Porod plot

The specific surface area determined from the Porod plot is approximately 10 m /g. This is a very low value and can be attributed to the absence of the Porod limit. 

The relative isotropic scattering intensity /(/ ), (h=4n/ -sm 0), 20=scattering angle, A=0.154 nm) was recorded by use of a Kratky plant. From the tightly packed arrangement of pores and walls the Porod plot of the scattering curve, see Fig. 3, results. Furthermore, for the isotropic two-phase material, for a fixed range order (here L=30 nm), y r) follows via Eq. (1). 

The NA measurement yields a mean chord length of the pores 7=10.8 nm, via Eqs. (2). By use of this value, it follows a normalized Porod plot, defined by... 

Figure 3 Normalized Porod plot P (h) of the scattering intensity 1(h). The resolution limit in real space is marked in the subfigure.

Fig. 18. Porod plot Is versus s of the equatorial SAXS curve of crazes II in polycarbonate...

Rgure 1.11 Porod plots of scattering data of silicates polymerized under a variety of conditions (from a study by Schaefer and Keefer ) (a) two-step acid-catalyzed tetraethoxysilane (TEOS) system (b) two-step acid-and base-catalyzed TEOS system (c) one-step base-catalyzed system TEOS system (W = 1) (d) one-step base-catalyzed system TEOS system (VP = 2) (e) aqueous silicate system. LUDOX . VP is the water/ silica ratio. 

The behavior of I q) at high q is often analyzed using a Porod plot versus q ). 

From equation (8-11), I q)q is expected to become asymptotically constant in the high q limit. Porod plot is used in order to determine (i) the asymptotic value of I q)q and, from it, the interface surface area, S (ii) eventual positive or negative deviations from Porod s law. For example, statistical density fluctuations in the phases produce an additional and constant scattering intensity and thus a deviation of Porod s law, evidenced by a positive slope ofthe linear part ofPorod s plots. On the other hand, a smooth (not sharp) transition in the electron density between the two phases leads to a negative slope (Ruland, 1971). 

A) Slit-smeared Porod plot for spinnable silicate precursor (W2) measured by SAXS. K=iniX(sm0/2). (B) Corresponding SiNMR spectrum. From Drinker and Assink [32]. 

SAXS Porod plots of porous silicates. (A) Aerogel prepared from a particulate precursor, p uik 0.09g/cm . (B) Aerogel prepared from a particulate precursor, Pbuik = 0.21 g/gm. (C) Xerogel prepared from a single-step based-catalyzed hydrolysis of TEOS (r = 2) followed by drying at room temperature [18). 

SANS Porod plot of silica aerogels prepared by single-step hydrolyses of TMOS (r = 4) under neutral (N) or basic ((B) 5 x 10 M NH4OH) conditions followed by supercritical extraction of methanol. Numbers in parentheses are the corresponding bulk densities. The oxidation (heat) treatment for the 0.36-g/cm sample was performed at 500°C 126). 

A) Porod plot for the two-step acid-catalyzed silicate precursor (W1). K is the scattering vector4n/ (sinfl/2). From Schaefer and Keefer [29], (B) Corresponding Si NMR spectrum. From Assink [31]. 

Porod plot of film prepared from monosized particulate silicate precursors (-lO-nm diameters). From Drinker et al. [20]. 

Figure 2. SAXS patterns of PET at 200°C. Hieline (—)istheraw/f9Jversus9 data and has a wetdk peak at qmax tutd a strong optum at low q. The effects of diffuse scattering from the thermal background at high q ate obvious in the uncorrected data, (o) I(q)gp versus q, and their corrected values, ( ) (I(q)-h) versus q. The inset is ite Porod plot used to calculate die thermal backhand fiom the slope of the linear portion in versus...

The quantity/], arises from density fluctuations and Kp is the Porod constant Note the strong positive deviation in the Porod plot (f(q)q versus q ) illustrated as an inset to Figure 2. Positive deviations from Porod s law are caused by thermal density fluctuations [6]. A regression analysis of tiie linear part of the curve gives values of = 0.0S and = 356. A most be subtracted from the raw intensity data before both dj and the invariant are calculated. 

As a result, a plot of the type log ql(q) versus (f-, often designated as a Porod plot, allows determination of r. ... 

Fig. 6.7 Porod plot showing different power-law decays that are common in SAXS interface with a well-defined electron density gradient blue curve), Porod surface scattering red), internal electron density fluctuations black), surface fractal green), Gaussian coil purple), and mass fractal agglomerate orange)...

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