Normal-stress coefficients difference

For common liquids, the viscosity is a material constant which is only dependent on temperature and pressure but not on rate of deformation and time. For polymeric liquids, the situation is much more complicated viscosities and normal stress coefficients differ with deformation conditions. Because polymer melts are viscoelastic their flow is accompanied by elastic effects, due to which part of the energy exerted on the system is stored in the form of recoverable energy. For this reason the viscosities are time and rate dependent polymer melts are viscoelastic. 

The coordinates (x, y, z) define the (velocity, gradient, vorticity) axes, respectively. For non-Newtonian viscoelastic liquids, such flow results not only in shear stress, but in anisotropic normal stresses, describable by the first and second normal stress differences (oxx-Oyy) and (o - ozz). The shear-rate dependent viscosity and normal stress coefficients are then [1]... 

Another often used representation of the viscoelastic flow behavior utilizes normal stress coefficients P/ = Ni/y. Figure 10 depicts flow curves of a family of PAA/water solutions differing in concentrations and therefore in their viscosities. Normalized by the zero-shear viscosity fiQ and by a constant shear rate /q shear stress value of to= 1 N/m they produce master curves for viscosity and the normal stress coefficient. The preparation... 

Figure 10 Flow curves of a family of polyacrylamide (PAA)/water-solutions of different concentrations and viscosities. Left side normalized viscosity curves id/po = f y/yo), right side normalized stress coefficients ( i + 2 2) ol o =f2 Source From Ref. 13.

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. 

N2 values are always lower than Nj values, see e.g. [40]. Therefore for many processes taking into consideration only Nj will suffice. The normal stress differences are independent of the direction of flow and, in laminar flow (low y), are proportional to y2. In following p = x/y for a Newtonian fluid, normal stress coefficients ipi = Nj/y2 and ip2 = N2/y2 are occasionally used. Their dependence on the shear rate i j(y) describes the non-linear viscoelastic behavior of the fluid. 

Here are the components of the stress tensor as defined in rheology Tn—T22 is the first normal stress difference and T21 the shear stress, equal to Nt and rxsh, respectively. Hence, from dynamic mechanical measurements it is possible to determine the zero shear first normal stress coefficient Fq0 and zero shear viscosity y0. 

Experimental determination of viscosity and normal stress coefficients Since the range of rj may extend from 10 2 to 1011 N s/m2 and the normal stress coefficients, Tj and Tj, cover also a wide range of values, a number of different experimental techniques have been developed to cover this wide range. Some methods are listed in Table 15.2. 

These equations show that both the shear stress and the first normal stress difference gradually increase from zero to a steady state value for —>oo. This results in values for the viscosity and the first normal stress coefficient equal to... 

In Sect. 15.4 it was shown how the shear thinning behaviour of the viscosity could be described empirically with the aid of many suggestions found in literature. It was not mentioned there that the first normal stress coefficient also shows shear thinning behaviour. In this Sect. 15.5 it became clear that also the extensional viscosity is not a constant, but depending on the strain rate upon increasing the strain rate qe the extensional viscosity depart from the Trouton behaviour and increases (called strain hardening) to a maximum value, followed by a decrease to values below the zero extensional viscosity. It has to be emphasised that results in literature may show different behaviour for the extensional behaviour, but in many cases this is due to the limited extensions used,... 

FIG. 15.46 Viscosity, 77, and first normal stress difference, Nh of Vectra 900 at 310 °C as functions of shear rate, according to Langelaan and Gotsis (1996). The first normal stress coefficient, Yi, is estimated from N, by the present author. ( ) Capillary rheometer ( ) and ( ) cone and plate rheometer ( ) complex viscosity rj (A) non-steady state values of the cone and plate rheometer. Courtesy Society of Rheology. 

It is amazing that in many cases the viscosity and the first normal stress differences are reported together it would be wise to compare the viscosity and the first normal stress coefficient, Th, because these both are material properties. For that reason in Fig. 15.46, the present author also incorporated Ll. It also seems to reach a Region II. 

Fiber-reinforced polymer systems, 38 Fickian diffusion, 665 Fick s law, 663,684 Field flow fractionation, 20 Filled polymers, 38 First normal stress coefficient, 545 difference, 640 First-order transition, 27,152 Flame-retardant additives, 861 Flammability, 847 Flashing, 804 Flash line region, 807 Flexibility of a chain molecule, 246 Flexible polymer molecules, 706 Flexural deformation under constant load, 825 Flexural formulas, 826 Flexural rigidity, 877 Floor temperature, 751 Flory-Huggins... 

