Fractal particle

A primary sol particle in an acid-cataly2ed 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 stmcture reaches a fractal dimension around 2.3 at the gelation point. 

Figure 7.12 Sequence of CLSM images recorded during melting of a 24-hour-old fractal particle gel formed by quenching a 1 wt% gelatin + 7 wt% oxidized starch system from 40 to 24 °C (a) immediately before heating (b) network rearrangement and coarsening (c) network break-up (d) aggregate melting into polydisperse liquid droplets. Reproduced from Firoozmand el al. (2007) with permission.

When a particle s surface is not uniformly smooth like that of a sphere but has a texture like that of broccoli or catdiflower, the particle is said to be fractal [12]. With the two-dimensional projection of a fractal particle, we find that the perimeter of the particle increases without limit as the size of the ruler used as a measure decreases in length. The circumference, C, estimated with a ruler of size x is proportional to... 

The properties of fractal ceramic powders depend on their fractal dimensions. The density, p, of a fractal particle depends on its radius, R, and the fractal dimension ... 

Fractal particles modify external phenomena in the surrounding space (e.g., a conducting fractal modifies the electric field around it, an absorbing fractal modifies the concentration of the diffusing species around it, and a fi ctal immersed in a fluid modifies fluid flow around... 

Another interesting property of fractals is the intersection probability between two fractals. The number of intersections, M i, between two fractal particles 1 and 2, both of size R, are placed independently in the same region of space is... 

Porosity within a particle is a manifestation of the shape of a particle. Fractal particles will have internal porosity as a result of their shapes. Fractal particles with low fractal dimensions (i.e., <2.0) will have a broad pore size distribution, where the largest pore approaches the size of the aggregate. Fractal particles with large fractal dimensions (i.e., >2.0) will have narrower pore size distributions with most of the porosity occurring at a size much smaller than that of the aggregate. Calcination of metal salt particles or metal hydroxides to produce oxides is another common method to produce internal porosity. In the gas evolution that takes place in transformation to the oxide, pores are opened up in the particle structure. The opening of pores in a hydrous alumina powder can increase its surface area from 0.5 m /gm (its external area) to 450 mVgm (its internal pore area). 

The porosity can be controlled in two ways. The first method is based on the scaling of mass Mf and size tf of the mass fractal particles. Since density equals mass/volume, the density pf of a mass fractal object varies in three-dimensional space as ... 

There is, however, one basic principle, which derives from the fact that fractal clusters are scale invariant. When a gel is formed, clusters of size Rg make bonds with each other via strands at the periphery of the cluster, and the average number of the strands involved per cluster will not depend on cluster size. Since the apparent surface area of a cluster scales with R2 the number of junctions between clusters per unit area of cross section of the gel will scale with R 2. By using the equation for Rg we arrive at the following equation for the shear modulus of a fractal particle gel ... 

A general conclusion may be that fractal particle gels follow some simple scaling laws that provide much insight. However, it needs much more than a knowledge of the values of (p, D, and ae to predict quantitatively the mechanical properties. For every gel type, careful study at a range of conditions is needed to obtain a full picture. 

For soya protein gels made at a low pH (3.8), relations between log B and log c, as well as between log G and log c (for a = 2, i.e., stretched strands), D values of 2.3 resulted. In many cases, however, the relations do not agree well with those for fractal particle gels. In conclusion, fractal aggregation is generally essential in obtaining a heat-set protein gel, but the final structure is mostly not fractal any more. 

Filler Particles. We will consider gels containing particles that are much larger than the pores in the gel network. The primary gel (i.e., without particles) may be a polymer gel, a fractal particle gel as obtained with casein, or a heat-set protein gel. We will consider the effect of filler particles on gel properties, especially the elastic modulus, which has been studied best. The following factors are known to affect the properties. 

TABLE 17.5 Comparison of Some Properties of a Typical Polymer Gel, a Typical Fractal Particle Gel, and a Heat-Set Protein Gel. 

The physical significance of this last relationship is that, as the fractals grow larger, they become less dense. Even though the mass concentration of oxide precursor is constant, the volume concentration of fractal particles is increasing because of the decreasing density of the fractals themselves ... 

K. -H. Naumann, H. Bunz, Van dta Waals inhaactions between fractal particles. J. Aerosol Sci. 24... 

In our laboratory, we have recently conducted gelation studies of silica nanopardcles in microgravity during the STS-95 space shuttle mission (28). Stable silica nanoparticle dispersions may be form either by polymerization of silicic acids in an aqueous system or through hydrolysis and condensation of silicon alkoxides (the sol-gel or Stober route). Comparison of small-angle x-ray scattering (SAXS) measurements of Ludox, a commercial aqueous silicate with acid- and base-catalyzed alkoxides shows that only aqueous silicate sols are uniform, whereas alkoxides generate fractal particles. As Brinker and Scherer point out (29), these results illustrate that sols derived from aqueous silicates are... 

Fumed silica is "glass soot" made by burning silicates in flames. It is known to be mass-fractal particles, i.e. submicron-size aggregates composed of random, weakly branched strings of 100 A silica spheres, with a great deal of internal porosity. 

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