The Cross model

we will consider one equation that describes the whole curve this is called the Qoss model, named after Malcolm Cross, an ICl rheologist who worked on dyestuff and pigment dispersions. He found that the viscosity of many suspensions could be described by the equation of the form 

If we make various simplifying assumptions, it is not difficult to show that the Qoss equation can be reduced to Sisko, power-law and Newtonian behaviour, see below. There is another Cross-like model which uses the stress rather than the shear rate as the independent variable, it has been called the Ellis or sometimes the Meter model, and for some specific values of the exponent, it has been gives other names for an exponent of unity it has been called the Williamson or Dougherty and Krieger model, while for an exponent of two it has been called the Phihppoff model, etc. 

The Carreau model is very similar to the Cross model, but with the whole of the bottom line within brackets, i.e. (1 + (K y )2)m/2 the two are the same at very low and very high shear rates, and only differ slightly at Ky 1. 

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Under conditions of steady fully developed flow, molten polymers are shear thinning over many orders of magnitude of the shear rate. Like many other materials, they exhibit Newtonian behaviour at very low shear rates however, they also have Newtonian behaviour at very high shear rates as shown in Figure 1.20. The term pseudoplastic is used to describe this type of behaviour. Unfortunately, the same term is frequently used for shear thinning behaviour, that is the falling viscosity part of the full curve for a pseudoplastic material. The whole flow curve can be represented by the Cross model [Cross (1965)] ... 

The disadvantage of the power law model is that it cannot predict the viscosity in the zero-shear viscosity plateau. When the zero-shear viscosity plateau is included, a nonlinear model must be specified with additional fitting parameters. A convenient model that includes the zero-shear viscosity and utilizes an additional parameter is the Cross model [30] ... 

If some or all of this curve is present, the models used to fit the data are more complex and are of two types. The first of these is the Carreau-Yasuda model, in which the viscosity at a given point (T ) as well as the zero-shear and infinite-shear viscosities are represented. A Power Law index (mi) is also present, but is not the same value as n in the linear Power Law model. A second type of model is the Cross model, which has essentially the same parameters, but can be broken down into submodels to fit partial data. If the zero-shear region and the power law region are present, then the Williamson model can be used. If the infinite shear plateau and the power law region are present, then the Sisko model can be used. Sometimes the central power law region is all that is available, and so the Power Law model is applied (Figure H. 1.1.5). 

The models used are typically either the Cross model or the Carreau-Yasuda model (UNIT Hl.l), if a complete curve is generated. A complete curve has both plateaus present (zero and infinite shear see Figure HI.1.4). 

The viscosity and the shear dependency of the viscosity both increase with growing effective volume fraction of particles. For very low shear rates there is usually a low shear plateau jo and for high shear rates a high shear plateau joo exists. These features are not described by the power model, but a more elaborate model such as the cross model is needed ... 

Isothermal flow curves are often summarized by simple empirical models to understand fabrication performance. The Cross model [39,40], given by Equation (1), is well-suited for fitting SAN copolymer data as seen in Figure 13.4. 

Both the Carreau and the Cross models can be modified to include a term due to yield stress. For example, the Carreau model with a yield term given in Equation (2.16) was employed in the study of the rheological behavior of glass-filled polymers (Poslinski et al., 1988) ... 

Apparent viscosity-shear rate data of food polymer dispersions have been reviewed by Launay et al. (1986), Lopes da Silva and Rao (1992) and others. The general log versus log y curve, discussed in Chapter 2, has been used to characterize food polymer dispersions. For example, Lopes da Silva et al. (1992) found that both the modified Carreau and the Cross models, wherein the infinite shear viscosity was considered to be negligible, described the apparent viscosity-shear rate data of locustbean (LB)... 

Carreau devised another general equation to model the shear thinning behavior, which has become a popular alternative to the Cross model ... 

Mathematical models, which can predict the shape of a flow curve of a shear thinning material including lower and upper Newtonian regions, require at least four parameters. The Cross model is one such model ... 

Similarly, the two models could be reduced by neglecting the lower Newtonian region. This has been done by Sisko for the Cross model in order to account for the rheological properties of grease in bearings ... 

Considering that in many cases the viscosity measured is significantly smaller than that of the lower Newtonian region (i] upper Newtonian region (i] > i]oo), the Cross model can be further simplified ... 

