The equivalence of time and temperature

In the glass to rubber transition, time and temperature are related through the speed of movement of the molecules undergoing internal rotation. We now have to quantify that relationship. The key is that the transition is observed when the rates (times) of the molecular rotation process and those of the observation are equal. 

Starting with the molecular movement, the rate of a molecular process crossing an activation energy barrier can be related to temperature through the Arrhenius equation. 

Then if the molecular and macroscopic rates (times) are equal at the transition, the molecular rate can be replaced by the observation rate or reciprocal time and, of course, T must be replaced by Tg.Then 

Rate of stress/strain = reciprocal observation time = y4exp(—AT / g) 

This is important in rubber technology because it means that a rubber that is above its transition temperature in some low rate, long time use could be below the transition temperature, and so become a breakable glass in some high rate, very short time use. 

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In this section we resume our examination of the equivalency of time and temperature in the determination of the mechanical properties of polymers. In the last chapter we had several occasions to mention this equivalency, but never developed it in detail. In examining this, we shall not only acquire some practical knowledge for the collection and representation of experimental data, but also shall gain additional insight into the free-volume aspect of the glass transition. 

From the data in Fig. 4.8b, estimate the shift factors required to displace the data at 0 = 0.5 (consider only this point) so that all runs superimpose on the experiment conducted at 128 C at 0 = 0.5. Either a ruler or proportional dividers can be used to measure displacements. Criticize or defend the following proposition Whether a buffered aqueous solution of H2O2 and 1. containing small amounts of S2O3 and starch, appears blue or colorless depends on both the time and the temperature. This standard general chemistry experiment could be used to demonstrate the equivalency of time and temperature. The pertinent reactions for the iodine clock are... 

The WLF equation can be widely applied, and demonstrates the equivalence of time and temperature, the so-called time-temperature superposition principle, on the mechanical relaxations of an amorphous polymer. The equation holds up to about 100° above the glass transition temperature, but after that begins to break down. 

Similar curves can be constructed for creep or dynamic mechanical test data of amorphous polymers. Because of the equivalence of time and temperature. 

Fortunately for linear amorphous polymers, modulus is a function of time and temperature only (not of load history). Modulus-time and modulus-temperature curves for these polymers have identieal shapes they show the same regions of viscoelastic behavior, and in each region the modulus values vary only within an order of magnitude. Thus, it is reasonable to assume from such similarity in behavior that time and temperature have an equivalent effect on modulus. Such indeed has been found to be the case. Viscoelastic properties of linear amorphous polymers show time-temperature equivalence. This constitutes the basis for the time-temperature superposition principle. The equivalence of time and temperature permits the extrapolation of short-term test data to several decades of time by carrying out experiments at different temperatures. 

Figure 1.15. The storage and loss moduli plotted against absolute temperature. Through the equivalence of time and temperature (see Section 1.5.7), the abscissa may also be taken as log t or its inverse, which is equal to minus log frequency.

For a satisfactory understanding of the viscoelastic behaviour of polymers data are required over a wide range of frequency (or time) and temperature. The number of experiments required ean sometimes be reduced by using either the equivalenee of creep, stress relaxation and dynamic mechanical data (described in Chapter 4) or the equivalence of time and temperature as variables (to be discussed in Chapter 6). Nevertheless a variety of techniques need to be combined to eover a wide range of both time and temperature. 

Accumulation of long-term data for design with plastics can be very inconvenient and expensive. The equivalence of time and temperature allows information about mechanical behavior at one temperature to be extended to longer times by using data from shorter time studies at higher temperature. It should be used with caution, however, because the increase of temperature may promote changes in... 

Time-temperature superposition (tTs) was carried out for these multi-temperature multi-frequency tests based on Williams-Landel-Ferry (WLF) relationship. It considers the equivalency of time and temperature in the context of free volume theory for an activated flow process in viscoelastic materials such as PET. It has been found the tTs holds for the whole temperature/frequenQr range. The master curve generated from tTs is shown in Figure 13 for the 10 wt. % bamboo-PET composite at the 25.0 C reference temperature. The time-temperature superposition shift factor follows Arrhenius temperature dqrendence according to the expression ... 

This equation provides a fundamental relationship between the effects of time and temperature on a transition mechanism. Time and temperature appear to be equivalent in their effect on the behaviour. At a fixed time scale, T, e.g. 1 sec, the transition temperature, T, is proportional to the energy of activation, AU. The transition temperature can be expressed in the time scale by ... 

Many measurements have been carried out with a large number of, sometimes very complicated, techniques. A shorter route towards a global knowledge over such a broad range of time and frequency, is supplied by the overall equivalence of time and temperature. In its simplest form, this equivalence can be expressed via the assumption that the effect of temperature on a molecular process often follows Arrhenius law ... 

