|Fig. 1.-Scheme of the sedimentation procedure. Sedimentation rate of silica particles for two different values of the particle size is also shown. Solid lines correspond to a fit to Stokes law|
After synthesis, a colloidal suspension (with a typical concentration of 1010 particles/cm3) is left to settle under gravity for several weeks (sedimentation time depends on particle size). Fig. 1 illustrates both the settling process and the sedimentation rate. Fitting the sedimentation velocity to Stokes law yields a very accurate estimation of the particle diameter, which is in excellent agreement with the diameter observed by TEM. This means that particles do not stick together, but rather sediment as single particles. It also indicates that they behave as hard spheres with no interaction. There are two main conditions that guarantee that a sedimentation gives rise to a good quality crystalline structure: A) the suspension concentration should be low to allow single particle behaviour (1-3) and, therefore, to meet Stokes behaviour and B) brownian and gravitational energy should be of the same order. Under these conditions, particles arriving at the growing sedimentary surface may sample many possible sites before they stop at the position of minimum potential energy (4). When all the particles have settled, the sediment is carefully dried and later removed from the substrate. It constitutes a so-called "green" or as-grown opal sample.
|Fig. 2.-AFM 3D image of a detail of the free surface of an artificial opal. A detailed analysis of AFM images shows that natural sedimentation of silica spheres follows the Edwards-Wilkinson scaling equation|
Atomic force microscopy (AFM) can be used to characterise these samples as they need no further treatment, as would be the case with other microscopies. Besides, it may yield very valuable information on the growth process as will be explained next. A typical AFM 3D image of the free surface of a green sample showing a triangular arrangement can be seen in Fig. 2. It shows that information on the topography of the samples can be obtained from AFM images. Fast Fourier transform of longer domains confirms a perfect order in a length range up to ca. 60 m m. At longer range, domains with defects (vacancies, and dislocations) are observed. A deep analysis of the surface roughness extracted from the AFM images reveals that the interface properties, for length scales between the nanosphere diameter (d) and the average domain size (D), can be well described by simple scaling theory equations. The study of the sedimentation of these colloidal particles could be useful as they can be regarded as a scaled-up model for atomic crystal growth with charges, sizes and length scales 102-105 times larger than atomic species. These particles constitute "ions" whose interaction, in the case of aqueous solutions, can be screened by either OH- or H3O+ species (5). The understanding of growth phenomena is of technological interest in techniques as Molecular Beam Epitaxy, Chemical Vapour Deposition, etc. The mechanism of crystal growth on a flat solid substrate determines the dynamics and structure of the growing interface. The interface evolution can be modelled by means of the dynamic scaling theory applied to AFM surface profiles on different time scales. This theory predicts that the root mean square roughness (interface width), x (L,h), for a scan length L and average thickness h, scales, when h® 0, as:
x (L,h)µ hb
whereas for h® Ą it scales as:
x (L)µ La
b and a being the dynamic and static surface roughness exponents, respectively. The growth process should involve local effects such as stochastic noise, site-dependent growth, and surface relaxation. One of the simplest models for phase growth is the random deposition model with surface relaxation (6). In this model each particle falls along a randomly selected single column toward the surface until it reaches the top of the interface wherefrom it "diffuses" along the surface to a finite distance stopping when it reaches the position with the lowest height. This simple model is well described by the Edwards-Wilkinson (EW) continuum equation (1):
where n is the surface tension term and h is the stochastic noise, which is a function of time and space. V is the constant average velocity of the propagating surface. For a 3D system this equation predicts a =0 and b =0. In such case, correlations decay logarithmically and the following behaviour is expected:
x µ (log<h>)0.5 for 1 < h < L
x µ (log(L))0.5 for 1 < L < h
|Fig. 3.- x versus L, semilogarithmic plot for a 750 µm thick film. The continuous line indicates the best x µ (log(L))0.5 fit for d < L < D. The dashed curve corresponds to the power spectral density of the same AFM data|
This oversimplified growth model is hardly expected to be applicable to real systems, as it has been shown that the EW equation can be reduced to a perfect equilibrium problem (8), whereas surface growth is typically an irreversible process (9). Thus, it is not surprising that only a few experimental systems have been reported to follow the EW equation. It has nevertheless been recently found that this is the case for the growth of interfaces generated by natural sedimentation of SiO2 colloidal particles (10). In Fig. 3 we show a semilogarithmic plot of the roughness x versus the length of the AFM scanning window L. The x vs. L behaviour for d<L<D supports the ability of the EW interface equation to describe the solid surface produced by natural sedimentation of spheres.