Scattering and Reflection Techniques

8/23/2019 Scattering and Reflection Techniques

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Chapter 12

Scattering and Reflection Techniques

Robert Richardson

12.1 Introduction

Scattering of radiation is an essential tool for the colloid scientist. This chapter aims tointroduce the use of scattering techniques in colloid science. For more detailed information,there are several books that are based on the science that can be done using particular typesof radiation [13]. There are also several books that describe the structure determinationfor crystals and glasses where the focus is the arrangement of atoms rather than the largerscale structures of interest to colloid scientists. Nevertheless, these are useful resources forinformationofrelevancetocolloidalsystems[4].Themostusefulbooksforacolloidscientistseeking to consolidate this introduction are those that describe the scattering methodsapplied to colloids and polymers [5].

Colloids are generally dispersions of particles with dimensions between 1 nm and 1 m,i.e. between 10 and 10 000 A (since 10 A = 1 nm). There are many experimental techniquesfor characterising particles in this size range and their approximate ranges of sensitivityare shown in Figure 12.1 and the strengths and weaknesses of the different techniques arediscussed below.

Electron microscopy (discussed in Chapter 13) can cover the whole range of sizes andgivesverydetailedresults.Theonlylimitationisthatingeneralthesampleisnotinitsnaturalequilibrium state. A vacuum is generally necessary so a colloidal dispersion would be frozenor its particles extracted and dried for ex situ electron microscopy. To overcome theselimitations, there are continuing developments in this field and environmental scanningelectron microscopy is capable of operating under a liquid vapour while cryo-transmissionelectron microscopy seeks to freeze in an equilibrium structure by very rapid cooling.

There are several techniques for particle size determination based on sedimentation.These give useful but rather limited information on particles in an equilibrium state. Anexample is the ultracentrifuge that can cover a wide range of particle size but experiments arequite demanding. Simple measurements of solution viscosity can also be used to estimateparticle size.

Light scattering and small angle X-ray and neutron scattering by the particles in a dis-persion are the main topic of this chapter. These scattering techniques cover the entire sizerange of interest and are capable of measuring the dimensions of particles in situ. Thesemethods may also give more detailed information on the internal structure of particles and

their interactions in a dispersion. It is this ability to determine the equilibrium structure


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Scattering and Reflection Techniques 229

Particle size in ngstrom

X-ray and neutron scattering

Light scattering and PCS

Ultracentrifuge

Electron microscope

N.B. 1 = 0.1 nm

10 100 10001 10000

Figure 12.1 Size ranges covered by different sizing techniques.

of colloids and surfaces in detail using a non-invasive probe that gives scattering methodsimportance in colloid and surface science.

12.2 The Principle of a Scattering Experiment

The basic scattering experiment is very simple. As shown in Figure 12.2, a monochromatic(i.e. single wavelength) beam is brought onto a sample.

The intensity of the scattered radiation is measured as a function of the scattering anglewhich we will label as but note there are other conventions. The important variable,however, is the scattering vector, Q, whose magnitude is related to the scattering angle andwavelength:

Q=4 sin /2

(12.1)

Sample

Detector

Incident beam

Wavelength,

Transmitted beam

Scattered beam

Scattering vector Q

q

Figure 12.2 Schematic diagram of a scattering experiment.


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230 Colloid Science: Principles, Methods and Applications

In principle, one could measure some scattering with two different wavelengths and plotthe intensity versus Q and get the same features of the curves at the same Q values. Thedistances probed in an experiment are inversely proportional to Q(i.e. distance 2 /Q).That means for large scale structures (e.g. 100 A or 10 nm) we need a small Q(i.e. Q 0.06

A1). To achieve small Qin a scattering experiment we need a suitable combination of largewavelength and small scattering angle. For light scattering, a wavelength that is comparableto or larger than the size of the scattering particles is generally used. For X-ray and neutronscattering a low scattering angle is generally used.

12.3 Radiation for Scattering Experiments

Table 12.1 summarises the properties of visible light, X-rays and neutrons for scatteringexperiments from colloidal dispersions.

Visible light has a wavelength of 400600 nm and is suited to particle sizes above 0.01malthough smaller particles can be detected. It is scattered by particles that have a differentrefractive index from that of the surrounding solvent.

X-rays (like visible light) are electromagnetic radiation. The useful wavelength range isshorter than about 0.2 nm because longer wavelengths tend to be adsorbed strongly. Smallangle X-ray scattering is suited to probing distances in the 1 nm1m range. For X-rays,it is the difference in mean electron density between a particle and the solvent that scattersthe radiation. They are therefore excellent for dispersions of high atomic number materialsin low atomic number solvents (e.g. metals or oxides in water). They are less good fordispersions of organic materials in aqueous solvents because the electron densities of the

two materials are similar.Neutrons are particles but have an associated wavelength. The useful wavelength range

is about 0.12 nm. Longer wavelength neutrons are difficult to produce with sufficientintensity. Small angle neutron scattering is suited to probing distances in the 1 nm1mrange. Neutrons do not generally interact with the electrons in atoms but they are scatteredby an interaction with the nuclei. The scattering is not dependent on the atomic number, sothey often offer a better optionthan X-rays for scattering from a dispersion of one material inanother of similar atomic number. It is also possible to get different scattering from the sameelement by using isotopes. Hydrogen/deuterium substitution is particularly useful in colloidscience. For instance, organics dispersed in heavy water (D2O) scatter neutrons strongly.

