Diffusive transport in porous polymers

Saltzman, W. M., Pasternak, S. H., and Langer, R., Micro-structural models for diffusive transport in porous polymers, in Controlled-Release Technology, ACS Symposium Series 348... 

Microstructural Models for Diffusive Transport in Porous Polymers... 

The results from this analysis can now be used to construct geometrically accurate models of the diffusive transport in porous polymers. Previous models of diffusion in these polymers have used an empirically determined tortuosity factor as a lumped parameter to account for the retardation of release by all mechanisms (7-8). 

The problem of non-steady state flow-which is characteristic of absorbency in polymers-is analogous to the non-steady state diffusion of solutes in a porous material [2]. Therefore, much of this chapter focuses on the solution of the equations for non-steady state diffusion In a porous medium. The most challenging aspect of the general problem of transport in porous polymers is relating the microscopic characteristics of the pore space (porosity, tortuosity, connectivity) to the macroscopic property of interest (permeability or diffusion coefficient). This chapter describes some of the methods that can be used to relate microstructure to transport. While most of the models presented are based on the general problem of diffusion in porous polymers, they can be adapted to explore the mathematically equivalent problem of absorbency in polymers. 

Vol. 1 Polymer Engineering Vol. 2 Filtration Post-Treatment Processes Vol. 3 Multicomponent Diffusion Vol. 4 Transport in Porous Catalysts... 

Problems involving transport through porous media occur in many disciplines. Although the most frequently studied individual problem is the movement of fluid through porous soils or rocks, examples of transport in porous media occur in many distinct types of systems. For example, tissues In the body are composed of cells and extracellular regions, two phases which often differ dramatically in resistance to diffusion of solutes. In recent years -as researchers have learned more about the structure of heterogeneous materials like soils, porous polymers, and animal tissues--the application of techniques developed to understand transport in porous media increase. 

A fundamental difference exists between the assumptions of the homogeneous and porous membrane models. For the homogeneous models, it is assumed that the membrane is nonporous, that is, transport takes place between the interstitial spaces of the polymer chains or polymer nodules, usually by diffusion. For the porous models, it is assumed that transport takes place through pores that mn the length of the membrane barrier layer. As a result, transport can occur by both diffusion and convection through the pores. Whereas both conceptual models have had some success in predicting RO separations, the question of whether an RO membrane is truly homogeneous, ie, has no pores, or is porous, is still a point of debate. No available technique can definitively answer this question. Two models, one nonporous and diffusion-based, the other pore-based, are discussed herein. 

The main emphasis in this chapter is on the use of membranes for separations in liquid systems. As discussed by Koros and Chern(30) and Kesting and Fritzsche(31), gas mixtures may also be separated by membranes and both porous and non-porous membranes may be used. In the former case, Knudsen flow can result in separation, though the effect is relatively small. Much better separation is achieved with non-porous polymer membranes where the transport mechanism is based on sorption and diffusion. As for reverse osmosis and pervaporation, the transport equations for gas permeation through dense polymer membranes are based on Fick s Law, material transport being a function of the partial pressure difference across the membrane. 

Polymer electrolyte fuel cell (PEFC) is considered as one of the most promising power sources for futurist s hydrogen economy. As shown in Fig. 1, operation of a Nation-based PEFC is dictated by transport processes and electrochemical reactions at cat-alyst/polymer electrolyte interfaces and transport processes in the polymer electrolyte membrane (PEM), in the catalyst layers consisting of precious metal (Pt or Ru) catalysts on porous carbon support and polymer electrolyte clusters, in gas diffusion layers (GDLs), and in flow channels. Specifically, oxidants, fuel, and reaction products flow in channels of millimeter scale and diffuse in GDL with a structure of micrometer scale. Nation, a sulfonic acid tetrafluorethy-lene copolymer and the most commonly used polymer electrolyte, consists of nanoscale hydrophobic domains and proton conducting hydrophilic domains with a scale of 2-5 nm. The diffusivities of the reactants (02, H2, and methanol) and reaction products (water and C02) in Nation and proton conductivity of Nation strongly depend on the nanostructures and their responses to the presence of water. Polymer electrolyte clusters in the catalyst layers also play a critical... 

