Model free volume

If the free volume fraction of the ith component is /, then we have 

For a two-component model, Takayanagi derived the following relationships for the glass temperatures  

One of the most important conclusions associated with the Takayanagi model is that the distribution function of the free volume fraction Fj f) is evaluated by the following equation based on the specific volume-temperature curve  

This approach is based on the forces originating from the electric field gradient that is induced optically. The main ingredients to be included take into account that experimental results showed no volume change in the film during the surface deformation, and this deformation is caused by lateral movement (in large scale) of polymer chains. The forces behind this movement are 

It is also known that SRGs arise from surface-initiated processes.The free polymer surface can be treated as a thin, mobile layei with a viscous drag between layers dominated by the force f x,y,z). The limit superficial velocity, v, due to this force is  

can be obtained from the continuity equation, under the boundary condition that v x,y,d) = 0, as  

For a Gaussian beam, linearly polarized with the field in the direction x, f x,y,z) has only an x component, as can be seen in Equation 14.2. Therefore, the surface deformation induced by the beam, in the approximation of small amplitudes, and also assuming that the intensity I is a function of x only in Equation 14.4, is given by  

The surface deformation caused by a circularly polarized Gaussian beam, according to the above analysis, is  

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The free volume model seems to be more adequate to describe the plasticization behaviour of the systems of lower amine content. According to Eq. (5), the higher is the change of the expansion coefficient the lower is the influence of the diluent volume fraction. The three TGDDM-DDS mixtures cured with 20, 30 and 50 PHR of hardener were characterized l2) by changes of the expansion coefficient at the glass transition, respectively, of 0.63, 1.08 and 2.94x 10 3 °C l. The more dense and stiffer resin crosslinked with 50 PHR of DDS should be, in principle, the less... 

The action of a muscle is a consequence of electrochemically stimulated conformational relaxation processes that occur along every electroactive chain inside a polymeric film. A free-volume model dependent on the... 

Generally, the values of the scaling exponent are smaller for polymers than for molecular liquids, for which 3.2 < y < 8.5. A larger y, or steeper repulsive potential, implies greater influence of jamming on the dynamics. The smaller exponent found for polymers in comparison with small-molecule liquids means that volume effects are weaker for polymers, which is ironic given their central role in the historical development of free-volume models. The reason why y is smaller... 

Studies of the effect of permeant s size on the translational diffusion in membranes suggest that a free-volume model is appropriate for the description of diffusion processes in the bilayers [93]. The dynamic motion of the chains of the membrane lipids and proteins may result in the formation of transient pockets of free volume or cavities into which a permeant molecule can enter. Diffusion occurs when a permeant jumps from a donor to an acceptor cavity. Results from recent molecular dynamics simulations suggest that the free volume transport mechanism is more likely to be operative in the core of the bilayer [84]. In the more ordered region of the bilayer, a kink shift diffusion mechanism is more likely to occur [84,94]. Kinks may be pictured as dynamic structural defects representing small, mobile free volumes in the hydrocarbon phase of the membrane, i.e., conformational kink g tg ) isomers of the hydrocarbon chains resulting from thermal motion [52] (Fig. 8). Small molecules can enter the small free volumes of the kinks and migrate across the membrane together with the kinks. 

In polymer electrolytes (even prevailingly crystalline), most of ions are transported via the mobile amorphous regions. The ion conduction should therefore be related to viscoelastic properties of the polymeric host and described by models analogous to that for ion transport in liquids. These include either the free volume model or the configurational entropy model . The former is based on the assumption that thermal fluctuations of the polymer skeleton open occasionally free volumes into which the ionic (or other) species can migrate. For classical liquid electrolytes, the free volume per molecule, vf, is defined as ... 

Figure 2 The temperature dependence of the self-diffusion coefficient of 2,3-dimethyl-butane predicted by Cohen and Turnbull s free volume model. (From Ref. 25.)...

D Turnbull, MH Cohen. Free-volume model of the amorphous phase Glass transition. J Chem Phys 34 120-125, 1961. 

The configurational entropy model describes transport properties which are in agreement with VTF and WLF equations. It can, however, predict correctly the pressure dependences, for example, where the free volume models cannot. The advantages of this model over free volume interpretations of the VTF equation are numerous but it lacks the simplicity of the latter, and, bearing in mind that neither takes account of microscopic motion mechanisms, there are many arguments for using the simpler approach. 

FREE-VOLUME MODEL FOR SELF-DIFFUSION IN LIQUIDS... 

Larry Duda and Jim Vrentas were the first to systematically study the diffusion of small molecules in molten polymers, formulate a free volume-based theoretical model, and elucidate the sharp dependence of the diffusion coefficient on temperature and concentration.2 Figure 8.8 shows diffusivities of toluene in polystyrene as a function of concentration and temperature. The values were computed using the Vrentas and Duda (17) free volume model and, as shown, coincide well with available data. 

Summing up, the two-phase model is physically consistent and may be applied for designing industrial systems, as demonstrated in recent studies [10, 11], Modeling the diffusion-controlled reactions in the polymer-rich phase becomes the most critical issue. The use of free-volume theory proposed by Xie et al. [6] has found a large consensus. We recall that the free volume designates the fraction of the free space between the molecules available for diffusion. Expressions of the rate constants for the initiation efficiency, dissociation and propagation are presented in Table 13.3, together with the equations of the free-volume model. 

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