Tortuous path model

This phase has presumably mobility and therefore gas permeability properties differing from fhe bulk. Based on fhis consideration, the oxygen permeation through the confined polymer phase was tentatively assumed to be negligible with respect to the bulk one, and a modified version of the tortuous path model of Gusev and Lusti taking into account the total non permeable volume (silica plus volume of confined polymer chains) was derived (Figure 11.27). 

Barrier properties of a rubber matrix are remarkably improved thanks to clay addition. The tortuous path model is proposed to explain this phenomenon. In a NR/Mt nanocomposite prepared from emulsion blending, 1, 2 and 3 phr of clay led to more than 35% and to about 45% and 50% reduction of oxygen permeability, respectively. " 3 phr of OC (Mt/ didodecyl methyl amine) gave a 50% reduction of the oxygen permeability and a 40% reduction of toluene absorption at 20 °C. About 10% and 15% reduction of oxygen permeability were obtained with 5 and 10 phr of OC, respectively, and 30% reduction of toluene absorption was achieved with 15% OC, at 30 °C. ... 

The barrier properties of a number of polymer-clay nanocomposites can be predicted reasonably well by a simple tortuous path model first proposed by Nielsen [1]. In the simple two-dimensional model, the clay nanoparticles act as impermeable barriers to the migration of gas molecules. The equation for the model is ... 

Figure 4.1 Schematic of the tortuous path model showing the increased path due to the high-aspect ratio clay plates.

The experimental data on barrier performance of polymer-clay nanocomposites can be divided into three categories. The first group includes systems that fit the tortuous path model reasonably well. The second category includes composites that exhibit relative gas permeabilities that are inferior to the tortuous path model predictions. The last group includes composites that exceed the performance predicted by the model. The following discussion will give specific examples of each of these categories with explanations for each type of behavior. 

Examples of nanocomposites that fit the tortuous path model... 

Examples of nanocomposites that fit the tortuous path model relatively well include work by Shah et al. [2] with low-density polyethylene (LDPE) and ionomer, and by Gatos and Karger-Kocsis [3] with a nanocomposite of nitrile butyl rubber (NBR) and organofluorohectorite... 

Figure 4.4 Relative permeability for LDPE nanocomposites compared to the tortuous path model for an aspect ratio of 120.

The second polymer studied by Shah was an ionomer (Surlyn 8945). Figure 4.5 contains the relative permeabilities for the ionomer nanocomposite compared to the tortuous path model predictions for an aspect ratio 120. The agreement between the tortuous path model and experimental is quite good. The generally accepted aspect ratio for this particular clay is about 150, which is close to the 120 utilized in the figures. The clay in the ionomer appears to be more fully exfoliated, which is consistent with the higher percent increase in modulus measured for the ionomer nanocomposite relative to the FDPE. 

Figure 4.7 Relative permeability for an EVA nanocomposite compared to the tortuous path model using aspect ratios of 200 and 100.

A second group of nanocomposites mentioned earlier is characterized by exhibiting negative deviations from the tortuous path model. These account for another significant number of composites. In these composites this negative deviation can be explained by one of two possible mechanisms. The first is the one mentioned in the previous example where tactoids predominate in the composite. In these cases the aspect ratio of the tactoids has not been determined but the data is compared to the tortuous path equation predictions utilizing the aspect ratio of the completely exfoliated clay. The solution in these cases is to utilize X-ray diffraction and electron microscopy to determine the true aspect ratio of the tactoids. This was done in the previous example. 

The experimental data utilizing the measured aspect ratio should be the basis for comparison. An example of this type of composite is data reported by Zhong et al. [4] on a nanocomposite of ethylene vinyl acetate (EVA) with montmorillonite. Eigure 4.7 contains the experimental relative permeabilities for this composite compared to the curves predicted by the tortuous path model utilizing aspect ratios of 200 and 50. The 200 aspect ratio represents the fully exfoliated nanoclay and the 50 represents... 

The last class of nanocomposites includes a group that exhibit relative gas permeabilities that exceed the performance predicted by the tortuous path model. This behavior would indicate that the nanoparticles have altered the fundamental behavior of the polymer. 

In order to explain the permeability of tin-filled polymer systems, the tortuous path model was developed by Nielsen. Particulates are impermeable and are responsible for tortuous diffusion path for gas or water molecules. The efficiency of the fiUer in reducing permeation depends on its concentration, orientation and aspect ratio. If the filler platelets are aligned perpendicularly to the diffusion direction, the Nielsen model can be adequately represented by Equation (12.4) ... 

Water vapor transmission data (for polylactic acid, PLA), fitted to the tortuous path" model [13]. 

Duan and Thomas [13] went further to extend the model of the tortuous path effect of crystallinity (obtained via the DSC) on another polymer, polytetraethylene, PET and obtained data as plotted in Figure 12.3 and concluded that the water vapor permeability decreased linearly with the percentage of crystallinity. They found out that the measured values of WVTR decreased linearly with increasing crystallinity of the PLA from 0% to 50% and they readily explained this observation in terms of the effect of crystallinity on the solubility of water vapor in polymers, i.e., that water is insoluble in the crystalline regions and so the solubility coefficient (5) of a semi-crystalline polymer is equal to the solubility coefficient of the amorphous fraction. They therefore concluded that the tortuous path model could also be used to explain the water permeability characteristics of PET. 

Duan Z, Thomas NL. Water vapour permeability of poly(lactic acid) crystallinity and the tortuous path model. J Appl Phys 2014 115(6) 064903-9. 

As reported by several authors and as proved by our studies on the transport behaviour, nanoeomposites exhibit substantial improvements in gas and vapour permeability. This phenomenon has traditionally been explained by a tortuous path model where the high aspect ratio platelets acts as physical barrier for diffusion of molecules. ... 

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