188 Nuclear Instruments and Methods in Physics Research B34 (1988) 188-202

North-Holland, Amsterdam

SMALL-ANGLE X-RAY SCATI’ERING STUDIES OF HETEROGENEOUS SYSTEMS USING SYNCHROTRON RADIATION TECHNIQUES

Ashley N. NORTH and John C. DORE

Physics L..aboraloty, Universiry of Kent at Canterbury Canterbury, Ken: CT2 7NR. England

Richard K. HEENAN

Neutron Division, Rutherford Appleton Laboratory Chilton. Didcot, Oxon. OXII OQX, England

Alan R. MACKIE and Andrew M. HOWE

Institute o/Food Research Norwich, Colney Lane, Norwich, Norfolk, NR4 7UA, England

Brian H. ROBINSON

Chemical Laboratory, University of Eust Anglia, Norwich, Norfolk, NR4 7TJ. Englund

Colin NAVE

Daresbury Laboratory, Daresbwy. Warrington, Cheshire, WA4 4AD, England

Received 12 November 1987 and in revised form 15 April 1988

The high intensity X-ray beams from synchrotron radiation facilities can be used to study the structural characteristics of heterogeneous systems. The present paper reports the use of this technique for SAXS measurements on a particulate system which

shows aggregation properties (microemulsion). a concentrated particulate system showing repulsive ordering, a colloidal suspension

(platinum) and two porous solids (a silica gel and an oil-bearing shaly rock). The advantages in the use of synchrotron radiation for

general SAXS studies concerning research topics in chemical physics and materials science are discussed in relation to other methods.

Further developments are also considered for applications to new areas of investigation and the complementary nature of SAXS and

SANS measurements is established.

1. Introduction

Small angle scattering (SAS) techniques with X-rays (SAXS) or cold neutron (SANS) have developed into one of the most important methods for studying the structural properties of materials on a length scale of 10 A to 0.1 pm. The range of interest extends from topics in basic physics to applied biology and covers many aspects of detailed information that would be difficult to obtain by any other means. Although the principles of small-angle scattering have been known for many years, it has been the increasing power of the primary beams and the improvement of instrumentation that has contributed to the rapid increase in utilisation over the last decade. Early SAXS studies were usually conducted on laboratory-based equipment but the advantages of central facilities with highly specialised equipment was soon real&d in the context of neutron instruments

0168-583X/88/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

using research reactors. More recently, the advent of synchrotron radiation sources has led to a second phase in the use of X-ray techniques.

Much of the work has naturally been focussed on the use of small-angle scattering techniques in the field of the biological sciences. It is the purpose of this paper to examine the impact of the new facilities on the separate areas of chemical physics and materials science. The current status of synchrotron-based equipment is as- sessed in terms of the information that can be obtained from the measurements and the limitations inherent in the method. Several examples are chosen to illustrate the main points and the likely development of new experimental procedures is considered. A critical com- parison of X-ray (SAXS) and neutron (SANS) results suggests that a combination of the two methods will often be more effective than the use of one experimental technique in isolation.

A.N. North et al. / X-ray scattering studies of heterogeneous system 189

2. Basic principles and -Ro is the Guinier Radius,

2. I. Theoretical fmma~ism

The essential theoretical relationships have been pre- sented in various review papers [l-3) and only a short description will be given here to define the required terms. The scattering amplitude u(Q) for a scattering vector Q may be written, in general, as

R&= j/‘~(r) dr

(7)

j p(r) dr V

For a sphere of radius R, the particle form factor P(Q) may be written exactly as

a(Q) = jVp(r) exp(iQ.R) d3r, (1)

where p(r) represents the distribution of coherent scattering length density throughout the sample volume. In the case of an isotropic sample where orientation effects are averaged, the scattering intensity, Z(Q), may be expressed as

Z(Q)= la(Q)l’=4~r~~p~(r) r’ydr (2)

and is radially symmetric; the scattering vector is de- fined by the experimental conditions as

[

2 ZP(Q)=P(Q)=9(Ap)2V2 sinQR-$$wsQR . 1 (8) 21.2. Z~ter~~ing spheres

In the case of concentrated dispersions of particles there is an additional contribution from inter-particle interference effects (resulting from a time-averaged order of the particle positions) such that the observed SAS intensity maybe written as

4&Q) = Gus, (9)

where S(Q) is the inter-particie structure factor given by the transform of the centre-centre pair correlation function, g,(r), i.e.

5”(Q) - 1= $fr[&(‘) - I] sin Qr dr, 00)

where pn is the particle number density. The determina- tion of S(Q) or g,(r) relates immediately to the strength and shape of the central particle-particle interaction potential, G(r). If the particles are not all equivalent the observed scattering becomes

bx(Q> = PtQ)(S(Q)h (11) where

Q= Fsinf,

where h is the wavelength of the incident beam and B is the angle of scatter.

The observed SAS intensity provides the required information about p(r) but it is rarely possible to use inverse transform methods to determine this function directly due to limitations in the Q-range of the experi- mental measurements. The interpretation is therefore based on the use of specified functions with parameters defined by some model representation There are a num- ber of different cases which are relevant to systems based on chemical physics or materials science research.

2. I. 1. Zsolated particles The SAS signal for a particle of arbitrary shape in a

homogeneous medium may be written as

I(Q)=/v[~(‘)-~M] exr4iQ.r) d3r, (4

where pM is the coherent scattering length density of the medium and I’ is the vohune of the particle. For an assembly of N homogeneous particles of uniform scattering density pP, with random orientation, the ob- served intensity at low Q-values is given by the Guinier expression [l]

b(Q) = N(&j2 ev( -R&Q”/3) QRG s 1, (5)

where Ap is the scattering density contrast,

AP = ( PP - PM) (6)

(S(Q)) = 1+ Ka(Q))12 (Ia(Q)12>(S(Q)-1)

and (a (Q )) and 1 (a(Q)> / ’ are o~entational averages.

