Temperature dependence of viscosity

The temperature dependence of liquid viscosity is usually expressed by one of the following models:

Exponential model

${\displaystyle \mu (T)=\mu _{0}\exp(-bT)}$ 

where T is temperature and ${\displaystyle \mu _{0}}$  and ${\displaystyle b}$  are coefficients. See first-order fluid and second-order fluid. This is an empirical model that usually works for a limited range of temperatures.

Arrhenius model

The model is based on the assumption that the fluid flow obeys the Arrhenius equation for molecular kinetics:

${\displaystyle \mu (T)=\mu _{0}\exp({\frac {E}{RT}})}$ 

where T is temperature, ${\displaystyle \mu _{0}}$  is a coefficient, E is the activation energy and R is the universal gas constant. A first-order fluid is another name for a power-law fluid with exponential dependence of viscosity on temperature.

WLF model

The Williams-Landel-Ferry model or WLF for short is usually used for polymer melt,s or other fluids that have a glass transition temperature.

The model is:

${\displaystyle \mu (T)=\mu _{0}\exp \left({\frac {C_{1}(T-T_{r})}{C_{2}+T-T_{r})}}\right)}$ 

where T-temperature, ${\displaystyle C_{1}}$ , ${\displaystyle C_{2}}$ , ${\displaystyle T_{r}}$  and ${\displaystyle \mu _{0}}$  are empiric parameters (only three of them are independent from each other).

If one selects the parameter ${\displaystyle T_{r}}$  based on the glass transition temperature, then the parameters ${\displaystyle C_{1}}$ , ${\displaystyle C_{2}}$  become very similar for the wide class of polymers. Typically, if ${\displaystyle T_{r}}$  is set to match the glass transition temperature ${\displaystyle T_{g}}$ , we get

${\displaystyle C_{1}\approx }$ 17.44

and

${\displaystyle C_{2}\approx }$ 51.6°K.

Van Krevelen recommends to choose

${\displaystyle T_{r}=T_{g}+43}$ °K, then

${\displaystyle C_{1}\approx }$ 8.86

and

${\displaystyle C_{2}\approx }$ 101.6°K.

Using such universal parameters allows one to guess the temperature dependence of a polymer by knowing the viscosity at a single temperature.

In reality the universal parameters are not that universal, and it is much better to fit the WLF parameters from the experimental data.