Normal stress differences measurement

As the flow accelerates into the gaps around the cylinder, it possesses a greater relative amount of extension. Ultimately, at distances far downstream from the cylinder, the flow is expected to relax back toward a parabolic profile. In these plots, the symbols represent the measured velocities and the solid curves are the results of a finite element, numerical simulation. The constitutive equation used was a four constant, Phan-Thien-Tanner mod-el[193], which was adjusted to fit steady, simple shear flow shear and first normal stress difference measurements. The fit to the velocity data is very satisfactory. 

This section describes two common experimental methods for evaluating i], Fj, and IG as functions of shear rate. The experiments involved are the steady capillary and the cone-and-plate viscometric flows. As noted in the previous section, in the former, only the steady shear viscosity function can be determined for shear rates greater than unity, while in the latter, all three viscometric functions can be determined, but only at very low shear rates. Capillary shear viscosity measurements are much better developed and understood, and certainly much more widely used for the analysis of polymer processing flows, than normal stress difference measurements. It must be emphasized that the results obtained by both viscometric experiments are independent of any constitutive equation. In fact, one reason to conduct viscometric experiments is to test the validity of any given constitutive equation, and clearly the same constitutive equation parameters have to fit the experimental results obtained with all viscometric flows. 

The rheological behaviour of the two polymers was determined using classical techniques of rheometry, already described in Chapter II. 1 (rotational and capillary rheometers for shear viscosity and first normal stress difference measurements CogsweU method for the elongational viscosity). 

The normal stress difference measures the difference in the normal stresses in the direction of elongation and that normal to it. The magnitudes of the stresses of a particle or point are not the same in the various directions of... 

The Maxwell relaxation time is commonly evaluated from the first normal stress difference measurement. 

Baird, D. G. 2008. Primary Normal Stress Difference Measurements for Polymer Melts at High Shear Rates in a Slit-Die Using Hole and Exit Pressure Data. J. Non-Newtonian Fluid Mech., 148, 13-23. 

Figm 31 Steady-state shear stress os and the first normal stress difference measured at room temperature for a 1 1 blend of PBMS (M= 1.0 X105, VO=107 Pa s) and PB (M= 1.8 X104, VO=97.8 Pas). Data taken from Takahashi, Y. Kitade, S. Kurashima, N. Noda, I. Polym. J. 1994,26,1206 with permission. 

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... 

Description of normal stress measurements on a practical but complex material, paint, is available (150). More recent pubHcations (151—154) give the results of investigations of normal stress differences for a variety of materials. These papers and their references form a useful introduction to the measurement of normal stress differences. 

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. 

Here we describe the strain history with the Finger strain tensor C 1(t t ) as proposed by Lodge [55] in his rubber-like liquid theory. This equation was found to describe the stress in deforming polymer melts as long as the strains are small (second strain invariant below about 3 [56] ). The permanent contribution GcC 1 (r t0) has to be added for a linear viscoelastic solid only. C 1(t t0) is the strain between the stress free state t0 and the instantaneous state t. Other strain measures or a combination of strain tensors, as discussed in detail by Larson [57], might also be appropriate and will be considered in future studies. A combination of Finger C 1(t t ) and Cauchy C(t /. ) strain tensors is known to express the finite second normal stress difference in shear, for instance. 

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. 

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... 

A special advantage of this method is that the high shear rate range becomes available. It appears that one can measure nu — n33 up to the critical shear stress, at which extrusion defect (melt-fracture) occurs. On the other hand, entrance effects can also be studied, when the windows are located sufficiently close to the entrance. With the aid of the stress-optical coefficient, the corresponding normal stress difference can be... 

There are some interesting points to be noted. First, it seems that also for polymer melts the normal stress differences (fin — fi22) and (fin—fi33) are practically equal. (Similar results have been obtained for melts of several polyethylenes.) Second, for the investigated polystyrene a practically quadratic dependence of nn — n33 on the shear stress is found up to the point of the inset of an extrusion defect. It is noteworthy that Fig. 1.9 shows no quadratic dependence of Pjd vs Ds, as would be expected for a second order fluid. Third, the measurements in the cone-and-plate apparatus have to be stopped at a shear stress at least one... 

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,...

L-100 (Mw — 1.4 0.1 x 10 , Mw/Mn — 2.2)4 in medicinal white oil as a rather high viscous solvent (1.50 poise at 25° C). In this figure the directly measured shear recovery (s ) (open triangles) is plotted against shear stress pzl of the preceding shear flow. From flow birefringence measurements (in a coaxial cylinder apparatus) and normal thrust measurements (in a cone-and-plate apparatus) values of normal stress difference (pn — p22) were calculated. These values were transformed with the aid of eq. (2.12) into recoverable shears s. The full circles (from... 

Kuo Y, Tanner RI (1974) Use of open-channel flows to measure the second normal stress differences Rheol Acta 13 931... 

Measurable normal stress differences, N = txx — ryy and N2 = tvv — tzz are referred to as the first and second normal stress differences. The first and second normal stress differences are material dependent and are defined by... 

The second normal stress difference is difficult to measure and is often approximated by... 

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.

In industry there are various ways to qualify and quantify the properties of the polymer melt. The techniques range from simple analyses for checking the consistency of the material at certain conditions, to more complex measurements to evaluate viscosity, and normal stress differences. This section includes three such techniques, to give the reader a general idea of current measuring techniques. 

For systems where the stress-optical rule applies, birefringence measurements offer several advantages compared with mechanical methods. For example, transient measurements of the first normal stress difference can be readily obtained optically, whereas this can be problematic using direct mechanical techniques. Osaki and coworkers [26], using a procedure described in section 8.2.1 performed transient measurements of birefringence and the extinction angle on concentrated polystyrene solutions, from which the shear stress and first normal stress difference were calculated. Interestingly, N j was observed to... 

Transient birefringence measurements were used by Larson et al. [112] to test the validity of the Lodge-Meissner relationship for entangled polymer solutions. This relationship states that the ratio of the first normal stress difference to the shear stress following a step strain is simply Nx/%xy - y, where y is the strain. Those authors found the relationship was valid, except for ultrahigh molecular weight materials. 

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