2.1.3. Aggregating particles Under certain conditions there may be a strong

attractive interaction which leads to the formation of clusters. The effective structure factor for this situation has a peak at low Q so that the shape of S(Q) is markedly different from that arising in normal disper- sions. The transient and reversible formation of large particle aggregates occurs as a precursor to phase sep- aration and has been conventionally treated within the formalism of critical scattering [4] such that

&i,(Q>= K 1+t2Q2’

(12)

where 5 is a correlation length related to the cluster size (or density fluctuations) and K is a susceptibility which

190 A. N. North er al. / X-ray scarrering srudies 01 helerogeneous sysfems

depends on the strength of the interaction. Both t and TV show scaling behaviour

f(X)=&)X-“; lc=#rOx-y

with critical constants Y and y, where

(13)

x= i i

T-7;: T,

(14)

for a critical temperature, LrC. More recently an alterna- tive form has been proposed based on the fractal struc- ture of the aggregate [S] and is described in section 2.1.5.

2. i ,4. Porow solids

Porous solids consist of a rigid framework of material containing voids with a connecting network of channels or pores. The geometrical characterisation of a dis- ordered pore structure poses considerable theoretical problems but recent use has been made of develop- ments based on a fractal representation [S]. Under these circumstances the spatial distribution is regarded as a self-similar so that the pair correlation function scales according to the expression

g(r) =R(r/r,)D”. (15)

where r, is a basic length parameter and DM is a fractal dimension~ty representing the ~st~bution of mass in the system. It can then be shown (71 that the SAS intensity follows a power law relation

1(Q) = KQ-‘M, (16)

where D, may be determined directly from the slope

of a In I(Q) vs tn Q graph. Any physical system will only show self-similarity over a restricted scale range so this behaviour is limited to Q-values in the intermediate range dependent on the cluster size. t. and the di- mensions of the pores, d, (i.e. l/l 5 Q I l/d).

At high Q-values the asymptotic behaviour is governed by Pored’s law

I~(Q)IQ-~ - zQ-4. (17)

where V/A represents the surface to volume ratio and the slope of a log I(Q) against Q plot is - 4. However there may also be a region depending on the surface roughness which causes a deviation from Pored’s Iaw. These features may also be represented by a fractal description [7] such that

r(Q) = kQ+Ds’ 08)

where D, is the fractal dimensionality of the surface. The transition from mass fractal to surface fractal is reasonably well defined if the corresponding length scales are sufficiently well separated so that there are usually several different regions of interest in the I(Q) data. This area of study is of current theoretical interest

[8] and it is likely that further developments will emerge over the next few years.

2.1.5. Aggregation processes

Although the static structure of porous networks or

aggregates may be of basic interest, there are many cases where the process of formation itself may help in providing information on the evolution of the spatial characteristics. For these systems it is less important to obtain a detailed interpretation of the SAS data and the main concern is the way in which the lobs(Q) function changes with time or other external parameters. The most obvious example of this behaviour is for irreversi- ble aggregation such as gel precipitation [9]. the setting of cement [lo], flocculation of colloids eg. silica [l l] and gold f12j. or sedimentation processes in colloids 1131. The characteristic time for these processes ranges from minutes to days and the rates may be nonuniform. The SAS technique provides a noninvasive method for prob- ing the changes and can yield information on particle size for nucleation and growth or alternatively define the progressive devefopment of network structures.

The question of small cluster size requires a separate treatment since the length scales characterising the par- ticle diameter and the cluster size may be sufficiently close to make the determination of D, and E from eq. (16) difficult. A modified form of the equation can be derived 1141 in which the parameter values have the same meaning but the predictions may no longer have a linear region to a In-In plot. The structure factor for the case of a finite-size cluster can be written as 1141

x sin(( D, - 1) tan-‘(QO), (19)

where T(X) is the gamma function. This method is particularly useful for cases where the cluster is in dynamic equilibrium with its surrounding as in the case of droplet aggregation within a microemulsion system that is close to a phase transition. The three parameters, r,,, (particle size), 1: (cluster size) and D, can be evaluated for each of the measurements and the change in values studied as the transition is approached by variation of temperature, pressure, concentration, com- position or other variable.

The different cases described in the above sections show that useful information can be extracted from a wide range of materials if the scattered beams are capable of providing sufficient statistical precision to define the shape of the scattering pattern over a suffi- ciently wide Q-range. The types of pattern obtained for each system are schematically illustrated in fig. 1. The question of count rate, resolution and Q-range for dif-

A.N. North et al. / X-ray scattering studies of heterogeneous system 191

In Ita In t(Q:

b

d -2.50 -2.00 -1.50 -1.00 -0.50

In Q

Fig. 1. a) Theoretical scattering curve for a dilute system of noninteracting*spheres of radius 60 A (In Z(Q) vs (Q). b) Theoretical scattering curve for a concentrated suspension of interacting particles of 60 A radius (In Z(Q) vs (Q). c) Theoretical scattering curve for a system of 60 A particles undergoing aggregation or clustering. The enhancement of the scattered intensity at low Q-values is predicted by both the fractal and Omstein-Zemike formalism (In Z(Q) vs (Q). d) Theoretical scattering curve for a system showing

both a finite sized mass fractal and surface fractal structure (in Z(Q) vs In Q).

ferent instrumental conditions is discussed in the fol- lowing section.

velocity selector [Pi]. The wavelength resolution is nor- mally in the region of - 10% but can be reduced to 5% for special applications. The spatial resolution of the

2.2. Experimentd criteria

The basic layout of a conventional small angle scattering instrument is shown in fig. 2. The monochro- matic input beam is collimated to illuminate a finite beam spot at the sample position and the scattered radiation is measured by a one- or two-dimensional multidetector. In the case of neutron scattering the incident wavelength is typically in the range 5-15 A and may be varied by altering the rotational speed of a

mceodrunatcr

Fig. 2. A schematic diagram of a small angle scattering system, SAXS or SANS.

