Ultrasmall-angle X-ray scattering studies of heterogeneous systems using synchrotron radiation techniques

Nuclear Instruments and Methods in Physics Research B47 (1990) 283-290 North-Holland

ULTRASMALL-ANGLE X-RAY SCATTERING STUDIES USING SYNCHROTRON RADIATION TECHNIQUES

283

OF HETEROGENEOUS

SYSTEMS

Ashley N. NORTH and John C. DORE Physics Laboratory,

University of Kent at CanterbuT,

Canterbury, Kent, CT2 7NR, England

Alan R. MACKIE and Andrew M. HOWE AFRC Institute of Food Research Notwich, Colney Lane, Norwich, Norfolk, NR4 7VA, England

J.

HARRIES

Daresbury Laboratory, Daresbuty,

Warrington,

Cheshire, WA4 4AD, England

Received 25 May 1989 and in revised form 16 January 1990

Ultrasrn~l”~~e X-ray scattering (USAXS) studies of heterogeneous systems are reported. The twin-crystal diffractometer developed by Bonse and Hart was used in combination with the high-intensity X-ray beams from a synchrotron radiation source. With this experimental configuration, scattering profiles were obtained over a Q-range of 5~10~~
1. Introduction Small-angle scattering techniques using X-rays (SAXS) and neutrons (SANS) have, in recent years, developed into one of the most important methods for studying the structural properties of materials on a length scale of 1 nm to 0.01 pm [1,2]. For a broad range of interests, from basic physics to applied biology, many aspects of detailed information lie in the larger dimension region of 0.1-1.0 pm. Light scattering techniques [3] are widely used to study systems in the range 0.1-1.0 pm. However, light scattering is often of limited use due to the systems being highly turbid (and/or highly absorbing) in the visible region of the electromagnetic spectrum. X-rays and neutrons interact weakly with matter and are consequently a useful probe of systems that are optically turbid and highly absorbing. In order to use X-rays and neutrons to observe structure on length scales in the range 0.01-1.0 pm it is necessary to measure the scattered intensity at much smaller angles than conventional SAXS and SANS instruments can achieve. Bonse and Hart [4] proposed a twin crystaf diffractometer system for X-rays which allowed scattering measurements to be made at angles as low as 20”. By using multiple reflections of the incident white beam, 0168-583X/90/$03.50

from the faces of a channel cut in each crystal, a monochromatic and well collimated beam is produced, with high angular resolution. Instruments similar in design have been developed for both laboratory based X-ray sources (USAXS) [5] and neutron sources (USANS) [6]. Both systems have suffered from a very low intensity incident beam at the sample and consequently low scattered intensities. This has imposed restrictions on the experimental technique, such that only strongly scattering samples could realistically be studied. In small angle scattering, high angular resolution and a truly monochromatic beam are not always required and may be sacrificed in order to increase the available scattered intensity. In the case of USAXS lower angular resolution and broader wavelength beams may be produced using curved crystals and fewer reflections 171. However, the combination of high intensity synchrotron radiation and the Bonse-Hart system greatly enhances the range of systems that may be studied without using such techniques. The USAXS system at the Daresbury Laboratory Synchrotron Radiation Source (SRS) is such a combination and preliminary work on this instrument has focussed on the use of USAXS in the field of biological sciences [8]. This paper reports exploratory studies for a variety of samples from the area of chemical physics. A study of a dilute latex suspension of

0 1990 - Elsevier Science Publishers B.V. (North-Holland)

A.N. North et al. / Ultrasmall-angle X-ray scattering

284

monodisperse spherical particles (diameter 210 nm) gives an estimate of the scope and problems involved in the technique. Work on a series of ~croemulsion samples, in which the particles are believed to cluster producing characteristic lengths > 10 nm, is compared with results from SAXS and dynamic light scattering. Measurements on oil-bearing shales (porous solids) are also discussed. Detailed studies of these shales have already been made using SAXS and SANS and show interesting features up to the lower Q-value limit available, the use of USAXS enables the extent of these features to be gauged.

