Structure and interactions in aqueous colloidal dispersions of oxide particles

Structure and interactions in aqueous colloidal dispersions of oxide particles

Colloids Elsevier and Surfaces, 18 (1986) Science Publishers B.V., 207-221 Amsterdam 207 - Printed in The Netherlands STRUCTURE DISPERSIONS AND...

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Colloids Elsevier

and Surfaces, 18 (1986) Science Publishers B.V.,

207-221 Amsterdam

207 -

Printed

in The Netherlands

STRUCTURE DISPERSIONS

AND INTERACTIONS IN AQUEOUS OF OXIDE PARTICLES

J.D.F.

and M. SCANLON

RAMSAY

Chemistry

Division,

AERE,

(Received

4 April 1985;

Harwell,

accepted

Oxfordshire

COLLOIDAL

OX1 1 ORA (United

in final form 29 July

Kingdom)

1985)

ABSTRACT The structure and interactions in two different types of aqueous dispersion of colloidal silica have been investigated by small angle neutron scattering and static light scattering. In the first type of dispersion containing discrete, approximately spherical particles (diameter < 20 nm), the ordering and strength of double layer interaction has been obtained by modelling the scattering results using the solution of the mean spherical approximation (MSA) of Hansen and Hayter. Dispersions of the second type contained aggregates of primary particles of similar size range which were prepared by dispersing pyrogenic silica powders in water. The scattering behaviour of these dispersions, which was markedly different, has been analysed on the basis of the recent theory of fractal structure.

INTRODUCTION

In the sol/gel process, an aqueous colloidal dispersion of particles is converted to a porous composite solid by removing water from the system [ 1). An underlying feature of this process, which has recently been established, is the relationship between the characteristics of the sol and the microstructure of the gel, which includes its surface and porous properties, as has recently been described [2, 31. It is evident that an understanding of this link is invaluable where oxide gels having predetermined properties are required for applications as adsorbents, catalysts, and ceramics for example. Although several techniques have been used to characterise oxide sols, static light scattering (SLS) and small angle neutron scattering (SANS) measurements have been particularly valuable in the determination of the structure and interactions in these systems [3, 41. These two techniques are complementary, as will be demonstrated; SANS is, however, ideally suited for studies on concentrated colloidal dispersions. In this paper, we will demonstrate the value of both SLS and SANS in probing the structure of two contrasting types of oxide sol, which are illustrated schematically in Fig. 1. In the first type we will consider a system composed of discrete, almost spherical particles which are nearly mono-

208

dispersed; such a system is typified by [email protected] silica sols, as has been described previously [2, 4, 51. The second type, which is more complex, is composed of aggregates of primary particles having a diameter somewhat similar to those of Ludox. A sol of this type can be prepared by dispersing pyrogenic oxide (e.g. silica, alumina) powders in water for example [6]. Dispersions of the latter type are typical of a wide range of colloid systems occurring as floes and particle clusters, which are important in a variety of natural and commercial processes. The importance of such systems has indeed stimulated considerable efforts to define floe structure, particularly in terms of computer simulations which model flocculation processes [7lo]. The assessment of these models has, however, been difficult, due mainly to a lack of suitable techniques which can probe floe structure. Here we will demonstrate how the properties of aggregates can be determined from scattering measurements and thence defined in terms of fractal structure. I