192 A. N. North et al. / X-ray scattering studies of heterogeneous systems

multidetector is usually at the 5 mm level so that the scattering angles are well-defined except for regions close to the beam stop. There are many instruments in operation with various characteristics for which perfor- mance details are well documented [15]. The wide choice of wavelength and often large sample-to-detector dis- tances gives an available Q-range of about 0.001-0.3 A-‘.

The use of laboratory-based equipment for SAXS studies has been described in various review articles [l-3]. A major difficulty with the use of such equipment is the smearing of the scattering pattern due to a broad distribution of wavelengths and geometrical effects aris- ing from slit collimation, The relatively recent develop- ment of high-intensity X-ray beams from synchrotron radiation sources has now led to a dramatic change in the performance of SAXS experiments. The basic prin- ciples were outlined in an article by Stuhrmann [16] in 1980 concerning the early use of synchrotron radiation for studies of macromolecules in solution; a brief tech- nical comparison of SAXS and SANS instrument per- formance was also included but the relative merits have now been influenced by further technical development during the last few years.

The data presented in section 3 has been primarily collected on the SAXS instrument installed on line 7 of the Synchrotron Radiation Source (SRS) at the Dares- bury Laboratory. The incident white beam falls on a focussing mirror followed by a Ge(ll1) focussing monochromator set to produce a wavelength of 1.608 A (the Co K, edge) with a band pass of 3 X 10W3 which is adequate for most most SAXS experiments [17]. The beam is focussed onto the detector where it produces a spot of 0.4 x 4 mm, with a flux of 4 x 10” photons s-l mmP2, several orders of magnitude greater than that available from the most powerful laboratory-based X- ray generators. The position of the focus may be varied between 2 to 6 m from the monochromator by adjusting the setting of the mirror and monochromator. The small beam size and large sample to detector distances result in negligible smearing of the scattered intensity except at the lowest Q-values [17]. Sample to detector distances of between 0.5 to 3.5 m give an available Q-range of 0.005 to 0.30 A-‘. Anionisation chamber placed after the sample position can be used to evaluate the absorp- tion in the sample by measuring the relative intensities of the beam with and without a sample present. This allows the subtraction of “background” and “empty cell” scattering from the total scattering with sample in the beam. The main origin of background counts in the detector is due to parasitic scattering from the mono- chromator, the focussing mirror, and the beam line before the sample position. General background contri- butions along the beam line are reduced by the use of absorbing collimators of variable slit-width. The sample cells must not absorb nor scatter X-rays too strongly.

ELECTRON BEAM

Focuss$=d

SAMPLE

DETkCTOR

Fig. 3. A schematic diagram of SAW instrument showing focussing optics and detector.

Normally the cells have plano-parallel walls of thin quartz, mica, or glass; in the case of liquid samples it is possible to use thin walled glass capillaries. The experi- ments reported in section 4 were undertaken using cells with flat mica walls of thickness - 40 pm. These cells reduce the beam intensity by approximately 15%.

The type of detector used for recording the intensity pattern is dependent on the characteristics of the sam- ple being studied. In many biological applications with crystalline or fibre samples there is a strong orienta- tional dependence of the scattered profile which re- quires the use of an area detector. For the applications in solution studies or powders where there is no prefer- ential orientation the scattering pattern is azimuthally symmetric, so that a linear detector is sufficient to provide the required information. This is normally mounted in a vertical position (fig. 3) since the focussed spot size is typically 0.3 mm in the vertical plane and 3 mm in the horizontal plane. The detector used to collect the data presented in this paper was a linear, delay-line position-sensitive detector. The active length of the de- tector is electronically divided into approximately 500 channels in a range of 100 mm with a spatial resolution of 0.3 mm. Although an overall count rate of - 100 kHz can be handled there are substantial dead-time losses which can lead to a distortion of the scattering profile due to nonlinearity and it was found desirable to restrict the count to less than 50 kHz.

In small-angle X-ray scattering studies a transmis- sion setting of the sample is used so that there can be significant loss of statistical accuracy in the recorded data due to absorption within the sample or container material. The optimum thickness, for maximum statisti- cal accuracy is given by t,nt = (~J~J’ where pabs is the linear absorption coefficient of the sample which may be evaluated by a relative measurement of the beam transmission for a full and empty container. Typi- cal values of t,rt are given in table 1.

A. N. North et al. / X-ray scatlering studies of heterogeneous sysrems 193

Table 1 p,-values. absorption coefficients and optimum path lengths of materials studied in this paper

Material Fabs

[mm-‘] Lp*

Imml

Hz0 9.1 1.1 0.88 Heptane 6.6 0.33 3.3 Glycerol 11 1.0 0.99 Platinum 144 470.0 0.002 Silica 26 8.0 0.13 NaCl 22 18 0.06 AOT 10 1.9 0.53

The measured intensity, l&$(Q), can show a large variation in magnitude across the Q-range of interest, particularly for fractal systems. In order to obtain good counting statistics at all angles it is often useful to run two experiments with cells of different thickness and/or two detector positions. An optimum thickness is best for the close detector position as this is the situation where high statistical accuracy is probably required for a wide range of Q-values. If there is strong scattering at small angles it is sometimes desirable to mask out this region of the detector or to use a thin absorbing foil in order to reduce the overall count-rate so as to collect good data at b&h Q-values. Multiple scattering can also be important in these cases so that m~urements with a reduced sample thickness are preferred and different datasets can sometimes be combined to give more de- tailed information on the corrections. In the far detector position the count-rate is lower due to the reduction in solid angle but the Q-resolution is increased so that a more accurate me~urement of the shyly-v~ng fea- tures in lobs{ Q) can be achieved over a reduced Q-range. If necessary the detector can be resited in an off-set position to measure the scattered profile at larger Q-val- ues but there is then a need to calibrate the channel positions as the beam-centre is no longer directly visible on the detector. Following these procedures the data files are readily combined to give the overall scattering pattern.

