Effective diffusivity, polymer fractionation modeling

The models most frequently used to describe the concentration dependence of diffusion and permeability coefficients of gases and vapors, including hydrocarbons, are transport model of dual-mode sorption (which is usually used to describe diffusion and permeation in polymer glasses) as well as its various modifications molecular models analyzing the relation of diffusion coefficients to the movement of penetrant molecules and the effect of intermolecular forces on these processes and free volume models describing the relation of diffusion coefficients and fractional free volume of the system. Molecular models and free volume models are commonly used to describe diffusion in rubbery polymers. However, some versions of these models that fall into both classification groups have been used for both mbbery and glassy polymers. These are the models by Pace-Datyner and Duda-Vrentas [7,29,30]. 

Most spraying processes work under dynamic conditions and improvement of their efficiency requires the use of surfactants that lower the liquid surface tension yLv under these dynamic conditions. The interfaces involved (e.g. droplets formed in a spray or impacting on a surface) are freshly formed and have only a small effective age of some seconds or even less than a millisecond. The most frequently used parameter to characterize the dynamic properties of liquid adsorption layers is the dynamic surface tension (that is a time dependent quantity). Techniques should be available to measure yLv as a function of time (ranging firom a fraction of a millisecond to minutes and hours or days). To optimize the use of surfactants, polymers and mixtures of them specific knowledge of their dynamic adsorption behavior rather than equilibrium properties is of great interest [28]. It is, therefore, necessary to describe the dynamics of surfeictant adsorption at a fundamental level. The first physically sound model for adsorption kinetics was derived by Ward and Tordai [29]. It is based on the assumption that the time dependence of surface or interfacial tension, which is directly proportional to the surface excess F (moles m ), is caused by diffusion and transport of surfeictant molecules to the interface. This is referred to as the diffusion controlled adsorption kinetics model . This diffusion controlled model assumes transport by diffusion of the surface active molecules to be the rate controlled step. The so called kinetic controlled model is based on the transfer mechanism of molecules from solution to the adsorbed state and vice versa [28]. 

A modified Cahn-Hilliard (CH) model [114] is used for the theoretical analysis of the impact of thermal diffusion on phase separation by taking into account an inhomogeneous temperature distribution, which couples to a concentration variation via the Soret effect. The Flory-Huggins model is used for the free energy of binary polymer-mixtures. The composition is naturally measured in terms of volume fraction 0 of a component A, which can be related to the weight fraction c by... 

Paul et al. (25) observed that for polymer volume fractions less than 0.8, the functional dependence of the diffusion coefficients on the polymer volume fraction was, generally, in accordance with Equation 40. Muhr and Blanshard (26) provide additional supporting data on different polymers than those reported by Paul et al, Roucls and Ekerdt (27) measured the diffusion of cyclic hydrocarbons in benzene-swollen polystyrene beads their diffusion coefficients satisfy the general form of Equation 40. The effective dlffuslvltles of organic substrates in crossllnked polystyrene reported by Marconi and Ford (17) also follow trends predicted in Equation 40. In the absence of experimental data, it appears that Equation 40 provides a reasonable, and the simplest, means to estimate D for use in detailed modeling or in estimation methods such as Equation 38. Equation 40 was used by Dooley et al. (11) in their study of substrate diffusion and reaction in a macroreticular sulfonic acid resin which involved vapor phase reactants. 

The constancy of the effective diffusion coefficient In the substrate transport with reaction model. Equation 26 versus Equation 27, depends on two factors. First, if the volume fraction of polymer is not constant with time, radial position in the gel, or extent of reaction, then D is influenced by the relation given in Equation 40. Dooley ct al. (11) present an example of this in their study. Second, if the substrate s diffusion coefficient in the solvent alone is dependent on substrate concentration at the range of concentrations in the reaction system, then D is Influenced similarly according to Equation 39. The assumption of a constant diffusion coefficient in the substrate transport with reaction model must always be justified. 

Figure 2.34 simulates the copolymerization of ethylene and propylene in the presence of hydrogen. Ethylene, being the faster comonomer, has a much steeper radial concentration profile than propylene. In the same way, hydrogen reacts much more slowly and also diffuses rather fast and therefore has a flat radial concentration profile. The effect of these profiles on the CLD and CCD is clear polymer made near the surface of the particle will have higher molecular weight and ethylene fraction than the polymer made near the center of the particle. This modeling approach was first proposed by Soares and Hamielec for a version of the PFM [70]. 

Within the context of this model, c and f are adjustable constants that vary from polymer to polymer. The ratio f /c is a crude measure of the average interchain separation. As polymer chain stif ess increases and free volume decreases, c should increase (20), and as fractional free volume increases, /c should increase. This theory is based on four hypotheses (1) the solution-di sion mechanism (equation 2) governs gas transport (2) diffusion obeys an Arrhenius expression (equation 8) (3) the linear free energy relation (equation 12) is valid and (4) the effect of penetrant size on activation energy is given by equation 13. 

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