Bergaya F. Handbook of Clay Science

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magnitude of negative charge and the cation exchange capacity (CEC) of clay min-

erals. This is because the substitution of Si

for Al

in the tetrahedral sheet, and

for Mg

,Fe

, etc. in the octahedral sheet, is at the origin of the (permanent)

negative charge in the phyllosilicate structure. XPS is capable of distinguishing be-

tween four- and six-coordinated Al

ions. As Wagner et al. (1982) pointed out, the

sum of the Al 2p binding energy and Al KL

Auger kinetic energy (modiﬁed Auger

parameter for Al) for four-coordinated Al

in silicate minerals is smaller than that of

six-coordinated Al

. In other words, the extra-atomic relaxation for four-coordinated

is smaller than that for six-coordinated Al

.Recently,Evans and Hiorns (1996)

examined the cleaved surface of chlorite single crystals by angle-resolved XPS. From

the magnitude of anisotropy in the XPD patterns they were able to estimate the

content of tetrahedrally coordinated Al

as a percentage of total Al

ions.

REFERENCES

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1998. XPS analysis of polyacrylamide adsorption to kaolinite, quartz and feldspar. Surface

and Interface Analysis 26, 518–523.

Anthony, M.T., Seah, M.P., 1984. XPS: energy calibration of electron spectrometers. 1 – An

absolute, traceable energy calibration and the provision of atomic reference line energies.

Surface and Interface Analysis 6, 95–106.

Ash, L.A., Evans, S., Hiorns, A.G., 1987. Cation ordering in lepidolite and biotite studied by

X-ray photoelectron diffraction. Clay Minerals 22, 375–386.

Briggs, D., Seah, M.P. (Eds.), 1990. Particle Surface Analysis, 2nd edition, vol. 1, Auger and

X-ray Photoelectron Spectroscopy. Wiley, Chichester.

Cocke, D.L., Vempati, R.K., Loeppert, R.H., 1994. Analysis of soil surfaces by X-ray pho-

toelectron spectroscopy. In: Amonette, J.E., Zelazny, L.W. (Eds.), Quantitative Methods in

Soil Mineralogy. Soil Science Society of America, Madison, WI, pp. 205–235.

De Stasio, G., Capozi, M., Lorusso, G.F., Baudat, P.A., Droubay, T.C., Perfetti, P.,

Margaritondo, G., Tonner, B.P., 1998. MEPHISTO: performance tests of a novel syn-

chrotron imaging photoelectron spectromicroscope. Review of Scientiﬁc Instruments 69,

2062–2066.

De Stasio, G., Gilbert, B., Frazer, B.H., Nealson, K.H., Conrad, P.G., Livi, V., Labrenz, M.,

Banﬁeld, J.F., 2001. The multidisciplinarity of spectromicroscopy: from geomicrobiology

to archaeology. Journal of Electron Spectroscopy and Related Phenomena 114–116,

997–1003.

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G., Tonner, B.P., 1999. MEPHISTO spectromicroscope reaches 20 nm lateral resolution.

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Dutta, N.C., Iwasaki, T., Ebina, T., Hayashi, H., 1999. A combined X-ray photoelectron and

Auger electron spectroscopic study of cesium in variable-charge montmorillonites. Journal

of Colloid and Interface Science 216, 161–166.

Ebel, M.F., Ebel, H., Zuba, G., Wernisch, J., 1983. Calibration of an X-ray photoelectron

spectrometer by means of noble metals. Surface and Interface Analysis 5, 170–172.

Chapter 12.4: X-ray Photoelectron Spectroscopy876

Evans, S., Hiorns, A.G., 1996. Angle-resolved X-ray photoelectron studies of cleavage in

chlorites. Clays and Clay Minerals 44, 398–407.

Evans, S., Raftery, E., 1980. X-ray photoelectron studies of titanium in biotite and phlogopite.

Clay Minerals 15, 209–218.

Evans, S., Raftery, E., 1982. X-ray photoelectron diffraction studies of lepidolite. Clay Min-

erals 17, 443–452.

Garwood, G.A. Jr., 1995. Scanning X-ray photoelectron microscopy (SXPEM). In: Hubbard,

A.T. (Ed.), The Handbook of Surface Imaging and Visualization. CRC Press, Boca Raton,

FL, pp. 687–704.

Gier, S., Johns, W.D., 2000. Heavy metal-adsorption on micas and clay minerals studied by

X-ray photoelectron spectroscopy. Applied Clay Science 16, 289–299.