3. Examples

A microemulsion is a thermodynamically-stable dis- persion which is often described as consisting of dis- crete droplets of water (oil) in a continuous oil (water) medium. The droplets are stabihsed by the presence of amp~philic molecules (surfactants) which form a iayer at the oil-water interface as shown schematically in fig. 4. A surfactant frequently used in studies of microemul-

sions is Aerosol-OT which readily forms a stable system without the need for a fourth component (co-surfactant). Microemulsions are characterised by their R-value which is given by [18]

R_ w201 --

[AOT] ’

where [x] is the molar concentration of material x. The value of R can be directly related to droplet radius, r 118). This is an important feature which influences the behaviour of the dispersion and is usually within the range lo-200 A depending on composition and other external parameters such as temperature and pressure. The total volume fraction of water and AOT (+r = (pw + ~1~~) also has an important influence on the be- haviour of the dispersion.

Typical scattering curves for a low concentration R = 20, water-in-heptane microemulsion (cpr < 0.05) are shown in fig. 5 obtained by X-ray and neutron small angle scattering. The lack of minima in the scattering profile compared to theoreticat scattering profiles (fig. la) is an indication that there is a wide variation in droplet sizes within the microemulsion and/or extensive fluctuations in the shape of the assumed spherical par- ticle. Analysis of the data using a modified version of eq. (8) allows particle polydisperity to be estimated i.e.

/ rmaxZ(Q, r)n(r) dr

I(Q) =: InUn (rmrxn(r) dr ’

where Z(Q, r) is the spherical particle scattering func- tion [given by eq. (8)] for a particle of radius r and nf r) is the particle size distribution function as a function of radius (an alternative approach is to use a particle size distribution function which is dependent on the volume of the particle). Data analysis using eq. (20) shows the particle size distribution function to be skewed distribu- tion weighted to the smaller particle sizes with a mean particle radius of - 37 A and a width of 30%. A second interesting feature of the X-ray scattering profile is the slow fall off in scattering intensity at high Q-values when compared to the SANS scattering profile. Analy- sis of the data Guinier plot of the two sets of data

Fig. 4. The structure of water in oil microemulsion stabilised by surfactant.

AX. North et al. / X-ray scattering studies of heterogeneous systems

I /

a ,

In I(Q

C

‘

.

.

,

.

.

.

.

0.00 0.05 0.10 0.15 0.20 0.25

o.30c&~5

I(Q

b

1

c

in KQ

d

0.00 0.02 Cl.04 0.06 0.08 0.10 0.12 0.14 0.16

Q/B;’

1 c *. i

‘. . . . . . . . * .

‘

. . .

. . . . -.. .

l .

=. ,.’ * ‘._. *n

. . - .-. -. .

18

0.00 0.05 0.10 0.15 0.20 0.25

o.30c+c~i35

Fig. 5. a) Z(Q) vs Q for SAXS from a O.OSM AOT water in oil microemulsion. b) SANS profile from 0.05M AqT water in oil ~cr~mulsion. c) Guinier plot of the data shown in (5a) showing two linear regions giving Guinier radii of Rsx = 32 A and R,, = 10

A. d) Guinier plot of SANS data shown in (Sb) showing only one distinct linear region.

shows two regions of linearity for X-rays but only one for neutrons. The difference appears to arise from the effective change in p(r) particularly due to the rela- tively large contribution from ionic species (surfactant headgroup in particular) in the SAXS data. The mea- surements can be interpreted in terms of the presence of “almost-dry” reverse micelles formed as aggregates of surfactant molecules; these droplets are a consequence of the a~egation/sep~ation dynamics of the micro- emulsion [19].

The above analysis assumes that the concentration of droplets is sufficiently small for inter-particle inter-

ference terms to be neglected. However as the droplet concentration is increased, the inter-particle effects con- tribute to the observed pattern and it becomes necessary to use eq. (9). Fig. 6 shows the scattering from a R = 20 water-in-heptane microemulsion as the concentration of AOT is increased from 0.05M to 0.6M, the total volume fraction of water and AOT +r is increased by the same ratio keeping the value of R fixed. The 0.6M AOT suspension clearly shows a peak in the structure factor term of eq. (9). The mean particle separation may be calculated from the peak position to be - 60 A. The extraction of the interference effects is complicated due

A.N. North et al. / X-ray scattering studies of heterogeneous system 195

a.. . . - , .

. * . . . - .

. . . . . . . . a- *.:*:.

_* . . v I

. . .* * .

. . . . ...*’ v* * : . : ’

.I.’ .:::r, ~“.~1lllllll*,*,,,,,,,,,,,

Fig. 6. SAXS scattering from varying concentrations of surfac- tant of water-m-oil with microemulsions v 0.3M AOT, A 0.4M

AOT, n 0.5M AOT, + 0.6M AOT.

to the presence of the “dry” micelles and the possibility that the droplet size distribution changes with con- centration despite the use of a fixed R-value. If the interaction potential between the water-coated droplets is assumed to be represented by a hard sphere potential a second peak in the structure factor would be expected. The absence of this peak in the measurements suggests that a softer potential is required to explain the scatter- ing profile.

