Fractals characterisation

At first, equilibrium time was increased in order to try to eliminate the low-pressure deviation. This proved to be impossible. The upward shift of the isotherm predicted in Fig 2 was only observed when the initial equilibrium times were very short. Once a reasonable equilibrium time was reached (in the order of several hours per point) the isotherm stayed identical. Even for equilibrium times as high as 12 hours per point the isotherm did not shift towards the Dubinin-Radushkevich line. From this, it was concluded that the deviation for low partial pressures was not an experimental artifact. In fact, this has already been suggested by several authors [6,7], but little attention has been given to these results. Partly because their observations and conclusions were mainly directed towards the fractal characterisation of porous materials. 

Blacher S, Heinrichs B, Sahouli B, Pirard R and Pirard J-P. Fractal characterisation of wide pore range catalysts Application to Pd-Ag/Si02 xerogels. Journal of Colloid and Interface Science 2000 226 123-130... 

The series is indeed being well received, and is growing prosperously. New volumes, on fractal properties of soil particles and physical techniques for micro/nanoparticle characterisation respectively, are in an early stage of preparation. As with all books in the series, these volumes will present critical reviews that reflect the current state of the art and provide guidelines for future research in the field. 

Barret, A. M., Normand, M.D., Peleg, M. and Ross, E. (1992). Characterisation of the jagged stress-strain relationships of puffed extrudates using the Fast Fourier Transform and fractal analysis. J. Food Sci. 57, 227-232, 235. 

Kaye BH (1986) Image analysis procedures for characterising the fractal dimension of fine particles. Proc Part Technol Conf Nurnberg... 

Bolton, R. G. Boddy, L. (1993). Characterisation of the spatial aspects of foraging mycelial cord systems using fractal geometry. Mycological Research, 97, 762-8. 

Earin, D. Avnir, D. The fractal nature of molecule-surface chemical activities and physical interactions in porous materials. In Characterisation of Porous Solids Unger, K.K., Rouquerol, J., Sing, K.S.W., Krai, H., Eds. Elsevier Science Amsterdam, 1988 421 32. 

In some specific cases other parameters can be considered as important in the characterisation of membrane morphology like the surface roughness, pore anisotropy and porous network connectivity [16,17]. Concepts of percolation and fractal geometry are also of interest to better describe the statistical and random structures of many porous solids [14,18,19]. 

The techniques given in Table 4.2 are well established and have been sub-divided into those which are described as either static or dynamic. We feel this distinction is of particular importance in the characterisation of the porous structure of membranes. Here the performance is determined by the complex link between the structural texture and transport behaviour. An insight into this complexity is frequently provided by dynamic techniques, which are not restricted by the limited quantity of membrane material and are sensitive to the active pathways through the porous structure. Further developments are required in this area both in the improvement of existing techniques and introduction of new techniques. Progress will also come from advances in the theory and modelling of flow behaviour in such porous media, which involve percolation theory and fractal geometry for example. With the refinement of such... 

Fractal dimension, D is considered as an effective number that characterises the irregular electrode surface. The term has been related to physical quantities such as mass distribution, density of vibrational stages, conductivity and elasticity. If we consider a 2-D fractal picture in its self-similar multi-steps, one can draw various spheres of known radii at various points of its structure and may thus count the number of particles, N inside the sphere by microscope, following relation will then hold good ... 

Yet another important property of fractals which distinguishes them from traditional Euclidean objects is that at least three dimensions have to be determined, namely, d, the dimension of the enveloping Euclidean space, df, the fractal (Hausdorff) dimension, and d the spectral (fraction) dimension, which characterises the object connectivity. [For Euclidean spaces, d = d = d this allows Euclidean objects to be regarded as a specific ( degenerate ) case of fractal objects. Below we shall repeatedly encounter this statement] [27]. This means that two fractal dimensions, d( and d are needed to describe the structure of a fractal object (for example, a polymer) even when the d value is fixed. This situation corresponds to the statement of non-equilibrium thermodynamics according to which at least two parameters of order are required to describe thermodynamically nonequilibrium solids (polymers), for which the Prigogine-Defay criterion is not met [28, 29]. 