First we will look at the factors that determine the intensity in a classic light scatteringexperiment.

Table 12.1 Properties of radiation for scattering experiments

Radiation Visible light X-rays Neutrons

Type Electromagnetic wave Electromagnetic wave Particle/wave

Wavelength, 400600 nm 0.010.2 nm 0.012.0 nm

Distances probed >0.01 m nm to m nm to m

Scattered by Refractive index Electron density Nuclear scattering

variations of properties


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12.4 Light Scattering

Light scattering has been used for decades to determine particle sizes. The intensity oflight scattered by a suspension of small particles (i.e. particle diameter wavelength) isdetermined by

I(Q) = kc M(1+ cos2 ) (12.2)

where the four factors are the concentration, c, the molar mass of the particles, M, a col-lection of constants, k, and the polarisation factor (1 + cos2 ). The polarization factorresults from the physics of the scattering process and is usually removed by an experimentalcorrection. It contains very little information about the sample. This equation was derivedby Lord Rayleigh in 1871. An interesting sideline is that the factor k contains the inversefourth power of the light wavelength which means that short wavelengths are scatteredmuch more strongly than long ones. Hence the sky is blue and the sun appears yellowor red.

When light of wavelength 500 nm is incident on small particles (20 nm radius) theanisotropy of the scattering only results from the polarisation factor. Forward and backscat-tering intensities are therefore equal as shown in the polar plot of scattered intensity inFigure 12.3. If the constant, k, and concentration, c, are known the particle mass may bedetermined.

However, for larger particles (40 nm) the scattered intensity develops an asymmetry be-tween forward and backward scattering. This is a particle size effect and is the basis forthe determination of particle size from scattering. The origin of this effect is shown inFigure 12.4. If a particle radius is comparable with the wavelength of the radiation, there

will be a different path length from source to detector for rays scattered by different parts

1000

500

0

500

1000

2000 1000 0 1000 2000

Is

I0

40 nmparticles

20 nm (small)particles

q

Figure 12.3 Polar plot of forward and backscattering intensities.


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Out of phase

In phase

Particle

Q0

Detector

I(Q)

Figure 12.4 Scattering from a 40 nm particle.

of the particle. Consider two rays scattered from opposite sides of the particle. They havedifferent source to detector distances. This path difference means that the two rays arriveat the detector somewhat out of phase. They interfere destructively so there is less intensitydetected. In fact the degree of destructive interference tends to increase as the scatteringangle increases so the intensity tends to decrease with increasing angle or Q.

The rate at which the intensity changes with angle (or Q) depends on the size of theparticle. However, the size of a particle is a vague concept. More precisely, the intensitydepends on the radius of gyration of the particle. The radius of gyration, RG, depends onthe dimensions and shape of a particle. In general it may be calculated for any shape by an

integral over the volume of a single particle:

RG =

1

V

V

r2 dV (12.3)

where ris the distance from the centre of gravity and Vis the volume of the particle. Therelationship of RG to the dimensions of some simple shapes is given in Figure 12.5.

For large particles, there is an additional factor in the equation that governs the scatteredintensity:

I(Q) = kcM(1+ cos2 )(1 (Q RG)2/3+ ) (12.4)


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Circular ring Thin disc Sphere Thin rod

a

12

lR

G=

2

aR

G= aR

G=

5

3aR

G=

l

a

Figure 12.5 Some simple particle shapes and their radii of gyration.

This new factor depends on the radius of gyration and is approximated by (1 (Q RG)2/3).

Hence the dependence of the scattered intensity on scattering angle or Q may be used todetermine the radius of gyration of particles in a dilute dispersion. This theory (known asthe RayleighGansDebye theory) is applicable where particle diameter remains less thanthe wavelength. For larger particles the scattering pattern becomes extremely complex andMie theory applies. This is covered in more advanced texts such as [3].

12.5 Dynamic Light Scattering

There is another light scattering technique that operates in a different way. It relies on the

fact that particles in a dispersion are moving by diffusion. When a particle scatters a photonof light, there is a small exchange of energy between the photon and the particle. The particlemaygainenergyfromorloseenergytothephotonandthephotonsenergyshiftsaccordingly.This is the same process (Doppler shifting of frequency) that is used in radar speed traps.The frequency of the radar is changed by reflection from a moving vehicle.

