First normal-stress difference

Polyolefin melts have a high degree of viscoelastic memory or elasticity. First normal stress differences of polyolefins, a rheological measure of melt elasticity, are shown in Figure 9 (30). At a fixed molecular weight and shear rate, the first normal stress difference increases as MJM increases. The high shear rate obtained in fine capillaries, typically on the order of 10 , coupled with the viscoelastic memory, causes the filament to swell (die swell or... 

Fig. 9. First normal stress differences of polypropylene of different molecular weight and distribution (30) see Table 4 for key. To convert N /m to...

Fig. 22. Shear viscosity Tj and first normal stress difference (7) vs shear rate 7 for a low density polyethylene at 150°C (149), where (Q) — parallel plate ...

A sliding plate rheometer (simple shear) can be used to study the response of polymeric Hquids to extension-like deformations involving larger strains and strain rates than can be employed in most uniaxial extensional measurements (56,200—204). The technique requires knowledge of both shear stress and the first normal stress difference, N- (7), but has considerable potential for characteri2ing extensional behavior under conditions closely related to those in industrial processes. 

Non-linear viscoelastic flow phenomena are one of the most characteristic features of polymeric liquids. A matter of very emphasised interest is the first normal stress difference. It is a well-accepted fact that the first normal stress difference Nj is similar to G, a measure of the amount of energy which can be stored reversibly in a viscoelastic fluid, whereas t12 is considered as the portion that is dissipated as viscous flow [49-51]. For concentrated solutions Lodge s theory [52] of an elastic network also predicts normal stresses, which should be associated with the entanglement density. 

Fig. 19. Shear stress and first normal stress difference plotted as a function of shear rate for different molar masses, and b at different concentrations of polystyrene in toluene... 

The corresponding first normal stress difference N t) = tu(t) — t22( ) as predicted from Eq. 4-2... 

Fig. 13. Shear stress t12 and first normal stress difference N1 during start-up of shear flow at constant rate, y0 = 0.5 s 1, for PDMS near the gel point [71]. The broken line with a slope of one is predicted by the gel equation for finite strain. The critical strain for network rupture is reached at the point at which the shear stress attains its maximum value...

Quotient of the first normal stress difference N ) and the square of the shear rate (y) in the limit of zero shear rate... 

Various investigations have considered the effects of titanate treatments on melt rheology of filled thermoplastics [17,41]. Figure 10, for example, shows that with polypropylene filled with 50% by weight of calcium carbonate, the inclusion of isopropyl triisostearoyl titanate dispersion aid decreases melt viscosity but increases first normal stress difference. This suggests that the shear flow of the polymer is promoted by the presence of titanate treatment, and is consistent with the view that these additives provide ineffective coupling between filler particles and polymer matrix [42]. 

Fig. 10. Viscosity and first normal stress difference vs. shear stress for polypropylene (at 200 °C) filled with calcium carbonate (50 wt%) with and without a titanate coupling agent (TTS ) (O, ) pure polypropylene (PP) (A,A) PP/CaC03=50 50 (by wt.) ( , ) PP/CaC03=50 50 with TTS (1 wt%). The open symbols were obtained from a cone and plate instrument and the closed symbols from a slit/capillary rheometer.

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... 

In Sections 1.2 and 1.3 only the measurements of shear stress p and of first normal stress difference (/>u — p22) have been discussed. To... 

In Chapter 1 the validity of a stress-optical law has been presumed. Furthermore, it has been shown for several polymer systems that this law is, at least approximately, valid and that the second normal stress difference (p22 — p33) must be very small compared with the first normal stress difference (pn — 22). In the present chapter some theoretical considerations of a more general character will be reviewed in order to indicate reasons for this special behaviour of flowing polymer systems. Some additional experimental results will be given. 

As is well-known, this pair of expressions will not be valid for the most general case of a second order fluid, since p22 — tzi must not necessarily vanish for such a fluid. Eq. (2.9) states that the first normal stress difference is equal to twice the free energy stored per unit of volume in steady shear flow. In Section 2.6.2 it will be shown that the simultaneous validity of eqs. (2.9) and (2.10) can probably quite generally be explained as a consequence of the assumption that polymeric liquids consist of statistically coiled chain molecules (Gaussian chains). In this way, the experimental results shown in Figs. 1.7, 1.8 and 1.10, can be understood. 

Fig. 2.1. First normal stress difference (pu—p22) as a function of shear rate q and doubled storage modulus 2 G as a function of angular frequency for a poly-dimethyl siloxane (M = 536,000) at a measurement temperature of 20° C. (o) (A n/C) cos 2y,...

With this quantity one obtains a simple relation between reduced first normal stress difference (pu — p2J)jv k T and reduced shear stress p2ilv k T. For this purpose, velocity gradient q is eliminated from eqs. (3.37) and (3.38). This relation reads ... 

This treatment will not be discussed in further detail, as a knowledge of distinct relaxation times is not required for the present purpose. Interest is focussed on the relation between reduced first normal stress difference and reduced shear stress, as expressed by eq. (3.41). The simplest way to evaluate this equation for a polydisperse polymer has been given by Peterlin (76). This procedure has extensively been used by Daum (32, 73) in his experimental investigations. 

Figure 2.31 Reduced first normal stress difference coefficient for a low density polyethylene melt at a reference temperature of 150°C.

The material functions, k i and k2, are called the primary and secondary normal stress coefficients, and are also functions of the magnitude of the strain rate tensor and temperature. The first and second normal stress differences do not change in sign when the direction of the strain rate changes. This is reflected in eqns. (2.51) and (2.52). Figure 2.31 [41] presents the first normal stress difference coefficient for the low density polyethylene melt of Fig. 2.30 at a reference temperature of 150°C. 

For example, Figs. 2.43 and 2.44 present the measured [55] viscosity and first normal stress difference data, respectively, for three blow molding grade high density polyethylenes along with a fit obtained from the Papanastasiou-Scriven-Macosko [59] form of the K-BKZ equation. A memory function with a relaxation spectrum of 8 relaxation times was used. 

Figure 2.44 Measured and predicted first normal stress difference for various high density polyethylene resins at 170° C.

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