Glasauer, S.M., Hug, P., Weidler, P.G., Gehring, A.U., 2000. Inhibition of sintering by Si during

the conversion of Si-rich ferrihydrite to hematite. Clays and Clay Minerals 48, 51–56.

Graveling, G.J., Ragnarsdottir, V.K., Allen, G.C., Eastman, J.R., Brady, P.V., Balsley, S.D.,

Skuse, D.R., 1997. Controls on polyacrylamide adsorption to quartz, kaolinite, and feld-

spar. Geochimica et Cosmochimica Acta 61, 3515–3523.

Hochella, M.F. Jr., 1995. Mineral surfaces: their characterization and their chemical, physical

and reactive nature. In: Vaughan, D.J., Pattrick, R.A.D. (Eds.), Mineral Surfaces. The

Mineralogical Society Series 5. Chapman & Hall, London, pp. 17–60.

Ilton, E.S., Moses, C.O., Veblen, D.R., 2000. Using X-ray photoelectron spectroscopy to

discriminate among different sorption sites of micas: with implications for heterogeneous

reduction of chromate at the mica–water interface. Geochimica et Cosmochimica Acta 64,

1437–1450.

Ilton, E.S., Veblen, D.R., Moses, C.O., Paeburn, S.P., 1997. The catalytic effect of sodium and

lithium ions on coupled sorption–reduction of chromate at the biotite edge-ﬂuid interface.

Geochimica et Cosmochimica Acta 61, 3543–3563.

Johns, W.D., Gier, S., 2001. X-ray photoelectron spectroscopic study of layer charge mag-

nitude in micas and illite–smectite clays. Clay Minerals 36, 355–367.

Kawai, J., Hayakawa, S., Kitajima, Y., Gohshi, Y., 1995. X-ray absorption and photoelectron

spectroscopies using total reﬂection X-rays. Analytical Sciences 11, 519–524.

Kordesch, M.E., 1995. Photoelectron emission microscopy. In: Hubbard, A.T. (Ed.), The

Handbook of Surface Imaging and Visualization. CRC Press, Boca Raton, FL, pp. 581–596.

McIntyre, N.S., Zetaruk, D.G., 1977. X-ray photoelectron spectroscopic studies of iron

oxides. Analytical Chemistry 49, 1521–1529.

Mosser, C., Mosser, A., Romeo, M., Petit, S., Decarreau, A., 1992. Natural and synthetic

copper phyllosilicates studied by XPS. Clays and Clay Minerals 40, 593–599.

Parﬁtt, R.L., Van der Gaast, S.J., Childs, C.W., 1992. A structural model for natural siliceous

ferrihydrite. Clays and Clay Minerals 40, 675–681.

Paterson, E., Swafﬁeld, R., 1994. X-ray photoelectron spectroscopy. In: Wilson, M.J. (Ed.),

Clay Mineralogy: Spectroscopic and Chemical Determinative Methods. Chapman & Hall,

London, pp. 226–259.

Seah, M.P., 1980. The quantitative analysis of surfaces by XPS: a review. Surface and In-

terface Analysis 2, 222–239.

Seyama, H., Soma, M., 1984. X-ray photoelectron spectroscopic study of montmorillonite

containing exchangeable divalent cations. Journal of the Chemical Society, Faraday

Transactions 1 (80), 237–248.

References 877

Seyama, H., Soma, M., 1985. Bonding-state characterization of the constituent elements of

silicate minerals by X-ray photoelectron spectroscopy. Journal of the Chemical Society,

Faraday Transactions 1 (81), 485–495.

Seyama, H., Soma, M., 1987. Fe 2p spectra of silicate minerals. Journal of Electron Spec-

troscopy and Related Phenomena 42, 97–101.

Seyama, H., Soma, M., 1988. Application of X-ray Photoelectron Spectroscopy to the Study

of Silicate Minerals. Research Report from the National Institute for Environmental

Studies, Japan, No. 111.

Seyama, H., Soma, M., 2003. Surface-analytical studies on environmental and geochemical

surface processes. Analytical Sciences 19, 487–497.

Seyama, H., Soma, M., Tanaka, A., 1996. Surface characterization of acid-leached olivines by

X-ray photoelectron spectroscopy. Chemical Geology 129, 209–216.

Soma, M., Churchman, G.J., Theng, B.K.G., 1992. X-ray photoelectron spectroscopic ana-

lysis of halloysites with different composition and particle morphology. Clay Minerals 27,

413–421.

