# TTS Factors Theories¶

## William-Landel-Ferry¶

### Summary¶

Time-temperature superposition based on a Williams-Landel-Ferry (WLF) equation with two parameters.

• Function
$\begin{split}\begin{eqnarray} \omega(T_r) &= & a_T \omega(T) \\ G(T_r) &= & b_T G(T) \\ \log_{10} a_T &= & \frac{-B_1 (T-T_r)}{(B_2+T_r)(B_2+T)} \\ b_T &= & \frac{\rho(T_r)T_r}{\rho(T)T} = \frac{(1+\alpha T)(T_r+273.15)}{(1+\alpha T_r)(T+273.15)} \\ T_g &= &T_g^\infty - \frac{C_{T_g}}{M_w} \end{eqnarray}\end{split}$
• Parameters
• $$T_r$$: Reference temperature to which the experimental data will be shifted.

• $$B_1$$: Material parameter, corresponding to $$C_1\cdot C_2$$, with $$C_1$$ and $$C_2$$ being the standard WLF material parameters.

• $$B_2$$: Material parameter, corresponding to $$C_2-T_r$$, $$C_2$$ being the standard WLF material parameter.

• logalpha: Decimal logarithm of the thermal expansion coefficient of the polymer at 0 °C.

• $$C_{T_g}$$: Material parameter that describes the dependence of $$T_g$$ with $$M_w$$.

• dx12: Fraction of 1-2 (vynil) units (valid for polybutadiene).

## Arrhenius Equation¶

### Summary¶

Arrhenius Equation

• Function
$a_T = \exp\left(\frac{E_a}{R} \left(\frac{1}{T} - \frac{1}{T_{ref}}\right) \right)$
• Parameters
• $$E_a$$: Activation Energy

• $$T_{ref}$$: Reference Temperature for the shift factors

• $$R$$: Gas Constant