Network-percolation effect

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... 

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. 

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. 

To discover new fitness peaks, the neutral network must be sufficiently extended, allowing neutral drift to effectively sample sequence space. A neutral network can be characterized by a mean fraction of neutral neighbors A (Reidys et al., 1997). If A exceeds a threshold A,., then the network is connected and dense, making it more likely the network percolates through sequence space. If A < Ac, the networks are partitioned into components. Using random graph theory, the threshold is derived analytically as... 

The aim of this chapter is to discuss in general terms the use of adsorption measurements for the characterization of mesoporous solids (i.e. adsorbents having effective pore widths in the approximate range of 2-50 nm). Our approach here is mainly along classical lines and is based on the concept of capillary condensation and the application of the Kelvin equation. However it is appropriate to include a brief discussion of the relevant aspects of network percolation and density functional theory. 

At present, it must be recognized that adsorption hysteresis may be generated in a number of different ways. In the context of the assessment of mesoporosity, we have seen that there are two major contributing factors (a) on the adsorption branch, the development of a metastable multilayer and the associated delay in capillary condensation (b) on the desorption branch, the entrapment of condensate through the effect of network-percolation. 

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. 

This section provides a systematic account of proton transport mechanisms in water-based PEMs, presenting studies of proton transport phenomena in systems of increasing complexity. The section on proton transport in water will explore the impact of molecular structure and dynamics of aqueous networks on the basic mechanism of proton transport. The section on proton transport at highly acid-functionalized interfaces elucidates the role of chemical structure, packing density, and fluctuational degrees of freedom of hydrated anionic surface groups on concerted mechanisms and dynamics of protons. The section on proton transport in random networks of water-filled nanopores focuses on the impact of pore geometry, the distinct roles of surface and bulk water, as well as percolation effects. 

The measured pore size distribution curves are frequently biased towards the small pore sizes due to the hysteresis effect caused by ink bottle shaped pores with narrow necks accessible to the mercury and wide bodies which are not. Meyer [24] attempted to correct for this using probability theory and this altered the distribution of the large pores considerably. Zgrablich et al. [25] studied the relationship between pores and throats (sites and bonds) based on the co-operative percolation effects of a porous network and developed a model to take account of this relationship. The model was tested for agglomerates of spheres, needles, rods and plates. Zhdanov and Fenelonov [26] described the penetration of mercury into pores in terms of percolation theory. 

Figure 9. Imbibition (the site problem) of the network with the site sizes re-distributed at random on the same network. The effect of having the sites randomly distributed is to move the percolation threshold to that of the diamond lattice. This means that the sites in the real network are not distributed randomly.

Poly(ethylene oxide). The synthesis and subsequent hydrolysis and condensation of alkoxysilane-terniinated macromonomers have been studied (39,40). Using Si-nmr and size-exclusion chromatography (sec) the evolution of the siUcate stmctures on the alkoxysilane-terniinated poly(ethylene oxide) (PEO) macromonomers of controlled functionahty was observed. Also, the effect of vitrification upon the network cross-link density of the developing inorganic—organic hybrid using percolation and mean-field theory was considered. 

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