Aggregation fractal particle

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). 

Fig. 41. Typical 2D fractal structure obtained by aggregation of particles in the journal bearing flow. Fractal dimension of the cluster is 1.54 (Hansen and Ottino, 1996b).

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.

Now various structures—for example, aggregates of particles in colloids, certain binary solutions, polymers, composites, and so on—can be conceived as fractal. Materials with a fractal structure belong to a wide class of inhomogeneous media and may exhibit properties differing from those of uniform matter, like crystals, ordinary composites, or homogeneous... 

The surface area per unit mass. A, of a fractal particle depends on the radius of the individual particles, r, the radius of the aggregate, R, and the fractal dimension ... 

FIGURE 6.21 Computer generated aggregate assuming particle-partide aggregation. Fractal dimension of 1.5 in two dimensions made to simulate a fractal dimension of 2.5 in three dimensions. Printed, with permission from Sutherland [72]. 

Diffusion-limited aggregation of particles results in a fractal object. Growth processes that are apparendy disordered also form fractal objects (30). Sol—gel particle growth has also been modeled using fractal concepts (3,20). The nature of fractals requires that they be invariant with scale, ie, the fractal must look similar regardless of the level of detail chosen. The second requirement for mass fractals is that their density decreases with size. Thus, the fractal model overcomes the problem of increasing density of the classical models of gelation, yet retains many of its desirable features. The mass of a fractal, Af, is related to the fractal dimension and its size or radius, R, by equationS ... 

Fig. 10 Schematic diagram showing that (A) for a compact aggregate, the particles on the edge will be shed first because of the lower interparticle forces because of less nearest neighboring particles. The final aggregate will have a lower boundary fractal dimension, although the structure compactness is preserved. (B) For a loose aggregate, after the rupture of the chain structure, some aggregates (shown in gray color) are more compact and also have lower boundary fractal dimension.

Fractal structures have been examined, in particular, in diffusion-controlled aggregation process (polymerization) [7-9], in colloids (aggregates of particles) [10-13], and in percolation clusters [1-3],... 

In particular, the analytical concentration seems alone to be a parameter quite unsuitable for this purpose. Moreover, the relationship of the mean activity coefQcient to the degree of association is exponential. Although a premature conclusion, it may be supposed that fractal aggregation (to particles) may be related to the mean activity coefQcient. 

Problem 7-24. Sedimentation of a Colloidal Aggregate. Colloidal particles often aggregate because of London-van der Waals or other attractive interparticle forces unless measures are taken to stabilize them. The aggregation kinetics are such that the aggregate formed has a fractal dimension Df, which is often less than the spatial dimension. The fractal dimension measures the amount of mass in a sphere of radius R, i.e., mass R D<. For a fractal aggregate composed of Aprimary particles of radius Op with mass mp, estimate the sedimentation velocity of the aggregate when the Reynolds number for the motion is small. What is the appropriate Reynolds number ... 

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. 

Particle size and aggregate fractal dimension were measured with a Malvern Instruments Mastersizer/E. A 100 mm lens was used to measure particle sizes between 200 nm and 110 im (the 45 mm lens allowed particle analysis from 50 nm to 80 pm). Measurements were taken immediately after filling the cell to avoid settling effects, and no stirring was applied. 

An efficient and practically convenient method of metal nanoparticles deposition has been developed which enables covering a large semiconductor surface area with nanoparticles of various metals. One can fabricate isolated nanoparticles, self-organized ensembles, both ordered and disordered, and also dense fractal aggregates. The particle size is in the range of 5-70 nm depending on the nature of metals. The shape of the nanoparticle ensembles is determined by a microrelief of the semiconductor surface. 

Hence, the aforementioned results have shown that nanofiller particle (aggregates of particles) chains in elastomeric nanocomposites are physical fractal within self-similarity (and, hence, fractality [41]) range of -500-1,450 nm. In this range, their dimension can be estimated accord-... 

As it has been noted earlier [45], the linearity of the plots, corresponding to Eqs. (6.23) and (6.25), and nonintegral value do not guarantee object self-similarity (and, hence, fractality). The nanofiller particle (aggregates of particles) structure low dimensions are due to the initial nanofiller particles surface high fractal dimension. 

In Fig. 1.2, the images of the nanocomposites studied, obtained in the force modulation regime, and corresponding to them nanoparticles aggregate, fractal dimension (d distributions are shown. As it follows Ifom the deduced values of dj = 2.40-2.48), nanofiller particles aggregates in the... 

Up to now we considered pol5meric fiiactals behavior in Euclidean spaces only (for the most often realized in practice case fractals structure formation can occur in fractal spaces as well (fractal lattices in case of computer simulation), that influences essentially on polymeric fractals dimension value. This problem represents not only purely theoretical interest, but gives important practical applications. So, in case of polymer composites it has been shown [45] that particles (aggregates of particles) of filler form bulk network, having fractal dimension, changing within the wide enough limits. In its turn, this network defines composite polymer matrix structure, characterized by its fractal dimension polymer material properties. And on the contrary, the absence in particulate-filled polymer nanocomposites of such network results in polymer matrix structure invariability at nanofiller contents variation and its fractal dimension remains constant and equal to this parameter for matrix polymer [46]. 

The Eq. (27) application gave an exact enough (within the limits of 6%) description of composites polymier matrix fractal dimension change in fractal space, created by filler particles (aggregates of particles) network [45]. 

---