The spectrum of light scattered is measured using the elegant technique of photon corre-lation spectroscopy (PCS). If the incident spectrum is monochromatic with frequency, 0,then the spectrum from a colloid generally has a Lorentzian shape and the width of the peakis determined by the diffusion coefficient of the particles, D, multiplied by Q2 as shown inFigure 12.6:

I() D Q2

( 0)2 + D Q2(12.5)

The hydrodynamic radius, a, may then be calculated from the diffusion coefficient usingthe StokesEinstein equation, provided the solvent viscosity, , is known (more details maybe found in [6]):

a=kT

6D(12.6)

Since the diffusioncoefficient is influenced by particleparticleinteractionsas well as viscousdrag, it is usually necessary to extrapolate the hydrodynamic radius to zero concentration.


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0

0.01

0.02

0.03

0.04

60 40 20 0 20 40 60

Frequency shift, 0 (rad s1)

Intensity

10Incident beamspectrum

Scattered beamspectrum

DQ2

(a)

a (nm)

Concentration

(b)

Figure 12.6 (a) Incident and scattered spectra for dynamic light scattering and (b) extrapolating thehydrodynamic radius to zero concentration.

It should be noted that the hydrodynamic radius is often greater than the radius determinedby static light scattering because it may include layers of bound solvent.

12.6 Small Angle Scattering

We now consider the techniques of small angle scattering of X-rays and neutrons (SAXSand SANS). A better term would be small Q scattering because the exact combination ofscattering angle and wavelength is not important.

Take a gold colloid for example. The scattering from such a sample is shown schematicallyin Figure 12.7. At small Q, the scattering is sensitive to the size and shape of the particles (aswe saw for light scattering) while at large Qthe scattering reflects the internal structure ofthe particles. For particles of a crystalline material the internal structure would give Braggpeaks. The Bragg peaks from colloidal particles are often broader than those from the samematerial in bulk form because of internal disorder and finite size effects on the diffraction.However it is the small Q scattering that is generally measured and interpreted in colloidscience.


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I(Q)

Q(1)0

Figure 12.7 Schematic of scattering from colloid of crystalline particles.

12.7 Sources of Radiation

X-rays may be generated in the laboratory using a sealed tube generator. A complete SAXSkit can be bought from several manufacturers. The intensity from a synchrotron source ismany orders of magnitude greater. Synchrotron radiation is produced tangentially when ahigh energy electron beam is deflected by a magnetic field. Facilities exist at the DaresburySRS, the ESRF at Grenoble and Diamond is under construction at the Rutherford AppletonLaboratory, Oxfordshire. More details may be found at the facility Web sites [79]. Neutronsin adequate quantities are only available from central facilities. In the United Kingdom wehave convenient access to the reactor at ILL, Grenoble [10] and the pulsed source at ISISat the Rutherford Appleton Laboratory [11]. A second source (Target Station 2) is under

construction at ISIS. Further details and links to neutron sources worldwide may be foundat the facility Web sites. The European Neutron Scattering Association gives informationfor accessing European Facilities and is actively involved in planning future sources [12].Neutrons produced by the fission process in a reactor have a high energy ( E 1 MeV) andby consequence of the de Broglie relationship,

(A) = 9.04/

E(meV) (12.7)

a very short wavelength ( 0.0003 A). This is useless for large scale structures. Fortunatelythe energy of neutrons may be moderated by passing them through a material to thermalise.The neutrons adopt the thermal energy of the moderator material. If its temperature is low,

the energy of the neutrons becomes low and so their wavelength becomes large. At the ILL,the cold moderator is liquid deuterium at 25 K and it turns out that a 25 K liquid deuteriummoderator gives a Maxwellian distribution of wavelengths peaked around 6 A. Such a coldsource is ideal for SANS. Figure 12.8 shows the moderator schematically and its wavelengthdistribution along with those from ambient and hot moderators. Pulsed neutron sourcesalso use moderators to generate neutrons in the useful wavelength range.

12.8 Small Angle Scattering Apparatus

The basic components of a small angle neutron scattering apparatus (SAXS is similar afterthe sample) are shown in Figure 12.9. The reactor core is surrounded by D 2O to reflect


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Cold source at ILL

25 K(cold)

2000 K(hot)

EC= 0.002 eV

lC= 6

Neutrons(cm

2A

1s

r

1)

()

EC

0.1 0.2 0.5 1.0 2.0 5.0 10 201011

1012

1013

1014

2

2

2

5

5

5

1 101 102 103

E(eV)

Figure 12.8 Schematic of a neutron moderator and the resultant wavelength distribution. Reprinted fromBee, M. (1988) Quasielastic Neutron Scattering. Institute of Physics Publishing Ltd, Bristol, with permission.

neutrons back and maximise flux. The cold source is used to maximise the useful fluxat 6 A. The velocity selector only passes a narrow(ish) band of neutron velocities. Sincevelocity, v, and wavelength, , are inversely proportional,

(A) = 3096/(v(ms1)) (12.8)

a narrow band of wavelengths is passed and so the radiation incident on the sample is nearlymonochromatic. The sample is typically 1 cm2 and 1 mm thick. A two dimensional positionsensitive detector collects scattered intensity. Software is often used to regroup the data asintensity versus Q. Evacuated tubes reduce background from air scattering.

