange ratio increases. The degree of dispersion decreased markedly ˚, with a small addition of polar acetone in toluene (hNi ¼ 4–5) and d ¼ 36 A but the nonlinearity of this effect implied a selective partitioning of the polar component to the interlayer. Hanley et al. (2001, 2003) studied, using SANS, dispersions in toluene and H/D toluene of commercial organoclay (Cloisite 15A) and organoclay prepared from cationic exchange of Mt with di-tallow and dioctadecyl dimethylammonium chloride. From the SAS fit, they obtained a layer thickness of around 2 nm and a mean layer per particle hNi ¼ 3–5. A suitable partial deuteration of the hydrocarbon solvent matches the contrast between the solvent and the clay mineral layer so that the scattering intensities arise from the surfactant layer.

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By means of SAXS, Pizzey et al. (2004) studied dispersions of organo-Mt of claytone AF and a dioctadecyl dimethylammonium bromide-treated Laponite in toluene or in a nematogenic compound liquid crystal K15. The model they used for SAXS fit considered also the existence of free single plates. Dispersion of 1% claytone in K15 provided 33% of single plates, but stack˚ as well, and so an inter-plate ings with hNi ¼ 5 1 and a periodicity of 37 A ˚ separation of 27 A, consistent with the model of two extended surfactant chains interlocking with a single of K15 dimers. In toluene, the delamination was better since most of the particles were free single plates (67%); there ˚. were also fewer plates in stacks hNi ¼ 2 and a higher basal spacing 38 A For Laponite dispersions, 90% of particles were delaminated for 1% Laponite in K15, 90% for 3% claytone in toluene and 100% for 1% Laponite in tolu˚ (in K15) ene, and in residual stacks, hNi ¼ 2, with a basal spacing of 27 A ˚ and 42 A (3% Laponite in toluene). King et al. (2007) using SAXS, SANS and USAXS related the structure of Cloisite 6A after dispersion in different solvents (p-xylene, chloroform, THF) to the rheological properties of the dispersion. Cloisite 6A is a commercial organoclay prepared with dioctadecyl dimethylammonium chloride. SAS fits were preferentially done on SAXS profiles. X-ray scattering arises mainly from the contrast between the metal oxide core and the surrounding organic surfactant tails. This core-contrast form factor accentuates the peak in S(q), whereas SANS data exhibit a significant incoherent background due to H-atoms obscuring the interlayer correlation peak. Dispersions with Cloisite contents ranging from 0.005 to 20 mass% in p-xylene showed a quite constant number of plates per stack hNi ¼ 1.9. Otherwise, USANS data indicate a transversal dimension R ˚ , which is much larger than that of an individual of plates of at least 32,000 A layer. These authors suggested a particle structure made of overlapping individual layers, with 80% of N ¼ 1 regions mixed with N 2 regions. This structure is consistent with gel formation at very low clay mineral contents.

2.6.3.1.1 SANS Matching Contrast A comparison between the neutron scattering contrast in H and H/D toluene allowed the estimation of the number of di-tallow moles that cover the surface of 1 mol of Naþ-Mt (Hanley, 2001, 2003). The SLD (scattering length density) of an organoclay particle may be obtained by performing SANS contrast variation experiments on dispersions of organoclay particles in various deuterated/protonated organic solvents such as p-xylene (Ho et al., 2001; King et al., 2007). The SLD of the aluminosilicate core may be theoretically calculated from the elemental analysis or experimentally by performing SANS contrast matching on the inorganic clay mineral immersed in various deuterated/protonated ratios of deionized water. Thus, the SLD of the organic phase inside the lamellar space may be deduced

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since the overall SLD of the organoclay particle is the volume-fractionweighted average of the SLD of the organic and inorganic parts. Similarly, from the overall organic phase SLD and the SLD of the dimethyl dioctadecyl ammonium (DDA) obtained from elemental analysis, King et al. (2007) estimated the solvent-to-DDA ratio in the organic phase. They showed that more than half of the interlayer space is occupied by trapped solvent when a commercial organoclay (Cloisite 6A) is immersed in p-xylene. After immersion in p-xylene, the DDA value reaches 0.7 mol starting from 1 mol in the dry powder. This value corresponds to the layer charge balance.

