Universal Williams-Landel-Ferry

The free volume theories state that the glass transition is characterized by an iso-free volume state, i.e. they consider that the glass temperature is the temperature at which the polymers have a certain universal free volume. The starting point of the theory is that the internal mobility of the system expressed as viscosity is related to the fractional free volume. This empirical relationship is referred to as the Doolittle equation. It is a consequence of the universal William-Landel-Ferry (WLF) equation and the Doolittle equation that the glass transition is indeed an iso-free volume state. The WLF equation, expressed in general terms, is ... 

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) ... 

A number of experimental methods exist that allow polymer solutions to be subjected to different shear rates or to oscillatory shear. Data obtained over a given range of shear rate, or frequency, are shifted to form a universal curve (as in the use of the Williams-Landel-Ferry equation, explained in Chapter 4). This can then be compared with the predictions of various models such as those proposed by Rouse or Zimm. The former assumes that there is minimal interaction between the solvent and the polymer, and is sometimes referred to as the free draining model. In reality, there is some interaction between the solvent and the polymer chain. This is addressed in the Zimm model, where the drag introduced by the solvent influences the motion of the chains. 

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... 

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. 

The fractional free volume f, which is the ratio of the free volume to the overall volume, occupies a central position in tr5nng to understand the molecular origins of the temperature dependence of viscoelastic response. The main assumption of the free-volume theory is that the fractional free volume assumes some universal value at the glass transition temperature. The Williams-Landel-Ferry (WLF) equation for the thermal dependence of the viscosity tj of polymer melts is an outgrowth of the kinetic theories based on the free volume and Eyring rate theory (35). It describes the temperature dependence of relaxation times in polymers and other glass-forming liquids above Tg (33-35). The ratio of a mechanical or dielectric relaxation time, Tm or ra, at a temperature T to its value at an arbitrary reference temperature To can be represented by a simple empirical, nearly universal function. 

However, because measurements are kinetically determined, this is a less accurate form of the equation. Very often it is observed that the measured shift factors, defined for different properties, are independent of the measured property. In addition, if for every polymer system, a different reference temperature is chosen, and ap is expressed as a function of T — rj, then ap turns out to be nearly universal for all polymers. Williams, Landel and Ferry believed that the universality of the shift factor was due to a dependence of relaxation rates on free volume. Although the relationship has no free volume basis, the constants and may be given significance in terms of free volume theory (Ratner, 1987). Measurements of shift factors have been carried out on crosslinked polymer electrolyte networks by measuring mechanical loss tangents (Cheradame and Le Nest, 1987). Fig. 6.3 shows values of log ap for... 

Williams, Landel and Ferry introduce their famous WLF-equation for describing the temperature dependence of relaxation times as a universal function of T and Tg... 

Equation (8.38), empirically formulated by Williams, Landel, and Ferry in the 1950s, is known as the WLF equation (15). Examples of the variation of Qqt with temperature are shown in Figure 8.15. The plots of T — Tq)/ (In flor) against T — Tq are straight lines whose slopes and intercepts are — l/Cj and —C2/C1, respectively. Though an analysis of limited data led to the postulation that and C2 were universal constants at Tg, this assumption was not supported when the results obtained for a wide variety of viscoelastic materials were considered. 

Williams, Landel, and Ferry (1955), who considered the a-relaxation first from a more empirical point of view by testing for universality of behavior over many amorphous polymers, determined that the best choices for Ci and C2 are... 

The free volume at reaches a universal value, which according to Fox and Flory (127) has been estimated to be about 0.02, and based on viscosity measurements by Williams, Landel and Ferry 131) is equal to about 0.025. Thus from Eq. (1), e /R 7 — In (0.02) = 4, a value which is useful in estimating e. Assuming that at 7 the holes are as good as frozen in. a, and are equal to the difference of their respective values just above and below 7 that is da, AP and AC, quantities which are experimentally accessible. The hole volume, V. can be... 

Williams, Landel, and Ferry found that C and Ci were similar for many amorphous polymers with Q = 17.44 and C2 = 51.6 in the temperature range between Tg and Tg -i-100 °C. The equation is referred to as the universal WLF equation when Q and C2 assume these values. While this equation is not truly universal, it was developed from a large database for various polymers. When the equation is written in this form, it is clear that Tg serves as a corresponding state for viscoelastic behavior. A plot of log aj versus (T - T) for the data of Fig. 5.15 is shown in Fig. 5.17 here 7] is the inflection point in the modulus curve at Tg. Each DMA vendor has software available that automates the TTS operations of curve shifting, determining values of aj, and fitting the data to the WLF equation. 

Williams, Landel and Ferry (1955) have given an empirical expression for a T)y applicable to a wide variety of polymeric materials, for temperatures not too different from the glass transition temperature Tg of the polymer. Their expression contains material-dependent parameters. In its simplest form, there is only one such parameter, the glass transition temperature Tg. This approximate universal expression is given by... 

In fact, the same increase in joint strength that is obtained with a simple viscoelastic adhesive on increasing the rate of debonding, can be achieved by a suitable reduction in test temperature. This is referred to as the principle of rate-temperature equivalence. For amorphous glass-forming liquids above their glass transition temperature Tg, Williams, Landel, and Ferry (WLF) proposed a universal relationship for the ratio of corresponding test rates at temperatures Tand Tgi ... 

Figure 12.14. Universal curve giving the variation of ax versus (T-TJ (in C), according to Williams, Landel and Ferry.

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