Fractal nature, aggregates

Consequences. The fractal nature of the aggregates implies two important properties ... 

The dependence of the elastic modulus on protein concentration has been used to establish the framework of fractal geometry. Bremer et al. (1990) indicated that a cluster of protein molecules would possess a fractal nature if the power-law dependence exists between the amount of floes in the cluster and the radius of the cluster. In addition, the magnitude of this power, signified as the fractal dimension, D, would be below 3. The elastic constant of protein aggregates could be described as a function of the aggregate volume fraction ... 

The fundamental assertion we make here regarding the fractal nature of cellulose microcrystalline aggregates is speculative. 

The mass fractal dimension gives a quantitative description of the spatial mass repartition within an aggregate, a higher fractal dimension being representative of more compact structures. Table 4.1 presents some fractal dimensions for some computer-generated and natural aggregates. The mass fractal dimension usually... 

The early fractal studies of humic materials alluded to the application of sample fractal dimensions for the description of aggregation mechanisms operant in these systems. In the first paper reporting a fractal nature for humic materials in solution, Osterberg noted that the observed fractal dimensions were indicative of an RLA process [26]. 

The fractal nature of bacterial aggregates (floes) has implications for transport processes, including movement of dissolved respiratory gases to and from the outside of the aggregate, and for movement of dissolved nutrient sources and metabolic products. Several methods have been employed to estimate fractal dimensions of floes in three dimensions. One approach is to measure light scattering of suspended floes [17,18]. The two-slopes method calculates fractal dimensions A from the slope of the cumulative size distribution for maximum length / and the slope of the cumulative solid volume F [17] ... 

The physical meaning of a proposed fractal model is not clearly apparent when the condition f ro is not obeyed. It is a general consensus that, in order to safely establish the fractal nature of an aggregate, the quotient t/r should be of the order of, or larger than 10. In addition, it must be remembered that power -dependences leading to D values smaller than 3 are also expected for non-fractal objects of low dimensionaUty such as narrow linear chains or thin platelets. Therefore, it is in general necessary to use independent arguments in order to provide additional support to a fractal model. 

It has been shown that, if the fractal nature of clusters is carefully considered, the Smoluchowski equation can express the cluster-cluster aggregation process precisely based on the comparison with the simulation by computer [14]. Assuming that the density of the cluster with the degree of pol3merization at time t is Cjff), the following relationship is obtained when there is no spatial interference ... 

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