Porod asymptote

This distortion has to be considered as Ifi is determined by choosing the value that maximizes the length of Porod s region. From the intercept the Porod asymptote Ap is determined (cf. Fig. 8.9). From the slope of the dashed line the width dz of the transition zone is computed. 

Here dc is the average thickness and 0 is the variance of the particle thickness distribution modeled by a Gaussian. Ap, is the ID Porod asymptote (cf. p. 125, Table 8.3). The particle thickness distribution considers polydispersity (cf. Chap. 1). 

The calibration process then involves measurement of the complete scattering curve of the secondary standard and the evaluation26 of k by determination of Porod s law with its asymptote Ap and the density fluctuation background Ipi, numerical extrapolation of the function s2 (/ (s) - Ipi) towards s = 07 and finally computation of the scattering power... 

The surface area per unit volume can be also measured in scattering experiments. For the fully developed system containing the domains of size L and the interface of the intrinsic width c (c -C L), the Porod law [1] predicts the following asymptotic for the structure factor S(k)... 

This asymptotic decrease in intensity in the high q tail is described in the Porod law region and arises when qR > 4, where R refers to the dimension of the scattering heterogeneity. 

The SAXS patterns at higher k-values display asymptotic linear slopes indicating a power law relation. For a sharp interfece with a smooth geometry, the intensity is expected to foUow a k power law in this Porod region. The observed slope varies systematically with pore size, membrane orientation and thickness. In some cases, exponents more negative than 4 are observed. This may arise from either angulosity or a non-sharp interface, and is discussed in more detail elsewhere [15]. 

Valuable information on crazes can be derived from their scattering behavior at large values of s. Following a derivation given by Porod it may be shown that a two-phase system with sharp phase boundaries and a cylindrically symmetric correlation function obeys the asymptotic law... 

The asymptotic form of the scattered intensity, derived by Porod and Debye for a two-phase system separated by a sharp interface, is given by ... 

Fig. 3.20. Form function H(q) of a Kratky-Porod chain in the infinite limit, according to ref. 20. Here fp is the persistence length and Nlp the length of the curve. The product y = N(qlp)2 H (q) is a function of (qlp) and the asymptot represented by a dashed line is given by the equation y=n(qlp) + 2/3.

Figure 5.6 Kratky plot of the scattering function (circles) calculated16 from an atomistic model of polyethylene. The curves shown are those calculated for a Kratky-Porod chain, a Gaussian chain, and an asymptotic thin rod. (After Kirste and Oberthur.15)...

The Porod law, given in Equation (5.71), shows the asymptotic behavior of I(q) at large q. A similar expression can be derived in terms of /(< ) Thus, substituting (5.71) into (5.170) gives... 

On the basis of this general scattering behavior Kratky and Porod [ 1 ] predicted long ago that a plot of k P k) against k should exhibit a "transition from the asymptotic plateau for a Gaussian coil to the asymptote for a rod. The transition point may be defined as the point of intersection of the two asymptotes. Then, k, the value of k at this point, is equal to G/irq. Thus, q can be estimated if the two asymptotes are experimentally observable so as to be able to determine k definitely. This method has been applied to several flexible polymers. However, actual scattering curves do not always exhibit a distinct plateau region. 

Apj, Porod s asymptote of the projected SAXS intensity, is the constant governing Porod s law. The non-ideal structure of the real two-phase system is described by Ipi and wt- Fluctuations of the electron density are considered by Ipi, the density fluctuation background. v>t is the width of the transition zone at the domain boundary. 

If the boundaries between the crystalline and the amorphous regions are sharp within the resolution limit of SAXS experiments, the asymptotic behavior of E(g) in the limit of large scattering angles is given by Porod s law... 

Kriste and Porod [22] and Tomita [23] worked out theories concerning the relationship between scattering and surface curvatures of the scattering objects. The asymptotic behavior of density correlation function y(r) at a small length scale r and that of scattering function I(q) at... 

The general properties of Fourier analysis tell us that the asymptotic trend, at high q, of the scattering intensity I q) is connected to the behavior of the y r) function at smaU r. The correlation fimction y(r), for two electron density systems, can be approximated at small r by (Porod, 1982) ... 

Porod s law applies to either dilute or concentrated systems of isolated nano-objects, provided they are not very thin sheets or very narrow cylinders in these particular cases, the asymptotic intensity is proportional to 1/ and l/q, respectively (Shull, 1947). For the... 

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). 

ILS experiments indicate a semi-rigid behavior for the PDA chains. Therefore we can expect to observe the form factor of the Porod-Kratky chain. More precisely, the q scattering behavior of a rod like molecule should be measured since the normalize form factor P( of an infinitely long worm-like chain has the asymptotic form ... 

So far in our consideration of small-angle diffraction behaviour we have not considered the effective of diffuse interfacial layers. Porod showed that the tail i.e. the asymptotic behaviour at high angles) of a scattering curve for an ideal two-phase system with sharp boundaries between the phases should have an intensity proportional to s " for a system studied with point collimation and proportional to s when studied with infinite slit collimation. 

The asymptotic behavior of SAXS from a two-phase system in form of an integer power law asymptote is known as Porod s law. In 3D at large s, we have... 

The mathematical treatment described so far applies to ideal two-phase systems with infinitesimal sharp density steps and perfectly constant densities within the two types of domains. While this situation can be approximated in very good approximation for certain types of samples, for example, porous silica prepared by calcination of a polymer template, in the majority of two-phase systems formed by polymers, the interface between the two domain is not sharp but fuzzy on a certain length scale, see Figure 2 for semicrystalline polymers or Figure 4 for block copolymers, and there are also density fluctuations within one or both of the two domains, deviating from the assumed constant density. Both types of effeas lead to deviations from the ideal asymptotic Porod behavior that may need to be taken into account in an SAXS analysis. 

Note that both finite interface faaors [28] and [29] are monotonically deaeasing funaions that lead to so-called negative deviations from Porod s law, that is, the correaed SAXS curve lies below the ideal s" asymptote. On the other hand, the effea of density fluauations within the supposedly constant density domains leads to an additive contribution to the SAXS curve, equivalent to a positive deviation from Porod s law. Note that both kinds of deviations can be present at the same time. 

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