Time-Temperature Superposition

Horizontal Shift Factor

Arrhenius

$$\log \left(a_T\right)=\frac{E_a}{R}(\frac{1}{T}-\frac{1}{T_0})$$

(733)

where:

• $E_a$ is the activation energy.
• $R$ is the universal gas constant.
• $T_0$ is the reference temperature in K.

Nahme

$$\log \left(a_T\right)=-\frac{E_a}{R{T}_0^2}(T-T_0)$$

(734)

Williams-Landel-Ferry (WLF)

$$\log \left(a_T\right)=\frac{{-C}_1(T-T_0)}{C_2+(T-T_0)}$$

(735)

where $C_1$ and $C_2$ are positive constants that depend on the material and the reference temperature. If the reference temperature is taken as the glass transition temperature $T_g$ then $C_1$ and $C_2$ are universal for a given material, and the temperature of the domain must be higher than the reference temperature. For any polymer, approximate values are $C_1\approx 15°C$ and $C_2\approx 50°C$ .

Vertical Shift Factor

The vertical shift factor $b_T$ can be calculated by the Rouse method:

where $T$ and $T_0$ are the given temperature and a reference temperature, respectively. When the simulation is isothermal or viscosity is constant, $b_T=1$ .

Newtonian Liquids

For non-isothermal simulations, the shift factors modify the viscosity as follows:

where ${\mu }_{ref}$ is the user-specified reference viscosity.

Generalized Newtonian Fluids

Power Law

The simplest model to describe a shear-rate dependent viscosity behavior is the power law.

The value of the power law exponent $n$ determines the class of the fluid:

$$\mu (\dot{\gamma },T)=\frac{a_T}{b_T}\cdot k{\left(a_T\dot{\gamma }\right)}^{n-1}$$

(738)

• $n=1\Rightarrow$ Newtonian fluid
• $n>1\Rightarrow$ Shear-thickening (dilatant) fluid
• $n<1\Rightarrow$ Shear-thinning (pseudo-plastic) fluid

Cross Fluid

The Cross Fluid is defined from [161] as:

$$\mu \left(\dot{\gamma },T\right)=\frac{a_T}{b_T}({\mu }_{\infty }+\frac{{\mu }_0-{\mu }_{\infty }}{1+{\left(\frac{a_T\dot{\gamma }}{{\dot{\gamma }}_c}\right)}^m})$$

(739)

where:

• $m$ is the Cross rate constant
• ${\mu }_0$ is the zero-shear viscosity, the viscosity at the Newtonian plateau
• ${\mu }_{\infty }$ is the infinite-shear viscosity
• $\dot{{\gamma }_c}$ is the critical shear strain-rate at which shear-thinning starts

Newtonian behavior is regained for $m=0$. As $m$ increases towards unity, the degree of shear-thinning becomes stronger.

Carreau-Yasuda Fluid

The non-Newtonian Generalized Carreau-Yasuda Fluid for viscosity comes from [158] and [219]. It is defined as:

$$\mu (\dot{\gamma },T)=\frac{a_T}{b_T}({\mu }_{\infty }+\left({\mu }_0-{\mu }_{\infty }\right){(1+{\left(\lambda a_T\dot{\gamma }\right)}^a)}^{\left(n-1\right)/a})$$

(740)

where:

• $n$ power constant
• $\lambda$ relaxation time constant
• $a$ parameter to control shear-thinning. When $a=2$ , the original Carreau model is regained.

Viscoplastic Fluids

Herschel-Bulkley

The Herschel-Bulkley model with time-temperature superposition is based on Eqn. (705).

$$\mu (\dot{\gamma },T)=\{\begin{array}{ccc}\frac{a_T}{b_T}\cdot {\mu }_0& ,\text{if}& \dot{\gamma }<\frac{{\tau }_0}{{\mu }_0}\\ \frac{a_T}{b_T}\cdot \frac{{\tau }_0+k{\left[a_T\right(\dot{\gamma }-\frac{{\tau }_0}{{\mu }_0}\left)\right]}^n}{a_T\dot{\gamma }}& ,\text{if}& \dot{\gamma }>\frac{{\tau }_0}{{\mu }_0}\end{array}$$

(741)

Viscoelastic Fluids

For viscoelastic fluids, both the polymer viscosity ${\mu }_0$ and the relaxation time constant $\lambda$ in Eqn. (711) to Eqn. (720) are functions of temperature, proportional to the temperature shift factors $a_T$ and $b_T$: