Free volume models limitations

Stern, Frisch, and coworkers have extended Fujita s free-volume model to the permeation of light gases (31-33) (see Figure 3) and binary gas mixtures (34,25) (see Figure 4) through polymer membranes. The extended model was found to describe satisfactorily the dependence of permeability coefficients on pressure and temperature for a variety of light gases in polyethylene, as well the dependence on composition for several binary mixtures in the same polymer. The validity of the extended model is limited to total penetrant concentrations of up to 20-25 mol-%. 

The concept of free volume has been of more limited use in the prediction of solubility coefficients although, Peterlin (H) has suggested that the solubility coefficient is directly proportional to the free volume available in the polymer matrix. In many respects, the free volume expressions closely resemble the relationships developed in the activated state approach. In fact for the case of diffusivity, the two models can be shown to be mathematically equivalent by incorporating thermal expansion models such as the one proposed by Fox and Flory (12). The usefulness of the free volume model however, lies in the accessibility of the fractional free volume, through the use of group contribution methods developed by Bondi (12.) and Sugden (li), for correlation of barrier properties of polymers of different structure as demonstrated by Lee (15.). ... 

A review of the literature reveals that previous finite-element analyses of adhesive joints were either based on simplified theoretical models or the analyses themselves did not exploit the full potential of the finite-element method. Also, several investigations involving finite-element analyses of the same adhesive joint have reported apparent contradictory conclusions about the variations of stresses in the joint.(24,36) while the computer program VISTA looks promising (see Table 1), its nonlinear viscoelastic capability is limited to Knauss and Emri.(28) Recently, Reddy and Roy(E2) (see also References 37 and 38) developed a computer program, called NOVA, based on the updated Lagrangian formulation of the kinematics of deformation of a two-dimensional continuum and Schapery s(26) nonlinear viscoelastic model. The free-volume model of Knauss and Emri(28) can be obtained as a degenerate model from Schapery s model. 

Because of the success of the empirical free-volume relations in describing the behavior of glassforming liquids, there have been many attempts since the Cohen and Turnbull free-volume model to quantify the concept and make the free-volume physics more than a convenient way to correlate data. The reader is referred to the literature for a general look at the various models and also for some specific developments. " However, due to space limitations, we limit our discussion to the cell model of Simha and Somcynsky " and the extensive developments of this model which have been carried out over the years by Simha and co-workers. 

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. 

In the model, the kinetic constants for propagation and termination are allowed to vary as a function of free volume, as suggested by Marten and Hamielec (16) and Anseth and Bowman (17). To account for diffusional limitations and still predict the non-diffusion controlled kinetics, the functional forms for the propagation and carbon-carbon termination kinetic constants are ... 

In this paper, the kinetics and polymerization behavior of HEMA and DEGDMA initiated by a combination of DMPA (a conventional initiator) and TED (which produces DTC radicals) have been experimentally studied. Further, a free volume based kinetic model that incorporates diffusion limitations to propagation, termination by carbon-carbon radical combination and termination by carbon-DTC radical reaction has been developed to describe the polymerization behavior in these systems. In the model, all kinetic parameters except those for the carbon-DTC radical termination were experimentally determined. The agreement between the experiment and the model is very good. 

This is a theoretical equation that was derived from free volume theory. If extruding materials at lower than normal temperatures, the higher sensitivity of the viscosity to temperature is an issue that needs to be considered. The engineering-based viscosity equation developed by Adams and Campbell [18] has been shown to hold for all nominal processing temperatures, from within a few degrees of Tg [26, 27] to conventional extruder melt temperatures. The Adams-Campbell model limiting shear temperature dependence is ... 

Figure 4.26 shows a cell model of the three phases. Gas in the upper region has a very low density and the molecules are free to fly around. When the vapor condenses into a liquid (shown lower right), the density is greatly increased so that there is very little free volume space the molecules have limited ability to move around, and they have random orientation that is, they can rotate and point in random directions. When the liquid freezes into a solid (shown lower left), the density is slightly increased to eliminate the void space, the molecules have assigned positions and are not free to move around, and there is now an orientation order that is, they cannot rotate freely and they all point at the same direction. 

The glass transition temperature and resulting free volume are calculated from the DiBenedetto equation 23.) or by an approach which incorporates the effect of crosslink density on the glass transition temperature (22). This part of the model is iterative in order to account for diffusion-limitations on the reaction rate in the later stages of thermoset cure 23). ... 

In this equation, known as the Yalkowsky-Rubino model, C is the molar concentration of the solute (component 3), and the (f> terms are the solute-free volume fractions of the solvents (components 1 and 2). This equation should be used only for compounds of limited solubility. 

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