Free-volume theories

A simple theory of free volume was formulated to explain the molecular motion and physical behavior of the glassy and liquid states of matter [87]. This theory has been widely accepted in polymer science because it is conceptually simple and intuitively plausible for understanding many polymer properties at the molecular level. The derived macroscopic properties from free volume perspective are fruitful with the assistance of quantum and statistical mechanical calculations. 

Batchinsky [88] considered fluidity in a liquid as being due to the presence of free volume, and developed the simple formula 

When Williams, Landel and Ferry [90] studied the temperature dependence of free volume, the result was the famous WLF equation  

The temperature dependence of free volume is more complex than that of occupied volume. Below T, the polymer chain segments are frozen and the free volume does not change with temperature, i.e., dVjdT=0(T T. As a result, the free volume below T and at T ean be written as  

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  

The universal Cj and constants are obtained by fitting the WLF equation to a wide variety of polymers. It must be noted the variation from polymer to polymer 

Cohen and Crest [91] extended the free volume theory by introducing the concept of percolation for particle diffusion in the liquid by focusing on the random distribution of free volume. The singularity at Ty in (10.24) represents the singularity induced by the percolation threshold. The free volume regions do not percolate below Ty, so the particle diffusion is limited as expected in the glassy region. Above Ty, the percolated network of free volume allows the particle diffusion to occur over the entire volume, which makes the system behave like a fluid. 

Drawbacks of the free volume approach should be mentioned. While the decrease of the free volume with temperature drop certainly explains the increase in the viscosity, it is rather difficult to explain in this approach the pressure dependence of the viscosity and negative dTg/dP observed in some SCLs (see, for example. Ref. [94]). Similarly, negative expansion coefficients are also untenable for the theory. 

It should be noted that what one measures in experiments is the difference in the entropy, and not the absolute entropy. Assuming that the entropy is zero at absolute zero in accordance with the Nemst-Planck postulate, one can determine the absolute entropy experimentally. However, it is well known that SCL is a metastable state, and there is no reason for its entropy to vanish at absolute zero [16]. Indeed, it has been demonstrated some time ago that the residual entropy at absolute zero obtained by extrapolation is a nonzero fraction of the entropy of melting [43 ], which is not known a priori. Therefore, it is impossible to argue from experimental data that the entropy indeed falls to zero, since such a demonstration will certainly require calculating absolute entropy though efforts continue to date [61, 62]. 

A simple way of expressing the concentration dependence of the diffusion coefficient has been given above in eq. V -119. A more quantitative approach is based on the free volume theory. 

Above the glass transition temperature, i.e. in the rubbery state, the mobility of the chain segments is increased and frozen microvoids no longer exist A number of physical parameters change at the glass transition temperanire and one of these is the density or specific volume. This is shown in figure V - 26 where the specific volume of an amorphous polymer has been plotted as a function of the temperature. 

The free volume may be defined as the volume generated by thermal expansion of the initially closed-pad molecules at 0 K. 

The observed or specific volume at a particular temperature can be obtained from the polymer density whereas the volume occupied at 0 K can be estimated from group contribution [27 8], 

Using the free volume concept based on viscosity, a fractional free volinne Vf 0.025 has been found for a number of glassy polymers and this value is now considered to be a constant (v = r Tg - Above T, the free volume increases linearly with teirperaQue according to 

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The Free Volume Theory. This extends the lubricity and gel theories and also allows a quantitative assessment of the plasticization... 

StoKes-Einstein and Free-Volume Theories The starting point for many correlations is the Stokes-Einstein equation. This equation is derived from continuum fluid mechanics and classical thermodynamics for the motion of large spherical particles in a liqmd. 

If the molecule moves without hindrance in a rigid-walled enclosure (the free enclosure ), as assumed in free volume theories, then rattling back and forth is a free vibration, which could be considered as coherent in such a cell. The transfer time between opposite sides of the cell t0 is roughly the inverse frequency of the vibration. The maximum in the free-path distribution was found theoretically in many cells of different shape [74]. In model distribution (1.121) it appears at a > 2 and shifts to t0 at a - oo (Fig. 1.18). At y — 1 coherent vibration in a cell turns into translational velocity oscillation as well as a molecular libration (Fig. 1.19). 

Back reflection of translational and rotational velocity is rather reasonable, but the extremum in the free-path time distribution was never found when collisional statistics were checked by computer simulation. Even in the hard-sphere solid the statistics only deviate slightly from Pois-sonian at the highest free-paths [74] in contrast to the prediction of free volume theories. The collisional statistics have recently been investigated by MD simulation of 108 hard spheres at reduced density n/ o = 0.65 (where no is the density of closest packing) [75], The obtained ratio t2/l2 = 2.07 was very close to 2, which is indirect evidence for uniform... 

