Percolation effect

The last, and less extensively studied field variable driving percolation effects is chemical potential. Salinity was examined in the seminal NMR self-diffusion paper of Clarkson et al. [12] as a component in brine, toluene, and SDS (sodium dodecylsulfate) microemulsions. Decreasing levels of salinity were found to be sufficient to drive the microemulsion microstructure from water-in-oil to irregular bicontinuous to oil-in-water. This paper was... 

In this section, we describe the role of fhe specific membrane environment on proton transport. As we have already seen in previous sections, it is insufficient to consider the membrane as an inert container for water pathways. The membrane conductivity depends on the distribution of water and the coupled dynamics of wafer molecules and protons af multiple scales. In order to rationalize structural effects on proton conductivity, one needs to take into account explicit polymer-water interactions at molecular scale and phenomena at polymer-water interfaces and in wafer-filled pores at mesoscopic scale, as well as the statistical geometry and percolation effects of the phase-segregated random domains of polymer and wafer at the macroscopic scale. 

This model, when applied to Nation as a function of water content, indicated a so-called quasi-percolation effect, which was verified by electrical impedance measurements. Quasi-percolation refers to the fact that the percolation threshold calculated using the single bond effective medium approximation (namely, x = 0.58, or 58% blue pore content) is quite larger than that issuing from a more accurate computer simulation. This number does not compare well with the threshold volume fraction calculated by Barkely and Meakin using their percolation approach, namely 0.10, which is less than the value for... 

F eOH FH20, and Fmgoh) for different solvated acidic polymers are presented in a way that allows some interesting comparisons and the calculation or estimation of the elements of the transport matrix Ljj. In many publications, these transport parameters are reported as a function of the solvent content and are expressed as the number of solvent molecules (i.e., water) per sulfonic acid group. Because of the importance of percolation effects in all considered transport coefficients, we have converted these solvent contents to solvent volume fractions, except for proton conductivities, as shown in Figures 17 and 18. 

Wilkinson D (1986) Percolation effects in immiscible displacement. Phys Rev A 34 1380-1391 Wilkinson D, WiUemsen JE (1983) Invasion percolation A new form of percolation theory. J Phys A 34 1380-1391... 

Bonny JD, Leuenberger H. Matrix type controlled release systems II. Percolation effects in non-swellabe matrices. Acta Pharm Helv 1993 68 25-33. 

Because the conductive filler is located into a single component of the blend, these materials show an onset in the electrical conductivity at very low filler loadings of 2-3%. These findings have been explained by a double percolation effect. The CNT filled blends show superior mechanical properties in the tensile tests and in impact tests (25). 

When a film is very thin, it may not be continuous, and conduction is subject to the percolation effect, whereby charge migrates by hopping or tunneling between island sites [50,51]. Such a process is activation controlled, and such thin films do not obey Ohm s law. The activation energy can be decreased by the presence of an applied electric field, making development of a rigorous theory difficult. The resistivity can be expressed by the relationship [5]... 

Case Study-Percolation Effects in Relation to Excipients Ratio and Granules Size... 

FIGURE 20.5 Cumulative particle size distribution pt= 0.62 for different ratios of the binary mixture lactose (L)/corn starch (MS). The granule diameter is critically linked to the concentration ratio (percolation effect ). (From Leuenberger, H., Usteri, M., Imanidis, G, and Winzap38ll,. Chem. Farrn 128, 54-61... 

A relatively simple pore structure of fairly uniform tubular pores would 1) expected to give a narrow Type HI hysteresis loop (see Figure 7.3) and in this cas the desorption branch is generally used for the analysis. On the other hand, if there i a broad distribution of interconnected pores it would seem safer to adopt the adsoif tion branch since the location of the desorption branch is largely controlled b network-percolation effects. If a Type H2 loop is very broad, neither branch canb used with complete confidence because of the possibility of a combination of effect (i.e. both delayed condensation and network-percolation). Furthermore, the condeii sate becomes unstable and pore emptying occurs when the steep desorption branch j located at a critical pjp° (i.e. at c. 0.42 for N2 adsorption at 77 K). 

So far it has been assumed that the adsorbent surface is homogeneous and that all the pores are of the same size and shape. In practice these conditions are rarely fulfilled. To arrive at the pore size distribution, it has been assumed that a porous adsorbent has an array of non-interacting pores (i.e. there are no network percolation effects) and that the distribution of pore widths can be described by a continuous function f(w). The experimental isotherm can then be regarded as a composite of isotherms for each group of pores. The amount adsorbed is presumed to be given by the general equation... 

The fact that the shape of the isotherms in Figure 10.7 has remained almost unchanged after the acid treatment is an indication that the mesopore structure was not altered to any signifiant extent. However, as pointed out in Chapter 7, this form of H2 hysteresis loop is not easy to interpret since it is associated with pronounced percolation effects in an irregular pore network. 

