SAXS Fractal Structure

In general we describe structuring of materials by means of domains. Frequently such domains are sufficiently smooth, and thus surface as well as volume and mass are well-defined parameters. If in Sect. 8.3.2 we would have deduced Porod s law mathematically, we would have handled domain surfaces, shades and the lengths of chords intersecting these domains (e.g., crystalline layers). 

What happens, if the surface of the domains becomes increasingly roughenedl In this case the shape of the diffuse tail of the SAXS is, again, modified. According to Ruland [140,142] an additional diffuse background is emerging. 

Characteristic for a fractal structure is self-similarity. Similar to the mentioned pores that cover all magnitudes , the general fractal is characterized by the property that typical structuring elements are re-discovered on each scale of magnification. Thus neither the surface of a surface fractal nor volume or surface of a mass fractal can be specified absolutely. We thus leave the application-oriented fundament of materials science. A so-called fractal dimension D becomes the only absolute global parameter of the material. 

Experimentally accessible is D by means of scattering methods [144]. The corresponding fractal analysis of scattering data is gaining special attractivity from its intriguing simplicity. In a double-logarithmic plot of I (x) vs. s the fractal dimension is directly obtained from the slope of the linearized scattering curve. It follows from the theory of fractals that 

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In situ SAXS investigations of a variety of sol-gel-derived silicates are consistent with the above predictions. For example, silicate species formed by hydrolysis of TEOS at pH 11.5 and H20/Si = 12, conditions in which we expect monomers to be continually produced by dissolution, are dense, uniform particles with well defined interfaces as determined in SAXS experiments by the Porod slope of -4 (non-fractal) (Brinker, C. J., Hurd, A. J. and Ward, K. D., in press). By comparison, silicate polymers formed by hydrolysis at pH 2 and H20/Si = 5, conditions in which we expect reaction-limited cluster-cluster aggregation with an absence of monomer due to the hydrolytic stability of siloxane bonds, are fractal structures characterized by D - 1.9 (Porod slope — -1.9) (29-30). 

Table 20.8. Mass fractal structural parameters for aerogels as determined by SAXS...

Kjems J K, Ircltoft T, Richter D et al (1986) Neutron-Scattering from Eractals. Physica B C 136 285-290 Aristov Y I, Lisitsa N, Zaikovski VI et al (1996) Fractal structure in base-catalyzed silica aerogels examined by TEM, SAXS and porosimetry. Reaction Kinetics and Catalysis Letters 58 367-375... 

Other. The curves show a homogeneous gel structure in the range of SAXS (Fig. 9). These aluminosilicate alcogel samples can be characterized by a fractal structure. The linear region on the log-log plot with a slope of about -2.0 indicates a mass fractal. The gel structures obtained from aluminum isopropoxide or nitrate are very similar to that of pure silica alcogel. The incorporated A1 atoms modify the elementary units rather than the fractal structure [13]. 

In one of the earhest ex situ SAXS studies on formation of titania from Ti(IV) precursors, it was demonstrated that titania oligomers were built of primary particles aggregated into larger self-similar, mass fractal structures of Df 1.6 [48]. The fractal dimension is essentially an adjustable parameter [85] by varying the acid/alkoxide ratio, stable sols (Df= 1.5-2), transparent (D =2) and turbid gels (Df = 2.9), and precipitates (Df= 3) can be obtained. 

SANS is useful in examining pore structures in nanoporous materials, or water in fractal networks (Li et al. 1994). Examples of aggregation-state analyses include studies of silica and titania nanoparticles (Hyeon-Lee et al. 1998), catanionic surfactants (Brasher and Kaler 1996) and Ti02-Si02 and Zr02-Si02 sol-gel materials (Miranda Salvado et al. 1996). Particle size distributions obtained via SANS, SAXS and TEM on nanophase Ti02-V02 catalyst particles are compared by Albertini et al. (1993). 

A primary sol particle in an acid-catalyzed sol has radius between 1 and 2 nm (3). The secondary fractal particle has a radius, R, of 5 to 20 nm as seen from saxs (3). For the TMOS-based sols investigated by saxs, increases with time, as does the Guinier radius, R. The structure reaches a fractal dimension around 2.3 at the gelation point. 

In the present paper we extend the range of scattering angles as far as five degrees and combine SAXS measurement with wide-angle x-ray diffraction measurements. Me report what we believe to be the first complete SAXS curves for cotton and Valonia cellulose. We also demonstrate how the fractal concept can be applied to explain the microcrystallite structure in cellulose. 

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