Correspondence time-temperature

The procedure described above is an application of the time-temperature correspondence principle. By shifting a set of plots of modulus (or compliance) versus time (or frequency) at any temperature (subscript 1) along the log t axis, we obtain the value of that mechanical property at another time and temperature (subscript 2). Using the shear modulus as an example, the time-temperature correspondence principle states... 

So Figures 13-78 and 13-80 are our idealized representations. This correspondence is obviously interesting and important, but we will defer a discussion of the molecular origin of this until later. First we want to explore this time-temperature correspondence and the various regions of viscoelastic behavior in a little more detail. 

A fundamental characteristic of the so-called thermorheologically simple systems is that consecutive isotherms have similar habits, so they overlie each other when they are shifted horizontally along the log t axis. In other words, the time-temperature correspondence principle holds. This property in creep experiments can be expressed by the relation (2,3)... 

By applying the time-temperature correspondence principle to /"(co) and / (co) at low frequencies, one obtains... 

If the time-temperature correspondence principle holds, Eq. (8.35) suggests that the temperature dependence of the shift factor in the time (or frequency) domain can be written as... 

The time-temperature correspondence principle holds not only for the viscosity but also for the normal stresses. In the latter case, however, the... 

In thermorheological simple systems, the time-temperature correspondence principle holds. Chapter 8 gives examples of isotherms for compliance functions and relaxation moduli. The shift factors are expressed in terms of terminal viscoelastic parameters, and the temperature dependence of the shift factors is interpreted in terms of the free volume and the WLF equation. The chapter outlines methods for determining the molecular weight between entanglements, and analyzes the influence of diluents and plasticizers on the viscoelastic functions. 

In actual long term applications of polymers, however, it is well known that chemical reactions occur which actually change the viscoelastic properties of the material while it is in use. In addition, environmental factors such as exposure to solvents or even water, while not always chemically modifying a material, can have a profound influence on its viscoelastic properties in much the same way as a true chemical transformation. If predictions based exclusively on time-temperature correspondence were to be successful, the rates of all of these processes would have to vary with temperature in exactly the same m inner as does the viscoelastic spectrum. While this might be approximately true in certain special cases, it is usually not so. Thus, a more general theoretical framework is necessary to predict the properties of simultaneously chemically reacting and physically relaxing networks. 

Figure 8. Verification of time-temperature correspondence for sample i>, CO), data at 28°C CX), data at 42°C where the time scale has been sh ted by log Or =...

Having examined some of the basic manifestations of the phenomenological aspects of viscoelasticity in Chapter 2, along with the efforts to model this behavior in Chapter 3, we now shift our emphasis to the results of experiments on polymers, and the interpretation of these results in terms of the accepted molecular mechanisms. We then explore the important idea of time-temperature correspondence and demonstrate the close relationship between these two variables in determining the viscoelastic responses of polymers. 

The time-temperature correspondence principle states that there are two methods to use to determine the polymer s behavior at longer (or shorter) times than those covered by a stress-relaxation experiment run at 7j. First, one may improve the experiment to measure directly the response at longer (shorter) times. For the longer times, however, this procedure rapidly becomes prohibitively time-consuming because the change is so slow (note that Figure 4-5 is plotted on a log scale). (For the shorter times, the limitations are equipment related, e.g., transducer response time, problems with instrument and sample inertia, etc.) An alternative, according to the time-temperature... 

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