If the flow is nonisothermal, the temperature dependence of the viscosity must be taken into account along with the shear-rate dependence. The viscosity law can be factorized as follows:
(10–16) |
where is the Arrhenius law (or one of the other available laws) and is the viscosity law at some reference temperature (as computed by one of the shear-rate-dependent laws described above).
The Arrhenius law is given as
(10–17) |
where is the ratio of the activation energy to the thermodynamic constant, and is a reference temperature for which . The temperature shift is set to 0 by default, and corresponds to the lowest temperature that is thermodynamically acceptable. Therefore and are absolute temperatures. They can also be defined relative to a non-absolute temperature scale, in which case corresponds to the absolute zero temperature in the current temperature scale.
Two versions of the Arrhenius law are available: one based on shear rate and one based on shear stress, as described below. Both methods provide similar, although not identical, results for most polymers.
Not all polymers strictly follow Equation 10–16. While this requires detailed measurements of the viscosity as a function of the temperature, most polymers follow a viscosity-temperature dependence that is slightly different from Equation 10–16.
Consider a typical viscosity curve such as the one shown in Figure 10.1: Typical Viscosity Curve. Curve A represents the viscosity at a reference temperature, say . This viscosity curve, which has been modeled using a Bird-Carreau law, presents a plateau zone for low shear rates.
If the polymer were to follow Equation 10–16, curve A would translate at a different temperature, say 200°, into curve B. This is the behavior modeled with either the Arrhenius or the approximate Arrhenius laws. However, sometimes the viscosity at 200° follows curve C; that is, not only is the absolute value of the viscosity decreased, but also the point of departure from the constant-viscosity regime moves toward the right in the diagram. This is precisely what the “shear stress" version of the temperature-dependence laws models, using the following equation:
(10–18) |
The law of Equation 10–18 corresponds to a vertical shift of the viscosity curve in a viscosity-shear-stress diagram, so it is referred to as the Arrhenius shear-stress law.
Note that a power-law model does not reflect the difference between Arrhenius shear rate and Arrhenius shear stress because of the absence of a transition zone.
The approximate Arrhenius law is written as follows:
(10–19) |
The behavior described by Equation 10–19 is similar to that described by Equation 10–17 in the neighborhood of . Equation 10–19 is valid as long as the temperature difference is not too large. As for the Arrhenius law, two versions are available: one based on shear rate and one based on shear stress, as described above.
Another definition for comes from the Fulcher law [13]:
(10–20) |
where , , and are the Fulcher constants. The Fulcher law is used mainly for glass.
The Williams-Landel-Ferry (WLF) equation is a temperature-dependent viscosity law that fits experimental data better than the Arrhenius law for a wide range of temperatures, especially close to the glass transition temperature:
(10–21) |
where and are the WLF constants, and and are reference temperatures.
The WLF law described above is based on shear rate. As for the Arrhenius law, there is also a version of the WLF law based on shear stress. In this version, the viscosity is computed from Equation 10–18, with computed from the WLF law, Equation 10–21. As for the Arrhenius shear-stress law, an increase in temperature will result in a shifting of the viscosity curve downward and to the right.
For the mixed-dependence law (which can be used only in conjunction with the log-log law for shear-rate dependence), the function is written as
(10–22) |
where is computed from the log-log law (Equation 10–14) and
(10–23) |
In this equation, , , and are the coefficients of the polynomial expression, and is the lowest temperature that is thermodynamically acceptable, with respect to the current temperature scale. Typically, if the units for temperature are Kelvin, will be 0; if the units for temperature are Celsius, will be –273.15.