Reduced variables shift factor

Note that subtracting an amount log a from the coordinate values along the abscissa is equivalent to dividing each of the t s by the appropriate a-p value. This means that times are represented by the reduced variable t/a in which t is expressed as a multiple or fraction of a-p which is called the shift factor. The temperature at which the master curve is constructed is an arbitrary choice, although the glass transition temperature is widely used. When some value other than Tg is used as a reference temperature, we shall designate it by the symbol To. 

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. 

The flat appearance of the E" curve is due to the compressed nature of this particular nomograph scale. Both functions appear to fit equally well and therefore satisfy the criteria of curve shape and shift factor consistency for using the reduced variable time-temperature superposition. Additionally, the criterion of reasonable values for a-j- is satisfied by virtue of using the "universal" WLF equation. 

The procedure by which the nomograph is generated is not limited to the WLF equation. Since it is based on the reduced variable concept, any superposition equation that results in the calculation of a temperature shift factor may be used to calculate the needed data to create the master curve and subsequent nomograph. The software can easily be modified to calculate and display a master curve on some other superposition equation. 

The paradigm shift from qualitative observation to quantitative measurement is the cornerstone of science. This shift was advocated by Sir Francis Bacon in the 17th century who insisted that repeated observations of a phenomenon were the first step in a scientific methodology that included experimentation (Jardine 2011). The objective of an experiment might be to test a hypothesis, maximize or minimize a process operation, improve a formulation, reduce variability, compare options, etc. Experiments may be carefully controlled measurements in a laboratory in which extraneous factors are minimized or they may be observational studies of nature, such as an epidemiological study, or field studies. 

Strain-Induced Dilatation. An alternative view of yield in polymers comes from the fact that a tensile strain induces a hydrostatic tension in the material and a corresponding increase in the sample volume. This in turn translates to an increase in the free volume, which increases the polymer mobility and effectively lowers the glass-transition temperature (Tg) of the polymer (alternatively it can be looked upon as increasing the free volume to the value it would have at the normal measured Tg). The increased mobility results in a lowering of the yield stress. Rnauss and Emri (35) used an integral representation of nonlinear viscoelasticity with a state-dependent variable related to free volume to model the yield behavior, with the free volume a function of temperature, time, and stress history. This model uses the concept of reduced time (see VISCOELASTICITY), where application of a tensile stress causes a volume dilatation and consequently causes the material time scale to change by a shift factor related to the magnitude of the applied stress. Yield occurs because the free-volume shift factor causes the molecular mobility to increase in such a way that yield can occur. 

At finite concentrations, the frequency dependence of G and G" — (aris can be examined directly without scaling the coordinates as in preceding figures. However, measurements at different temperatures may be combined by reduction to a standard temperature To G and G" — coris are multiplied by Toco/Tc and u> is multiplied by a shift factor aj which is given by rjo Vs)ocT/ t]o — Vs)coTq. Here r]o is the steady-flow (vanishing shear rate) viscosity and the subscript 0 otherwise refers to the reference temperature. This is the method of reduced variables which will be discussed fully in Chapter 11 with explanation of its rationale, and affords an extension of the effective frequency range. 

For some polymers and polymer solutions the temperature and frequency dependence of the components of the dielectric constant can be analyzed by reduced variables to determine a function br analogous to the shift factor ot of Sections A and 133-136 when the temperature dependence of br follows... 

The inclusion of values in Table 1 l-III derived from dynamic bulk viscoelastic measurements implies the concept that the relaxation times describing time-de-pendent volume changes also depend on the fractional free volume—consistent with the picture of the glass transition outlined in Section C. In fact, the measurements of dynamic storage and loss bulk compliance of poly(vinyl acetate) shown in Fig. 2-9 are reduced from data at different temperatures and pressures using shift factors calculated from free volume parameters obtained from shear measurements, so it may be concluded that the local molecular motions needed to accomplish volume collapse depend on the magnitude of the free volume in the same manner as the motions which accomplish shear displacements. Moreover, it was pointed out in connection with Fig. 11 -7 that the isothermal contraction following a quench to a temperature near or below Tg has a temperature dependence which can be described by reduced variables with shift factors ay identical with those for shear viscoelastic behavior. These features will be discussed more fully in Chapter 18. 

An example of nonlinear stress relaxation is shown in Fig. 16-17, where the ratio of time-dependent tensile stress to tensile strain is plotted logarithmically against time for different strains for cellulose monofilaments. (In this case the structure is no doubt preoriented.) The differences can be interpreted as due to a decrease in relaxation times with increasing stress, and the curves can be combined approximately into a composite curve by plotting with reduced variables, with a shift factor Os which decreases very rapidly with increasing strain. It is doubtful, how-ever, 2 whether the latter can be entirely related to fractional free volume in crystalline polymers as it is for amorphous polymers (Section Cl of Chapter 15). 

The more general phenomenon of adhesion between a rubbery polymer and a different surface is frequently studied by tests of the force required to peel a thin layer of polymer from a rigid substrate. 2,53 at different peel rates and different temperatures can be combined by reduced variables with shift factors given by the WLF equation, indicating that here as in friction the process is controlled by rates... 

Ferry went to Harvard University in 1937 and worked there in a variety of posts, including as a Junior Fellow, until he joined the University of Wisconsin in 1946. He was promoted to Full Professor in 1947 His extensive measurements of the temperature dependence of the dynamic mechanical properties of polymers led to the concept of reduced variables in rheology. His demonstration that time-temperature superposition applied to many systems is the basis for the rational description of polymer rheology. He measured the dynamic response over a very wide range of frequency. One of the fruits of this work is the Williams-Landel-Ferry (WLF) equation for time-temperature shift factors. 

In an early paper on the subject Szepe and Levenspiel [refe 10) introduced the notion of separability. The equations (1), (2) and (3), In which the deactivation function is a variable factor multiplying the initial rate of a reaction, correspond to their definition of separability. It may be useful to remind here that any kinetic treatment assuming ideal surfaces or accepting an average activity for the catalytic sites is bound to lead to such a form, provided, of course, there is no shift in rate determining steps. The question whether the deactivation is separable or not reduces to the question generally encountered in kinetic studies is it necessary to account explicitly for non uniform activity of the catalytic sites ... 

In most the cases, r(T) is a monotonous function, moreover, it is reducing with temperature increase. Keeping in mind that other variables in the expression (35) can be assumed as temperature-independent at least at low temperatures, one can see that the relaxation attenuation has maximum at a certain temperature T = Tm-Similar to the resonant attenuation, its position is shifted due to the the factor /T with respect to T = T which is defined in this case with the condition... 

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