Volume diffusion model

An account of the mechanism for creep in solids placed under a compressive hydrostatic suess which involves atom-vacancy diffusion only is considered in Nabano and Hemirg s (1950) volume diffusion model. The counter-movement of atoms and vacancies tends to relieve the effects of applied pressure, causing extension normal to the applied sU ess, and sluinkage in the direction of the applied sU ess, as might be anticipated from Le Chatelier s principle. The opposite movement occurs in the case of a tensile sU ess. The analysis yields the relationship... 

The model of diffusion of hard spheres is applicable to interpret self-diffusion in liquids which behave according to the van der Waals physical interaction model (56). This might be the case for simple dense fluids at high temperature, T Tg, but it is an oversimplified model for the real diffusion of small organic penetrants in polymers. The functional relationships derived in the model of hard-spheres have been reinterpreted over course of the time, leading to a series of more sophisticated free-volume diffusion models. Some of these models are presented briefly below. 

One of the simplest early free-volume diffusion models was formulated in (51,52,60). The concept of this model was considered an advance, because some of the parameters required to describe the concentration dependence of the diffusion coefficient could be obtained from the physico-chemical properties of the polymer and penetrant. The relation proposed for the calculation of the thermodynamic diffusion coefficient, DT, was (51,60) ... 

In order to develop a consistent free-volume diffusion model, there are some issues which must be addressed, namely i) how the currently available free-volume for the diffusion process is defined, ii) how this free-volume is distributed among the polymer segments and the penetrant molecules and iii) how much energy is required for the redistribution of the free-volume. Any valid free-volume diffusion model addresses these issues both from the phenomenologic and quantitative points of view such that the diffusion process is described adequately down to the microscopic level. Vrentas and Duda stated that their free-volume model addresses these three issues in a more detailed form than previous diffusion models of the same type. Moreover, it was stated that the model allows the calculation of the absolute value of the diffusion coefficient and the activation energy of diffusion mainly from parameters which have physical significance, i.e. so-called first principles . In the framework of this model the derivation of the relation for the calculation of the self-diffusion coefficient of the sol-... 

To conclude this section, it may be interesting to mention what was concluded recently in (17) on the future of the free-volume diffusion models . .. However, phenomenological transport models based on free-volume concepts are likely to become obsolete during the coming decade, due to the development of computational techniques of simulating polymer microstructures . The development of such techniques and their results are discussed in Section 5.2. 

Local density fluctuations occur in penetrant polymer systems both above and below Tg. It is then reasonable to expect that a free-volume diffusion model should also provide an adequate description of the diffusion of small penetrants in glassy polymers. To reach this goal the free-volume model for diffusion of small penetrants in rubbery polymers, second part of Section 5.1.1, was modified to include transport below Tg (64,65,72,91-93). 

Thus, both the surface diffusion and volume diffusion models produce similar (R,a) curves. The application of the volume diffusion model has been somewhat limited (see, for example, the recent review by Bennema [33]). 

The BCF volume diffusion model has already been discussed in Sect. 5.1.2. The classical approach is to consider the diffusion of lattice ions from the... 

Figure 4.7 illustrates how the available free volume for transport increases with increasing temperature (Ff = Vj - V ), and the remarkable change when passing the Tg of a polymer [50]. According to the free volume diffusion model, the diffusion of molecules depends on the available free volume as well as sufficient energy to overcome polymer-polymer attractive forces. The specific volume at a particular temperamre can be obtained from the polymer density, whereas the volume occupied at 0 K can be estimated from group contributions. Details on this theory may be found in relevant handbooks, textbooks, and numerous publications [25,48-52]. 

Preexponential factor in free volume diffusivity model, Eq. (69)... 

At low supersaturations the BCF equation approximates to 7 oc but at high supersaturations Roc a. In other words, it changes from a parabolic to a linear growth law as the supersaturation increases. The volume diffusion model proposed by Chernov (1961) gives the same result. The general form of these expressions is shown in Figure 6.8. 

In the volume diffusion model, developed by Wagner [5], it was proposed that bulk diffusion of the less noble (LN) component to the dealloying front was the mechanism by which the selective dissolution reaction proceeded into the alloy. It was shown later by Cook and Hilliard [6] that bulk diffusion at room temperature was too slow to account for the rates of dealloying that were observed. 

Fig. 2. Shrinkage isotherms for Alcoa A-14 plotted according to the volume diffusion model.

Fig. 5. Shrinkage isotherms for pure and titania-doped Linde Cl.O alumina plotted according to the volume diffusion model. T = 1300°C. Data from Bagley (12). 

At subcritical potentials a single oxidation process with participation of only electronegative component takes place on the surface of the alloy. The surface layer is saturated with nonequilibrium defects (mainly vacancies), maintains morphological stability and represents a diffusion zone in which the atomic fraction of the noble component gradually increases as we approach the interface with the solution [6-9], According to the volume-diffusion model [10, 11], the formation of such zone is limited by the time-dependent interdiffusion of alloy components for the vacancy mechanism. 

Due to the universality in all glasses, physical aging can be theoretically explained in a straightforward way based on the free-volume concept. As proposed by Struik, This is the basic and rather obvious idea that the transport mobility of particles in a closely packed system is primarily determined by the degree of packing of the system or by its inverse measure, viz. the free volume [2]. The idea could date back to 1943 when Alfrey et al. proposed that the isothermal aging below Tg can be attributed to the diffusion of free volume holes from the interior of polymers into the surface [34]. This free volume diffusion model (FVDM) was developed by Curro et al. [35] to quantitatively analyze the volume relaxation experiments of poly(vinyl acetate) [36, 37]. The motion of free volume holes can be described by a diffusion equation ... 

Zielinksi and Duda used a free-volume diffusion model to predict diffusion constants for mixtures of common solvents and polymers, while Rauch and Kohler carefully studied the interaction of polystyrene and toluene. Both demonstrated that polymer diffusion increases as the Tg decreases upon addition of solvent. In other words, polymer chains gain mobility at room temperature when swollen with sufficient quantities of solvent. Mori et al demonstrated that addition of nonselective solvent toluene to PS-b-PI led to depression of the polystyrene block s Tg to below room temperature with 25% or more nonselective good solvent. " ... 

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