Relaxation pattern

The transition strongly affects the molecular mobility, which leads to large changes in rheology. For a direct observation of the relaxation pattern, one may, for instance, impose a small step shear strain y0 on samples near LST while measuring the shear stress response T12(t) as a function of time. The result is the shear stress relaxation function G(t) = T12(t)/ < >, also called relaxation modulus. Since the concept of a relaxation modulus applies to liquids as well as to solids, it is well suited for describing the LST. 

Rubbery materials beyond the gel point have been studied extensively. A long time ago, Thirion and Chasset [9] recognized that the relaxation pattern of a stress r under static conditions can be approximated by the superposition of a power law region and a constant limiting stress rq at infinite time ... 

Microphase separated systems are also known to yield a physical network which results in the self-similar relaxation pattern at an intermediate state... 

The dynamic properties depend strongly on the material composition and structure. This is not included in current theories, which seem much too ideal in view of the complexity of the experimentally found relaxation patterns. Experimental studies involving concurrent determination of the static exponents, df and t, and the dynamic exponent, n, are required to find limiting situations to which one of the theories might apply. 

Silane and hydrogen show relaxation patterns with the same characteristic time t, however, inverse signs. The fragmentation of silane induced by collisions with electrons, yields molecular hydrogen in an order of magnitude faster than the time resolution of the mass spectrometry setup, i. e. faster than 1 ms. Two possible pathways of silane fragmentation can be regarded ... 

The exposure can be either in air or a liquid chosen to simulate service conditions. Commonly, ring test pieces are chosen for liquid exposure, so simulating the geometry of practical seals and giving a relatively large surface area to volume ratio so that equilibrium swelling is reached reasonably quickly. It should be noted that the swelling effect of the liquid will affect the relaxation pattern measured and an increase in stress may be seen over a limited time period if there is a volume increase. 

For systems that exhibit slow anomalous transport, the incorporation of external fields is in complete analogy to the existing Brownian framework which itself is included in the fractional formulation for the limit a —> 1 The FFPE (19) combines the linear competition of drift and diffusion of the classical Fokker-Planck equation with the prevalence of a new relaxation pattern. As we are going to show, also the solution methods for fractional equations are similar to the known methods from standard partial differential equations. However, the temporal behavior of systems ruled by fractional dynamics mirrors the self-similar nature of its nonlocal formulation, manifested in the Mittag-Leffler pattern dominating the system equilibration. 

It has been claimed that reactions in proteins can, as an approximation, be formulated within the Kramers reaction theory of barrier crossing [106]. The highly nonexponential relaxation pattern can now be explained by our model,... 

The characteristic changes brought about by fractional dynamics in comparison to the Brownian case include the temporal nonlocality of the approach manifest in the convolution character of the fractional Riemann-Liouville operator. Initial conditions relax slowly, and thus they influence the evolution of the system even for long times [62, 116] furthermore, the Mittag-Leffler behavior replaces the exponential relaxation patterns of Brownian systems. Still, the associated fractional equations are linear and thus extensive, and the limit solution equilibrates toward the classical Gibbs-B oltzmann and Maxwell distributions, and thus the processes are close to equilibrium, in contrast to the Levy flight or generalised thermostatistics models under discussion. 

The Mittag-Leffler function [44-46] can be viewed as a natural generalization of the exponential function. Within fractional dynamics, it replaces the traditional exponential relaxation patterns of moments, modes, or of the Kramers survival. It is an entire function that decays completely monotoni-cally for 0 < a < 1. It is the exact relaxation function for the underlying multiscale process, and it leads to the Cole-Cole behavior for the complex... 

Most conspicuous modifications of the dynamic mechanical response spectra of PHEMA and related polymers are brought about by incorporation of low-molecular weight compounds (Fig. 13). Along with alterations of parameters (temperature, height, shape) of the peaks characteristic of a dry polymer, usually a new diluent peak appears. (The relaxation patterns of various polymethacrylates are not modified by diluents in a unique way but several modes can be distinguished as mentioned before.) A remarkable feature... 

The second type of the diluent-induced alterations in the relaxation pattern is characteristic of e.g. PMAAc149 (Fig. 16), PAAm136 (Fig. 17), and PHEA127 (Fig. 18). Incorporation of water (no other diluent has been used) accounts for a continuous decrease in the temperature and increase in the area of the existing / or /T loss peak. Since no new peak is produced, it seems reasonable to infer that the mobility of diluent molecules sets in along with the partial rotation of the COOR side chains. However, it is possible to... 

A similar idea can also be the basis of the relaxation pattern (25) when these two different types of fractal evolution coexist for two subspaces (p(.q() and ipvq ) of the total statistical system phase space (p. q). Here the total distribution function pf (p,q] t) is the product of two statistically independent distribution functions p (py, y f) and p (p, q t) ... 

In Section II.B we have classified several types of non-Debye relaxation and have mentioned a few particular approaches that have been developed in order to explain the origins of these relaxation patterns. We shall now discuss a model that considers one particular case of nonexponential relaxation. 

The alkaline earth nuclei in their ionic solvated state are expected to follow the quadrupolar relaxation pattern established for their Group I counterparts. However, there is one predictable distinguishing feature. Owing to their larger effective charge the alkaline earths are more... 

Thus, it can be concluded that the relaxation time distribution concept applies to Debye-like relaxation (even though its frequency dependence may be smeared-out), whereas it becomes inapplicable for still slower relaxation patterns. In the latter situation, the distribution of relaxation times over a selfsimilar, fractal ensemble seems a physically more reasonable assumption. As is well known, the fractality of geometrical objects implies their non-integer dimension however, a more exact definition of the fractal concept with respect to the ensemble of relaxation times is in order. 

Thus, the crossover from a strictly exponential to an anomalous relaxation pattern can be associated with the change of a continuous distribution of relaxation times (a = 1) into a fractal-like one (0 < a = df < 1). 

Mours M., Winter H. H. Relaxation patterns of nearly critical gels. Macromolecules 1996 29 7221-7229. 

To Start we describe an extension an extension of Eq. (A. 15) due to Machlup and Onsager [4], We begin by writing down a model equation of thermodynamic relaxation patterned after the particle mechanics eq. (2.7). 

The similarity to the A A state bonding representation is conspicuous (the only difference lying in the half-occupancy of Uq in the A state vs. Uq in the C state), and the adiabatic relaxation patterns might be expected to be similar. The similarity of initial A- vs. C-state relaxation is also suggested by the small-angle torsional dependence in Figure... 

The fact that one and the same conformer may show a different relaxation pattern upon excitation to different electronlc/vlbratlonal states. 

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