Master Curve Construction
Purpose
In Section 3.4.2. Temperature dependence of the relaxation spectrum the relationship between the temperature and frequency of rheologically simple materials was discussed. Without going into the theoretical details, this section describes how a frequency spectrum over a large frequency range at a reference temperature can be constructed from different measurement curves. The curve obtained is known as a master curve.
Sample
Unfilled and uncross-linked SBR
Conditions
Measuring cell: DMA/SDTA861e with the shear sample holder
Sample preparation: The SBR was pressed to a 1-mm thick film. Cylinders of 4-mm diameter were punched out and mounted in the shear sample holder with 10% predeformation.
DMA measurement: The measurement was performed in the frequency range 100 mHz to 1 kHz under isothermal conditions at different temperatures between –50 °C and 100 °C. Maximum force amplitude 5 N; maximum displacement amplitude 10 Pm; offset control zero.
Master curve of the shear modulus at a reference temperature of –10 °C.
Master curve of the shear compliance at a reference temperature of 10 °C.
Evaluation
From the measurement curves in Section 3.4.2. Temperature dependence of the relaxation spectrum, all the storage and loss moduli are displayed as a function of frequency at every temperature. The curves measured at temperatures lower than the reference temperature are shifted to higher frequencies in such a way that the individual curves of the storage modulus and the loss modulus overlap to the greatest possible extent with the corresponding composite curves so formed. In the same way, the curves measured at higher temperatures are shifted to lower frequencies. The basic principle behind the shifting of the curves is the time-temperature superposition principle previously discussed. As a result of these operations, one obtains curves of both components of the complex modulus over a frequency range of 10-8 to 1011 Hz.
The second diagram shows the corresponding master curve construction for the complex compliance obtained from the same measurement. The shift of the individual curve sections corresponds to that in the construction of the master curve for the modulus.
Interpretation
At low frequencies, both the storage and the loss moduli have about the same value of 30 kPa. The material is in the flow range. Flow relaxation is the cause of the G" peak at about 106 Hz. Afterward, the storage modulus exhibits the rubbery plateau with a modulus value that is a little less than 1 MPa. The corresponding frequency range is between 10 and 10 Hz. The storage modulus then shows a step of about 3 decades that coincides with a peak in the loss modulus. This is the main relaxation (glass transition) with a characteristic frequency of about 300 Hz (frequency at the maximum of the G" peak). At higher frequencies the storage modulus is almost constant at about 800 MPa. The loss modulus decreases linearly in the log-log presentation. At frequencies above 10 Hz, regions appear in the G" curve that cannot be brought to overlap through shifting. These regions may have been influenced by the cooling conditions. They could be interpreted as the beginning of secondary relaxation (β-relaxation).
The master curve of the compliance behaves similarly to that of the modulus except that J' is large at low frequencies, and in the main relaxation region J' becomes smaller with increasing frequency. The peak of the flow relaxation in the J" curve is overlapped by the linear increase in the flow range. The maximum of the J" peak is shifted to lower frequencies in the main relaxation region compared to the G" peak. The characteristic frequency is 0.2 Hz. If one compares the maximum frequency of the G" peak with that of the J" peak, one obtains a difference of about 3 decades of frequency.
Conclusions
With rheologically simple materials, it is possible to construct a master curve based on the time-temperature superposition principle that shows the entire relaxation spectrum. A detailed analysis of the resulting curves allows one to study molecular dynamics and to check whether the dynamic material properties are suitable for a particular application.
Master Curve Construction | Thermal Analysis Handbook No.HB425 | Application published in METTLER TOLEDO TA Application Handbook Elastomers, Volume 1