## 2 Polymer Nanocomposites for Membranes Applications
Polystyrene (PS)/clay hybrid systems have been prepared via in situ polymerization of styrene in the presence of (1) unmodified Na-MMT clay; (2) MMT modified with zwitter-ionic cationic surfactant octadecyldimethyl betaine (C\({}_{18}\)DMB) and (3) MMT modified with polymerizable cationic surfactant vinylbenzyldimethyldodecylammonium chloride (VDAC) [113]. For all composites, an accurately proportional decrease in the permeability and diffusion of oxygen with the addition of clay was noted. The effect of clay on the oxygen solubility was significantly less. Solubility data changes were declared being within the range of experimental error of 10% with the addition of clay. These facts are demonstrated in Figure 7, presenting a plot of relative oxygen permeability \(\mathrm{P_{c}}/\mathrm{P_{0}}\) versus relative oxygen diffusivity \(\mathrm{D_{c}}/\mathrm{D_{0}}\), where \(\mathrm{P_{c}}\) and \(\mathrm{D_{c}}\) are the permeability and the diffusivity of the filled polymer, and \(\mathrm{P_{0}}\) and \(\mathrm{D_{0}}\) are the permeability and the diffusivity of PS. The assumption was made about the invariance of the internal solubility of the matrix \(\mathrm{S_{0}}\) in the presence of mineral layers. Therefore, the polymer/clay composite solubility \(\mathrm{S_{c}}\) can be expressed as \(\mathrm{S_{c}}=\mathrm{S_{0}}\) (\(1-\phi_{\mathrm{m}}\)), where \(\phi_{\mathrm{m}}\) is the volume fraction of mineral phase. The authors suggested that changes in the solubility of oxygen in composites were comparable, since composites containing no more than 10% of the mineral phase were studied. The nanocomposites PS/VDAC-MMT and PS/C18 DMB-MMT showed significantly lower relative permeability (diffusion) comparing with the PS/Na-MMT composite containing untreated clay in the case of samples with a similar clay content. In about 3% wt/wt of clay, only 4% decrease of oxygen permeability was found by authors in the case of PS/VDAC-MMT, 8% in the case of PS/C\({}_{18}\)DMB-MMT, and for PS/Na-MMT, changes of permeability were negligible. With about 9% wt/wt of clay, 18% decrease of oxygen permeability was found, 42% for PS/C\({}_{18}\)DMBMT, and only about 2% for PS/Na-MMT conventional composite. Different continuum models have been discussed to predict the permeability of plates-filled composites. Many nanocomposite systems presented in the literature can be described using models of this type, wherein only the shape and volume of the filler fraction are required to predict the properties [1]. These models usually consider random placement (random only in two  
Figure 6: (**a–d**) Schematic diagrams showing dispersion of the Na-MMT silicate layers of the slurry into PA6 during compounding [108]. directions) parallel plates arranged perpendicularly to the direction of penetration of substances. In Reference [114], the orientation factor was introduced. The prediction of a significant decrease in permeability of nanocomposites was given for cases of the use of fillers with a high aspect ratio. Transport properties of polymers can be significantly altered due to changes in the diffusion pathway of penetrant molecules by introducing inorganic plates with a sufficient aspect ratio [1; 106]. Although composites based on nanoparticles are often subject to the predictions of continuum mechanics, there are situations when nanocomposites can exhibit properties being not expect for the filling with larger particles [1]. In the case of polymers loaded with plate-like inorganic fillers, including clays, semi-empirical models proposed by [PERSON] [115] and by [PERSON] [116; 117] are used to describe the gas barrier [1]. Three model PS/clay hybrid systems presented above (Figure 7) with very different layer morphologies were considered. [PERSON] et al. made a comparative analysis of the simulations of [PERSON] and [PERSON] [113]. It was demonstrated that [PERSON]'s formula is more applicable to composite materials filled by discs located at a distance greater than the radius of disk R (the dilute mode) [1].