Doolittle empirical equation

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

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

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) ... 

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. 

The values obtained are still much higher than the estimates from the Will-iams-Landel-Ferry (WLF) equation. This is an empirical equation, but it ean also be derived from free-volume considerations by starting with a description of the viseosity of the system, based on an empirical equation proposed by Doolittle to describe the temperature dependence of viscous flow... 

The appearance of the glass transition results from the reduction of molecular mobility as the temperature falls, slowing the collapse of free volume. We now introduce the concept that the mobility at any temperature depends primarily on the free volume remaining, so that the rates of both bulk and shear deformations can be advantageously expressed in terms of Vf rather than T as the independent variable. This principle was applied long ago to the shear viscosity of simple liquids by Batchinski, and more recently by Doolittle with an empirical equation which was found to represent with high accuracy viscosities of ordinary liquids of low... 

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 ... 

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... 

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... 

The constants correspond to coo = C, B = yVflEf, and Tq = T o (7b is the Vogel temperature). With Cohen and Turnbull delivered this free-volume model, a theoretical justification of the empirical VFTH equation and the equivalent Williams-Landel-Ferry (WLF) [Williams et al., 1955] equation as well as of the empirical free-volume models of viscosity [Fox and Flory, 1950 Doolittle, 1951]. 

Because SCFs typically have lower density and higher compressibility than a pure polymer melt, dissolution of the SCF into the polymer melt results in swelling of the polymer. This in turn leads to an increase in free volume of the mixture, so transport properties such as viscosity and diffusion coefficient can be significantly enhanced. The semi-empirical Doolittle equation [128,129] predicts that the zero-shear rate viscosity, of a polymer is exponentially related to the fractional free volume,/, via ... 

Usually, the WLF equation is derived from the Doolittle equation of viscosity and that of the thermal expansion with a constant coefficient [13]. However, these equations are rather empirical and must be derived theoretically. 

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 WLF equation can be derived from the empirical Doolittle equation that relates viscosity to fractional free volume together with assumption that the free volume is linearly related to the temperature [1, p 287]. Equation 4.68 is generally used when the temperature is at least one hundred degrees above the glass transition temperature, while the WLF Eq. 4.69 usually provides a better fit of data at temperatures closer to Tg. Both of the above equations are basically empirical, and one should not expect them to be strictly obeyed by any material. 

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