Doolittle viscosity equation

The Doolittle viscosity equation (Doolittle 1951) was the pioneering equation for most free-volume theories... 

The WLF equation was originally based purely on empirical observations. It can, however, be derived from free volume theory starting from the empirical Doolittle viscosity equation ... 

Simple free-volume theories such as Doolittle s equation (Doolittle, 1951) suggest that the viscosity of liquids varies with the exponential of the fractional free volume. Viscoelastic scaling theories based on the free-volume... 

This explains the success of the Doolittle empirical equation for the viscosity of low molar mass hydrocarbon liquids ... 

At the time of development of free volume theory, two important empirical equations of viscosity were known. They are the Doolittle (1951) equation (3.01) and the Vogel, Tamman and Fulcher (VTF) equation (3.02) (Vogel, 1921, Fulcher, 1923, Tammann and Hesse, 1926), which are given below. 

Doolittle, A. K., and Doolittle, D. B., Newtonian flow V. Further verification of the free-space viscosity equation, J. Appl. Phys., 28,901-905 (1957). 

The free volume is related to the polymer viscosity rj according to Doolittle empirical equation (Doolittle 1951), as given by... 

At this juncture, we introduce Doolittle s viscosity equation [18], based on experimental data for monomeric hquids, that relates the viscosity to the free volume through... 

The Doolittle-type equation of viscosity tj is derived by replacing the activation energy E of translation of a segment by (1 /3)// and it follows that... 

Williams, Landel, and Ferry (WFF) observed that if Tr is set to Tg, the variation of log flr with T — Tr is similar for a wide variety of polymers [10]. They rationalized this in terms of the molecular response, starting with Doolittle s equation [Eq. (41)] for the viscosity, where A and B are constants. f is the fractional free volume, equivalent to the unoccupied volume divided by the total volume of the polymer (the occupied volume includes that necessary to accommodate thermal vibrations). 

Fig. 7.6 Specific or relative volume vs. temperature for an amorphous polymer. Doolittle s equation for the viscosity of a liquid is,...

The following relationship between viscosity of a polymer melt and its free volume can be written on the lines of Doolittle s equation [80] as... 

We note that the Doolittle/WLF equations predict a rapid increase of viscosity with decreasing free volume (or as is approached). Another empirical equation used to describe this is the... 

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]... 

The three most important factors in the equation are the viscosity and the thermodynamic parameters G and Gm- The viscosity can be approximated between the liquidus temperature, Tuq, and the liquid-+glass transition temperature, Tg, by a Doolittle expression involving the relative free volume (Ramachandrarao et al. 1977) while G can be calculated using the relationship... 

Investigating the viscosity of a homological series of liquid normal paraffins, Doolittle [84] pointed out that the direct relationship between viscosity ( resistance to flow ) and free volume ( relative volume of molecules per unit free space ) is an intuitive hypothesis and the experimental dependence is described better by a logarithmic equation... 

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) ... 

The free volume theory of glass transition is based on Doolittle s empirical assumption (29), which states that the viscosity, q, at T > Tg is related to the free volume fraction by the equation... 

Early work of Doolittle [38] on the viscosity (77) of nonassociated pure liquids such as n-alkanes led to an equation of the form... 

Here, three unknowns and two equations are shown, which can be solved by assuming for the constant B a value of unity [41], consistent with the viscosity data of Doolittle. Then fg = 0.025, and a/ = 4.8x10 °K . Sharma et al [37] found a/ = 3.2x10° °K . ... 

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. 

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