Time-temperature-superposition

Time-temperature superposition is of interest in two contexts. For the experimentalist it is the basis of a technique for substantially increasing the range of times or frequencies over which linear behavior can be determined. And for the polymer scientist, it may provide additional information about molecular structure. It was Ferry [1] who first provided a scientific basis for this procedure. The essence of the concept is that if all the relaxation phenomena involved in G t) have the same temperature dependency, then changing the temperature of a measurement will have the same effect on the data as shifting the data horizontally on the log(time) or log(frequency) axis. Let us say that a change in the temperature from a reference value Tg to a different temperature T has the following effect on all the relaxation times  

The factor is thus a horizontal shift factor that can be used to shift data taken at a temperature 

T along the time axis so that they will equal data taken at the reference temperature, Tg. This leads to the definition of a reduced time for use in making a temperature-independent master curve, where this time is defined as follows  

When oscillatory shear data are being shifted, the frequency shift factor is simply a, and the reduced frequency is  

The dependence of the time shift factor on temperature is of great interest, and several empirical expressions have been proposed to describe it. The two that are most commonly used are the Arrhenius dependence and the WLF dependence, which are given by Eqs. 4.68 and 4.69, respectively. 

(3-40), each t, and G,- depend on absolute temperature T. For t,, the most important temperature dependence is not the explicit T dependence, but instead the indirect depen-dence of the friction coefficient on T. Note that this dependence is the same for each t,. 

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 

As a second simple example, consider how the zero-shear viscosity t q shifts with temperature. Since tjo — limaj .o G jo), we have-------------------------------------- 

The dependence of the shift factor aj on temperature can often be fit to an empirical expression known as the WtF (Williams-Landel-Ferry) equation (Williams et al. 1955 Ferry 1980)  

The relaxation spectrum can also be affected by changes in pressure, p, especially if these pressure changes are large— that is, hundreds of atmospheres (Ferry 1980 Tanner 1985). As with temperature, a simple shifting procedure can be used to account for pressure effects. Thus the shift factor aj p is both temperature-and pressure-dependent. 

Note that the dimensions of the angular frequency w are time The angular frequency thus corresponds to a reciprocal time scale. For dynamic (oscillating) deformations, then, the Deborah number is 

Staring at the curves for a while indicates that they appear to be sections of one continuous curve, chopped up, with the sections displaced along the log-time axis. That this is indeed so is shown in Fig. 18.21. Here, 25 C has arbitrarily been chosen as a reference temperature Tq and the curves for other temperatures shifted along the log time axis to line up with it. The data below 25°C are shifted to the left (shorter times) and those above 25°C are shifted to the right (longer times), giving a master, curve at 25 C. 

Sometimes, the relaxation moduli at each temperature T are corrected to the reference temperature To by multiplying by the ratio To/J before superposing. 

Shifting a constant-temperature curve along the log-time axis corresponds to dividing every value of its abscissa by a constant factor (it is immaterial what kind of scale is used for the ordinate). This constant factor, which brings a curve at a particular temperature T into alignment with the one at the reference temperature Tq, is known as the temperature shift factor aji 

For temperatures above the reference temperature, it takes less time to reach a particular response (the material responds faster, Le., has a shorter relaxation time), so is less than one, and vice versa. The logarithm of the experimentally determined temperature shift factor is plotted as a function of temperature in Fig. 18.21. 

Remembering that the modulus is a function of time as well as temperature leads us to wonder about the parallels between modulus measured as a function 

Again it should be emphasized that there are two aspects to time-temperature superposition (1) empirical superposition of data gathered at 

Division by the temperature corrects for the changes in modulus due to the inherent dependence of modulus on temperature, while division by the density corrects for the changing number of chains per unit volume with temperature variation. 

When constructing a master curve, one arbitrarily picks a reference temperature, T0. (Instead of T0, many authors choose to use Tr to signify the reference temperature.) The modulus at any time t, which one would observe at the temperature T0 in terms of the experimentally observed modulus values at different temperatures T, is therefore given as 

It should be clear that any temperature might have been chosen as the reference temperature. If r3 had been chosen, for example, only two of the shifts (T and T2), would have been to the left with shift factors larger than 1.0, while two of the shift factors (T4 and T5) would have been to the right with shift factors less than 1.0. In fact, one does not need to use one of the experimental temperatures as the reference temperature any value within the temperature range can be used simply by interpolation.5 

The creep rate under shear loading increases with both time and temperature. Similar curves are obtained for tensile creep and other fibre angles. Using a transformation equation, it was possible to superpose these curves to form a master curve. 

