Volume, Doolittle free

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),... 

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 ... 

Generally, the VTF behaviour of all transport properties may be understood from the free volume concept introduced by Doolittle (1951) and further developed by Cohen and Turnbull (1959). Essentially, any diffusing species is depicted as encaged by the nearest atoms in a cell... 

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... 

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 ... 

Covacs6 proposed another definition of the fractional free-volume. As the extrapolated volume of supercooled liquids reaches the crystalline state, vc, at temperature rc > 0 K, it is possible to compare this critical temperature with rg , the limit glass temperature at infinitely slow cooling. Free-volume in the crystalline state is assumed to be zero. In this case the value /g according to Doolittle may be compared with /g, c, which characterizes the excess free-volume of glass as compared with the crystalline state ... 

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 WLF relationship can be derived from the Doolittle relationship that links the mobility M to the free volume fraction ... 

The usual free volume interpretation of Cfand Cf is questionable Cf = BD/fg and Cf = fg/Aa, where BD is the Doolittle constant, fg is the free volume fraction at Tg and Aa = oq — ag is the free volume expansion coefficient. It appears impossible to... 

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. 

Le volume libre est defini par les divers auteurs de deux manures essentiellement differentes. Pour les uns ce paramfetre de la structure liquide est rattache la mobility des configurations, et derive de l applica-tion de la relation (27) de WLF, ou plus gdndalement de celle de Doolittle (eq. 24), aux rdsultats des mesures des paramfetres viscoelastiques des polymdes. Pour les autres, le volume libre rdsulte d une extrapolation plus ou moins judicieuse du volume du liquide (surfondu), ou de celui du verre, au zero absolu. Afin de distinguer les deux concepts, nous appellerons le premier, volume libre de relaxation (relaxation free volume), et le second volume libre d extrapolationi>, bien que ce terme ne soit pas tr s heureux. 

The decrease of En with increasing T may be explained by the extra free volume created by thermal expansion. This was suggested by Batchinski in 1913 already. Several attempts have been made to formulate a joint temperature function for polymer melts and rubbery amorphous polymers on this basis. Doolittle (1951) formulated the equation ... 

An alternative explanation of the VFT model (28) is based on the free volume concept introduced by Fox and Floury [66-68] to describe the relaxation kinetics of polystyrene. The main idea behind this approach is that the probability of movement of a polymer molecule segment is related to the free volume availability in a system. Later, Doolittle [69] and Turnbull and Cohen [70] applied the concept of free volume to a wider class of disordered solids. They suggested a similar 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... 

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-... 

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... 

Since the lower the free volume the larger the relaxation time, the effect of the pressure on x can be interpreted in terms of the Doolittle equation. Accordingly,... 

The Doolittle equation [Eq. (8.130)] can be combined with the assumed linear temperature dependence of free volume [Eq. (8.131)] to get the WLF equation, so-named for Williams, Landel, and Ferry, who first applied it to polymer melts in 1955 ... 

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. 

Equation (5) was introduced as a simple consequence of the Doolittle equation, which is the basic artifact of free volume models for the glass transition." However, its final form with the volume related fragility strength coefficient Dy was only recently proposed. Eq. (6) was postulated ad hoc, by analogy to eq. (4), since the pressurization was coincided with densification. °... 

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