The power-law model is the most extensively used shear-mte model for thermosets and has been used for unfilled (Ryan and Kamal, 1976, Kascaval et al., 1993, Riccardi and Vazquez, 1989) and filled (Ryan and Kamal, 1976, Knauder et al., 1991) epoxy-resin systems. Sundstrom and Burkett (1981) showed that there was a good fit of the viscosity of diallyl phthalate to the Cross model. The viscosity of polyesters has been modelled by Yang and Suspene (1991) using a Newtonian model. The WLF model has been used by Pahl and Hesekamp (1993) for a moderately filled epoxy-resin system. Rydes (1993) showed that the viscosity of DMC polyesters followed a power-law relationship at high shear rate. 

Figure 8.5 CoiH)arison of shear-thinning behaviors of Thermx LNOOl at 345°C and PET at 285°C. The high shear rate data (>100 sec ) for LCP were measured by a capillary rheometer, and those for PET were extrapolated from the dynamic data at low shear rates using the Cross model (from Seo [18], reprinted with permission from Society of Plastics Engineers).

In comparing the Carreau and Cross models on a variety of commercial-grade polymer melts, Hieber and Chiang (1992) found that the Cross model provides a better overall fit for the shear-rate dependence. The Cross model has been widely used in injection molding modeling. 

For most structured liquids at high shear rates, rjo rjoo and Ky l,and then it is easy to show that the Cross model simplifies to the Sisko equation... 

A simple but versatile model that can cope with a yield stress, yet retain the proper extreme of a finite zero and infinite shear-rate viscosity is the Cross model with the exponent set to unity, so... 

Figure 12 The Cross model with m = 1 and typical values ofthe parameters, showing the apparent yield stress region and upper and lower Newtonian regions.

Figure 13 The Cross model with m=, plotted linearly to show Bingham-type behaviour.

Fig. 13.23. Shear thinning behavior of an ABS polymer melt at three different temperatures. Broken lines are power law fits and solid lines represent the Cross model, (from C. W.

Newtonian regimes are nevertheless widely observed in polymer melts in the high and low shear rate limits, where the viscosities are designated by and respectively. This is reflected by the empirical expressions widely used in engineering practice to describe the response to steady shear flow, an example being the Cross model [Eq. (55)], which reduces to the well known power law of Eq. (56) when tj tj , with n = l/(m+l) between 10 and 20 for most polymer melts. 

Another possible candidate for X may be the reciprocal of the shear rate at which the shear viscosity attains 80% of the zero-shear value in a shear viscosity vs. shear rate plot (the so-called flow curve ). This has been shown [24] to be the point of maximum curvature of the flow curve for the Cross model (see further... 

There are several phenomenological models available to correlate the rate dependence of suspension viscosity. The Cross model deseribed below is a general model that requires four parameters to describe the dependence of viscosity on the shear rate of a suspension ... 

The yield point descrihed hy the above models is a calculated value. The true yield point of the material is difficult to measure and depends on the sample shear history and measurement technique. It is sometimes more useful to describe low shear paste behavior by the zero shear viscosity using the Cross model. StUl, yield point is of practical importance in describing certain engineering apphcations. 

FIGURE 8.92 Zero-shear viscosity versus vol% solids as calculated from the Cross model. 

Typically % so when Uho) = y is very small, goes to %. At intermediate y the Cross model has a power law region... 

The Cross model has been used to fit the data sets shown in Figures 2.4.1 and 2.1.2. The parameters used are shown in Table 2.4.1. 

This is equivalent to the Cross model with a fifth fitting parameter, Carreau model (Bird et al., 1987, p. 171). 

In the melt flow curve (viscosity versus shear rate), a Newtonian plateau at very low shear rate is usually considered as the ZSV. For polymers of very low molecular weight, this ZSV can be obtained directly from the shear rheology experiment (Dealy and Wissburn 1990). However, this is really difficult to obtain for high molecular weight polymers as well for LCPs. From the experimental data presented in Fig. 4.5, it was impossible to determine the ZSV, since there was no distinct plateau detected at a lower shear rate up to 0.01 s . Therefore, a modified Cross model (4.6) was used to determine the zero shear viscosity of the LCPs. Data from low and high shear rates were combined to fit into the Cross model predictions (Fig. 4.5). The model prediction showed good agreement with experimental data. 

They noted that the Cross model could also be fitted to viscosity data for similar polymers into which a small level of long chain branching has been introduced but that the presence of the branches caused a departure from Eq. 10.55. They proposed the use of this departure as an indicator of the level of long-chain branching in such materials. To this end, they defined the Dow Rheology Index, DRI, as follows ... 

Equation 16C.2 is called the Power Law and Equarion 16C.3 is called the Cross Model. Plots of these two models are shown in Figure 16C.2. In general, the Cross Model does a better job of describing the actual viscosity versus shear rate data observed on commercial polymers. Now we have a mathematical model that describes the actual viscosity versus shear rate profile of mbber materials. 

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