FR characteristic functions

The overall in-phase and out-of-phase characteristic functions on the RHS of Eqs. 1 and 2 depend on the theoretical models describing the overall kinetic processes involved in the system. By fitting the experimental data on the LHS with the theoretical models on the RHS of the equations, diffusion coefficients and some other kinetic parameters which are included in the overall FR characteristic functions can be extracted. 

In all these investigations, very small input amplitudes were used to justify the use of a linear technique for investigation of generally nonlinear adsorption systems. One of the main drawbacks of the classical, linear FR method for investigation of adsorption kinetics is that, in a number of cases, same shapes of linear FR characteristic functions are obtained for different kinetic mechanisms, and the method is reduced to estimation of kinetic parameters of an assumed model. One example of such behavior is the same shape of characteristic curves obtained for adsorption governed by micropore, pore, or pore-siuface diffusion mechanisms [28]. Another very characteristic example is the case of bimodal characteristic functions (with two maxima of the out-of-phase and two inflection points of the in-phase characteristic functions [31,33,35]). These results were shown to fit equally well to three different kinetic models [19,33,35]. As it will be shown in Section III, the nonlinear FR method can overcome this problem. 

The problem of estimation of the kinetic parameters from linear FR characteristic functions, has been solved long ago, for simple isothermal kinetic models [15]. The process time constant can be estimated from the extremum of the so-called out-of-phase function [15], which is identical to the negative imaginary part of the first-order particle FRF Fi,p(o)) [28]. 

Allahverdyan and Nieuwenhuizen " consider Thomson s formulation of the second law and argue that the variable of the quantum FRs in previous studies should not be considered to be the work. They discuss different definitions of the work in quantum systems and argue for introduction of a new definition. Esposito and Mukamel present transient and steady state FRs as well as JE by developing a quantum mechanical trajectory, and then carrying out a derivation in a similar way to that used for stochastic dynamics. Talkner and Flanggi recently derived a quantum version of the Crooks FR using a characteristic function for the work obtained by Talkner et al ... 

The overall in-phase and out-of-phase characteristic functions rely on the kinetic mechanisms involved in the FR system. As mentioned in Sect. 3.1, various theoretical models have been developed to cover the multi-kinetic processes taking place in the system. Some commonly used models are summarised in the following sections. 

Figure 8 displays some typical FR data of Ci - Ce n-alkanes diffusing in coffin shaped crystals of silicalite-1 (40 x 40 x 260 p,m ). All the spectra in Fig. 8a-f,l can be fitted by the theoretical in-phase and out-of-phase characteristic function curves of the single diffusion model described by Eqs. 3-6, implying that only a simple, single diffusion process is involved in these systems. The diffusivities calculated from the best fit are presented in Fig. 9 and Tables 1 and 2. Equations 5 and 6 were applied since the channel framework structure of sihcahte-1 is comprised of near circular (0.54 x 0.56 nm)... 

Fig. 8 FR spectra of methane (a), ethane (b), propane (c), n-butane (d), n-pentane (e) and n-hexane (f) in sUicalite-l (cf. [65]). (n.o) indicate the experimental in-phase and out-of-phase KSqm characteristic functions, respectively. A single diffusion process model was used to fit the data in (1), while the non-isothermal diffusion model was used to fit the data in (2) except (f, 2) which was fitted using the two independent diffusion processes model. Solid lines denote the theoretical overall characteristic functions, and dash and dash-dot lines denote the theoretical diffusion processes occurring in the straight channels (dash) and the sinusoidal channels (dash-dot). Note 1 Torr= 133.33 Pa...

Fig. 10 FR spectra of benzene in sUicalite-l (cf. [65]). Continuous lines are the fits of the theoretical single diffusion process model (a-c) and the two diffusion processes model (d-f). The symbols ( , o) are the experimental in-phase and out-of-phase characteristic function data, respectively. Note 1 Torr = 133.33 Pa...

The sound absorption coefficient, a, is increased when the dynamics of the chemical system are of the same order of magnitude as the frequency of the sound wave,41 and experimentally this quantity is measured as a function of frequency of the ultrasonic sound wave (Fig. 4). When the frequency of the sound wave is of the same order as the frequency for the relaxation process, effects due to relaxation of the equilibrium give rise to characteristic changes in the quantity a//2, where a is the sound absorption coefficient measured at frequency /40 The variation of a with frequency, /, has an inflection point at the relaxation frequency of the system, fr, which is related to 1/t, where r is the relaxation time (1/t = 27i/r).40,41 The expression relating the quantity... 

Many nanoparticles (oxides, hydroxides, POSS, metallic phosphates, catalysts residues,...) have been developed for various purposes corresponding to various functional properties of polymers and composites. Some of them have been evaluated as potential FRs owing to their characteristics... 

The increasing demand for FR PVC-U applications has prompted work in this area. The use of functional fillers can cause a dramatic deterioration of physical properties at the addition levels necessary to achieve the requirements of flame retardancy and smoke emission. An evaluation has shown zinc hydroxystannate to give the best overall FR and SS characteristics without adversely influencing important physical properties (129). The addition of low levels of zinc borate and/or ATH was noted to improve flame retardancy and reduce smoke density in a PVC-U formulation (265). 

A detailed description of the batch type FR system used in the present study has been given previously [6]. The FR functions measured were fitted by in-phase and out-phase characteristic... 

A variety of specialty resins with increased Tg is available, albeit at higher prices. Modified FR-4 materials with higher functionality offer the best combination of improved Tg at a reasonable price. Further improvements in Tg and other characteristics can be obtained with bis-maleimide triazine (BT), GETEK, cyanate ester, and polyimide, but at greater price penalties. 

The elements on the left part are related to the Dynamic Testing phase [8] starting from system Functional Requirements Specifications (FRS), written in natural language, the system dynamic model is manually defined, and the Automatic Test Generation is calibrated on this model. In the ATG phase the Reduction Rules (deriving from the specifications themselves and/or from one s own domain characteristics) are helpful to reduce the number of the tests which have to be executed. The Reduction Rules, indeed, define the input variables which don t affect the output ones therefore, these input variables will be set to default values to test the output ones, saving hence time and costs during the test execution phase [9]. 

Fig. 1. FR spectra of neopentane sorption on MWCNT-81 at 50 C and 133 Pa. Each measurement was made on 200 mg CNT. Symbols correspond to the in-phase ( ) and out-of-phase (o) components of the experimentally determined response fimctions. Full lines are the best-fit eharaeteristic functions. FR rate spectra of parallel processes, represented by dotted lines, were summarized to get the characteristic FR function of the sample, comprising 150-600-pm particles.

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