Soma, M., Seyama, H., Yoshinaga, N., Theng, B.K.G., Childs, C.W., 1996. Bonding state of

silicon in natural ferrihydrites by X-ray photoelectron spectroscopy. Clay Science 9,

385–391.

Wagner, C.D., Gale, L.H., Raymond, R.H., 1979. Two-dimensional chemical state plots: a

standardized data set for use in identifying chemical states by X-ray photoelectron spec-

troscopy. Analytical Chemistry 51, 466–482.

Wagner, C.D., Passoja, D.E., Hillery, H.F., Kinisky, T.G., Six, H.A., Jansen, W.T., Taylor,

J.A., 1982. Auger and photoelectron line energy relationships in aluminum–oxygen and

silicon–oxygen compounds. Journal of Vacuum Science & Technology 21, 933–944.

Chapter 12.4: X-ray Photoelectron Spectroscopy878

Handbook of Clay Science

Edited by F. Bergaya, B.K.G. Theng and G. Lagaly

Developments in Clay Science, Vol. 1

879

Chapter 12.5

SMALL-ANGLE SCATTERING TECHNIQUES

D. TCHOUBAR

AND N. COHAUT

Expert CRT-Plasma Laser, Orle

ans, F-45160 Olivet, France

CRMD, CNRS-Universite

d’Orle

ans, F-45071 Orle

ans Cedex 2, France

With the possible exception of those who live in very dry deserts, people can see

small-angle scatteri ng (SAS) by water droplets when they observe a moon halo

predicting rain for the next day. Indeed, any medium showing matter density ﬂuc-

tuations can give rise to observable SAS when it is placed in the path of a radiation

beam, provided the medium is adequately transparent and the ﬂuctuations in matter

density are at appropriate scales with respect to the radiation wavelengths. This

chapter is concerned with scattering by a heterogeneous medium in the form of a

colloidal dispersion of small particles in a solvent.

The last 50 years have seen the development of small-angle scattering 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 with the radiation wavelength (Table 12.5.1).

The different techniques, shown in Table 12.5.1, are generally complementary,

taking into account their respective wavelengt hs. For light scattering experiments,

the choice of sampl es is more restricted by the necessity for transparency, and

problems of turbidity and multiple scattering. In all cases involv ing 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 above. ‘‘Zooming’’ on the central part of the diffraction patterns

enables a proﬁle analysis to be made.

DOI: 10.1016/S1572-4352(05)01031-7

12.5.1. 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 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 parame ter

s ¼ q=2p ¼ 1=d (where d is the Bragg distance) in place of q. For particles in dis-

persion, the scattered intensity is expressed by

IðqÞ¼I

F k

PðqÞð1Þ

where I(q) is the number of photons (or neutrons) counted per unit of solid angle

and time; I

the intensity of the incident beam, i.e., the total number of photons

(or neutrons) irradiating the sample per unit of time; F the volume fraction of

particles that are immersed in the incident beam; and k the scattering length contrast

of the particle (see below). The interference function, P(q), is the most impor-

tant parameter of Eq. (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.

A. Scattering Length and Particle Scattering Length Contrast

It is generally assumed that the unit domain size (scattering unit) is large in com-

parison with the interatomic distances. On this assumption, the angular extension of

Table 12.5.1. Scattering techniques in relation to particle dimension (nm) (from Glatter,

1991)

Lorenz-Mie

Raleigh - Debye - Gans

Ultra Small Angle Scattering

Elastic Light Scattering

Fraunhofer diffraction

Small Angle Scattering

(SAXS and SANS)

(X- Ray and Neutrons)

Quasi-Elastic Light Scattering

Dimension d(nm)

10 10

Chapter 12.5: Small-Angle Scattering Techniques880

the SAS pattern is limited to a q

max

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, b

. Depending on the interaction mode

of the radiation with the mate rial, b

is speciﬁc to each technique.

Neutron Scattering

The neutrons normally used (Cotton, 1991) have an energy of about 10

eV and their

wavelengths are 0.1 nmolo2 nm. The neutrons interact with the nuclei of the atoms

of the particles and the dispersion medium. For each nucleus, the scattering length,

b, ca nnot yet be theoretically computed but only experimentally evaluated. The b

values were tabulated (Bacon, 1975). The values can either be positive or negative,

and differ from one isotope to another. The difference in value between hydrogen

(0.374 10

–12

cm) and deuterium (+0.667 10

–12

cm) is an important point.