The critical indices estimated from these relations fall into the admissible ranges of variation P = 0.39-0.40, V = 0.8-0.9, and t = 1.6-1.8, determined in terms of the percolation model for three-dimensional systems. The researchers [7] noted that not only numerical values but also the meanings of these values coincide. Thus the index P characterises the chain structure of a percolation cluster. The 1/p value, which serves as the index of the first subset of the fractal percolation cluster in the model considered [7], also determines the chain structure of the cluster. The index v is related to the cellular texture of the percolation cluster. The 2/df index of the second subset of the fractal percolation cluster is also associated with the cellular structure. By analogy, the index t defines the large-cellular skeleton of the fractal percolation cluster. The relationship between the critical percolation indices and the fractal dimension of the percolation cluster for three-dimensional systems and examples of determination of these values for filled polymers are considered in more detail in the book cited [7]. Thus, these critical indices are universal and significant for analysis of complex systems, the behaviour of which can be interpreted in terms of the percolation theory. 

Fractal objects are characterised by the following relationship between the mass M (or density p) and the linear scale of measurement L [3] ... 

Secondly, polymers are known to possess multilevel structures (molecular, topological, supermolecular, and floccular or block levels), the elements of which are interconnected [43, 44]. In addition, an external action on a polymer can induce the formation of new (secondary) structural elements — cracks, fractured surfaces, plastic flow regions, etc. These primary and secondary structural elements as well as the processes forming them are characterised by miscellaneous parameters therefore, only empirical correlations have been obtained, at best, between these parameters. If each of the above-mentioned elements (processes) is described by a standard parameter, for example, fractal dimension, one can derive analytical equations relating them to one another and containing no adjustable parameters. This is very significant for the computer synthesis of structure and for the prediction of properties and behaviour of polymeric materials during performance. Note that fractal analysis has been used successfully to describe the phenomena of rubber elasticity [16, 45, 46] and fluidity [25, 47-49]. 

Yet another factor which also influences the shape of the cluster is whether or not attractive interactions are present. As the temperature increases, the attractive interactions diminish. For example, no interactions of this sort are found in isolated macromolecular coils in good solvents at high temperature [77]. Family [9] defined this state of a statistical fractal as an uncoiled state because in this case, it is characterised by the smallest fractal dimension. 

Fractal analysis allows consideration of the surface structure of the filler particles, which are characterised by its fractal dimension (d ) and by the self-similarity interval. Because the polymer structure is also described in the framework of the fractal analysis, it becomes possible to consider the interaction between the filler surface and polymer matrix, including the interfacial layers, based on the analysis of their fractal dimensions. Application of the model of irreversible aggregation allows description of the processes of aggregation of the filler particles in a particulate filled composites. This aggregation causes changes... 

The structure of the filler particle surfaces and of the polymer surface characterised hy their fractal dimensions, affects the interfacial adhesion in composites. To explain the structural effect let us introduce the concept of the accessibility of the sites on these surfaces to form adhesion joints (physical or chemical). As a first approximation the degree of such accessibility may be defined as a difference of the fractal dimensions of two surfaces. The higher is this difference the lower is the accessibility of the surface and the less is the adhesion [21]. Suppose that the filler particle has a very rough surface with dimensions which are close to the Euclidean dimension d = 3 (for example, AI2O3 particles) [33], whereas the polymer surface is very smooth, i.e., dp = d = 2. In this case the contact between two surfaces is possible only at the apexes of the rough surface of the filler and the result could be very low adhesion. In other words, the disparity of the dimensions determines the inaccessibility of the greater part of the filler particle for the formation of adhesion bonds [21]. 

The value of the fractal dimension of the surface of a fractal object is taken to being equal to the corresponding parameter of its structure [21] and because of it for the polymer matrix may be taken as fractal dimension of the polymer dp. In such a way the degree of accessibility of the filler particles may be characterised by one more parameter ... 

This chapter considers the reasons for a variation of microgel structure characterised by its fractal dimension, D, formed in the cure of epoxy resin systems. Quantitatively, change of D during the increase of reaction time is well described within the framework of mechanism of aggregation cluster - cluster. The fractal space, in which the reaction curing proceeds, is formed by a structure of the greatest cluster in system. 

It has been found that the basic elements of structure of initial and deformed polymers are homogeneous fractals that can be characterised by their fractal dimension. Examples of such elements are macromolecular coil, supermolecular organisation as seen in the cluster structure and a stable crack in film samples of polymers. These examples are given as examples of the possible variants of the term "multifractality with reference to polymers. 

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