Fractal heterogeneous

Accordingly, we expect a power law behavior G,0 (O/Op)3 5 of the small strain elastic modulus for 0>0. Thereby, the exponent (3+df [j)/(3—df)w3.5 reflects the characteristic structure of the fractal heterogeneity of the filler network, i.e., the CCA-clusters. The strong dependency of G 0 on the solid fraction Op of primary aggregates reflects the effect of structure on the storage modulus. 

The established concepts predict some features of the Payne effect, that are independent of the specific types of filler. These features are in good agreement with experimental studies. For example, the Kraus-exponent m of the G drop with increasing deformation is entirely determined by the structure of the cluster network [58, 59]. Another example is the scaling relation at Eq. (70) predicting a specific power law behavior of the elastic modulus as a function of the filler volume fraction. The exponent reflects the characteristic structure of the fractal heterogeneity of the CCA-cluster network. 

Wheatcraft, S.W., G.A. Sharp, and S.W. Tyler. 1991. Fluid flow and solute transport in fractal heterogeneous porous media, p. 695-722. In J. Bear and M.Y. Corapcioglu (ed.) Transport processes in porous media. Kluwer Acad., Dordrecht, The Netherlands. 

It is easier to understand if the heterogeneous structure of gels is divided into two kinds of heterogeneity [21]. The fractal heterogeneous gel structure model is the case where multiple fractal percolation clusters connect and form networks as shown in Fig. 7(b). In this model, there are... 

D. Avnir, ed.. The Fractal Approach to Heterogeneous Chemistry, Wiley, New York, 1989. 

Avnir D (ed) 1989 The Fractal Approach to Heterogeneous Chemistry Surfaces, Colloids, Polymers (New York Wiley)... 

Different kinds of heterogeneity can be imagined. In the most simple case only a few differing structural entities are found to coexist without correlation inside the volume irradiated by the primary beam. In this case it is the task of the scientist to identify, to separate and to quantify the components of such a multimodal structure. In an extreme case heterogeneity may even result in a fractal structure that can no longer be analyzed by the classical methods of materials science. 

B. Sapoval, M. Rosso, and J. -F. Gouyet, in The Fractal Approach to Heterogeneous Chemistry, Ed. by D. Avnir, Wiley, New York, 1989, p. 227. 

It was shown, that the conception of reactive medium heterogeneity is connected with free volume representations, that it was to be expected for diffusion-controlled sohd phase reactions. If free volume microvoids were not connected with one another, then medium is heterogeneous, and in case of formation of percolation network of such microvoids - homogeneous. To obtain such definition is possible only within the framework of the fractal free volume conception. 

Keywords Imidization, nanofiller, reactive medium, heterogeneity, fractal free volume. 

Figure 4. The dependence of heterogeneity exponent h of reactive medium on relative fractal volume J for PAA solid state imidization. The notation is the same, that in figure 2.

In figure 1 the kinetic curves of reesterification reactions without catalyst and in the presence of TBT are shown. The attention is draw by itself both quantitative and qualitative differences of these Q(t) curves. The quantitative difference is expressed by much faster growth Q at t increase due to catalyst presence that was expected. The qualitative change is reflected in the Q(t) curve form change. If in the absence of TBT linear dependence was obtained, which indicates on the reaction proceeding in Euclidean (homogeneous) space [7], then in TBT presence a typical curvilinear 0(1) dependence was obtained with reaction rate dQ / dt decrease with t increase. Such reactions are typical for heterogeneous (fractal)... 

Daoud M, Martin JE (1989) In Avnir D (ed) The fractal approach to heterogeneous chemistry. WUey, New York... 

Schmidt, P. W. in The Fractal Approach to Heterogeneous Chemisty Surfaces, Colloids, Polymers Avnir, D. Ed. John Wiley Chichester 1989. 

A mathematically definable structure which exhibits the property of always appearing to have the same morphology, even when the observer endlessly enlarges portions of it. In general, fractals have three features heterogeneity, setf-similarity, and the absence of a well-defined scale of length. Fractals have become important concepts in modern nonlinear dynamics. See Chaos Theory... 

A reaction occurring at an interface of two phases. Some heterogeneous processes also display homogeneous aspects for example, a particular reaction may occur in only one of the system s phases (e.g., the rapid dissolution of a gas into a liquid followed by a particular reaction). See Fractal Reaction Kinetics... 

PHYSICAL ORGANIC CHEMISTRY NOMENCLATURE HETEROGENOUS CATALYSIS HETEROGENOUS REACTION FRACTAL REACTION KINETICS Heteroleptic,... 

Feder, J., Fractals Physics of Solids and Liquids, Plenum Press, New York 1988 Avnir, D., Ed., The Fractal Approach to Heterogeneous Chemistry, John Wiley Sons, New York, 1989. 

Avnir, D. (1989) The fractal approach to heterogenous chemistry. J. Wiley, New York Avnir, D. Farin, D. Pfeifer, P. (1983) Chemistry in noninteger dimension between two and three. II. Fractal surfaces at adsorbens. J. Chem. Phys. 79 3566-3571 Avotins, P.V. (1975) Adsorption and coprecipitation studies of mercury on hydrous iron oxide. Ph.D. Thesis, Stanford University, California, 124 p. 

Farin, D. Avnir, D. (1989) The fractal nature of molecule-surface interactions and reactions. In Avnir, D. (ed.) The fractal approach to heterogenous chemistry. Wiley, New York, 271-294... 

Pfeifer, P. Avnir, D. (1983) Chemistry in noninteger dimensions between two and three. I. Fractal Theory of heterogeneous surfaces. 

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