2. Theory 2.1. Scattering

from particles

Only a brief summary of the theoretical relationships will be given since they have been widely presented in various review articles [9,10]. The small angle scattering profile from an inhomogeneous system may be written as

Z(Q>=f'(Q)s(Q)t

0)

where Q, the scattering vector is defined by the experimental conditions as 471 Q = -sinx

8 2’

= 9( Ap)‘Y2

sin QR - QR cos QR

(QN3

=(Ap)2exp(-q),

1 '

(3)

(4)

min Q-O

where R, is the Guinier radius which for spherical particles is R = R&.

(5)

In systems in which there is some degree of ordering of the particles there is a contribution to the scattering profile from the interparticle structure factor, s(Q), which results from interparticle interference terms and is given by the transform of the centre-centre pair correlation function g(r), i.e. S(Q)

particles

Under certain conditions there may be a strong attractive interaction which leads to the formation of clusters. In many systems the transient and reversible formation of large aggregates occurs as a precursor to phase separation and this has been treated within the formalism of critical scattering [ll]. More recently an alternative form of the structure factor for aggregates has been proposed, based on fractal formalism [12] in which the aggregate is treated as exhibiting self-similarity. Teixeira gives the structure factor to be [13] 1 ‘(‘)=l+

(QR)"

Xsin[(I)

= 1 + 4vprp,im[g(r)

- l]rzs$$dr

Dr( D - 1) (1+1jQ252)(D-1/2)

- 1) tan-‘(Q[)j,

(7)

where r(x) is the gamma function, D is the mass fractal dimensionality of the cluster, R is the individual particle radius and 6 is the correlation length of the aggregate. If the correlation length is very large (4 -+ co) then

(8)

If Q + 0 then for fractal systems with an upper cut-off length scale, eq. (7) reduces to a generalised Guinier approximation and the cluster size may be derived from a Guinier plot [14] providing Qc < 1. 2.3. Porous solids

2

where Y is the volume of the particle. At low Q values R(Q) may be rewritten to give the Guinier expression:

P(Q)

2.2. Aggregating

s(Q)aQ-".

where h is the wavelength of the incident beam and 8 is the angle of scatter. The particle form factor P(Q) is a function of particle shape and orientation as well as the scattering density contrast between the particles and the homogeneous phase ( Ap). For a homogeneous sphere of radius R, P(Q) may be written as [9] P(Q)

where pn is the particle number density. In the case of a dilute system of noninteracting particles the structure factor will approximate to unity for all Q-values.

(6)

Porous solids consist of a network of pores and connecting channels in a rigid framework of material. The form of the scattering from networks of pores which are geometrically similar to aggregates of particles is identical, according to the principle of reciprocity, to that of the aggregates [15]. Consequently, it is possible to envisage the pore networks as fractal systems exhibiting self-similarity [HI. The dimensions of the network are of the order of microns compared to angstroms for the pore dimensions, and as such eq. (8) is valid. Many porous systems have pores with highly irregular surfaces and these may also be considered from a fractal viewpoint. The surface fractal dimensionality, D,, can be shown to give a structure factor of the form

WI S(Q)

_

Q-(6-%)

for Q > l/R. It is possible for porous media to exhibit both volume and surface fractal dimensionality [17].

A.N. North et al. / Ultrasmall-angle

Since the two fractal regimes exist on different length scales, fractal surfaces at lengths less than the pore dimension (R) and volume fractals on length scales greater than R, then the scattering from the two regimes are observed at different Q-ranges and are often distinct.