NCl

\

Table 2 Mean particle radius rc. correlation length 5, and fractal dimensionality D, of water-in-dodecane microemulsion close to a phase transition

+T ‘0 I DM

0.03 30.5 91 1.69 0.07 32.7 156 1.66 0.11 32.2 175 1.52 0.16 30.8 135 1.49 0.20 30.3 105 1.59 0.25 30.2 62 1.76

The behaviour of a microemulsion system close to a phase transition is of considerable interest since the single phase may undergo a gradual aggregation of droplets prior to separation into two microemulsion phases of different droplet concentration. This phenom- enon has been compared to the critical behaviour of a vapour-liquid equilibrium [20] and leads to a corre- sponding increase in scattering at low Q due to long- range concentration fluctuations in the system. During recent years there has been much theoretical and experi- mental investigation of aggregating systems using fractal theories [14]. The enhanced scattering may be sucess- fully analysed using eqs. (8) (9) and (19) for fractal aggregates: Fig. 7 shows data fitted using this structure factor, collected from water in dodecane microem- ulsions (R = 18.2) in which the concentration of AOT was varied. The R = 18.2 composition path lies just inside the single phase microemulsion stability region and closely follows a coexistence curve (with the two- phase region) of the phase diagram. The fitted correla- tion lengths 6 (table 2) are in good agreement with those obtained from photon correlation spectroscopy [5]. A maximum in the value of [ is observed at approximately $r = 0.12, such maxima have been ob- served in a variety of water-in-oil microemulsion sys- tems for +r = 0.1 to 0.2 [21,22]. This maximum correla- tion length is consistent with increased attractive inter- actions between droplets leading to a liquid/gas phase separation of the single phase microemulsion at the critical point [23]. An increase in attractive interactions between droplets reduces the fractal dimensional&y, D M, of aggregates in both the diffusion limited and cluster-cluster aggregation models [24]. This is reflected in the SAXS results (table 2) with the lowest fractal dimensionalities corresponding to the larger correlation lengths. The use of fractal theories in the analysis also gives a value for D, which shows the expected varia- tions as the temperature is varied [25].

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14

CA-’ Fig. 7. The enhanced scattering from a microemulsion system close to a phase transition, fitted using fractal formalism [eq.

(19)l.

3.2. Colloidal suspensions

In addition to the liquid-liquid dispersions discussed in the previous section, SAXS techniques can be readily

196 A.N. North et al. / X-ray scattering studies of heterogeneous systems

adopted for solid suspensions such as solutions of finely divided metal particles. Such systems have been of interest for many years, in particular they provide a large active surface area and have many applications in catalysis. Two different platinum containing systems were studied using synchrotron-based SAXS. The first consisted of a colloid formed of small platinum par- ticles suspended in water prepared by chemical reduc- tion of hexachloroplatinate by sodium citrate. The volume fraction of platinum particles was kept below 0.05 in order to keep multi-scattering of the X-rays to a minimum. The second system was based on an R = 20, water-in-heptane microemulsion stabilised by AOT, in

NQ

“‘-i:l;..-l:--.i_

\ “_-.Ar\_

which small platinum particles are initially formed within the water core of the droplet by allowing the reduction process to be initiated within the aqueous region of the droplet. In order to ensure a complete reduction of the hexachloroplatinate an excess of reduc- ing agent, hydrazine, was used.

Metals have a relatively large electron density com- pared to water and other commonly used materials (see table 1). As a result platinum colloids are relatively high intensity scatterers of X-rays. It can also be seen from table 1 that platinum is a strong absorber of X-rays at a wavelength of 1.608 A and the optimum sample thick- ness t,nt is therefore very sensitive to platinum con-

I(Q

Fig. 8. a) The scattering profile from a platinum colloid. b) The scattering profile from a platinum in microemulsion system fitted using an Omstein-Zernike formalism [eq. (12)]. c) As in fig. 8b but fitted using a bimodal particle distribution.

A.N. North et al. / X-ray scattering studies of heterogeneous systems 197

centration. SAXS data from the two platinum suspen- sions are shown in fig. 8. The data for the conventional preparation were fitted using eq. (8) for isolated spheres, and required a small polydispersity (- 5X0 to be intro- duced into the fitting function (fig. 8a) with a mean particle radius of 18 A. In the case of the platinum formed in the microemulsion two different models were used to interpret the data which shows enhanced scattering at low Q-values when compared to the stan- dard water-in-oil microemulsion. As was seen in the last section enhanced scattering at low Q-values is often associated with phase transitions and the first model used was the Omstein-Zernike representation of critical scattering [eqs. (9) and (12)J (fig. 8b). The implication of this model is that the excess reducing agent and/or the by-products of the reaction produce a shift towards a phase transition in the microemulsion system. This could arise from the hydrazine entering the water-AOT inter- face and changing the physical or chemical characteris- tics of the interface. The second model used was a bin&al distribution of particlz radii (fig. 8~). The smaller radius of mean value 33 A; approximates to that for water droplets in a R = 20, water-in-oil microemul- sion in the absence of platinum (see earlier section on heterophase systems) whilst the second radius was ap- proximately three times larger (95 A). After the initial formation of small platinum particles within the drop- lets, the pool-exchange mechanism 1261, in which drop- lets fuse and break apart, allows the platinum particles to aggregate. These larger particles may be associated with only a small amount of water but are likely to be surrounded by a surfactant coat [27]. A statistical analy- sis of the minimised x2 and the residuals favours neither model, however evidence from other techniques [27] favours the bimodal radii model.

3.3. Porous solids

A wide range of natural and manufactured materials are porous solids. They contain a large number of open pores within their internal volume and are therefore less dense on the macroscopic scale than their non-porous counterparts. An example of a naturally occuring por- ous solid is the oil bearing shaly rocks in which the oil is dispersed in the pores of the rock. An understanding of pore shapes, size distribution and anisotropies are re- quired in order to characterise the mechanism of oil migration from source rocks of relatively low porosity to reservoir rocks. An example of shaly rock is the Bakken shale which has a bdoad pore size distribution [28] with a peak around 50 A and a long tail extending to much higher values.

