Steady-state shearing

Figure A3.1.5. Steady state shear flow, illustrating the flow of momentum aeross a plane at a height z.

Normal Stress (Weissenberg Effect). Many viscoelastic fluids flow in a direction normal (perpendicular) to the direction of shear stress in steady-state shear (21,90). Examples of the effect include flour dough climbing up a beater, polymer solutions climbing up the inner cylinder in a concentric cylinder viscometer, and paints forcing apart the cone and plate of a cone—plate viscometer. The normal stress effect has been put to practical use in certain screwless extmders designed in a cone—plate or plate—plate configuration, where the polymer enters at the periphery and exits at the axis. 

Rheo-Optical Observations. The rheo-optical observations were made with a WILD stereoscope equipped with a custom made rotary optical stage using polarized light. The rotary optical stage described in detail elsewhere (9) was used for both the steady state shear and dynamic test conditions. 

Rheological Properties Measurements. The viscoelastic behavior of the UHMWPE gel-like systems was studied using the Rheometric Mechanical Spectrometer (RMS 705). A cone and plate fixture (radius 1.25 cm cone angle 9.85 x 10" radian) was used for the dynamic frequency sweep, and the steady state shear rate sweep measurements. In order to minimize the error caused by gap thickness change during the temperature sweep, the parallel plates fixture (radius 1.25 cm gap 1.5 mm) was used for the dynamic temperature sweep measurements. 

Steady state shear viscosity measurements indicate a power-law type relation for the variation of the shear viscosity with shear rate even in the lower shear rate range between 10"1 to 1 sec"l. The results at higher shear rates were questionable due to the slip between the sample and cone and plate fixtures. 

For the limiting cases, N 0 and k oo, Eqs. (12-47) and (12-49) reduce to equations derived by Taylor (T2, T3) for fluid particles in steady-state shear or hyperbolic flows, i.e.,... 

Figure 7.9 Effect of pectin (DE = 76%) on (a) creaming of protein-stabilized emulsions (11 vol% oil, 0.6 wt% protein, 0.28 wt% pectin, I = 0.01 M) containing (A) asi-casein (pH = 7), (A) p-casein (pH = 7), and ( ) o i-casein (pH = 5.5) and (b) steady-state shear viscometry of casein-stabilized emulsions (40 vol% oil, 2 vt% protein). Apparent shear viscosity at 22 °C is plotted against stress pH = 7.0, / = 0.01 M, (A) -casein, (A) p-casein, ( ) ocsi -casein + 0.5 wt% pectin, ( ) p-casein + 0.5 wt% pectin, ( ) p-casein + 1.0 wt% pectin, (O) as[-casein + 1.0 wt% pectin pH = 5.5,1 = 0.01 M, (x) ocsi -casein, (O) as[-casein + 0.5 wt% pectin, ( ) oc -casein + 1.0 wt% pectin. Reproduced from Semenova (2007) with permission.

The symbols Nt and N2 denote the normal stress functions in steady state shear flow. Symmetry arguments show that the viscosity function t](y) and the first and second normal stress coefficients P1(y) and W2(y) are even functions of y. In the... 

Behavior in steady state shearing flows likewise provide values of these parameters (83) ... 

At sufficiently low shear rates the shear stress should decay at the termination of steady state shear flow according to the equation from linear viscoelasticity ... 

The total amount of shear recoil after steady state shear flow at sufficiently low shear rates is related to Je° ... 

According to the theory of linear elastico-viscous behaviour (47) the steady-state shear viscosity t] and the steady-state shear compliance Je depend in the following way on the shear relaxation modulus G (t), where t is here the time of the relaxation experiment ... 

Lodge s expression for the steady-state shear viscosity reads (10) ... 

This means that for a linear elastico-viscous liquid, the steady-state shear compliance Je must be equal to the constrained shear recovery sx which follows on a steady shear flow at unity shear stress. From this one deduces the following relation between and Je ... 

The corresponding value of the reduced steady-state shear compliance reads ... 

Unfortunately, Fixman has not yet given a value for the reduced steady-state shear compliance. However, from a comparison of eqs. (3.60a), (3.64) and (3.66) the impression is obtained that the theory of Ptitsyn and Eizner overestimates the influence of the excluded volume on 0 and JeR. As will be shown in the experimental section of this chapter, this impression is supported by flow birefringence measurements on solutions in 0- solvents and in good solvents. 

This is the well-known front factor of the reduced steady-state shear compliance as quoted by Ferry (113). In this form polydispersity factor p can be compared with the diverse molecular weight averages, as obtained from equilibrium ultracentrifugation, light-scattering and osmometry. 

Fig. 4.3. Reduced steady-state shear compliance J,B vs. volume fraction p or concentration c, according to a combination of eq. (4.10) with eqs. (4.6) and (4.9), respectively (smoothed curves). Figures below full lines... MjMp-values, figures above dashed lines. . . M/if-values (theoretical curves)...

Fig. 4.4. Concentration dependence of reduced steady-state shear compliance J,B for a series of anionic polystyrenes, as mostly provided by Pressure Chem. Corp., Pittsburgh, Pa. Except for the solutions of the three lowest concentrations of S 111 (Dow Chem. Corp.), which were prepared with methyl (4-bromo-phenyl) carbinol and used at various temperatures, all solutions were prepared with mono-bromo-benzene and used at 25° C (32). Measurement temperatures for the melts varied from 196 to 240° C (59). For the molecular weights of the polymers see Table 4.1...

Fig. E3.5 Steady-state shear viscosity rj and first normal stress coefficient i, obtained from dynamic measurements versus shear rate for a low-density polyethylene melt, melt I. [H. M. Laun, Rheol. Ada, 17, 1 (1978).]...

J m(t — t )y 1](t, t) dt. Consider a fluid with a single relaxation time, 20, and modulus, Go, and with h(y) = e y. Calculate the steady-state shear viscosity function... 

Fig. 11.18 (a) The steady state shear rate and (b) shear stress-dependent viscosity of carbon black... 

In steady-state shear, when the only component of the velocity gradient tensor differs from zero is z/12, equation (9.19) is followed by... 

Anisotropy in a Simple Steady-State Shear Flow 209... 

Let us consider the anisotropy of polymer system undergoing simple steady-state shear. This situation can be realised experimentally in a simple way (Tsvetkov et al. 1964). The quantity measured in experiment are the birefringence An and the extinction angle x which are defined by formulae (10.19) and (10.20), correspondingly, through components of the relative permittivity tensor. 

Steady state shear-dependent behaviour is discussed in Volume 1, Section 3.7.1. 

Before discussing theoretical approaches let us review some experimental results on the influence of flow on the phase behavior of polymer solutions and blends. Pioneering work on shear-induced phase changes in polymer solutions was carried out by Silberberg and Kuhn [108] on a polymer mixture of polystyrene (PS) and ethyl cellulose dissolved in benzene a system which displays UCST behavior. They observed shear-dependent depressions of the critical point of as much as 13 K under steady-state shear at rates up to 270 s Similar results on shear-induced homogenization were reported on a 50/50 blend solution of PS and poly(butadiene) (PB) with dioctyl phthalate (DOP) as a solvent under steady-state Couette flow [109, 110], A semi-dilute solution of the mixture containing 3 wt% of total polymer was prepared. The quiescent... 

As can be seen, determination of the plateau modulus as a function of blend composition yields AGR, which is closely related to AN,. Thus, the free-energy change under steady-state shear is... 

Figure 10. Shear viscosity as a function steady-state shear rate for poly(sty-rene-b-butadiene-b-styrene) at 150°C (after Ref. 47)...

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