Figure 12.9 Schematic diagram of a small angle neutron scattering apparatus.


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On a pulsed neutron source, such as ISIS, the velocity selector is unnecessary since thetime of flight or velocity of the neutrons can be measured. From this the speed and hencewavelengthcan be calculated. Since a white beam is used the large intensity losses associatedwith monochromatisation are avoided. Although intrinsically weaker, pulsed sources tend

to make efficient use of the neutrons.Figure12.10showstheNG3SANSapparatusatNIST.D11andD22,theSANSinstruments

at ILL, have similar layout. Other reactor based facilities also have similar instrumentation.The SANS apparatus at ISIS is LOQ and more instruments are to be built on the secondtarget station.

12.9 Scattering and Absorption by Atoms

The amplitude of the scattering by an atom is characterised by its scattering length, b. For

X-ray scattering, b is proportional to the atomic number, z. (Actually b= z ae whereae = 2.85 10

15 m, which is the scattering length for one electron.) For neutrons, thescattering length is a nuclear property and it varies irregularly with atomic number and alsodepends on isotope. Table 12.2 shows the scattering lengths and absorbtion cross sectionsfor some atoms. We see that hydrogen and deuterium have very different scattering lengths.Physically, a positive or negative scattering length is related to the phase shift of the waveduring scattering but we do not need to be concerned over the origin of the sign. We justuse it.

The absorption cross sections indicate how strongly an element absorbs the radiation. Ab-sorptionof X-raysincreasesverystronglywithatomicnumber,socellsforX-rayexperiments

are made of low-z materials. Neutrons tend to be absorbed less, so sample containment isnot a problem. There are useful exceptions such as cadmium which can be used for shieldingand beam definition apertures.

12.10 Scattering Length Density

In small angle experiments (i.e. low Q) the distances probed are generally much greaterthan inter-atomic spacings, so the technique is sensitive to changes in scattering lengthdensity over distances of up to 1000 A rather than the scattering by individual atoms.

Scattering length density, , is therefore a very useful concept because it can be used to

Table 12.2 Some scattering lengths and absorption cross sections [13]

Species bN1015m bX(atQ= 0)10

15m N(abs)1028m2 X(abs)10

28m2

H 3.74 2.85 0.28 0.73

D 6.67 2.85 0.0 0.73

C 6.65 17.1 0.003 92

O 5.83 22.8 0.0 306

Cd2+ 3.7 131.1 >103 9400


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Figure 12.10 The NG3 small angle neutron scattering apparatus at the NIST Centre for Neutron Research,Gaithersburg, MD, USA. It shows the shielding around the incident beam in the foreground, the sample

position and the large vacuum tank containing the detector in the background.


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Table 12.3 Scattering length densities of some materials

Material N (105 A2) X (10

5 A2)

H2O 0.05 0.94D2O 0.64 0.94

(CH2)n 0.06 0.65

(CD2)n 0.61 0.65

describe the scattering from a large volume (such as a particle) without having to specifythe position of every atom. The scattering length density of a material, , is calculated fromthe product of the number density of each atom type, Nj, and its scattering length, bj. Theproducts for different types of atom are then summed.

For neutrons:

N=

Njbj (12.9)

where bj is the scattering length of an atom for neutrons.For X-rays:

X = ae

Njzj (12.10)

where ae is the scattering length of an electron for X-rays (ae = 2.85 105 A) and zj is

the atomic number.

The scattering length densities for some materials are shown in Table 12.3. It is worthnoting that for neutrons the scattering length density depends on the isotope. This givesthe experimentalist an important tool because it is possible to vary the isotopic con-tent of a sample, for instance by switching from H2O to D2O as solvent, in order toemphasise some aspect of the scattering without changing the chemistry of the sampleappreciably.

12.11 Small Angle Scattering from a Dispersion

A simple picture of a dispersion is a number of identical particles suspended in a matrix (thesolvent) as shown in Figure 12.11. For a dilute dispersion the inter-particle distance will beable to take almost any value and so inter-particle interference effects are eliminated. Theobserved scattering intensity, I(Q), then depends only on the four factors in the followingequation:

I(Q) = (P M)2 NPV

2P P(Q) (12.11)

where (P M) is the contrast in scattering length density between a particle and thematrix, NP is the number of particles in the sample, VP is the volume of a particle and P(Q)is the particle form factor which is defined by the size and shape of the particle.


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rP rM

Figure 12.11 Simple picture of a dispersion of homogeneous particles in a matrix.