3.1. The USAXS station The USAXS diffractometer uses two channel cut silicon monoliths, in an arrangement proposed by Bonse and Hart [4]. Repeated Bragg diffraction, using four reflections of the primary beam from the (111) plane on opposite sides of the channel is used. Multiple reflection narrows the intrinsic reflection curves and more importantly suppresses the long-reaching tails of the Bragg reflection curve (if the single reflection curve is R(8) then P reflections would produce a curve of the form RP(t?) for a large perfect crystal). Consequently the beam divergence is reduced every time it is reflected from the channel wall, leading to a monochromatic beam with a divergence of only a few seconds of arc P81. The first crystal of the USAXS instrument (fig. 1) collimates the incident beam and the second crystal (which can be rotated in the plane of the paper) is used to analyse the radiation scattered by the sample. The monochromatic beam emerges from the first crystal and is incident upon the sample situated at the centre of rotation of the second crystal. It is clear that if the second crystal is rotated from its position parallel to the first crystal only the radiation scattered by the sample to the same angle can pass through the second crystal to the detector. An ionisation chamber positioned after the sample allows the corrections for incident beam intensity and sample absorption to be made. A solid-state detector is used to collect radiation passed by the analyser crystal. The energy resolution of the detector (- 10%) is used to eliminate contributions from harmonics and other reflections passed by the crystals. The collimation achieved with a channel cut crystal, often referred to as angular collimation, produces a

X-ray scattering

primary beam with very small angular divergence and large spatial dimensions (4 mm by 10 mm). All other techniques require narrow entrance slits, which in turn, results in small spatial dimensions, to achieve the same magnitude of beam divergence. A quantitative experimental comparison of the twin crystal system and slit collimation system was presented by Kratky and Leopold [19]. The USAXS system produces a constant intensity and fixed resolution whilst the slit col~ma~on system yields an increasing intensity for decreasing resolution so that for small length scale systems (< 50 nm) conventional SAXS techniques are usually preferred to USAXS. The USAXS technique is used, so far, in only a very few special problems. Collimation in the USAXS camera is in one direction only (perpendicular to the (111) planes in the crystal) and therefore corresponds to the standard “slit geometry” arrangement used with a standard laboratory X-ray source. The camera is thus ideally suited to long period one-dimensional oriented structures [20] such as biological fibres 181.The radial intensity dist~bution of radially symmetric diffraction patterns, such as those obtained with solution scattering (and all the samples of interest in this paper) may be obtained by the use of desmearing techniques. 3.2. Data reduction Smearing of the collected pattern arises from two sources. A slit-width smearing due to the lack of collimation parallel to the crystal planes and a slit-height smearing due to the instrument resolution in the direction of collimation, Theoretically, instrument resolution in the direction of collimation is appro~ately 3” of arc. It is possible to measure the resolution experimentally by measuring the intensity transmitted by the two crystals as the second crystal is rotated (the measured pattern is often refered to as the rocking curve of the two crystals) which gives a resolution of 6”. This is not insignificant at the very low angles at which measurements are being made. However, in order to simplify data reduction in these exploratory measurements the rocking curve will be ignored. The angular range of measurement of the USAXS instrument is small compared to that subtended by the “slit dimensions” and corrections for the slit-width smearing may be applied using an infinite-slit geometry. The method of correction used for this work is based on that proposed by Vonk in 1971 [21]. The slit smeared intensity I, at a Q-vector Q from the primary beam is given by the relation [22]

A(Q) = j:_ WMQ’) Fig. 1. Schematic diagram of the Daresbury USAXS instru-

285

dy,

(9)

where Z(Q’) is the true ‘epinhole” intensity curve and D(y) represents the intensity distribution of the primary

beam in the direction perpendicular This may be written as

to the collimation.

This relation may be approximated relation

by the summation 0.50.

01) 0.70.

where i = serial -lumber of obse~ation 1, and j = serial number of the interval AQ’ along the desmeared intensity curve I. If M observations were made and AQ = AQ’ eq. (11) may be regarded as a system of M linear equations with m unknowns

(12)

0.60.

