Scaling limit disordered models

Small-angle neutron scattering (SANS) can be applied to food systems to obtain information on intra- and inter-particle structure, on a length scale of typically 10-1000 A. The systems studied are usually disordered, and so only a limited number of parameters can be determined. Some model systems (e.g., certain microemulsions) are characterized by only a limited number of parameters, and so SANS can describe them fully without complementary techniques. Food systems, however, are often disordered, polydisperse and complex. For these systems, SANS is rarely used alone. Instead, it is used to study systems that have already been well characterized by other methods, viz., light scattering, electron microscopy, NMR, fluorescence, etc. SANS data can then be used to test alternative models, or to derive quantitative parameters for an existing qualitative model. 

Disordered porous media have been adequately described by the fractal concept [154,216]. It was shown that if the pore space is determined by its fractal structure, the regular fractal model could be applied [154]. This implies that for the volume element of linear size A, the volume of the pore space is given in units of the characteristic pore size X by Vp = Gg(A/X)°r, where I), is the regular fractal dimension of the porous space, A coincides with the upper limit, and X coincides with the lower limit of the self-similarity. The constant G, is a geometric factor. Similarly, the volume of the whole sample is scaled as V Gg(A/X)d, where d is the Euclidean dimension (d = 3). Hence, the formula for the macroscopic porosity in terms of the regular fractal model can be derived from (65) and is given by... 

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 ... 

We close this introduction with a final remark about the modelling of the failure. In a real situation, failure takes place in solid samples which are, by nature, continuous in space. However, many studies (numerical and experimental) have been made on lattices. In all these studies, it is an implicit assumption that one can replace a continuous solid by a lattice. For example, a conducting solid can be described by a lattice in which the bonds between sites are identical resistors. It is a very common practice in percolation type models of disordered solids. We stress that this transformation (continuous solid to lattice) defines a particular length scale the length of the unit cell of the lattice. This implies that defects appear by discrete steps and this does not correspond always to real situations. We shall see later how to remove this limitation. 

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