Single-relaxation-time mechanism

Figure B2.5.2. Schematic relaxation kinetics in a J-jump experiment, c measures the progress of the reaction, for example the concentration of a reaction product as a fiinction of time t (abscissa with a logaritlnnic time scale). The reaction starts at (q. (a) Simple relaxation kinetics with a single relaxation time, (b) Complex reaction mechanism with several relaxation times x.. The different relaxation times x. are given by the turning points of e as a fiinction of ln((). Adapted from [110].

Before discussing tire complex mechanical behaviour of polymers, consider a simple system whose mechanical response is characterized by a single relaxation time x, due to tire transition between two states. For such a system, tire dynamical shear compliance is [42]... 

Figure C2.1.14. (a) Real part and (b) imaginary part of tire dynamic shear compliance of a system whose mechanical response results from tire transition between two different states characterized by a single relaxation time X.

The stress-strain relations for viscoelastic materials are reviewed. The simplest case of intrinsic absorption in polymers is a molecular relaxation mechanism with a single relaxation time. However, the relaxation mechanisms which lead to absorption of sound are usually more complicated, and are characterized by a distribution of relaxation times. Under causal linear response conditions the attenuation and dispersion of sound in a... 

As with mechanical relaxation, the single-relaxation-time model is inadequate for polymers and the Cole-Cole plots are not semicircles. Figure 7-... 

In Debye s model of dielectric relaxation, the polarisation process has a single relaxation time. The model has both electrical circuit and viscoelastic model analogues (Fig. 12.14). The electrical circuit is the dual of the mechanical model, because the voltages across the capacitor and resistor... 

A distribution of dielectric relaxation times is observed for most materials analogous to what is found for mechanical relaxation processes. The single relaxation time used in equations 5.30 and 5.31 is therefore replaced by an integral, or... 

The Narayanaswamy-Moynihan model (NMM), discussed in detail by Donth and Hempel [49], is an improvement on this approach. Instead of postulating a simple exponential relaxation mechanism governed by a single relaxation time [47], the nonexponential stractural relaxation behavior and the spectmm effect were studied. It was assumed that the whole thermal history Tit) started from a thermodynamic equilibrium state where 7 (to) = 7 (to)- Tool s fictive temperature is defined by... 

Measurements of didectric and mechanical loss as a function of frequency have shown that rdaxation is not a simple exponential process in polymers and hence cannot be expressed in terms of a single relaxation time. The dispersion in didectric constant and storage modulus is spread over a wide frequency range and the loss curves exhibit broad maxima. The dispersion regions in the frequency spectrum are direcdy rdated to the transition phenomena discussed in the preceding section. The reason is that the molecular motions required for changes in the liquid configuration are quite similar to those responsible for didectric and mechanical rdaxation. 

In order to illustrate the typical nonlinear mechanical response of wormlike micelles under steady shear flow, we chose to focus on the cetylpyridinium (CPCl)/sodium saUcylate (NaSal) system. It is often considered as a model system since it follows the right scaling laws for the concentration dependence of the static viscosity and plateau modulus [32]. Moreover, for concentrations ranging from 1 to 30wt. %, the samples behave, in the linear regime, as almost perfect Maxwellian elements with a single relaxation time Tr and a plateau modulus Go- This system has been... 

A direct connection between a- and P-relaxations also follows from recently found linear correlation between the logarithm of the P-relaxation time at T = rg[logTp(Tg)] and the coefficient b in the Kohlrausch a-relaxation correlation function exp [(—f/Xa) ] for dielectric and mechanical relaxations [117]. A simple relaxation process with a single relaxation time corresponds to ft = 1, and the P-relaxation event was considered by this author as the primary ( primitive ) segmental motion in the cooperative a-process. 

Dielectric experiments that involve studies of the relaxation function (r) are denoted as time-domain experiments, while those related to the complex permittivity function e (co) are considered dynamic experiments. The latter have the advantage of introducing an experimental timescale ( l/(o), which, when compared to the different intrinsic timescales of the system (the relaxation time x), provides useful information on the molecular level. In terms of the single-relaxation-time model of Debye (1921,1929), the complex permittivity for a dipolar mechanism can be written as follows ... 

Figure 6.2. Schematic representation of the signal ascribed to a single-relaxation-time (Debye-type) mechanism (Su = 2, e, = 12, and x = ls) (a) e and e" versus log(cor) (b) M, M" and tan 5 versus log(cor).

If the chemical structure of a polymer is altered during a viscoelastic experiment—in particular, if a cross-linked network is subjected to a reaction which increases or decreases the number of network strands while it is being investigated in the rubbery zone of viscoelastic behavior—the apparent mechanical properties will be profoundly influenced. For example, scission of the network strands will cause stress relaxation at constant strain" (Fig. 14-12) or creep under constant stress. Formally, if a single first-order chemical reaction is responsible, the relaxation may be described by a single relaxation time which is the reciprocal of the chemical rate constant, instead of the broad spectra which are characteristic of the usual mechanical processes. 

The real and imaginary parts of the susceptibility of HAT6 are shown in Fig. 7 (a and b), and the temperature dependence of e is shown in Fig. 7(c) [38]. The frequency response can be explained by assuming a Debye law with, surprisingly, a single relaxation time (=5 x 10 s). This process is most probably due to the rotation of the side chain around the oxygen bond. Numerical simulations have shown this mode to have a relatively low activation energy (effective overall thermal dipole can appear. 

Several rheological studies showed that solutions of telechelic polymers submitted to oscillatory shear exhibit a Maxwellian behavior (single relaxation time). ° ° This behavior was explained by assuming that the main stress relaxation mechanism is the exit of one hydrophobe from the... 

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