(al DILUTE SOL

i-l_--

ibl CONCENTRATED SOL

~~~

Fig. 1. Diagram depicting the formation aggregated and (II) aggregated ~01s.

(cl GEL

1 of gels of low and high porosity

from:

(I) un-

EXPERIMENTAL

Materials

Concentrated silica sols (Ludox SM and HS) containing discrete, almost were spherical particles (mean diameters, d,, 12 and 17 nm, respectively) obtained commercially (Du Pont de Nemours & Co.) from designated sample batches (S2 and S3) which have been studied previously [2, 41. These stock sols were dialysed against dilute NaNO, solutions of different ionic strength and controlled pH [4]. Sols used for neutron scattering experiments were prepared in deuterium oxide (> 99% DzO) to reduce the background incoherent scattering. Colloidal dispersions of silica and

209

alumina, containing aggregates of particles, were prepared by adding ultrafine powders, which had been produced by flame hydrolysis, to water (D,O for neutron scattering) [6]. These powders were obtained commercially (Degussa, F.R.G.) and had different primary particle sizes, corresponding to [email protected] silica grades 130, 200 and 380, and Alumina C. Particle sizes, as determined from the specific surface areas of the powders assuming a spherical particle shape, were 21, 14, 7 and 14 nm, respectively. The dispersions derived from these powders are designated S130, S200, S380 and A(C) in the text. In the text, the concentration of silica dispersions is given on a mass/volume basis; this can be converted to volume fractions using the density of 2.2 g rn.l-’ for the amorphous silica particles.

Methods Small angle neutron

sea ttering

Measurements were made as described previously [4, 51 at wavelengths of 6 and 10 A on samples contained in silica cells (path lengths 1 and 2 mm), using the spectrometer installed in the PLUTO reactor at AERE, Harwell. The data were analysed using standard programmes to normalise counter efficiencies and to correct for sample self-absorption and incoherent background [5, 131. Absolute scattered intensities, expressed as macroscopic [dZ /dR ] cob, were obtained using a light-water standard. cross-sections, Here, [dC/dQ] is a measure of the fraction of neutrons scattered into an element of solid angle dS2 [ 131.

Static light scattering Measurements were made (Sofica, model 4200) using polarised light at two different wavelengths (546 and 365 nm, respectively) as described previously [ 111. The scattered intensities on an absolute basis (viz. Rayleigh ratios) were determined using a benzene standard for calibration. THEORY

Scattering

from discrete colloidal particles

The scattering behaviour of colloidal dispersions of the above type has been extensively described previously [12-141. Consequently, it is only necessary here to indicate the complementary features of SLS and SANS, and draw attention to more recent developments in the analysis of SANS data, which can give a unique insight into the structure and interactions in concentrated dispersions of particles.

Neutron

scattering

The coherent macroscopic scattering cross-section for colloidal dispersion of identical particles is given by [ 131 :

a concentrated

210

[

dc 1 dS2

P(Q) s(Q)

cob = (P, - PS)* V;nP

where Q is the scattering

vector,

defined

(1) as

I&I = 47r sin 8/A

(2)

for a scattering angle 20 and wavelength A; pP and ps are the mean scattering length densities of the particles and solvent, respectively, VP is the volume of each particle, IZ~ is the particle number density, and P(Q) is the particle form factor, which for spheres of radius R is given by

P(Q) =

3[sin

(QR) -

QR cos (QR)]

(3)

Q3R3

(Frequently, it is customary to express the scattered intensity on an arbitrary or relative scale as I(Q) rather than on an absolute basis as dX /da. These two terms are, however, simply related by I(Q) = F.dC /da

(4)

where F is a constant determined by the sample size and instrumental configuration.) The structure factor, S(Q), is determined by the nature of the particle interaction potential. In very dilute dispersions (viz. np --f 0) containing widely separated non-interacting particles, S(Q) = 1. The spatial distribution of the particles as a function of the mean interparticle separation, r, is given by the particle pair-distribution function g(r), and is related to S(Q) by the Fourier transform

g(r) - 1 =

2+ P

7

[S(Q)

-

11 Q*

sin (QR) QR

dQ

(5)

O