12.12 Form Factor for Spherical Particle

The form factor may be calculated by integration over the volume of a particle of any shape.For a spherical particle, the formula given below is obtained [14]:

P(Q) =3(sin QRSQRS cos QRS)

(QRS)3

2(12.12)

where RS is the radius of the sphere.This formula is plotted in Figure 12.12 for two values of the sphere radius. It demonstrates

several of the important general features of form factors (NB the log scale)

r At Q= 0 it has a value of 1.r Initially it decreases with increasing Q.r For smaller particles the function is more stretched out in Qand vice versa. In fact it is a

function ofQRS.r Maxima and minima appear at higher Q.

12.13 Determining Particle Size from SANS and SAXS

There are two complementary approaches to determining the particle dimensions in a dilutedispersion.

r By calculating the scattering from an assumed particle shape (e.g. a sphere) and varyingthe parameters (e.g. radius, number of particles) until a good agreement is found betweenmeasured data and model. If no agreement is found, then we assume another shape and


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8

6

4

2

0

0 0.05 0.1 0.15

Q(1)

log10

(P)

50

200

Figure 12.12 Form factor of spherical particle.

repeat the fitting process. This is usually done with a least squares fitting program. It is auseful method but the black-box approach may lead to incorrect conclusions regardingthe shape.

r The Guinier law relates the lowQslope of the scattering to the radius of gyration of the

particle and makes no assumptions regarding the particle shape.

12.14 Guinier Plots to Determine Radius of Gyration

It turns out that at lowQ(Q< 1/RG ) the scattering from a dilute dispersion is insensitive tothe shape of the particles. The intensity, I(Q), only depends on contrast, number of particles,particle volume and the radius of gyration as shown in this approximate equation, knownas Guiniers law [15]:

I(Q) = 2 NPV2P exp(Q

2 R2G/3) (12.13)

The radius of gyration was introduced in Section 12.4 for light scattering and is a veryconvenient quantity for characterising the size of a particle. Figure 12.13 shows a Guinierplot of natural log of intensity versus Q2. It has a slope ofR2G/3, So it is therefore possibleto determine RG without assuming a shape. Note that caution is required because theapproximation is only valid for Q< 1/RG.

In the next few sections we look at variations and extensions of this basic type of mea-surement. Much more detail is available in specialised texts [16, 17].


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Q2

3R

Slope2G

=

Intercept = ln(NP2VP)

ln I(Q)2

Figure 12.13 Showing the straight line behaviour on a Guinier plot at low Q.

12.15 Determination of Particle Shape

At Q> 1/RG the shape of the particle does have a major influence on the particle formfactor and hence the shape of the scattering from a dilute suspension. This can be seen mostclearly on a loglog plot of the particle form factor for a sphere, a thin (i.e. 5 A thick) discand a thin (i.e. 5 A radius) rod as shown in Figure 12.14. This shows a characteristic regionwith a slope of 1 for the rod and 2 for the discs. At Q 1/(the dimension of the particle),Porods law (discussed later) applies. The particle sizes in this example have been chosen tohave the same radius of gyration (100 A), so the form factor is the same for all three in theregion belowQ 1/RG where the scattering obeys the Guinier law.

12.16 Polydispersity

Polydispersity does not greatly affect the lowQslope but it tends to smear out the maximaand minima at higher Qas shown in Figure 12.15. This can be visualised by averaging formfactors with slightly different values of the sphere radius, RS. The trial and error (fitting)method can be used to deduce the degree of polydispersity.

12.17 Determination of Particle Size Distribution

There are other computer methods for extracting particle size distributions. For instancethere is a maximum entropy approach where the smoothest particle size distribution con-sistent with the scattering curve is determined [18].


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Sphere,radius = 129

Thin disc,radius = 141

Thin rod,

length = 346

4

3.5

3

2.5

2

1.5

1

0.5

0

3 2

log10 (Q(1))

log10

(P)

1 0

Figure 12.14 Form factors for different particle shapes with same radius of gyration.

2

8

6

4

0

0 0.05 0.1 0.15

Q(A1)

Log10

(p)

Monodisperse

10% polydisperse

Figure 12.15 Effect of polydispersity on form factor.


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0

20

40

60

80

100

120

0 10 20 30 40

Radius ()

Numberofparticles

0

0.1

0.2

0.3

0.00 0.05 0.10 0.15 0.20 0.25

Q(

1

)

ln(I(a.u.))

(a)

(b)

Figure 12.16 (a) SAXS and (b) particle size distribution determined using maximum entropy method.

Figure 12.16a shows an example of SAXS from partly hydrolysedzirconiumchloride which

forms polynuclear ions in solution.The maximum entropyparticle size distributionis shownin Figure 12.16b. The particles appear to have a radius of 10 A with a small proportion oflarger particles (possibly dimers).

12.18 Alignment of Anisotropic Particles

Fornon-sphericalparticlesitisadvantageoustoaligntheparticlesforasmallanglescatteringmeasurement. The two characteristic dimensions may then be determined by analysing thescattering in the two perpendicular directions on the detector. For instance, worm-likemicelles may be aligned by shearing in a couette as shown schematically in Figure 12.17.


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Aligned rodmicelles

Stator

Incidentbeam

Rotor

Figure 12.17 Schematic of scattering from shear aligned sample.

