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 is presented at the IRIS graphics portal, Figure 1, and 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 to generate an optimal overlay, Figure 3. The shifts occurs with respect to a chosen reference temperature, which is conveniently chosen near the center of the experimental temperature range. In this case, data are referenced to T = 181°C

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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 set of master curves G'() and (G''() . The angular-frequency range increased by several decades while preserving the material response. The shift factors are shown in Figure 4. The horizontal shift dominates, but some materials also vertically shift. For PS this vertical “modulus shift factor” does not contribute, bT(T) =1 about.

Figure 4: Horizontal temperature shift factors aT(T) and vertical shift factor b(T) with reference temperature (T = 181°). Editing was supported by icon-based graphics shown here as a screenshot on the right figure..

Once the shift factors have been established, IRIS can immediately generate equivalent master curves at any desired reference temperature without repeating the experiment, Figure 5.

Figure 5: left figure: Shifted master curves G'() and (G''(), referenced to 160°C, 180°C , 200°C , 220°C, and 240°C. Right figure : Booij Palmen plot

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.

Figure 6: Complex viscosity examples: isotherm complex viscosities, Winter plot, master curves, shifted master curves