Terminal behavior

This result shows that the highest modes of response have the shortest relaxation times and influence the initial response of the sample. Conversely, the longest relaxation time is ti, which we can identify with the terminal behavior of the sample. For example, in Fig. 3.9 the final collapse of the modulus at long times occurs at Ti. An example will show how we can use this idea. 

In close analogy to the PCL based nanocomposites, the terminal zone dependence of G and G" for the 2 and 5 weight % samples, show non-terminal behavior with power-law dependencies for G and G" much smaller than the expected 2 and 1 respectively. Furthermore, like the PCL based nanocomposites, there also appears to be a gradual decrease in the power-law dependence of G and G" with increasing silicate loading. 

A dimensionless quantity called the Deborah number, De, is defined as the fluid s characteristic relaxation time t divided by a time constant tf characterizing the flow (Reiner 1964). Thus, De = t///. In an oscillatory shearing flow, for example, we might take tf to be the inverse of the oscillation frequency (o, and then De = xo). At high Deborah number, the flow is fast compared to the fluid s ability to relax, and the fluid will respond like a solid, to some extent. Thus, in an oscillatory shearing flow, when De = cur 3> 1 the complex modulus is solid-like, while when De = 1 a liquid-like terminal behavior is... 

The behavior of PS-PI is similar in some respects to that of PEP-PEE. Figure 13-15 shows G as a function of frequency for a range of temperatures that includes Todt = 152°C, for a PS-PI sample studied by Winey et al. (1993a, 1993b). Similar behavior has been observed by others (Gupta et al. 1996 Han et al. 1995 Pinheiro et al. 1996). As with PEP-PEE, terminal behavior is observed for T > Todt and nonterminal behavior is obtained when T drops below this temperature. For PS-PI, one does not find the anomalous low-frequency behavior observed in the disordered state near Todt for PEP-PEE. 

Figure 13.15 Reduced storage modulus versus reduced frequency arco for a lamellae-forming polystyrene-polyisoprene diblock copolymer (M = 22,000) at temperatures above the order-disorder transition temperature Todt = 152°C, and quenched to temperatures below it. The disordered samples show terminal behavior, and the ordered (but unoriented) ones show nonterminal behavior. (Reprinted with permission from Patel et al.. Macromolecules 28 4313. Copyright 1995, American Chemical...

As with lamellae-forming samples, when the hexagonal phase forms after a quench from the disordered state, terminal behavior in G and G" is replaced by nonterminal behavior (see Fig. 13-13). For hexagonal cylinders, the low-frequency behavior is generally more elastic than for lamellae the apparent power-law exponent in G oc tu is usually somewhat lower (n 0.0-0.4) for the former than for the latter (n 0.5) (Gouinlock and Porter 1977 Morrison et al. 1990 Winter et al. 1993 Almdal et al. 1992). This is perhaps not surprising, since ordered hexagonal cylinders are solid-like (i.c., positionally ordered) in two dimensions and lamellae are positionally ordered in only one. 

As with cylinder- and lamellae-forming block copolymers, the rheological behavior of block copolymers that form spherical domains depends on whether or not the domains possess macrocrystalline order. If the domains are disordered, then the low-frequency moduli show terminal behavior typical of ordinary viscoelastic liquids that is, G and G" fall off steeply as the frequency becomes small (Watanabe and Kotaka 1983, 1984 Kotaka and Watanabe 1987). When the spherical domains are ordered, however, elastic behavior is observed at low frequency that is, G approaches a constant at low frequency, and a yield stress is observed in steady shearing. 

In contrast to lamellar microdomains, where removal of defects and alignment of domains can, in principle, restore terminal behavior to G and G", even after alignment, G for three-dimensionally ordered spherical domains remains that of a soft solid, albeit with a much lower modulus than that of the unoriented material (Koppi et al. 1994). 

Fig. 23 (a) Frequency-dependent linear viscoelastic moduli (G closed symbols, G" open symbols) of a colloidal star with nominal values f = 12Sarms and = SOkgmol at a concentration 5 wt% in -tetradecane and different temperatures (circles 40°C, squares. 5O C. triangles 55°C). A liquid-lo-solid transition is marked between 50 and 55°C. Lines with slopes 1 and 2 indicate terminal behavior of G" and G, respectively. Inset The temperature dependence of the hydrodynamic radius /fh of the same star, indicating swelling, (b) Respective moduli for the same system at 4() C in two different solvents, n-decane (circles, solid-like behavior) and n-tetracane (triangles, Uquid-Uke behavior) [26]... 

Figure 7(d) shows that stress relaxation is less sensitive to the model details. Indeed, all curves collapse onto a master curve, providing that the stress is multiplied by N and plotted against i/tr. In these units, the terminal behavior is well described by exp(-2tliR) as given by eqn [31]. More details can be noticed if this plot is multiplied by Vt/ra as shown in Figure 9. In particular, one can see deviations below the Rouse curve for the semiflexible model and deviations upward for the multichain models. As was demonstrated in Reference 4, the later deviations are due to glassy modes, resulting from collisions with other chains. They strongly depend on density and eventually diverge near the glass transition density.

The above remarks do not invalidate the free-volume explanation that was brought forward to account for the termination behavior of oligomers. However, on the other hand, the simplicity of the approach does not form a sound foundation to validate it either and thus requires further investigations. 

It is known that polymer chains are fully relaxed and exhibit characteristic homopolymerlike terminal flow behavior, resulting in that the flow curves of polymers being expressed by the power law G oc a/ and G" oc a> [93-95]. Krisnamoorti and Giannelis [96] reported that the slopes of G (a) and G"(a) for polymer/layered silicate nanocomposites were much smaller than 2 and 1, respectively, which are the values expected for linear homodispersed polymer melts. They suggested that large deviations in the presence of a small quantity of layered silicate were caused by the formation of a network structure in the molten state. The slopes of the terminal zone of G and G for the PEN/CNT nanocomposites are presented in Table 7. This result indicated the non-terminal behavior with the power-law dependence for G and G of the PEN/CNT nanocomposites the flow eurves of the PEN/CNT nanocomposites can be expressed by a power law of G [Pg.64]

The interpretation of termination behavior in the TD region is far more complicated, as monomer conversion is no sufficient measure for characterizing radical mobility. Average polymer size or, even better, size distribution and the extent of branching should additionally be known. The situation may be referred to as history-dependent kinetics, which basically says that kt depends on the characteristics of the polymerizing system, in particular on the properties of the polymer. These properties are determined by the specific mode of polymerization by which a particular conversion has been reached. The pathway thus affects kt under TD conditions via the polymeric structure given by the preceding polymerization. 

Dr. Carl Pfaffmann and his collaborators presented a good example of the psychobiological perspective on the study of food preferences in animals ( ). They pointed out that the key questions concern the variables which lead an animal to initiate, maintain and terminate behavior directed towards nutritive and non-nutritive substances. They classified these variables as nhysiological and behavioral. 

The shear storage modnlns (G ) as a function of frequency is shown in Figure 5. All of the samples show evidence of terminal behavior as the frequency decreases to about 5 rad/s. At lower frequencies, a second terminal zone appears, similar to that observed in LCPs and filled systems (Note that the shear loss modulus (G ) displays terminal behavior over this entire frequency range). 

The slope of G (with respect to co) as a function of number of cycles is shown in Figure 6. This shows that the sample processed ten times departs from terminal behavior at a higher frequency than the other samples, as shown by a smaller slope at 79.4 rad/s. However, it also begins to approach its second terminal region at higher frequeneies than the other samples, as shown by a larger slope at 0.126 rad/s. 

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