Porod equation

In the rod limit, Le = L, and de = d. On the other hand, in the coil limit, Le - V6de, if we assume the chain to be in the unperturbed state. It is to be noted that the axial ratio of the fuzzy cylinder is greater than unity even in the coil limit. At intermediate N, L and the axial ratio Le/de may be calculated, respectively, from the Kratky-Porod equation [102,103] for the (unperturbed)... 

For a non-ideal (pseudo) two-phase system having interface layer, the overall scattering will show negative deviation from Porod s law (see the curve 11 in Fig.l) and Debye s theory (see the curve II in Fig.2), thence the Porod equation (1) becomes [6,9]... 

In eq 1 vd2o is the volume of a water molecule. Hence, can be estinmted as 115 5 and the intercept is consistent with a radius for die polar core of a dry micelle of around 11 A. This head group area Afj can also be calculated from high Q SANS data with the Porod equation, as described elsewhere (13, 86). These results indicate that fluoro-surfactants at the water-COa interface adopt a lower packing density than a related hydrocarbon surfactant (AOT) at analogous water-alkane interfaces (86). 

The application of the Porod equation or of the Debye-Bueche approach are particularly attractive because they offer the possibility to evaluate the interfacial area between the phases of the blend, and they are probably the only way to quantify such feature in polymer blends and composites. In fact, when the two polymers are mixed together in a blend, traditional methods based on the adsorption of small molecules, i.e. the BET approach, are inapplicable. Image analysis of TEM micrographs can in principle be an option, but it is extremely time consuming and it suffers from a number of limitations, such as dependence on sample preparation, on projection effects, and on image defocus. The validity of SAXS for the study of interpenetrating networks has been shown for several systems. ... 

The total scattered intensity can be obtained by integrating different regions of the 2-D pattern. Usually, focus is posed on the diameters along the equatorial and meridional directions. This allows to monitor the development of structural modifications during the deformation. Jansen et al integrated the 2-D patterns and then applied the Porod equation, quantitatively characterizing the size of the crazes which form as a consequence of tensile stress. 

Lumpkin, D jardin and Zimm38 used Eq.(13) to study the effect of a pore size distribution, but did not assume that . a- for each pore. Instead, they calculated -(a.) for each pore size a. by solving the Kratky-Porod equation l. However, as explained above, one cannot obtain Eq.(13) from the very general Eq.(lO) in this case therefore, their results, which showed that the pore size distribution has a strong effect on the electrophoretic mobility, are not reliable for quantitative comparisons. 

According to the Porod law [28], the intensity in the tail of a scattering curve from an isotropic two-phase structure havmg sharp phase boundaries can be given by eqnation (B 1.9.81). In fact, this equation can also be derived from the deneral xpression of scattering (61.9.56). The derivation is as follows. If we assume qr= u and use the Taylor expansion at large q, we can rewrite (61.9.56) as... 

This equation is the same as equation (B 1.9.81). which is temied the Porod law. Thus, the scattered intensity/ (q) at very large q values will be proportional to the q tenn, this relationship is valid only for sharp... 

The least recognized fonns of the Porod approximation are for the anisotropic system. If we consider the cylindrical scattering expression of equation (B 1.9.61). there are two principal axes (z and r directions) to be discussed... 

This equation is the Porod law for the large-angle tail of the scattering curve along the equatorial direction, which indicates that the equatorial scattered intensity I q is proportional to in the Porod... 

Basic Equations. Scattering according to Porod s law [18,137] is a consequence of phase separation in materials. In a two-phase system (e.g., a semicrystalline polymer) every point of the irradiated volume belongs to one of two distinct phases (in the example to the crystalline phase or to the amorphous phase). In a multiphase system there are more than two distinct phases. 

Equations. For a ID two-phase structure Porod s law is easily deduced. Then the corresponding relations for 2D- and 3D-structures follow from the result. The ID structure is of practical relevance in the study of fibers [16,139], because it reflects size and correlation of domains in fiber direction . Therefore this basic relation is presented here. Let er be50 the direction of interest (e.g., the fiber direction), then the linear series expansion of the slice r7(r)]er of the corresponding correlation function is considered. After double derivation the ID Fourier transform converts the slice into a projection / Cr of the scattering intensity and Porod s law... 

