Free volume diffusion model

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). 

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)... 

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. " ... 

Steady-state behavior and lifetime dynamics can be expected to be different because molecular rotors normally exhibit multiexponential decay dynamics, and the quantum yield that determines steady-state intensity reflects the average decay. Vogel and Rettig [73] found decay dynamics of triphenylamine molecular rotors that fitted a double-exponential model and explained the two different decay times by contributions from Stokes diffusion and free volume diffusion where the orientational relaxation rate kOI is determined by two Arrhenius-type terms ... 

The lower cycle represents the chemical changes occurring during polymerization and relates them to the free volume of the system. In general, free volume of a polymer system is the total volume minus the volume occupied by the atoms and molecules. The occupied volume might be a calculated van der Waals excluded volume [139] or the fluctuation volume swept by the center of gravity of the molecules as a result of thermal motion [140,141]. Despite the obscurity in an exact definition for the occupied volume, many of the molecular motions in polymer systems, such as diffusion and volume relaxation, can be related to the free volume in the polymer, and therefore many free volume based models are used in predicting polymerization behavior [117,126,138]. 

Lustig, S.R. and Peas, N.A. (1987). Solute and penetrant diffusion in swellable polymers. 7. A free volume based model with mechanical relaxation. J. Alied Polymers Sci., 43, 533-549. 

Since the time dependence of variations in magnetization, dipolar or quad-rupolar coupling, and chemical-shift anisotropy are inherent parts of the NMR phenomenon, it is a natural tool for studying molecular motions. These relationships are usually well understood, and NMR data can therefore be used to test dynamic models. Conformations, free volume, diffusion, and mechanical properties have all been correlated with NMR observations. This section will first review mobile systems (solution, melts, and elastomers), and then discuss solids. 

Free Volume Diffusion and Annihilation (FVDA) Model... 

To evaluate the time-dependent function, X(t), a simple model of diffusion is proposed. Starting from Langmuir adsorption theory, we consider that liquid molecules having diffused into the elastomer are localized on discrete sites (which might be free volume domains). In these conditions, we can deduce the rate of occupation of these sites by TCP with time. Only the filhng of the first layer of the sites situated below the liquid/solid interface at a distance of the order of the length of intermolecular interaction, i.e., a few nanometers, needs to be considered to estimate X(t). 

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. 

Cohen and Turnbull [20,21] laid down the foundation for the free volume concept in modeling self-diffusion in simple van der Waals liquids. They considered that the volume in a liquid is composed of two parts, the actual volume occupied by the liquid molecules and the free volume surrounding these molecules opened up by thermal fluctuation. Increasing temperature increases only the free volume and not the occupied volume. The average free volume per molecule, vf, can be 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.)...

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 19 Free volume plot for co(polyether)polyurethane membranes at 37°C for model drugs. D is the diffusion coefficient in cm2/sec, and H is the hydration expressed as (wet volume — dry volume)/wet volume. (Reproduced with permission from Ref. 59.)...

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),... 

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. 

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