*,50

0.40 i

0.00

0.10

0.20

0.30

0.40

0.60

0.50

l~

where uj = D(Qj> AQ which is readily solved for lP In the following treatment D
4. Resuhs 4. I. Latex spheres

A suspension of chlorinated latex particles was supplied by ICI (Runcorn) plc. A dilute suspension (5% volume fraction) was produced such that the particles were noninteracting allowing eq. (1) to be used with S(Q) = 1 over the Q-range studied. Chlorine is a strong absorber of X-rays at the wavelength used in the experiment, however, the increase in the contrast between the particles and the aqueous medium obtained by chlorination of the particles is such that thin samples are strong scatterers and data collection times are silently reduced (by approximately a factor of 5) when compared to a similar system of polystyrene spheres. The latex was first studied on a SAXS station (2.1) of the

0.70

d-1

Fig. 2. Small-angle scattering from chlorinated ICI latex of radius 105 nm (A) with a theoretical scattering curve for particles of the same dimension ( -). Rue to smearing and polydispersity effects, the minima in the intensity distribution are nonzero.

SRS. Over twenty maxima were observed in the scattering profile from which the particles were calculated to have a radius of 110 nm. The large number of oscillations in the scattering profile indicate that the particles in the latex are highly monodisperse. Fig. 2 shows the SAXS data and a theoretical calculation (including smearing and polydispersity effects) of the scattering distribution which indicates that the first two orders of the scattering are “lost” behind the beam stop of the SAXS instrument. One of the major areas of interest in colloid science is the ~t~action potential between particles, To measure the structure factor resulting from this potential it is necessary to measure the scattering profile over the first order of the scattering pattern. fn order to do tbis for particles of radius > 40 nm, it is necessary to use the USAXS instrument. The USAXS instrument collects data point by point and is thus relatively slow compared to SAXS data collection (for which position sensitive detectors are widely used}. The data presented in fig. 3 were collected in three separate runs of approximately 45 min each. Also shown in the same figure is the measured background. Fig. 4 shows the data after subtraction of the backwood and wrrection for slit-width smearing. A Guinier plot of the data reveais a linear region corresponding to a radius of 107 nm which is in good agreement with SAXS measur~ents. The desmeared data show significant ripples at high Q-values, resulting from the errors in the original data propagating through the desmearing process. As is to be

281

A.N. North et al. / Ultrasmall-angle X-ray scattering

distribution, especially in the regions where minima are observed. The lack of deep minima is readily explained by introducing a small polydispersity of particle size into the theoretical calculation. It may also prove necessary in the case of larger particles to take into account the smearing effect due to the finite width of the rocking curve (slit-height) of the two crystals. 4.2. Microemulsions

Fig. 3. Raw USAXS spectra (A) for the chIorinated ICI latex, including background scatterer (B) which increases rapidly at very low Q-values. expected, an attempt to solve all the linear equations associated with the desmearing routine (eq. (12)) results in ripples of greater amplitude than those obtained using the Vonk method. A comparison of the desmeared data with a theoretical scattering distribution (fig. 4) shows that the desmeared data deviates significantly from the theoretical

Microemulsions are thermodynamically stable dispersions of water in oil which are stabilised by the presence of a surfactant coat between the water and oil domains [23]. The phase diagrams of such three-component systems are highly complex [24] and the interpartitle behaviour close to a phase transition is of great interest [25]. Such systems have been widely studied using SAXS [25] and SANS [ll], where enhanced scattering is found at low Q-values [ll]. This is believed to be due to the clustering of the droplets prior to phase transition [ll]. An analysis of SAXS data from such a system has been recently reported using a fractal formalism [25]. The correlation length 4 of the cluster increases as the temperature T approaches the critical temperature T,. As the cluster size increases the scattering profile narrows, that is a larger number of photons are scattered at lower Q-values. USAXS measurements were made on a series of such microemulsions in order to study the finite sized clusters in greater detail than was possible using SAXS and photon correlation spectroscopy (PCS). The USAXS data from a variety of microemulsions was desmeared and the background scattering (the scattering from a cell containing only heptane) was subtracted. The corrected data are shown in fig. 5. As is shown in section 2.1 the Guinier radius of the cluster may be extracted provided Q[ =K 1. Guinier plots of the data have been made and the resulting radii are given in table 1 along with the cluster correlation lengths from SAXS and PCS studies [25]. The results from the three experiments are seen to be in satisfactory agreement. The microemulsion work is an ongoing project and measurements much closer to the phase transition will Table 1 A comparison of correlation lengths for clustering in a microemulsion using different experimental techniques

Fig. 4. Scattering from ICI latex desmeared and background subtracted. Theoretical curve shows the expected scattering from a 210 MI diameter particle. The rapid oscillations in the corrected data are a result of the desmearingprocess.