Equation (8.59) defines the ID interference function of a layer stack material. G (s) is one-dimensional, because p has been chosen in such a way that it extinguishes the decay of the Porod law. Its application is restricted to a layer system, because misorientation has been extinguished by Lorentz correction. If the intensity were isotropic but the scattering entities were no layer stacks, one would first project the isotropic intensity on a line and then proceed with a Porod analysis based on p = 2. For the computation of multidimensional anisotropic interference functions one would choose p = 2 in any case, and misorientation would be kept in the state as it is found. If one did not intend to keep the state of misorientation, one would first desmear the anisotropic scattering data from the orientation distribution of the scattering entities (Sect. 9.7). 

Scattering and Disorder. For structure close to random disorder the SAXS frequently exhibits a broad shoulder that is alternatively called liquid scattering ([206] [86], p. 50) or long-period peak . Let us consider disordered, concentrated systems. A poor theory like the one of Porod [18] is not consistent with respect to disorder, as it divides the volume into equal lots before starting to model the process. He concludes that statistical population (of the lots) does not lead to correlation. Better is the theory of Hosemann [158,211], His distorted structure does not pre-define any lots, and consequently it is able to describe (discrete) liquid scattering. The problems of liquid scattering have been studied since the early days of statistical physics. To-date several approximations and some analytical solutions are known. Most frequently applied [201,212-216] is the Percus-Yevick [217] approximation of the Ornstein-Zernike integral equation. The approximation offers a simple descrip-... 

From their light-scattering measurements Holtzer, Benoit, and Doty (126) concluded that the short-range interactions control the dimensions of cellulose nitrate chains, and they discussed their results in terms of the worm-like chain model of Kratky and Porod (142), obtaining a persistence length of about 34.7 A. In Fig. 21 these data are shown as a plot of (S yjMw against Mw. The open circles are the experimental points and the broken curve is that calculated from the equations for the worm-like chain model. The theoretical curve is claimed to reproduce the data to within the probable experimental error in all but two cases. 

Porod chain [41] with random conformations and a mean square end-to-end distance given by Equation 2 [42] ... 

Apb is the scattering length density difference, Q is Porod s invariant, and Y the mean chord length. For the calculation of Yo(r) we approximated I(q) hy a cubic spline. The equations used for the calculation of " pore and " soUd are to be found in [8,30,39-41,47]. Analytical expressions for the descriptors of RES were published in [10,11,13,42,43]. In its most simple variant, the stochastic optimization procedure evolves the two-point probability S2 (r) of a binary representation of the sample towards S2(r) by randomly excWiging binary ceUs of different phases, starting from a random configuration which meets the preset volume fractions. After each exchange the objective function... 

Porod s law provides the second equation which is required in addition to Eq. (1) to evaluate both Vj and D from the scattering data. It may be shown that a two-phase system with sharp phase bounderies and a cylindrically symmetric correlation function obeys the asymptotic law... 

The mass-fractal dimension of the ramified agglomerates is determined from the slope of the weak power law decay in between the power law regimes that follow Porod s law (Equation 10.9) ... 

The regime for which q > tr, where the length scale of the scattering experiment can resolve below the fractal scaling regime and see the individual monomers. In this regime S(q) q, which is known as Porod s law. This feature is not included in Equation 14.35 but may be included by multiplying Equation 14.35 by the form factor, i.e., the normalized differential cross section, for the monomer. 

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.

The length /P, known by some as Porod s length of inhomogeneity, is thus a measure of the average size of the heterogeneities present in the system. Equation (5.83) shows that for any ideal two-phase system, the correlation function T (r) decays exponentially, at least for small r, and /P plays the role of the correlation length. Taking the slope of the correlation function (5.83) at r = 0, we obtain... 

The main theoretical results derived from the ideal two-phase model, i.e., Equation (5.70) relating the invariant Q to the phase volumes and the Porod law (5.71), are no longer valid and need be modified when in the two-phase system the phase boundaries are diffuse. To see the modifications necessary to these theoretical results, we represent by p r) the scattering length density distribution in a two-phase material with diffuse boundaries and by p d(r) the density distribution in the (hypothetical) system in which all the diffuse boundaries in the above have been replaced by sharp boundaries. The two are then related to each other25 by a convolution product... 

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... 

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