Making Use Of Models: The Cross Model
The rheological model based on the Cross Equation is one of the most popular in use today.
It can be found on virtually every research rheometer software packages and it can be used to extract some meaningful numbers from the “full” viscosity vs shear rate profile such as you see below.
Typical viscosity profile for a dispersion, polymer solution or melt
So what is the Cross Equation and how can we use it?
Here is the Cross Equation, giving viscosity as a function of shear rate:
η0 is the Zero Shear Viscosity, the magnitude of the viscosity at the lower Newtonian plateau. It is a critical material property and can prove valuable in making assessments of suspension and emulsion stability, estimates of comparative polymer molecular weight and tracking changes due to process or formulation variables etc.
η∞ is the Infinite Shear Viscosity. This tells us how our product is likely to behave in very high shear processing situations such as blade, knife and roller coating.
The parameter "m" is known as the (Cross) Rate Constant. It is dimensionless and is a measure of the degree of dependence of viscosity on shear rate in the shear-thinning region. A value of zero for m indicates Newtonian behaviour with m tending to unity for increasingly shear thinning behaviour.
"C" is known as the Cross Time Constant (or sometimes the Consistency) and has dimensions of time. The reciprocal, 1/C, gives us a critical shear rate that proves a useful indicator of the onset shear rate for shear thinning.
Applications of The Cross Model
m and 1/C can be related to texture, application properties, pumping, mixing and pouring characteristics and many other everyday flow processes which often occur in the shear thinning region of the fluid’s flow behaviour. Below you can see some examples of how flow curves with varying C and m parameters would look.
So, the Cross model provides us with a simple way of quantifying the “full” viscosity/shear rate profile for a shear thinning fluid.
One word of warning: It is easy to get confused between the Cross model parameters C and m and the Power Law Model parameters K and n. They are not the same although they all describe the shear-thinning part of the profile. To make matters worse various rheometer software packages use differing conventions and nomenclature to denote these parameters!
If you want some advice on choosing from these or other rheological models please don’t hesitate to get in touch.