Porod invariant

A useful parameter for the analysis of porous materials, which can be derived from the application of the Porod law, is the Porod invariant (PI). PI is defined as [82]... 

Figure 4.- Relative Porod Invariant values (calculated relative to the maximum Porod Invariant value for each sample) corresponding to the measurement carried out across the fiber diameter for the samples CFC50 and CFS48.

Similarly to the previous section, the relative Porod Invariant corresponding to each measurement for the two carbon fibers has been calculated and plotted versus beam position in Figure 5. 

Since all the parameters on the righthand side of this expression should be constant with activation, except for (j)s. the value of the Porod invariant should follow the behavior of < >s(l- s)- Thus, division of the scattering intensities by this factor can be used to correct for the variation of ( )s. It is emphasized, however, that this is only an approximation for a number of reasons, including the fact that when l(q) varies with q to an exponent less than -2 (as at high q for the current data), the integral diverges [3], and accurate values of the Porod invariant cannot be obtained. 

Figure 4. SAXS curves for saran char from Figure 3), divided by the corresponding Porod invariants.

Scattering data for a cellulose char sample activated isothermally at 425°C are presented in Figure 6. These data show that this carbon, like the saran char, initially exhibits a significant amount of microporosity, as indicated by the plateau centered at about q = O.lA". The Porod invariants for these data initially increase and then decrease with activation, also like the saran char. The scattering data, corrected by the PI values, are presented in Figure 7, which shows somewhat different behavior than for the saran char. 

The Porod invariants for the data in Figure 8 increase monotonically with activation. No maximum was observed as for the saran char and unpromoted cellulose char samples. The... 

From the scattering results, the Porod invariant (PI), which is a parameter related with the porosity development, was estimated for each scattering measurement [77]. From these calculations, the pore distribution across the fiber diameter could be deduced. The results showed that the scattering profiles, as a... 

By immersing the sample in water at room temperature for several hours, we found that the PIP-Mg and PB-Ti samples did not swell significantly. However, a very thin layer on the surface became opaque. The SAXS maximum was displaced to lower q values by 7% and 3%, respectively. The most dramatic effects, however, were found for the samples of PB-Mg (Figure 4) neutralized to 80% and 100%. In these specimens, the swelling appeared uniform and the peak sharpened dramatically and was shifted to lower q s by as much as 25%. As expected, the water uptake was accompanied by an increase in domain size from 6.2A to 11.5A for PB-Mg (100%) and from 5.8A to 13.6A for PB-Mg (80%) with a very slight increase in Ed. The Porod invariant Q and zero-order scattering also decreased. 

A characteristic parameter for the surface scattering regime is the Porod invariant Q, which should be derived only from that part of the scattering that is due to the primary particles (cf. Sect. 2.3.4.4 Eq. (2.25)). This parameter is proportional to the average aggregate volume ... 

Once having obtained an analytical description of the scattering curve, it is possible to calculate the Porod invariant Q (cf. Eq. (2.25)) from the primary particle contributions... 

On the other hand, the Porod invariant Q (equation (8-12)) for a dilute set of identical nano-objects (occupying a total volume fraction 0, very small) becomes Q,= 8tt — po) V. Thus the volume Vi can be derived, regardless the object shape, from the quotient 7(0)/ as follows... 

This statistical model leads to remarkably single results for advanced stages of phase separation, when the two phases have nearly reached the final electron concentration and volume fractions. Under this condition, the integral of the scattering intensity or Porod invariant Q (equation (8-12)) becomes essentially time independent and the pure coarsening process starts. The results of computer simulations for the coarsening regime (Lebowitz, 1982) indicate that the time dependent structure function, the different moments S(q, t) and S t), and normalized moments q t) defined by... 

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