Fractal structure of silica colloids revisited

We present extensive computer simulations of a modified Eden model on a tetragonal lattice. This model is able to reproduce the crossover from mass fractal to surface fractal structures observed experimentally in the clustering process of partially hydrolysed silica particles. One of the most interesting phenomena in random growth is that systems with only shortrange interactions can form aggregates whose large-scale structures are statistically well defined. Many of these structures can be described as mass fractals or surface fractals. [1, 2]. For mass fractals, the mass (M) and the corresponding radius of gyration (R) obey the scaling relationship: M ∼ Rf where Df is the fractal dimension; 1 6 Df < d. In contrast to mass fractals, the mass of a surface fractal scales with the radius in a Euclidean fashion (Df = d), but the surface area S now increases with the radius more rapidly: S ∼ Rs ; d − 1 6 Ds < d. Scattering experiments are used to quantitatively determine the structure of such systems. For mass fractals, the scattering intensity as a function of wavevectors, I (k), shows a powerlaw behaviour for large k given by [3] I (k) ∼ k−Df except for the case Df → d, when instead we obtain the asymptotic form I (k) ∼ k−(d+1) which is Porod’s law [4, 5]. For surface fractals, the scattering from the bulk is parallel to the incident beam, and the scattering at finite angles arises only from the surface. In this case, the scattering intensity follows a power law [6] I (k) ∼ kDs−2d . Thus, scattering measurements can distinguish between mass fractals, whose scattering curves, when plotted on a log–log scale, will be straight lines with slopes lying between −1 and −3, for a three-dimensional system, while the scattering curves for surface fractals will have slopes larger than −3. Ceramic precursors are a typical example of systems that exhibit a variety of random structures. These range from highly ramified fractal objects to homogeneous colloidal particles with fractally rough surfaces [7, 8]. A prototypical example which we now describe in detail is a particular silica species, of chemical formula (SiOC2H5)4−n(OH)n, 1 6 n 6 4, that grows in solution by condensation of partially hydrolysed silicon tetraethoxide (TEOS). 0305-4470/96/030533+08$19.50 c © 1996 IOP Publishing Ltd 533