Correlation functions of amorphous multiphase systems

Ciccariello, S.; Cocco, G.; Benedetti, Alvise; Enzo, Stefano

doi:10.1103/physrevb.23.6474

Cited by 68 publications

(84 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We will consider now a system consisting of p homogeneous phases divided by sharp interfaces. This means that the differences of the atomic number densities can be described as [40] …”

Section: Multiphase Systemsmentioning

confidence: 99%

See 1 more Smart Citation

Structure analysis of multiphase systems by anomalous small-angle X-ray scattering

Tatchev¹

2008

Philosophical Magazine

View full text Add to dashboard Cite

“…We will consider now a system consisting of p homogeneous phases divided by sharp interfaces. This means that the differences of the atomic number densities can be described as [40] …”

Section: Multiphase Systemsmentioning

confidence: 99%

“…Small-angle scattering of three phase systems were treated first by Peterlin [38]. Multiphase systems were considered in terms of "stick probability functions" [39][40][41]. The main concern was the evaluation of the correlation function for three phase system.…”

Section: Introductionmentioning

confidence: 99%

Structure analysis of multiphase systems by anomalous small-angle X-ray scattering

Tatchev¹

2008

Philosophical Magazine

View full text Add to dashboard Cite

“…The interphase surface areas obey (Goodisman & Brumberger, 1971) Sij/4 V = (dPit/dr)r = o (6) while the second derivative of Pv at r = 0 is related (Ciccariello, Cocco, Benedetti & Enzo, 1981) to the angularity of the corresponding surface. It is also useful to write (Ciccariello & Benedetti, 1985) ~(r)= E ~ijQij …”

Section: ~' Pij(r)(ni-nj) 2 V (5) Y(r) = 1 --~ Q~iq~j(n I __ Nj)2 Umentioning

confidence: 99%

Small-angle X-ray scattering analysis of catalysts: comparison and evaluation of models

Brumberger¹,

Delaglio²,

Goodisman³

et al. 1986

J Appl Cryst

View full text Add to dashboard Cite

Small-angle X-ray scattering (SAXS) can be used to obtain interphase surface areas of a system, such as a supported-metal catalyst, composed of internally homogeneous phases with sharp interphase boundaries. Measurements of SAXS for samples of porous silica, alumina, platinum on silica, and platinum on alumina are reported. A variety of models and forms for the correlation function, the Fourier transform of which gives the X-ray scattering, are considered, and theoretical and measured intensities are compared. A criterion of fit for comparing models with different numbers of parameters is proposed. For the two-phase (unmetallized) systems the 'Debye-random' model must be rejected. Modifications of the Debye (exponential) correlation function are also not particularly good compared to an exponential-plus-Gaussian form, not derivable from a physical model, and forms based on Voronoi cell models. Since intensities can be fit to experimental error with a five-parameter correlation function, it seems incorrect to ascribe significance to the result of fitting a function with six or more parameters. It is shown that values for the single interphase surface area can be obtained independently of a model. However, fitting intensities using a modelbased correlation function gives information about the structure of the system. The two-cell-size Voronoi and the correlated Voronoi cell models are useful in this regard. For the systems containing metal, fiveparameter correlation functions again suffice to fit intensities. However, for three-phase systems a model or physical assumption is necessary to obtain values for the three surface areas from X-ray scattering intensities. The area of the surface between support and void is quite insensitive to the assumptions employed and the metal-support surface area somewhat less so, but values for the metal-void surface area $23 are consistent only to one significant figure from model to model. If the support in the three-phase catalyst is known to be unchanged from support in the absence of metal, a 'support-subtraction' model can be used to obtain reliable values for $23. In the present systems, the assumption does not seem to be borne out.

show abstract

“…The CLD contains the complete structural information obtainable from non-normalized SAS intensities of statistically isotropic twophase systems. The behavior of g r in the vicinity of small r is de®ned by the general structure of the interface (edges, vertices, curvature; Ciccariello et al, 1981;Ciccariello & Benedetti, 1982;Ciccariello, 1993;Sobry et al, 1994;Ciccariello & Sobry, 1995). Singularities in g and g H at ®nite values of r are related to particularities of the surface in the vicinity of minimal or maximal diameter of monodisperse particles in dilute solution (Wu & Schmidt, 1974) or of the structural unit in periodic dense two-phase systems, respectively.…”

Section: Introductionmentioning

confidence: 99%

Analysis of chord-length distributions

Bürger

Ruland

2001

Acta Cryst Sect A

View full text Add to dashboard Cite

A closed-form analytical solution for the inversion of the integral equation relating small-angle scattering intensity distributions of two-phase systems to chord-length distributions is presented. The result is generalized to arbitrary derivatives of higher order of the autocorrelation function and to arbitrary projections of the scattering intensity (including slit collimation). This inverse transformation offers an elegant way to investigate the impact of certain features, e.g. singularities, in the chord-length distribution or its higher-order derivatives on the scattering curve, e.g. oscillatory components in the asymptotic behavior at a large scattering vector. Several examples are discussed.

show abstract

Correlation functions of amorphous multiphase systems

Cited by 68 publications

References 9 publications

Structure analysis of multiphase systems by anomalous small-angle X-ray scattering

Structure analysis of multiphase systems by anomalous small-angle X-ray scattering

Small-angle X-ray scattering analysis of catalysts: comparison and evaluation of models

Analysis of chord-length distributions

Contact Info

Product

Resources

About