This method consists of taking creep data at high temperatures and short times and constructing a master curve. This could then be used to allow extrapolation to lower temperatures and longer times and hence give an indication of the creep response. It may only apply for certain load conditions and specific material combinations and needs to be validated. 

If creep is a design constraint then suitable component tests will need to be undertaken. BS 5480 recommends that materials should only be 

Sullivan, J.L. (1990) Creep and physical aging of composites. Composites Science and Technology, 39. 

There are many aspects to consider, which include type of loading, load duration, load introduction, loading rate, temperature and environment. The major influences are considered in turn with a suitable generic illustration. 

It has been shown throughout this chapter that the properties of plastics are dependent on time. In Chapter 1 the dependence of properties on temperature was also highlighted. The latter is more important for plastics than it would be for metals because even modest temperature changes below 100°C can have a significant effect on properties. Clearly it is not reasonable to expect creep curves and other physical property data to be available at all temperatures. If information is available over an appropriate range of temperatures then it may be possible to attempt some type of interpolation. For example, if creep curves are available at 20°C and 60°C whereas the service temperature is 40°C then a linear interpolation would provide acceptable design data. 

If creep curves are available at only one temperature then the situation is a little more difficult. It is known that properties such as modulus will decrease with temperature, but by how much Fortunately it is possible to use a time-temperature superposition approach as follows  

It was shown earlier that the variation of creep or relaxation moduli with time are as illustrated in Fig. 2.9. If we now introduce temperature as a variable then a series of such curves will be obtained as shown in Fig. 2.58. In general the relaxed and unrelaxed modulus terms are independent of temperature. The remainder of the moduli curves are essentially parallel and so this led to the thought that a shift factor, aj, could be applied to move from one curve to another. 

It may be seen from Fig. 2.59 that the two modulus curves for temperatures T1 and T 2 are separated by a uniform distance (log aj). Thus, if the material behaviour is known at Ti, in order to get the modulus at time, t, and temperature 

it would be necessary to use a time (t/ar) as shown in Fig. 2.59, in relation to the TI data. This means that 

Both the Rouse and Zimm models, as well as other molecular models to be discussed in Chapter 9, tacitly assume that the relaxation time associated with each mode has the same temperature dependence. Each mode s relaxation time is the product of temperature-independent factors and the 

First of all, the temperature dependence of all relaxation times is controlled by the ratio of friction coefficient and absolute temperature [see Eq. (8.15)]  

The temperature dependence of the modulus at any relaxation time r is proportional to the product of the polymer mass density p and absolute 

Viscosity is the product of relaxation time and the modulus at the relaxation time [Eq. (7.120)]. The temperature dependence of viscosity is proportional to the product of liquid density and friction coefficient  

The ratio of viscosity and density is the kinematic viscosity, which is directly measured in gravity-driven flows. The kinematic viscosity has the same temperature dependence as the friction coefficient. The density of polymer melts weakly decreases as temperature is raised, imparting a weak temperature dependence to the modulus at any relaxation time r. The temperature dependence of the viscosity of polymer melts is dominated by the dependence of the friction coefficient. Near the glass 

Viscoelastic functions depend on both temperature and time. For many polymers, the logarithmic plot of a viscoelastic function at the temperature T may be obtained from that at the temperature Tq by shifting the curve along the logarithmic time axis by the amount of log (T)- This procedure is called time-temperature superposition. The ability to superpose viscoelastic data is known as thermorheological simplicity. Thermorheological simplicity demands that all the molecular mechanisms involved in the relaxation process have the same temperature dependencies. 

Although empirical in nature, the WLF equation has a theoretical justification based on the free-volume idea, as described in Ferry (1980). At first, the WLF constants Cf and Cf were thought to be universal for all polymers. The universal 

If some experimental data for ut are available, one may use an arbitrary reference temperature Tq to replace Tg and rewrite the equation as 

If the material data follow the WLF equation, then a plot of — T — 7o)/log Uj-versus T — Tq) should be a straight Une, and Ci and C2 can be worked out from the slope and intercept. 

Dynamic mechanical analysis is an extremely powerful and widely used analytical tool, especially in research laboratories. In addition to measuring the temperature of the glass transition, it can be used to study the curing behavior of thermosetting polymers and to measure secondary transitions and damping peaks. These peaks can be related to phenomena such as the motion of side groups, effects related to crystal size, and different facets of multiphase systems such as miscibility of polymer blends and adhesion between components of a composite material [24]. Details of data interpretation are available in standard texts [1,2,25]. In the next section, we consider time-temperature superposition, which is another very useful apphcation of dynamic mechanical data. 

