Percolated network

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. 

A systematic study of the adsorption of nitrogen by packed assemblages of spheroidal particles was undertaken by Adkins and Davis (1986, 1987). After the consideration of various pore filling models, it was concluded that the desorption process can be adequately described by the instability of a Kelvin, hemispherical meniscus in the neck (i.e. the window) of the structure and the adsorption process can be viewed as a delayed Kelvin condensation in the largest dimension of the void structure. This reasoning is consistent with the network-percolation theory of hysteresis, which is discussed in Section 7.5. 

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). 

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. 

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. 

In that region, the EMA predicted Pr curve decreases linearly with I s, while the network solution exhibits a non-linear behavior and reaches a higher percolation threshold, Vsc - This is because Ksr predicted by the network model, corresponds to the theoretical fbc predicted by percolation theory ( c 1.5/r, [9]), while Vsc found by EMA corresponds to fhc"=2/z [10]. A similar picture is presented in figure 2b, where Pr is plotted as a function of the fraction of the open pores, //,. It can be seen that for all r, near the percolation threshold, EMA shows a linear decrease of Pr with /. On the other hand, network results indicate that, in the same region, Pr decreases with / according to a power law. For an infinitely large network percolation theory states ... 

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. 

Berkowitz, B., and C. Braester. 1991. Dispersion in sub-representative elementary volume fracture networks Percolation theory and random walk approaches. Water Resour. Res 27 3159-3164. Berkowitz, B., and R.P. Ewing. 1998. Percolation theory and network modeling applications in soil physics. Surv. Geophys. 19 23-72. 

The simulations of conductivity in random resistor-capacitor (RC) networks confirmed freqnency dependence of Eqnation (65) (Panteny et al., 2005). In random networks percolated resistors lead to the DC plateau at low frequencies. In contrast, non-percolated resistors generate negative deviation of conductivity from DC plateau and non-monotonons decrease of conductivity at low frequencies. These regular or irregular deviations from DC plateau are in polymer-salt systems due to electrode polarization. At high frequencies random network simulations recovered the power-law dependency of conductivity as in Equation (65). Generally, one may say conductivity in random RC networks will be preferably determined by resistors at low frequencies and by capacitors at high frequencies. 

This functional dependence, which accounts for the interplay of increasing lEC, proton dilution, and random network percolation, gives rise to a nonmonotonic dependence of on lEC andXy. Upon increasing lEC, is observed to pass through... 

D. Sokolowska, A. Krol-Otwinowska, J. K. Moscicki, Water-network percolation transitions in hydrated yeast, Phys. Rev. E 70 (2004) 052901-052904. 

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