Doolittle viscosity

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 ... 

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]... 

Loutfy and coworkers [29, 30] assumed a different mechanism of interaction between the molecular rotor molecule and the surrounding solvent. The basic assumption was a proportionality of the diffusion constant D of the rotor in a solvent and the rotational reorientation rate kOI. Deviations from the Debye-Stokes-Einstein hydrodynamic model were observed, and Loutfy and Arnold [29] found that the reorientation rate followed a behavior analogous to the Gierer-Wirtz model [31]. The Gierer-Wirtz model considers molecular free volume and leads to a power-law relationship between the reorientation rate and viscosity. The molecular free volume can be envisioned as the void space between the packed solvent molecules, and Doolittle found an empirical relationship between free volume and viscosity [32] (6),... 

Doolittle AK (1952) Studies in Newtonian flow III. The dependence of the viscosity of liquids on molecule weight and free space (in homologous series). J Appl Phys 23(2) 236-239... 

The importance of free volume effects in diffusional processes at a molecular level should be further emphasized. An empirical relationship between viscosity and free volume was proposed by Doolittle ... 

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... 

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... 

Free Volume Theory. Free volume theory suggests that the glass transition temperature is observed for polymers when their viscosity approaches that of their liquid state. Following a derivation based on the Doolittle expression for polymer viscosity (r ) as a function of free volume (Eisenberg, 1984)... 

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... 

As pointed out by Doolittle, the relationship between the viscosity of liquids and their free volume remained for a long time only an intuitive hypothesis though it described quite well numerous experimental results. A theoretical approach to the solution of the problem of the relationship between the viscosity of liquid and its free volume was generalized for the first time by Eyring [85] in terms of the absolute reaction rates theory. The formulas obtained by Eyring pointed to a qualitative relationship between viscosity and the ratio of the volume occupied by liquid molecules C to the volume occupied by holes through which molecules jump to the neighboring position ... 

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]... 

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

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) ... 

Batchinski (1913), and more recently Doolittle and Doolittle (1957), developed an empirical relationship between the viscosity and the free volume, from which the relaxation time Ty can be extracted ... 

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... 

A salient property of pol3mieric liquids is their glass forming ability, and this has played a central role in attempts to account for the temperature dependence of the viscosity. Recent developments on the dependence of the viscosity on temperature stem largely from the observation of Doolittle (5. 78, 79, 80, 81) that viscosity data on normal paraffins could be correlated by the expression... 

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. 

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. 

By an extended and careful examination of n-alkanes over a long range of temperature at atmospheric pressure, Doolittle [31] found the following expression suitable for the viscosity ... 

The theoretical models can be applied to the pressure dependence of viscosity if appropriate physical interpretations can be assigned to the various quantities in the equations. The significant-structure expression (Eqn 4-40) becomes physically intractable at elevated pressures because of difficulties with the dependence of fii) on pressure. Matheson [33] assumed the Doolittle model to be valid and wrote the temperature dependence of viscosity as... 

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...

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