Williams-Landel-Ferry equation glass transition

The most popular and widely used formula for the function aT(T) is the Williams-Landel-Ferry equation, which is quite adequate for amorphous polymers above the glass transition temperature ... 

An alternative approach to describe nucleation from the amorphous state utilizes the glass transition temperature (Tg) concept (Williams et al. 1955 Slade and Levine 1991). Based on this approach, molecular mobility below Tg is sufficiently limited to kinetically impede nucleation for very long times. Amorphous systems, at temperatures above Tg, nucleate at a rate depending on the temperature difference above Tg. Williams et al. (1955) suggested that the rate of nucleation increases rapidly at temperatures just above Tg according to a kinetic expression given by the WLF (Williams-Landel-Ferry) equation. 

Empirical Relationship - Empirical relationships correlating glass transition temperature of an amorphous viscoelastic material with measurement temperature and frequency, such as the William Landel Ferry equation (17) and the form of Arrhenius equation as discussed, assume an affine relationship between stress and strain, at least for small deformations. These relationships cover finite but small strains but do not include zero strain, as is the case for the static methods such as differential scanning calorimetry. However, an infinitely small strain can be assumed in order to extend these relationships to cover the glass transition temperature determined by the static methods (DSC, DTA, dilatometry). Such a correlation which uses a form of the Arrhenius equation was suggested by W. Sichina of DuPont (18). 

The free-volume parameters are again obtained by fitting viscosity versus temperature data using either the adopted Doohttle expression (low-molecular-weight species) or the Williams-Landel-Ferry equation (polymers). The glass transition temperature, Tg, is as reported in the literature or can be estimated from the melting temperature. 

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) ... 

Another important result is the similarity of the temperature variation of the correlation time r, associated with conformational jumps, and observed for all the polymers considered except polyisobutylene, to the predictions of the Williams-Landel-Ferry equation for viscoelastic relaxation, which indicates that the segmental motions observed by NMR belong to the glass-transition phenomenon. Moreover, the frequency of these intramolecular motions is mainly controlled by the monomeric friction coefficient of the polymer matrix. 

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 ... 

At temperatures T > (melting temperature), the dependence of viscosity on temperature is controlled by the Arrhenius equation. In most materi als, in the temperature range from to (glass transition temperature), the temperature decrease results in an increase of activation energy ( ), which relates to the fact that molecules do not move as individuals, but in a coordinated maimer. At T > Tg, viscosity is satisfactorily described by the so called VTF (Vogel Fulcher Tammany) equation ijj. = A.exp D.Tq/(T Tq) or WLF (Williams—Landel—Ferry) equation Oj. = exp [Cjg.(T—Tg)]/[C2g (T-Tg)], where ijj, = viscosity at temperature T, j. = ratio of viscosities at T and Tg, or the ratio of relaxation times r and tg at temperatures T and Tg and A, D, Tg, Cjg and are constants. Parameters and are considered universal... 

Differences in the rheological response of the acid and ester copolymers can be successfully interpreted on the basis of differences in their glass transition temperatures. This is done using a WLF (Williams-Landel-Ferry) equation... 

In Ngai and Plazek s publications, they determined that numerous sets of data could be reconciled with the well known Williams-Landel-Ferry Equation. If the glass transition temperature (Tg) was utilized as the reference temperature, then ... 

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 Arrhenius equation holds for many solutions and for polymer melts well above their glass-transition temperatures. For polymers closer to their T and for concentrated polymer and oligomer solutions, the Williams-Landel-Ferry (WLF) equation (24) works better (25,26). With a proper choice of reference temperature T, the ratio of the viscosity to the viscosity at the reference temperature can be expressed as a single universal equation (eq. 8) ... 

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. 

As the segmental dynamics probed by relaxation measurements of P(4HB) and the 3HB and 4HB units in two P(3HB-eo-4HB) samples followed the Williams-Landel-Ferry empirical equation [86], they are considered to be involved in the glass-transition phenomena and so the decrease in Tg values... 

The same free volume also occurs in the Williams-Landel-Ferry dynamic glass-transition-temperature equation (see Section 10.5.2). The free-volume fraction is independent of the polymer type. It has a value of about 2.5% (Table 5-7). 

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]. 

In this equation, a is the conductivity, A is a constant proportional to the number of carrier ions, B is a constant, and To is the temperature at which the configurational entropy of the polymer becomes zero and is close to the glass transition temperature (Tg). The VTF equation fits conductivity rather well over a broad temperature range extending from Tg to about Tg +100 K. Equation [3.2] is an adaptation of the William-Landel-Ferry WLF relationship developed to explain the temperature dependence of such polymer properties as viscosity, dielectric relaxation time and magnetic relaxation rate. The fact that this equation can be applied to conductivity implies that, as with these other properties, ionic... 

In the above description of local motions, characterizes the segmental modes. In order to know whether these segmental motions observed by NMR in bulk at temperatures well above the glass-transition temperature belong to the glass-transition processes, it is of interest to compare the variations of Ti as a function of temperature with the predictions of the Williams-Landel-Ferry (WLF) equation [19]. The WLF equation describes the frequency dependence of the motional processes associated with the glass-transition phenomena. It can be written as [20]... 

The temperature dependence of the relaxation time (r) of polymers in the glass transition region cannot be described by the Arrhenius equation as the In r versus 1/T plot is not linear. This means that the motional activation energy is not a constant but a function of temperature. In this situation, the temperature dependence of the relaxation time can be well described by the William-Landel-Ferry (WLF) equation as follows ... 

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 ... 

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