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).