Time-temperature shift

As an alternative to the Arrhenius equation, property curves can be shifted along the time axis to create a master curve known from presentations of physical effects, such as creep at various temperatures. This approach is feasible when the property curves at different temperatures look similar. The shift of temperature dependent individual curves to a master curve is called time-temperature shift or time-temperature superposition. 

The time compression corresponding to a time-temperature shift is then expressed using a time shift factor o,. For any component property 

Mathematically, shifting the property curve means that the time at each temperature is multiplied by a temperature-dependent temperature shift factor and that the property value is divided by the same factor. 

The Arrhenius-equation describes the shift factors by a linear equation  

To simplify the presentation in diagrams and enhance mathematical handling, the exponential function is converted to the common logarithm. Then  

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The standard methods for testing creep, the elongation and possible rupture of a plastic under sustained load, are ISO 899-1 [34] for tension and ISO 899-2 [35] for flexure. Tests last typically for 1,000 hours or six weeks. Tests at higher temperatures may be required either because of a higher service temperature or to provide a prediction of longer term behaviour by time-temperature shifting. 

It is common practice, when similar materials are being compared, to ignore the shape of the curve and to take the time for the property to reach some percentage, say 50%, of its initial value. This may be expedient but is clearly less satisfactory than modelling the curve and could be extremely misleading if materials with substantially different curves are compared. If a time/temperature shift method is used to model the effect of temperature (see below) no function to describe the change of property with time need be assumed. 

Time-temperature shifting is used widely in the prediction of creep and creep-rupture in polyester geosynthetics. Creep of oriented polyesters is expressed by a linear or quadratic increase of strain with log (lifetime). The lifetime under constant load is expressed by the semilogarithmic formula ... 

Figure 9.1 Schematic prediction of creep modulus of polyester fibres from time-temperature shifted data (based on information from [5])...

Creep at different temperatures can be represented by separate bundles of isochrones. A simple time-temperature shift could mean that for higher temperatures shorter time values could be written at each curve. It is, however, easier to transform the stress scale. Sometimes this is possible with sufficient accuracy in those cases only one bundle of isochrones is given with different stress scales for a number of temperature levels (Figure 7.9) (see also Qu. 7.13). 

Deduction of Shift Factors. Time-temperature shift factors for the blends were obtained by shifting the experimental relaxation isotherms to the calculated master curves (10). The temperature and time dependence of the shift factors of the 75/25 and 50/50 blends are represented in Figures 10a and 10b at t = 10 sec and t = 1000 sec for a reference temperature of 140°C. The empirically determined shift factors of the pure components are given in these figures by dotted lines their temperature dependence is of the WLF type. 

Mechanical Characterization of Sulfur-Asphalt. The serviceable life of a pavement comes to an end when the distress it suffers from traffic and climatic stresses reduces significantly either the structural capacity or riding quality of the pavement below an acceptable minimum. Consequently, the material properties of most interest to pavement designers are those which permit the prediction of the various forms of distress—resilient modulus, fatigue, creep, time-temperature shift, rutting parameters, and thermal coefficient of expansion. These material properties are determined from resilient modulus tests, flexure fatigue tests, creep tests, permanent deformation tests, and thermal expansion tests. 

Time-Temperature Shift Characteristics. Temperature is a major environmental influence on viscoelastic pavement response. The VESYS IIM program can handle material properties as a function of temperature variations. The computer input command BETA relates the time-temperature shift factor, au to the temperature variable for the pavement materials. This relationship is given by ... 

The same shift factor, a-j-, must superpose all of the viscoelastic functions. One must first perform the time-temperature shift on one of the viscoelastic functions and determine the values of the WLF constants. The same constants must then be applied to the other viscoelastic functions to determine their consistency in shifting the data. This process may need to be repeated several times in order to determine the best set of... 

Figure 7 - Empirically determined time-temperature shift factors for two blends of 97% 1,2-PBD with PIP. The concentration of the PIP is as indicated.

Time-temperature shifting is also extremely useful in practical applications, since it allows one to make predictions of material response for time scales either much longer... 

Figure 3.12 (a) Storage compliance ot poIy(n-octyl methacrylate) versus frequency measured at 24 temperatures, given in degrees Celsius, (b) Master curve of J versus reduced frequency (Mj obtained from the data of (a) by time temperature shifting, fc) Temperature-dependence of the shift... 

For polyethylene, the glass-transition temperature is far below the crystallization temperature and time-temperature shifting satisfies an Arrhenius form, with activation energy Ea = 6.5 kcal/mol for high-density polyethylene. 

Figure 5.7 Frequency dependences of the storage (Q) and loss (-h) moduli for poly(dimethylsilox-ane) (PDMS) samples whose reactions were quenched at the times indicated (see Fig. 5-6). The data are time-temperature-shifted to the reference temperature T gf of 34°C, and they are shifted additionally by an amount A on the logarithmic axis to keep the curves from overlapping. The vertical shift factors bf are given by p T sf)T d/ p T)T), where p is the density. (From Winter and Chambon 1986, with permission from the Journal of Rheology.)...

Figure 13.13 Reduced storage modulus G and dynamic viscosity rj = G /w as functions of reduced frequency uto) for a cylinder-forming polystyrene-polybutadiene-polystyrene triblock copolymer with block molecular weights of 7000-43,000-7000. The curves are time-temperature-shifted to a reference temperature of 138°C the open symbols were obtained in the low-temperature ordered state the closed symbols were obtained in the high-temperature disordered state. (From Gouinlock and Porter 1977, reprinted with permission from the Society of Plastics Engineers.)...

Figure 13.14 Storage modulus G as a function of reduced frequency for a symmetric PEP-PEE sample at temperatures above Toot = 96°C (open symbols), and below it (filled symbols), time-temperature shifted to obtain superposition atcu > o)c with a reference temperature of 35 °C. (Reprinted with permission from Rosedale and Bates, Macromolecules 23 2329. Copyright 1990, American Chemical Society.)...

Figure 13.20 Master curves of G Tq/T ) versus reduced frequency ajco for the PS-PI block copolymer described in Fig. 13-15 in the high-temperature disordered state and in the low-temperature ordered state flow-aligned into either the parallel or perpendicular directions. The arrows show the time-temperature-shifted frequencies at which the parallel (sideways arrow) and perpendicular (up arrow) orientations are achieved by large-amplitude shearing. The frequency ajcoc 300 secri is the reduced frequency below which G in the disordered and quenched states no longer superpose. (Reprinted with permission from Patel et al., Macromolecules 28 4313. Copyright 1995, American Chemical Society.)...

Assuming that the WLF equation does indeed describe the time-temperature shifts, the complete viscoelastic response of any polymer under any experimental conditions may be obtained from knowledge of any two of the following three functions the master curve at any temperature, the modulus-temperature curve at any time, and the shift factors relative to some reference temperature. For example, suppose we are given the constants Cj, and C2 for a polymer whose master curve is known. (The values given for C, and C2 are those that result from fitting equation (4-6) to the aT vs. T data.5) For simplicity, we can assume that the master curve is at the same reference temperature as that in the WLF equation, perhaps Tg. Suppose it is desired to calculate the 10-second modulus-versus-temperature curve for this polymer. 

Figure 7. Experimentally determined time-temperature shift of the cohesive fracture energy for model solid propellent...

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