Fuoss-Kirkwood equation

Table 2.2 Parameters of Fuoss-Kirkwood equation for 8 and y relaxations of PCHPM. ...

Sub-glass relaxations for crystalline and amorphous polymers in the frequency domain are described by the empirical Fuoss-Kirkwood equation (35)... 

Sub-glass relaxations fit this equation. Plotting cosh [straight lines from whose slopes (= mEJPi) the evolution of the parameter m with the frequency of the isochrones can be evaluated. The Fuoss-Kirkwood equation also allows determination of the relaxation strength of sub-glass absorptions. 

Equation (5.3.6) reflects the relationship between Steeltjes and Fourier transforms. Using the Fuoss-Kirkwood equations... 

An alternative description of a non-ideal relaxation process commonly applied to polymeric systems is the Fuoss-Kirkwood equation. It is usually used when only the loss factor can be measured precisely. 

The relaxation process associated with the dynamic glass transition, the a relaxation, and the P relaxation, as a shoulder of the a relaxation can be observed in all these figures. At low temperatures another relaxation labeled as y relaxation can be observed. In the case of PCHpM, the maximum of the y relaxation is well away from the temperature range. Heijboer and Pineri [36,57] have reported that the maximum for this polymer is at about 100 K for 1 Hz. In the case of PCHpMM and PCOcM, the y relaxation can be observed which may be analyzed by using the Fuoss-Kirkwood (F-K) equation ... 

Two relaxation processes, called 8 and y can be observed, the last as a shoulder of the low temperature side of the f i relaxation [37], Assuming symmetry for these two relaxations the analysis according to Fuoss-Kirkwood empirical equation (2.35) can be performed as in previous systems [36,37,69], Due to the presence of the /3 relaxation an exhaustive analysis of this systems allow to obtain important information about these relaxation processes. 

For this reason in this system he deconvolution of the three relaxations is performed using the addition of two Fuoss-Kirkwood [69] equations for 8 and y relaxations and a power law for the low temperature side of the f) relaxation as follows... 

The analysis in the subglass region allow to fit the loss factor permittivity by empirical equations. As in systems previously analyzed a reliable model to represent secondary relaxation is that of Fuoss-Kirkwood [9], Assuming that the two overlapped contributions for 8 and y relaxation are additive Sanchis and coworkers [64] have proposed the following equation ... 

Many symmetrical loss relaxation curves fit the so-called Fuoss-Kirkwood (FK) equation given by... 

There were several attempts to generalize the Debye function like the Cole/Cole formula (Cole and Cole 1941) (symmetric broadened relaxation function), the Cole/Davidson equation (Davidson and Cole 1950, 1951), or the Fuoss/Kirkwood model (asymmetric broadened relaxation function) (Fuoss and Kirkwood 1941). The most general formula is the model function of Havriliak and Negami (HN function) (Havriliak and Negami 1966,1967 Havriliak 1997) which reads... 

The inversion of transform (5.3.2) and the determination of L (t ), when the analytical dependence Y j is known, have been considered previouslyThe results were based on the well-known inversion equations of Kirkwood and Fuoss (see also Ref. ) establishing a relationship between L(t ) and the corresponding dynamic compliance function Z(ico). Indeed, the reduced complex dynamic compliance corresponding to a distribution L j) is given by... 

Fuoss and Kirkwood have obtained equations identical with Eqs. 31 and 32 without introducing, explicitly, the exponential decay function. Like Debye they reasoned as if the problem were mainly one of diffusion by Brownian motion under the influence of an external force. Treating this problem as a Sturm-Liouville equation, they developed /i, into a complete set of orthogonal functions fx- A relaxation time tx is associated with each of these functions. 

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