Equations Doolittle

L. L. Blyler and T. K. Kwei [39] proposed the direct opposite (to 4). In their reasoning, they proceeded from the known and generally acceptable Doolittle equation, which puts liquid viscosity in exponential dependence on the inverse value of the free volume of the latter. According to [39], gas has a volume of its own, the value of which it contributes to the free volume of the polymer when it dissolves therein as a result, viscosity falls. The theoretical formula obtained by the authors was experimentally confirmed in the same work. The authors measured pressure values at the entrance of cylindrical capillaries, through which melts of both pure polyethylene, and polyethylene with gas dissolved in it, extruded at a constant rate. 

According to free-volume interpretations, the rate of molecular motions is governed entirely by the available unoccupied space ( free volume ). Early studies of molecular liquids led to the Doolittle equation, relating the viscosity to the fractional free volume, / [23,24]... 

Combination of this equation with the Doolittle equation (8.13) yields... 

The above equation suggests that the Doolittle equation (13) has to be modified to include the nonequilibrium contribution. When 5 = 0, Equation 4 can be written in the form of the WLF equation which is known to be valid for T > Tg. In order to... 

As an example of the concentration dependence of viscoelastic properties in Fig. 16.11 the shear creep compliance of poly(vinyl acetate) is plotted vs. time for solutions of poly(vinyl acetate) in diethyl phthalate with indicated volume fractions of polymer, reduced to 40 °C with the aid of the time temperature superposition principle (Oyanagi and Ferry, 1966). From this figure it becomes clear that the curves are parallel. We may conclude that the various may be shifted over the time axis to one curve, e.g. to the curve for pure polymer. In general it appears that viscoelastic properties measured at various concentrations may be reduced to one single curve at one concentration with the aid of a time-concentration superposition principle, which resembles the time-temperature superposition principle (see, e.g. Ferry, General references, 1980, Chap. 17). The Doolittle equation reads for this reduction ... 

Dissociation energy of the weakest bond, 764 Dissolution of polymers, 696 Distortional waves, 506 Distribution of molecular mass, 17 Ditonic system, 66 Doolittle equation, 621 n-doping, 338 p-doping, 338... 

Several well-known equations are available for interpreting the temperature dependence of viscosity, diffusion coefficient, and other relaxation rates for T > Tg. The Doolittle equation [18], the WLF equation [19], the Vogel-Fulcher equation [20], and the Adam-Gibbs equation [21] can be expressed in the same form. They are known to fit well with the relaxation data of liquids in equilibrium. The universal functional form is [20]... 

We discussed the nature of glass transition only qualitatively in the section on thermal properties (Chapter 10). We did, however, mention a couple of essentially empirical equations that describe the viscosity of a fluid. One such is the Doolittle equation, which we rewrite here in a somewhat different form (Equation 13-103) ... 

B. Obtain an equation for the shift factor, av (Equation 13-106 in the text) from the Doolittle equation. 

Since the lower the free volume the larger the relaxation time, the effect of the pressure on x can be interpreted in terms of the Doolittle equation. Accordingly,... 

The Doolittle equation [Eq. (8.130)] can be combined with the assumed linear temperature dependence of free volume [Eq. (8.131)] to get the WLF equation, so-named for Williams, Landel, and Ferry, who first applied it to polymer melts in 1955 ... 

Equation (5) was introduced as a simple consequence of the Doolittle equation, which is the basic artifact of free volume models for the glass transition." However, its final form with the volume related fragility strength coefficient Dy was only recently proposed. Eq. (6) was postulated ad hoc, by analogy to eq. (4), since the pressurization was coincided with densification. °... 

With this result in hand, we may now return to the theoretical rationalization of the form of the WLF equation. The starting point is the semiempirical Doolittle equation for the viscosity of a liquid... 

Open-ended) Investigate the applicability of the Doolittle equation to a simple fluid with the objective of showing that temperature per se has no influence on viscosity. To approach this problem, find high-accuracy viscosity and specific-volume data in, for example, the Handbook of Chemistry and Physics. Compare these data with the predictions of the Doolittle equation, carefully noting any systematic discrepancies. 

Turnbull and the Doolittle equations, both of which are essentially free-... 

Figure 4-19. Variation of (P) with pressure. The points o represent experimental values of (P) for cis-decahydronaphthalene. Points are experimental values for n-pentadecane. Solid curves are calculated from the modified Doolittle equation, dashed curves from the modified Cohen-Turnbull equation. Data by Hogenboom, Webb and Dixon [151.

Fig. I. Sketch of the logarithm of viscosity tj (in poise) with reciprocal temperature (when the liquid is cooled from the liquid to the glassy state). Curve a corresponds to Arrhenius behavior, 7 ->0. Curves b and c show the typical form for simple molecular glass formers. Curves b and c correspond to the Doolittle equation, where the free volume ty oc T— Tqh 0 at the high temperature and Vj cc T— at low temperatures. In curve 7 0, and in...

The original work on the free-volume model by Cohen and Turnbull - showed that the fluidity obeyed the Doolittle equation (2.8). We show that the percolation ideas developed in this paper give rise to the same equation for the fluidity. 

In fitting the experimental data for tj to the Doolittle equation, Vj is successfully approximated as proportional to for the high-temperature,... 

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