Vogel equation

The theory predicts an iso-free-volume state at T , which is equal to T0 of the Vogel equation, but an examination of the calculated values of /g does not indicate that Tg is an iso-free-volume state. 

T being the experimental temperature and To the reference temperature to which the data is shifl d. The WLF may be reduced to the Vogel equation which describes the viscosity of molten glasses and supercooled liquids ... 

An alternative description such as the Vogel equation introduces a temperature =Tq - C2 which is a constant for a given high polymer species. 

Evidence for the failure of the three-parameter Vogel equation to represent data on rj as a function of T covering a wide range in these variables is provided by data on 1,3.5 tri-(a-naphthyl)-benzene (780a) and on glucose (170). The data on the former cannot be correlated over the entire temperature interval studied by the Vogel relation Eq. (2.4) with constant a and To. nor by a modified Vogel equation with A oc exp (EjR T) with E positive and less than about 10 kcal/mol (P, 76, 77, 707). The... 

Fig. 5. Log r] versus F — Fg for l,3,5-tri( -naphthyl) benzene o polystyrene, Z = 375, and polystyrene, = 10.5, . The light line continuing below the data points for F — Fg > 80 is an extension of the Vogel equation calculated with parameters and F found for tri(ot-naphthyl) benzene for F— Fg < 50...

A revised Vogel equation with new parameters A, a, and To can be used to fit the versus T data in Fig. 5 at higher temperature, but values of a and Tq so obtained are larger than those observed near Tg. Similar observations have been reported by Barlow, Lamb, and Matheson 29) from studies on some supercooled liquids over the approximate range 50° [Pg.277]

It is useful to display the two equivalent forms of the Vogel equation employed previously, together with the analogous equation suggested by the free volume concept. Thus, we have the relations... 

It is apparent that the free volume concept leads to a relation identical to the Vogel equation. This does not insure, of course, that the parameters B an so defined need have any fundamental significance, but it does provide at least a plausible insight into the nature of the rj T) dependence. 

In the following, we shall attempt to obtain better estimate for oc, and thus for B, and point out the experimental difficulties inherent in such an endeavor. The estimates for B and fg so obtained wiU be examined for any discernible dependence on molecular structure. Similar studies of the Vogel equation for isolated systems have been undertaken by Williams (223) and more recently by Miller 153, 154, 155, 156). 

The second calculation is more involved. The only values of Co listed in Table 2 that are relatively free of error in setting W = 0 are those for poly(dimethyl siloxane) and poly(tetramethyl-p-silphenylene silo-xane) since for these polymers Tq 0 and the modified Vogel expression reduces to the simpler Vogel equation. Unfortunately, these materials are also studied far from their T, a and T,... 

This result is known as the Vogel equation. (See also Problem 4-6.) For polystyrene, the Vogel temperature T2 is expected to be around 50 °C... 

M=Na, Li) complexes also gives rise polyiodide containing conductors. Conductivity results for MEEP based ionic conductors have been fit effective by the Bendler-Schesinger form of the Vogel equation which indicates that the defect-diffusion approach is a viable model for the observed... 

Thermally Stimulated Current (TSC) studies allow us to investigate the transition spectra of amorphous polymers. The relaxation modes observed around and above the glass transition T have common features (1) The TSC peak isolated around Tg corresponds to a distribution of relaxation times following an Arrhenius equation. The width of the distribution characterizes the distribution of the order parameter. (2) The TSC peak observed some 50° above Tg is well described by a Fiilcher-Vogel equation. This mode, which can also be distributed, has been associated with the dielectric manifestation of the liquid-liquid transition (Ty). 

Table I. Comparison of the parameteis of the Fulcher-Vogel equation with the empirical WLF values. 

Above the glass franjifion, a well-resolved TSC peak is observed at T (see Fig. 6) for samples with > M. The analysis of these TSC peaks shows that they are characterized by relaxation tiiiies following a Fulcher-Vogel equation, eq. (7). The corresponding parameters are listed in Table III,... 

Table III. Temperature position (t-.ii) and parameters of the Fulcher-Vogel equation for the TSC peaks associated with the liquid-liquid transition. 

Figure 13 shows the relaxation map of oriented polystyrene. Hie elementary processes isolated between 9VC and 102°C are characterized by relaxation times following a compensation law, eq. (8) they have the same relaxation time (t. = 0.11 s) at the compensation temperature = 145°C. This mode is the dielectric manifestation of the glass transition. The elementary peak isolated at 13S°C is well-described by a Fulcher-Vogel equation, eq. (7), with a critical temperature Too = 50"C, a thermal expansion coefficient of the free volume of 1.7 x lO" and a pre-exponential factor of 0.74 s. 

The long time mode corresponds to the liquid-liquid transition. The relaxation time follows a Fulcher-Vogel equation, the mobility is frozen at a critical temperature, T. Such behavior is characteristic of the vitreous state since it has also been observed in inorganic glasses and even in spin glasses. The departure of from Tg is given by the empirical WLF value in polymers like polystyrene it may be very different in odier polymers like poly(cyclohexyl methacrylate). This departure is also dependent upon the thermodynamic history of the polymer. 

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