HI. We believe this to be a useful indication that network-percolation effects are not playing a major role in the emptying of the mesopores (i.e. on the desorption branch). Thus, the narrow and almost vertical loops in Figure 12.8 are more likely to be associated with delayed condensation rather than the more complex percolation pore-blocking phenomena (see Chapter 7). Of course, this is to be expected in view of the non-intersecting tubular pore structure of the model MCM-41. 

In the original IUPAC classification, the hysteresis loop was said to be a characteristic feature of a Type IV isotherm. It is now evident that this statement must be revised. Moreover, we can distinguish between two characteristic types of hysteresis loops. In the first case (a Type HI loop), the loop is relatively narrow, the adsorption and desorption branches being almost vertical and nearly parallel in the second case (a Type H2 loop), the loop is broad, the desorption branch being much steeper than the adsorption branch. These isotherms are illustrated in Figure 13.1 as Type IVa and Type IVb, respectively. Generally, the location of the adsorption branch of a Type IVa isotherm is governed by delayed condensation, whereas the steep desorption branch of a Type IVb isotherm is dependent on network-percolation effects. 

A long-standing problem is the interpretation of the hysteresis loop. For many years the desorption branch was favoured for pore size analysis, but this practice is now considered to be unreliable. There are three related problems (a) network-percolation effects (b) delayed condensation and (c) instability of the condensate below a critical p/p°. 

Nitrogen adsorption/condensation is used for the determination of specific surface areas (relative pressure < 0.3) and pore size distributions in the pore size range of 1 to 100 nm (relative pressure > 0.3). As with mercury porosimetry, surface area and PSD information are obtained from the same instrument. Typically, the desorption branch of the isotherm is used (which corresponds to the porosimetry intrusion curve). However, if the isotherm does not plateau at high relative pressure, the calculated PSD will be in error. For PSD s, nitrogen condensation suffers from many of the same disadvantages as porosimetry such as network/percolation effects and pore shape effects. In addition, adsorption/condensation analysis can be quite time consuming with analysis times greater than 1 day for PSD s with reasonable resolution. 

The attractiveness of surface/pore characterization via NMR spin-lattice relaxation measurements of pore fluid lies in the potential advantages this technique has as compared to the conventional approaches. These include rapid analysis, lower operating costs, analysis of wet materials, no pore shape assumption, a wide range of pore sizes can be evaluated (0.5 nm to >1 /im), no network/percolation effects and the technique is non-destructive. When determining specific surface areas, NMR analysis does not require out-gassing and has the potential for on-line analysis of slurries. 

The predicted values obtained from above equation for ethyl butyrate-ethyl isovalerate system is depicted in Figure 4. The predicted values obtained from eq. (18) are somewhat lower than actual adsorbed amounts obtained from experiments. This due to the fact that actually pure component adsorption of ethyl butyrate occurs in those pores not accessible to ethyl isovalerate, an effect that enhances ethyl butyrate adsorption. The error is also higher with the increase of concentration of the larger component (ethyl isovalerate) as expected. On the other hand when percolation effect and accessibility factor was appropriately considered the predictions were more accurate as seen in Figure 3. 

Figure 4. Adsorption isotherm of ethyl butyrate in the presence of ethyl isovalerate on Filtrasorb-400 activated carbon and predicted values neglecting percolation effects at (a) 303.15 K, (b) 308.15 K, and (c) 313.15 K.

The above observations have been interpreted within the framework of two distinct models, one involving trapping/detrapping of the photogenerated electrons [345, 346] and the other based on electron diffusion (or field-assisted diffusion) not attenuated by electron localization [347, 348]. The millisecond transit times also mean that the transit times are very long compared with equilibration of majority carriers in a bulk semiconductor or electron-hole pair separation within the depletion layer of a flat electrode. The slow transport is rationalized by a weak driving force and by invoking percolation effects [338]. 

Figure 6. Graph showing the comparison of experimental (points) and theoretical (lines) curvatures for the case of three rods non-symmetrically arranged (/ , =11 =2R ). For ( > less than 67° there is only a single meniscus. Between < ) =67° and (j> =74° there are two menisci with a single sided arc meniscus separating them. This arc meniscus acts downwards in one tube and upwards in the other. Above ( ) =74° there are two independent arc menisci. At < ) =72°. if the larger tube (the one on the left) were a pore w hich did not contain a meniscus (because of percolation effects) then the smaller tube would contain a meniscus with a curv ature of 7.7 (instead of 6.7). The sequence of drainage can thus affect the cun ature associated with a pore.

Static methods Mercury intrusion Laplace (Washburn) Cylindrical 5 nm-15 pm Pore size distribution (including dead-end pores) Porosity Outgassed (dry) samples. Measurement of pore entrance. Destructive method. For small pore sizes damage of the porous structure may occur. Network percolation effects derived. 

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