In this problem you will use the principle of time temperature superposition $($TTS$)$ to shift data $($G$\text{'}$ vs frequency$)$ for a polymer sample that was obtained at a variety of temperatures, in order to create a master curve. The first step is to obtain a reduced $G^{\text{'}}(G_r^{\text{'}}),$ as defined below. Empirically and later on through theory, it has been found that $G^{\text{'}}$ is proportional to $nkT($where $n$ is the number of moles, $k$ is the Boltzmann constant, and $T$ is the temperature in $K).$ Thus, the reduced variable can be defined as:

$G_r^{\text{'}}=\frac{G^{\text{'}}}{\frac{T}{T_o}}$ where $T_{o}is$ a chosen reference temperature

Once the data has been converted into reduced variables, then the shift factor at each temperature can be calculated to shift the data onto one curve. The following figure shows an example of how thie chiftine san he dono

Therefore, $log{}_T$ is simply the distance that the $log{G}^{\text{'}},vs.log$ curve has shifted horizontally:

Use the lowest temperature as the reference temperature. Plot $log{G}_r^{\text{'}}vs.log$ for the two lowest temperatures. Determine the value of $log{}_T$ required to collapse the higher temperature data set onto the data set at the reference temperature. Repeat this process for the data all temperatures until the master curve is completed. Plot $log{G}_r^{\text{'}}vs.log{}_T$ for all temperatures.

After completing the TTS$,$ answer the following questions:

Does the WLF theory describe your shift factor, ${}_T,$ as a function of temperature? If so$,$ what are $C_1$ and $C_2?$

What is the purpose of doing the time$-$temperature$-$superposition? What advantage does it give us when conducting a rheology experiment?

The data are given in the table below:

$\backslash$table$[[,G^{\text{'}}(dy\frac{n}{c}{m}^2)$