Normal stress differences oscillatory

J. A. Kornfield, G. G. Fuller, and D. S. Pearson, Third normal stress difference and component relaxation spectra for bidisperse melts under oscillatory shear, Macromolecules, 24, 5429 (1991). 

In Equation 3.116, is rigorously defined as [(an - 022)I(S 2 > 1 is the sum of a constant term and two oscillating terms, accounted by ijr[ and y i is the strain rate amplitude. Equations 6 to 8 suggest that oscillatory shear stress data are related to oscillatory primary normal stress difference data (Ferry, 1980). Youn and Rao (2003) calculated values of (co) for starch dispersions is applicable to oscillatory shear fields. 

Steady shear flow measnrements, however, can measure only viscosity and the first normal stress difference, and it is difficult to derive information abont fluid structure from such measurements. Instead, dynamic oscillatory rheological measurements are nsed to characterize both enhanced oil recovery polymer solutions and polymer crosslinker gel systems (Prud Homme et al., 1983 Knoll and Pmd Homme, 1987). Dynamic oscillatory measurements differ from steady shear viscosity measnrements in that a sinusoidal movement is imposed on the fluid system rather than a continnons, nnidirectional movement. In other words, the following displacement is imposed ... 

In a dynamic experiment, a small-amplitude oscillatory shear is imposed to a molten polymer confined in the rheometer. The shear stress response of the polymeric system can be expressed as in Equation 22.14. In this equation, G and G" are dynamic moduli related to the elastic storage energy and dissipated energy of the system, respectively. For a viscoelastic fluid, two independent normal stress differences, namely, first and second normal stress differences can be defined. These quantities are calculated in terms of the differences of the components of the stress tensor, as indicated in Equation 22.15a and 22.15b, and can be obtained, for instance, from the radial pressure distribution in a cone-and-plate rheometer [5]. Some other experiments used in the determination of the normal stress differences can be found elsewhere [9, 22] ... 

Some of the manifestations of viscoelasticity are delayed relaxation of stress after cessation of flow phase shift between stress and strain rate in oscillatory shear flow shear thinning (decrease of viscosity) at shear rates exceeding the reciprocal of the longest relaxation time and normal stress differences in shear flow, whose magnitudes are related to the relaxation time spectrum. A very convenient observation for experimentalists is that there is a close similarity between the shear viscosity and first normal stress difference as functions of shear rate and the corresponding parameters, complex viscosity and storage modulus, as functions of frequency in a small amplitude oscillatory shear. 

Larson and co-workers [104] have carefully studied the rheology of lyotropic polypeptide liquid crystals. The shear-rate dependence of rj has been examined for isotropic and anisotropic (liquid-crystalline) solutions (Fig. 5.29). Additionally, the first and second normal-stress differences Ni and N2 have been shown to exhibit very unusual behaviors e.g. N2 is an oscillatory function of the shear rate (Fig. 5.31). The observations are in qualitative agreement with extensions of Doi s theory. (In a disclination-ridden mesophase it is necessary to average over... 

Data is provided on the complete rheological characterisation, relative to steady-state shear stress and first and second normal stress differences as a function of shear rate, as well as the complementary oscillatory data, see figxires 13 and 14. The measured extensional properties are also available, see figure 15 [7]. 

If steady shear flow of a viscoelastic liquid is abruptly halted, the normal stress differences a — azi and shear stress an (Fig, 1 -5). Also, for sinusoidally varying shear strains, each normal stress difference is a sinusoidal function of time, with oscillatory components at twice the frequency of the shear strain about a nonzero mean value. Some of these relations are given in Chapter 3. Other time-dependent experimental patterns of strain or stress history evoke various characteristic nonlinear phenomena. 

A sinusoidally varying shear strain rate with small amplitude, such that 721 = 721 cos cor, evokes a sinusoidally varying normal stress difference symmetry considerations because of the proportionality to 721, is predicted by the phenomenological models previously quoted.5 -54 jj,g oscillatory stress difference is superposed on a constant stress and both are proportional to 721 if 721 is small. The coefficient is now defined as the ratio (ai i — ff22)/(72i) - It is the sum of a constant term and two oscillating terms ... 

Thus oscillatory measurements of the primary normal stress difference give the same information as oscillatory shear stress measurements. Oscillatory measurements of the secondary normal stress difference, however, would provide additional... 

Polymer solutions are viscoelastic. This was shown by Eisenschitz and Philippoff [67] as early as 1933 using oscillatory experiments. Polymer solutions exhibit normal stresses and complex viscoelastic properties [66,68 to 70] (see Section 1.3.5). Tanner [70] and Ide and White [66] have sought to determine the dependence of normal stresses upon concentration in polymer solutions (see Section 1.3.5). They considered at the principal normal stress difference Nj at low shear rates. 

In this section, we present the rheological behavior of linear flexible homopolymers (i.e., without side-chain branching). We first present methods for obtaining temperature-independent plots for shear viscosity ( ) and first normal stress difference (N ) in steady-state shear flow and for dynamic moduli (G" and G") in oscillatory shear flow. We then discuss the effect of molecular weight and molecular weight distribution on the rheological behavior of linear flexible homopolymers. 

Besides being used as a calibration device, concentric cylinders can be used to find the first normal stress difference (c e —O to investigate orientation effects in shearing and oscillatory ffow. The measurements provide shear orientation information for both the rigid and flexible types of microstructure. 

A.4 Estimate the Primary Normal Stress Difference from Dynamic Data. Use Eq. 3.148 and the dynamic oscillatory data given in Appendix A.3, Table A.8 to estimate Ni for LLDPE at 170 °C. Compare your values with those given in Appendix A.3, Table A.7. 

As an extension of the dynamic test in the linear regime, a large-amplitude oscillatory shear (LAOS), y(t) = /o sin or, is often applied to a material to measure the oscillating shear stress (Ts (and sometimes the first normal stress difference, too). In the stationary state, cTs under LAOS is contributed not only from the fundamental harmonics (oscillating at the angular frequency of strain, cu) but also from the higher order (mostly odd) harmonics and can be expressed as... 

The first of these relations was noted by Lodge [(46), Eq. (6.43)] and by Williams and Bird (77) as a result of the study of two different empirical constitutive equations. Later Spriggs (70) obtained Eqs. (7.22) and (7.23) from the Coleman and Noll (21) theory of second-order viscoelasticity. Eqs. (7.22) and (7.23) indicate that no additional information about fluids can be obtained from normal stress oscillatory measurements than has not already been obtained by shear stress oscillatory data. Eq. (7.24) seems to be new and is probably specific to rigid dumbbell suspensions. 

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