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TUTORIAL   Time– Temperature Superposition

Small-amplitude oscillatory shear (SAOS) measurements are performed as frequency sweeps over the angular-frequency range ( ) at a series of temperatures (T1, T2, T3, ….). The rheometer combines all measurements into a single composite data file containing the dynamic moduli G'( ,T) and G''( ,T). A typical data set, Figure 1, will be used to demonstrate the phenomenon.

Figure 1: Composite SAOS data (G'( ,T) and G''( ,T)) for a commercial Monsanto polystyrene, PS.

IRIS automatically separates the composite data into individual isothermal frequency sweeps, one for each experimental temperature, Figure 2.

Figure 2: Isothermal SAOS frequency sweeps at (T = 132, 145, 161, 181, 219, 269°C.

Each isothermal data set is then shifted horizontally to the reference temperature, which is conveniently chosen near the center of the experimental temperature range, Figure 3. The horizontal shifts determines the temperature shift factors aT.

Figure 3: Horizontal shifting of the isothermal SAOS data to (T= 181°C.  Progressive stages of shifting demonstrate the expansion of the frequency window.

The shifted data merge smoothly into a single master curve, extending the accessible angular-frequency range by several decades while preserving the material response, Figure 4.

Figure 4: Master curves G'( ) and (G''( ) referenced to T = 181°C

The corresponding horizontal temperature shift factor aT(T) are determined simultaneously, Figure 5. Some materials also vertically shift, but for PS this vertical “modulus shift factor” does not contribute, bT(T)  =1.

Figure 5: Horizontal temperature shift factors aT(T) with reference temperature (T = 181°)

Once the shift factors have been established, IRIS can immediately generate equivalent master curves at any desired reference temperature without repeating the experiment, Figure 6.

Figure 6: Shifted master curves G'( ) and (G''( ), referenced to 160°C, 180°C , 200°C , 220°C, and 240°C

The master curves provide the starting point for many subsequent IRIS analyses, including relaxation spectrum determination, constitutive model fitting, prediction of rheological properties beyond the experimental frequency window, and quantitative comparison of different materials.

The Booij-Palmen plot, Figure 7,  eliminates the temperature dependence as show.

Figure 7: Booij-Palmen plot as it unites the master curves G'( ) and (G''( ) of figure 5 into a temperature – independent format. Solid line for tand and dashed line for 2d/p.