This expression defines the fractal dimension, average density function given by g(r) = < P(r’)p(r’
D, in terms of the ensemble
+ r) >/P(r)
where p (r’) is the density It also follows that
g(r) 01 rtDed)
Furthermore, it can be shown that the scattered I(Q)] is given [ 121 by the general expression:
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
power law rela-
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
(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  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 
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:
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 . RESULTS
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  . 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
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
_I ~ 0025
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
Results from concentration Cont. 0.137 0.266 0.550
fitting the MSA at pH - 8
on S3 silica sols of different
0 (J.IC cm-*)
* 0 (mV
9.8 9.3 8.4
1.01 0.71 0.66
28 26 19
35 27 21
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
Fig. 4. Dependence of electrophoretic mobility, dilute (< 10.’ g cm-3) silica sol S3 ([Na’] = 10.’
u, and zeta mol dm-‘).
p, on pH for a
obtained by conductimetric titration (> 2 /.LCcme2) . These differences have been discussed in detail elsewhere  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 , 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.
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 . 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
plotted against Q for S3 sols of similar concentrations pH, (a) 7.2 and (b) 4.5, both dialysed against 10.’
0.19 g cm-‘) drne3 NaNO,.
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 , 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
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
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-‘.
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.
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 . 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
Light scattering Alumina C
s130 s200 S380 A(C?
1 -6 3 x lo-‘-3 8 x 10-‘--8X 1 x 10-h-5
range (g ml-‘) x 10.’
x 1o-3 lo-* x 10-h
after passing through
D -1.7 - 1.7-1.8-1.5 filter.
-1 -1 <1 -2
-3.7 -3.9 -3.0 -4.0
x x x x
10.’ lo-) 10-l 10.’
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
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.