Modulus spectrum

F(i) is the underlying modulus spectrum for that system. As noted above, since the time scale of relaxation is so broad, results are best depicted on a logarithmic time scale. To do this, one needs the contribution to the modulus associated with or lying in the time interval between In T and In T 4- Jin T this incremental contribution to the modulus is designated as... 

The continuous function II( n T) [often simply given the symbol H(r) as in this chapter) is the continuous relaxation spectrum. Although called, by long-standing custom, a spectrum of relaxation times, it can be seen that H is in reality a distribution of modulus contributions, or a modulus spectrum, over the real time scale from 0 to < or over the logarithmic time scale from - to +. ... 

The contribution of the crystalline high melting nylon 6 blocks and soft-block hard-phase separation are also reflected in the resistance to heat sag exhibited by NBC. Table IV shows the heat sag at 163°C as these values are related to the flexural modulus. Even at the very low end of the modulus spectrum, sag values were quite low. 

Another strong indication of the entanglement effect is the observation of a clear plateau in the linear relaxation modulus amj storage modulus spectrum when the polymer has a sufficiently high molecular... 

As shown in Fig. 11.9, the storage-modulus spectrum of sample A is well described by the Rouse theory Eq. (11.9) with the molecular weight distribution (VFiAi(M) and W2A2 M)) shown in Fig. 11.8(a). In the calculation... 

Fig. 11.9 Comparison of the measured storage-modulus spectrum (o and ) and that calculated from the Rouse theory (solid line) for sample A. The dashed line indicates the separation of the contributions from the G1 and GIO components. The arrow at 1/ti(1) indicates the frequency that is the reciprocal of the relaxation time of the first Rouse mode of the G1 component calculated from Eq. (7.57) with K = x 10, wheresis the arrow at l/ri(2) indicates the same for the GIO component.

Fig. 11.12 Comparison of the measured (dots) storage-modulus spectrum and the calculated (line) from the Rouse theory for sample FI.

The impedance and modulus spectra for this model in general consist of two semicircles. The diameters of these are gf. gi in the impedance and cf , c in the modulus spectra. The time constants of the two phases are defined as Ti = Ci/gi and 2 = C2/g2. If these time constants differ as a result of differences in c, then the arcs will be well resolved in the impedance spectrum. If they differ as a result of g, they will be resolved in the modulus spectrum (Hodge et al. [1976]). In practice, good resolution is not obtained in both Z and M spectra. 

Figure 4.1.9. Simulated impedance and modulus spectra for a two-phase microstructure, based on the effective medium model. Values of the input parameters are given in Table 4.1.1. (a, b) Spectra for a matrix of phase 1 containing 25% by volume of spheres of phase 2. Resolution is achieved in the modulus spectrum (b) but not the impedance spectrum (a), (c, d) Spectra for a spherical grain of phase 2 surrounded by a grain boundary shell of phase 1. The ratio of shell thickness to sphere radius is 10" Resolution is achieved in the impedance spectrum (c) but not the modulus spectrum (d).

Selected values of 0 are given in Table 4.1.2. A simulated modulus spectrum for Eq. (23) is shown in Figure 4.1.10. The parameters used to generate this are listed in Table 4.1.3. A plot of complex resistivity (not shown) displays only the low-frequency arc, with a very small distortion at high frequency. 

For the parameters chosen, there are two readily resolvable arcs in the modulus spectrum Figure 4.1.10. The low-frequency arc corresponds to the low-conductivity continuous phase and is apparently a perfect semicircle with its center on the real axis. The high-frequency arc corresponds to the discontinuous phase and is composed of three relaxations corresponding to the three possible orientations of the ellipsoids. In Figure 4.1.10 these are not well resolved, but cause the arc to be nonideal. The two-phase dispersion can be represented by either of the two circuits in Figure 4.1.11 the values shown in Table 4.1.4 are the results of NLLS fits to the specttum. 

Evaluation of this expression generates a spectrum rather similar to that for the Maxwell-Wagner model, but with a different weighting of the two phases. Figure 4.1.12 shows a modulus spectrum for the same input parameters as those that were used to produce the spectrum in Figure 4.1.9fc. 

Figure 4.1.12. Modulus spectrum for the Bruggeman asymmetric dispersed phase model with CTi = 1 X 10" Scm , cTi = 1 X 10" Scm" and X2 = 0.3. The calculation was made using the series expansion Eq. (25) due to A. Sihvola Labels indicate log(f).

Figure 4.1.45. Modulus spectrum obtained at 150°C for a single crystal of NaCl CdCl2, of which a micrographs is shown in Figure 4.1.44. (Reprinted with permission from N. Bonanos and E. Lilley, Conductivity Relaxation in Single Crystals of Sodium Chloride Containing Suzuki Phase Precipitates, J. Phys. Chem. Solids, 42, 943-952. Copyright 1981 Pergamon Journals Ltd.).

The two phases have representative time constants = RjCj and = R C, if these time constants differ as a result of differences in capacitance, then the arcs will be well resolved in the impedance spectrum. If two time constants are significantly different as a result of resistance differences, the modulus spectrum will resolve the arcs [2]. However, in practice good resolution is difficult to obtain. For example, when high-conductivity suspended matter reaches a certain volume fraction O, the impedance spectrum is often unable to resolve the time constants, and the complex modulus spectrum is preferred in order to see two arcs [4, p. 200]. 

The impedance analysis can be further refined by looking at the modulus representation of the bulk lubricant response. The modulus spectrum sometimes allows determining an additional MHz range relaxation with characteristic capacitance of 20pF/cm and resistance in high ohms to low kohms in "model lubricant systems" with abnormally large excesses (15 to 20 percent) of polar additive [19]. This MHz featiue was identified as orientational... 

Below Tg, there often exist several more dispersion peaks of the loss modulus spectrum, where some additional dissipation of the applied mechanical energy takes place. The positions of these peaks are called sub-Tg or secondary transition regions. They are called the P-transition temperature, y-transition temperature, and so on, in the descending order, as the a-transition of an amorphous material is the glass-to-rubber transition. The storage modulus spectrum is usually quite insensitive to the secondary transitions virtually no changes are observed at transition temperatures corresponding to the peaks of a loss modulus spectrum. 

Certain macroscopic properties of materials sometimes show major changes near the secondary transitions detected by the loss modulus spectrum. It is well known that the toughness of certain resins, such as polycarbonate, suddenly disappears and the material becomes brittle below the... 

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