Crossover modulus

Zeichner and coworkers [33,34] developed a measure of the breadth of the molecular weight distribution that is based on the curves of storage and loss moduli versus frequency. Based on data for a series of polypropylenes made by Ziegler-Natta catalysts and degraded by random chain scission, they found that the polydispersity index, i.e., the ratio Af /M , was related to the crossover modulus, G, as shown in Eq. 5.8... 

The crossover modulus is the value of G (and G") at the frequency where the two moduli are equal, i.e, where the curves of G co) and G (o) cross (see Fig. 5.5). An objective identification of the crossover point can be made by fitting the points on both curves in the vicinity of this point with cubic splines. Bafna [35] has pointed out that this empirical correlation is based on data for a single family of polypropylenes, i.e. a group of polymers having similar molecular weight distributions, and that it cannot be expected to be valid for other groups of polymers. 

Wu [49] proposed estimating G° as the storage modulus at the frequency where tan 5= minimum, but often there is no minimum in the data. Using this method, Fuchs et al. [38] reported a modest effect of the tacticity of PMMA on the plateau modulus. They also found that Eq. 5.4 fitted their viscosity data with a = 3.4 but that the constant K depended on tacticity. Plazek et al. [50] studied the effect of tacticity on the creep behavior of PMMA and noted that absorbed water is always a problem in making measurements on polar polymers. Wu [51] later proposed an empirical equation between the ratio (G Gq) of the plateau modulus to the crossover modulus, G, (where G = G") and the polydispersity index. 

The open-loop transfer function is third-order type 2, and is unstable for all values of open-loop gain K, as can be seen from the Nichols chart in Figure 6.33. From Figure 6.33 it can be seen that the zero modulus crossover occurs at a frequency of 1.9 rad/s, with a phase margin of —21°. A lead compensator should therefore have its maximum phase advance 0m at this frequency. Flowever, inserting the lead compensator in the loop will change (increase) the modulus crossover frequency. 

Place ujm at the modulus crossover frequency of 2 rad/s and position the compensator corner frequencies an octave below, and an octave above this frequency. Set the compensator gain to unity. Flence... 

Figure 6.35 shows the Bode gain and phase for both compensated and uncompensated systems. From Figure 6.35, it can be seen that by reducing the open-loop gain by 5.4dB, the original modulus crossover frequency, where the phase advance is a maximum, can be attained. 

Required modulus attenuation is 12 dB. This reduces the modulus crossover frequency from 1.4 to 0.6 rad/s. 

There are also some far-fetched proposals for the LST a maximum in tan S [151] or a maximum in G" [152] at LST. However, these expectations are not consistent with the observed behavior. The G" maximum seems to occur much beyond the gel point. It also has been proposed that the gel point may be reached when the storage modulus equals the loss modulus, G = G" [153,154], but this is contradicted by the observation that the G — G" crossover depends on the specific choice of frequency [154], Obviously, the gel point cannot depend on the probing frequency. Chambon and Winter [5, 6], however, showed that there is one exception for the special group of materials with a relaxation exponent value n = 0.5, the loss tangent becomes unity, tan Sc = 1, and the G — G" crossover coincides with the gel point. This shows that the crossover G = G" does not in general coincide with the LST. 

Now these expressions describe the frequency dependence of the stress with respect to the strain. It is normal to represent these as two moduli which determine the component of stress in phase with the applied strain (storage modulus) and the component out of phase by 90°. The functions have some identifying features. As the frequency increases, the loss modulus at first increases from zero to G/2 and then reduces to zero giving the bell-shaped curve in Figure 4.7. The maximum in the curve and crossover point between storage and loss moduli occurs at im. 

Fig. 128 Temperature dependence of the loss modulus, in the fi-a crossover region for PMMA and various CMIMx copolymers (from [79])...

The polyethenes prepared with catalyst 2 (Fig. 3a) have greatly elevated elastic modulus G values due to LCB compared to the linear polymers shown in Fig. 3b. LCB also shifts the crossover point to lower frequencies and modulus values. The measured complex viscosities of branched polymers (see also Table 2) are more than an order of magnitude higher than calculated zero shear viscosities of polymers having the same molecular weight but a linear structure. The linear polymers have, in turn, t] (0.02 radvs)... 

Fig. 14. Correlation of gel time and time of modulus crossover of thermosetting resins. Reproduced with permission from Journal of Applied Polymer Science, 27, 572,1982. Copyright 1982 John Wiley Sons, Inc.

Recall from Chapter 5 that the crossover concentration (p Ki jb [Eq. (5.36)] denotes the boundary between semidilute and concentrated solutions. For 0 > 0 chains are nearly ideal in concentrated solutions, whereas for 0 < 0 chains are swollen on intermediate scales. Network modulus and equilibrium swelling depend on the relative value of preparation and fully swollen concentrations (0o and l/Q) with respect to the crossover concentration 0. Since the swollen concentration is always lower than the preparation concentration (l/Q < 0o) there are three... 

In Fig. 7.17, the dry modulus of various PDMS networks is plotted as a function of their equilibrium swelling in toluene. A single curve results for both model networks (open symbols) made by end-linking linear chains with two reactive ends and networks with intentionally introduced defects in the form of dangling ends (filled symbols) made by end-linking mixtures of chains with one and two reactive ends. The data are fit to Eq. (7.91) as the solid lines in Fig. 7.17 and their intersection determines the crossover concentration 0 = 0.2, which is typical for good solvents. 

The two asymptotic temporal power-laws of MCT also affect the frequency dependence of G" in the minimum region. The scaling function Q describes the minimum as crossover between two power laws in frequency. The approximation for the modulus around tlie minimum in the quiescent fluid becomes [38]... 

The effect of fillers on the gel point of thermosets has not been studied extensively. Ng and Manas-Zloczower (1993) used isothermal dynamic time tests to measure the crossover point for a silica-filled epoxy resin. They noted a decrease in gel time with increasing filler loading. Metzner (1985) also noted that the storage modulus and loss modulus increased by different amounts with filler loading. Therefore, the gel-point tests for highly filled systems must involve knowledge of the effect of filler characteristics at various levels. 

Ways to increase membrane durability have been examined by various researchers across the country. Maurtiz et al. examined the use of metal-oxide metal particles to increase the properties of the membrane. A titanium isopropoxide (Figure 11.4) addition to Nation membranes generates quasi-network particles this improves membrane modulus and dimensional stability [17], In addition, the titanium matrix reduces fuel crossover and minimizes chemical degradation. Table 11.2 shows the increase in modulus along with stress and strain and stress changes after the addition of the titanium matrix [17], With a 20% load of the titanium matrix, performance criteria remain comparable. Acid functionality remains intact however, water uptake is reduced as volume inside clusters is occupied. Conductivity is reduced due to chain mobility [17],... 

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