Theory Doolittle free volume

The temperature dependence of the dynamic viscosity t of a liquid close to its glass temperature Tg can be described by the Vogel-Tammann-Fulcher (VTF) equation [43-45] or by the Theory of free volume introduced by Doolittle [46 8], Cohen and Turnbull [49, 50]. An exponential dependence from the reciprocal temperature 1/T is found (see (8.8)). 

These authors were the first FGSE workers to make extensive use of the concept of free volume 42,44) and its effect on transport in polymer systems. That theory asserts that amorphous materials (liquids, polymers) above their glass transition temperature T contain unoccupied volume randomly distributed and in parcels of sufficient size to permit jumps of small molecules — and of polymer jumping segments — to take place. Since liquids have a fractional free volume fdil typically greater than that, f, of polymers, the diffusion rate both of diluent molecules and (uncrosslinked and unentangled) polymer molecules should increase with increasing diluent volume fraction vdi,. The Fujita-Doolittle expression 43) describes this effect quantitatively for the diluent diffusion ... 

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)... 

The self-diffusion of benzene in PIB [36], cyclohexane in BR [37] and toluene in PIB [38-40] has been investigated by PFG NMR. In addition more recently Schlick and co-workers [41] have measured the self-diffusion of benzene and cyclohexane mixtures in polyisoprene. In the first reported study of this kind, Boss and co-workers [36] measured the self-diffusion coefficients of benzene in polyisoprene at 70.4 °C. The increase in Dself with increasing solvent volume fraction could be described by the Fujita-Doolittle theory which states that the rate of self-diffusion scales with the free volume which in turn increases linearly with temperature. At higher solvent volume fractions the rate of selfdiffusion deviates from the Fujita-Doolittle theory, as the entanglement density decreased below the critical value. 

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 ... 

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-... 

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... 

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. 

Other theories within the general framework of the free-volume concept have been advanced. They include the works of Kumins and Roteman (1961), Bueche (1953), Barrer (1957), DiBenedetto (1963), DiBenedetto and Paul (1964), Wilkens and Long (Wilkens, 1957 Wilkens and Long, 1957), and Vasenin (1960). Part of the reason that these theories are not so popular lies in the fact that their predictions of D c) were no better than those obtained by Fujita and Doolittle. In addition, most of these other theories concentrated on the temperature dependence of the intrinsic mobility, which is less important compared to its concentration dependence. In spite of the predictive limitations of the free-volume theory, it is applicable at the widest concentration range, and certainly it is the best theory to use for modeling diffusion limited behavior of polymerization systems. 

The free volume theory is based on the concept introduced by Doolittle which describes the nonlinear behaviour of the viscosity of a liquid as Tg is approached ... 

Sufficient free volume is essential for these processes. An increase in Vf results in an increase in the number of conformational changes in the chains enhancing the chain relaxation capability (CRC) (Brostow and Macip 1989 Doolittle 1951 Akinay et al. 2001). Brostow and Macip (1989) defined CRC as the amount of external energy dissipated by a relaxation process in a unit of time and unit mass of the polymer. The excess energy that cannot be dissipated by relaxational processes can go into destructive processes (irreversible changes in response to an applied force and/or temperature such as bond fracture and plastic deformation). The theory of the chain relaxation capability is discussed in detail in several references (Akinay et al. 2002 Brostow and Kubat 1996 Brostow 2000 Brostow et al. 2000 Brostow et al. 1999a, b). 

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

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

Specifically, on the basis of free-volume theory, Doolittle (1951) has related the zero-shear viscosity (jjq) of an amorphous polymer to relative (fractional)... 

Many fiber properties are related to the free volume available in the amorphous phase of the fibers. For example, according to free volume theory, viscosity can be related to the fraction of free volume by the Doolittle equation ... 

---