Rheology steady shear flow

The viscosity function rj (referred to as the steady shear viscosity), the primary and secondary normal stress coefficients ij/, and respectively, are the three viscometric functions which completely determine the state of stress in any rheologically steady shear flow. They are defined as follows ... 

Woods,M.E., Krieger,I.M. Rheological studies on dispersions of uniform colloidal spheres. I. Aqueous dispersions in steady shear flow. J. Colloid Sci. 34,91-99 (1970). 

The conclusion is that Lodge s rheological constitutive equation results in relationships between steady shear and oscillatory experiments. The limits y0 0 (i.e. small deformation amplitudes in oscillatory flow) and q >0 (i.e. small shear rates) do not come from Lodge s equation but they are in agreement with practice. These interrelations between sinusoidal shear deformations and steady shear flow are called the relationships of Coleman and Markovitz. 

Another peculiar property of LCPs is shown in Fig. 15.47, where the transient behaviour of the shear stress after start up of steady shear flow is shown for Vectra A900 at 290 °C at two shear rates. We will come back to this behaviour in Chap. 16 for lyotropic systems where this behaviour is quite common and in contradistinction to the transient behaviour of conventional polymers, as presented in Fig. 15.9. This damped oscillatory behaviour is also found for simple rheological models as the Jeffreys model (Te Nijenhuis 2005) and according to Burghardt and Fuller, it is explicable by the classic Leslie-Ericksen theory for the flow of liquid crystals, which tumble, rather than align, in shear flow. Moreover, it is extra complicated due to the interaction between the tumbling of the molecules and the evolving defect density (polynomial structure) of the LCP, which become finer, at start up, or coarser, after cessation of flow. 

Kulicke, W.-M. and Porter, R. S. 1980. Relation between steady shear flow and dynamic rheology. Rheol. Acta 19 601-605. 

Haugen P, Tung MA, Runikis JO. Steady shear flow properties, rheological reproducibility and stability of aqueous hydroxyethylcellulose dispersions. Can ] Pharm Sci 1978 13 4-7. 

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. 

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... 

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). 

The rheological behavior of these materials is still far from being fully understood but relationships between their rheology and the degree of exfoliation of the nanoparticles have been reported [73]. An increase in the steady shear flow viscosity with the clay content has been reported for most systems [62, 74], while in some cases, viscosity decreases with low clay loading [46, 75]. Another important characteristic of exfoliated nanocomposites is the loss of the complex viscosity Newtonian plateau in oscillatory shear flow [76-80]. Transient experiments have also been used to study the rheological response of polymer nanocomposites. The degree of exfoliation is associated with the amplitude of stress overshoots in start-up experiment [81]. Two main modes of relaxation have been observed in the stress relaxation (step shear) test, namely, a fast mode associated with the polymer matrix and a slow mode associated with the polymer-clay network [60]. The presence of a clay-polymer network has also been evidenced by Cole-Cole plots [82]. 

Rheology in steady shear flow shear viscosity... 

Rheology in Steady Shear Flow Shear Viscosity... 

RHEOLOGICAL PROPERTIES OF LIQUID CRYSTALLINE POLYMERS Steady Shear Flow... 

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"). 

Vinckier, I., Moldenaers, P., and Mewis, J. (1996) Relationship between rheology and morphology of model blends in steady shear flow. J. Rhed., 40 (4), 613-631. 

We now introduce the major rheological material functions, with illustrations provided by typical experimental results. Figure 7.15 depicts data obtained for low density polyethylene under steady shear flow conditions, employing a cone-and-plate rheometer. Curves display both the shear rate dependence of the viscosity, with similar results as in Fig. 7.1, and the shear rate dependence of the first normal stress difference. The stresses arising for simple shear flows may be generally expressed by the following set of equations... 

Except for a brief review of the basic concepts of viscosity, steady shear flow and dynamic flow, the details of rheological measurements and analysis wiU be left to the noted references. 

This little exercise is significant. It tells us that one of the four key rheological phenomena laid out in the introduction to these chapters on constitutive relations-normal stresses in steady shear flows-cannot be explained by any function of the rate of the deformation tensor. On the other hand, almost any function of B, the Finger tensor, does generate proper shear normal stresses. We will wait until Chapter 4 to pursue this reasoning further. 

Besides the sinusoidal oscillations, transient tests such as stress relaxation, start-up of steady shear flow, and cessation of steady shear flow are also important in the rheological characterization of polymeric liquids. The instrument limitations and features, and their effect on the data obtained in some types of transient tests, are examined in this section. Also, we shall illustrate the use of linear viscoelastic transformation to obtain, for example, stress relaxation data from tests such as start-up of steady shear flow and sinusoidal oscillations. 

---