This is usually made of silica which is transparent to neutrons with a gap of 1 mm or lessbetween the external rotor and the internal stator. Nematic liquid crystals may be alignedby applying a magnetic field [19].

12.19 Concentrated Dispersions

For concentrated dispersions, rays scatteredfrom different particles will interfere. This inter-particle interference is accounted for by a term called the structure factor, S(Q), as shownin Figure 12.18:

I(Q) = (P M)2 NPV

2P P(Q)S(Q) (12.14)

For dilute dispersions, S(Q) = 1. For concentrated dispersions it is an oscillatory functionand it can be used to determine how the particles pack together [20]. Again least-squaresmodel fitting is used to determine parameters such as the closest distance of approach of two

particles (the hard sphere repulsion radius). For charged particles (e.g. micelles) the surfacecharge and screening length may be determined by model fitting [21]. One potential pitfallwhen using the Guinier law to determine radius of gyration is that the slope of the plot isonlyR2G/3 ifS(Q)= 1. For samples not very dilute, this may not be correct and analysisusing the Guinier law leads to an incorrect value of RG.

Figure 12.19 shows SAXS from overbased detergents dispersed in oil. These are used asan engine oil additive. They are calcium carbonate particles stabilised by surfactant. Sincethe surfactant and the oil have very similar electron densities which are different from thatof the CaCO3 core, the scattering is dominated by the more electron dense core. For theconcentrated dispersion, the peak position and shape may be analysed to give the hardsphere radius. On dilution, the peak disappears (S(Q) tends to 1) and a Guinier plot can beused to determine the core radius.


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Q

Q

S(Q)

Dilute

Concentrated

1

0

I

(Q

)

Figure 12.18 S(Q) and I(Q) for dilute and concentrated dispersions.

12.20 Contrast Variation using SANS

For SANS the use of contrast variation gives access to more detailed structural informationand it is particularly usefulfor composite particles.Consider such a particle which consists of

acoreandarelativelythincoatingasshowninFigure12.20.Therewillbethreecontributions:scattering from the core, scattering from the coating and scattering that comes from both.This can be modelled but it is complex and there is a tendency for the core scattering todominate (because core has more volume than coating) so the coating structure is difficultto extract.

To use contrast variation we first arrange for the solvent to have the same scattering lengthdensity as the coating. For an aqueous medium, this is done by choosing the correct ratioof H2O and D2O. The coating is now contrast matched and the only scattering is from thecore, so radius of gyration of the core, RG, can be determined by a Guinier plot.

Now we arrange for the solvent to have the same scattering length density as the core. Thecore is now contrast matched and the only scattering is from the coating. The thickness ofthe coating RT can now be determined using a version of Guiniers law that applies to the


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(a)

(b)

0

0.2

0.4

0.6

0 0.1 0.2 0.3 0.4 0.5

Q(1)

I(a.u.)

3

4

5

6

0.00 0.05 0.10 0.15 0.20

Q2 (2)

ln(I(a.u.)

)

Figure 12.19 SAXS from concentrated and dilute calcium carbonate dispersions.

scattering from anisotropic, plate-like objects at Q> RG [22]

I(Q) 1

Q2exp(Q2 R2T) (12.15)

so the coating thickness may be determined.

12.21 High Q Limit: Porod Law

We now consider the form of the scattering at Qwell above the Guinier region. Since thedistance probed is inversely proportional to Q, very high Qmeans short distances and the


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(c)

(a) (b)

M

2

1

M=

2

rM

=r1

Figure 12.20 Contrast variations from a composite particle.

only scattering comes from the step in scattering length density at the surface of the particlesin a dispersion as shown schematically in Figure 12.21.

There are no inter-particle effects because the distances probed are again very muchshorter than the particle separations. This high Qscattering decays as the fourth power of

Scattering is sensitive toregion of dimension Q1

Q1

lnQ

ln(I) Guinierregion

Slope~4

Porodregion

Figure 12.21 Distance probed by high Qand the corresponding Porod region in the scattering curve.


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2=S

D

3Rarea

3S

DArea R2 Area R3

Smooth surface Rough porous surface

R

R

Figure 12.22 Fractal surfaces.

Qand its strength depends only on the contrast, 2, and the amount of surface area Sinthe sample:

I(Q) 2S2 Q4 (12.16)

This is known as Porods law [23]. The scattering intensity can therefore be used to measuresurface area in powders, dispersions, etc.

Porods law is modified if the surfaces are not smooth. The nature of a surface can becharacterised by its surface fractal dimension, DS.Thisconceptcanbeunderstoodasfollows.Consider a sphere of radius, R, on a smooth surface. As the radius of the sphere increases,

the area of surface in the sphere increases as the second power of the radius. Hence for asmooth surface, DS = 2. For a very rough, porous surface the surface area inside the spherewill increase as the third power of the radius. Hence for a rough surface, DS = 3. In generalthe surface fractal dimension will lie between these extremes: 2 < DS < 3. Figure 12.22shows the two extremes schematically.