as has been described previously for oxide sols, where the effects of changing concentration, leading to gel formation, have been investigated [2]. The form of S(Q), which may be determined experimentally, can also provide a valuable insight into the interactions and stability of a dispersion which are influenced by the surface charge on the particles, as will be described shortly. Light scattering For incident unpolarised light, the time-averaged expressed in an equivalent form to Eqn (l), is given by R,

= K* MC P(Q)S(Q)

scattered

intensity,

(6)

where R, is the Rayleigh ratio, M is the molecular weight of the particles with a mass concentration c, and K* is an optical constant given by K” = 2n2 ;,

(dx/dc)2 At4 N-’

(7)

211

where K,, is the refractive index of the solvent, dK/dc is the refractive index increment and X0 and N are the wavelength of the incident light and Avogadro’s number, respectively. Information on the interactions between colloidal particles is obtained from S(Q), which can be derived from Eqn (6) by a simple normalisation procedure [ 41:

S(Q) = [Kg /cl /i&j /AC + o Colloid particle

(8)

interactions

The derivation of S(Q) is of particular interest because it can be related to the effective pair potential resulting from the double-layer interaction between colloidal particles. This aspect has attracted considerable theoretical and experimental interest recently, especially in regard to concentrated dispersions [15] and will not be considered in detail here. In essence, however, a theoretical S(Q), generated from models based on well-established interaction potentials, is compared with that determined experimentally, using standard computer fitting routines. Two models which have frequently been applied are the hard-sphere (HS) potential [4,16, 171, and the mean spherical approximation (MSA), in which the effects of screened Coulombic forces can be taken into account. Since the latter potential comprises a hard core and soft tail, it is evidently more appropriate to colloidal dispersions, where double-layer interactions are often dominant. For spherical particles, where KR < 1, as is here, the form of this potential, U(r) is given by [lS]: r<2R

U(r) = 03,

(9)

U(r) = 47reOeRZ\Ir~exp [-K

(r - 2R)] /r,

r> 2R

(10)

Here, Eqn (10) is the standard DLVO type potential, where \I’0 is the surface potential, E is the dielectric constant of the solvent, e. the permittivity.of vacuum, and K the Debye-Hiickel inverse screening length, which is given by

2ciNe’ K=

___

[ eoekT

1

‘/1 (11)

where ci is the ionic strength of the solution, and N is Avogadro’s number. The surface charge on the particle zP is related to \Ilo by the approximation ‘-PO= zp/4nee&(1

+

KR)

(12)

The application of this procedure will be illustrated tained with silica sols containing discrete particles.

Scattering

here

by results

ob-

from particle aggregates

A recent approach based on the concept

for defining of the fractal

the structure of particle aggregates is dimension [ 191. This concept has been

212

developed theoretically using computer simulations of aggregate formation [9, 10, 201. Aggregates so formed have the property of self-similarity, that is the gross structure of the aggregate is the same on length scales greater than that of the range of interaction between the individual particles within the aggregate. The Hausdorff or fractal dimension, D, of an object can be defined as: N(r) = NorD

(13)

where N(r) is the number of spherical particles contained in a radius r about any point in the object, and No is a constant. For the case of a solid mass for example, D = 3 and N,, is 4n/3. Furthermore, for this simple case, D, and the Euclidian dimension, d, of free space will be identical (viz. D = d). However, for fractal objects, such as aggregates, .D < d. This nonintegral dimensionality, d, thus implies that the average density of particles in an aggregate will decrease as the volume sampled is increased. In effect it can be shown that the average number of particles, N(a), within a volume of radius a, say, will vary as UD, which is expressed as aDo:N(a)=y

d%/

(14)

This expression defines the fractal dimension, average density function given by g(r) = < P(r’)p(r’

D, in terms of the ensemble

(15)

+ r) >/P(r)

where p (r’) is the density It also follows that

at position

r’.

g(r) 01 rtDed)

(16)

