William-Landel-Ferry shift factor

William-Landel-Ferry shift factor E Young s modulus... 

Several attempts have been made to superimpose creep and stress-relaxation data obtained at different temperatures on styrcne-butadiene-styrene block polymers. Shen and Kaelble (258) found that Williams-Landel-Ferry (WLF) (27) shift factors held around each of the glass transition temperatures of the polystyrene and the poly butadiene, but at intermediate temperatures a different type of shift factor had to be used to make a master curve. However, on very similar block polymers, Lim et ai. (25 )) found that a WLF shift factor held only below 15°C in the region between the glass transitions, and at higher temperatures an Arrhenius type of shift factor held. The reason for this difference in the shift factors is not known. Master curves have been made from creep and stress-relaxation data on partially miscible graft polymers of poly(ethyl acrylate) and poly(mcthyl methacrylate) (260). WLF shift factors held approximately, but the master curves covered 20 to 25 decades of time rather than the 10 to 15 decades for normal one-phase polymers. 

For transport in amorphous systems, the temperature dependence of a number of relaxation and transport processes in the vicinity of the glass transition temperature can be described by the Williams-Landel-Ferry (WLF) equation (Williams, Landel and Ferry, 1955). This relationship was originally derived by fitting observed data for a number of different liquid systems. It expresses a characteristic property, e.g. reciprocal dielectric relaxation time, magnetic resonance relaxation rate, in terms of shift factors, aj, which are the ratios of any mechanical relaxation process at temperature T, to its value at a reference temperature 7, and is defined by... 

The shift factor is modeled either as a modified Williams-Landel-Ferry (WLF) equation, or as a best fit to the general form of the Equation [20-25]... 

For amorphous polymers which melt above their glass transition temperature Tg, the WLF equation (according to Williams, Landel, Ferry, Eq. 3.15) with two material-specific parameters q and c2 gives a better description for the shift factors aT than the Arrhenius function according to Eq. 3.14. 

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) ... 

T > To are shifted to longer times, and measurements for T < Tq aie shifted to shorter times. A well-defined reduced curve means the viscoelastic response is thermorheologically simple (Schwarzl and Staverman, 1952). It represents log Jp(t) at To over an extended time range. The time scale shift factors aj that were used in the reduction of the creep compliance curves to obtain the reduced curve constitute the temperature dependence, ar is fitted to an analytical form, which is often chosen to be the Williams-Landel-Ferry (WLF) equation (Ferry, 1980),... 

The fourth step is to plot the shift factors Aj against temperature (Figure 2.15c). This representation of the time-temperature superposition characteristic of viscoelastic materials has been extensively analysed with the well-known Williams-Landel-Ferry (WLF) relationship, at temperatures above T ... 

Using 25°C as our reference temperature again, a William Landel Ferry (WLF) equation is fitted to our shift factors as... 

In Figure 12.4 the shift factor log Uj has been plotted vs. temperature. There are two horizontal scales, since the first corresponds to 20°C and the second to the glass transition temperature of the PET-rich phase in the PLC, that is Tg = 62 C. Incidentally, the Tg evident in Figure 12.4 agrees well with values obtained by several other techniques [13,39]. The broken line in the figure has been calculated from the Williams-Landel-Ferry (WLF) equation. The large deviation from experimental values (circles) was expected, since Ferry [37] states that their equation works well around Tg + 50 K, while here an attempt was made to use it below Tg. There are also other problems with the WLF equation, as discussed by Brostow in Chapter 10 of reference [38]. 

Williams-Landel-Ferry equation that relates the value of the shift factor, ax (associated with time-temperature superposition of viscoelastic data), required to bring log-modulus (or log-compliance) vs. time or frequency curves measured at different temperatures onto a master curve at a particular reference temperature. To, usually taken at 50 °C above the glass transition temperature (To = Tg + 50 °C) ... 

The master curve in the form of stiffness versus frequency can be created by fitting the experimentally determined shift factors to a mathematical model. With a multifrequency measurement, frequencies beyond the measurable range of the DMA can be achieved by using the superposition method based on the Williams-Landel-Ferry (WLF) equation [60, 61]. For a temperature range above the T, it is generally... 

To calculate the time-temperature superposition shift factor by using the WLF (Williams-Landel-Ferry) equation for polymers at temperatures less than 100°C (232 F) above their Tg [2], use... 

Once the shift factor has been determined for a large enough number of curves, the shift factors themselves may be fitted to a model. This allows the determination of shift factors and master curves at arbitrary temperatures. For viscosity curves taken within 100°C of a material s glass transition temperature (Tg), the WLF (Williams-Landel-Ferry) equation [17] is used ... 

Vogel temperature) for the Williams-Landel-Ferry (WLF) shift factor where... 

The ratio of characteristic times at two temperatures T> To, aT=r(T)lT(To), (shift factor) consequently follows the Williams-Landel-Ferry (WLF) equation... 

In this case, an apparent activation energy is determined, and it has higher values than secondary relaxations 100-300 kJ/mol for urethane-soybean oil networks (Cristea et al. 2013), 200-300 kJ/mol for polyurethane-epoxy interpenetrating polymer networks (Cristea et al. 2009), more than 400 kJ/mol for semicrystalline poly(ethylene terephtalate) (Cristea et al. 2010), and more than 600 kJ/mol for polyimides (Cristea et al. 2008, 2011). The glass transition temperature is the most appropriate reference temperature when applying the time-temperature correspondence in a multifrequency experiment. The procedure allows estimation of the viscoelastic behavior of a polymer in time, in certain conditions, and is based on the fact that the viscoelastic properties at a certain tanperature can be shifted along the frequency scale to obtain the variation on an extended time scale (Brostow 2007 Williams et al. 1955). The shift factor is described by the Williams-Landell-Ferry (WLF) equation ... 

Typical behavior is shown in Fig. 3.12(a), where the storage modulus in the plateau and terminal regions for a commercial polystyrene melt is plotted against frequency at several temperatures [17]. A reference temperature is selected, in this case To = 160 °C, and best-fit scale factors for data obtained at other temperatures are determined empirically to form Fig. 3.12(b). The timescale can shift very rapidly, as indicated by the plot of ar versus T in Fig. 3.13 [17]. The Williams-Landel-Ferry (WLF) equation, introduced in Chapter 2 and shown for this particular sample and choice of reference temperature in Fig. 3.13, describes rather well the temperature dependence of aj for most polymer melts and concentrated solutions. 

To) (where is the empirical shift factor and is the reference temperature to which the data are shifted), shown in the inset, implies that the results are consistent with the Williams-Landel-Ferry (WLF) equation, known to describe viscoelastic behavior in many amorphous polymers (14) ... 

There are two common ways of representing the temperature dependence of the shift factor. One is as an exponential in reciprocal absolute temperature [exp( /Rr)], as done with the polystyrene data this functionality is usually vahd at temperatures well above the glass transition temperature, which is 104 °C for the polystyrene. The polystyrene data are fit with E/R = 9,300 K a value of order 4,000 K is typical of polyethylene, reflecting the wide variation in this parameter. The other commonly used functionahty is the Williams-Landel-Ferry (WLF) equation, which is written... 

The free volume concept has been successfully used for describing the glass transition phenomenon and the Pick s law of diffusion tor polymers. One of such successful examples is the Williams-Landel-Ferry(WLF) equation [72,73], which provides the relationship between the time-temperature superposition shift factor and temperature. The free volume is considered to be the holes resulting from the packing void or irregularity of polymeric molecules. This concept will be continuously used in this section for deriving the viscosity equation ol both polymer melts and polymer solutions. 

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. 

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