Power law relaxation behavior

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... 

Power law relaxation behavior is also expected (or has already been found) for other critical systems. Even molten polymers with linear chains of high molecular weight relax in a self-similar pattern if all chains are of uniform length [61]. 

Fig. 3. Evolution of relaxation modulus of a cross-linking polymer as shown with five samples of increased cross-link density. Parameter is the reaction time distance from the gel point t — tc). The values are calculated from dynamic mechanical data (13). The power law relaxation is limiting behavior for the liquid and the solid. One of the samples is very close to the critical gel. However, at very long times it deviates from the power law behavior. It is still a fluid.

A wide variety of polymeric materials exhibit self-similar relaxation behavior with positive or negative relaxation exponents. Positive exponents are only found with highly entangled chains if the chains are linear, flexible, and of uniform length [61] the power law spectrum here describes the relaxation behavior in the entanglement and flow region. 

One of the most popular applications of molecular rotors is the quantitative determination of solvent viscosity (for some examples, see references [18, 23-27] and Sect. 5). Viscosity refers to a bulk property, but molecular rotors change their behavior under the influence of the solvent on the molecular scale. Most commonly, the diffusivity of a fluorophore is related to bulk viscosity through the Debye-Stokes-Einstein relationship where the diffusion constant D is inversely proportional to bulk viscosity rj. Established techniques such as fluorescent recovery after photobleaching (FRAP) and fluorescence anisotropy build on the diffusivity of a fluorophore. However, the relationship between diffusivity on a molecular scale and bulk viscosity is always an approximation, because it does not consider molecular-scale effects such as size differences between fluorophore and solvent, electrostatic interactions, hydrogen bond formation, or a possible anisotropy of the environment. Nonetheless, approaches exist to resolve this conflict between bulk viscosity and apparent microviscosity at the molecular scale. Forster and Hoffmann examined some triphenylamine dyes with TICT characteristics. These dyes are characterized by radiationless relaxation from the TICT state. Forster and Hoffmann found a power-law relationship between quantum yield and solvent viscosity both analytically and experimentally [28]. For a quantitative derivation of the power-law relationship, Forster and Hoffmann define the solvent s microfriction k by applying the Debye-Stokes-Einstein diffusion model (2)... 

In the case of polymer samples, it is expected that, at the temperatures and frequencies at which the rheological measurements were carried out, the polymer chains should be fully relaxed and exhibit characteristic homo-polymer-like terminal flow behavior (i.e., the curves can be expressed by a power-law of G oc co2 and G" oc co). 

It is known that in some cases the modulus representation M (oo) of dielectric data is more efficient for dc-conductivity analysis, since it changes the power law behavior of the dc-conductivity into a clearly defined peak [134]. However, there is no significant advantage of the modulus representation when the relaxation process peak overlaps the conductivity peak. Moreover, the shape and position of the relaxation peak will then depend on the conductivity. In such a situation, the real component of the modulus, containing the dc-conductivity as an integral part, does not help to distinguish between different relaxation processes. 

In NMR, it is possible to distinguish glass formers that do show a /1-process (type B) from those that do not show a /1-process (type A), but exhibit a power-law behavior (excess wing) in the dielectric loss. However, it remains an open question whether a clear-cut physical difference exists between the two relaxation processes. In view of gradual variations in the -relaxation behavior among different glass... 

The relaxation times, t, and corresponding moduli, G, constitute what is called the distribution or spectrum of relaxation times. The relaxation spectrum given in Eq. (3-40) is a distinctive feature of the Rouse model that can be tested experimentally. A simple type of rheological experiment from which this spectrum can be obtained is small-amplitude oscillatory deformation, discussed in Section 1.3.1.4. In this test, at low frequencies, (o < /x, the Rouse model predicts the usual terminal relaxation behavior G — Gco rf, and G" = Gcnxi. More significantly, at higher frequencies, where co is in the range 1/t, oj < 1/tat, the Rouse model predicts a power-law frequency dependence of G and G" ... 

Slow relaxation of water was further discussed in more broad contexts. In particular, since the IS picture is expected to become appropriate as the temperature is decreased, special attention has been paid to water in supercooled states, and dynamics in supercooled states has been investigated in relation to applicability of the mode-coupling theory [51]. It was found that the bond lifetime of individual molecules obeys the thermal process, whereas the bond correlation function shows power-law behavior [52,53], The behavior below or above the temperature at which the mode-coupling theory can be applied was also studied and the transition between IS structures, which is just the network rearrangement dynamics just mentioned above, has clearly been identified in supercooled regions. 

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