For a fractal surface, Porods law is extended by changing the power from 4 to (6 DS)and so the fractal dimension may be extracted from the slope of a loglog plot of the high

Q scattering as indicated in Figure 12.22. Note that for a smooth surface, Porods law isrecovered.

ln(I(Q)) (6 DS)ln(Q) (12.17)

Figure 12.23 shows the high Qscattering from a sample of porous glass (Vycor). When itis dry, the slope of the loglog graph is 3.3, indicating a surface fractal dimension of 2.7(i.e. quite rough). On exposing it to vapour (a halogenated solvent with similar scatteringlength density to glass) the slope is 3.9, indicating a surface fractal dimension of 2.1 (i.e.nearly perfectly smooth). The conclusion is that the pores have been filled in by capillarycondensed vapour of the same scattering length density as the glass, so the surface appearssmooth.

The concept of fractals has many applications. For instance adsorbed polymer layersand aggregates of particles may be characterised by fractal dimensions. A more detaileddiscussion of scattering from surface and mass fractals may be found elsewhere [2426].


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Dry porous glass, slope = 3.3DS= 2.7

Porous glass + vapour,slope = 3.9; DS= 2.1

0.05

0.15

0.25

0.35

0.45

0.24 0.22 0.2 0.18 0.16

ln(Q(1))

ln(I(a.u.)

)

Figure 12.23 SAXS from porous glass in dry state and exposed to vapour.

12.22 Introduction to X-ray and Neutron Reflection

The reflectivity technique is a recently developed method for studying the structure ofmacroscopic surfaces [27]. We have seen in Section 12.20 that it is possible to characterise

surfaces of particles using small angle scattering by contrast matching the cores of theparticles to the solvent.However reflectionfrommacroscopicsurfaces has several advantagesas compared to studying surfaces of particles in dispersions. These include:

r Reflection is not restricted to stable dispersions.r Reflection is not restricted to core contrast matched conditions.r It is more precise because surface contribution to the scattering is separated out experi-

mentally as a specular reflection.r It is relatively simple to calculate reflectivity from a smooth surface exactly.

The big disadvantage is that at present several square centimetres of flat surface are required.This is simple for the liquidair interface but more difficult for solidair, solidliquid andliquidliquid interfaces.

12.23 Reflection Experiment

The reflectivity method, shown schematically in Figure 12.24, is very simple in principle.A well-collimated monochromatic beam of X-rays or neutrons is brought onto the surfaceand the intensity of the reflected beam is measured. The angle of incidence is scanned tovaryQ. On a pulsed source, the Qvariation can be done by measuring the time of flight ata fixed angle so that Q is varied through the range of wavelengths, . The reflectivity as afunction ofQis determined and interpreted in terms of the surface structure. It is possible


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/2 /2z

Figure 12.24 Principle of reflection experiment.

to purchase an X-ray reflectometer, and X-ray and neutron reflectometers are also availableat the central facilities mentioned above.

The reflectivity from a surface R(Q) may be calculated exactly using a method originally

developed for the optics of multi-layers [28]. However, for the purposes of understandingreflectivity results, the kinematic approximation is very useful [29].

In this approximation there are two factors. The first factor is the reflectivity that wouldbe observed from an ideally smooth sharp interface where the change in scattering lengthdensity between the two media is . It is a Q4 decay. The second factor results from anysurface structure such as an adsorbed layer or diffuseness of the interface. It is the Fouriertransform squared of the scattering length density gradient perpendicular to the surface (i.e.the zdirection):

R(Q) =1622

Q4 1

(z)

z

ei Qzdz2

=162

Q4

(z)

z

ei Qzdz2

(12.18)

12.24 A Simple Example of a Reflection Measurement

As an example of neutron reflectivity, consider a monolayer of a deuterated surfactantadsorbed at the surface of water as shown schematically in Figure 12.25. The water can bemade invisible to neutrons by using 8% by volume D2O so that its scattering length densityis zero and it contrast matches to air.

The scattering length density profile is then a simple block shape and it can be shown

from standard Fourier transform results [30] that the surface structure factor has a cosineform:

R(Q) 162

Q42F2(1 cos Qd) (12.19)

where F is the scattering length density of the surfactant film and d is its thickness.If the reflectivity from such a system is plotted as RQ4, the rapid decay is removed from

the data. The position of the first maximum of the cosine is easily measured and hence thelayer thickness, d, may be determined:

d= /QMAX (12.20)


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z

Air

Null water

D-surfactant

8% D2O + 92% H2O

z

r d

rF

Figure 12.25 Deuterated surfactant adsorbed at the surface of null water and the corresponding scat-tering length density profile.

