Fractal-type exponent

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. 

These two extreme models cannot be distinguished from a poor autocorrelation curve with a logarithmic time scale. Conventional analyses for cases with anomalous diffusion use alternative types of equation with fractal consideration with an exponent on D or t as... 

Determination of D is the first step in studying macromolecular coils by fractal analysis. D is usually estimated by finding the exponents in the Mark-Kuhn-Houwink type equation, which relate the characteristic viscosity [r ], the translational diffusion coefficient Dq, or the rate sedimentation coefficient Sq) with the molecular weight (M) of polymers [3] ... 

In Fig. 21 the kinetic curves conversion degree—reaction duration Q-t for two polyols on the basis of ethyleneglycole (PO-1) and propylene-glycole (PO-2) are adduced. As it was to be expected, these curves had autodecelerated character, that is, reaction rate was decreased with time. Such type of kinetic curves is typical for fractal reactions, to which either fractal objects reactions or reactions in fractal spaces are attributed [85], In case of Euclidean reactions the linear kinetics (i> =const) is observed. The general Eq. (2.107) was used for the description of fractal reactions kinetics. From this relationship it follows, that the plot Q t) construction in double logarithmic coordinates allows to determine the exponent value in this relationship and, hence, the fractal dimension value. In Fig. 3.22 such dependence for PO-1 is adduced, from which it follows, that it consists of two linear sections, allowing to perform the indicated above estimation. For small t t 50 min) the linear section slope is higher and A =2.648 and for i>50 min A =2.693. Such A increase or macromolecular coil density enhancement in reaction course is predicted by the irreversible... 

Let us consider this question for the most often applied in practice reactive medium type, namely, polymer solutions. The general fractal description of reaction in solution kinetics is given by the Eq. (26). In its turn, the reaction rate 9 can be obtained by the Eq. (26) ddferentiation by time t, that gives the Eq. (69). The comparison of exponents in the Eqs. (108) and (69) allows to obtain the following equality [178] ... 

Note that the critical exponents do not take mean-field values, even when the fractal dimension of the lattice becomes greater than 4. This is clearly because of the special structure of the n-simplex lattice, where, even though the fractal dimension becomes greater than 4 for large n, the spectral dimension remains below 2, and probability of intersection of paths of random walks remains large. Also that the non-analytical dependence of the type in the critical exponents on the lattice cannot be obtained... 

We note that the leading correction to asymptotic value of the exponent is proportional to 2 — dt>. The next correction term is of order l/log6, and is proportional to 2—Db, where )(, = 2 — e is fractal dimension. These are like the e-expansions, except that there are several inequivalent definitions of dimension for fractals. Thus there are several e s, and the exponents on fractals may require a multi-variable e-expansion. Interestingly, at higher orders, corrections to scaling to the finite-size scaling functions f x), g x) etc. would give corrections to the exponents of the type 1/6. These are of the type exp(—A/e), and such correction terms are not calculable within the conventional e-expansions framework. 

The fractal exponent is thus significantly smaller than that of a simple random walk, and the structure of the coil is considerably more open. This result is in good agreement with experiment, as shown by the example of polystyrene dissolved in benzene, at sufficiently low concentrations that the polymers cannot penetrate each other (which would totally change the type of solution). 

Formally speaking, we postulate the existence of a walk fractal exponent D, giving the time t required to cover a distance r by a relation of the type t(r), where (and it amounts to the same thing)... 

It was found subsequently that, although fractal geometry produces CPE behavior, in practice there is no relation between the CPE exponent and fractal dimensions [333, 334]. Qualitatively, however, higher fractal dimensions lead to smaller values of different type of fractals like Cantor bars [335-338] or Sierpihski carpets [339-341], for which different relations hold. This means that the impedance technique does not allow for the determination of the surface fractal dimension. Such information can be obtained by the analysis of current-time curves in the presence of diffusion to the surface [323, 324,342-344]. 

We close this section by noting that the relations between the scaling exponent and the spectral dimension are very general. We will meet them again in Sect. 9.3, in the study of regular hyperbranched fractals. It is also noticeable that the inclusion of hydrodynamic interactions into the dynamic picture leads to the loss of scaling for Sierpinski-type polymers in the intermediate regime [116,117]. 

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