Furthermore, it can be shown that the scattered I(Q)] is given [ 121 by the general expression:

I(Q)

= J

sin (Qr)

4rr2g(r) ~

Qr

amplitude

d3r

such that I(Q) and g(r) are related by a Fourier transform. It thus follows .that I(Q) will also obey a corresponding tion given by:

I(Q) a Q-D

[or intensity,

(17)

power law rela-

(18)

Evidently, Eqn (14) will only apply for a certain range of length, r, which for aggregates will be approximately between the diameter of the individual particles and the overall size of the aggregate. Correspondingly, the fractal power law observed from scattering measurements [viz. Eqn

213

(IS)] will hold for a range of Q, which is determined in reciprocal space by these limiting values of r. For small r, comparable with the primary particle size (viz. high Q), the power law expected would then be dominated by surface fractal structure. Thus for scattering samples composed of two components (such as aggregates of particles dispersed in a liquid, and porous solids) which are separated by a smooth interface, it can be shown that [21] I(Q) cc [email protected]+) where S is the surface area of the interface and D is the surface dimension of 2. This leads t,o the familiar Porod law relation [22]

(19) fractal

I(Q) a SSQ-~

(20) In certain extreme cases, however, where the surface is irregular, or has curvature on a scale smaller than reciprocal Q space, D may become greater than 2 [21, 231. This can lead to power law exponents in Eqn (20) which are smaller than -4. Such a situation may arise with very small colloidal particles (diameter < 100 A) as will be demonstrated here. Experimental measurements of scattering from colloidal aggregates or clusters of particles in aqueous dispersions may thus, in principal, provide an insight into the uniformity size, and openness of their structure, as will be demonstrated subsequently. The latter feature is implicit in Eqn (16) which leads to the relationship between the radius of gyration, RG, of a cluster containing N, particles:

N, aR,

D

(21)

Differences in D which reflect the openness of aggregates have indeed been demonstrated by computer simulations, where two idealised models have been considered. Thus for the case of diffusion-limited aggregation (DLA), Meakin [lo] has derived a value of D of 2.5 whereas, for a situation involving the aggregation of clusters of comparable size (CA), a dimensionality, D, of - 1.75 is obtained [24]. RESULTS

AND DISCUSSION

Discrete particle dispersions Small angle neutron sea ttering The dependence of scattered neutron intensity, I(Q), on momentum transfer, Q, for S3 silica sols of different concentration, which have been dialysed against NaNO, solutions having a fixed pH (- 8) and ionic strength (5 X 10e3 mol dme3), are shown in Fig. 2. The features demonstrated are typical of those described previously [2] . Thus, the development of the maximum in I(Q) is caused by interference effects and implies that the particles are not arranged at random but have some short-range ordering

214

due to interparticle repulsion. This is indicated by the movement of the maxima to higher values of Q with increasing sol concentration, which reflects a reduction in the equilibrium separation distance, rgcrlmax, between the particles. Values of rg(r)max in Table 1 are those derived by Fourier transforming [cf. Eqn (5)] the fitted S(Q) obtained with the MSA model. The continuous lines in Fig. 2 show the calculated product K*P(Q)S(Q), obtained from the MSA model, which has been fitted to the experimental data by least squares refinement. The corresponding S(Q) derived from the model is shown in Fig. 3; the surface charge density, CJ,and potential, \Iro are given in Table 1. The value of the particle radius, R, obtained from the fitted P(Q) is also shown and is in satisfactory agreement with that of - 8.5 nm as measured by electron microscopy. The effective surface potential, \IIO, is approximately half the zeta potential as measured by laser electrophoresis (see Fig. 4) on much more dilute S3 sols under similar pH conditions but lower ionic strength. Values of effective surface charge, u, are furthermore markedly lower than those

. LO

_I ~ 0025

l*..

l.

9. oJo25

0

0'

0025

%

005

Q Ia-1

Fig. 2. Small angle neutron scattering of S3 sols of different concentrations dialysed against 5 X 10.) mol drn-’ NaNO, solution: (0) 0.137; (m) 0.266 and (4) 0.550 g cm-‘; continuous lines are the theoretical fits to the data. TABLE

