Frequency dependence, isothermal

In situ frequency dependent electromagnetic-impedence measurements provide a sensitive, convenient, automated technique to monitor the changes in macroscopic cure processing properties and the advancement of the reaction in situ in the fabrication tool. This chapter discusses the instrumentation, theory, and several applications of the techniques, including isothermal cure, complex time—temperature cure, resin film infusion, thick laminates, and smart, automated control of the cure process. 

The line forms, described by Eqs. (371) and (373), are illustrated in Figs. 45 and 46. In the first one (Fig. 45) we compare the loss (a) and absorption (b) for the isothermal, Gross, and Lorentz lines see solid, dashed, and dashed-and-dotted curves, respectively. These curves are calculated in a vicinity of the resonance point x = 1. In Figure 45c we show the frequency dependences of a real part of the susceptibility the three curves are extended also to a low-frequency region. The collisions frequency y and the correlation factor g are fixed in Fig. 45 (y = 0.4, g = 2.5). 

Figure 45. Frequency dependence of an imaginary (a) and of real (c) parts of the susceptibility (b) absorption coefficient versus x. All quantities are nondimensional. Calculation for the isothermal (solid curves), Gross (dashed curves), and Lorentz (dashed-and-dotted curves) lines. The normalized collision frequency, y, is 0.4, and the correlation factor, g, is 2.5.

Figure 46. Evolution of frequency dependencies of loss (a, c) and of real part of the susceptibility (b, d) stipulated by change of the correlation factor g (nondimensional quantities). In Figs, (a, b) g = 2.5 and in Figs, (c, d) g — 2. Isothermal line (solid curves) and Gross line (dashed curves). Curves 1, 3 for y = 0.4 and curves 2, 4 for y = 0.8.

Dielectric relaxation of complex materials over wide frequency and temperature ranges in general may be described in terms of several non-Debye relaxation processes. A quantitative analysis of the dielectric spectra begins with the construction of a fitting function in selected frequency and temperature intervals, which corresponds to the relaxation processes in the spectra. This fitting function is a linear superposition of the model functions (such as HN, Jonscher, dc-conductivity terms see Section II.B.l) that describes the frequency dependence of the isothermal data of the complex dielectric permittivity. The temperature behavior of the fitting parameters reflects the structural and dynamic properties of the material. 

Major source of errors in LIvtv evaluation are the incomplete data of the frequency dependence of the dielectric permeability. Donners [263] has shown that Ylvw values derived from e(i ) where is the dielectric permeability of the imaginary axis of frequency [221,256], are not very reliable at thicknesses smaller than 10 nm. Since the Il(/ ) isotherms of thicker films exhibit such a disagreement as well, the inaccurate calculation of IW cannot be considered to be as an important reason. This is confirmed by the fact that Ylvw calculations employing another method for estimation of the dielectric permeability [259] does not show considerable differences. 

Isotherms at several temperatures showing the frequency dependence of the real component / (co) of the complex compliance / (coi) of a viscoelastic material are plotted on a double logarithmic scale in Figure 8.9 (7). At high temperatures and low frequencies, J (co) decreases slightly with increas-... 

In most of the cases, an ultrasonic wave propagates adiabatically, so the (20) looks more naturally its right-hand side represents the adiabatic (non-relaxed) modulus and non-adiabatic contribution to the dynamic modulus. Recall that the relaxed (or isothermal) modulus should be regarded as quasi-static one. Figure 1 shows the frequency-dependent factor of non-adiabatic contribution as function of cox. One can see that transformation from isothermal-like to adiabatic-like propagation occurs in the vicinity cox = 1. The velocity of ultrasound is increased in this region, while the attenuation reaches its maximum value. 

Figure 15. Frequency dependence of shear viscosity for the system GB(3, 5, 2, 1) (TV = 576) at several densities along the isotherm at temperature T = 1. The inset shows the low-frequency data (co) in a semilog plot. (Reproduced from Ref. 121.)...

Figure 11 shows the frequency dependence of the dielectric loss, normalized with respect to the peak position and height of the a-process, for four different annealing times during the isothermal annealing at Ta = 425 K for stacked thin films of 18-nm thick P2CS layers. The dielectric loss data in the frequency domain at... 

Fig. 20 Frequency dependence of A after isothermal aging at three different aging temperatures Ta = 339.0 squares), 319.5 (circles), and 300.4 K (triangles) for 30 h with d = 3.7 nm...

Figure 22 shows Arrhenius plots of the relaxation times for the a- and approcesses obtained by the peak frequency of the dielectric loss e" for P2CS thin films with d = 3.7 nm. The peak frequencies/ and/a, are evaluated fi-om the frequency dependence of the dielectric loss due only to the a- and approcesses that are reproduced from Eq. (12) with best-fit parameters. The peak frequency of As"(f, Ta) after isothermal aging at a given aging temperature Ta for 30 h and with... 

Finally, the amplitude and frequency dependence of the TMDSC response is of interest. In Fig. 4.115, the dependence of the reversing, specific heat capacity of polycaprolactone with the repeating unit (-CH2)5-CO-0 is displayed as a function of modulation amplitude close to the melting peak [37]. The experimentation involved quasi-isothermal TMDSC at 334 K. Within the experimental error, no amplitude-dependence of the reversing specific heat capacity is seen, as is expected for a linear... 

The slow, irreversible cold crystallization is followed in Fig. 6.53 for more than 10 days with quasi-isothermal TMDSC to a fixed value of RAF. At the end of the crystallization there is no frequency dependence of the heat-capacity. The crystallization and the glass transition to the RAF occur simultaneously (see also Fig. 6.18). 

Saruyama Y (1999) Quasi-isothermal Measurement of Frequency Dependent Heat Capacity of Semicrystalline Polyethylene at the Melting Temperature using Light Heating Modulated Temperature DSC. Thermochim Acta 330 101-107. 

Figure 21 Frequency dependence of the in-phase (x ) and out-of-phase (x") magnetic susceptibility, xt and Xs are the isothermal and adiabatic susceptibility, respectively.

G. Frenning, A. K. Jonsson, A. L. Larsson, and M. Strpmme [2003] Theoretical Investigation of Ion Conduction in Three-Layered Ion-Conductor Systems Derivation of the Isothermal Transient Ionic Current and Frequency-Dependent Impedance, J. Appl. Phys. 94,... 

The adsorption isotherm of 2,3-dimethylpyridine (2,3-DMP) indicates that 2,3-DMP is adsorbed at the mercury electrode in two states (I and II). Adsorption state II is limited towards negative potentials by sharp, strongly frequency-dependent needle peaks. State II represents a condensed film of perpendicularly oriented molecules [133]. 

In addition to the adiabatic or isothermal difference, acoustically determined elastic constants of polymers differ from static values because polymer moduli are frequency-dependent. The deformation produced by a given stress depends on how long the stress is applied. During the short period of a sound wave, not as much strain occurs as in a typical static measurement, and the acoustic modulus is higher than the static modulus. This effect is small for the bulk modulus (on the order of 20%), but can be significant for the shear and Young s modulus (a factor of 10 or more) (5,6). 

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