Examples of synthetic porous solids are the silica gels which can be manufactured with a range of pore sizes. Such gels are used for purification (absorption of H,O, COz etc. from gases), separation, and as catalytic

Table 3 Scattering length densities, pX and p,,, for various liquids

Liquid

H2O

CC14 CHCI, CH,BrCI C-I%,

CH,OH

p;P cm-21 P;Or” cm-‘]

9.3 - 0.56 13 6.3 12 2.8 15 2.4 7.9 - 0.48 7.4 7.4

supports. ,Spherisorb is a well characterised silica gel, which is widely used as a catalytic support and as a drying agent. It has an average pore size of 90 A (diameter) with a pore distribution of between 55 and 110 A. The p, value of silica is 260 X lo9 cm-’ com- pared with that of 10’ cm-* for air. Such high contrast and large volume leads to high count rates at the detector. Samples of 0.25 mm thickness produce count rates of the order of 80 kHz. Multiple scattering is often a problem with systems that scatter the incident radia- tion strongly and can significantly distort the measured scattering profile often resulting in inaccurate interpre- tation of the data. In these cases the use of very thin cells 5 0.25 mm reduces the significance of multiple scattering but may introduce other problems if there is nonu~fo~ packing of powdered sample materials. The scattered intensity may also be reduced by filling the pores with liquids such as H,O, Ccl,, which have p,-values of a similar magnitude to the silica. Table 3 gives the scattering densities for a variety of liquids and solids. This method of reducing the contrast is obvi- ously effective only if the pore network is completely filled and could possibly be used in a self-consistent manner to determine further information about the restricted volume which is inaccessible to molecules of varying size. No work has yet been reported on the use of this method which may also have applications for studies of fract~ly-rout surfaces.

Examples of the small angle scattering profile from the Bakken shale and Spherisorb are given in fig. 9 along with In I(Q)-ln Q plots of the data. In the case of the Bakken shaly rock there is a linear relationship over the entire range of the measurement with a slope of -3.55. Such a slope would suggest that the surface of the pores was fractally rough with a fractal dimension, D, of 2.45. The slope is linear over the accessible Q-range and it is possible to estimate the average size of the pores to be > 50 A and the characteristic length scale of the surface to be less than 7 A. Since the Q-range does not cover very small values (> 0.02 A), it is not possible to obtain information concerning the mass (or volume) scaling of the pore. A similar experi- ment with SANS also produces a straight line but of significantly different slope (- 3.67). This is in agree-

198 A. N. North et al. / X-ray scattering studies of heterogeneous systems

a b

In Cl

c J -2.00 -1.80 -1.60 -1.40 -1.20 -1.w -0.80

In Q

Fig. 9. a) SAXS scattering profile for Spherisorb fitted with a pore size distribution of 75 to 110 A. b) ln I(Q) vs In Q plot of the Sperisorb data showing the Qe4 dependence of the scattered intensity at higher Q-values. c) In I(Q) vs In Q plot for the Bakken

shaly rock showing the linear region related to fractal surfaces.

ment with work by Mildner et al. [6] who suggest that water or hydrocarbons adsorbed at the pore interface result in different contrast profiles for X-rays and neu- trons. The fractal dimensional&y represented by the slope is a measure of the surface roughness. Any hydro- carbon liquids in the pores are likely to be present at the interface leading to a variation in the contrast profile as seen by X-rays and neutrons. However, since X-rays and neutrons probe different chemical species they are likely to be scattered from different volumes depending on molecular orientation, sire and composi- tion so giving different surface fractal dimensionalities.

In the case of Spherisorb the slope of the In I( Q)-ln Q plot at high Q-values is - 4 which corre- sponds to Pored’s law and suggest a smooth interface. However, Rojanski et al. [29] have shown that for some porous silica gels a QT4 dependence is also characteris- tic of a self-filling pore with fractal dimensionality of approximately 3 where the surfaces are so rough and irregular that they are space-filling. The measured data may be fitted as shown in fig. 9a using eq. (8) with a Gaussian-shaped pore-size distribution with a mean pore diameter of 92 A which is in good agreement with the value supplied by the manufacturer [30].

A.N. North et al. / X-ray scattering studies of heterogeneous systems 199

4. Discussion

The previous sections have illustrated some ways in which small-angle scattering of synchrotron radiation can be used to provide useful information in chemical physics and materials science. In order to set these techniques into a suitable context it is necessary to compare the method with alternative approaches that can be used to give the same or related information. One of the major gains in the use of synchrotron source is the high intensity of the primary beam. This means that satisfactory counting rates can be obtained from weak scatterers or alternatively that the beam optics can be adapted to provide a suitable range and resolution in the scattering vectors appropriate to the sample under investigation. Most laboratory-based SAXS equipment is carefully adapted for special applications and usually involves a compromise in the loss of intensity resulting from the tighter collimation required in high resolution experiments. Many different geometrical arrangements dependent on either slit or point collimation are used but almost all slit systems have some degree of smearing and require complex deconvolution routines in the data analysis. Conversely, the use of point collimation can eliminate this difficulty but only at the expense of a drastic reduction in scattered intensity. A SAXS instru- ment on an X-ray beam line at a synchrotron source will probably be used for a wide variety of experiments in a relatively short time period so that precise optimis- ation of the beam optics for each set of conditions is impracticable. Fortunately this is not a problem, as the resolution is much less sensitive to instrument parame- ters because the primary X-ray beam from the circulat- ing electron beam has such a small angular divergence. It is therefore possible to work with a fixed wavelength instrument and to cover a wide range of Q-values by using several sample-to-detector distances. The change in beam-optics requires only a small variation in the curvature of the focussing mirror and monochromator for the different detector positions. As a result of this arrangement, the instrument has a very wide dynamic- range, and relatively simple adjustments can be made to position the detector to give the required resolution for selected regions of Q-value from 10e3 to 0.3 A-‘. This feature is particularly important for samples which have a large variation in scattered intensity with Q-value such as the those exhibiting fractal properties. In many cases, the asymptotic behaviour of the 1(Q) curve is of particular interest, usually in the form of a Q4Z( Q) plot and its relation to the Porod limit (section 2.1.4). This expression requires the precise measurement of a small and rapidly decaying signal and is clearly influenced by the signal-to-background ratio at relatively high Q-val- ues. The excellent beam optics and high intensity of the synchrotron-based X-ray beam are shown to their main advantage in this region so that measurements can be

