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Figure: Dialog box Rheological data

Fluids can be divided into two groups regarding to their flow characteristics (see figure):

• Newtonian fluids
• Non-Newtonian fluids

The following law is valid for Newtonian fluids: $$\tau = \eta \cdot \dot{\gamma}$$

with shear stress τ, viscosity η and shear rate γ̇. This law states that the shear stress and shear rate are proportional to each other, with the viscosity being the proportionality factor. In the case of polymeric fluids, or melts, this flow behavior only occurs at very low shear rates and, in some cases, at very high ones. Deviations are manifested in so-called pseudo plasticity, dilatancy or the presence of a yield point.

In the shear rate ranges that prevail in practice, the flow behavior of polymer melts is characterized as pseudo plastic. This describes a flow behavior which deviates from that of Newtonian fluids, where the viscosity is no longer constant but highly dependent on the shear rate.

Figure: Material properties

The figure shows the basic viscosity profile as a function of the shear rate. Where the shear rate range is not too large, it is possible to describe this behavior by the empirical power law according to OSTWALD and DE WAELE: $$\eta = K \cdot \dot{\gamma}^{n-1}$$

here n is the exponent of the power law and K the power law coefficient. The simple structure of this equation means that almost all flow problems that can be solved for Newtonian fluids can be treated analytically. A straight line is also obtained for the power law when plotted on a double-logarithmic scale. As can be seen from the figure, a corresponding exponent n must be calculated for different curve segments.

A formulation derived from the purely thermally activated process (Arrhenius law) has proved successfully for describing the temperature dependence of the viscosity with the aid of the power law coefficient.

$$\eta = K \cdot \dot{\gamma}^{n-1}$$

Constant $K_{0,T}$ corresponds to the viscosity at a shear rate of and the reference temperature T0 = 0°C describes the temperature dependence of the viscosity.

Figure: Basic viscosity function profile for a polymer

Figure: Approximation of the viscosity function in ranges with straight-line sections

The CARREAU law supplies a better description over broad areas of the viscosity function. This is particularly true for materials which display a pronounced transition from the Newtonian to the structural-viscosity range:

where A is the zero viscosity, B the reciprocal transitional shear rate and C the gradient (C = n-1).

Figure: Determination of the Carreau parameters

• Carreau Yasuda-Law:

$$\eta = \eta_{\infty} \cdot a_T + \frac{A \cdot a_T - \eta_{\infty} \cdot a_T}{(1 + (B \cdot a_T \cdot \dot{\gamma})^a)^{\frac{1-c}{a}}}$$

• Polynomial Law:

$$\ln(\eta) = a_0 + a_1 \cdot \ln(\dot{\gamma}) + a_{11} \cdot \ln^2(\dot{\gamma}) + a_2 \cdot T + a_{22} \cdot T^2 + a_{12} \cdot \ln(\dot{\gamma}) \cdot T$$

• Power Law:

$$\eta = K \cdot a_T \cdot \dot{\gamma}^{n-1}$$

Allowance is made for the temperature dependence by means of the temperature shift factor aT, which can be established from the WLF equation:

• WLF ($T_B, T_S$):

$$\lg(a_T) = \frac{C_1 \cdot (T_B - T_S)}{C_2 + (T_B - T_S)} - \frac{C_1 \cdot (T - T_S)}{C_2 + (T - T_S)}$$

TB, TS are specified, C₁ = 8,86, C₂ = 101,6

OR:

• WLF ($C_1, C_2$):

$$\ln(a_T) = -\frac{C_1 \cdot (T - T_B)}{C_2 + (T - T_B)}$$

C₁, C₂, $T_B$ are specified

with:
$T_B$ = reference tmperature
$T_S$ = standard temperature
T = current Temperature

• Arrhenius (E, $T_B$):

$$K = K_{0T} \cdot \exp\left[\frac{\Delta E}{R} \cdot \left(\frac{1}{T} - \frac{1}{T_0}\right)\right]$$

with:
E = Activation energy
R = General gas constant
K₀T = consistance at T₀
T₀ = reference Temperature

The Carreau law can be used for the correct description of the specific material behavior of polymers over a wide shear rate and temperature range. Given that this law is restricted in its analytically applications, from the shear rates and temperatures which prevail in the process the coefficients of the power law will be calculated in SIGMA. The input mask Rheological data gives three different options for the description of the rheological behavior of the polymer melt. In the Rheological data input mask (see figure), there are three different formulation forms to choose from for describing the rheological behavior of polymer melts. The difference of the laws is that they describe the temperature dependency of the viscosity in different ways. This distinction was introduced in order to ensure a problem-free input independent to the origin of the data (CAMPUS, BAYMAT, VISCOSITY). In case that Carreau-WLF data are taken from the BASF VISCOSITY database it is necessary to select the switch position Carreau-WLF (C1, C2). In the event of that the data are derived from BAYER BAYMAT select the Carreau-WLF (TB, TS) option. In this case SIGMA sets the two constants C1 and C2 on 8.86 and 101.6. Then they cannot be edited any longer.