1

Results from concentration Cont. 0.137 0.266 0.550

(g cm-‘)

fitting the MSA at pH - 8

model

to SANS

data obtained

on S3 silica sols of different

R (nm)

0 (J.IC cm-*)

* 0 (mV

rgwmax

9.8 9.3 8.4

1.01 0.71 0.66

28 26 19

35 27 21

(nm)

215

6

8

S(Q), for S3 sols of different concentrations Fig. 3. Structure factors, to scattering data shown in Fig. 2. Concentrations are 0.137, 0.266 for (a), (b) and (c), respectively.

derived from fits and 0.550 g cmm3

PH

Fig. 4. Dependence of electrophoretic mobility, dilute (< 10.’ g cm-3) silica sol S3 ([Na’] = 10.’

u, and zeta mol dm-‘).

potential,

p, on pH for a

obtained by conductimetric titration (> 2 /.LCcme2) [25]. These differences have been discussed in detail elsewhere [5] and can perhaps be ascribed to the adsorption of counter-ions (Na+) at the silica surface. This feature, which has already been noted to a lesser extent with latex particles [17], may arise because the silica particles have a diffuse surface with a high adsorption capacity for counter-ions. It is thus possible that the values of \kO determined are lower than the zeta potential because the counter-ion adsorption becomes more pronounced as the sol concentration is increased.

216

Another aspect which cannot be overlooked, however, concerns the assumptions inherent in the MSA model, in particular the use of the DebyeHiickel potential to describe double-layer interactions in concentrated dispersions. The gradual increase in the negative charge on the surface of silica, which is demonstrated in Fig. 4 by the electrophoresis results obtained with dilute S3 sols, is, however, reflected by changes in the scattering behaviour for concentrated sols of different pH, illustrated in Fig. 5. Thus, the sharper peak obtained at the higher pH of 7.2 indicates a stronger repulsive interaction than that at pH 4.5; this arises from a greater effective surface charge, as confirmed by the u values obtained from the fits, which are 0.18 and 0.35 ~.ICcm-*, respectively. Light sea ttering The range of Q covered by light scattering is considerably lower than that of SANS (- 8 X 10e4 to 4 X 10m3 a-‘). This has two implications: Firstly, we would expect that over this range P(Q) would be close to unity since QR << 1 [cf. Eqn (3)]. Secondly, the S(Q) for a particular sol concentration should have reached a limiting value (S(Q)+ u ), as is indicated in Fig. 3. It follows that the measured scattered intensity, R, versus Q, will be almost constant, as has been demonstrated previously [4]. Because of the attenuation of light and multiple scattering effects, measurements on colloids are usually restricted to much lower concentrations, c, than possible with SANS. However, with very small particles, which scatter weakly, these restrictions are less severe. This is illustrated in Fig. 6, where

2 Fig. but

5. I(Q) different

a-1

0.03

0.06

plotted against Q for S3 sols of similar concentrations pH, (a) 7.2 and (b) 4.5, both dialysed against 10.’

(mol

0.19 g cm-‘) drne3 NaNO,.

217

values of S(Q)Q_~, obtained from Eqn (8), are plotted for a silica S2 sol (diameteter 12 nm) up to a concentration of 8 X 10V2 g ml-‘. The importance of the interactions between the electrical double layers of the sol particles is apparent from the decline in S(Q),, which is particularly steep for the sol of highest pH. Although the features illustrated here will be considered in more detail subsequently [26], it is evident that, in dilute colloidal dispersions containing very small particles, interactions can still be considerable since the interparticle separations are comparable with the Debye-Hiickel screening distances, K-~, even when the ionic strength of the supporting electrolyte reaches - 5 X 10m3 mol dmm3 (viz. K -’ = 4.3 nm).

Particle aggregates Small angle neutron

sea ttering

The scattering behaviour of dispersions containing particle aggregates are typified by those shown for two different concentrations of Aerosil 200 silica in Fig. 7, curves (i) and (ii). In both of these, I(Q) decays with a power law of Qm3.’ above a Q of - 2.5 X 10e2 a -‘. Such a decay corresponds closely to that of the Porod law, discussed earlier, and demonstrates that

0

5

10

c /10m2g cmm3

Fig. 6. Dependence of S(Q)o+e (o) - 9; (0) 5-6. Concentration

on concentration, C, for S2 silica sols of different pH: of electrolyte is similar at each pH: [Na’] = 10m4 dm-‘.

