Time-temperature superimposed

Figure 1. Time-temperature superimposed master curves of storage and loss moduli as functions of reduced frequency. The four networks belong to the D family. Numbers indicate wt. % dicumyl peroxide.

The time-temperature superposition, implying that the functional form does not appreciably depend on temperature (see e.g. [34, 111]). For instance, mechanical or rheological data corresponding to different temperatures can usually be superimposed if their time/frequency scales are shifted properly taking a given temperature Tr as reference. 

The scaling the functional shape hardly depends on temperature. Curves corresponding to different temperatures superimpose in a single master curve when they are represented against a reduced time variable that includes a T-dependent shift factor. 

Fig. Z4 (a) Temperature ramp at a frequency a> = lOrads (strain amplitude A = 2%) for a nearly symmetric PEP-PEE diblock with Mn = 8.1 X 104gmol l, heating from the lamellar phase into the disordered phase. The order-disorder transition occurs at 291 1 °C, the grey band indicates the experimental uncertainty on the ODT (Rosedale and Bates 1990). (b) Dynamic elastic shear modulus as a function of reduced frequency (here aT is the time-temperature superposition shift factor) for a nearly symmetric PEP-PEE diblock with Mn = 5.0 X 1O g mol A Shift factors were determined by concurrently superimposing G and G"for w > and w > " respectively. The filled and open symbols correspond to the ordered and disordered states respectively. The temperature dependence of G (m < oi c) for 96 < T/°C 135 derives from the effects of composition fluctuations in the disordered state (Rosedale and Bates 1990). (c) G vs. G"for a PS-PI diblock with /PS = 0.83 (forming a BCC phase) (O) 110°C (A) 115°C ( ) 120°C (V) 125°C ( ) 130°C (A) 135°C ( ) 140°C ( ) 145°C. The ODT occurs at about 130°C (Han et at. 1995).

If stress relaxation curves are obtained at a number of different temperatures, it is found that these curves can be superimposed by horizontal shifts to produce what is called a master curve .42 This concept of time-temperature equivalence is very important to understanding and predicting polymer behavior. As an example, a polymer at very low... 

The Time-Temperature Superposition Principle. For viscoelastic materials, the time-temperature superposition principle states that time and temperature are equivalent to the extent that data at one temperature can be superimposed upon data at another temperature by shifting the curves horizontally along the log time or log frequency axis. This is illustrated in Figure 8. While the relaxation modulus is illustrated (Young s modulus determined in the relaxation mode), any modulus or compliance measure may be substituted. 

The second important consequence of the relaxation times of all modes having the same temperature dependence is the expectation that it should -bp possible to superimpose linear viscoelastic data taken at different temperatures. This is commonly known as the time-temperature superposition principle. Stress relaxation modulus data at any given temperature Tcan be superimposed on data at a reference temperature Tq using a time scale multiplicative shift factor uj- and a much smaller modulus scale multiplicative shift factor hf. 

Demonstration of the time-temperature superposition principle, using oscillatory shear data (G, filled circles and G", open diamonds) on a PVME melt with M — 124000 gmol. The right-hand plot shows the data that were acquired at the six temperatures indicated, with Tg = - 24°C chosen as the reference temperature. All data were shifted empirically on the modulus and frequency scales to superimpose, constructing master curves for G and G" in the left-hand plot. Data and... 

When a TV camera is installed on the microscope, a video cassette recorder can be used to make a permanent record of observations. This is especially valuable when the temperature and time are superimposed on the image of the sample allowing review as a function of time and temperature. This can be done a frame at a time if necessary. 

Time-temperature superposition is applicable to a wide variety of viscoelastic response tests, as are creep and stress relaxation. We illustrate the principle by considering stress relaxation test data. As a result of time-temperature correspondence, relaxation curves obtained at different temperatures can be superimposed on data at a reference temperature by horizontal shifts along the time scale. This generates a simple relaxation curve outside a time range easily accessible in laboratory experiments. This is illustrated in Figure 14.13 for polyisobutylene. Here, the reference temperature has been chosen arbitrarily to be 25°C. Data obtained at temperature above 25°C are shifted to the right, while those obtained below 25°C are shifted to the left. 

The shape of the mastercurve is related to the polymer microstructure. That for polystyrene at 100 °C (Fig. 7.5b) shows a transition from a glassy compliance at Is to a rubbery one at times exceeding 10 s. It continues to 10 ° s, so it can be used for extrapolation to times longer than those accessible by experiment. Time-temperature superposition for semi-crystalline polymers, such as polyethylene, may be successful for a limited temperature range, i.e. 20°C-80°C. As polyethylene starts to recrystallise if heated within 50 °C of T, and residual stresses may start to relax, data for higher temperatures will not superimpose. 

Figure 13 tests another prediction of the Rouse model, the time-temperature superposition property. Again, a representative example is shown, t.e., the correlation function of the third Rouse mode. As the theory anticipates, it is indeed possible to superimpose the simulation data, obtained at different temperatures, onto a common master curve by rescaling the time axis. The required scaling time, T3, is defined by the condition pp(r3) = 0.4. The choice of this condition is arbitrary. Since the Rouse model predicts that the correlation function satisfies equation (10) for all times, any other value of pp(t) could have been used to define T3. This scaling behavior is in accordance with the theory. However, contrary to the theory, the correlation functions do not decay as a simple exponential, but as... 

In many applications, plastic parts carry reasonably constant mechanical loads over periods up to few years. The polymer will creep during the lifetime of the part. At moderate load levels, long-term prediction of creep from short-term tests is possible, because the viscoelastic response of polymers (creep, stress relaxation) measured at different temperatures superimpose when shifted along the time axis [24]. 

The time-temperature correspondence principle is known to be widely applicable to polymers other than PLCs, e.g. [37] or [38]. It was therefore of interest to test the applicability of the principle to our PLC. For this purpose, the experimental creep data are presented in Figure 12.2 in logarithmic coordinates. It can be seen that the shape of these curves suggests that a master curve may be constructed by shifting experimental curves along the log time axis until they all superimpose on one curve for the chosen reference temperature. Figure 12.3 presents such a master curve for the reference temperature of 20 C. As can be seen in... 

The polymer SBR is mechanically simple, and it was found that the results for G for a given substrate could be superimposed using the Williams, Landel and Ferry (WLF) technique (see Viscoelasticity - time-temperature superposition). The master curves for each substrate were close to being parallel to each other and to a similar curve for cohesive fracture of the SBR (Fig. 1). 

Time-temperature equivalence in its simplest form implies that the viscoelastic behaviour at one temperature can be related to that at another temperature by a change in the time-scale only. Consider the idealized double logarithmic plots of creep compliance versus time shown in Figure 6.7(a). The compliances at temperatures T and T2 can be superimposed exactly by a horizontal displacement log at, where at is called the shift factor. Similarly (Figure 6.7(b)), in dynamic... 

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