Volume fraction

Correlation length (A) SAXS [25]

PCS [25]

USAXS

0.03 0.07

91 156

104 160

95 147

0.11 0.16 0.25

175 135. 62

203 136 46

161 127 63

AN. Nortk et al. / ~ltr~rna~l-Angie

X-ray scattering

7.00

105I(Q) * I

6.00

* * 5.00

*

*

4.00

3.00

2.00

1.00

0.00

0.03

0.06

0.09

0.12

0.15

10 QIB;’ Fig. 5. Corrected USAXS data from microemulsions.

-3.50

-2.50

-3.00

-2.00

-1.50

Q Fig. 6. log Z(Q) vs log Q showing the linear dependence of the scattering from B&ken shale in the Q-range covered by the USAXS instrument. log

depend on the use of USAXS and PCS, the correlation length being outside the range measurable by SAXS. 4.3. Porous solkis The structure of porous systems is of great importance due’to the prominent role of manufactured porous media as catalytic sites. In addition, natural porous materials are of interest, e.g. a wide range of shales contain natural oil and gas deposits. The recovery of oil from the shales and the use to which catalytic supports may be put will depend on the shape, size and surface characteristics of the pore. In addition the nature and structure of the interconnecting network are of importance. A wide range of meas~ements have already been made using light scattering, SAXS and SANS on silica gels and shales [l&27,28]. In particular it has been shown that many shales have pores with fractal-type surfaces whilst the aggregation process of the gel formation produces volume-fractal-like structures. In any physical system the validity of the fractal interpretation is limited to a finite size regime. In the case of fractal surfaces the lower limit may be on the atomic scale whilst the upper limit is the distance over which the surface exists, such that in the case of a sphere the maximum extent would depend on the radius of the sphere. On a scale larger than the dimensions of the sphere no sofas-frac~-tie dependency should be observed in the scattering profile. Bakken shale is an oil-bearing shale in which the pore surface is believed to be fractal [27]. The average size of a pore in this shale is approximately 12 nm. SAXS and SANS studies of this system have always shown a linear dependence of log Z(Q) on log Q for the

range of Q-values over which the m~surements were made. Studies using the USAXS technique to probe length scales much greater than 10 nm are therefore of interest. USAXS data, corrected for background and desmearing, from the Bakken shale are presented (in fig. 6) in the form of a log Z(Q)-log Q plot. As in the SAXS and SANS experiments, a linear region is observed over the full extent of the measuring range (3.5 x low4 < Q ( 1.4 X 10F2). The slope of this linear region is, within experimental error, the same as that taken from SAXS data by Mildner et al (table 2) [17] (this SAXS work was repeated on the sample used for the USAXS experiment in order to confirm the result 1281). Table 2 also shows similar results for a second shale sample, Utah Clay. From these results we surmise that the fractal surface extends over larger dimensions than was expected from the apparent pore size. This may be due to a large number of large pores within the system or the complexity of the pore network. An alternative explanations is that the linear dependence arises not from fractal surfaces but a very wide pore size distribution, as was proposed for lignite coals by Bale and Schmidt [I6].