To model a porous electrocatalyst we may consider a second type of mass transport (in addition to diffusion) locally within the electrode, i.e., a mass transport resistance between the electrode surface and the solution. This situation may arise, for example, when the electrode surface is covered by a thin layer of polymer electrolyte or as in a fuel cell electrode in which the electrocatalyst is also covered by a thin water layer. 

Polymerization of styrene in microemulsions has produced porous solid materials with interesting morphology and thermal properties. The morphology, porosity and thermal properties are affected by the type and concentration of surfactant and cosurfactant. The polymers obtained from anionic microemulsions exhibit Tg higher than normal polystyrene, whereas the polymers from nonionic microemulsions exhibit a lower Tg. This is due to the role of electrostatic interactions between the SDS ions and polystyrene. Transport properties of the polymers obtained from microemulsions were also determine. Gas phase permeability and diffusion coefficients of different gases in the polymers are reported. The polymers exhibit some ionic conductivity. 

Retention of ionic species modifies ionic concentrations in the feed and permeate liquids in such a way that osmotic pressure or electroosmotic phenomena cannot be neglected in mass transfer mechanisms. The reflexion coefficient, tr, in Equations 6.4 and 6.5 represents, respectively, the part of osmotic pressure force in the solvent flux and the diffusive part in solute transport through the membrane. One can see that when a is close or equal to zero the convective flux in the pores is dominant and mostly participates to solute transport in the membrane. On the contrary when diffusion phenomena are involved in species transport through the membrane, which means that the transmembrane pressure is exerted across an almost dense stmcture. Low UF and NF ceramic membranes stand in the former case due to their relatively high porous volume and pore sizes in the nanometer range. Recendy, relevant results have been published concerning the use of a computer simulation program able to predict solute retention and flux for ceramic and polymer nanofiltration membranes [21]. 

The method of preparation also influences the properties of the film. Cast films of varying properties can be prepared by variation inter alia of the solvent power of the casting solution containing the polymer, although the complex processes involved in film formation are not yet fully understood. It is clear, however, that the conformation of the polymer chains in concentrated solution just prior to solvent evaporation will determine the density of the film, and the number and size of pores and voids. Dmg flux through dense (nonporous) polymer membranes is by diffusion flux through porous membranes will be by diffusion and by transport in solvent through pores in the film. With... 

Structural models emerge from the notion of membrane as a heterogenous porous medium characterized by a radius distribution of water-filled pores. This structural concept of a water-filled network embedded in the polymer host has already formed the basis for the discussion of proton conductivity mechanisms in previous sections. Its foundations have been discussed in Sect. 8.2.2.1. Clearly, this concept promotes hydraulic permeation (D Arcy flow [80]) as a vital mechanism of water transport, in addition to diffusion. Since larger water contents result in an increased number of pores used for water transport and in larger mean radii of these pores, corresponding D Arcy coefficients are expected to exhibit strong dependencies on w. 

Membrane Separations. Separation processes using polymeric membranes (30) have become important techniques because of their simplicity and low consumption of energy in comparison to alternatives such as distillation. Membranes for ultrafiltration are porous, and no diffusive transport actually occurs through the polymer itself. However, for separation at the molecular level, diffusion through the polymer provides a possible mechanism for selective passage of the desired small molecule. Reverse osmosis for desalination of water can occur by this mechanism, and large commercial processes using this technique are now in operation. 

The permeabilities of different components in a membrane depend on the mechanism by which the components are transported. For example, in homogeneous polymer membranes, the various chemical species are transported under a concentration or pressure gradient by diffusion. The permeability of these membranes is determined by the diffusivities and concentrations of the various components in the membrane matrix and the transport rates are, in general, relatively slow. In porous membrane structures, however, mass is transported under the driving force of a hydrostatic pressure difference via viscous flow and, in gen-... 

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