Tortuous path effect

Platelet additives may impart color and luster, metallic appearance, or a pearlescent effeet and also can strongly affect permeation properties. Most of these additives have little or no permeation through them, so when a film contains particulate additives, the permeating molecule must follow a path around the particulate additive as shown in Figure 1.19. This is often called a tortuous path effect. 

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. 

In a packed column the individual solute molecules will describe a tortuous path through the interstices between the particles and some will randomly travel shorter routes than the average and some longer. The multi-path effect is diagramatically depicted in Figure 5. 

The tortuous-path and barrier theories consider the effects of the media on the electrophoretic mobility in a way similar to the effect of media on diffusion coefficients discussed in a previous section of this chapter. The tortuons-path theory seeks to determine the effect of increased path length on electrophoretic mobility. The barrier theory considers the effects of the barrier or media conductivity on the electrophoretic mobility. 

It is relevant to note that tortuosity defined by Equation (57) is by no means the same as that defined by (JLJL), where L, is the effective tortuous path length... 

Although there are a lot of data in the literature regarding diffusion coefficients in liquids or then calculation from molecular properties (Appendix I, Section 1.2), it is not the case for diffusion coefficients in solids, where the phenomena appearing are more complex. In solids, the molecule may be forced to follow a longer and tortuous path due to the blocking of the cross-sectional area, and thus the diffusion is somehow impaired. Several models have been developed to take into consideration this effect in the estimation of diffusion coefficients, leading, however, to a variety of results. 

Both Knudsen and molecular diffusion can be described adequately for homogeneous media. However, a porous mass of solid usually contains pores of non-uniform cross-section which pursue a very tortuous path through the particle and which may intersect with many other pores. Thus the flux predicted by an equation for normal bulk diffusion (or for Knudsen diffusion) should be multiplied by a geometric factor which takes into account the tortuosity and the fact that the flow will be impeded by that fraction of the total pellet volume which is solid. It is therefore expedient to define an effective diffusivity De in such a way that the flux of material may be thought of as flowing through an equivalent homogeneous medium. We may then write ... 

The molecular diffusivity D must be replaced by an effective diffusivity De because of the complex internal structure of the catalyst particle which consists of a multiplicity of interconnected pores, and the molecules must take a tortuous path. The effective distance the molecules must travel is consequently increases. Furthermore, because the pores are very small, their dimensions may be less than the mean free path of the molecules and Knudsen diffusion effects may arise Equation 10.170 is solved in Volume 1 to give equation 10.199 for a catalyst particle in the form of a flat platelet... 

To apply Fick s laws to solute fluxes in sediments, adjustments have to be made to these equations to account for the negative interference effects that sediment particles have on the diffusion of solutes in pore waters (Lerman, 1979 Berner, 1980). For example, tortuosity, defined as the length of the tortuous path that a solute travels around particles across a distance across a certain depth interval can be described by the following equation (Berner, 1980 Krom and Berner, 1980a) ... 

Additionally, due to the presence of solid material the volume in which diffusion can take place is reduced by a factor Sp, the particle porosity. The tortuous path increases the diffusion length for a molecule relative to the spatial coordinate by a factor tp. The effective diffusivity can therefore be expressed as... 

It would be inappropriate in the present context to attempt to review in detail the tortuous path followed by the many investigations of waterweakening that have been made since the initial discovery of the effect. Suffice to say that there is, as yet, no direct independent evidence for an... 

Equation 2.1 defines the flux in a bulk liquid or gas phase (with unit porosity or in a straight capillary). The effect of tortuous paths has to be considered in a porous medium. We use the effective diffusion coefficient, D, to replace Do in Eq. 2.1 to consider the effect of tortuosity. The relationship between D, and Do may be defined (Childs, 1969) using Eq. 2.4,... 

Crystals or filler particles that do not sorb the penetrant will obviously be Impermeable to them and, thus, reduce transport rates in the composite. Figure 4 illustrates this effect for matrices in which such particles are arranged in an ordered and in a random manner. The rate of transport in such systems will be slower than when such particles are absent because of the reduced area for transport and the resulting more tortuous path for permeation. The effective diffusion coefficient for such cases will be a factor K smaller than in the pure amorphous material D. The permeability coefficient in a crystalline polymer should then be given by... 

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