Doolittle’s 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... 

The authors of the cluster theory draw the conclusion that the theory affords a sufficiently rigorous theoretical derivation of Doolittle s equation (72). Verification of the free volume theory advanced by Cohen and Grest was carried out by Hiwatari using computer simulation [97], showed that glass transition in liquids can really be described in terms of the percolation theory, the value of Pcr in this case being close to 0.2. Unlike Cohen and Grest s assumptions, however, this transition is not accompanied by a drastic change in the fluidity of the liquid near Per-... 

If we consider, from the same standpoint, Doolittle s equation (72) whose validity was proved by Cohen and Grest [96], we see that by taking into account the variables introduced into Eq. (80), we can write it as follows ... 

Equations for structural relaxation are presented. At the same time, a theoretical justification of extending Doolittle s equation from the equilibrium liquid to nonequilibrium glassy states is discussed. 

In miscible blends, the flow behavior depends on free volume, entanglements, and specific interactions. From Doolittle s equation, assuming additivity of the occupied volume and non-addi-tivity of the free volume, Steller and Zuchowska [1990] derived ... 

Curro et al. [1981] followed a similar procedure in studies of the aging kinetics of poly(methyl methacrylate) (PMMA). For predicting the shift factors of aging experiments at 23°C, the authors computed from PVT the free-volume function, h = h P, V), and then substituted these into Doolittle s equation (6.63). The resulting prediction agreed with the experimental values, contrasting with the inadequacy of the WLF relation. Next, the polymer aging process was modeled as a diffusion of free volume [Curro et al., 1982]. 

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... 

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... 

Fox and Flory first postulated that the liquid-glass transition resulted from the decrease of the free volume of the amorphous phase below some critical value. The subsequent derivation of Doolittle s fluidity equation within the free-volume model is based on four simple assumptions, which can be worded as follows ... 

The WLF equation for aj has been rationalized in terms of Doolittle s firee volume theory [17]. According to this theory... 

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... 

Equation 7.6 can be developed from Doolittle s concept of free volume of a liquid. In Chapter 4 it was noted that the specific volume varies with temperature during quenching as shown in Fig. 4.27 and can be used to identify the degree of crystallinity as well as the melt temperature, T. ... 

Doolittle s torsional viscosimeter was essentially a damped, oscillating Couette viscometer, and Doolittle chose as the unit of viscosity the "number of degrees of retardation between the first and second complete arcs." There is no mention In this paper of the Hagen-Poiseuille law, the Navier-Stokes equations, the treatise of Lamb (1879) or the work of Basset (1888), and it would appear that Doolittle was unaware of the studies of fluid mechanics carried out in the previous century and half. Nearly two decades later, Gillet (1909) began his paper entitled "Analysis and Friction Tests of Lubricating Greases" with the comment ... 

As stress relaxation is a manifestation of viscous flow, the WLF equation can also be applied to polymer viscosity, and comparison with the Doolittle equation shows that BI2.303 f = 17.44. If the value of B is unity, which is demonstrated by some of Doolittle s data, then the value of /is 0.025. This indicates that the free volume fraction at Tgof any polymer is 2.5%. 

Whittaker and Robinson( ) contain a general discussion, but the most satisfactory method is due to M. H. Doolittle. We set out the equations as (A ), (B ) and (C ) in a table as Table 10.2. Inserting the numerical values quoted above, we get (A"), (B"), (C"). It will be understood that all the figures occurring in the column headed Ca are effectively the coefficients of Ca in the successive equations, and similarly for Cb and Cc The right hand sides have been multiplied by 10 temporarily to avoid the occurrence of excessively large numbers of O s after the decimal points. 

Assuming that there is a direct link between the shift factor ar and the ratio of the frequencies of the controlling molecular process. Equation (7.23) is identical in form to Equation (7.21). Note also that Equation (7.22) is Bueche s analogy to the Doolittle equation (7.20). 

As noted previously, the WLF equation (26) and the Doolittle equation are found to be same if one treats the free volume as varying linearly with temperature. If we adopt Ferry s notation that the fractional free-volume is /= vjv and the temperature dependence of / is... 

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