Taking into account the atomic composition of the scattering unit, b

may be

computed from the relation: b

¼ S

where n

indicates the number of atoms of

type i,andb

their scattering lengths. The summation ðS

Þ is performed on all types

of atoms constituting the scattering unit.

If the particles are homogeneous, the particle scattering length density, r

, is deﬁned

as the ratio b

where v

is the scattering unit volume. For particles that are dispersed

in a solvent, the solvent scattering length density, r

, must also be computed.

The scattering length contrast, k, of the particle is deﬁned by the difference

r

), while the scattering power of the dispersed particle is given by:

¼ Fðr

 r

ð2Þ

X-ray Scattering

The X-ray photons have an energy of about 10

eV and their wavelengths, l, are

distributed about 0.05 nmolo0.5 nm (Guinier and Fournet, 1955). Their electro-

magnetic interactions with matter involve the Z electrons of the atom electronic

shell. Therefore, the scattering length for one atom is b ¼ Zb

, where b

( ¼ +0.282 10

–12

cm) is the scattering power of an electron and can be computed

theoretically. In that case, the scattering power of the disper sion only depends on the

electron density of the particle and that of the solvent, and on the scattering power of

free electron, I

( ¼ b

¼ FI

ðr

 r

ð3Þ

where r

and r

are respectively the particle and the solvent electron density.

Light Scattering

The visible light photons have an energy of about 10 eV. Their elect romagnetic

interaction with matter is macroscopic and induces an electrical ﬁeld giving rise to

electric dipoles in the medium. The scattering length is expressed as a function of the

12.5.1. Principles 881

polarizibility, a, of the medium, which, in turn, is related to the refraction indexes of

the particles and the solvent. The theoretical expression of the scattering power of

the dispersion depends on the size of the particles, in compari son with the wave-

length of the light. This expression, therefore, depends on the approximations used

in the theoretical treatment of the patterns. Table 12.5.1 illustrates the range of

validity of the different method s of SALS. The corresponding theories were dis-

cussed by Hofer (1991).

B. Quantitative Deﬁnitions of the Interference Function, PðqÞ

Dilute Suspension of Homogeneous Particles without any Interactions

In the case of a statistically isotropic dispersion, the particles can adopt all possible

orientations. The interference function, P(q), from Eq. (1), is expressed by the fol-

lowing Fourier transform:

PðqÞ¼4p

pðrÞ

sin qr

dr with pðrÞ¼r

gðrÞð4Þ

For monodisperse particles, V

is the particle volume and g(r), the characteristic

function of the particle (Porod, 1982). It is deﬁned as the probability to ﬁnd two

points at the distance r within a particle.

The intensity i(q) scattered by only one particle is called the particle shape

function:

iðqÞ¼4p

gðrÞ

sin qr

dr ð5Þ

One property is of interest: for q ¼ 0, ið0Þ¼V

and therefore Pð0Þ¼ðV

. If the

parameter k and the particle volume fraction, F, is known, the particle mass can be

determined from I(0) in the general express ion of Eq. (1).

Scattering by dilute suspensions of monodisperse or weakly polydisperse particles

may be analys ed by comparing the experimental curve proﬁles with theoretical i(q).

Expressions for i(q) were determined for all geometrical shapes, including such

complex conﬁgurations as polyme r blobs (Guinier and Fournet, 1955; Glatter and

Kratky, 1982).

Properties of the Shape Function i(q)

The i(q) proﬁle for hom ogeneous particles is characterized by three ‘q-ranges’

(Fig. 12.5.1). Range (1) depends only on particle size. As shown by Guinier and

Fournet (1955), i(q) can be approximated by

iðqÞﬃ1 

þﬃexp

ð6Þ

Chapter 12.5: Small-Angle Scattering Techniques882

where R

is the gyration radius of the particle. q-range (1) is limited by the condition

¼ 1. The corresponding Guinier plot ln i(q) vs. q

shows a linear decrease whose

slope is equal to R

/3. Extrapolation to q ¼ 0 yields i(0). The q-range (2) depends

on particle shape. For a cylinder of diameter 2R and height 2 H,

iðqÞ¼V

p=2

sinðqH cos jÞ

qH cos j



ðqR sin jÞ

qR sin jðÞ



sin j dj ð7Þ

where V is the volume of a particle, J

, the ﬁrst order Bessel function and the

integration variable j, the angle between the scattering vector q and the cylinder

axis. For very thin disks, Eq. (7) is approximated by

ðqÞﬃ

exp

ð8aÞ

If the thickness of the disk is very small in comparison with its radius,

ðqÞﬃ

ð8bÞ

This applies to montmorillonite layers. If the clay mineral is totally dispersed, the

plot of log i(q) vs. log q shows a linear decrease, with of slope of 2 within the

q-ranges (2) and (3) (Fig. 12.5.1).