However, it is now worth pointing out that the difference in ax and aN imply that the initial model should be replaced in shear by a pair of two independent correlations for shear stress (eq. 50a and 50c) or for first normal stress coefficient (eq. 50b and 50d). But at this point some questions arise concerning the choice of the proper value (bt or un) to be used in any other flow situation. Though it is possible to imagine equation (49) Including some variation of a with flow history or invariants, it could hardly be different in two equations for the same flow kinematics. 

Figure 1-1 First Normal Stress Coefficient Data of Starch Dispersions with Different Concentrations as a Function of Shear Rate (Genovese and Rao, 2003). Abbreviations cwm, cross-linked waxy maize tap, tapioca gran, granule.

According to Fig. 6-15, this Peclet number is low enough to be in the low Peclet-number limit, where the normal stress differences are quadratic in the shear rate, and hence the first normal stress coefficient is a constant. This constant can be obtained from the low-shear-rate portion of Table 6-1. Since p is large (p oo), this table gives... 

The ability to homogenize resins with widely different molecular weight can be exemplified by UHMWPE/HDPE blends. Addition of high MW polymer is expected to increase G , G , T), and the first normal stress coefficient, Pj. For the linear polymers, these parameters at... 

Alternatively, from steady shearing experiments, which yield rio directly in the limit of low shear rate, the steady-state recoverable compliance can be obtained from the first normal stress coefficient, which is the ratio of the first normal stress difference to the square of the shear rate, measured at low shear rate... 

The ability to homogenize resins with widely different molecular weight can be exemplified by UHMWPE/HDPE blends. Addition of high MW polymer is expected to increase G, G", t, and the first normal stress coefficient, Pj. For the linear polymers, these parameters at low deformation rates, rjo and Pjo, are proportional to M , and Mw , respectively. Thus, the elasticity is more sensitive to the high MW fractions. For this reason, the frequency dependence of the storage modulus ratio G (blend)/G (PE), at 200 °C, for HOPE and its blends with 3 wt% UHMWPE was measured. The blends prepared in TSE had the worst... 

Figure 9.20(a) and (b) show the two normal stress coefficients hi and — F2 (solid lines) and their ratio — 2/ (broken line) as a function of log)>. The ratio turned out to be virtually constant over a wide range of y, though we have no specific analytical reason for this. Both coefficients are monotonic functions of y and show only a minor quantitative difference. 

Using a filler size distribution, the normal stress difference of the filled polymer system can be altered. The relative primary normal stress coefficient is reduced when a bimodal distribution is used and gives the lowest values at about 10-30% volume fraction of the small particles. 

The viscoelastic equivalents to viscosity—the stress divided by the shear rate—are the so-called first and second normal-stress coefficients, F,and These are given by the first and second normal-stress differences divided by the shear rate squared, so... 

Particular examples of the first normal-stress difference Ni or its coefficient are shown in figures 12 - 29, where a comprehensive collection of examples is given for polymer solutions and melts, as well as an emulsion. These are compared with either the equivalent shear stress or the viscosity. All the different possible combinations of shear and normal-stress difference, and viscosity and normal-stress coefficient are displayed to show the way that results are presented in the rheological literature. The figures are set out to illustrate overall behaviour, with especial emphasis on low shear-rate and mid-range (i.e. power-law) behaviour. 

The normal stresses c,/ cannot be specified on an absolute basis because of the arbitrary hydrostatic pressure in equation 59, but their differences are predicted by continuum mechanics and several molecular theories and can be measured. The primary and secondary normal stress differences are defined as (Th — an and 022 — an respectively. Data are often expressed in terms of the primary normal stress coefficient... 

At very low shear rates, the normal stress coefficients 1,0 and 2,0 are also independent of 721 i.e., the normal stress differences are proportional to 721. At higher shear rates, 1 and 2 are observed to decrease. The course of stress relaxation after cessation of steady-state flow and the magnitude of the steady-state compliance J° are also strongly affected at high shear rates. In general, description of these phenomena requires more complicated constitutive equations than the single-integral models mentioned above. 

It is evident that the Giesekus model can quantitatively describe the shear thinning behavior of the entangled solutions of rod-shaped micelles. The decrease of the viscous resistance is caused by the alignment of the anisometric aggregates in the streaming solutions. Similar conclusions can be drawn from measurements of the first normal stress difference. This parameter is often represented in terms of the first normal stress coefficient ... 

Steady state shear viscosity and primary normal stress coefficient for low density polyethylene melt T and from the Kaye-Bernstein, Kearsley, Zapas (K-BKZ) equation wim the double exponential damping function, eq4.4.13 (solid lines) and with the single exponential, eq4.4.12 (dotted Une). Data at different temperatures have been shifted to one master curve by ar T). Replotted from Laun (1978). 

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