The amplitude of the cosine oscillations is governed by the scattering length density of thefilm which can therefore be determined:

F= (R Q4)MAX

642 (12.21)Since the scattering length density depends on the total scattering length of a surfactantmolecule,

molecule b,andthevolumeitoccupies,theareapermolecule, A,maybecalculated

from the scattering length density of the film:

A=

molecule

b

Fd(12.22)

where the summation is over the scattering lengths of all the atoms in one molecule.Figure 12.26 shows the reflectivity of d-behenic acid spread on water plotted as R Q4.The

data were taken using the CRISP reflectometer at the ISIS neutron source [31].The position of the maximum indicates that the layer is 24 A thick and since the totalscattering length of the molecule is 441 105 A1, the area per molecule is determined as23 A2. This simple example shows how the two most important characteristics of the sur-factant layer may be determined. The method may be extended to cope with more complexinterfaces and to determine more detailed structural information.

12.25 Conclusion

We have discussed how light, X-ray and particularly neutron scattering can give usefulinformation on the structure of colloids and surfaces. Although the techniques do not give


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Qmax=p/d

1

0

1

2

3

4

5

0.0 0.1 0.2 0.3 0.4

Q(1)

108

RQ4

(

4)

Figure 12.26 Neutron reflectivity of d-behenic acid spread on water.

real space images, the interpretation of scattering and reflection data in terms of structureis reasonably direct. The data are generally obtained from the samples in an equilibriumstate and so are free from artefacts introduced by sample preparation or by the invasivenature of the probe. Hence the methods outlined are widely used in colloid and surface

science. Potential users of the methods are encouraged to consult some of the books inthe References section where more detailed introductions may be found. The Web pagesof the central facilities are also a useful source of further information, particularly aboutinstrumentation.

References

1. Kostorz, G. (ed.) (1979) Neutron scattering, Treatise on Materials Science and Technology, vol.15.

Academic Press, New York.

2. Als-Nielsen, J. and McMorrow, D. (2001) Elements of Modern X-ray Physics. Wiley, New York.3. Kerker, M. (1969) The Scattering of Light and other Electromagnetic Radiation. Academic Press,

New York.

4. Guinier, A. (1994) X-ray Diffraction in Crystals, Imperfect Crystals and Amorphous Bodies. Dover,

New York.

5. Lindner, P. and Zemb, Th. (eds) (2002) Neutron, X-ray and Light Scattering Methods Applied to

Soft Condensed Matter. North-Holland, Amesterdam.

6. Pusey, P.N. and Taugh, R.J.A. (1982) In R. Pecora (ed.), Dynamic Light Scattering and Velocimetry:

Applications of PCS. Plenum, New York.

7. http://www.srs.ac.uk/srs/

8. http://www.esrf.fr/

9. http://www.diamond.ac.uk/10. http://www.ill.fr/


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11. http://www.isis.rl.ac.uk/

12. http://neutron.neutron-eu.net/

13. http://www.ncnr.nist.gov/resources/n-lengths/

14. Lord Rayleigh, (1911) Proc. R. Soc. (London) A, 84, 25.

15. Guinier, A. (1939) Ann. Phys., 12, 161.16. Feigin,L.A.andSvergun,D.I.(1987) Structure Analysis by Small Angle X-rayand Neutron Scattering.

Plenum, New York.

17. Glatter, O. and Kratky, O. (1982) Small Angle X-ray Scattering. Academic Press, New York.

18. Potton, J.A. Daniell, G.J. and Rainford, B.D. (1986) Inst. Phys. Conf. Ser. No. 81, Institute of Physics

Publishing, Bristol, Chapter 3, p. 81.

19. Hayter, J.B. and Penfold, J. (1984) J. Phys. Chem., 88, 4589.

20. Ottewill, R.H. (1982) In J.W. Goodwin (ed.), Colloidal Dispersions. R.S. Chem. p. 143.

21. Hayter, J.B. and Penfold, J. (1983) Colloid. Polym. Sci., 261, 1022.

22. Kratky, O. and Porod, G. (1948) Acta. Phys. Austriaca, 2, 133.

23. Porod, G. (1951) Kolloid-Z. 124, 83.

24. Legrand, A.P. (ed.) (1998) Surface Properties of Silica. Wiley, New York.25. Bale, H.D. and Schmidt, P.W. (1984) Phys. Rev. Lett., 53, 596.

26. Allen, A. and Schofield, P. (1986) Inst. Phys. Conf. Ser., No 81, Institute of Physics Publishing,

Bristol, Chapter 3.

27. Bucknall, D.G. (1999) In R.A. Pethick, and J.V. Dawkins (eds), Modern Techniques for Polymer

Characterisation. Wiley, New York.

28. Penfold, J. and Thomas, R.K. (1990) J. Phys. Condens. Matter, 2, 1369.

29. Als-Nielsen, J. (1885) Z. Phys. B, 61, 411.

30. Champeney, D.C. (1973) Fourier Transforms and their Physical Applications. Academic Press, New

York.

31. Grundy, M.J., Richardson, R.M., Roser, S.J., Penfold, J. and Ward, R.C. (1988) Thin Solid Films,

159, 43.

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