218

the scattering behaviour is dominated by the total surface area of the particles in the dispersion, when Q begins to exceed considerably the inverse of the size of the primary particles. Similar behaviour was exhibited by the other Aerosil dispersions; the power law exponents, X, are given in Table 2. In the lower range of 4, there is no evidence of any interference maximum as was observed in dispersions of discrete particles of comparable size and concentration. In contrast, the scattered intensity continues to increase monotonically, albeit with a reduction in the power law slope (the broken line shown in Fig. 7 corresponds to an increase as Q-l.‘), until the lowest Q experimentally attainable (Qmin - 1 X lo-* A -‘) is reached.

Light scattering The intensity of light scattered from dispersions containing particle aggregates was considerably greater than that from discrete particle systems of similar size and concentration. This feature limited the upper concentration of measurement to a range between lo-* and 10e3 g ml-’ because of Results for such dilute Aerosil 200 multiple scattering and attenuation. silica dispersions (Fig. 7) are typical of those obtained with other Aerosil

ala-1 Fig. 7. Static light scattering and small angle neutron scattering of silica (S200) dispersions of different concentrations. Concentrations for light scattering are: (a) 3.4 x 10.“; (b) 6.8 x 1Om4; (c) 1.7 X 10.‘; and (d) 3.4 X 10m3 g ml“, respectively; symbols (0) and (0) correspond to measurements with h (nm) of 546 and 365. Concentrations (in D,O) for SANS are: (i) 0.10 and (ii) 0.23 g ml-‘, respectively; full lines correspond to slope of - -3.9; broken line to -1.7.

219

grades: These all show a power law increase in I(Q) with a power law exponent between -1.7 and -1.8, which is in agreement with the limiting behaviour of I(Q) at low Q observed in SANS. Such behaviour implies that the aggregates have fractal properties [cf. Eqn (18)], and that these aggregates remain intact, and probably associate as the concentration of the dispersions is increased. Indeed, there is evidence from SANS that the pcrous gels derived from these dispersions also retain fractal properties; this aspect will be considered in a subsequent paper. Values of the power law slope, corresponding to the fractal dimensionality D, for other dispersions are given in Table 2. It will be noted that, for the Aerosil silica dispersions, D is very similar, and falls within a range from - 1.6 to 2.0, suggesting that the aggregates have a relatively open structure, which is common to all three grades of Aerosil. This value of D is similar to that of 1.75 which has been predicted by computer simulation for a process termed cluster aggregation (CA), in which clusters are formed by the homogeneous aggregation of a collection of particles; these small clusters subsequently diffuse and stick together to give larger aggregates [24]. It would seem possible that a comparable structure may arise here as a result of particle aggregation processes which occur during the vapourphase production of Aerosil powders. Although such a possibility will require further study, it is evident that the scattering behaviour is significantly different to that predicted for a structure formed by diffusion-limited aggregation, which involves the accretion of single diffusing particles onto a seed aggregate. The latter structure results in a D of - 2.5, which implies a much slower fall-off in t.he particle density correlation function, whereas cluster-cluster aggregat.ion results in a highly ramified structure, as depicted in two dimensions in Fig. Ba. Furthermore, it is significant that electron microscopy of Aerosil powders shows particle aggregates containing chain-like formations. The overall size of the aggregates is indicated by the lower Q value where a departure from power law behaviour occurs, as illustrated schematically in Fig. 8b. Thus, a decrease in slope begins to occur in Fig. 7 at Q1 5 10m3 corresponding to a size, a,, of - 100 nm. This size corresponds to the range TABLE

2

Light scattering Alumina C

and

Sol

Concentration

s130 s200 S380 A(C?

1 -6 3 x lo-‘-3 8 x 10-‘--8X 1 x 10-h-5

‘Sol

neutron

scattering

range (g ml-‘) x 10.’

x 1o-3 lo-* x 10-h

after passing through

1.2~pm Millipore

results

for

dispersions

D -1.7 - 1.7-1.8-1.5 filter.

1.8 2.0

of

Aerosil

silicas

Q, (A-‘)

X