2.6.3.2 Clay–Polymer Nanocomposites (CPN) In combination with other morphological techniques, SAXS was extensively used to describe the morphology of CPN, including the dispersion level of clay mineral layers and also consecutive structural modifications in the polymeric matrix such as the degree of crystallinity and the lamellar long period. Depending on the existing interactions between the surfactant adsorbed on the aluminosilicate layer surface and the polymer chains, as well as on the preparation method (melt blending, solution blending or in situ polymerization), SAXS patterns exhibit a disappearance of the 001 reflection, which corresponds to an exfoliation of the clay mineral layer or, more often, a shift of this reflection to lower angles due to the swelling of the interlamellar space resulting from the intercalation of polymer chains. The number of layers per stack may also decrease because of an in situ delamination. Causin et al. (2005) assumed that, in CPN, the morphology may be described by high-density clay mineral layers alternating with low-density polymer layers. In this case, as the lateral width of the clay mineral particles is much larger than the thickness, SAXS patterns may be fitted with the scattered intensities obtained by the Fourier cosine transform of g(x), the 1D correlation function for a distorted lamellar model given by Vonk’s formula (Glatter and Kratky, 1982). A quantitative analysis of the dispersion in a copolymer poly(butylene succinate-co-adipate) (PBSA) of Cloisite 30B was reported by Bandyopadhyay and Sinha Ray (2010). The probability of finding neighbouring clay mineral particles in the PBSA matrix, as well as their thickness, was calculated using the generalized indirect Fourier transformation technique developed by Glatter and the modified Caille´ theory. The p(r) function obtained with this method and the electron density profile of the thickness cross-section allowed the estimation of the mean layer distance ( 4.5 nm) and the mean number of layers ( 5.7). Nawani et al. (2010) quantitatively described the orientation distribution of organoclays (up to 20 mass% of Cloisite 20A) in melt-pressed CPN films containing EVA copolymers.

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CPN exhibited intercalation/exfoliation clay mineral morphology where the intercalated structure possessed partial orientation parallel to the in-plane direction of the 1 mm thick film. The preferred orientation is visible with an edge-on direction of the X-ray beam (parallel to the film surface). With this configuration, Nawani et al. (2010) obtained from 2D SAXS analysis (X27C beamline at NSLS, Brookhaven National Laboratory) the orientation function of the perpendicular to the clay mineral layers and they deduced orientation parameters. The increase of the degree of orientation compared to the film surface is in agreement with the decrease of the measured gas permeability. When the thickness of the film is smaller than the beam size, the recently developed grazing-incidence small-angle X-ray scattering (GISAXS) technique is appropriate. The off-specular scattering can be analyzed for incidence angles close to the critical angle of total external reflection of the CPN, revealing both the lateral structure within the film and the structure normal to the substrate. GISAXS study are reported by Lutkenhaus et al. (2007) on the G1 beamline at the Cornell High-Energy Synchrotron Source with electrolyte films formed after N deposition cycles of trilayers: PE imine, PEO and Laponite, by the layer-by-layer (LbL) technique. In this case, GISAXS allows measuring the orientation of clay mineral particles within the thin trilayers PEI/Laponite/PEO, using the Hermans orientation parameter. The peaks from the LbL assemblies suggest periodic structure of clay mineral particles, which are adsorbed in multiple layers but without intercalation of the polymer between individual particles.

2.6.4

CONCLUSIONS

All scattering techniques are capable of making in situ observations of finely divided materials without causing any damage to the samples. Rheo-SAXS and Rheo-SANS experiments are good examples of the development of in situ measurements observed during the last decade. SAXS, SANS and SALS, when used together, allow a multi-scale sample exploration to be done (from nanometre to several tens of micrometres) under static and dynamic conditions. In many cases, the macroscopic behaviour of materials can be explained on the basis of the structure of the system in the nanometre dimension. SALS. This technique can be used to investigate dispersions at a large length scale and thus provide a relatively large overview of the samples than do the other techniques. But the range of samples that can be studied by SALS is limited because dispersions have to be transparent to light. Nevertheless, the use of SALS was probed for clay mineral colloid studies in combination with DLS and rheological measurements. The Fraunhofer scattering range (Table 2.6.1) is interesting for granulometric measurements, which can be made in conjunction with sedimentation experiments, as illustrated by the work of Pabst et al. (2000) on kaolins.

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SANS. The primary advantage of neutrons is their very weak absorption by clay mineral dispersions and powders. The samples can be as thick as 1 cm, whereas they cannot be thicker than 0.2–0.5 mm for SAXS. Thus, dry clay mineral samples can be investigated by SANS, and particular SAS experiments such as Rheo-SANS can be performed. The added advantage of using neutrons is the possibility to change the scattering contrast by varying the D:H ratio in aqueous or organic media. SAXS. Many SAS studies were performed by X-rays because X-ray techniques preceded neutron scattering and the scattering contrast of clay mineral layers is high for X-rays. Unlike neutrons, X-rays do not give any background of inelastic scattering at small angles, and the high brilliance of thirdgeneration synchrotron allows very short time acquisition ( milliseconds) compatible with the study of quick transformation kinetics. The matching method based on the anomalous scattering near the absorption edges is a useful feature. A number of synchrotron radiation centres with specially designed spectrometers were developed, making the technique available to non-specialists. Otherwise, the improvements in the technical performance of laboratory SAXS equipment make them useful for some routine SAXS structural studies. It seems that, in future, the use of the GISAXS will be generalized to study clay-mineral-containing thin films, as GISAXS is compatible with synchrotron sources and also laboratory apparatus. This non-exhaustive review of SAS techniques reveals the possibility and potential of using these methods to characterize clay mineral dispersions. The SGT in such systems, for example, is now reasonably well understood. Sample preparation, however, must be carefully controlled if consistent results are to be obtained, and valid data interpretation is to be achieved. SAS techniques have long been used to study aqueous colloidal dispersions of ‘pure’ clay minerals. It is only relatively recently that other clay mineral systems, notably organoclays and smectite–polymer nanocomposites, have begun to be investigated by scattering methods.

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