In the literature there is only one serious attempt to develop a detailed mechanistic model of free radical polymerization at high conversions (l. > ) This model after Cardenas and 0 Driscoll is discussed in some detail pointing out its important limitations. The present authors then describe the development of a semi-empirical model based on the free volume theory and show that this model adequately accounts for chain entanglements and glassy-state transition in bulk and solution polymerization of methyl methacrylate over wide ranges of temperature and solvent concentration. 

A useful model should account for a reduction of kt and kp with increase in polymer molecular weight and concentration and decrease in solvent concentration at polymerization temperatures both below and above the Tg of the polymer produced. For a mechanistic model this would involve many complex steps and a large number of adjustable parameters. It appears that the only realistic solution is to develop a semi-empirical model. In this context the free-volume theory appears to be a good starting point. 

Flory (11) improved the notation and form of Prigogine s expressions, and it is essentially the Flory form of Prigogine s free-volume theory that is of most use for design purposes. The Flory work (11) leads to an equation of state which obeys the corresponding-states principle ... 

It is also useful to note that other approaches to describe diffusion in solvent-polymer systems have been developed using free-volume theory [408-410]. 

Vrentas, JS Duda, JL, Diffusion in Polymer-Solvent Systems. I. Reexamination of the Free-Volume Theory, Journal of Polymer Science Polymer Physics Edition 15, 403, 1977. Vrentas, JS Duda, JL, Diffusion in Polymer-Solvent Systems. II. A Predictive Theory for the Dependence of Diffusion Coefficients on Temperature, Concentration, and Molecnlar Weight, Journal of Polymer Science Polymer Physics Edition 15, 417, 1977. 

In addition to temperature and concentration, diffusion in polymers can be influenced by the penetrant size, polymer molecular weight, and polymer morphology factors such as crystallinity and cross-linking density. These factors render the prediction of the penetrant diffusion coefficient a rather complex task. However, in simpler systems such as non-cross-linked amorphous polymers, theories have been developed to predict the mutual diffusion coefficient with various degrees of success [12-19], Among these, the most notable are the free volume theories [12,17], In the following subsection, these free volume based theories are introduced to illustrate the principles involved. 

Yasuda et al. [64] developed a free volume theory describing the diffusion... 

JS Vrentas, JL Duda. Diffusion in polymer-solvent systems. I. Reexamination of the free volume theory. J Polym Sci, Polym Phys Ed 15 403-416, 1977. 

Numerous models have been proposed to interpret pore diffusion through polymer networks. The most successful and most widely used model has been that of Yasuda and coworkers [191,192], This theory has its roots in the free volume theory of Cohen and Turnbull [193] for the diffusion of hard spheres in a liquid. According to Yasuda and coworkers, the diffusion coefficient is proportional to exp(-Vj/Vf), where Vs is the characteristic volume of the solute and Vf is the free volume within the gel. Since Vf is assumed to be linearly related to the volume fraction of solvent inside the gel, the following expression is derived ... 

Figure 14 The free volume theory of Yasuda and coworkers holds for the diffusion of acetaminophen in swollen 10 X 4 poly(lV-isopropyl acrylamide) gel. (Adapted from Ref. 176.)...

Yasuda s free volume theory [57] has been proposed to explain the mechanism of permeation of solutes through hydrated homogeneous polymer membranes. The free volume theory relates the permeability coefficients in water-swollen homogeneous membranes to the degree of hydration and molecular size of the permeant by the following mathematical expression ... 

Joshi and Topp [58] used Yasuda s free volume theory [57] to explain their... 

Equation 39 has the structure proposed for the rate constants on the basis of the free volume theory (1,5,9). From this, it would be expected that the models developed from the free volume theory would be very successful in predicting both, the rate behaviour and the molecular properties at high conversions. The reason why these models have been only partially successful stems from the... 

The rate parameters follow similar conversion trajectories. Therefore, the rate constants and the initiator efficiency can be modelled with the same equation. An equation of the form of equation 39 is suggested. The theoretical Justification for the form of equation 39 stems from the free volume theory. 

Analysis of mixture models, established techniques, 61 Analysis of styrene suspension polymerization continuous models, 210-211 efficiency, 211,212f,213 free volume theory, 215,217 initiator conversion vs. 

It would be an advantage to have a detailed understanding of the glass transition in order to get an idea of the structural and dynamic features that are important for photophysical deactivation pathways or solid-state photochemical reactions in molecular glasses. Unfortunately, the formation of a glass is one of the least understood problems in solid-state science. At least three different theories have been developed for a description of the glass transition that we can sketch only briefly in this context the free volume theory, a thermodynamic approach, and the mode coupling theory. 

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