On a logarithmic plot, the storage modulus at reference temperature 7 and frequency ojg equals the storage modulus at temperature T and frequency a  

However, because the difference between log and log is a constant equal to loga, where Uj- is called the temperature shift factor, we have 

Because we can use linear viscoelastie theory (see Chapter 14 and Ref 1) to relate one viscoelastic fimction to another, the use of a limited amoimt of data along with the time-temperature superposition principle makes it possible to 

The impact strength of an article depends on the inherent molecular stmcture of the grade used and the morphology arising from the processing conditions. Changes in the 

One of the major reasons for the failure of PP artefacts is the brittle failure. This is mainly caused by the incorrect selection the PP grade, particularly the use of PP homopol5mier in place of copol5mier or use of wrong material at the moulding floor. Infrared microscopy and gel permeation chromatography can quickly identify the source of the problem. 

Williams, Landel, and Ferry (WFF) observed that if Tr is set to Tg, the variation of log flr with T — Tr is similar for a wide variety of polymers [10]. They rationalized this in terms of the molecular response, starting with Doolittle s equation [Eq. (41)] for the viscosity, where A and B are constants. f is the fractional free volume, equivalent to the unoccupied volume divided by the total volume of the polymer (the occupied volume includes that necessary to accommodate thermal vibrations). 

Equation (41) is based on the idea that the greater /(, the greater the molecular mobility (owing to reduced crowding), and the lower fj. For T Tg, /( is given by Eq. (42), where otf is the coefficient of thermal expansion of the fractional free volume and fg is the fractional free volume associated with the glass transition. 

If the T dependence of all the relaxation times is assumed to be that of fj in Eq. (41), the shift factor for the scaled viscoelastic functions is given by Eq. (43), and hence Eq. (44) follows. 

Equation (44) is the well-known WLF equation. Universal values of the various physical parameters in Eq. (44) lead to Cf = 17.44 and C = 51.26 K [10]. These are of the same order of magnitude as Cf and C obtained empirically (34 and 80 K for PMMA, for example), and indeed, time-temperature superposition has been found to work well for a wide range of single-phase polymers, with the proviso that it begins to break down for the relatively fast vibrational modes characteristic of the glassy state [12]. Moreover, although superposition may work for T Tg, at temperatures above about 7 + 50 K the shift factors tend to show an Arrhenius dependence rather than following the WLF equation. 

FIGURE 16.11 D3fnamic mechanical properties of poly(methyl methacrylate) [8]. The data were obtained with a torsion pendulum at about 1 cycle/s. 

One of the big challenges in developing materials that are expected to have a lifetime of years is developing laboratory tests to prove that the materials will last. One way to accelerate aging is to conduct tests at higher temperatures. Anyone who has ever wrestled 

FIGURE 16.12 Time-temperature superposition for NBS polyisobutylene. Adapted from Tobolsky and Catsiff [17]. 

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The time-temperature superpositioning principle was applied f to the maximum in dielectric loss factors measured on poly(vinyl acetate). Data collected at different temperatures were shifted to match at Tg = 28 C. The shift factors for the frequency (in hertz) at the maximum were found to obey the WLF equation in the following form log co + 6.9 = [ 19.6(T -28)]/[42 (T - 28)]. Estimate the fractional free volume at Tg and a. for the free volume from these data. Recalling from Chap. 3 that the loss factor for the mechanical properties occurs at cor = 1, estimate the relaxation time for poly(vinyl acetate) at 40 and 28.5 C. 

The successflil appHcation of time—temperature superposition (159) for polystyrene foam is particularly significant in that it allows prediction of long-term behavior from short-term measurements. This is of interest in building and constmction appHcations. 

Fig. 49. Illustration of the time—temperature superposition principle as based on stress—relaxation data for polyisobutylene (299,300). To convert Pa to...

Hamiltonian does not give rise to any crystalline order in the system. By employing models hke this, the quench-rate and chain-length dependence of the glass transition temperature, as well as time-temperature superposition, similar to experiments [23], were investigated in detail. 

Another important characteristic aspect of systems near the glass transition is the time-temperature superposition principle [23,34,45,46]. This simply means that suitably scaled data should all fall on one common curve independent of temperature, chain length, and time. Such generahzed functions which are, for example, known as generalized spin autocorrelation functions from spin glasses can also be defined from computer simulation of polymers. Typical quantities for instance are the autocorrelation function of the end-to-end distance or radius of gyration Rq of a polymer chain in a suitably normalized manner ... 