made over a range that would be totally inaccessible to laboratory-based SAXS equipment. This particular characteristic has not yet been widely exploited but can be applied to a range of topics in surface science involving microemulsions, colloids, porous solids, powders, etc. The variation of external parameters such as temperature or the observation of time-varying phe- nomena require a relative intensity measurement so that difficulties inherent in absolute calibration can often be circumvented.

Small angle neutron scattering (SANS) has devel- oped rapidly over the last decade and established a wide user community even though it requires central facili- ties. The demand for beam-time on instruments such as the Dll diffractometer at the Institut Laue-Langevin has increased dramatically and applications are typi- cally a factor of five or more above the available beam- time. Neutrons have several-well-known advantages over X-rays for structural characterisation of certain materi- als. Apart from a few notable exceptions, the absorption is much lower so that thicker samples and containers can be used. This factor is particularly significant for solution scattering and therefore provides an excellent basis for a wide variety of chemical and biological studies. Another important feature is the difference in sign of the coherent scattering lengths of hydrogen and deuterium which allows variation of the contrast profile p,(r) by means of selective H/D isotopic substitution. The principle of contrast-matching is often used to simplify the scattering profile so that specific informa- tion can be more readily extracted from the observa- tions. SANS is therefore more versatile in its appli- cations than SAXS when judged in a present context. However, there are many different opportunities for extending X-ray techniques, and a similar kind of devel- opment could be anticipated over the next decade with the emphasis on SAXS rather than SANS, some likely areas of development are considered in the next section.

In general, the contrast profile for neutron and X-ray experiments [i.e. pN(r) and p,(r)] will be different so that the information obtained from the measurements will also be different. It is therefore impossible to make a clearcut statement concerning preference for SANS or SAXS in any specific case since it will strongly depend on the nature of the problem under investigation. Several broad categories can be defined which may help to indicate the relative advantages but it is inevitable that there will be some exceptions. The high penetration of neutrons is clearly beneficial for materials consisting of heavy elements (i.e. high Z-values) and for thick sam- ples. This means that neutrons will normally be pre- ferred for metallic or alloys systems and will in some instances, be effective in probing magnetic properties by polarisation methods. The SANS and SAXS techniques are therefore more comparable for systems comprising light elements (Z < 20) such as hydrogen, carbon,

2ocl A.N. North et al. / X-ray scattering studies of heterogeneous systems

oxygen etc. which includes most of the topics involving the liquid-state. The systematic increase in X-ray scattering due to variation of electron density ensures that SAXS is advantageous for the study of dispersed heavy elements in aqueous or other solutions. The im- portant field of metallic colloids and other inorganic particulate suspensions falls clearly into this category. This distinction is less obvious for heterophase liquid systems such as microemulsions. The X-ray contrast between water and oil is small (table 3) but for neutrons the contrast can be made large by appropriate deutera- tion of various components. It appears that this effect is partially compensated by the higher beam intensity of the synchrotron X-rays so that comparable counting rates are achieved for similar samples. An interesting feature of recent X-ray studies on microemulsions is the detection of “almost-dry” micelles. The information obtained from the combination of both SAXS and SANS data in this case was greater than that from a single measurement. This example of complementarity will probably be present in other situations and indi- cates that the two techniques should not simply be regarded as alternatives.

The use of neutrons for hydrogenous systems is significantly affected by the presence of incoherent scattering whereas this contribution is negligible in the case of X-rays. Under these conditions, SAXS has ad- vantages for the measurement of small signals in the high Q-value range since the background scattering contribution from the sample is much lower. The effec- tive Q-range of X-ray studies therefore extends to higher values and will be particularly useful for investigation of fractal systems and for studies in the Porod regime. It is interesting to note that complementarity may also be important in the characterisation of porous materials where surface roughness is present. Mildner and Hall [6] have shown that the apparent fractal dimensionality of geological rock samples is different for neutrons and X-rays and that this variation can be ascribed to surface details resulting from occluded hydrocarbons at the shaly rock interface; further developments in this type of experiment are to be expected.

Another difference between SAXS and SANS ex- periments concerns the instrument geometry and selec- tion of a monochromatic beam. Most SANS instru- ments use a monochromatic beam with a wavelength of 5-15 A produced by transmission through a variable speed velocity-selector which usually has a band-pass of 10%. This poor resolution can be improved to 5% but is normally adequate for most applications where the slope of the intensity pattern [dI/(Q)/dQ] is not too large. These conditions are not satisfied in the low Q-value regime for critical scattering so that SAXS studies are preferred. Furthermore, the neutron beam is usually spread over a circular aperture of 10 mm diameter so that precise temperature control is difficult to achieve to

better than f 0.1“ C. The smaller beam size available on the SAXS instruments (0.4 x 4 mm* at the specimen) is obviously a major improvement for investigations close to a phase transition. If the scattering is sufficiently intense the system can be used for dynamic studies with a controlled variation of temperature across the transi- tion region. Little work has yet been done in this research area but the possibilities are already evident.