Auvray and Auroy (1991) discussed the scattering by thin microemulsion layers.

Their ﬁrst conclusion is that layer curvature does not change the scattering behav-

iour and always decreases as q

2

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).

i(q)

(1) Size range

Guinier’s law

(3) Surface range

Porod’s law

(2) Shape range

Shape function

-1

q (Å

-1

)

 /H

Fig. 12.5.1. The three ranges of the SAS scheme related to the geometrical properties of dilute

particle dispersions.

12.5.1. Principles 883

For compact particles, the external part of scattering, i(q), varies as sq

4

, where s is

the speciﬁc surface area.

Particle Size Distribution

If the particles are not equal (polydisperse), i(q) is replaced by its average, hiðqÞi:

hiðqÞi ¼

ðqÞð9Þ

where a

is the particle weigh t fraction of size m, and i

(q), its respective scattering.

The mean particle volume hV

i replaces V

in Eq. (4).

This case is illustrated by the SANS study of Hall et al. (1985) on dilute sus-

pensions (0.8%,w/v) of allophane particles. The spherical shape of these particles is

particularly well suited for determining their size distribution by SAS techniques.

The analysis was performed by comparing experimental SANS spectra with sim-

ulated ones.

Heterogeneous Particles

The p(r) function not only depends on particle size and shape but also on its intern al

structure. In this case the product k

i(q), indicating the scattering power of one

particle, is replaced by the structure factor F(q)

. If the particles are fractal clusters,

their mass fractal dimension a is generally a non-integer, varying within 1oao3

(Mandelbrot, 1967, 1977). The scattered intensity varies as the power law i(q)Eq

a

within the q-ranges (2) and (3) in Fig. 12.5.1 (Schaefer et al., 1985). The limit value of

a ¼ 3 for uncompacted particles, observed for some clay gels, is not yet clearly and

quantitatively explained.

In case of homogeneous particles but with fractal surfaces, a generalized Porod’s

law was derived by Bale and Schmidt (1984),asbyi(q)Eq

ds6

, where ds is the fractal

dimension of the surface. For a smooth surface, ds ¼ 2 and the classic Porods’ law is

recovered.

Interacting Identical Particles

The function P(q) also takes into accou nt interparticle interactions (Guinier and

Fournet, 1955). In the simplest case of spherical particles P(q) ¼ F(q)

G(q) and the

scattered intensity is expressed by

IðqÞ¼I

F F ðqÞ

GðqÞð10Þ

where F(q) is the struc ture factor of one particle; G(q) the interparticle interference

function that only de pends on particle–particle interactions.

If F(q) is known, G(q) can be obtained from the relation

GðqÞ¼IðqÞ=ðI

F F ðqÞ

Þð11Þ

Chapter 12.5: Small-Angle Scattering Techniques884

W(r), the inverse Fourier trans form of G(q), is the radial function distribution

giving the probability of ﬁndin g 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

could be app lied only if the external parts of the experimental scattering can

be assimilated into an extrapolatable law such as Porod’s law for largest q values,

or into 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.

C. Special Problems for Clay Minerals

For natural smectites the situation is not really simple, even when we deal with dilute

dispersions.

(1) Because of the layer structure of particles and their great anisotropy, the con-

dition of statistical isotropy for the dispersions is not so evident and needs to be

speciﬁed in each case. For locally oriented systems, Eqs. (10) and (11) have to be

resolved taking into account the particle orientation.

(2) Because of the small layer thickness, a continuum of scattering could exist be-

tween the 000 and the 001 reﬂections. The interlayer distances can be distributed

within a wide range of dimensions from about a nanomete r to several micro-

meters. Indeed, the limit between WAD and low-angle scattering is not wel l

deﬁned in some cases.

(3) The particles in the same sample have different thicknesses.

In case of interacting anisotropic particles, the correlati on 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.

D. 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 dis-

persions’’, in many publications. Following Mering (1949), we prefer using the no-

tation G(q), the ‘interference function’; to avoid confusion wi th the ‘‘particle

structure factor’’ F(q).

12.5.1. Principles 885