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. 

Since we are interested in this chapter in analyzing the T- and P-dependences of polymer viscoelasticity, our emphasis is on dielectric relaxation results. We focus on the means to extrapolate data measured at low strain rates and ambient pressures to higher rates and pressures. The usual practice is to invoke the time-temperature superposition principle with a similar approach for extrapolation to elevated pressures [22]. The limitations of conventional t-T superpositioning will be discussed. A newly developed thermodynamic scaling procedure, based on consideration of the intermolecular repulsive potential, is presented. Applications and limitations of this scaling procedure are described. 

Time-temperature superposition [10] increases the accessible frequency window of the linear viscoelastic experiments. It applies to stable material states where the extent of reaction is fixed ( stopped samples ). Winter and Chambon [6] and Izuka et al. [121] showed that the relaxation exponent n is independent of temperature and that the front factor (gel stiffness) shifts with temperature... 

Time-temperature superposition at the gel point does not let us distinguish between the vertical and the horizontal shift, since the spectra are given by... 

Time-temperature superposition. Because of the relatively strong relaxations in the frequency range at room temperature (300 K), oscillation measurements were also performed at 345, 390 and 435 K in addition the D networks were measured at 265 K. 

The terminal region of EPDM networks can be extended by time-temperature superposition. 

To get accurate distributions of relaxation or retardation times, the expetimcntal data should cover about 10 or 15 decades of time. It is impossible to get experimental data covering such a great range of times at one temperature from a single type of experiment, such as creep or stress relaxation-t Therefore, master curves (discussed later) have been developed that cover the required time scales by combining data at different temperatures through the use of time-temperature superposition principles. 

There are two superposition principles that are important in the theory of Viscoelasticity. The first of these is the Boltzmann superposition principle, which describes the response of a material to different loading histories (22). The second is the time-temperature superposition principle or WLF (Williams, Landel, and Ferry) equation, which describes the effect of temperature on the time scale of the response. 

Figure 8 WLF time-temperature superposition applied to stress-relaxation data obtained at several temperatures to obtain a master curve. The master curve, made by shifting the data along the horizontal axis by amounts shown in the insert for r> is shown with circles on a line.

Time is the major (actor in determining the mechanical properties of a polymer. This is seen directly in creep and stress-relaxation experiments. These tests cover long periods of time, so that they are sensitive to the types of molecular motions that require long times. Tfrey give little direct information on the types of molecular motion that take place at short times. However, by using the time-temperature superposition principle and the WLF equations, access to these short times can be achieved even though they may not easily be attainable by direct experimentation. 

Time of flight (TOF), 75 660-661 Time-of-flight (ToF) mass analyzers, 24 109 Time of flight diffraction (TOFD), 79 486 Time-of-flight instrumentation, in particle counting, 78 150—151 Time-of-flight-SIMS technique, 24 109 Time-resolved fluorimetry, 74 148-149 Time-resolved spectra, analysis of, 74 613 Time standards, 75 749—750 Time-temperature parameters (TTP), 73 471, 478, 479 creep properties and, 73 480 Time-temperature superposition, 27 746-747... 

In the interval between 198 K and 253 K, the form of the structural relaxation does not change114 as is evidenced by the success of the time-temperature superposition shown in Figure 21. One can also see from this figure that an additional regime intervenes between the short-time dynamics (first 10% of the decay at the lowest temperatures) and the structural relaxation (last 80% of the decay). We will identify this regime as the MCT (3-regime... 

Figure 5 Typical velocity relationship of kinetic friction for a sliding contact in which friction is from adsorbed layers confined between two incommensurate walls. The kinetic friction F is normalized by the static friction Fs. At extremely small velocities v, the confined layer is close to thermal equilibrium and, consequently, F is linear in v, as to be expected from linear response theory. In an intermediate velocity regime, the velocity dependence of F is logarithmic. Instabilities or pops of the atoms can be thermally activated. At large velocities, the surface moves too quickly for thermal effects to play a role. Time-temperature superposition could be applied. All data were scaled to one reference temperature. Reprinted with permission from Ref. 25. 

Time-temperature superposition is frequently applied to the creep of thermoplastics. As mentioned above, a simple power law equation has proved to be useful in the modelling of the creep of thermoplastics. However, for many polymers the early stages of creep are associated with a physical relaxation process in which the compliance (D t)) changes progressively from a lower limit (Du) to an upper limit (DR). The rate of change in compliance is related to a characteristic relaxation time (x) by the equation ... 

Time-temperature superposition is performed in the same empirical manner as for creep. 

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