Table 2 Sample

Fractal dimensional&y USAXS

SAXS

Baken shale Utah clay

2.27 2.56

2.35 2.55

A.N. North et al. / Ultrasmall-angle

5. Discussion

The results of the experiments presented in the previous sections have clearly illustrated the usefulness of the USAXS technique using synchrotron radiation in the area of chemical physics. The work on the chlorinated latex sample clearly demonstrated the effectiveness of the technique for the study of colloidal systems with large particle dimensions. The microemulsion and porous solid experiments produced novel results which could not be achieved with coventional SAXS or SANS. Dynamic light scattering gives similar results for the almost-transparent microemulsion systems but is of little or no use in investigating the structure of shales. Current approximations, (considering only the “slit effect” for desmearing corrections) do not give a full representation of the data. New work must take into account the convolution of the scattering curve with the rocking curve of the two crystals. In 1965, Bonse and Hart [29] suggested the used of a third crystal in the USAXS instrument optics. The third crystal would be used for collimation in the plane perpendicular to the collimation of the current system, producing a point source instrument. Until recently a third crystal would have meant an unacceptably large decrease in the available radiation. However, the High Brightness Lattice, recently installed at the SRS, Daresbury produces a beam which is collimated to a greater extent (smaller in dimensions but brighter) and so less photons would be lost due the third crystal. Such a development would greatly reduce the smearing effect observed in measurements on a two system and so lead to a reduction in the increasingly large errors produced in the desmearing of the data. SAXS and SANS are often seen as complementary techniques and in the same way so must USAXS and USANS [9]. With USANS the intensity of the incident beam at the sample is very low compared to a USAXS instrument on a synchrotron radiation source. USANS is therefore a technique that should only be considered if the sample is strongly absorbing for X-rays or if the desired information may only be obtained by isotopic substitution. Isotopic substitution is a widely used technique in neutron scattering. Different isotopes of the same element have different scatttering length densities and so the scattering contrast of a system may be changed without altering the chemical composition of the sample. No simple technique for changing the scattering contrast exists for X-rays, one method is to use elements with higher atomic numbers but this requires a change in chemical composition. Anomalous-small-angle X-ray scattering (ASAXS) may be used to change scattering contrasts for X-rays without changing the chemical composition. In general the atomic form factor for X-rays may be written as

X-ray scattering

289

f=fo +f’ +.f”, where f0 is the mean atomic scattering and f’ and f” are the real and imaginary terms of the absorption coefficient. At wavelengths well removed from any absorption edges both f’ and f” are small compared to fa, whilst close to an absorption edge both terms increase. Therefore by performing measurements at carefully chosen wavelengths it is possible to change the scattering contrast in a known way. Such experiments could be readily undertaken using USAXS instrumentation, the wavelength of the beam at the sample being a function of the angle of reflection within the first channel cut crystal.

6. Conclusions USAXS using synchrotron radiation in a new techniques, which although at a preliminary stage of development has already produced interesting results. The work reported here clearly shows the potential that the technique offers for the investigation of structure in systems with characteristic sizes in the range 0.1 to 1 km. In recent years there has been a rapid expansion of facilities for SANS and synchrotron based SAXS. With the introduction of synchrotron based USAXS and its future development it would seem that scientific interest in the USAXS technique will grow rapidly, particularly in the field of colloid science. We would like to thank the Synchrotron Radiation Facility Committee of the Science and Engineering Research council (SERC) for their support of this work. One of us, A.N.N. would like to thank the Agricultural and Food Research Council and the SERC for financial support which made this work possible.

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X-ray

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X-ray scattering

[20] C. Nave, G.P. Diakun and J. Bordas, Nucl. Instr. and Meth. A246 (1986) 609. [21] C.G. Vonk, J. Appl. Cryst. 4 (1971) 340. [22] 0. Glatter, in: Small-angle X-ray Scattering, eds. 0. Glatter and 0. Kratky (Academic Press, London, 1980) p. 119. [23] J.Th. Overbeek, Faraday Disc. 66 (1978) 7. [24] M. Kotlarchyk, S.H. Chen and J.S. Huang, Phys. Rev. A28 (1983) 508. [25] A.N. North, J.C. Dore, A. Katsikides, J.A. MacDonald and B.H. Robinson, Chem. Phys. Lett. 132 (1986) 541. [26] D.W. Schaefer and K.D. Keefer, Phys. Rev. Lett. 56 (1986) 2199. [27] D.W. Schaefer and K.D. Keefer, Phys. Rev. Lett. 53 (1984) 1383. [28] A.N. North, unpublished results. [29] U. Bonse and M. Hart, Z. Phys. 181 (1965) 151.