Photon correlation spectroscopy (PCS), sometimes referred to as quasi-elastic light scattering (QES) is also frequently used to study heterophase liquid systems and can yield valuable results on temperature variation of the correlation lengths, E(T), but does not contain the same degree of detail as SAXS [5]. However the con- venience of PCS measurements suggests that this is another research area where the two techniques can be usefully combined to obtain a more complete picture than either one could independently provide.

The above examples suggest that SAXS studies based on a synchrotron X-ray beam have unique features which can be advantageously employed in a wide variety of research fields. A full comparability study is impossi- ble at this stage since the technique is only in its initial stages of development. It is, however, already clear that much will be gained by additional work and that several specific areas can be more effectively exploited than has been achieved so far.

5. Further developments

Since the measurements reported in this paper were made there has been a further improvement in oper- ation of the SRS at the Daresbury Laboratory. The machine was shut down in September 1986 for the installation of the High Brightness Lattice which has decreased the area of the source by a factor of 10. This will give improved resolution in a scattering pattern. With suitable optics the decrease in source size can be used to produce extra flux on the specimen while retain- ing adequate resolution in the scattering pattern. A new small angle scattering station at the SRS uses this approach to provide a gain factor of 3 or 4 which can be converted to shorter running times or used to probe weak scatterers. Area detectors, or detectors which pro- vide circular integration of a solution scattering pattern, are being developed. These will give much shorter run- ning times than the linear detector used in the present studies. Within a long time period, the European syn- chrotron Research facility (ESRF), to be built in Greno- ble, will offer even greater increases of photon flux into the 1990s. It is clear from these comments that the counting-rates obtained on current synchrotron facili- ties can be increased by factors of over 100. In this respect SAXS measurements which presently take a few minutes will be made in seconds if adequate detectors

A. N. North et al. / X-ray scattering studies of heterogeneous systems 201

and data-handling systems can be built. The main ad- vantages will be in the study of dynamic processes with time scales in the 10-3-103 s range. There are many possible applications in the field of chemical physics since this opens up the area of reaction kinetics using methods such as the “stopped-flow” technique. The observation of phase transitions, aggregation, demixing, viscous flow in porous materials and other surface re- lated phenomena will be accessible to real-time study for reversible and irreversible process. At this stage it is difficult to fully appreciate the wealth of detailed in- formation that these measurements could provide.

The higher effective count-rate will also open up new avenues for the extension of SAXS techniques. The incident X-ray beam has a continuous wavelength spec- trum and it is merely an experimental convenience to operate at fixed wavelength. The tunability of the source means that the variation of the coherent scattering amplitude near an absorption edge can be utilised in much the same way as for anomalous X-ray diffraction and EXAFS. The change in monochromator setting would therefore lead to a variation in contrast profile similar to that of isotopic substitution in neutron scattering. The method would would probably be re- stricted to relatively high atomic weight materials (Z > 35) and therefore most applicable to metallic colloids or heavy ion distributions in solution. Applications in elec- trochemistry would also seem likely. Anomalous small- angle X-ray scattering (ASAXS) [31] would simply re- quire a method of modulating the incident energy without perturbing the beam geometry. The principles are already well known from EXAFS instrumentation and a hybrid apparatus could already be built to study this problem.

Another extension of the SAXS technique is already under development on line 2.2 of the SRS at the Daresbury Laboratory based on the Bonse-Hart tech- nique [32]. The instrument consists of two channel cut silicon crystals with the sample situated between them. The first crystal produces a finely-collimated monochro- matic beam by four reflections within the crystal and the second similar crystal is rotated about the sample position in order to measure the scattered intensity distribution. The precise definition of the X-ray beam direction means that the system can be used for high angular-resolution studies at very low scattering angles. Some measurements on biological samples consisting of muscle fibres have already been reported [32,33]. Pre- liminary measurements have also been made for several of the samples presented in this paper and a more detailed description will be given later [34]. The low Q-value range can be extended down to - lo-’ A-’ by this technique although a full evaluation of the mea- sured intensity distribution will require careful analysis due to “slit-smearing” resulting from the fact that col- limation is only obtained in one dimension. The initial

results suggest that this ultralow small-angle scattering (USAXS) apparatus has an important contribution to make in a regime where neutron scattering is limited in its effectiveness due to very low count rates. The im- proved resolution is gained at the expense of count-rate since the scattering profile must be determined in a point-by-point manner. However there is a partial com- pensation through the increased scattering intensity from fractal solids or aggregation clusters at very low Q-val- ues so that typical counting times are of the order of 30 min or less for a scan over the Q-range of interest. Several alternative geometries are also feasible and there is considerable scope for further improvement. The main application will be probing heterogeneous systems with length scales up to a few microns, particularly where electron microscopy and PCS measurements can- not be used. The USAXS instrument can also be readily adapted for anomalous scattering studies as the choice of wavelength is dependent on the setting of the chan- nel-cut crystals.

6. Conclusions

The present results obtained with SAXS instrument on line 7.3 of the Synchrotron Radiation Source at the Daresbury laboratory have demonstrated some valuable applications in the fields of chemical physics and materials science. Comparison with alternative tech- niques has shown that there are considerable ad- vantages in using the SRS technique. It is also clear that these initial experiments can be further improved by various developments currently in operation or planned for the future. The last decade has seen a rapid expan- sion in the facilities for SANS studies with a corre- sponding spread in the range of applications. It now seems that the scene is set for similar development in SAXS instrumentation and the expected range of scien- tific interest is bound to grow as the facilities for this type of work become accessible to a wider user com- munity.

The experimental work was carried out as part of a research programme supported by the Synchrotron Radiation Facility Committee of the SERC. One of us. (A.N.N.) wishes to acknowledge a research fellowship from the AFRC covering the period when the work was done.

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