Theoretical maximum extension ratio

Fig. 16. Experimental extension ratio of crazes in various hranopolymers and copolymers plotted against the theoretical maximum extension ratios of a single entangled chain and the entanglement network, and A.,., respectively...

Fig. 8 a, b. Experimental extension ratio X of crazes as a function of the theoretical maximum extension ratio of a single strand in the network (entanglements and crosslinks) a crosslinked polystyrene (open circles) and poly(pmethylstyrene) (closed circles) and b various entangled homopolymers and copolymers (open circles), and polymer blends of PS and PPO (closed circles)... 

Figure 14.20 Experimental extension ratios of crazes, X, in homo- and copolymers versus the theoretical maximum extension ratio (From Ref. 29.)...

Deformation behavior of amorphous polymers is well understood by using the concept of strand network density. Whether crazing or shear deformation is observed depends on the total network density, and the extension ratio of each deformation zone is proportional to the theoretical maximum extension ratio. Macroscopic shear deformation, leading to ductile behavior under tension, can be achieved by using thin layers whose thickness is below the critical value, even for normally brittle polymers like PS. The observed extension ratio of macroscopic shear zones is the same as that observed for localized shear zones. 

Figure 25 shows that the tensile behavior of PMMA and PS, stretched some 10-30 °C below Tg, is very similar to that of PC. Like PC both PMMA and PS exhibit extensive stress-whitening at high extension ratios, k", where the peak or inflection of the stress-strain curve is observed. In Table 3, the value of k" measured for PMMA is compared to the maximum extension ratio, k , of chains between entanglement points. The fact that k" = k provides further evidence that the cavitation in PMMA must be attributed to the same deformation mechanism as discussed in Section 4.2 for PC. The value of k" = kikj" = 6.8 measured for pre-oriented PS is higher than the theoretical value of k = 4.1 included in Table 3. This result is not... 

Scale-up of low pressure extruders usually begins in the laboratory with testing on smaller equipment. After extensive experimentation with the formulation and equipment, an optimal set of parameters is defined which includes information on the material s bulk density (before and after extrusion), the extrusion rate, the power consumption during extrusion, and the product s temperature rise. An efficiency factor is then determined by ratioing the actual extrusion rate obtained on the small equipment to the calculated theoretical maximum extrusion rate. Efficiency factors are in the range of 5-35 % for axial, 15-55 % for radial, and 35-85 % for dome extruders. This efficiency factor is then applied to the theoretical extrusion rates of the industrial extruder. Many manufacturers of extruders will also include an application related experience factor for the determination of a safe but reasonable expected extrusion rate. Fig. 8.35 depicts relative levels of extrusion pressure and shear that are applied by the various low pressure extrusion equipment. 

Since hydrocarbon chains in the Uquid state are never fully extended, an effective chain length, /etf, can be defined that gives the statistically most likely extension as calculated by the same procedure used for the calculation of polymer chain dimensions. For a chain with n c = 11, the ratio of /max to /ett will be approximately 0.75. In the micellar core, because of restrictions imposed by the attachment of the hydrocarbon tail to the head group bound at the surface, the mobility of the chains may be significantly limited relative to that of bulk hydrocarbon chains. The presence of kinks or gauche chain conformations, which may be imposed by packing considerations, will result in a calculated Imax amounting to only about 80% of the theoretical maximum. 

Also consistent with theoretical predictions was the observation that the ratio An/fp was essentially constant, independent of extension and reaction time. Here = (/)—l)/(Z)-l-2) is the polyene segment orientation function which follows from eqn. (11), noting that the absorbance is a maximum in the direction of the segment axis and that the small perpendicular component of absorbance is negligible. Assuming that the constant vklue A /f0 = may also be applied to the pure PVC, we may write. 

As noted, once the material has been drawn into the craze it appears to stabilize, and little, if any, further extension takes place. As with cold drawing, the fibrillar material is seen to take on a natural draw ratio, Xn, the value of which is characteristic of the individual polymer. Kramer (116,141) found that the natural draw ratio correlated well with the maximum theoretical extension expected for an entangled network, Xmax ... 

Here Geo is the equilibrium modulus of the unfilled polymer, is the volume fraction of filler, and , is a maximum volume fraction corresponding to close packing, which may be between 0.74 and 0.80. For < 0.70, this equation is equivalent to the result of a theoretical formulation by van dcr Pocl (which can be evaluated only numerically) relating the shear and bulk moduli of a composite with spherical particles to the shear and bulk moduli and Poisson s ratios of the two component materials. The derivation of van dcr Poel has been corrected and simplified by Smith.For a hard solid in a rubbery polymer, the ratio of the shear moduli is so large that the result is insensitive to its magnitude. An example is shown in Fig. 14-13 for data of Schwarzl, Brcc, and Nederveen for nearly monodisperse sodium chloride particles of several different sizes embedded in a cross-linked polypropylene ether. Extensive comparisons of data with equation 18 have been made by Landcl, -"- - who has also employed an alternative relation ... 

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