-1 -1 <1 -2

-3.7 -3.9 -3.0 -4.0

x x x x

10.’ lo-) 10-l 10.’

and

220

of self-similarity, whereas the effective size of the aggregates will be somewhat larger due to a few particle chains extending beyond this range [lo]. From this it is evident that the average number of particles in an aggregate may exceed lo3 [cf. Eqn (21)]. In dispersions of alumina C, a, is significantly smaller (see Table 2), and can be reduced further by filtration (using Millipore filters of different pore size). It is also notable that the value of D is lower, indicating a more open aggregate structure than in the Aerosil dispersions. These differences may reflect a change in the structure of the powders produced in the vapour-phase hydrolysis process, however this possibility will require further investigation. Preliminary investigations have also shown that the value of D can give an indication of the interactions within an aggregate. Thus, under pH conditions where the surface charge is enhanced, D tends to decrease which implies a more open aggregate structure. Such effects also correlate with rheological studies made on more concentrated dispersions which indicate considerable interaction between particle aggregates. These investigations are, however, beyond the scope of the present paper. CONCLUSIONS

This investigation has illustrated how a combination of light scattering and small angle neutron scattering techniques can provide details of both the structure and interactions in two contrasting types of colloidal disper-

-

Range of fractal self slmllarlty

log Q Fig. 8. Schematic representation of a particle aggregate between approximately a, and a,. The form of the in (b).

(a) having a range of self similarity scattering expected is as depicted

221

sion. A knowledge of these differences in structure are important, firstly for an understanding of the factors which control the stability and rheology of these systems. Secondly, as has recently been described [2, 31, the structure and interactions in relatively dilute colloidal dispersions may predetermine the microstructure of the porous gels which finally result from the progressive concentration of the ~01s. ACKNOWLEDGEMENTS

We are indebted to Dr J. Penfold for help in the application of the MSA model and gratefully acknowledge discussions on fractals with Dr P. Schofield. REFERENCES 1 2 3

4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

R.L. Nelson, J.D.F. Ramsay, J.L. Woodhead, J.A. Cairns and J.A.A. Crossley, Thin Solid Films, 81 (1981) 329. J.D.F. Ramsay and B.O. Booth, J. Chem. Sot. Faraday Trans. 1, 79 (1983) 173. R.G. Avery and J.D.F. Ramsay, in M. Che and G.C. Bond (Eds), Adsorption and Catalysis on Oxide surfaces, Studies in Surface Science and Catalysis, Vol. 21, Elsevier, Amsterdam, 1985, p. 149. J.D.F. Ramsay, R.G. Avery and L. Benest, Faraday Discuss. Chem. Sot., 76 (1983) 52. J. Penfold and J.D.F. Ramsay, J. Chem. Sot. Faraday Trans. 1, 81 (1985) 117. J.D.F. Ramsay, Oxide Sols, B.P. 1567003; USP4389385, October 1976. M.J. Vold, J. Colloid Sci., 18 (1963) 684. D.N. Sutherland, J. Colloid Interface Sci., 25 (1967) 373. T.A. Witten and L.M. Sander, Phys. Rev. B, 27 (1983) 5686. P. Meakin, Phys. Rev. A, 27 (1983) 1495. J.D.F. Ramsay, S.R. Daish and C.J. Wright, Faraday Discuss. Chem. Sot., 65 (1978) 65. A. Guinier and G. Fournet, Small-Angle Scattering of X-rays, Wiley, New York, NY, 1955,5 pp. et seq. B. Jacrot, Rep. Prog. Phys., 39 (1976) 911. R.H. Ottewill, in J.W. Goodwin (Ed.), Colloidal Dispersions, Royal Society of Chemistry, London, 1982, pp. 197. E. Dickinson, in D.H. Everett (Ed.), Colloid Science, Vol. 4 (Specialist Periodical Reports), The Royal Society of Chemistry, London, (1983) pp. 150. A. Vrij, J. Colloid Interface Sci., 90 (1982) 110. D.J. Cebula, J.W. Goodwin, G.C. Jeffrey, R.H. Ottewill, A. Parentich and R.A. Richardson, Faraday Discuss. Chem. Sot., 76 (1983) 37. J.P. Hansen and J.B. Hayter, Mol. Phys., 46 (1982) 651. B.B. Mandelbrot, Fractals, Form, Chance and Dimension, W.H. Freeman, San Francisco, CA, 1977. S.R. Forrest and T.A. Witten, J. Phys. A, Math. Nucl. Gen., 12 (1979) 109. H.D. Bale and P.W. Schmidt, Phys. Rev. Lett., 53 (1983) 596. G. Porod, Kolloid Z., 124 (1951) 83. P. Schofield, personal communication, 1984. M. Kolb, R. Botet and R. Julien, Phys. Rev. Lett., 51 (1983) 1123. G.H. Bolt, J. Phys. Chem., 61 (1957) 1166. J.D.F. Ramsay and M. Scanlon, unpublished work.