Walk dimension

The above-defined df and dt are structural parameters characterizing only the geometry of a given medium. However, when we are interested in processes like diffusion or reactions in disordered media, we need functional parameters, which are associated with the notion of time in order to characterize the dynamic behavior of the species in these media. The spectral or fracton dimension ds and random-walk dimension dw are two such parameters, and they will be defined in Section 2.2. 

To characterize the dynamic movement of particles on a fractal object, one needs two additional parameters the spectral or fracton dimension ds and the random-walk dimension dw. Both terms are quite important when diffusion phenomena are studied in disordered systems. This is so since the path of a particle or a molecule undergoing Brownian motion is a random fractal. A typical example of a random fractal is the percolation cluster shown in Figure 1.5. 

The random-walk dimension dw is useful whenever one has a specific interest in the fractal dimension of the trajectory of the random walk. The value of dw is exclusively dependent on the values of df and ds ... 

Mazo (1998) studied Taylor dispersion in fractal media and found that the proportionality constant between the spatial spreading of a solute pulse and the time depended on both the fractal dimension of the medium and the dimension of the random walk through it. In normal diffusion the average particle position is directly proportional to the time. Diffusion in fractal media is anomalous with proportional to f2/dt, where dt is the random walk dimension. 

It is interesting to compare eq. (15) with the results obtained on finitely ramified fractals by means of Green function renormalization [9-10]. It has been shown that the fractional uptake curve for a structure possessing fractal dimension dj, walk dimension d, and adsorbing from a reservoir at constant concentration c through an exchange manifold B (which represents the permeable boundary for treuisfer) possessing fractal dimension d scales as... 

As far as transport properties of a fractal structure are concerned, the mean square displacement (MSD) of a particle follows a power law, (r ) where r is the distance from the origin of the random walk and is known as the random walk dimension. In other words, diffusion on fractals is anomalous, see Sect. 2.3. Recall that for normal diffusion in three-dimensional space the MSD is given by (r ) = 6Dt. For fractals, dy, > 2, and the exponent of t in the MSD is smaller than 1. We introduce a dimensionless distance by dividing r by the typical diffusive... 

The mean-field result for this ratio is given in terms of the random walk dimension df ... 

When the actual experimental temperature used is equal to 6, xi = 1/2, at which point all excess contributions to the solution thermodynamics disappear and the solution exhibits ideal behaviour since the second virial coefficient has a value of zero. At this point the excluded volume effects that cause an expansion of the polymer molecule are exactly balanced by the unfavourable polymer-solvent interactions and the molecule adopts imperturbed, random walk dimensions. The influence on polymer dimensions and the highly detailed theories of polymer configuration in relation to the excluded volume parameter are beyond the scope of this book but are extensively covered by Yamakawa (1971) and to some extent by des Cloizeaux and Jannink (1990). 

This corresponds to the classical Flory result for the dimensions of a swollen chain in a good solvent for the case P = 1 the chain remains swollen but the degree of swelling decreases as P increases, until the mushroom reaches the ideal, unswollen random walk dimensions which occurs when P =... 

In a melt of homopolymers, the excluded volume interaction is effectively screened. There is no tendency for a chain to swell beyond the ideal random-walk dimension. Only the prefactor, or more precisely the persistence length, is governed by the very local monomer-monomer interactions. The mean-square end-to-end distance of a chain of length N has the form... 

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