The Cross Equation

The Cross equation assumes that a shear-thinning fluid has high and low shear-limiting viscosity (16) (eq. 4), where a and n are constants. 

Lai et al. [100] proposed the use of the Dow Rheology Index (DRI) as an indicator for comparing branching level in industrial polymers. For a linear polymer molecule, like unbranched polyethylene, the viscosity of the polymer as a function of the applied shear rate is given by the Cross equation [84,100],... 

The estimates do not even come close to satisfying the cross equation restrictions. The parameters in the cost function are extremely large, owing primarily to rather severe multicollinearity among the price tenns. 

Carrean-Yasnda Model (full curve) Cross Model (full curve) For portions of the Cross equation that predict portions of the complete flow curve h-ih, 1 ho-1 . [1 + (k1y)2P 2 110-11 =(KiT Tl-11 1 oo... 

This equation is similar to the Cross equation [3] for polymers to be discussed later. The Cross equation is also one of the key constitutive equations for ceramic suspensions at hig solids loading. 

At the percolation limit, the rheological behavior of the suspension changes from Newtonian to either the Cross equation with a low shear limit viscosity or the Bingham plastic equation with an apparent yield... 

FIGURE 12S Schematic of the low shear viscosity of TiOj as a function of pH. Near the zero point of charge (ZPC) the rheology is non-Newtonian for dilute suspensions, conforming to the Cross equation, which suggests that aggregation is responsible for this increase in viscosity. Away from the ZPC, the rheology is Newtonian for dilute suspensions. 

The Peclet number gauges the magnitude of the departure from equilibrium configuration of the particles. (Note that the rotational Peclet number, Pe, for a sphere has nearly the same definition only 6 is replaced by 8.) As such the Peclet number can be used in the Cross equation to determine the value of > .[= 8], giving... 

If we are to use a direct analogy of suspension rheology to the Cross equation derived for polymer solutions, we should consider that the... 

For the Cross equation, which is valid for monodisperse suspensions of hard spheres, the shear stress, t,, is given by... 

The apparent viscosity (tja) of the solution can be correlated with shear rate (y) using the Cross (Equation 2.14) or the Carreau (Equation 2.15) equations, respectively. 

Jones and co-workers (72, 88) found that the suspension viscosity variation with shear rate can be fitted fairly well by the Cross equation, equation 12, with m = 0.5 — 0.84. Both the low and high shear limit relative viscosities, Mro, Mroo> can be expressed by the Quemada s equation with 0max = 0.63 and 0.71, respectively. 

Doublier and Launay (1981) have calculated the four parameters of the Cross equation for guar gum by a computerized nonlinear regression... 

The drop in viscosity with increasing shear rate was described well [110] for all systems investigated by means of the Cross equation [Eq. (16)] that was derived for the flow of pseudoplastic systems [111],... 

Newtonian flow. This is called the infinite shear or upper Newtonian viscosity, denoted as T]. This flow behaviour can be accurately modelled using the Cross equation [11] ... 

This equation goes to a zero-shear viscosity as 0 and to a power law for k 3> i "" in the latter case, corresponds to the consistency index, K. The parameters appear to correlate with changes in molecular structure for polyolefins. The Cross equation, in which a = l - n, works well for many polymers, including polyesters and polyacetals. 

In many situations, t]o rioc Ky l,and 77,is small. Then the Cross equation (with a simple change of the variables K and m) reduces to the well-known power-law (or Ostwald-de Waele) model, which is given by... 

The following empirical expression, the Cross equation (Cross 1965, 1969), has also been found to be very useful to describe, with reasonable accuracy, the shear-rate dependence of viscosity of molten polymers ... 

A) 200 and ( ) 220. The solid curve represents the Cross equation, for which the following numerical values were used (1) i)o = ... 

Table 6.2 Parameters appearing in the Cross equation for some molten polymers...

In this chapter, we have presented the rheological behavior of homopolymers, placing emphasis on the relationships between the molecular parameters and rheological behavior. We have presented a temperature-independent correlation for steady-state shear viscosity, namely, plots of log ri T, Y) r](jiT) versus log or log j.y, where Tq is a temperature-dependent empirical constant appearing in the Cross equation and a-Y is a shift factor that can be determined from the Arrhenius relation for crystalline polymers in the molten state or from the WLF relation for glassy polymers at temperatures between and + 100 °C. 

It is important to note that this model contains no characteristic time. It thus implies that the power-law parameters are independent of shear rate. Of course such a model cannot describe the low-shear-rate portion of the curve, where the viscosity approaches a constant value. Several empirical equations have been proposed to allow for the transition to Newtonian behavior over a range of shear rates. It was noted in the discussion of the Weissenberg number earlier in this chapter that the variation of 77 with y implies the existence of at least one material property with units of time. The reciprocal of the shear rate at which the extrapolation of the power-law line reaches the value of tiq is such a characteristic time. Models that can describe the approach to tIq thus involve a characteristic time. Examples include the Cross equation [64] and the Carreau equation [65], shown below as Eqs. 10.55 and 10.56 respectively. 

Plumley-Karjala etal. [69] evaluated the ability of the models presented above to describe data for a large number of linear and branched metallocene polyethylenes. They found that the Cross equation gave a good fit to the data and that adding parameters did not lead to a... 

Viscosity models are sometimes used to estimate the zero-shear viscosity when no experimental data are available at shear rates sufficiently low that the viscosity is constant. However, this is an unreliable procedure, as there is no fundamental basis for any of these equations, and the resulting value of t/q should be deemed at best a rough estimate. For example, Kataoka and Ueda [70] found that the Cross equation yielded extrapolated values of t]q that were about 50% less than measured values. 

Several methods of estimating LCB levels using linear viscoelastic data are described in Section 5.12. A technique based on the shape of the viscosity curve has been proposed for single-site polyethylenes with low levels of LCB. Lai etal [80] found that for strictly linear polyethylenes prepared using single-site catalysts, the Cross equation (Eq. 10.55) gives a reasonably good fit to viscosity data. They further showed that the characteristic time A of the cross equation is proportional to the zero shear viscosity for these materials ... 

Zero-shear viscosity, rjo, was obtained by fitting experimental data with the Cross equation ... 

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