The kinetic nature of the glass transition should be clear from the last chapter, where we first identified this transition by a change in the mechanical properties of a sample in very rapid deformations. In that chapter we concluded that molecular motion could simply not keep up with these high-frequency deformations. The complementarity between time and temperature enters the picture in this way. At lower temperatures the motion of molecules becomes more sluggish and equivalent effects on mechanical properties are produced by cooling as by frequency variations. We shall return to an examination of this time-temperature equivalency in Sec. 4.10. First, however, it will be profitable to consider the possibility of a thermodynamic description of the transition which occurs at Tg. 

Product Heat Treatment. Equivalent heat treatment for destmction of microorganisms or inactivation of enzymes can be represented by plotting the logarithm of time versus temperature. These relationships were originally developed for sterilization of food at 121.1°C, therefore the time to destroy the microorganism is the V value at 121.1°C (250°F). The slope of the curve is and the temperature span is one log cycle. The heat treatment at 131°C for one minute is equivalent to 121.1°C for 10 minutes (Fig. 10). 

Clearly, the maximum degree of simplification of the problem is achieved by using the greatest possible number of fundamentals since each yields a simultaneous equation of its own. In certain problems, force may be used as a fundamental in addition to mass, length, and time, provided that at no stage in the problem is force defined in terms of mass and acceleration. In heat transfer problems, temperature is usually an additional fundamental, and heat can also be used as a fundamental provided it is not defined in terms of mass and temperature and provided that the equivalence of mechanical and thermal energy is not utilised. Considerable experience is needed in the proper use of dimensional analysis, and its application in a number of areas of fluid flow and heat transfer is seen in the relevant chapters of this Volume. 

Fig. 3.14. The data is for a very broad range of times and temperatures. The superposition principle is based on the observation that time (rate of change of strain, or strain rate) is inversely proportional to the temperature effect in most polymers. That is, an equivalent viscoelastic response occurs at a high temperature and normal measurement times and at a lower temperature and longer times. The individual responses can be shifted using the WLF equation to produce a modulus-time master curve at a specified temperature, as shown in Fig. 3.15. The WLF equation is as shown by Eq. 3.31 for shifting the viscosity. The method works for semicrystalline polymers. It works for amorphous polymers at temperatures (T) greater than Tg + 100 °C. Shifting the stress relaxation modulus using the shift factor a, works in a similar manner.

Storage Charts of time and temperature of storage are important to control the increased levels of degradedness [6], Shelf life is defined as the amount of time in storage that a product can maintain quality and is equivalent to the time taken to reach 90% of the composition claim or have 10% degradation. The availability of an expiration date is assumed under specified conditions of temperature. Based on zero- and first-order reaction calculations, Connors et al. [45] show the estimation methods to determine the shelf life of a drug product at temperatures different from the one specified under standard conditions. 

Note that Equation 8 or 9 represents an equivalence between frequency and temperature, which can be expressed as a time-temperature equivalence. The Arrhenius equation is found to be most applicable at lower temperatures. At higher temperatures, a better representation of the equivalence between frequency and temperature is given by the WLF (Williams-Landel-Ferry) equation, which can be written as... 

The aforesaid extrapolations make use of a time-temperature superposition principle which is based on the fact that time and temperature have essentially equivalent effects on the modulus values of amorphous polymers. Figure 3.19 shows modulus data taken at several temperatures for poly(methyl methacrylate) [8]. Because of the equivalent effect of time and temperature, data at different... 

Plots of time and temperature provide a basis for assessing fire severity. The procedure compares fires with different temperature histories to standard time-temperature profiles. A test fire is equivalent in severity to the standard when the areas under the time-temperature curves are equal. Barriers, such as walls, floors, ceilings, and doors, must be able to withstand the desired fire severity. 

The principle of TTS lies in the equivalency of time (frequency) and temperature. Due to various limitations, one cannot carry out experiments at conditions such as at very low frequencies and very high temperatures or vice versa. TTS is used to obtain data at different conditions to save experimental time. The viscoelastic data of one temperature can be related to the higher or lower temperature using a shift factor a ) to the right side, or to the left side of the time axis using a reference temperature (T f). A fully overlapped curve can be obtained for any reference temperature this is called a master curve . It is also widely accepted that a minor vertical shift factor may also be applied to more accurately model master curves. 

Then, the simple thermorheological behaviour says that the stress relaxation function at any temperature is obtained in the log scale by shifting the curve by log[ (T)]. A reverse operation consists of determining the behaviour of the glass and shifting the creep functions (so-called equivalence between time and temperature) to obtain the master curve evolution with time at a reference temperature (Section 6.4.3 Figure 6.24). 

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