Self-similarity and scaling laws

Power law relaxation is no guarantee for a gel point. It should be noted that, besides materials near LST, there exist materials which show the very simple power law relaxation behavior over quite extended time windows. Such behavior has been termed self-similar or scale invariant since it is the same at any time scale of observation (within the given time window). Self-similar relaxation has been associated with self-similar structures on the molecular and super-molecular level and, for suspensions and emulsions, on particulate level. Such self-similar relaxation is only found over a finite range of relaxation times, i.e. between a lower and an upper cut-off, and 2U. The exponent may adopt negative or positive values, however, with different consequences and... 

This form neglects the self-scattering term appropriate for the ktr— regime, but which is irrelevant in a continuum-of-sites description. Equation (3.1) very accurately describes the exact Gaussian continuum model. In particular, it exactly reproduces the )k = 0 value and the self-similar intermediate scaling regime, d> k) = 2 k(Ty for R k (r. In real space, this selfsimilar behavior corresponds to power law, or critical-like, correlations, a (r) acr". This is a polymeric effect associated with the ideal random... 

The second contribution to g(r) in Eq. (3.2) is called the correlation hole effect by deGennes and is associated with the longer wavelength universal aspects of chain connectivity and interchain repulsive forces. On intermediate length scales it has a critical power law form due to chain conformation self-similarity, and this simple analytic form remains an excellent representation even for chemically realistic models when intersite separations exceed an intrinsic (/V-independent) distance of the order of 3-5 site diameters... 

Intensive research works have been carried out by Winter et al., over many years to elucidate the pewer law observed for the sol-gel transition, which is shown in Section 2.1.3. (Winter, 1997) The concepts of mechanical self-similarity, and further topwlogical selfsimilarity as its extension, have been introduced by their studies, where the same properties are found at different length scales within the observed network. Afterwards, their research results have been accepted broadly. Many works follow to find the experimental results of various polymer gels to obey power law relaxation. With these research results, theoretical works in relation to sol-gel transition is discussed in the following three sections. 

The Universal Hopkinson-Cranz and Sachs Laws of Blast Scaling have both been verified by experiment. These laws state that self-similar blast (shock) waves are produced at idendcal scaled distances when two explosive charges of similar geometry and the same explosive composition, but of different size, are detonated in the same atmosphere [49]. 

MeV required in proton-therapy for an effective treatment of deep seated tumors [26]. Fuchs and co-authors have proposed a scaling law [27], allowing the necessary laser parameters to produce proton beams of interest for such applications to be estimated. In their work, best suited to hundreds of fs/some ps duration laser pulses, they use the self-similar fluid model proposed by Mora [28] giving the following estimate for the maximum FWD proton energy ... 

Irrespective of the origin of fractals or fractal-like behavior in experimental studies, the investigator has to derive an estimate for df from the data. Since strict self-similarity principles cannot be applied to experimental data extracted from irregularly shaped objects, the estimation of df is accomplished with methods that unveil either the underlying replacement rule using self-similarity principles or the power-law scaling. Both approaches give identical results and they will be described briefly. 

Therefore, the power-law behavior itself is a self-similar phenomenon, i.e., doubling of the time is matched by a specific fractional reduction of the function, which is independent of the chosen starting time self-similarity, independent of scale is equivalent to a statement that the process is fractal. Although not all power-law relationships are due to fractals, the existence of such a relationship should alert the observer to seriously consider whether the system is self-similar. The dimensionless character of a is unique. It might be a reflection of the fractal nature of the body (both in terms of structure and function) and it can also be linked with species invariance. This means that a can be found to be similar in various species. Moreover, a could also be thought of as the reflection of a combination of structure of the body (capillaries plus eliminating organs) and function (diffusion characteristics plus clearance concepts). 

A useful (also extreme) counterpart to the also idealized linear geometry is fractal geometry which plays a key role in many non-linear processes.280 281 If one measures the length of a fractal interface with different scales, it can be seen that it increases with decreasing scale since more and more details are included. The number which counts how often the scale e is to be applied to measure the fractal object, is not inversely proportional to ebut to a power law function of e with the exponent d being characteristic for the self-similarity of the structure d is called the Hausdorff-dimension. Diffusion limited aggregation is a process that typically leads to fractal structures.283 That this is a nonlinear process follows from the complete neglect of the back-reaction. The impedance of the tree-like metal in Fig. 76 synthesized by electrolysis does not only look like a fractal, it also shows the impedance behavior expected for a fractal electrode.284... 

Fractals can be considered as disordered systems with a non-integral dimension, called the fractal dimension. An important property of fractal objects is that they are self-similar, independent of scale. This means that if part of them is cutout, and then this part is magnified, the resulting object will look exactly the same as the original one. The other distinct property of a fractal is the power law or scaling behavior, where the property and variable of a system are related in the following manner ... 

We have demonstrated miscible blends of PET-PHB/PEI can be formed by rapid solvent casting from the mixed solvent of phenol and tetrachloroethane. The miscibility was confirmed by the systematic movement of Tg in the DSC studies. However, the blend is unstable and undergoes thermally induced phase separation with a miscibility window reminiscent of LCST. The dynamics of spinodal decomposition is non-linear in character and obeys the power law with kinetic exponents of -1/3 and 1 in accordance with the cluster dynamics of Binder and Stauffer as well as of Furukawa. In the temporal scaling analysis, the structure function exhibits universality with time, suggesting temporal self-similarity of the system. 

---