Error percolation

Gilks WR, Audit B, De Angelis D, Tsoka S, Ouzounis CA. Modeling the percolation of annotation errors in a database of protein sequences. Bioinformatics 2000 18 1641-9. 

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 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. 

To test this theory, the room temperature conductivity of "Nafion" perfluorinated resins was measured as a function of electrolyte uptake by a standard a.c. technique for liquid electrolytes (15). The data obey the percolation prediction very well. Figure 9 is a log-log plot of the measured conductivity against the excell volume fraction of electrolyte (c-c ). The principal experimental uncertainty was in the determination of c as shown by the horizontal error bars. The dashed line is a non-linear least square law to the data points. The best fit value for the threshold c is 10% which is less than the ideal value of 15% for a completely random system. This observation is consistent with a bimodal cluster distribution required by the cluster-network model. In accord with the theoretical prediction, the critical exponent n as determined from the slope of... 

Figure 9. Log-log plot of conductivity vs. excess volume fraction (c-Co) of the aqueous phase. Typical error bars for the determination of volume fraction are shown. The corresponding errors for conductivity are much smaller and omitted for clarity. The straight line is a fit of the percolation prediction Equation 1 to the data with n = 1.5, Co = 0.10, and

The gas at incipient fluidization percolates through the solid particles, creating a liquidlike phase referred to as the emulsion phase. Although this so-called two-phase theory (Toomey and Johnstons, 1952) is not entirely accurate, it is generally valid (within acceptable error limits) and has served as the basis for several fluidized-bed reactor models. These models indicate that conversions increase due to mass transfer between the emulsion and bubble phases. 

Under conditions where no phase separation or percolation can occur, termination of the square pulse is followed by a double exponential decay (t / and T. y) of the birefringence. The forward and reverse relaxations are found to be symmetrical, i.e., t, = t, and An, = Art-i within experimental error [41]. In the faster process the induced dipoles of the droplets (individually and as constituents of clusters) rapidly collapse, the shape of the droplets reverts to spherical, and/or the droplets randomize in their orientation. If the field-free fast relaxation is interpreted as the ellipsoid-to-sphere structural relaxation of the droplets, the bending modulus k of the surfactant monolayer can be estimated from the measured t / [49]. Depending on the polydispersity assumed, the values found in the range k = 0.4-1.0 kT are consistent with those obtained from the static birefringence A o [7,9]. 

The stretched exponent u depends essentially on the temperature (Fig. 26). At 14°C, u has a value of 0.5. However, when the temperature approaches the percolation threshold = 27°C, u reaches its maximum value of 0.8, with an error margin of less than 0.1. Such rapid decay of the KWW function at the percolation threshold reflects the increase of the cooperative effect of the relaxation in the system. At temperatures above the T, the value of the stretched exponent v decreases, and indicates that the relaxation slows down in the interval 28°-34°C. At temperatures above 34°C, the increase in u with the rise in temperature suggests that the system undergoes a structural modification. Such a change implies a transformation from anZ.2 phase to lamellar or bicontinuous phases (132, 133). On the other hand, the temperature behavior of the fractal dimension Dp (Fig. 26) shows that below the percolation... 

FIGURE 23.7 Effect of pressure upon thiophene conversion during hydrotreatment. Simulation temperature 553 K. (a) Thiophene conversion at network centerplane. (b) Fraction of pores on a percolating path to the pellet surface. The line is shown to guide the eye error bars represent the spread in thiophene conversion values resulting from simulations performed upon 5 separately generated pore networks. (From Wood, J., Gladden, L.F., and Keil, F.J., Chem. Eng. ScL, 57, 3047-3059, 2002. With permission.)... 

EJE for PC is adduced. As one can see, both indicated equations give a good enough correspondence with the experiment their average discrepancy makes up 5.6% in the Eq. (15.7) case and 9.6% for the Eq. (15.10). In other words, in both cases the average discrepancy does not exceed an experimental error for mechanical tests. This means, that both considered methods can be used for PC elasticity modulus prediction. Besides, it necessary to note, that the percolation relationship (the Eq. (15.7)) qualitatively describes the dependence E better, than the empirical relationship... 

Specifically for gelation, we will discuss in Sect. C.V. various modifications of the simple percolation model of Fig. 1 and check if the exponents diange. In most cases, they do not in particular, the lattice structure (simple cubic, bcc, fee, spinels ) is not an important parameter since different lattices of the same dimensionality d give the same exponents within narrow error bars. More importantly, percolation on a continuum without any underlying lattice structure has in two and three dimensions the same exponents, within the error bars, as lattice percolation. In the classical Flory-Stockmayer theory which does not employ any periodic lattice structure, the critical exponents are completely independent of the functionality f of the monomers or the space dimensionality d. But if the system is not isotropic or if the gel point is coupled with the consolute point of the binary mixture solvent-monomers , the exponents may change as discussed in Sect. D. 

For percolation, scaling and hyperscaling is used whenever it determines an exponent more accurately than direct data. Rational numbers indicate (presumably) exact results, numbers with a decimal point are numerical extrapolations with an estimated error typically of the order of one unit in the last digit given. Data from earlier reviews and recent research ... 

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