Extinction, angles

In Fig. 1.2 the extinction angle curves as functions of the applied shear rate are given. In Fig. 1.3, the corresponding birefringence curves are shown. These results are chosen, because the mentioned solution... 

Fig. 1.2. Extinction angles % vs. shear rate q for a 2 wt. per cent solution of anionic polystyrene Sill in methyl 4-bromo-phenyl carbinol at the indicated temperatures...

This behaviour of the extinction angles is in accordance with eq. (1.3), if % = In fact, for a second order fluid the first normal stress difference increases with the square of the shear rate, whereas the shear stress increases with the first power of this rate (constant viscosity). As a consequence, it follows from eq. (1.3) that cot 2% increases linearly. From this fact the above mentioned linear behaviour of the extinction angle curve is deduced, since... 

Fig. 1.4. Evidence for the validity of the stress-optical law [Philippoff (8, 9)]. 15 per cent solution of poiyisobutylene B-100 in decalin. (A) extinction angle % and flow birefringence An vs. shear stress pa at 30° C. (o) and An at 50° C. ( ) calculated % according to eq. 1.3 from cone-and-plate measurements at 30° C. ( , An sin2 at 30 and 50° C, respectively...

To check whether eq. (1.5) is valid also for fluids of more general properties than those of the just described solution, flow birefringence, extinction angle and shear stress should be measured in a sufficient range of shear rates. When a plot is made according to eq. (1.5), a straight line should result. The fact that the shear rate is eliminated, indicates the quasistatic character of the stress-optical consideration. 

Instead of checking the second stress-optical relation, viz. eq. (1.6), Philippoff preferred to use eq. (1.3), assuming % = %. That this assumption is justified, can be seen in the same Fig. 1.4, In this figure also the extinction angle is plotted against the shear stress. Orientation angles calculated with the aid of eq. (1.3), fit rather well on the extinction angle curve. The normal stress difference (pn — p22) has been measured in the way explained in the previous section. Similar results were published later by the same author for solution of carboxy methyl celluose in water (33) and for S 111 in Aroclor (34) a chlorinated biphenyl. 

Fig. 1.6. Doubled extinction angle 1% ( a) and doubled orientation angle 2 (o) vs. shear rate q for a polypropylene melt [36) at 210° C...

This relation has first been proposed by Philippoff (9). It becomes particularly suitable on condition that the stress-optical law, eq. (1.4), is valid. In this case dynamic measurements can be compared with the extinction angle of flow birefringence. 

Fig. 2.4. Doubled extinction angle 2% vs. shear rate (open circles) and loss angle <5 vs. angular frequency (closed circles) for the melt of anionic polystyrene Sill at a measurement temperature of 196° C [Wales, Den Otter (56)]...

Anionic polystyrene Sill seems to be a suitable polymer for this type of investigation. Some solution properties of this polymer have already been discussed in the previous chapter. Fig. 2.4 gives some properties of the melt of this polymer at a measurement temperature of 196° C (56). In this figure the doubled extinction angle 2% as a function of shear rate q (open circles) is compared with the loss angle d as a function of angular frequency (closed circles). In accordance with eq. (2.22) the initial slopes of these curves coincide. This coincidence, however, appears to persist even into the non-linear part of the functions. Such a persistence... 

After this rather extensive discussion of the experimental verification of eqs. (2.11), (2.20) and (2.22), where the interrelations of steady flow properties with dynamic properties and shear recovery are scrutinized, a few words should be added with respect to stress-relaxation after cessation of steady shear flow. As pointed out above, Lodge predicted a decrease of the extinction angle during stress relaxation (4). That this... 

Fig. 2.7. Drop off of doubled extinction angle 2"/ during stress-relaxation after cessation of steady shear flow according to Wales (59). Measurements on the melt of a high-density polyethylene (Marlex 6002) at a measurement temperature of 147° C. Shear rate of the steady shear flow q = 0.06 sec-1...

This equation will be particularly useful for the comparison of extinction angle and shear stress measurements. For the purpose, the second eq. (1.4) is used in the form ... 

It is noticed that the correction for the solvent birefringence (A n0) occurs only in eq. (3.44b) and not in eq. (3.44a). As was already pointed out in Chapter 2, the extinction angle of a low molecular weight liquid is, practically under all conditions, equal to 45 degrees. From this it is clear that the cosine of the doubled extinction angle of the solvent vanishes, whereas its sine becomes equal to unity. This corresponds to the fact that the solvent yields no contribution to the normal stress effect [cf. eq. (1.6)]. 

As a consequence, the corrected value %c for the extinction angle as used above reads ... 

When a low viscous solvent must be used in combination with a rather low molecular weight of the polymer, measurements are restricted to low /3-values, due to the discussed onset of turbulent flow. As in such a case the extinction angle % does not deviate very much from 45 degrees within the regime of laminar flow, it must be measured with a high absolute accuracy to furnish a reliable value for cos 2% or cot 2% [cf. eq. (3.42) or (3.44a)]. Measurements on a polydisperse sample become more reliable under such conditions due to the fact that cot 2 is increased by the polydispersity factor [eqs. (3.75a) and (3.83a)]. Examples for such a behaviour will be discussed in Section 3.8.3. 

IUPAC.-Working Party on "The Relationship of Performance Characteristics to Basic Parameters of Polymers . This sample has been investigated in transdecalin at 160° C. Fig. 3.8 shows the extinction angle curves obtained for the three indicated concentrations as functions of A linear extrapolation at various /Sy.urvalues was possible and led... 

Fig. 3.8. Extinction angles % vs. reduced shear rate for solutions of a high density polyethylene fraction (Dow Chem. Corp.) in transdecalin at 160° C (75). The concentrations are indicated near the curves in g/100 cm. Open and closed symbols indicate repeat measurements. The dotted line gives the extinction angle vs. at zero concentration, as obtained by linear extrapolation at several...

As a final remark it may be mentioned that the discussed polypropylene melts do not at all behave like second-order fluids in the range of shear rates and angular frequencies accessible to measurement. This is shown in Fig. 4.6. In this figure the doubled extinction angle 2 is plotted... 

Fig. 4.6. Doubled extinction angle 2y (closed triangles) and doubled orientation angle 2% (open circles and triangles) as function of shear rate q, and loss angle 6 as a function of angular frequency (closed circles, connected by dashed lines) for the melts of two polypropylene samples. Data of samples are given in Table 3.3. Measurement temperature 210° C (36)...

Fig. 5.2. Extinction angle x vs. shear rate q for solutions of a polystyrene fraction (M = 3.3 X 10°) in dioxane according to Frisman and Tsvetkov (138). The numerals at the curves indicate concentrations in g/100 cm3...

---