# Barriers and Chemistry in a Bottle: Mechanisms in Today's Oxygen Barriers for Tomorrow's Materials
## 3 Passive Barriers
### Composites  
Multilayered technologies all require specific equipment, adding up to the total cost. Many research groups are therefore focusing on the improvement of the passive barrier in monolayer systems. The main polymer will therefore need to be intrinsically modified. Blending in a barrier polymer such as the nylon poly(m-xylene adipamide) (MXD6) has been proven to be effective as an oxygen barrier at concentrations starting from 10 wt % [88; 106]. MXD6 is immiscible in PET, forming micron-sized islands. The oxygen that attempts to enter the bottle will encounter a high tortuosity, increasing the diffusion pathway. The effect on solubility is minimal, but the diffusion can be halved in a blend of 10 wt % MXD6 [106]. Several research groups have attempted to formulate a relation between modifier concentration and permeability. This led to variable mathematical models that have been widely discussed in other reviews. The most basic model equation is only briefly presented here and the reader is referred to more in-depth reviews for further information [107; 108; 109]. Most models are derived from [PERSON]'s equation for electrical conduction through a heterogeneous medium. Herein, a tortuosity factor \(\tau\) can be introduced to account for the increase in path length [110; 111; 112]. This factor can be related to the volume fraction of an additive present as spherical particles by following Equation (8) [80; 106]. \[\frac{\mathbf{P_{Blend}}}{\mathbf{P_{PET}}}=\mathbf{1}+\ \frac{3\Phi}{ \frac{\mathbf{R}+2}{\mathbf{R}-1}-\Phi}=\ \frac{\mathbf{1}-\Phi}{\tau} \tag{8}\]  
With \(\mathrm{P_{blend}}\) the permeability of the blend, \(\Phi\) the volume fraction of the barrier polymer and \(\mathrm{R}\) the ratio \(\mathrm{P_{i}}/\mathrm{P_{PET}}\), \(\mathrm{P_{i}}\) being the permeability of the pure barrier polymer. In both films and bottles, a stretch is induced on the blend when processed. This will alter the particle configuration to an ellipsoidal structure with the longitudinal axes following the stretch direction. Diffusion through the wall occurs perpendicular to this stretch, so the particles appear much larger to the diffusing compound, providing the most efficient increase of tortuosity. This factor needs to be considered in the equation. Additionally, the equation can be simplified as \(\mathrm{P_{i}}\) tends to be much lower than \(\mathrm{P_{PET}}\) in order to be relevant. R in \begin{table}
\begin{tabular}{c c c} \hline \hline
**Coating Layer** & **O\({}_{2}\) Permeability (cm\({}^{3}\) mm m\({}^{-2}\) day\({}^{-1}\) atm\({}^{-1}\))** & **Source** \\  & **RH 50\%, 23\({}^{\circ}\)C** & \\ \hline Epoxy-amine & 0.2–1.3 & [97] \\ Diamond-like Carbon (DLC) & 0.005 & [103] \\ Silicon oxide (SiO\({}_{\mathrm{x}}\)) & 1–\(3\times 10^{-5}\) & [104; 105] \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Oxygen permeabilities of common coatings in packaging. Equation (8) will then approximate the value of 0. Nielsen has implemented both alterations to provide Equation (9) as an estimation for the tortuosity in blends [113]. \[\frac{\mathbf{P}_{\mathbf{Blend}}}{\mathbf{P}_{\mathbf{PET}}}=\ \frac{\mathbf{1}-\Phi}{ \mathbf{1}+\frac{\alpha\Phi}{2}}=\ \frac{\mathbf{1}-\Phi}{\tau}\ \mathbf{with}\ \boldsymbol{\alpha}=\frac{3}{2} \boldsymbol{\lambda}^{3} \tag{9}\]  
With \(\lambda\) the axial aspect ratio (axial length/hoop length of the ellipsoid). Other models are often adapted to more specific conditions. For example, [PERSON]'s model is most fitted for polymer blends with very similar permeabilities [114]. In PET, the popularity of nylon MXD6 yields from it owning a similar refractive index as PET. The immiscibility of MXD6 in unstretched PET will not lead to light refraction. However, stretching will induce a change in refractive indices in a ratio that is different for each polymer and will lead to a high haze-level in the product. This diminishes the attractiveness of the product to the customer [88,115].