Steady shear flow measurement

Quantitative evidence regarding chain entanglements comes from three principal sources, each solidly based in continuum mechanics linear viscoelastic properties, the non-linear properties associated with steady shearing flows, and the equilibrium moduli of crosslinked networks. Data on the effects of molecular structure are most extensive in the case of linear viscoelasticity. The phenomena attributed to chain entanglement are very prominent here, and the linear viscoelastic properties lend themselves most readily to molecular modeling since the configuration of the system is displaced for equilibrium only slightly by the measurement. 

Studies have been made of the stresses produced in several non-steady flow histories. These include the buildup to steady state of a and pu — p22 at the onset of steady shearing flow (355-35 ) relaxation of stresses from their steady state values when the flow is suddenly stopped (356-360) stress relaxation after suddenly imposed large deformations (361) recoil behavior when the shear stress is suddenly removed after a steady state in the non-linear region has been reached (362) and parallel or transverse oscillations superimposed on steady shearing flow (363-367). Experimental problems caused by the inertia and compliance of the experimental apparatus are much more severe than in steady state measurements (368,369). Quantitative interpretations must therefore still be somewhat tentative. Nevertheless, the pattern of behavior emerging is suggestive with respect to possible molecular flow mechanisms. 

Fig. 2.7. Drop off of doubled extinction angle 2"/ during stress-relaxation after cessation of steady shear flow according to Wales (59). Measurements on the melt of a high-density polyethylene (Marlex 6002) at a measurement temperature of 147° C. Shear rate of the steady shear flow q = 0.06 sec-1...

Traditionally, shear viscosity measurements are used to rheologically characterize fluids. Eigure 6.1 shows the principle for shear viscosity measurement this figure shows a steady shear flow field between two parallel plates, one of which is moving with a velocity v. The measured quantities are the velocity of the top plate, the separation gap d, and the force in the direction of shear experienced by the stationary plate. Equation 6.1 is used to calculate the shear viscosity of the fluid, and the shear rate is calculated y = v/d (velocity/distance between the two parallel plates). Shear rate is also called velocity gradient. We can see that this shear rate or velocity gradient is constant. In this case, the displacement (strain) is... 

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 ... 

Steady-state shear rheology typically involves characterizing the polymer s response to steady shearing flows in terms of the steady shear viscosity (tj), which is defined by the ratio of shear stress (a) to shearing rate y ). The steady shear viscosity is thus a measure of resistance to steady shearing deformation. Other characteristics such as normal stresses (Ai and N2) and yield stresses (ffy) are discussed in further detail in Chapter 3. 

We next turn our attention to the relaxation of shear and normal stresses after a steady shearing flow is suddenly stopped. Before time t=0 the fluid is in a state of steady shear flow with a velocity profile vx = K0y, vy = 0, v2 = 0. After time t = 0 the fluid is completely motionless, but the shear and normal stresses decrease gradually to zero. This flow was analyzed for rigid dumbbells by Giesekus (12) and later by Bird, Warner, and Ramakka (10). Schremp, Ferry, and Evans (69), Benbow and Howells (2), and Huppler et al. (36) have measured the time-decay of the stresses in an experiment which closely approximates the above-described idealized problem. 

Parallel-plate rheometers are often more useful for studying rheology of filled polymers or composite materials, particularly when the size of the fillers is comparable fo fhe disfance between the truncated cone and the surface of the plate. Again, the torque M and the normal force N tending to separate the two plates are measured. In steady-shear flow, the shear rate and the shear stress at the edge of the disks located atr = R are given by... 

Here t, 4, and 4 2 are three important material functions of a nonnewtonian fluid in steady shear flow. Experimentally, the apparent viscosity is the best known material function. There are numerous viscometers that can be used to measure the viscosity for almost all nonnewtonian fluids. Manipulating the measuring conditions allows the viscosity to be measured over the entire shear rate range. Instruments to measure the first normal stress coefficients are commercially available and provide accurate results for polymer melts and concentrated polymer solutions. The available experimental results on polymer melts show that , is positive and that it approaches zero as y approaches zero. Studies related to the second normal stress coefficient 4 reveal that it is much smaller than 4V and, furthermore, 4 2 is negative. For 2.5 percent polyacrylamide in a 50/50 mixture of water and glycerin, -4 2/4 i is reported to be in the range of 0.0001 to 0.1 [7]. 

Direct measurements of and indicate a parallel dependence of both these functions plotted vs. ( ), even when these have a sigmoidal form. Considering the steady shear flow of a two-phase system, it is generally accepted that the rate of deformation may be discontinuous at the interface, and it is more appropriate to consider variation of the rheological functions at constant stress than at constant rate, i.e., = Nj(Oj2). 

For Newtonian lipid-based food systems, it is sufficient to measure the ratio of shearing stress to the rate of shear, from which the viscosity can be calculated. Such a simple shear flow forms the basis for many rheological measurement techniques. The rheological properties resulting from steady shear flow for variety of food systems have been studied by many laboratories (Charm, 1960 Holdsworth, 1971 Middleman, 1975 Elson, 1977 Harris, 1977 Birkett, 1983 Princen, 1983 Shoemaker and Figoni, 1984 Hermansson, 1994 Kokini et al., 1994, 1995 Morrison, 1994 Pinthus and Saguy, 1994 and Meissner, 1997). 

Viscoelastic data of relatively high concentrated solutions were measured in oscillating and steady-shear flow at room temperature (ca. 25C) by using Fluid Spectrometers RFS-8500 and RFSn, and a Mechanical Spectrometer RMS-800 of Rheometrics, Inc., with the cone-and-plate and the co-axial cylinder. At relatively low concentrations most of 7 data was measured with capillary viscometers of Maron-Krieger-Sisko type at 2511, and J data at two low concentrations were obtained from flow birefringence measurements[12] for comparison. To examine the degradation of the samples after the measurements of viscoelastic properties, we measured [ 7 ] to confirm that there was no degradation. 

The theory of flow of materials with yield values dates to the work of Schwedoff [S8] of the University of Odessa in 1890. Schwedoff designed and built a coaxial cylinder instrument to measure the shear viscosity of gelatin suspensions. For steady shear flows he correlated his data with... 

Owing to experimental difficulties, steady-state shear measurements of Ni and 0 2 are relatively rare. Their rate of shear gradients, Ni/y,r] = 012/y usually show a similar dependence ]322]. The value of the complex viscosity ist] >t]. In the steady shear flow of a two-phase system, the stress is continuous across the interphase, but the rate of deformation is not. Thus, for polymer blends, plots of the rheological functions versus stress are more appropriate than those versus rate, that is, a Ni = Ni oi2) plot is similar to G = G (G"). 

The non-chiral main chain LC polyester also studied for their shear alignment. The chain-folded lamellae formation was observed. X-ray measurement carried out on the main-chain LC polyesters to observe how smectic liquid crystal gets oriented under steady shear flow. When the temperature is increased the mesogens within... 

---